Games and Economic Behavior 130 (2021) 543–568 Contents lists available at ScienceDirect Games and Economic Behavior www.elsevier.com/locate/geb Value-free reductions ✩ David Pérez-Castrillo ∗ , Chaoran Sun ∗ Universitat Autònoma de Barcelona and Barcelona School of Economics (BSE), Spain a r t i c l e i n f o a b s t r a c t Article history: We introduce the value-free (v-f ) reductions, operators that map a coalitional game played Received 24 July 2020 by a set of players to another “similar” game played by a subset of those players. We Available online 14 October 2021 propose properties that v-f reductions may satisfy, we provide a theory of duality, and we characterize several v-f reductions (among which the value-free version of the reduced JEL classification: C71 games proposed by Hart and Mas-Colell, 1989, and Oishi et al., 2016). Unlike reduced games, introduced to characterize values in terms of consistency, v-f reductions are not Keywords: defined in reference to values. However, a v-f reduction induces a value. We characterize v- Coalitional games f reductions that induce the Shapley, the stand-alone, and the Banzhaf values. We connect Reduced games our approach to the theory of implementation. Finally, our new approach is a valuable Axiomatization tool to provide new characterizations of values in terms of consistency. We present new Consistency characterizations of the Banzhaf and the stand-alone values. Shapley value © 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC Duality BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction We consider environments where a set of participants can collaborate to obtain and share surplus, that is, we study coalitional games with transferable utility (TU games). In such environments, we look at the consequences of removing some players from the game. In the new game faced by the remaining participants, the worth of each coalition of players is a function of the strategic possibilities of all the players in the initial game. This problem is relevant in many economic contexts. For instance, when a group of shareholders leave a company, the remaining shareholders reorganize the ownership among themselves. The process through which the outstanding share- holders acquire the shares of the leaving shareholders will determine the strategic environment where they will interact from then on, that is, the worth of each possible coalition in the new environment. Thus, in this paper, we look at TU games from a new perspective. We study “operators” that map a TU game played by a set N of players to another, similar but “reduced” game, played by a subset of N. We propose properties that such functions may satisfy, and we use these properties to characterize several operators. Our research question is different but related to the search for consistency properties of values for TU games.1 Before continuing with the contribution of our paper, it is ✩ We thank Ramon Caminal, Alfredo Contreras, Ezra Einy, Itzhak Gilboa, Roweno Heijmans, Ori Haimanko, Chenghong Luo, Inés Macho-Stadler, Andreu Mas-Colell, Jordi Massó, Abraham Neyman, David Wettstein, Eyal Winter, the seminar participants at MicroLab at Universitat Autònoma de Barcelona, ENTER Jamboree at Tilburg University, Brown bag seminar at Ben-Gurion University, and Symposium of the Spanish Economic Association for their helpful comments. We also thank the associate editor and two reviewers for their insightful comments. We gratefully acknowledge financial support from the Ministerio de Economía y Competitividad and FEDER (through the Severo Ochoa Programme for Centres of Excellence in R&D, CEX2019-000915-S, and the project PGC2018-094348-B-I00), Generalitat de Catalunya (2017SGR-711 and FI fellowship), and ICREA under the ICREA Academia programme. * Corresponding authors. E-mail addresses: david.perez@uab.es (D. Pérez-Castrillo), chaoran.sun@barcelonagse.eu (C. Sun). 1 In this respect, the closest papers to ours are Hart and Mas-Colell (1989) and Oishi et al. (2016). https://doi.org/10.1016/j.geb.2021.09.009 0899-8256/© 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 worthwhile to discuss the relationship between this line of research and our approach. To that aim, we first briefly describe the consistency requirement. Consider a value for TU games, that is, a function that associates a payoff to every player in every game. Starting from a TU game with a set of players N, we can define a reduced game among the players of any N   N. The worth of a coalition in the reduced game takes into account the payoffs that the players in the coalition give, according to the value, to the players who are removed, that is, to the players in N \ N  . Hence, the characteristic function of the reduced game depends on the original characteristic function and the solution in question. The value is consistent if a player in N  obtains the same payoff in the initial game and in the reduced game. There are several possibilities to define a reduced game depending on how the removed players are compensated. In particular, Hart and Mas-Colell (1989) (HM) and Oishi et al. (2016) (ONHF) define two different reduced games. They use them to characterize the Shapley value as the only value that is standard for two-person games (that is, it divides the surplus equally between the two players) and consistent. In contrast to the previous literature on consistency, we study operators that reduce games without reference to any value. We refer to them as value-free reductions (v-f reductions, for short). For any TU game with a set of players N and any N  ⊆ N, a v-f reduction generates a game played by N  . A simple example is the subgame v-f reduction, which assigns each coalition in the reduced game the same worth as in the initial game. Our interest lies in the analysis of the reduction processes, that is, in the v-f reductions. We propose properties that one may ask any such v-f reduction to satisfy. In this paper, we study v-f reductions that satisfy four properties. First, we request that a v-f reduction is “well defined,” in the sense that how players in N \ N  are removed to arrive at a game with a set of players N  should not matter. The game played by the set N  should be the same if the players in N \ N  have been removed one by one, all simultaneously, or in any other sequence. We call this property path independence. The second property is the additivity of the v-f reduction. Reducing two games through an additive v-f reduction and then summing the corresponding reduced games and directly reducing the sum of the games gives the same result. The other two properties are related to the presence of null players in the initial game. The contribution of a null player to any coalition is zero. Hence, it seems reasonable that they play no role in a v-f reduction. We require that if a player is a null player in the initial game, he should still be a null player after a v-f reduction. We call this property the permanent null player. Moreover, if a null player is removed from the game, then the worth of the coalitions should not change, a property that we call the null player out property. Path independence, additivity, permanent null player, and null player out do not suffice to identify a unique v-f reduction. But, by including alternative “invariance” properties, we characterize several v-f reductions. Each invariance property states how changes in the worth of coalitions of the same size affect the reduction of the game. First, we characterize the subgame v-f reduction using an axiom that requires that an increase in the worth of the grand coalition should not affect the reduction of a game, a property that we call grand-coalition invariance. Second, we consider the four previous properties plus the invariance axiom that states that the reduced game is im- mune to changes in the players’ strategic prospects derived from an identical increase or decrease in all the stand-alone coalitions. The axiom requires that if the worth of each stand-alone coalition, say, increases by the same amount, then this change should not affect how the game is reduced. Interestingly, these five axioms characterize a unique v-f reduction that corresponds to the HM v-f reduction, that is, the value-free version of the reduction method proposed by HM. To continue our analysis of the properties of v-f reductions, we propose a duality theory for them. We define the dual of a v-f reduction as the v-f reduction of the dual of the game. We show that the ONHF v-f reduction (that is, the value-free version of the ONHF reduction method) is dual to that of the HM v-f reduction. We also show that our basic properties of path independence, additivity, permanent null player, and null player out are all self-dual properties, in the sense that they are satisfied by a v-f reduction if and only if they are satisfied by the dual of the v-f reduction. We use the duality theory to characterize the ONHF v-f reduction by using the invariance axiom that is dual of the one in the characterization of the HM v-f reduction. According to this new axiom, the reduction of a game should be immune to an identical increase or decrease in the worth of all the coalitions that include all the players except one. We note that, given a v-f reduction, then any (initial) game can unambiguously be reduced to a game played by just one player, say player i ∈ N. We can interpret the worth of coalition {i} (the only non-empty coalition) in this reduced game as the benefit or cost to be distributed to this player in the initial game. Repeating this process for every player in N allows us to define a value for the initial game. Thus, a v-f reduction “induces” a value. We show that the subgame v-f reduction induces the stand-alone value and, as one may expect, the HM and the ONHF v-f reductions induce the Shapley value. Moreover, we can connect our approach to the previous literature on consistency because any value induced by a path-independent v-f reduction (such as the subgame, the HM, and the ONHF v-f reductions) is consistent relative to that reduction. We also link our approach to the theory of implementation. Indeed, we use the players’ payoffs obtained at the Pérez- Castrillo–Wettstein bidding mechanism (a mechanism that implements the Shapley value, see Pérez-Castrillo and Wettstein, 2001) to propose another v-f reduction. We characterize the new v-f reduction by an alternative invariance axiom and show that it also induces the Shapley value. Moreover, we apply our duality theory again and characterize the dual of that v-f reduction. The existence of this dual PW v-f reduction prefigures the existence of a new PW-style bidding mechanism (see Sun, 2020, for the analysis of such a mechanism). Thus, the connection of our approach to the theory of implementation –by constructing the v-f reduction of an extensive-form game and then finding its dual– could help enrich the literature of the Nash program. It can suggest new mechanisms that are “dual” of existing mechanisms. 544 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Our four basic axioms can lead to characterizations of v-f reductions that induce additive values other than the stand- alone and the Shapley values. We use them as part of the characterization of a v-f reduction that induces the Banzhaf value (Banzhaf, 1964). Although we do not use them in our characterizations, we discuss the properties of anonymity and linearity. Anonymity of a v-f reduction requires that a player’s name does not matter in the reduction of the game. It has two implications: (a) the worth of the coalitions in the reduced game does not depend on the names of the players in the initial game but only on their contributions to coalitions, and (b) the v-f reduction itself depends not on the names of the removed players but only on their contributions. The notion of anonymity is unrelated to the other axioms. In fact, our basic properties do not imply anonymity. However, all the v-f reductions that we study satisfy anonymity of the process. They also satisfy linearity, which is additivity plus homogeneity. We have based some of our examples of v-f reductions on existing reduced games, which were introduced to study the internal consistency of values. We also propose the reverse process. That is, given a v-f reduction, we can find a reduced game whose v-f version coincides with the v-f reduction. Through this process, we provide a new characterization of the Banzhaf value as the only value consistent relative to a new reduced game and standard for two-player games. We provide a similar characterization for the stand-alone value. In addition to Hart and Mas-Colell (1989) and Oishi et al. (2016), several authors have used the consistency principle to characterize values for TU games.2 Among others, Sobolev (1975) defines a distinct reduced game and axiomatize, together with other axioms, the Shapley value. Noticeabley, Davis and Maschler (1965) define a reduced game which turns out crucial in Peleg’s (1985, 1986) axiomatizations of the core and the prekernel, and Sobolev’s (1975) axiomatization of the prenu- cleolus. Moulin (1985) defines a reduced game and axiomatizes three families of choice methods in the framework of the “quasi-linear social choice problem.”3 Tadenuma (1992) also employs this reduced game to provide another axiomatization of the core. The analysis of our paper may shed light on the discussion on the use of consistency relative to a reduced game when comparing different solutions for cooperative games. On that matter, Maschler (1990) advocates that the choice between two solution concepts that can be characterized by the same set of basic properties plus consistency relative to a reduced game (reduced games that are different for the two concepts) boils down to the examination of the reduced games. There are two strands of research related to this view. The first strand is pursued by Chang and Hu (2007), who propose a criterion to “distinguish” two different solutions through two different reduced games. The second strand includes Driessen and Radzik (2003), Yanovskaya and Driessen (2002), and Yanovskaya (2004), which characterize reduced games directly. Our approach is closer to the second strand since we adopt a pure axiomatic approach. The remainder of the paper is organized as follows. In Section 2, we recall basic concepts, including the definition of reduced games. In Section 3, we introduce our central concept of a value-free reduction, together with a list of properties that a v-f reduction may satisfy. In Section 4, we develop a duality theory for v-f reductions. In Section 5, we provide an axiomatic characterization of several v-f reductions, we discuss the properties of anonymity and linearity, and we make a comment on non-additive v-f reductions. In Section 6, we use our approach to characterize the Banzhaf and the stand- alone values through consistency. Logical independence of each property in the characterization of the HM v-f reduction is established in Section 7. In Section 8, we conclude the paper. All proofs are collected in the Appendix. 2. TU games, values, and reduced games Let an infinite set U represent the universe of the players. We restrict attention to games where the set of players constitutes a finite subset of U . We denote P f in (U ) the set of all finite subsets of U . A coalitional game with transferable utility (abbreviated as a TU game) is a vector ( N , v ) where N ∈ P f in (U ) is the set of players and v : 2 N → R satisfies v (∅) = 0. For S ⊆ N, v ( S ) represents the worth of the coalition S in the  game v. The class of all TU games with N as the set of players is denoted by G N . Thus, the set of all finite TU games is N ∈P f in (U ) G N .  A subgame of ( N , v ) ∈ G N is a game ( N  , v | N  ) ∈ G N for some N  ⊆ N, where v| N  ( S ) = v ( S ) for all S ⊆ N  . For a fixed set of players N, the set of all TU games G N may be viewed as a vector space. The zero vector of G N corresponds to the zero game ( N , 0) ∈ G N . The worth of the coalition S ⊆ N in ( N , 0) is 0( S ) ≡ 0. One particular subset of games that we will use as a basis for G N is the set of unanimity games, which are denoted by ( N , u T ) ∈ G N , for T ∈ 2 N \{∅}. The worth of the coalition S ⊆ N in ( N , u T ) is:  1 if S ⊇ T ; uT (S) ≡ 0 otherwise. Among the unanimity games, ( N , u N ) ∈ G N depicts a particularly simple situation: one unit of transferable utility is generated only when the grand coalition forms. 2 For introductions to the consistency principle in general, see Driessen (1991) and Thomson (2011). 3 See also Chang and Hu (2007). Thomson (2011) refers to this type as “complement-reduced games” since the complement of the players that stay in the reduced game is also involved in the reduced game. 545 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Cooperative game theory accords particular attention to the search of appealing solution concepts and their character- izations through desirable properties from the mathematics and/or economics points of view. Single-valued solutions for TU games are called values. A value allocates a payoff to each player in a game, for every possible game. Thus, a value ϕ prescribes, for each N ∈ Pfin (U ), each TU game ( N , v ) ∈ G N , and each i ∈ N, a payoff ϕi ( N , v ) ∈ R. The most prominent value is the Shapley value (Shapley, 1953), which is denoted by Sh henceforth4 :  t !(n − t − 1)! Sh i ( N , v ) = D i v ( T ), n! T ⊆ N \{i } for any ( N , v ) ∈ G N and for any i ∈ N, where D i v ( T ) ≡ v ( T ∪ {i }) − v ( T ) denotes the marginal contribution of player i to the coalition T ⊆ N \ {i }. Another solution concept which we will discuss later is the Banzhaf value (see Banzhaf, 1964, and Owen, 1975) which we henceforth denote by Ban:  1 Bani ( N , v ) = D i v ( T ). 2n−1 T ⊆ N \{i } We notice that, in contrast to the Shapley value, the Banzhaf value is not efficient in the sense that the sum of the outcomes obtained by the players need not be v ( N ). Two-player TU games constitute the most simple subclass of TU games. Unsurprisingly, several solution concepts for TU games prescribe the same payoff when restricted to this simple subclass. According to this prescription, in the game ({i , j }, v ) ∈ G {i , j } each player k ∈ {i , j } is assigned, on top of his stand-alone value, half of the surplus generated from the collaboration: 1 ϕk ({i , j }, v ) = v ({k}) + [ v ({i , j }) − v ({i }) − v ({ j })]. (1) 2 This is, in particular, the prescription of the Shapley value and the Banzhaf value for two-player games. Hence, it is com- monly said that a value ϕ is standard for two-player games if for each game ({i , j }, v ) ∈ G {i , j } , ϕ satisfies equation (1). For TU games with more than two players, solution concepts may be pinned down by imposing consistency relative to some reduced games. In the literature, reduced games are always associated with a solution concept as follows. Given a ϕ  value ϕ , a reduction ϕ is a function that associates each TU game ( N , v ) ∈ G N with a reduced game ( N  ,  N , N  ( v )) ∈ G N for any two finite sets of players N , N  such that N   N.5 That is, a reduction applied on a game with a set of players N specifies how to “reduce” the game if it were to be played only by a subset N  of N. Notice that the value ϕ appears in this ϕ  function  N , N  as a parameter, so that different values lead to different ways of “reducing” a game in G N to a game in G N . Now we can formulate the definition of consistency of a value relative to some reduction: Definition 1. The value ϕ is consistent relative to the reduction ϕ if for all N , N  ∈ Pfin (U ) such that N   N, all ( N , v ) ∈ G N , and all i ∈ N  ,   ϕi N  , ϕN ,N  ( v ) = ϕi ( N , v ). Consistency of ϕ means that the prescribed payoff for any player i ∈ N  in the initial game ( N , v ) according to the value ϕ is the same as that in the reduced game ( N  , ϕN ,N  ( v )) according to this value. We close this section with two examples of reductions: the HM reduction (see Hart and Mas-Colell, 1989) and the ONHF reduction (see Oishi et al., 2016). ϕ Definition 2. Given a value ϕ , the HM reduction  H M is defined by: ϕ   NH ,MN  ( v )( S ) ≡ v ( S ∪ ( N \ N  )) − ϕi ( S ∪ ( N \ N  ), v | S ∪(N \N  ) ), i∈N \N  for all S , N  , N ∈ Pfin (U ) such that S ⊆ N   N and all ( N , v ) ∈ G N . The interpretation of the HM reduction is as follows. Given a value ϕ , consider a game ( N , v ) ∈ G N that is reduced to be played by players in N   N. If a coalition S ⊆ N  is formed, then the players in S collaborate with all removed players in 4 We follow the convention by using uppercase letters to denote sets of players and letting the corresponding lowercase letters represent their cardinali- ties. For instance, the cardinality of N, N  , and T are n, n , and t. 5 We call the operator ϕ a reduction, even though the previous literature does not address such an operator abstractly. They propose the consistency property using reduced games, which are the images of a concrete reduction. 546 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 N \ N  , which yields a worth v ( S ∪( N \ N  )). However, each removed player i ∈ N \ N  is entitled to ϕi ( S ∪( N \ N  ), v | S ∪( N \ N  ) ), his “fair” share of the worth of the coalition S ∪ ( N \ N  ). Then, the coalition S has a claim to the residual, which defines the worth of coalition S in the HM reduced game. Hart and Mas-Colell (1989) characterize the Shapley value as the unique value that is consistent relative to the HM ϕ reduction  H M and that is standard for two-player games. Oishi et al. (2016) obtain a different characterization of the Shapley value through a reduction à la Hart and Mas-Colell by exploiting the self-duality of the Shapley value. To define the ONHF reduction, we first introduce the following notation: given a TU game ( N , v ) ∈ G N and S  N, we denote by ( N \ S , v S ) ∈ G N \ S the game defined by: v S ( T ) ≡ v ( T ∪ S ) − v ( S ), (2) for all T ⊆ N \ S. ϕ Definition 3. Given a value ϕ , the ONHF reduction  O N H F is defined by: ϕ    NO ,NNH F ( v )( S ) ≡ v ( S ) − ϕi ( N , v ) + ϕi ( N \ S , v S ), i∈N \N  i∈N \N  for all S , N  , N ∈ Pfin (U ) such that S ⊆ N   N and all ( N , v ) ∈ G N . In contrast to the HM reduction, the intuition of the ONHF reduced game (as acknowledged by Oishi et al., 2016) is more involved. To determine the worth of a coalition S ⊆ N  in an ONHF reduced game, we consider all the players in S together. Forming the coalition S entitles the players in the coalition to offer their joint collaboration to the rest of the players to play a new TU game ( N \ S , v S ) ∈ G N \ S . As defined above, in this new game any coalition T ⊆ N \ S is formed with the collaboration of S and T , which yields a worth v ( T ∪ S ). The coalition S is entitled to two payments. First, it receives v ( S ) in forming this game. Second, it makes a swap agreement with the removed players: the coalition S pays ϕi ( N , v ) to each player i ∈ N \ N  , which equals the amount  i deserves in the initial game, and it collects the sum of what these players receive in ( N \ S , v S ), which adds up to i ∈ N \ N  ϕi ( N \ S , v ). The net payoff for S after the two payments corresponds to S its worth in the ONHF reduced game. Oishi et al. (2016) show that the Shapley value is the only value that is consistent relative to the ONHF reduction ϕ  O N H F and that is standard for two-player games. 3. Value-free reductions: definition and axioms The existing literature takes the values as the main object of study and considers the reduced games associated with values to characterize particular values. By contrast, our approach takes the reductions as the primitive concept, analyzes properties of the reductions, characterizes some of them through the properties, and eventually uses the reductions to derive values. To develop our approach, we first formally introduce the concept of a value-free reduction, that is, a reduction that does not make any reference to a value. Definition 4. A value-free reduction (v-f reduction for short)  is a function that associates to each finite set of players N,  each TU game ( N , v ) ∈ G N , and each subset N  ⊆ N, a TU game ( N  ,  N , N  ( v )) ∈ G N .6 Because of the defining feature of v-f reductions, we must forsake the superscript ϕ from a generic v-f reduction. To illustrate the concept, we provide a first example of a v-f reduction. Example 1 defines sub , which we call the subgame v-f reduction.7 According to this operator, the value of any subset in the reduced game is the same as its value in the initial game.8 Example 1. We define the subgame v-f reduction sub by: sub N , N  ( v )( S ) ≡ v | N ( S ) = v ( S ),  for all S , N  , N ∈ Pfin (U ) such that S ⊆ N   N and all ( N , v ) ∈ G N . 6 We allow for the possibility that N  = N for convenience. 7 We refer to all the examples of value-free reductions as “v-f reductions” even though the use of “v-f” is not always necessary. 8 Myerson (1980) uses the subgame operator to define his famous balanced contributions property of the Shapley value. 547 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Any v-f reduction induces one-player v-f reduced games. That is, a game ( N , v ) ∈ G N can be reduced to n games ({i },  N ,{i } ( v )), for i ∈ N. This procedure provides the possibility of identifying the value of a player i in the game ( N , v ) as the worth of the coalition {i } in the v-f reduced game consisting of this player only. We propose the following definition of the value induced by a v-f reduction: Definition 5. The value ϕ  induced by a v-f reduction  is, for all ( N , v ) ∈ G N and all i ∈ N, ϕi ( N , v ) ≡ N ,{i} ( v )({i }). For instance, the value induced by the subgame v-f reduction is the stand-alone value: sub ϕi ( N , v ) = sub N ,{i } ( v )({i }) = v ({i }), because the prescribed payoff of the value induced by the subgame v-f reduction for all i ∈ N is v |{i } ({i }) = v ({i }). We now propose and explain some properties that v-f reductions may satisfy. We see v-f reductions as a way to remove players from a game while keeping the remaining players’ strategic prospect intact. Thus, we suggest properties that may be coherent with this view. We first introduce a minimum requirement of a well-behaved v-f reduction, the path-independence property: Axiom 1. A v-f reduction  is path independent if for all N 1 , N 2 , N 3 ∈ Pfin (U ) such that N 3 ⊆ N 2 ⊆ N 1 , then9 N2 ,N3 ◦ N1 ,N2 = N1 ,N3 . Path independence means that, for any game ( N , v ) ∈ G N , the way players in N \ N  are removed to reach the v-f reduced game of ( N , v ) with N  as the remaining players should be irrelevant. In particular, it should not matter whether a player’s removal precedes another player’s or if they are removed simultaneously. The only relevant information is the set of players who remain at the end. Reduced games were introduced in the literature to study the consistency of values. Then, it is natural to ask about the consistency of the value induced by a v-f reduction with respect to that reduction. Although Definition 1 refers to consis- tency relative to a reduced game (and not to v-f reduced games), the definition can be easily accommodated. Proposition 1 shows that the value induced by a v-f reduction is indeed consistent if the v-f reduction is path independent. Proposition 1. The value ϕ  induced by a path-independent v-f reduction  is consistent relative to . Our second axiom on v-f reductions is additivity: Axiom 2. A v-f reduction  is additive if for all N , N  ∈ P f in (U ) such that N  ⊆ N and all ( N , v 1 ), ( N , v 2 ) ∈ G N , then  N , N  ( v 1 + v 2 ) =  N , N  ( v 1 ) +  N , N  ( v 2 ). To put it in words, additivity means that if game ( N , v ) is the sum of two games ( N , v 1 ) and ( N , v 2 ), then directly reducing ( N , v ), and reducing ( N , v 1 ) and ( N , v 2 ) and then summing the corresponding reduced games, give the same result. We will use additivity in our characterizations. Since we use the concept of a linear v-f reduction later and in some of the proofs in the Appendix, we introduce linearity here. A v-f reduction is linear if it satisfies the axioms of additivity and homogeneity. Axiom 3. A v-f reduction  is homogeneous if for all N , N  ∈ P f in (U ) such that N  ⊆ N, all ( N , v ) ∈ G N , and all α ∈ R, then  N , N  (α v ) = α  N , N  ( v ). Homogeneity of a v-f reduction  means that the scale in which we measure the worth of the coalitions in a TU game does not influence how the game is reduced. Our next two axioms concern the consequences of the presence of “null players” in the game, that is, players who do not contribute to any coalition, on the reduced game. Before introducing the axioms, we formally define null players. Definition 6. A player i ∈ N is a null player in a TU game ( N , v ) ∈ G N if D i v ( S ) = 0 for all S ⊆ N \ {i }.   9 The symbol “◦” denotes the composition of two functions: for f : X → Y and g : Y → Z , g ◦ f (x) = g f (x) ∈ Z for all x ∈ X . 548 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Given that null players have no impact on the worth of any coalition, it may seem reasonable that they also have no impact on the reduction of games. Thus, we propose the following property: Axiom 4. A v-f reduction  satisfies the null player out property if for all N ∈ Pfin (U ), all i ∈ N, and all ( N , v ) ∈ G N such that player i is a null player in ( N , v ), then  N , N \{i } ( v ) = v | N \{i } . The null player out property means that if a null player is removed from the game, then his removal has no effect on the worth of coalitions in the game without him. The axiom reflects the idea that given that a null player has no influence on the game, the worth of any coalition should not change if the game is reduced because he is removed. Moreover, a null player should gain no influence after a reduction: Axiom 5. A v-f reduction  satisfies the permanent null player property if for all N , N  ∈ Pfin (U ) such that N  ⊆ N, all i ∈ N  , and all ( N , v ) ∈ G N such that player i is a null player in ( N , v ), then player i is also a null player in ( N  ,  N , N  ( v )). The interpretation of the permanent null player property is that if a player is a null player in the initial game, then he is still a null player after the removal of some other arbitrary players. In general, null player out and permanent null player properties reflect the rationale perceiving null players as irrelevant or redundant. Still, they are distinct axioms, as we will show in Section 7, where we analyze the logical independence of the axioms. Our last set of axioms provides alternative views of how the reduction of a game is affected by changes in the worth of coalitions of the same size. Indeed, it is conventional to postulate the monotonocity principle that a player’s strategic perspective should be monotonic with respect to the worth of the coalitions containing him (see, e.g., Young, 1985). In line with this principle, if we consider, for example, a symmetric game and we increase the worth of all coalitions of the same size by the same amount, then the enhancing strategic effects for the players may be entirely canceled out. This reasoning is akin to the disagreement convexity in Peters and Van Damme (1991) in the context of the bargaining problem: if each player’s disagreement point is increased properly, then the solution should not be changed. Our version of addition invariance properties borrows from ideas developed by Béal et al. (2015). In our formulation, we follow the terminology used in that paper, which we introduce here: Definition 7. Given the set of players N, for all k ∈ Z+ such that k ≤ n, and α ∈ R, the game ( N , w (k,α ) ) ∈ G N is defined as follows: for all S ⊆ N,  α if | S | = k; w (k,α ) ( S ) ≡ 0 otherwise. The game ( N , w (k,α ) ) is a useful tool to express an identical increase or decrease in the worth of all coalitions of size k in a TU game ( N , v ) as the addition of ( N , w (k,α ) ) to ( N , v ). We point out that the reduction of a game necessarily leads to losing some information contained in the characteristic function v since the domain of the reduced game is a proper subset of that of the initial game. Our first invariance axiom suggests discarding the information contained in the level of the worth of the coalitions of size one. The reduction may depend on the relative worth of the singletons, that is, whether the stand-alone coalition of one player has a higher or lower worth than the stand-alone coalition of other players. However, the axiom postulates that the reduction cannot depend on whether the worth of all the one-player coalitions is high or low. We can translate this idea to the property that if the worth of every coalition of size one in the unanimity game ( N , u N ) ∈ G N is increased or decreased by the same amount, then the reduction of the game u N should not change. Axiom 6. A v-f reduction  satisfies 1-addition invariance if for all α ∈ R and all N , N  ∈ Pfin (U ) such that N   N, then  N , N  (u N + w (1,α ) ) =  N , N  (u N ). Our second invariance axiom proposes an alternative property, in the same spirit as the previous one. It prescribes what happens after an increase or decrease in the worth of every coalition except the grand coalition, where the change in the worth is proportional to the number of players in the coalition. The axiom requires that the change does not affect the reduction of the unanimity game. Axiom 7. A v-f reduction  satisfies proportional addition invariance if for all α ∈ R and all N , N  ∈ Pfin (U ) such that N   N, then 549 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 n −1   N , N  (u N + w (k,kα ) ) =  N , N  (u N ). k =1 The previous two axioms share the view that the reduction of the game leads to the loss of information from the worth of coalitions smaller than the grand coalition. Our third invariance axiom takes the opposite view. It postulates that the worth of the grand coalition should not affect the reduction of the unanimity game. Axiom 8. A v-f reduction  satisfies grand-coalition invariance if for all α ∈ R and all N , N  ∈ Pfin (U ) such that N   N, then  N , N  (u N + w (n,α ) ) =  N , N  (u N ). Before we turn to the characterization of several v-f reductions in Section 5, we first propose a duality theory for v-f reductions in Section 4. We adapt the approach of Oishi et al. (2016). The main difference of our approach is that we take the v-f reductions as primitive, while Oishi et al. (2016) stick to the conventional view that takes the solution concepts as primitive and uses reduced games to characterize solutions in terms of consistency. We use our duality theory in two characterizations of Section 5. 4. Duality theory for value-free reductions We first recall the definition of the dual of a game and the dual of a value. For a TU game ( N , v ) ∈ G N , the dual of ( N , v ) is the game ( N , v ∗ ) ∈ G N , defined by: v ∗ ( S ) ≡ v ( N ) − v ( N \ S ), (3) for all S ⊆ N. For a value ϕ , the dual ϕ ∗ of ϕ is defined by the value: ϕ ∗ ( N , v ) ≡ ϕ ( N , v ∗ ), (4) for all ( N , v ) ∈ G and all N ∈ P f in (U ). N A value is self-dual if ϕ = ϕ ∗ . Examples of self-dual values include the Shapley value and the Banzhaf value. We now define the dual of a v-f reduction: Definition 8. The dual ∗ of a v-f reduction  is defined, for all N , N  ∈ Pfin (U ) such that N  ⊆ N, and all ( N , v ) ∈ G N , as  ∗ ∗N , N  ( v ) ≡  N , N  ( v ∗ ) . That is, consider a v-f reduction  and a game ( N , v ). The dual v-f reduction of ( N , v ) consists in first, applying  to the dual of ( N , v ), and then taking the dual of the reduced game. We already know that the dual operator for TU games is reflexive because ( v ∗ )∗ = v. The dual operator for v-f reductions is also reflexive, that is, (∗ )∗ = .10 If the v-f reduction is path independent, then we can relate the concepts of duality for values and for v-f reductions. Indeed, by recognizing that a one-player TU game coincides with its dual, we obtain the result that the concept of the dual of a value is compatible with the concept of the dual of a v-f reduction: Proposition 2. The value induced by a path-independent v-f reduction is dual to the value induced by the dual v-f reduction: ∗ ∗ (ϕ  ) = ϕ ( ) . (5) An immediate corollary of Proposition 2 is the following: Corollary 1. The value induced by a path-independent v-f reduction is self-dual if and only if it is also induced by the dual of the v-f reduction. We also define dual properties, or axioms, of v-f reductions. 10 This property holds because for all ( N , v ) ∈ G N and all N  ⊆ N: (∗ )∗N , N  ( v ) = (∗N , N  ( v ∗ ))∗ = (( N , N  (( v ∗ )∗ ))∗ )∗ =  N , N  ( v ), where the last equality uses twice that the dual operator for TU games is reflexive. 550 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Definition 9. Consider two properties P and P ∗ regarding v-f reductions. We say that property P is dual to property P ∗ if for all v-f reductions ,  satisfies P ⇐⇒ ∗ satisfies P ∗ . We say that a property is self-dual if it is satisfied by a v-f reduction if and only if it is satisfied by the dual of the v-f reduction: Definition 10. P is self-dual if P is dual to itself, that is, for all v-f reductions ,  satisfies P if and only if ∗ satisfies P. An important result, very helpful in the characterization of v-f reductions, is that the basic axioms that we use are all self-dual, as Proposition 3 states. Proposition 3. The axioms of additivity, null player out, permanent null player, and path independence of v-f reductions are all self- dual properties. 5. Characterization of several value-free reductions In this section, we use the axioms of additivity, null player out, permanent null player, and path independence to char- acterize several v-f reductions. Each characterization of a v-f reduction uses an additional invariance axiom. Before presenting our characterizations, we state an intuitive property that is common to the v-f reductions that are path independent and satisfy the axiom of null player out: a game will be unchanged after a reduction where no player is removed. We state this property in Remark 1, which we will use in the proofs of the characterizations. Remark 1. If a v-f reduction  satisfies null player out and path independence, then for all N ∈ P f in (U ) and all ( N , v ) ∈ G N , N ,N ( v ) = v . 5.1. Characterization of the subgame value-free reduction The subgame v-f reduction sub , defined in Example 1, satisfies our four basic axioms. Moreover, it is characterized with the help of the axiom of grand-coalition invariance (Axiom 8), which postulates that changes in the worth of the grand coalition should not influence the way in which the unanimity game is reduced. Theorem 1. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and grand-coalition invariance if and only if:  = sub . Given that the axiom of grand-coalition invariance emphasizes how difficult coordination is for players striving to achieve the worth of the grand coalition, since the worth of the grand coalition is irrelevant for the reduction, it is reasonable that it leads to the characterization of a v-f reduction where those outside the reduced set of players have no role: the worth of any subgame coincides with that in the initial game. 5.2. Characterization of the HM value-free reduction Next, we study the consequences of including the axiom of 1-addition invariance. It requires that an identical increase or decrease in the worth of all the one-player coalitions in a game should not affect the reduction of the game. Interestingly, 1-addition invariance together with our four basic axioms characterizes the value-free version of the most popular reduced game, the HM reduction (see Definition 2). We call this v-f reduction the HM v-f reduction and we denote it by  H M . We ϕ construct the HM v-f reduction by substituting ϕ = Sh in  H M . Example 2. We define the HM v-f reduction  H M by11 :   NH ,MN  ( v )( S ) ≡ v ( S ∪ ( N \ N  )) − Sh i ( S ∪ ( N \ N  ), v | S ∪( N \ N  ) ) i∈N \N  11 The second equality is implied by the efficiency of the Shapley value. 551 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568  = Sh i ( S ∪ ( N \ N  ), v | S ∪( N \ N  ) ), i∈ S for all S , N , N  ∈ Pfin (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . Theorem 2 states the characterization. It also stresses that, as one could expect, the HM v-f reduction induces the Shapley value. Theorem 2. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and 1-addition invariance if and only if:  = H M . Moreover,  H M induces the Shapley value. Theorem 2 provides a characterization of  H M that is particularly interesting because it is based on a property (the 1-addition invariance) which seems unrelated to the definition of the reduction. On the one hand, the idea behind the reduction of a game ( N , v ) ∈ G N to ( N  ,  N   , N  ( v )) is that the worth of a coalition S ⊆ N in ( N ,  N , N  ( v )) is computed HM HM taking into account that the players in S profit from the collaboration with every removed player i ∈ N \ N  , who is entitled to a compensation of Sh i ( S ∪ ( N \ N  ), v | S ∪( N \ N  ) ). On the other hand, the notion of 1-addition invariance concerns the effect of identical changes in the worth of the one-player coalitions.12 Therefore, Theorem 2 highlights that a characteristic property of the HM v-f reduction is that it is immune to changes in the strategic prospects of the players derived from the changes in their stand-alone worth, as long as the changes are identical for every player. We illustrate how the axioms in the theorem allow to pin down the HM v-f reduction in two-player games. Con- sider a v-f reduction  satisfying the axioms. Then, for an arbitrary game ({1, 2}, v ) ∈ G {1,2} , we can write v = α1 u {1} + α2 u {2} + w (1,α12 ) , where αi = 12 ( v ({i , j }) + v ({i }) − v ({ j })) for i , j ∈ {1, 2} such that i = j, and α12 = 12 ( v ({1}) + v ({2}) − v ({1, 2})). We use the properties to find, for example, {1,2},{1} ( v )({1}). First, since player 2 is a null player in ({1, 2}, α1 u {1} ), null player out implies that {1,2},{1} (α1 u {1} )({1}) = α1 u {1} ({1}) = α1 . Second, player 1 is a null player in ({1, 2}, α2 u {2} ). Hence, by permanent null player, {1,2},{1} (α2 u {2} )({1}) = 0. Third, 1-addition invariance and additivity im- ply that {1,2},{1} ( w (1,α12 ) )({1}) = 0 because the v-f reduction of ({1, 2}, w (1,α12 ) ) must be the same as the v-f reduction of the zero game. Finally, we use additivity to obtain {1,2},{1} ( v )({1}) = α1 = Sh1 ({1, 2}, v ) = {H1M ,2},{1} ( v )({1}). The same arguments can be used for games with more than two players. However, they must be complemented with the property of path independence. For instance, for three-player games, we need to compute terms such as {1,2,3},{1} (α12 u {1,2} )({1}). To do this computation, we need to use path independence to ensure that the previous reduction is equal to {1,2},{1} ({1,2,3},{1,2} (α12 u {1,2} ))({1}) which, since player 3 is a null player in ({1, 2, 3}, α12 u {1,2} ), is equal to {1,2},{1} (α12 u {1,2} )({1}) by null player out. Because of our analysis of the two-player games, we also know that this corre- sponds to {H1M ,2},{1} (α12 u {1,2} )({1}). The proof in the Appendix extends the previous arguments to show that the five axioms (and an induction argument) lead to Theorem 2. 5.3. Characterization of the ONHF value-free reduction In the previous subsection, we define the value-free version of the HM reduction. We can use the same method to define the value-free version of the ONHF reduction,  O N H F , which we will refer to as the ONHF v-f reduction: Example 3. We define the ONHF v-f reduction  O N H F by13 :    NO ,NNH F ( v )( S ) ≡ v ( S ) − Sh i ( N , v ) + Sh i ( N \ S , v S ) i∈N \N  i∈N \N    = Sh i ( N , v ) − Sh i ( N \ S , v S ), (6) i∈N  i∈N  \ S for all S , N  , N ∈ Pfin (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . Oishi et al. (2016) construct the ONHF reduced game as the dual of the HM reduced game. Hence, it is no surprise that  O N H F is the dual v-f reduction of  H M . We state this result as a corollary of the analysis developed by Oishi et al. (2016): 12 1-addition invariance together with additivity implies that  N , N  ( v + w (1,α ) ) =  N , N  ( v ) for any ( N , v ) ∈ G N .    i ∈ N \ N  Sh i ( N , v ) = v ( N ) − i ∈ N \ N  Sh i ( N \ S , v ) = v ( N \ S ) − 13 The two expressions for  O N H F are equivalent because i ∈ N  Sh i ( N , v ), S S  i ∈ N  \ S Sh i ( N \ S , v ), and v ( N \ S ) = v ( N ) − v ( S ). S S 552 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Corollary 2. The v-f reduction  O N H F is the dual of the v-f reduction  H M . As we proved in Section 4, additivity, null player out, permanent null player, and path independence are all self-dual properties. Given that they are satisfied by  H M ,  O N H F also satisfies these axioms. On the other hand, the property of 1-addition invariance, which is the additional axiom that characterizes  H M , is not self-dual. Proposition 4 states that the dual property of the 1-addition invariance is the (n − 1)-addition invariance axiom, defined as follows: Axiom 9. A v-f reduction  satisfies (n-1)-addition invariance if for all α ∈ R and all N , N  ∈ Pfin (U ) such that N   N,  N , N  (u N + w (n−1,α ) ) =  N , N  (u N ). Proposition 4. The dual of the 1-addition invariance axiom is the (n − 1)-addition invariance axiom. In conjunction with the interpretation of the 1-addition invariance property provided in the previous section, there is a dual interpretation of the (n − 1)-addition invariance property. This axiom requires discarding the information contained in the level of the worth of the coalitions of size (n − 1) (instead of the information contained in the level of the worth of the coalitions of size 1). Theorem 3 provides our characterization of  O N H F . It can be thought of as a dual theorem to Theorem 2 as it gives a characterization of the dual of  H M through the dual properties of the axioms used in Theorem 2. The theorem also states that the O N H F v-f reduction induces the Shapley value. Theorem 3. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and (n − 1)-addition invariance if and only if  = O N H F . Moreover,  O N H F induces the Shapley value. Theorems 2 and 3 together reveal a distinctive difference between  H M and its dual,  O N H F . Whereas the HM v-f reduction postulates that the strategic prospects of the agents should not change after an identical modification in the worth of every stand-alone coalition (the 1-addition invariance property), the ONHF v-f reduction considers that the players’ strategic prospects should not change after an identical modification in each player’s maximum compensation (the (n − 1)- addition invariance property). 5.4. Value-free reductions inspired by the bidding mechanism We have characterized v-f reductions that bear some relationship to existing reduced games. In the current subsection, we propose and characterize two new v-f reductions. They link our approach to the theory of implementation. Indeed, the first v-f reduction is based on the out-of-equilibrium payoffs obtained at the Pérez-Castrillo–Wettstein (PW) bidding mechanism (see, Pérez-Castrillo and Wettstein, 2001), which implements the Shapley value. The second v-f reduction is the dual of the first. Thus, we start by explaining the bidding mechanism, and its equilibrium. j In the PW bidding mechanism, each player j ∈ N in a game ( N , v ) ∈ G N makes a bid b i ∈ R to each player i = j. The player with the highest total net bid (the difference between a player’s total bid to the others minus the sum of the bids the others make to him) is chosen as the proposer (let’s denote him by α ). The proposer α pays the bids to the rest of the players and makes them an offer to join him. If the proposal is accepted, then α pays the offers that he has made to the other players (in addition to the bids that he has already paid), forms the grand coalition, and receives the worth v ( N ). If the proposal is rejected, then α is removed from the game and obtains the worth of his stand-alone coalition v ({α }). The rest of the players, that is, the set N \ {α }, keep the bids and play the same game again among them. j At the subgame perfect equilibrium of the bidding mechanism, any player j ∈ N bids b i = Sh i ( N , v ) − Sh i ( N \ { j }, v | N \{ j } ) to each player i = j and the proposer α makes an offer that is accepted (see, Pérez-Castrillo and Wettstein, 2001). The offer submitted to the players in N \ {α } makes them indifferent between accepting the offer and playing the new game among them (because this is the continuation outcome of the mechanism in case of rejection). That is, the offer to each player is the payoff that this player would obtain in the “reduced game” where the set of players is N \ {α }. In this reduced game, the assets of any coalition S ⊆ N \ {α } are composed by two elements: the worth of the coalition and the sum of the bids that the players in S collect from α , that is, v ( S ) + i ∈ S bα i = v ( S ) + i ∈ S ( Sh i ( N , v ) − Sh i ( N \ {α }, v | N \{α } )). If we continue deleting players, we obtain the extension of the previous formulae for the reduced game played by any N   N (which corresponds to a situation where the players in N \ N  were proposers in the bidding mechanism with their proposals being rejected and with their bids being collected). This way, we define the following v-f reduction: 553 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Example 4. We define the PW v-f reduction  P W by:    NP W , N  ( v )( S ) ≡ v ( S ) − Sh i ( N  , v | N  ) + Sh i ( N , v ), (7) i∈ S i∈ S for all S , N  , N ∈ Pfin (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . Theorem 4 shows that  P W is characterized in a similar way to Theorems 1, 2, and 3. It uses the alternative property of proportional addition invariance, which we have described in Section 3 (see Axiom 7). Theorem 4. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and proportional addition invariance if and only if  = P W . Moreover,  P W induces the Shapley value. PW Theorem 4 also identifies the value ϕ  induced by the path-independent v-f reduction  P W . Given that the PW bidding mechanism implements the Shapley value, it is unsurprising that the value induced by the reduction is also the Shapley value. On the other hand, nothing in the bidding mechanism suggests that the equilibrium bids are related to the size of the coalitions. Therefore, the characterization of the PW v-f reduction owing to the axiom of proportional addition invariance provides a new perspective on the out-of-equilibrium payoffs of the players in the bidding mechanism. We now use the duality theory developed in the previous section to provide and characterize another v-f reduction, the ∗ dual of  P W , which we denote by  P W . To that end, we first identify the dual of the proportional addition invariance (since the other axioms used in the characterization of Theorem 4 are self-dual). The proportional addition invariance prescribes that a change in the worth of every coalition (except for the grand coalition) that is proportional to the size of the coalition, should not affect the strategic possibilities of the players, hence it should not affect the reduction of the unanimity game either. The reverse-proportional additional invariance axiom proposes that the reduction should not be affected if the worth of every coalition is changed in reverse proportion to their size. Axiom 10. A v-f reduction  satisfies reverse-proportional addition invariance if for all α ∈ R and all N , N  ∈ Pfin (U ) such that N   N, then n −1   N , N  (u N + w (k,(n−k)α ) ) =  N , N  (u N ). k =1 Proposition 5. The dual of the proportional addition invariance axiom is the reverse-proportional addition invariance axiom. ∗ Theorem 5 provides the characterization of  P W , which we formally define in Example 514 : ∗ Example 5. We define the PW∗ v-f reduction  P W by: ∗       NP W , N  ( v )( S ) ≡ v ( S ∪ ( N \ N )) − v ( N \ N ) − Sh i ( N  , v N \ N ) + Sh i ( N , v ), (8) i∈ S i∈ S for all S , N  , N ∈ Pfin (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . Theorem 5. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and reverse-proportional addition invariance if and only if ∗  = P W . ∗ Moreover,  P W induces the Shapley value. We note that Example 5 and Theorem 5 suggest the possible existence of a PW-style bidding mechanism such that its subgames on the off-equilibrium path would correspond to the dual PW v-f reduction. Sun (2020) constructs such a mechanism and shows that it implements the Shapley value. 14 ∗ See the Appendix for the derivation of the expression for  P W . 554 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Taken together, Theorems 2 to 5 provide additional evidence that the Shapley value is a solution concept with strong properties. Indeed, it is induced by v-f reductions that are characterized by very diverse invariance properties. We can use an operator that reduces a game so as to keep the same players’ strategic possibilities after an identical change in the worth of all the one-player coalitions or of all maximum possible compensations; or after a change that is proportional to the number of players in any subcoalition, or that is reverse to the number of players in any subcoalition. The Shapley value is attained after any of those different reductions. 5.5. A value-free reduction inducing the Banzhaf value The objective of this subsection is to illustrate how to use our approach to characterize v-f reductions that induce solution concepts different from the Shapley value, or the stand-alone value. In particular, we propose a v-f reduction that induces the Banzhaf value, which we introduced in Section 2. Dragan (1996) proposes a reduced game which is implicitly defined by a functional equation to axiomatize the Banzhaf value.15 In contrast, we propose a v-f reduction that is based on the same basic axioms used in our previous characteriza- tions, to which we add a new axiom that we call the “maximum ignorance” property: Axiom 11. A v-f reduction  satisfies maximum ignorance if for all N ∈ Pfin (U ) such that | N | ≥ 2, all i ∈ N, all α ∈ R, and all S ⊆ N \ {i }, α  N , N \{i } (α u N )( S ) = u N ( S ∪ {i }). 2 The maximum ignorance property takes the view that when player i is removed from the scene, he is still able to exert influence on the rest of the players, but his influence is uncertain. The resulting reduced game is a game of the remaining players contingent on the removed player’s behavior. However, unlike for instance the HM v-f reduction, the model analyst is totally ignorant of the removed players’ behavior. So the predicted distribution should be the one with the maximum entropy, which is, player i independently chooses to join or leave with equal probability (for an introduction to the principle of maximum entropy, see e.g., chapter 11 of Jaynes, 2003). Then ( N  ,  N , N  (u N )) can be interpreted as the resulting expected game. The reduction that we propose is given in the next example. We call it the Banzhaf v-f reduction. Example 6. We define the Banzhaf v-f reduction  Ban by:  1  NBan , N  ( v )( S ) ≡  [ v ( S ∪ T ) − v ( T )], (10) 2n−n T ⊆N \N  for all S , N  , N ∈ Pfin (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . We can interpret the Banzhaf v-f reduction as follows. Consider a game ( N , v ) ∈ G N that is reduced to be played by players in N  ⊆ N. The players in the coalition S ⊆ N  can collaborate with any subset T of the set of removed players N \ N  . Then, they obtain a worth of v ( S ∪ T ) but they have to compensate the players in T with the worth of their coalition v ( T ). Each of the possible coalitions T ⊆ N \ N  has the same probability of being available. Therefore, the worth of a coalition S ⊆ N  in ( N  ,  N Ban , N  ( v )) is the simple average of the marginal worth that S can add to the worth of the coalitions T ⊆ N \ N .  Ban Theorem 6 provides an axiomatic characterization of  Ban . It also postulates that ϕ = Ban. Theorem 6. A v-f reduction  satisfies additivity, null player out, permanent null player, path independence, and the maximum igno- rance property if and only if  =  Ban . Moreover,  Ban induces the Banzhaf value. 15 The reduced game ϕ proposed by Dragan (1996) is implicitly defined as follows: for all S , N , N  ∈ P f in (U ) such that S ⊆ N   N and all ( N , v ) ∈ G N ,    Bani ( S ∪ ( N \ N  ), v | S ∪( N \ N  ) ) − ϕi ( S ∪ ( N \ N  ), v | S ∪(N \N  ) ). ϕ Bani ( S ,  N , N  ( v )| S ) ≡ (9) i∈ S i ∈ S ∪( N \ N  ) i∈N \N  555 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 5.6. The axioms of anonymity and linearity In this subsection, we discuss two additional properties that v-f reductions can satisfy: anonymity and linearity. One sensible property that many values satisfy is anonymity, which requires that the players’ names are irrelevant for the value they obtain in the game. We can propose an axiom for v-f reductions in the same spirit. The axiom of anonymity for v-f reductions requires that the name of the players does not matter in the reduction of the game. To formally define the axiom, let σ : N → U be an injection. For ( N , v ) ∈ G N , we define σ v ∈ G σ [ N ] by σ v ( T ) ≡ v (σ −1 ( T )) for all T ⊆ σ [ N ]. Axiom 12. A v-f reduction  satisfies anonymity if for all S , N  , N ∈ Pfin (U ) such that S ⊆ N  ⊆ N, all ( N , v ) ∈ G N , and all injections σ : N → U , then σ [ N ],σ [ N  ] (σ v )(σ [ S ]) =  N , N  ( v )( S ). (11) Anonymity of a v-f reduction implies that the contribution of a player in the reduced game depends not on his name but on his contribution in the initial game. It also implies that if two players in the initial game are identical in terms of their contribution, then the reduced game if one of them is removed should be the same if the other is removed. We notice that although anonymity refers to the way games are reduced according to v-f reductions, it has implications for the prescribed payoff that equal players obtain in the induced value. In fact, if we substitute both N  and S with {i } in Axiom 12, we have  N ,{i } ( v )({i }) = σ [ N ],{σ (i )} (σ v )({σ (i )}), which is, ϕi ( N , v ) = ϕσ(i ) (σ [ N ], σ v ). Therefore, anonymity of a v-f reduction  implies anonymity of its induced value ϕ  . We state this result in Proposition 6. Proposition 6. If a v-f reduction  satisfies anonymity, then the induced value ϕ  satisfies anonymity as well. None of the axioms used in the characterizations provided in Theorems 1 to 6 is related to the idea of anonymity. However, Proposition 7, whose proof is immediate, shows that all of the v-f reductions characterized in our paper satisfy the axiom of anonymity. ∗ Proposition 7. The v-f reductions  Sub ,  H M ,  O N H F ,  P W ,  P W , and  Ban satisfy anonymity. Given that all the characterizations use the axioms of additivity, null player out, permanent null player, and path in- dependence, one may think that these axioms imply anonymity. Moreover, like the aforementioned axioms, we can easily check that anonymity is a self-dual property. However, Example 7 satisfies our four basic properties although it does not satisfy anonymity. Example 7. Given X ⊆ U , the v-f reduction  X is defined by   NX , N  ( v )( S ) ≡ Sh i ( S ∪ (( N \ N  ) ∩ X ), v | S ∪(( N \ N  )∩ X ) ) i∈ S   − Sh i ( N  ∪ (( N \ N  ) ∩ X ), v | N  ∪(( N \ N  )∩ X ) ) + Sh i ( N , v ), (12) i∈ S i∈ S for all S , N  , N ∈ P f in (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . Proposition 8. The v-f reduction  X satisfies additivity, null player out, permanent null player, and path independence for any X ⊆ U . However, it does not satisfy anonymity. Finally, let us mention that all the v-f reductions that we have characterized also satisfy linearity, that is, they are homogeneous (see Axiom 3). As anonymity, homogeneity is not implied by our four basic axioms. The construction of an example requires the use of a Hamel basis and we provide it in the Appendix. 5.7. A comment on non-additive value-free reductions Like the classical axiomatization of the Shapley value, our axiomatizations of v-f reductions rely on additivity. It is imme- diate that the value induced by an additive v-f reduction is necessarily additive as well. Hence, all the values characterized by v-f reductions that satisfy our basic axioms are additive, as is the case for the Shapley, the stand-alone, and the Banzhaf values. 556 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 However, we can also consider non-additive v-f reductions. Such reductions can induce non-additive values, such as the prenucleolus PN (Schmeidler, 1969).16 We illustrate here this possibility by adapting the Davis-Mascher (DM) reduced game, which allows characterizing the prenucleolus in terms of consistency (Sobolev, 1975). We start by formally defining the prenucleolus.  Denote by X ( N , v ) the set of preimputations of the game ( N , v ), that is, x ∈ X ( N , v ) if x ∈ R N and i ∈ N xi = v ( N ). For  each preimputation x ∈ X ( N , v ) and each coalition S ⊆ N, we denote by e S (x) ≡ i ∈ S xi − v ( S ) the “excess” of coalition S at x. Also, we denote by e (x) ≡ (e S (x)) S ∈2N \{ N ,∅} the vector of excesses, where the entries are arranged in increasing order. Finally, for x, y ∈ X ( N , v ), we denote by e (x) lx e ( y ) if the vector e (x) is lexicographically superior to e ( y ).17 We can now define the prenucleolus: PN ( N , v ) ≡ {x ∈ X ( N , v ) :  y ∈ X ( N , v ) s.t. e ( y ) lx e (x)}. We construct the v-f version of the DM reduced game as we did for the HM and ONHF v-f reductions: We take the original reduced game, and we substitute the generic value used in that game by the particular value that it helps to characterize. For the DM, we use the prenucleolus: Example 8. We define the DM v-f reduction  D M by: ⎧  ⎪ v ( N ) − i ∈ N \ N  PN i ( N ⎪ , v) if S = N   N , ⎪ ⎨max T ⊆N \N  v ( S ∪ T ) − i ∈ T PN i ( N , v ) if ∅ = S  N   N ,  ND ,MN  ( v )( S ) ≡ ⎪ ⎪ v(S) if N  = N , ⎪ ⎩ 0 if S = ∅, for all S , N  , N ∈ P f in (U ) such that S ⊆ N ⊆ N and all ( N , v ) ∈ G . N The v-f reduction  D M is not additive and, as one can expect, it induces the prenucleolus.18 Moreover, it satisfies null player out and permanent null player, as well as anonymity. However, we do not have a characterization of the DM v-f reduction. 6. New characterizations of the Banzhaf and the stand-alone values We have based some of our examples of v-f reductions on existing reduced games, which were introduced to study the internal consistency of values. In this section, we consider the reverse process. We take a v-f reduction  that is defined   without reference to an existing reduced game. We look for value-reductions ϕ such that  = ϕ , where ϕ results ϕ from substituting the value ϕ induced by the v-f reduction  in  . This process may identify reduction games ϕ  that  would allow the characterization of solution concepts using consistency properties (as in Hart and Mas-Colell, 1989, and Oishi et al., 2016). We conduct such a reverse process by introducing a new reduced game that, following the terminology used in Lehrer ϕ (1988), we call the “amalgamating reduced game.” We denote it by  A . It is inspired by the definition of the Banzhaf v-f reduced game  . It satisfies that if we substitute ϕ for the Banzhaf value Ban (which is the value induced by  Ban ) in Ban ϕ Ban  A , then  A =  Ban (see the Appendix). ϕ The definition of  A requires some preliminaries. For ( N , v ) ∈ G N and S ∈ 2 N \ {∅}, we may “amalgamate” the coalition S into one player and denoted him by S̄ . Formally, we define the S-amalgamated game (( N \ S ) ∪ { S̄ }, v S ) (Lehrer, 1988) by:  v (( T \ { S̄ }) ∪ S ) if S̄ ∈ T , v S (T ) ≡ v (T ) otherwise. ϕ Then we define the amalgamating reduced game (A reduction, for short)  A as follows: ϕ Definition 11. Given a value ϕ , the A reduction  A is defined by:   ϕ ϕ S̄ (( N \ N  ) ∪ { S̄ }, ( v | S ∪(N \N  ) ) S ) if S ∈ 2 N \ {∅},  NA , N  ( v )( S ) ≡ 0 if S = ∅, for all S , N  , N ∈ Pfin (U ) such that S ⊆ N   N and all ( N , v ) ∈ G N . 16 Schmeidler (1969) defines the nucleolus for 0-monotonic TU games. Throughout our paper, we focus on v-f reductions defined on an unrestricted domain, so we consider the prenucleolus rather than the nucleolus. The latter is empty for those TU games with an empty set of imputations. 17 That is, there exists t  ∈ {1, 2, ..., 2n − 2} such that et (x) ≥ et ( y ), for t = 1, . . . , t  and et  (x) > et  ( y ). DM  18 Indeed, ϕN,{i } ( N , v ) = ND ,{Mi } ( v )({i }) = v ( N ) − j ∈ N \{i } PN j ( N , v ) = PN i ( N , v ), where the last equality follows from efficiency of the prenucleolus. 557 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 To interpret the A reduction, consider a value ϕ , a TU game ( N , v ), and a coalition N  ⊆ N. Similar to the HM reduced game, every non-empty coalition S ⊆ N  collaborates with all removed players in N \ N  , and players in N  \ S exert no influence. However, in the A reduction, the coalition S is treated as an individual player. Moreover, while in the HM reduced game, the worth of S is the residue after paying up those players in N \ N  , the worth of S in the amalgamating reduced game is what is due for S as a single player according to ϕ . ϕ The reduced game  A allows the characterization of the Banzhaf value in a parallel manner as Hart and Mas-Colell (1989) and Oishi et al. (2016) characterize the Shapley value: The Banzhaf value is the only value that is consistent relative to the A reduction and that is standard for two-player games. We establish this result in Theorem 7. Theorem 7. Let ϕ be a solution. Then: ϕ (i) ϕ is consistent relative to  A ; and (ii) ϕ is standard for two-player games; if and only if ϕ is the Banzhaf value. The Banzhaf value is not the only value that is consistent relative to the A reduction. The stand-alone value is also consistent relative to the amalgamating reduced game. Interestingly, it can also be characterized by consistency plus the behavior of the value in the two-player games. Theorem 8. Let ϕ be a solution. Then: ϕ (i) ϕ is consistent relative to  A ; and (ii) ϕ coincides with the stand-alone value for two-player games; if and only if ϕ is the stand-alone value. Theorems 7 and 8 provide new characterizations of the Banzhaf and the stand-alone values. They also highlight that, once the “right” consistency requirement is applied, they only differ in their prescriptions for two-player games. 7. Logical independence In this section, we show that our characterization of the HM v-f reduction is minimal in the sense that none of the characterizing properties can be deduced from the rest. Each time we leave out one axiom, we can find examples of v-f reductions satisfying the remaining four properties. First, as we have already shown in Theorems 1, 3 and 4, the subgame v-f reduction, the ONHF v-f reduction and the PW v-f reduction satisfy all the axioms but 1-addition property. Examples 9, 10, 11, and 12 show that the axioms of null player out, permanent null player, additivity, and path independence are not redundant either. Example 9 (No null player out). Let ¬ N P O be the v-f reduction defined by: ¬ NPO N , N  ( v )( S ) ≡ 0, for all S , N  , N ∈ P f in (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . The v-f reduction ¬ N P O satisfies additivity, permanent null player, path independence, and 1-addition invariance, but it does not satisfy null player out. Example 10 (No permanent null player). Let ¬ P N P be the v-f reduction defined by:  0 S =∅ ¬ PNP N , N  ( v )( S ) ≡ v ( S ∪ ( N \ N  )) otherwise, for all S , N  , N ∈ P f in (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . The v-f reduction ¬ P N P satisfies additivity, null player out, path independence, and 1-addition invariance, but it does not satisfy permanent null player. Example 11 (No additivity). Let ¬ A be the v-f reduction defined by:   NH ,MN  ( v ) if Sh i ( N , v ) = 0 for all i ∈ N \ N  ¬ A N ,N  ( v ) ≡ ¬ NPO N ,N  ( v ) otherwise, for all N , N  ∈ P f in (U ) such that N  ⊆ N and all ( N , v ) ∈ G N . The v-f reduction ¬ A satisfies null player out, permanent null player, path independence, and 1-addition invariance, but it does not satisfy additivity. 558 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Example 12 (No path independence). Let ¬ P I be the v-f reduction defined by:  2v ( S ) if n = n = 1 ¬ PI N , N  ( v )( S ) ≡  NH ,MN  ( v )( S ) otherwise, for all S , N  , N ∈ P f in (U ) such that S ⊆ N  ⊆ N and all ( N , v ) ∈ G N . The v-f reduction ¬ P I satisfies additivity, null player out, permanent null player, and 1-addition invariance, but it does not satisfy path independence. 8. Conclusion In this paper, we introduce the notion of the value-free reduction of a coalitional game with transferable utility. A v-f reduction of a game describes the change in the worth of the coalitions in a TU game when some players leave the game.19 Thus, this new concept allows us to study TU games from a new perspective, focusing on the properties that a v-f reduction may or may not satisfy. A v-f reduction induces a value. One may say that the value somehow reflects the properties of the v-f reductions that induce it. We consider additive v-f reductions that are path independent and satisfy properties that indicate that null players must still be treated as null players when any such reduction is applied. These properties by themselves do not pin down a unique v-f reduction. Moreover, they do not identify a unique value induced by the reductions either. We define v-f reductions that satisfy all the previous properties and induce either the Shapley value, or the Banzhaf value, or the stand-alone value.20 To characterize each of the examples of v-f reductions that we have defined, we use an additional axiom to ensure that the players remaining in the reduced game keep the same strategic perspective as in the initial game after a change in the worth of some particular coalitions. These are invariance properties. The exercises suggest that the Shapley value is a resilient value as it is induced by several v-f reductions, each characterized by a different invariance axiom. A duality theory for v-f reductions, which is also developed in this paper, helps in the proof of some of the characterizations. Moreover, we show that the duality theory can be helpful in the identification of new mechanisms that implement specific values. We also show that our new approach is a useful tool to provide new characterizations of values in terms of consistency. In this paper, we provide new characterizations of the Banzhaf and the stand-alone values. Appendix Proof of Proposition 1. We prove that the value ϕ  induced by a path-independent v-f reduction  is consistent relative to . For a given N we have  N  ,{i } ◦  N , N  =  N ,{i } for all N  ⊆ N and all i ∈ N  , by path independence. Therefore, for any      ( N , v ) ∈ G N , given that ( N  ,  N , N  ( v )) ∈ G N , we have ϕi N  ,  N , N  ( v ) =  N  ,{i }  N , N  ( v ) ({i }) =  N  ,{i } ◦  N , N  ( v )({i }) =  N ,{i } ( v )({i }) = ϕi ( N , v ). Hence, ϕ  is consistent relative to .  To prove Proposition 3, as well as Propositions 4 and 5 later, several properties of the mapping v → v ∗ are useful, which we state in Lemma 1: Lemma 1. The mapping v → v ∗ is additive. Moreover, if i ∈ N is a null player in ( N , v ) ∈ G N , then player i is also a null player in ( N , v ∗ ). Proof of Lemma 1. We check that v → v ∗ is additive: for all ( N , v ), ( N , w ) ∈ G N and all S ⊆ N, then ( v + w )∗ ( S ) = ( v + w )( N ) − ( v + w )( N \ S ) = ( v ( N ) + w ( N )) − ( v ( N \ S ) + w ( N \ S )) = ( v ( N ) − v ( N \ S )) + ( w ( N ) − w ( N \ S )) = v ∗ ( S ) + w ∗ ( S ). To see that if i is a null player in ( N , v ), then i is also a null player in ( N , v ∗ ) we have that for all S ⊆ N \ {i }, v ∗ ( S ∪ {i }) − v ∗ ( S ) = ( v ( N ) − v ( N \ ( S ∪ {i }))) − ( v ( N ) − v ( N \ S )) = v ( N \ S ) − v ( N \ ( S ∪ {i })) = 0.  Proof of Proposition 3. To verify that additivity is self-dual, we show that the mapping v → ∗N , N  ( v )( S ) is additive if  ∗  ∗ the mapping v →  N , N  ( v )( S ) is additive. Indeed, ∗N , N  ( v + w )( S ) =  N , N  (( v + w )∗ ) ( S ) =  N , N  ( v ∗ + w ∗ ) ( S ) =  ∗  ∗  ∗  N , N  ( v ∗ ) +  N , N  ( w ∗ ) ( S ) =  N , N  ( v ∗ ) ( S ) +  N , N  ( w ∗ ) ( S ) = ∗N , N  ( v )( S ) + ∗N , N  ( w )( S ), where the first equality follows from Definition 8, the second and fourth from the additivity of v → v ∗ (Lemma 1 in the Appendix), and the third from the additivity of . Therefore, additivity is self-dual. 19 Concerning this depiction, v-f reductions may be viewed as generalizations of the subgame operator by allowing the players who leave the game to influence the remaining players. For concepts where subgame plays a role, such as population monotonicity (Sprumont, 1990) and projection consistency (Funaki and Yamato, 2001), one can define and study versions where the subgame is replaced with a distinct v-f reduction. 20 It is also natural to think about v-f versions of the Moulin-Tadenuma reduced game (Moulin, 1985, and Tadenuma, 1992). Several linear values are consistent relative to this reduced game, such as the equal division value (that distributes the worth of the grand coalition equally among the players) and the zero value (that distributes zero to every player). Each of them can give rise to linear and path-independent v-f reductions inducing these two values, respectively. 559 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 We now check that null player out is self-dual. We show that if  satisfies the null player  out axiom, ∗ then ∗N , N \{i } ( v )( S ) = v ( S ) for all S ⊆ N \ {i } if i ∈ N is a null player in ( N , v ). Indeed, ∗N , N \{i } ( v )( S ) =  N , N \{i } ( v ∗ ) ( S ) = ( v ∗ | N \{i } )∗ ( S ) = v ∗ | N \{i } ( N \ {i }) − v ∗ | N \{i } (( N \ {i }) \ S ) = v ∗ ( N \ {i }) − v ∗ ( N \ ( S ∪ {i })) = v ∗ ( N ) − v ∗ ( N \ S ) = v ( S ), where the first equality follows from Definition 8, the second one holds because i is a null player in ( N , v ∗ ) according to Lemma 1, the third from the definition of the dual of a game, and the penultimate equality follows again from the fact that i is a null player in ( N , v ∗ ). Therefore, null player out is self-dual. We verify that the permanent null player property is self-dual by proving that if  satisfies this property and i ∈ N  is a null player in ( N , v ), then i is a null player in ( N  , ∗N , N  ( v )) as well. Let i ∈ N  be a null player in ( N , v ). Then, from Lemma 1, i ∈ N  is a null player in ( N , v ∗ ) and, by the permanent  null player ∗ property of , he is also a null player in ( N  ,  N , N  ( v ∗ )). Using Lemma 1 again, i is a null player in ( N  ,  N , N  ( v ∗ ) ), that is, in ( N  , ∗N , N  ( v )). Therefore, the permanent null player property is self-dual. Finally, we prove that path independence is self-dual by proving ∗N 2 , N 3 (∗N 1 , N 2 ( v )) = ∗N 1 , N 3 ( v ) if  is path indepen-   ∗ ∗ ∗  ∗  ∗ dent: ∗N 2 , N 3 (∗N 1 , N 2 ( v )) =  N 2 , N 3 (  N 1 , N 2 ( v ∗ ) ) =  N 2 , N 3 ( N 1 , N 2 ( v ∗ )) =  N 1 , N 3 ( v ∗ ) = ∗N 1 , N 3 ( v ), where the first and last equalities follow from Definition 8, the second from v ∗∗ = v, and the third from the assumption of the path- independence of . Therefore, path independence is self-dual.  Proof of Remark 1. Given N ∈ P f in (U ) and ( N , v ) ∈ G N , take any i ∈ U \ N. Define ( N ∪ {i }, w ) ∈ G N ∪{i } by w ( S ) ≡ v ( S \ {i }) for all S ⊆ N ∪ {i }. Notice that player i is a null player in ( N ∪ {i }, w ) and that the subgame of ( N ∪ {i }, w ) restricted to N is ( N , v ). Then for any v-f reduction  satisfying null player out and path independence,  N , N ( v ) =  N , N ( N ∪{i }, N ( w )) =  N ∪{i }, N ( w ) = v, where the first and the third equality follow from null player out and the second from path independence. Therefore,  N , N must be an identity function if  satisfies null player out and path independence.  Since every v-f reduction we will present satisfies null player out and path independence, we will not repeat the property established in Remark 1 in the proof of their corresponding theorems below. Proof of Theorem 1. It is immediate that the subgame v-f reduction satisfies all the stated properties. We now prove that if the v-f reduction  satisfies the five properties, then  = Sub . Notice  first that, under path independence, it suffices to show the equality restricted to one-player removal operators  N , N \{i } , for all N ∈ P f in (U ) and all i ∈ N. Second, by additivity, it suffices to establish the equality for each operator  N , N \{i } restricted to the set of all scalar   multiples of elements in a basis of G N . We choose the set of all scalar multiples of all unanimity games α u T T ∈2N \{∅},α ∈R . We show that  N , N \{i } (α u T ) =  N , N \{i } (α u T ) for all T ∈ 2 \ {∅}, all α ∈ R, and all i ∈ N by induction on n. We Sub N notice that since α u N = w (n,α ) , additivity and grand-coalition invariance imply that  N , N \{i } (α u N ) = 0 = α u N | N \{i } =  NSub , N \{i } (α u N ). Thus, we only need to check the equality of the remaining scalar multiples of elements in the basis, i.e.,   αu T T ∈2 N \{∅, N },α ∈R . Consider N = {i , j }, that is, n = 2. (a) When T = { j }, then {i , j },{ j } (α u { j } )({ j }) = α u { j } |{ j } ({ j }) by null player out, since i is a null player in ({i , j }, α u { j } ). (b) When T = {i }, then {i , j },{ j } (α u {i } )({ j }) = 0 = α u {i } |{ j } ({ j }) by permanent null player, since j is a null player in ({i , j }, α u {i } ). Hence,  N , N \{i } (α u T )( S ) = α u T | N \{i } ( S ) =  N , N \{i } (α u T )( S ) for all S ⊆ N \ {i }, all T Sub such that | T | = 1, all α ∈ R, and all N such that | N | = 2. Now we proceed to consider any N, and suppose that the induction property holds for any set with fewer than n players. (a) When i ∈ / T then i is a null player in ( N , α u T ), hence  N , N \{i } (α u T ) = α u T | N \{i } by null player out. (b) We show that  N , N \{i } (α u T )( S ) = α u T | N \{i } ( S ) for all S ⊆ N \ {i } when i ∈ T and T  N. Take any player j ∈ N \ T . Then, j is a null player in ( N , α u T ). Moreover, by the permanent null player property, j is also a null player in ( N \ {i },  N , N \{i } (α u T )). We consider two possibilities. (b1) First, if S ⊆ N \ {i , j }, then    N , N \{i } (α u T )( S ) =  N , N \{i } (α u T )| N \{i , j } ( S ) =  N \{i }, N \{i , j }  N , N \{i } (α u T ) ( S )   =  N \{ j }, N \{i , j }  N , N \{ j } (α u T ) ( S ) =  N \{ j }, N \{i , j } (α u T | N \{ j } )( S ), (13) where the first equality holds because S ⊆ N \ {i , j }; the second by null player out, given that j is a null player in the game ( N \ {i },  N , N \{i } (α u T )); the third by path independence; and the fourth by null player out, given that j is a null player in ( N , α u T ). We apply the induction argument to state that the last expression (which involves a reduction from a set of n − 1 players) is equal to  N \{ j }, N \{i , j } (α u T | N \{ j } )( S ) = α u T | N \{i , j } ( S ) = α u T ( S ), where the last equality holds Sub because S ⊆ N \ {i , j }. (b2) Second, if j ∈ S, then  N , N \{i } (α u T )( S ) =  N , N \{i } (α u T )( S \ { j }) because j is a null player in ( N \ {i },  N , N \{i } (α u T )). Now we apply equation (13) to S \ { j } and, by the same argument as in (b1),  N , N \{i } (α u T )( S ) = α u T ( S \ { j }), which is equal to α u T ( S ) since j is a null player in ( N , α u T ). Thus, if a v-f reduction satisfies the five properties, then it is equal to  Sub .  560 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Proof of Theorem 2. We verify the stated properties of  H M . First,  H M is the composition of three functions: the restric- tion operator, the Shapley value, and the summation operator. It is easy to check that the three functions are additive. Therefore,  H M is additive. Second, to verify that  H M satisfies null player out, let i ∈ N be a null player in ( N , v ) ∈ G N . Then,  N HM , N \{i } ( v )( S ) =   j∈ S Sh j ( S ∪ ( N \ ( N \ {i })), v | S ∪( N \( N \{i })) ) = j ∈ S Sh j ( S ∪ {i }, v | S ∪{i } ) = v ( S ∪ {i }) − Sh i ( S ∪ {i }, v | S ∪{i } ) = v ( S ∪ {i }) = v ( S ), where the third equality follows from the efficiency of the Shapley value, the fourth from the null player property of the Shapley value, and the fifth holds because i is a null player in ( N , v ).   Third, we check that  H M satisfies permanent null player.  Let i ∈ N be a null player in ( N , v ) ∈ G . Then, for all S ⊆ N \ N {i }, D i ( NH,MN  ( v ))( S ) =  NH,MN  ( v )( S ∪ {i }) −  NH,MN  ( v )( S ) = [ j ∈ S ∪{i } Shi (( S ∪ {i }) ∪ ( N \ N  ), v |( S ∪{i })∪(N \ N  ) )] − [ j ∈ S Shi ( S ∪    ( N \ N  ), v | S ∪(N \ N  ) )] = [ j ∈ S Shi (( S ∪ {i }) ∪ ( N \ N  ), v |( S ∪{i })∪(N \ N  ) )] − [ j ∈ S Shi ( S ∪ ( N \ N  ), v | S ∪(N \ N  ) )] = [ j ∈ S Shi ( S ∪    ( N \ N ), v | S ∪(N \ N  ) )] − [ j ∈ S Shi ( S ∪ ( N \ N ), v | S ∪(N \ N  ) )] = 0, where the third equality follows from the null player property of the Shapley value, and the fourth from null player out of the Shapley value (see Derks and Haller, 1999).  Fourth, we prove the path independence axiom. For any T ⊆ S, we can write  H M S ∪( N \ N  ), S ( v | S ∪(N \ N  ) )( T ) = i ∈ T Shi ( T ∪ ( N \ N  ), v | T ∪(N \ N  ) ) =  NH,MN  ( v )( T ) =  NH,MN  ( v ) | S ( T ), where the first equality holds because v | S ∪(N \ N  ) | T ∪( S ∪(N \ N  ))\ S ) = v | T ∪(( S ∪( N \ N  ))\ S ) = v | T ∪( N \ N  ) . Therefore:  NH ,MN  ( v )| S =  HS ∪( M N \ N  ), S ( v | S ∪( N \ N ) ),  (14)  NH ,MN  ( v )( S ) =  HS ∪( M N \ N  ), S ( v | S ∪( N \ N ) )( S ).  (15)  HM  We now claim that, given equations (14) and (15), the verification of path independence, that is,  N HM 2 ,N3 N1 ,N2 ( v ) ( S ) =  NH1M, N 3 ( v )( S ) for all N 1 , N 2 , N 3 , S ∈ P f in (U ) such that S ⊆ N 3 ⊆ N 2 ⊆ N 1 , is equivalent to verifying the condition only for S = N 3 , i.e.,    NH2M, N 3  NH1M, N 2 ( v ) ( N 3 ) =  NH1M, N 3 ( v )( N 3 ). (16) To prove the equivalence, we use (15), where we substitute N , N  and v by N 2 , N 3 and  N HM 1 ,N2 ( v ), to obtain    HM   NH2M, N 3  NH1M, N 2 ( v ) ( S ) =  HS ∪( M N 2 \ N 3 ), S  N 1 , N 2 ( v )| S ∪( N 2 \ N 3 ) ( S ). (17) Similarly, we substitute N , N  and S by N 1 , N 2 and S ∪ ( N 2 \ N 3 ) in (14), to obtain  NH1M, N 2 ( v )| S ∪( N 2 \ N 3 ) =  HS ∪( M N 2 \ N 3 )∪( N 1 \ N 2 ), S ∪( N 2 \ N 3 ) ( v | S ∪( N 2 \ N 3 )∪( N 1 \ N 2 ) ), i.e.,  NH1M, N 2 ( v )| S ∪( N 2 \ N 3 ) =  HS ∪( M N 1 \ N 3 ), S ∪( N 2 \ N 3 ) ( v | S ∪( N 1 \ N 3 ) ). (18) Using (18) in equation (17), we have    HM   NH2M, N 3  NH1M, N 2 ( v ) ( S ) =  HS ∪( M N 2 \ N 3 ), S  S ∪( N 1 \ N 3 ), S ∪( N 2 \ N 3 ) ( v | S ∪( N 1 \ N 3 ) ) ( S ). Then, the worth of coalition S ⊆ N 3 in the game resulting from two sequential reductions of ( N 1 , v ) (from N 1 to N 2 , then from N 2 to N 3 ) is equal to the worth of the grand coalition S in the game resulting from two reductions of ( S ∪ ( N 1 \ N 3 ), v | S ∪( N 1 \ N 3 ) ) (from S ∪ ( N 1 \ N 3 ) to S ∪ ( N 2 \ N 3 ), then from S ∪ ( N 2 \ N 3 ) to S). This property means that it suffices to verify that the worth of the grand coalition satisfies path independence, that is, that equation (16) holds for all possible games. To prove (16), we use the definition of  H M :      NH2M, N 3  NH1M, N 2 ( v ) ( N 3 ) = HM Sh i ( N 2 ,  N 1 ,N2 ( v )) = HM Sh i ( N 1 , v ) =  N 1 ,N3 ( v )( N 3 ). i∈N3 i∈N3 Therefore,  H M is path independent. , N  (u N + w (1,α ) ) =  N , N  (u N ) HM HM Finally, we verify the 1-invariance property of  H M . The axiom of additivity implies that  N  , N  ( w (1,α ) ) = 0. We show that  N , N  ( w (1,α ) )( S ) = 0 for all S ⊆ N . By definition of  , N  ( w (1,α ) )( S ) = HM HM HM HM if and only if  N , N   i∈ SSh i ( S ∪ ( N \ N  ), w (1,α )| S ∪( N \ N  ) ). Notice that ( S ∪ ( N \ N  ), w (1,α )| S ∪( N \ N  ) ) ∈ G S ∪( N \ N ) is a game where each player is symmetric  with each other. Then, the Shapley valueprescribes an equal share of the worth of the grand coalition S ∪ ( N \ N  ). Thus,  i ∈ S Sh i ( S ∪ ( N \ N ), w (1,α )| S ∪( N \ N  ) ) = 1  i ∈ S | S ∪( N \ N  )| w (1,α ) ( S ∪ ( N \ N )) = 0. Therefore, the HM v-f reduction satisfies the 1-addition invariance property. To show the reverse implication of the theorem, we first prove the following lemma: Lemma 2. For all N ∈ P f in (U ) such that | N | > 2, the set {u T | T  N , T = ∅} ∪ { w (1,1) } forms a basis of G N . 561 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Proof of Lemma 2. Take any N ∈ P f in (U ). To prove Lemma 2, we start by showing the following equality between games in G N :  (−1)n nu N = − w (1,1) + (−1)s−1 su S . (19) S ∈2 N \{∅, N } We show that the two functions in equation (19) are equal when evaluated at any T ⊆ N, by considering three different  cases: (a) If T  N and | T | = 1, then − w (1,1) ( T ) + S ∈2N \{∅, N } (−1)s−1 su S ( T ) = −1 + u T ( T ) = 0 = u N ( T ) = (−1)n nu N ( T ). For the other two cases, we use the following formula:  s(−1)s−1 = 0, (20) S ∈2 T \{∅}  for any T such that | T | > 1.21 Then, (b) for T  N such that | T | > 1, we can write: − w (1,1) ( T ) + S ∈2N \{∅, N } (−1)s−1 su S ( T ) =  s−1  S ∈2 T \{∅} s(−1) = 0 = u N ( T ) = (−1)n nu N ( T ). Finally, (c) for T = N, − w (1,1) ( T ) + S ∈2N \{∅, N } (−1)s−1 su S ( T ) =  s−1  s−1 S ∈2 N \{∅, N } (−1) s = (−1)n n + S ∈2 N \{∅} (−1) s = (−1)n n = (−1)n nu N ( T ). Given that equation (19) holds and the set {u T | T ⊆ N , T = ∅} forms a basis of G N , then the set resulting from replacing u N with w (1,1) in this basis spans G N , which proves Lemma 2.  We now continue with the reverse implication of Theorem 2. We prove that  =  H M if the v-f reduction  satisfies the five properties, using the same procedure as in the proof of Theorem 1. Because of path independence and additivity, it suffices to show that  N , N \{i } ( v ) =  NHM , N \{i } ( v ) for all i ∈ N and all v ∈ {α u T | T  N , T = ∅, α ∈ R} ∪ { w (1,α ) | α ∈ R} (see Lemma 2). First, if v = w (1,α ) , then additivity and 1-addition invariance imply  N , N \{i } ( w (1,α ) ) = 0 =  N HM , N \{i } ( w (1,α ) ) for all N ∈ P f in (U ) for all α ∈ R and all i ∈ N. Second, we show that  N , N \{i } (α u T ) =  N HM , N \{i } (α u T ) for all T ∈ 2 \ {∅, N }, all α ∈ R, and all i ∈ N by induction on n. N For N such that n = 2, the proof is identical to that of Theorem 1 since  H M and  Sub coincide for the proper subsets T of N and we did not use grand-coalition invariance in that part of the proof. Consider now any N and suppose that the induction property holds for any set with fewer than n players. (a) When i∈/ T then i is a null player in ( N , α u T ), hence  N , N \{i } (α u T ) = α u T | N \{i } =  NH,MN \{i } (α u T ) because both  and  H M sat- isfy null player out. (b) When i ∈ T and T  N, take any j ∈ N \ T . Player j is a null player in ( N , α u T ) and, under the permanent null player property, also in ( N \ {i },  N , N \{i } (α u T ). Therefore, (b1) if S ⊆ N \ {i , j }, then equation (13) holds by the same arguments as in the proof of Theorem 1. Using the induction argument, we have  N , N \{i } (α u T )( S ) =  N \{ j }, N \{i , j } (α u T | N \{ j } )( S ) =  NH\{ M j }, N \{i , j } (α u T | N \{ j } )( S ). Since j is a null player in ( N , α u T ) and  HM satisfies null player   \{ j }, N \{i , j } (α u T | N \{ j } )( S ) =  N \{ j }, N \{i , j }  N , N \{ j } (α u T ) ( S ) =  N , N \{i } (α u T )( S ). (b2) HM HM HM HM out and path independence, we have  N If j ∈ S, then  N , N \{i } (α u T )( S ) =  N , N \{i } (α u T )( S \ { j }) because j is a null player in ( N \ {i },  N , N \{i } (α u T )). Now we can apply equation (13) to S \ { j } and, by the same argument as in (b1),  N , N \{i } (α u T )( S ) =  N , N \{i } (α u T )( S \ { j }) = HM  NH,MN \{i } (α u T )( S ), where the last equality holds because j is a null player in ( N , α u T ). Therefore, if a v-f reduction satisfies the five properties, then it is equal to  H M . ,{i } ( v )({i }) = Sh i ({i } ∪ ( N \ {i }), v|{i }∪( N \{i }) ) = Sh i ( v ) for all ( N , v ) ∈ G and all i ∈ N. Therefore, HM N Finally, we notice that  N  HM induces the Shapley value.  Proof of Corollary 2. We prove that  O N H F is the dual of  H M . Indeed,  ∗  NO ,NNH F ( v ∗ ) ( S ) = NO ,NNH F ( v ∗ )( N  ) −  NO ,NNH F ( v ∗ )( N  \ S )     = Sh i ( N , v ∗ ) − Sh i ( N \ N  , ( v ∗ ) N ) − Sh i ( N , v ∗ ) i∈N  i∈N  \N  i∈N   N\S + Sh i ( N \ ( N  \ S ), ( v ∗ ) ) i ∈ N  \( N  \ S )    = Sh i ( N \ ( N  \ S ), ( v ∗ ) N \ S ) = Sh i ( S ∪ ( N \ N  ), ( v | S ∪( N \ N  ) )∗ ) i∈ S i∈ S  t    dxs d( xs ) d( (ts)xs ) 21 We check that (20) holds: S ∈2 T \{∅} s(−1)s−1 = S ∈2 T \{∅} sxs−1 x=−1 = S ∈2 T \{∅} dx x=−1 = S ∈2 T \{∅} dx x=−1 = s=1 dx x=−1 = d((1+x) −1) t dx x=−1 = t (1 + x)t −1 x=−1 = 0. 562 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568  = Sh i ( S ∪ ( N \ N  ), v | S ∪( N \ N  ) ), i∈ S where the first equality follows the definition of a dual game, the second one from the defining equation (6) of  O N H F , and the last equality from the self-duality of the Shapley value. To check the fourth equality, notice that, for all T ⊆ S ∪ ( N \ N  ),  on the one hand, v ∗ N \ S ( T ) = v ∗ ( T ∪ ( N  \ S )) − v ∗ ( N  \ S ) = [ v ( N ) − v ( N \ ( T ∪ ( N  \ S )))] − [ v ( N ) − v ( N \ ( N  \ S ))] = v ( N \ ( N  \ S )) − v ( N \ ( T ∪ ( N  \ S ))) = v ( S ∪ ( N \ N  )) − v (( S ∪ ( N \ N  )) \ T ); on the other hand, ( v | S ∪( N \ N  ) )∗ ( T ) = v | S ∪( N \ N  )  ( S ∪ ( N \ N  )) − v | S ∪(N \ N  ) (( S ∪ ( N \ N  )) \ T ) = v ( S ∪ ( N \ N  )) − v (( S ∪ ( N \ N  )) \ T ). Thus v ∗ N \ S = ( v | S ∪(N \ N  ) )∗ .  Proof of Proposition 4. Let  be a v-f reduction that satisfies 1-addition invariance. Our aim is to verify that ∗ satisfies (n − 1)-addition invariance. We first show that the dual of the game ( N , w (n−1,α ) ) is ( N , w (1,−α ) ). Indeed, w ∗(n−1,α ) ( S ) = w (n−1,α ) ( N ) − w (n−1,α ) ( N \ S ) = − w (n−1,α ) ( N \ S ) = w (1,−α ) ( S ) for all S ⊆ N.  ∗  ∗  Then, for any α ∈ R, we have ∗N , N  ( v + w (n−1,α ) ) =  N , N  (( v + w (n−1,α ) )∗ ) =  N , N  ( v ∗ + w ∗(n−1,α ) ) =  N , N  ( v ∗ + ∗  ∗ w (1,−α ) ) =  N , N  ( v ∗ ) = ∗N , N  ( v ), where the first and last equalities follow from Definition 8, the second from the additivity of v → v ∗ (see Lemma 1), the third equality follows from the fact that the dual of w (n−1,α ) is w (1,−α ) , and the fourth from 1-addition invariance of . Therefore, (n − 1)-addition invariance is dual to 1-addition invariance.  Proof of Theorem 3. The ONHF v-f reduction is dual to the HM v-f reduction. Then, by Proposition 3,  O N H F satisfies additivity, null player out, permanent null player, and path independence, because they are self-dual properties and  H M satisfies them. Similarly,  O N H F satisfies (n − 1)-invariance, which is dual to 1-addition invariance (see Proposition 4), because  H M satisfies 1-addition invariance. For the other direction, consider a v-f reduction  satisfying all the stated axioms. Then, the dual ∗ of  satisfies all the axioms stated in Theorem 2, which implies ∗ =  H M . Hence, the dual v-f reductions of ∗ and  H M , i.e.,  and  O N H F , must coincide, as we wanted to prove. Finally, Corollary 1 implies that  O N H F induces the Shapley value since it is a self-dual value.  Proof of Theorem 4. First, we verify that  P W satisfies all the stated properties. It is linear and hence additive, because it is the composition of linear functions. To show path independence, linearity ensures that it suffices to verify that the unanimity games satisfy the prop-    erty. Consider any T ∈ 2 N \ {∅}, then  NP W , N  (u T )( S ) = u T ( S ) − i ∈ S Sh i ( N , u T | N  ) + i ∈ S Sh i ( N , u T ) = u T | N  ( S ) −  |T ∩ S |  |T ∩ S | Sh i ( N  , u T | N  ) + i∈ S t . Notice that u T | N  = 0 if T  N  and i∈ S Sh i ( N  , u T | N  ) = t if T ⊆ N  . Thus we have, for all S ⊆ N  ,  u T |N  ( S ) if T ⊆ N  ;  NP W , N  (u T )( S ) = |T ∩ S | t if T  N  .   , N  (u T ) is equal to  N , N  (u T ) if T ⊆ N . Otherwise, each player in N \ T is Sub The previous expression implies that  NP W  PW 1 a null player in ( N ,  N , N  (u T )) and the rest of the players have a constant marginal contribution t to any coalition in ( N  ,  NP W , N  (u T )). Now we verify that  P W satisfies path independence. Take N 3 ⊆ N 2 ⊆ N 1 . First, if T ⊆ N 3 , then  NP W 2 ,N3 ( NP 1W, N 2 (u T )) = |T ∩ S |  NP W 1 ,N3 (u T ) = u T | N 3 by path independence of  Sub . Second, if T  N 3 , then  NP W 1 ,N3 (u T ) = t . There are two possi- |T ∩ S | bilities: (a) if T ⊆ N 2 , it is immediate that  NP 2W, N 3 ( NP W 1 ,N2 (u T )) = t ; (b) if T  N 2 , then for S ⊆ N 3 , it happens that   |T ∩ S |  NP W 2 ,N3 ( NP W 1 ,N2 (u T ))( S ) =  NP W 1 ,N2 (u T )( S ) − i∈ S Sh i ( N 3 ,  NP W 1 ,N2 (u T )| N 3 ) + i∈ S Sh i ( N 2 ,  NP W 1 ,N2 (u T )) = t − | T ∩t S | + | T ∩t S | = |T ∩ S | t , where the first equality follows from equation (7), and the terms in the second equality follow from (i) the expression |T ∩ S | 1 of the game  NP 1W, N 2 (u T )( S ) = t and its subgames, (ii) each player i ∈ T ∩ N 2 has a constant marginal contribution t in ( N 2 ,  NP W 1 ,N2 (u T )), and (iii) the rest of the players are null players in ( N 2 ,  NP W 1 ,N2 (u T )). Therefore, the PW v-f reduction is path independent. , N \{i } ( v ) = v | N \{i } for all ( N , v ) ∈ G such that i ∈ N is a null player in We verify the null player out property, i.e.,  NP W N   ( N , v ). We notice that for all S ⊆ N \ {i },  NP W , N \{i } ( v )( S ) = v ( S ) − j ∈ S Sh j ( N \ {i }, v | N \{i } ) + j ∈ S Sh j ( N , v ) = v ( S ), where the first equality follows from (7) and the second from Sh j ( N \ {i }, v | N \{i } ) = Sh j ( N , v ) if i is a null player in ( N , v ), i.e., null player out of the Shapley value. Therefore  P W satisfies null player out. As for permanent null player, let i ∈ N  be a null player in ( N , v ). Then, it is the case that for all S ⊆ N  \ {i },  NP W , N  ( v )(S ∪        {i }) −  N , N  ( v )( S ) = v ( S ∪{i }) − j ∈ S ∪{i } Sh j ( N , v | N ) + j ∈ S ∪{i } Sh j ( N , v ) − v ( S ) − j ∈ S Sh j ( N , v | N ) + j ∈ S Sh j ( N , v ) = PW   ( v ( S ∪ {i }) − v ( S )) − Shi ( N  , v | N  ) + Shi ( N , v ) = 0, where the third equality follows from the premise that i is a null player in ( N , v ) and its subgames and from the null player property of the Shapley value. Therefore  P W satisfies permanent null player. 563 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 n−1 We check the proportional addition invariance property. By additivity, it suffices to show that  NP W , N  ( k=1 w (k,kα ) )( S ) = n−1  , n  n−1 0. Indeed,  NP W ,N  ( k=1 w (k,kα ) )( S ) = s α − i∈ S Sh i ( N k=1 w | (k,kα ) N  ) + i∈ S Sh i ( N , k=1 w (k,kα ) ) = sα −  n  n−1    i∈ S Sh i ( N  , w (k,kα ) | N  ) + i ∈ S k=1 Sh i ( N , w (k,kα ) ) = sα − i ∈ S Sh i ( N  , w (n ,n α ) | N  ) = sα − i ∈ S nnα = 0, where k=1 the first equality follows from (7), the second from additivity, and the third and fourth equalities hold because the Shapley value of each player in a symmetric game is equal to an equal share of the worth of the grand coalition. To prove the reverse implication we need a previous lemma: n−1 Lemma 3. For all N ∈ P f in (U ) such that | N | > 2, the set {u T | T  N , T = ∅} ∪ { k=1 w (k,k) } forms a basis of G N . Proof of Lemma 3. Take any N ∈ P f in (U ). To prove Lemma 3, we first note that: n −1     nu N = u {i } − w (k,k) , (21) i∈N k =1  i ∈ N u {i } ( S ) = s for all S ∈ 2 \ {∅}. N which is easily seen by recognizing that Given n−1that equation (21) holds and { u T | T ⊆ N , T =  ∅} forms a basis of G N , then the set resulting from replacing u N k=1 w (k,k) on this basis spans G , which proves Lemma 3.  N with The proof that  =  P W if the v-f reduction  satisfies the five properties is very similar n−1 to the proof of Theorem 2. The only difference is in the proof that  N , N \{i } ( v ) =  NP W , N \{i } ( v ) for all i ∈ N, when v = k=1 w (k,kα ) . In this case, additivity n−1 n−1 , N \{i } ( k=1 w (k,kα ) ) for all N ∈ P f in (U ), all and proportional addition invariance imply that  N , N \{i } ( k=1 w (k,kα ) ) = 0 =  NP W α ∈ R, and all i ∈ N. Therefore, if a v-f reduction satisfies the five properties, then it is equal to  P W . Finally, regarding the value induced by  P W , we notice that, for all ( N , v ) ∈ G N and all i ∈ N,  NP W ,{i } ( v )({i }) = v ({i }) − Sh i ({i }, v|{i } ) + Sh i ( N , v ) = Sh i ( N , v ). Therefore,  P W induces the Shapley value.  Proof of Proposition 5. We prove that if  satisfies proportional addition invariance then ∗ satisfies reverse-proportional addition invariance. We first show that the dual of the game ( N , w (n−k,kα ) ) is ( N , w (k,−kα ) ), for k = 1, 2, . . . , n − 1. Indeed, w ∗(n−k,kα ) ( S ) = w (n−k,kα ) ( N ) − w (n−k,kα ) ( N \ S ) = − w (n−k,kα ) ( N \ S ) = w (k,−kα ) ( S ) for all S ⊆ N. n−1  n−1 ∗   −1 ∗ ∗ Then, for all α ∈ R, ∗N ,N  ( v + k=1 w (k,(n−k)α ) ) =  N , N  (( v + k=1 w (k,(n−k)α ) )∗ ) =  N , N  ( v ∗ + nk=1 w (k,(n−k)α ) ) =   −1 ∗  ∗  N , N  ( v ∗ + nk=1 w (k,−kα ) ) =  N , N  ( v ∗ ) = ∗N , N  ( v ), where the first and last equalities follow from Definition 8, ∗ the second from the additivity of v → v in Lemma 1, the third from the property that the dual of ( N , w (n−k,kα ) ) is ( N , w (k,−kα ) ), and the fourth from the proportional addition invariance of . Therefore, reverse-proportional addition in- variance is dual to proportional addition invariance.  ∗ Proof of the expression in Example 5. We prove that the expression for  P W corresponds to that provided in Example 5: ∗  P W ∗ ∗ ∗  ∗   NP W PW PW , N  ( v )( S ) =  N , N  ( v ) ( S ) =  N , N  ( v )( N ) −  N , N  ( v )( N \ S )   =[ v ∗ ( N  ) − Sh i ( N  , v ∗ | N  ) + Sh i ( N , v ∗ )] i∈N  i∈N    ∗  − [v (N \ S ) − Sh i ( N , v ∗ | N  ) +  Sh i ( N , v ∗ )] i∈N  \ S i∈N  \ S   ∗  ∗   ∗ =v (N ) − v (N \ S ) − Sh i ( N , v | N  ) + Sh i ( N , v ∗ ) i∈ S i∈ S     =[ v ( N ) − v ( N \ N )] − [ v ( N ) − v ( N \ ( N \ S ))] − Sh i ( N  , v ∗ | N  ) + Sh i ( N , v ∗ ) i∈ S i∈ S        ∗ ∗ = v ( S ∪ ( N \ N )) − v ( N \ N ) − Sh i N , ( v | N  ) + Sh i ( N , v ), i∈ S i∈ S where the first equality follows from Definition 8, the second from the defining equation (3) of a dual game, the third from (7), the fifth from (3) and the self-duality of the Shapley value, which also leads to the sixth equality.  Finally, we show that ( v ∗ | N  )∗ = v N \ N . Consider T ⊆ N  . By repeated application of the definition of dual game, for all T ⊆ N , ( v | N  ) ( T ) = v | N  ( N ) − v | N  ( N  \ T ) = v ∗ ( N  ) − v ∗ ( N  \ T ) = [ v ( N ) − v ( N \ N  )] − [ v ( N ) − v ( N \ ( N  \ T ))] =  ∗ ∗ ∗  ∗  v ( T ∪ ( N \ N  )) − v ( N \ N  ) = v N \ N ( T ).  564 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Proof of Theorem 5. The proof of this theorem is identical to that of Theorem 3.  Proof of Theorem 6. First, we verify that  Ban satisfies the properties. It satisfies linearity and hence additivity because it is the composition of linear functions. To verify the null player out property, let i ∈ N be a null player in ( N , v ). Then, for all S ⊆ N \ {i },  N , N \{i } ( v )( S ) = Ban  1 = 12 v ( S ) + 12 [ v ( S ∪ {i }) − v ({i })] = v ( S ), where the first equality follows from the defining equation T ⊆{i } 2 [ v ( S ∪ T ) − v ( T )] (10) and the second holds because i is a null player. Therefore  Ban satisfies null player out.  . Then, for all S ⊆ N  \ {i }, To verify that  Ban that i ∈ N1 is a null player in v ∈ G Ban N  satisfies permanent 1 null player, suppose  NBan ,N  ( v )( S ∪ { i }) =  T ⊆ N \ N n−n [ v (( S ∪ { i }) ∪ T ) − v ( T )] =  T ⊆ N \ N n−n [ v ( S ∪ T ) − v ( T )] =  N ,N  ( v )( S ), where the first 2 2 and last equalities follow from (10) and the second from the premise that i is a null player in ( N , v ). Therefore  Ban satisfies permanent null player. To verify maximum ignorance, we substitute N  = N \ {i } and W = N in equation (10), then  N , N \{i } (α u N )( S ) = Ban  1 T ⊆{i } 2 [ α u N ( S ∪ T ) − α u N ( T )] = 12 [α u N ( S ) − α u N (∅)] + 12 [α u N ( S ∪ {i }) − α u N ({i })] = α2 u N ( S ∪ {i }). Therefore  Ban satisfies maximum ignorance. To verify path independence, we use the following claim: Claim 1. For all N , N  , W , S ∈ P f in (U ) such that N  , W ∈ 2 N \ {∅} and S ⊆ N  ,   2−| W \ N | u W ( S ∪ ( W \ N  )) if S ∩ W = ∅,  NBan , N  (u W )( S ) = (22) 0 otherwise. To verify the claim, we have,  1  NBan , N  ( v )(u W )( S ) =  [u W ( S ∪ T ) − u W ( T )] 2n−n T ⊆N \N   1 =  [u W ( S ∪ T ) − u W ( T )] 2n−n T ⊆N \N  :T ⊇W \N   1 =  [u W ( S ∪ ( W \ N  ) ∪ T  ) − u W ( W \ N  )] 2n−n T  ⊆( N \ N  )\ W  1 =  [u W ( S ∪ ( W \ N  )) − u W ( W \ N  )] 2n−n T  ⊆( N \ N  )\( W \ N  )    2n−n −| W \ N | =  [u W ( S ∪ ( W \ N  )) − u W ( W \ N  )] 2n−n T  ⊆( N \ N  )\( W \ N  )  =2−|W \ N | [u W ( S ∪ ( W \ N  )) − u W ( W \ N  )]   2−| W \ N | [u W ( W \ N  ) − u W ( W \ N  )] = 0 if S ∩ W = ∅ = −|W \ N  | 2 u W ( S ∪ ( W \ N  )) if S ∩ W = ∅ where the first equality follows from (10), the fourth from u W ( S ∪ ( W \ N  ) ∪ T  ) = u W ( S ∪ ( W \ N  )) if T  ∩ W = ∅ and the first case of the seventh from the same reasoning, the second case of the seventh from u W ( W \ N  ) = 0 if S ∩ W = ∅, which implies N  ∩ W = ∅. Then, to check path independence, first notice that equation (22) is equivalent to   2−| W \ N | u W ∩ N  | N  if W ∩ N  = ∅  NBan , N  (u W ) = (23) 0 otherwise. Let N 1 , N 2 , N 3 , T ∈ P f in ( N ) such that N 3 ⊆ N 2 ⊆ N 1 and T ⊆ N 1 . To compute  N Ban 2 ,N3 ( NBan 1 ,N2 (u T )), there are three different possibilities to consider: (i) If T ⊆ N 3 , then  N Ban 2 ,N3 ( NBan 1 ,N2 (u T )) =  NBan 2 ,N3 (2−| T \ N 2 | u T ∩N 2 | N 2 ) = 2−| T \ N 2 |  NBan 2 ,N3 (u T ∩ N 2 | N 2 ) = 2−| T \ N 2 | · 2−|( T ∩ N 2 )\ N 3 | u ( T ∩ N 2 )∩ N 3 | N 2| N 3 = 2−| T \ N 3 | u T ∩ N 3 | N 3 =  N Ban 1 ,N3 (u T ), where the first, the third and the last equali- ties follow from equation (23) and the second from linearity. (ii) If T  N 3 and T ⊆ N 2 , then T ∩ N 2  N 3 . We have  NBan 2 ,N3 ( NBan 1 ,N2 (u T )) =  NBan 2 ,N3 (2−| T \ N 2 | u T ∩N 2 | N 2 ) = 2−| T \ N 2 |  NBan 2 ,N3 (u T ∩N 2 | N 2 ) = 2−| T \ N 2 | · 0 = 0 =  NBan 1 ,N3 (u T ). (iii) If T  N 2 , Ban then  N 2 ,N3 ( NBan 1 ,N2 (u T )) =  NBan 2 ,N3 (0) = 0 =  NBan 1 ,N3 (u T ). Therefore,  Ban satisfies path independence. We now prove the reverse implication of the theorem by showing that if the v-f reduction  satisfies the five properties, then  =  Ban . By path independence and additivity, it suffices to show the equality restricted to one-player operators 565 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568    N , N \{i } , for any N ∈ P f in (U ) and i ∈ N, restricted to a set of all scalar multiples of elements in a basis of G N . We choose   the set αu T T ∈2 N \{∅},α ∈R . We show that  N , N \{i } (α u T ) =  N , N \{i } (α u T ) for all T ∈ 2 \ {∅}, all α ∈ R, and all i ∈ N by induction on the number Ban N of players n. We notice that maximum ignorance implies that  N , N \{i } (α u N ) = α2 u N \{i } . Thus, we only need to check the re-   maining elements in the set, that is, the games α u T T ∈2N \{∅, N },α ∈R . The proof of this part is identical to the corresponding part of the proof of Theorem 1. Therefore, a v-f reduction that satisfies the five properties coincides with  Ban . Ban  Finally, we show that  Ban induces the Banzhaf value: ϕi ( v ) =  N ,{i } ( v )({i }) = T ⊆ N \{i } 2n1−1 [ v ( T ∪ {i }) − v ( T )] = Bani ( N , v ), where the second and the third equality follows from the defining equation (10). Therefore,  induces Ban.  Proof of Proposition 8. It is easy to see that  X is additive as a result of the linearity of the Shapley value. Moreover,  X =  P W if X = ∅ and  X =  H M if X = U . Equivalently,  NX , N  =  NP W   , N  if ( N \ N ) ∩ X = ∅;  N , N  =  N , N  if ( N \ N ) ∩ X HM X = N \ N  . Therefore, the reduction of a game from N to N \ { j } is different depending on whether the removed player j belongs to X or not. Hence,  X does not satisfy anonymity if X = ∅ and X = U . Finally,  X satisfies null player out and permanent null player if  X satisfies path independence, which we show next. |T ∩ S | For ease of notation, for each T ⊆ 2 N \ {∅}, let us define ( N , e T ) ∈ G N by e T ( S ) ≡ t for all S ⊆ N. It is easy to see that  NX , N  (e T ) = e T | N  . (24) X X   By linearity of  , it suffices to verify the path independence of  operating on a basis uT T ∈2 N \{∅} . We  need to consider three different cases of T : (i) If T ⊆ S ∪ (( N \ N ) ∩ X ), then  NX , N  (u T )( S ) = i ∈ S Sh i ( S ∪ (( N \    N  ) ∩ X ), u T | S ∪(( N \ N  )∩ X ) ) − i ∈ S Sh i ( N  ∪ (( N \ N  ) ∩ X ), u T | N  ∪(( N \ N  )∩ X ) ) + i ∈ S Sh i ( N , u T ) = i∈ S Sh i ( S ∪ (( N \ N  ) ∩   |T ∩ S | X ), u T | S ∪(( N \ N  )∩ X ) ) − i ∈ S Sh i ( S ∪ (( N \ N  ) ∩ X ), u T | S ∪(( N \ N  )∩ X ) ) + i ∈ S Sh i ( N , u T ) = t , where the second equal- ity follows from Sh i ( S ∪ (( N \ N  ) ∩ X ), u T | S ∪(( N \ N  )∩ X ) ) = Sh i ( N  ∪ (( N \ N  ) ∩ X ), u T | N  ∪(( N \ N  )∩ X ) ), i.e., the null player out of the Shapley value, and the last from equaltreatment of the Shapley value. (ii) If T   S ∪ (( N \ N  ) ∩ X )   and T ⊆ N ∪ (( N \ N ) ∩ X ), then  N , N  (u T )( S ) =  Sh ( S ∪ (( N \ N ) ∩ X ), u T | S ∪(( N \ N )∩ X ) ) −  i ∈ S Sh i ( N ∪ (( N \ X    i∈ S i   N  ) ∩ X ), u T | N  ∪((N \ N  )∩ X ) ) + i∈ S Sh i ( N , u T ) = − i∈ S Sh i ( N ∪ (( N \ N  ) ∩ X ), u |   T N ∪(( N \ N )∩ X ) ) + i∈ S Sh i ( N , uT ) = − i ∈ S Shi ( N , u T ) + i ∈ S Shi ( N , u T ) = 0, where the second equality follows from the premise T  S ∪ (( N \ N  ) ∩ X ), which implies that u T | S ∪(( N \ N  )∩ X ) = 0, and the third from null player out of the Shapley value and the premise that T ⊆ N  ∪ (( N \ N  ) ∩ X ) which imply that Sh i ( N  ∪ (( N \ N  ) ∩ X ), u T | N  ∪(( N \ N  )∩ X ) ) = Sh i ( N , u T ). Finally, (iii) if   T  N  ∪ (( N \ N  ) ∩ X ), then  NX , N  (u T )( S ) = i ∈ S Sh i ( S ∪ (( N \ N  ) ∩ X ), u T | S ∪(( N \ N  )∩ X ) ) − i ∈ S Sh i ( N  ∪ (( N \ N  ) ∩   |T ∩ S | X ), u T | N  ∪(( N \ N  )∩ X ) ) + i ∈ S Sh i ( N , u T ) = i ∈ S Sh i ( N , u T ) = t , where the second equality follows from the premise, which implies that u T | S ∪(( N \ N  )∩ X ) = 0 and u T | N  ∪(( N \ N  )∩ X ) = 0. To sum up, if T ⊆ N  ∪ (( N \ N  ) ∩ X ), for all S ⊆ N  ,  0 if T  S ∪ (( N \ N  ) ∩ X );  NX , N  (u T )( S ) = |T ∩ S | t if T ⊆ S ∪ (( N \ N  ) ∩ X ). The previous expression means that, if T ⊆ N  ∪ (( N \ N  ) ∩ X ), |T ∩ N  |  NX , N  (u T ) = u T ∩N  |N  , (25) t whereas if T  N  ∪ (( N \ N  ) ∩ X ),  NX , N  (u T ) = e T | N  . (26) Now we can verify that  N 2 , N 3 ( N 1 , N 2 (u T )) =  N 1 , N 3 (u T ) for all N 1 , N 2 , N 3 , S ∈ P f in (U ) such that S ⊆ N 3 ⊆ N 2 ⊆ N 1 and all T ⊆ N 1 . We have three possibilities: (c1) T ⊆ N 2 ∪ (( N 1 \ N 2 ) ∩ X ) and T ∩ N 2 ⊆ N 3 ∪ (( N 2 \ N 3 ) ∩ X ); (c2) T ⊆ N 2 ∪ (( N 1 \ N 2 ) ∩ X ) and T ∩ N 2  N 3 ∪ (( N 2 \ N 3 ) ∩ X ); (c3) T  N 2 ∪ (( N 1 \ N 2 ) ∩ X ) and T ⊆ N 3 ∪ (( N 2 \ N 3 ) ∩ X ). |T ∩N2 | |T ∩N2 | |T ∩N2 | |T ∩N2 ∩N3 | For (c1),  NX 2 , N 3 ( NX 1 , N 2 (u T )) =  NX 2 , N 3 ( t u T ∩N2 |N2 ) = t  NX 2 , N 3 (u T ∩N 2 | N 2 ) = t |T ∩N2 | u T ∩N2 ∩N3 |N2 |N3 = |T ∩N3 | t u T ∩ N 3 | N 3 =  NX 1 , N 3 (u T ), where the first and the third equalities follow from equation (25), the second from linear- ity of  X , and the last from the fact that the premise of (c1) implies that T ⊆ N 3 ∪ (( N 1 \ N 3 ) ∩ X ). |T ∩N | |T ∩N2 | X For (c2),  NX 2 , N 3 ( NX 1 , N 2 (u T )) =  NX 2 , N 3 ( t 2 u T ∩ N 2 | N 2 ) = t  N 2 , N 3 (u T ∩N 2 | N 2 ) = | T ∩tN 2 | e T ∩N 2 | N 3 = e T | N 3 =  NX 1 , N 3 (u T ), where the first equality follows from equation (25), the second from linearity of  X , the third from equation (26), the fifth from the fact that the premise of (c2) implies that T  N 3 ∪ (( N 1 \ N 3 ) ∩ X ). For (c3),  NX 2 , N 3 ( NX 1 , N 2 (u T )) =  NX 2 , N 3 (e T | N 2 ) = e T | N 2 | N 3 = e T | N 3 =  NX 1 , N 3 (u T ), where the first and second equalities follow from equation (26), the fourth from the fact that the premise of (c3) implies that T  N 3 ∪ (( N 1 \ N 3 ) ∩ X ). Therefore,  X is path independent.  566 D. Pérez-Castrillo and C. Sun Games and Economic Behavior 130 (2021) 543–568 Example of a v-f reduction that does not satisfy linearity. We construct a v-f reduction that satisfies additivity, null player out, permanent null player, path independence, but not homogeneity. We can invoke path independence to define  N , N  for any N  ⊆ N, once we will determine the functions taking the form  N , N \{k} such that k ∈ N. Moreover, it suffices to construct a non-homogeneous function {i , j },{i } : G {i , j } → G {i } that satisfies null player out, permanent null player and additivity. For concreteness, we let the rest of functions, i.e.,  N , N \{k} such that k ∈ N and | N | > 2 coincide with the subgame operator. Denote by Q the set of all rational numbers. To define a non-homogeneous additive function, we use the concept of R as a vector space over Q. A linear basis of this vector space is called a Hamel basis. Let H be a Hamel basis. Then for each  γ ∈ R, we can find a unique finite set of elements {x1 , . . . , xk } ⊆ H such that γ = kj=1 c j x j where c1 , . . . , ck ∈ Q \ {0}. Choose an arbitrary element y ∈ H. Then for each γ ∈ R, we can determine its corresponding coefficient (which is possibly zero) in the expression of γ , coefficient that we denote c (γ ). Thus we have a function c : R → Q defined by the projection γ → c (γ ). This function is additive but not homogeneous. Indeed, choose an arbitrary element y  ∈ H \ { y }, then α c ( y ) = y c (α y ) when α= y . Moreover, this function satisfies that c (0) = 0.22 Before defining {i , j },{i } , recall that for each ({i , j }, v ) ∈ G {i , j } , v can be expressed by α u {i} + β u { j} + γ u {i, j} for some α , β, γ ∈ R. Now we define {i, j},{i} ( v ) as follows: {i , j },{i } ( v )({i }) ≡ α + c (γ ), (27) where α , γ ∈ R are such that v = α u {i } + β u { j } + γ u {i , j } for some β ∈ R. Notice that if i is a null player in ({i , j }, v ) then v must take the form of β u { j } and that if j is a null player in ({i , j }, v ) then v must take the form of α u {i } . Therefore, {i , j },{i } satisfies null player out and permanent null player. Moreover, it is additive but not linear because the function c is additive but not homogeneous.  Ban =  Ban . Indeed, for S ∈ 2 N \{∅},  NA , N  ( v )( S ) = Ban S̄ (( N \ N  ) ∪{ S̄ }, ( v | S ∪(N \ N  ) ) S ) = Ban Proof of Theorem 7. We check that  A  1  1 D S̄ ( v | S ∪( N \ N  ) ) S )( T ) = T ⊆(( N \ N  )∪{ S̄ })\{ S̄ } 2n−n T ⊆ N \ N  n−n [( v | S ∪( N \ N  ) ) S )( T ∪ { S̄ }) − ( v | S ∪(N \ N  ) ) S )( T )] = 2  1 Ban T ⊆ N \ N  2n−n [( v ( T ∪ S ) − ( v ( T )] =  NBan A , N  ( v )( S ), where the equalities just follow the definitions of  , Ban, v S , and Ban  . ϕ Moreover,  Ban is path independent. Therefore, the Banzhaf value is consistent relative to  A . Also, it is immediate that it is standard for two-player games. ϕ For the other direction, we prove that if ϕ is consistent relative to  A and standard for two-player games, then ϕ = Ban. We do the proof by induction on the number of players | N |. It holds for | N | = 2 by standardness. Consider now ( N , v ) ∈ G N such that | N | > 2 and assume that ϕ = Ban for any game with less than | N | players. Take any i ∈ N. We first note that ϕ v(S) v ( S ∪ {i }) − v ({i })  NA , N \{i } ( v )( S ) = ϕ S̄ ({i , S̄ }, ( v | S ∪{i } ) S ) = + , (28) 2 2 where the second equality follows from standardness. To prove that ϕ j ( N , v ) = Ban j ( N , v ) for any j ∈ N, take any i ∈ N such that i = j. Then, ϕ j (N , v ) = ϕ j (N \   {i },  NA , N \{i } ( v )) = Ban j ( N \ {i },  NA , N \{i } ) = S ⊆N \{i , j} 2n1−2 [ NA , N \{i } ( v )( S ∪ { j }) −  NA , N \{i } ( v )( S )] = S ⊆N \{i , j } 2n1−2 [ v ( S ∪{ j }) ϕ ϕ ϕ ϕ 2 + v ( S ∪{i , j })− v ({i }) v(S) v ( S ∪{i })− v ({i })  1 v ( S ∪{ j })− v ( S ) v ( S ∪{i , j })− v ( S ∪{i })  1 2 − 2 − 2 ] = S ⊆N \{i , j} 2n−2 [ 2 + 2 ] = T ⊆N \{ j } 2n−1 [ v ( T ∪{ j }) − v ( T )] = Ban j ( N , v ), where the first equality follows from the consistency of ϕ , the second from the hypothesis that ϕ = Ban for games with n − 1 players, the third and the last from the definition of the Banzhaf value, and the fourth from the equation (28).  ϕ Proof of Theorem 8. It is easy to see by substituting the stand-alone value in  A that it coincides with the subgame v-f ϕ reduction sub . Moreover, sub is path independent. Thus, the stand-alone value is consistent relative to  A . For the other direction, we prove that ϕ j ( N , v ) = v ({ j }) for j ∈ N by an induction on the number of players | N |. It holds for | N | = 2 by condition (ii) of the theorem. Assume that the induction hypothesis holds for any game with less than n player, with n > 2, and consider ( N , v ) ∈ G N ϕ with | N | = n. For any i ∈ N, we have  NA , N \{i } ( v )( S ) = ϕ S̄ ({i , S̄ }, ( v | S ∪{i } ) S ) = v ( S ), where the second equality follows from (ii) of the theorem. Thus: ϕ ( N \ {i },  NA , N \{i } ( v )) = ( N \ {i }, v ). (29) Aϕ Consider now any j ∈ N, and take any i ∈ N such that i = j. 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