Consistency, anonymity, and the core on the domain of convex games

Transcription

1 Consistency, anonymity, and the core on the domain of convex games Toru Hokari Yukihiko Funaki April 25, 2014 Abstract Peleg (1986) and Tadenuma (1992) provide two well-known axiomatic characterizations of the core on the domain of balanced TU games. Peleg s characterization says that the core is the only solution that satisfies non-emptiness, individual rationality, super-additivity, and a reduced game property introduced by Davis and Maschler (1965). Tadenuma s characterization says that the core is the only solution that satisfies non-emptiness, individual rationality and a reduced game property introduced by Moulin (1985). In this note, we investigate what happens when the domain is restricted to the class of convex TU games. In particular, we show that (i) the core is not the only solution that satisfies Peleg s four axioms and anonymity; (ii) the core is the only solution that satisfies Peleg s four axioms, anonymity, and additional three axioms; and (iii) the core is not the only solution that satisfies Tadenuma s three axioms and anonymity. 1 Introduction The core (Gillies, 1959) is one of the most important solutions for cooperative games. It is important mainly because it satisfies many desirable We thank two anonymous referees and an associate editor for helpful comments and suggestions. We are responsible for any remaining errors. Faculty of Economics, Keio University. hokari@econ.keio.ac.jp School of Political Science and Economics, Waseda University 1

2 properties. In particular, it satisfies two kinds of reduced game properties, namely, max consistency (Davis and Maschler, 1965) and complement consistency (Moulin, 1985). 1 There are two well-known axiomatic characterizations of the core on the domain of balanced TU games based on each of these two axioms: (i) the core is the only solution that satisfies nonemptiness, individual rationality, super-additivity, and max consistency (Peleg, 1986); (ii) it is the only solution that satisfies non-emptiness, individual rationality and complement consistency (Tadenuma, 1992). 2 In this note, we investigate what happens when the domain is restricted to the class of convex TU games. Although the core satisfies Peleg s four axioms on this domain, it is not the only one. 3 It so happens that except for the core itself, all existing examples of such solutions violate anonymity. So, one may conjecture that an axiomatic characterization of the core might be obtained by adding anonymity to Peleg s three axioms. In this note, we disprove this conjecture. We also consider a similar problem for complement consistency. In particular, we show that the core is not the only solution on the domain of convex games that satisfies Tadenuma s three axioms and anonymity. 2 Definitions and questions Let N denote the class of non-empty and finite subsets of the set N of natural numbers. We use for strict set inclusion, and for weak set inclusion. There is an infinite set of potential players indexed by the members of N. Given N N, a transferable utility (TU) game for N is a function v : 2 N R with v( ) 0. A game v for N is convex (Shapley, 1971) if for all i N and all S, T 2 N with i S T, we have v(s) v(s \ i) v(t ) v(t \ i). Let V N vex denote the class of convex games for N. A game v is balanced if for all nonnegative-valued function δ : 2 N R + such that for all i N, S i δ(s) 1, we have v(n) S 2 N δ(s)v(s). 1 These two axioms are usually called DM-consistency and M-consistency, respectively. We use the terminology introduced by Thomson (1996) and call them max consistency and complement consistency because each name suggests how the underlying reduced games are defined in each case. 2 Voorneveld and van den Nouweland (1998) provide an axiomatization of the core which is closely related to Tadenuma s result. 3 Although this fact is widely known, we don t know any published or unpublished paper that mentions it. 2

3 Given a game v for N, the core of v, denoted C(v), is the set of vectors x R N such that i N x i v(n) and for all S N, i S x i v(s). It is well-known that a game is balanced if and only if its core is non-empty (Bondareva, 1963; Shapley, 1967). It is also well-known that every convex game is balanced (Shapley, 1971). Suppose that for all N N, a class V N of games for N is specified, and let V N N VN. A solution on V is a mapping that assigns to all N N and all v V N a set of vectors x R N with i N x i v(n). The core, as a mapping, is a solution on the class of balanced games. We use ϕ as a generic notation for solutions. Next, we define max consistency (Davis and Maschler, 1965) and complement consistency (Moulin, 1985). Each of these axioms says that the original choice should be confirmed in associated reduced games, obtained by imagining a subset of players leaving with their payoffs and reevaluating the situation from the viewpoint of the remaining players. The different definitions come from the various ways of performing this reassessment. Given N N, a game v for N, x R N, and N N, the max reduced game of v relative to x and N, denoted ˆr N x (v), is defined by setting for all S N, ˆr N x (v)(s) max T N\N [ v ( S T ) i T x i] if S N,, v(n) i N\N x i if S N, 0 if S. Max consistency: A solution ϕ satisfies max consistency if and only if for all N N, all v V N, all x ϕ(v), and all N N, we have ˆr N x (v) VN and x N ϕ(ˆr N x (v)). Given N N, a game v for N, x R N, and N N, the complement reduced game of v relative to x and N, denoted rn x (v), is defined by setting for all S N, r x N (v)(s) v ( S (N \ N ) ) i N\N x i if S, 0 if S. Complement consistency: A solution ϕ satisfies complement consistency if and only if for all N N, all v V N, all x ϕ(v), and all N N, we have rn x (v) VN and x N ϕ(rn x (v)). The following axioms apply to games with a fixed set of players. 3

4 Non-emptiness: For all v V N, ϕ(v). Individual rationality: For all v V N, all x ϕ(v), and all i N, we have x i v(i). Super-additivity: For all v, w V N with v + w V N, we have ϕ(v) + ϕ(w) ϕ(v + w). As mentioned above, on the domain of balanced games, (i) the core is the only solution satisfying non-emptiness, individual rationality, superadditivity, and max consistency (Peleg, 1986); and (ii) the core is the only solution satisfying non-emptiness, individual rationality and complement consistency (Tadenuma, 1992). On the domain of convex games, the core satisfies max consistency (Maschler, Peleg, and Shapley, 1972), as well as nonemptiness, individual rationality, and super-additivity. Given a strict ordering on N, consider the following solution ϕ, which picks for each convex game the marginal contribution vector with respect to : for all N N, all v V N vex, and all i N, ϕ i (v) v ( j N j i i ) v ( j N j i ). On the domain of convex games, this solution satisfies max consistency (Orshan, 1994; Núñez and Rafels, 1998; Hokari, 2005). Moreover, it satisfies non-emptiness, super-additivity, and individual rationality. This means that on the domain of convex games, the core is not the only solution that satisfies Peleg s four axioms. Clearly, the above solution violates the following axiom: Anonymity: For all N, N N with N N, all v V N, and all w V N, if there exists a bijection π : N N such that for all S N, w(π(i) i S ) v(s), then for all x ϕ(v), we have (x π 1 (j)) j N ϕ(w). As far as we know, other than the core itself, no anonymous solution on the domain of convex games that satisfies Peleg s three axioms can been found in the literature. So, the first question we would like to ask is the following: Question 1. On the domain of convex games, is the core the only solution that satisfies non-emptiness, individual rationality, super-additivity, max consistency, and anonymity? Now, let us consider complement consistency. It is trivial to show that the core satisfies this axiom on the domain of convex games. Given that the 4

5 core satisfies this property on the domain of balanced games, the only thing we have to check is whether the class of convex games is closed under the reduction operation underlying complement consistency. Lemma 1. On the domain of convex games, the core satisfies complement consistency. Proof. Let N N, v Vvex, N x C(v), and N N with N N. Since the core satisfies complement consistency on the domain of balanced games (Tadenuma, 1992), it is enough to show that rn x (v) is convex. Let i N and S, T 2 N be such that i S T. If S 2, then rn x (v)(t ) rx N (v)(t \ i) rx N (v)(s) + rx N (v)(s \ i) v(t (N \ N )) v((t \ i) (N \ N )) 0, v(s (N \ N )) + v((s \ i) (N \ N )) where the last inequality follows from the convexity of v. If S i, then r x N (v)(t ) rx N (v)(t \ i) rx N (v)(i) v(t (N \ N )) v((t \ i) (N \ N )) v(i (N \ N )) + j N\N x j v(t (N \ N )) v((t \ i) (N \ N )) v(i (N \ N )) + v(n \ N ) 0, where the last two inequalities follow from x C(v) and the convexity of v, respectively. So, the next question is: Question 2. On the domain of convex games, is the core the only solution that satisfies non-emptiness, individual rationality, and complement consistency? If it is not, what if anonymity is added to this list of axioms? 3 Results Consider the following solution ϕ on the domain of convex games: for all N N and all v Vvex, N let S(v) S 2 N \ N, x C(v), x i v(s), i S 5

6 and ϕ (v) x C(v) S S(v), i S x i > v(s). Non-emptiness of ϕ (v) can be checked as follows: if S S(v), then there exists y S C(v) such that i S ys i > v(s). Since C(v) is a convex set, one can obtain an element of ϕ (v) by taking a strict convex combination of these y S s. Essentially, ϕ is the relative interior of the core. Note that ϕ trivially satisfies individual rationality and anonymity. On the domain of balanced games, ϕ satisfies max consistency (Yanovskaya, 1999). Together with the fact that the core satisfies the property on the domain of convex games, the max consistency of ϕ on the domain of balanced games implies the max consistency of ϕ on the domain of convex games. We show that it also satisfies super-additivity. Lemma 2. On the domain of convex games, ϕ satisfies super-additivity. Proof. Let N N, v, w Vvex, N x ϕ (v), and y ϕ (w). Since the core is super-additive and ϕ is a subsolution of the core, we have x + y C(v + w). Let S S(v + w). Then there exists z C(v + w) such that i S z i > v(s) + w(s). Note that on the domain of convex games, the core is additive (Dragan, Potters, and Tijs, 1989). 4 Thus, there exist x C(v) and y C(w) such that z x +y. Thus, i S x i+ i S y i i S z i > v(s)+w(s) so that either S S(v) or S S(w). Since x ϕ (v) and y ϕ (w), either i S x i > v(s) or i S y i > w(s). Together with x C(v) and y C(w), this implies i S x i + i S y i > v(s) + w(s). We have the following result: Proposition 1. On the domain of convex games, the core is not the only solution that satisfies non-emptiness, individual rationality, super-additivity, max consistency, and anonymity. Next, let us consider Question 2. Our starting point is the solution ϕ, defined in Section 2, that picks for each game the marginal contribution vector with respect to a given ordering of players. Although ϕ itself does not satisfy complement consistency, we can enlarge it so that the resulting 4 The definition of additivity is obtained by replacing with in the definition of super-additivity. 6

7 (0, 0, 4) (3, 0, 1) (0, 3, 1) (4, 0, 0) (3, 1, 0) (1, 3, 0) (0, 4, 0) Figure 1: Let N 1, 2, 3 and v V N vex be such that v(1) v(2) v(3) 0, v(1, 2) 0, v(1, 3) v(2, 3) 1, and v(n) 4. Then, v(n) v(2, 3) v(n) v(1, 3) < v(n) v(1, 2). So, there are two strict orderings on N that satisfy condition (i) in the definition of ϕ (v): and Thus, ϕ (v) x C(v) x 2 0 or x 1 0. solution satisfies the axiom. Then we endogenize the strict ordering to make the resulting solution anonymous. Consider the following solution ϕ on the domain of convex games: for all N N, all v V N vex, and all x C(v), x ϕ (v) if and only if there exists a strict ordering on N such that (i) for all i, j N, if v(n) v(n \ i) < v(n) v(n \ j), then i j; (ii) for all i N, if j N j i, then x i v ( j N j i i ) v ( j N j i ). Again, since the marginal contribution vectors are in the core on this domain, ϕ satisfies non-emptiness. Note that it coincides with the core when N 2. Figure 1 illustrates a case in which ϕ (v) does not coincides with the core, and there are two strict orderings that satisfy condition (i) above. This solution trivially satisfies anonymity and individual rationality. We show that it also satisfies complement consistency. Lemma 3. On the domain of convex games, ϕ satisfies complement consistency. Proof. Let N, N N with N N, v V N vex, and x ϕ (v). By the definition of ϕ (v), there exists a strict ordering on N such that (i) for all i, j N, if v(n) v(n \ i) < v(n) v(n \ j), then i j; (ii) for all i N, if j N j i, then x i v ( j N j i i ) v ( j N j i ). 7

8 Since x C(v) and the core is complement consistent, we have rn x (v) VN vex and x N C(rN x (v)). We want to show that x N ϕ (rn x (v)). If N 2, then ϕ (rn x (v)) C(rx N (v)), and we are done. Suppose that N 3. Note that for all i N, since N \ i, we have r x N (v)(n ) r x N (v)(n \ i) v(n) x j v((n \ i) (N \ N )) + j N\N v(n) v(n \ i). Thus, if i, j N are such that j N\N x j r x N (v)(n ) r x N (v)(n \ i) < r x N (v)(n ) r x N (v)(n \ j), then i j. Let i N and S j N j i. Since v is convex, v((s i) (N \ N )) v(s (N \ N )) v(s i) v(s). If S N, then r x N (v)((s i) N ) r x N (v)(s N ) v((s i) (N \ N )) j N\N x j v(s (N \ N )) + v((s i) (N \ N )) v(s (N \ N )) x i. v(s i) v(s) Thus, x N ϕ (r x N (v)). So, we have the following answer to Question 2: j N\N x j Proposition 2. On the domain of convex games, the core is not the only solution that satisfies non-emptiness, individual rationality, complement consistency, and anonymity. Given Propositions 1 and 2, one may wonder what axioms could be added to either of these lists to obtain the core as the unique solution. 5 Here we 5 We thank an associate editor for prompting us to work on this question. 8

9 provide a partial answer: by adding the following three axioms to the list of axioms that appear in Proposition 1, we can single out the core. Homogeneity: For all v, w V N, all a R ++, and all x ϕ(v), if w αv, then αx ϕ(w). Closedness: For all v V N, ϕ(v) is a closed set. Converse max consistency (Peleg, 1986): For all N N with N 3, all v V N, and all x ϕ(v), if for all N N with N 2, we have ˆr N x (v) VN and x N ϕ(ˆr N x (v)), then x ϕ(v). Proposition 3. On the domain of convex games, the core is the only solution that satisfies non-emptiness, individual rationality, super-additivity, anonymity, homogeneity, closedness, max consistency, and converse max consistency. We don t know whether homogeneity is independent from others. In this sense, this axiomatization is not complete. However, since homogeneity itself is a desirable and innocuous property, one can properly say that Proposition 3 essentially describes the implications of all other axioms. Finally, although we have shown that two well-known axiomatizations break down if the domain is restricted to the class of convex games, we should mention that there is another axiomatization of the core on the domain of all TU games provided by Peleg (1986), which remains valid even on the domain of convex games. 6 It says that on this domain, the core is the only solution that satisfies max consistency, converse max consistency, and the additional axiom of unanimity, which says that the solution should coincide with the core in the two-person case. 6 We thank an anonymous referee for pointing out this fact. 9

10 References [1] Bondareva ON (1963) Some applications of linear programming methods to the theory of cooperative games (in Russian). Problemy Kibernetiki 10: [2] Davis M, Maschler M (1965) The kernel of a cooperative game. Naval Research Logistics Quarterly 12: [3] Dragan I, Potters J, Tijs SH (1989) Superadditivity for solutions of coalitional games. Libertas Mathematica 9: [4] Gillies DB (1959) Solutions to general non-zero-sum games. In: Tucker AW, Luce RD (Eds) Contributions to the Theory of Games. IV, Princeton University Press, pp [5] Hokari T (2005) Consistency implies equal treatment in TU games. Games and Econ Behavior 51:63 82 [6] Mascheler M, Peleg B, Shapley, LS (1972) The kernel and bargaining set for convex games. Int J of Game Theory 1:73 93 [7] Moulin H (1985) The separability axiom and equal sharing methods. J of Econ Theory 36: [8] Núñez M, Rafels C (1998) On extreme points of the core and reduced games. Annals of Operations Research 84: [9] Orshan G (1994) Non-symmetric prekernels. Discussion Paper 60, Center for Rationality and Interactive Decision Theory, Hebrew University of Jerusalem [10] Peleg B (1986) On the reduced game property and its converse. Int J of Game Theory 15: [11] Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J on Applied Mathematics 17: [12] Shapley LS (1967) On balanced sets and cores. Naval Research Logistics Quarterly 14: [13] Shapley LS (1971) Cores of convex games. Int J of Game Theory 1:

11 [14] Tadenuma K (1992) Reduced games, consistency, and the core. Int J of Game Theory 20: [15] Thomson W (1996) Consistent allocation rules. RCER Working Paper 418, University of Rochester [16] Voorneveld M, van den Nouweland A (1998) A new axiomatization of the core of games with transferable utility. Econ Letters 60: [17] Yanovskaya E (1999) Strongly consistent solutions to balanced TU games. Int Game Theory Review 1:

12 Appendix In this appendix, first, we prove Proposition 3, and then prove the claim that ϕ satisfies converse max consistency. Consider the following axioms: Zero-independence: For all v, w V N, all b R N and all x ϕ(v), if for all S 2 N, w(s) v(s) + i S b i, then x + b ϕ(w). Convex-valuedness: For all v V N, ϕ(v) is a convex set. Lemma 4. On the domain of convex games, if a solution satisfies individual rationality and super-additivity, then it satisfies zero-independence. Proof. Let ϕ be a solution on V vex that satisfies individual rationality and super-additivity. Let b R N and v, w Vvex N be such that for all S 2 N, w(s) v(s) + i S b i. Let x ϕ(v). We want to show x + b ϕ(w). For all S 2 N, let w (S) i S b i. Then w Vvex N and w v + w. Since w is additive and ϕ is individually rational, ϕ(w ) b. By superadditivity, x + b ϕ(v + w ) ϕ(w). Lemma 5. On the domain of convex games, if a solution satisfies homogeneity and super-additivity, then it is convex-valued. Proof. Let ϕ be a solution on V vex that satisfies homogeneity and superadditivity. Let N N, v Vvex, N x, y ϕ(v), and λ (0, 1). We want to show that λx + (1 λ)y ϕ(v). Note that λv, (1 λ)v Vvex N and v λv + (1 λ)v. By homogeneity, λx ϕ(λv) and (1 λ)y ϕ((1 λ)v). By super-additivity, λx + (1 λ)y ϕ(λv + (1 λ)v) ϕ(v). The following lemma is due to Peleg (1986). Lemma 6. On the domain of balanced games, if a solution satisfies individual rationality and max consistency, then it satisfies efficiency. A similar claim for convex games can be proved in exactly the same way as Peleg s proof of Lemma 6. So, its proof is omitted. Lemma 7. On the domain of convex games, if a solution satisfies individual rationality and max consistency, then it satisfies efficiency. Proof of Proposition 3. The core satisfies the seven axioms. Let ϕ be a solution on V vex that satisfies the seven axioms. We show that ϕ coincides with the core. Since both the core and ϕ satisfy converse max consistency, it is enough to show that ϕ coincides with the core in the two-person case. 12

13 x 2 1 α x 2 v(12) x 2 O α 1 x 1 v(2) O v(1) x 1 v(12) x 1 (a) (b) Figure 2: In (a), v(1) v(2) 0, v(12) 1, and ϕ(v) x R 1,2 0 x i α. In (b), x 1 α(v(12) v(2)) + (1 α)v(1), x 2 α(v(12) v(1)) + (1 α)v(2), and ϕ(v) x R 1,2 v(i) xi x i. By Lemmas 4, 5, and 7, ϕ is zero-independent, convex-valued, and efficient. Let N 1, 2 and v 0 Vvex N be such that v 0 (1) v 0 (2) 0 and v 0 (12) 1. Let α max (x1,x 2 ) ϕ(v 0 ) x 1. By efficiency and individual rationality, ϕ(v 0 ) is bounded. By closedness, ϕ(v 0 ) is a closed set. Thus, α is welldefined. By anonymity, (1 α, α) ϕ(v 0 ). By convex-valuedness, ϕ(v 0 ) is the interval connecting (α, 1 α) and (1 α, α). By non-emptiness, α [ 1, 1]. 2 By zero-independence, homogeneity, and anonymity, for all N N with N 2 and all v Vvex, N [ ϕ(v) x C(v) for all i N, x i v(i) α v(n) v(j)]. j N First, suppose that α 1. Then ϕ coincides with the standard solution 2 in the two-person case: for all N N with N 2, all v Vvex, N and all i N, ϕ i (v) v(i) + 1 [ v(n) ] v(j). 2 j N On the domain of convex games, the nucleolus (Schmeidler, 1969) is the only solution that coincides with the standard solution in the two-person case and 13

14 satisfies max consistency. 7 Thus, ϕ coincides with the nucleolus. However, the nucleolus violates super-additivity. Thus, α cannot be 1. 2 Next, suppose that 1 < α 1. Then, let N 1, 2, 3 and v 2 VN vex be such that v(1) v(2) v(3) v(2, 3) 0, v(1, 2) v(1, 3) 1, and v(n) 3. We show that if 1 < α < 1, then ϕ(v) is not convex. 2 Let x α ( ) 3α, 3α 3(1 α),, y α (4α 1, 2(1 α), 2(1 α)). First, we show that (x α 1, x α 2 ) ϕ(r1,2 xα (v)). Note that ˆr xα ˆr xα 1,2(v)(1) maxv(1), v(1, 3) x α 3 3(1 α) max 0, 1 2(2α 1) max 0, 2(2α 1), ˆr xα 1,2(v)(2) maxv(2), v(2, 3) x α 3 3(1 α) max 0, 0, 1,2(v)(1, 2) v(1, 2, 3) x α 3 3(1 α) 3 6α. 7 This result follows from the fact that the nucleolus coincides with the prekernel on this domain (Maschler, Peleg, and Shapley, 1972), and the prekernel satisfies converse max consistency. See Hokari (2005) for more detail. 14

15 Thus, x α 1 ˆr 1,2(v)(1) xα xα α [ˆr 1,2(v)(1, 2) ˆr 1,2(v)(1) xα ˆr 1,2(v)(2) ] xα [ ] 3α 2(2α 1) 6α α 2(2α 1) (α + 2)(2α 1) < 0, x α 2 ˆr 1,2(v)(2) xα xα α [ˆr 1,2(v)(1, 2) ˆr 1,2(v)(1) xα ˆr 1,2(v)(2) ] xα [ ] 3α 6α α 2(2α 1) α(1 2α) < 0. Hence, (x α 1, x α 2 ) ϕ(ˆr 1,2 xα (v)). Next, we show that (x α 1, x α 3 ) ϕ(ˆr 1,3 xα (v)). Note that ˆr xα 1,3(v)(1) maxv(1), v(1, 2) x α 2 max 0, 1 3α max 0, 1 2α 0, ˆr xα 1,3(v)(3) maxv(3), v(2, 3) x α 2 max 0, 3α 0, ˆr 1,3(v)(1, xα 3) v(1, 2, 3) x α 2 3 3α 3. 15

16 Thus, x α 1 ˆr 1,3(v)(1) xα xα α [ˆr 1,3(v)(1, 3) ˆr 1,3(v)(1) xα ˆr 1,3(v)(3) ] xα 3α α 3 0, x α 3 ˆr 1,3(v)(3) xα xα α [ˆr 1,3(v)(1, 3) ˆr 1,3(v)(1) xα ˆr 1,3(v)(3) ] xα 3(1 α) α 3 3(1 2α) < 0. Hence, (x α 1, x α 3 ) ϕ(ˆr 1,3 xα (v)). Next, we show that (x α 2, x α 3 ) ϕ(ˆr 2,3 xα (v)). Note that ˆr xα 2,3(v)(2) maxv(2), v(1, 2) x α 1 max 0, 1 3α max 0, 1 2α 0, ˆr xα 2,3(v)(3) maxv(3), v(1, 3) x α 1 max 0, 1 3α 0, ˆr 2,3(v)(2, xα 3) v(1, 2, 3) x α 1 3 3α 3. 16

17 Thus, x α 2 ˆr 2,3(v)(2) xα xα α [ˆr 2,3(v)(2, 3) ˆr 2,3(v)(2) xα ˆr 2,3(v)(3) ] xα 3α α 3 0, x α 3 ˆr 2,3(v)(3) xα xα α [ˆr 2,3(v)(2, 3) ˆr 2,3(v)(2) xα ˆr 2,3(v)(3) ] xα 3(1 α) α 3 3(1 2α) < 0. Hence, (x α 2, x α 3 ) ϕ(ˆr 2,3 xα (v)). By converse max consistency, x α ϕ(v). Next, we show that (y1 α, y2 α ) ϕ(r yα 1,2(v)). Note that ˆr yα 1,2 (v)(1) maxv(1), v(1, 3) yα 3 max0, 1 2(1 α) max0, 2α 1 2α 1, ˆr yα 1,2 (v)(2) maxv(2, v(2, 3) yα 3 max0, 2(1 α) 0, ˆr yα 1,2 (v)(1, 2) v(1, 2, 3) yα 3 3 2(1 α) 2α

18 Thus, [ ] y1 α ˆr yα 1,2 (v)(1) α ˆr yα 1,2(v)(1, 2) ˆryα 1,2(v)(1) ˆryα 1,2 (v)(2) 4α 1 2α + 1 α(2α + 1 2α + 1) 0, [ ] y2 α ˆr yα 1,2 (v)(2) α ˆr yα 1,2(v)(1, 2) ˆryα 1,2(v)(1) ˆryα 1,2 (v)(2) 2(1 α) α(2α + 1 2α + 1) 2(1 2α) < 0. Hence, (y1 α, y2 α ) ϕ(ˆr yα 1,2 (v)). Next, we show that (y1 α, y3 α ) ϕ(ˆr yα 1,3(v)). Note that ˆr yα 1,3 (v)(1) maxv(1), v(1, 2) yα 2 max0, 1 2(1 α) max0, 2α 1 2α 1, ˆr yα 1,3 (v)(3) maxv(3, v(2, 3) yα 2 max0, 2(1 α) 0, ˆr yα 1,3 (v)(1, 3) v(1, 2, 3) yα 2 3 2(1 α) 2α + 1. Thus, [ ] y1 α ˆr yα 1,3 (v)(1) α ˆr yα 1,3(v)(1, 3) ˆryα 1,3(v)(1) ˆryα 1,3 (v)(3) 4α 1 2α + 1 α(2α + 1 2α + 1) 0, [ ] y3 α ˆr yα 1,3 (v)(3) α ˆr yα 1,3(v)(1, 3) ˆryα 1,3(v)(1) ˆryα 1,3 (v)(3) 2(1 α) α(2α + 1 2α + 1) 2(1 2α) < 0. 18

19 Hence, (y1 α, y3 α ) ϕ(ˆr yα 1,3 (v)). Next, we show that (y2 α, y3 α ) ϕ(ˆr yα 2,3(v)). Note that ˆr yα 2,3 (v)(2) maxv(2), v(1, 2) yα 1 max0, 1 4α + 1 max0, 2(1 2α) 0, ˆr yα 2,3 (v)(3) maxv(3, v(1, 3) yα 1 max0, 1 4α + 1 0, ˆr yα 2,3 (v)(2, 3) v(1, 2, 3) yα 1 3 4α + 1 4(1 α). Thus, [ ] y2 α ˆr yα 2,3 (v)(2) α ˆr yα 2,3(v)(2, 3) ˆryα 2,3(v)(2) ˆryα 2,3 (v)(3) 2(1 α) α 4(1 α) 2(α 1)(2α 1) 0, [ ] y3 α ˆr yα 2,3 (v)(3) α ˆr yα 2,3(v)(2, 3) ˆryα 2,3(v)(2) ˆryα 2,3 (v)(3) 2(1 α) α 4(1 α) 2(α 1)(2α 1) 0. Hence, (y α 2, y α 3 ) ϕ(ˆr yα 2,3 (v)). By converse max consistency, y α ϕ(v). Let z α xα + y α. 2 19

20 We show that if 1 < α < 1, then 2 (zα 1, z3 α ) ϕ(ˆr 1,3 zα (v)). Note that ˆr zα 1,3(v)(1) maxv(1), v(1, 2) z2 α max 0, 1 3α 2() α(2α 1) max 0, 2() α(2α 1) 2(), ˆr zα 1,3(v)(3) maxv(3), v(2, 3) z2 α max 0, 0 3α 2() max 0, 3α (1 α) 2() 0, ˆr zα 1,3(v)(1, 3) v(1, 2, 3) z α 2 3 3α 2() 2α2 + 3α () Thus, z1 α ˆr 1,3(v)(1) zα zα α [ˆr 1,3(v)(1, 3) ˆr 1,3(v)(1) zα ˆr 1,3(v)(3) ] zα z1 α (1 α)ˆr 1,3(v)(1) zα αˆr 1,3(v)(1, zα 3) + αˆr 1,3(v)(3) zα 3α 2() + 2α 1 α(2α 1) (1 α) 2 2() α 2α2 + 3α + 4 2() (α 1)(2α 1). 2() Now, suppose that 1 2 z α 1 ˆr zα 1,3(v)(1) α < α < 1. Then zα [ˆr 1,3(v)(1, 3) ˆr 1,3(v)(1) zα ˆr 1,3(v)(3) ] zα > 0. This implies (z α 1, z α 3 ) ϕ(ˆr zα 1,3 (v)), and hence, zα ϕ(v), which contradicts convex-valuedness of ϕ. (The arguments above are illustrated in Figures 3 and 4 for the case of α 3 4.) 20

21 So, we conclude that α 1, which implies that ϕ coincides with the core in the two-person case. Let N N with N 3 and v Vvex. N We show that ϕ(v) C(v). Let x ϕ(v). Since ϕ is max consistent and ϕ coincides with the core in the two-person case, for all N N with N 2, we have ˆr N x (v) VN vex and x N ϕ(ˆr N x (v)) C(ˆrx N (v)). Since the core is conversely max consistent, we have x C(v). Thus, ϕ(v) C(v). Next, let y C(v). Since the core is max consistent and the core coincides with ϕ in the two-person case, for all N N with N 2, we have ˆr y N (v) V N vex and y N C(ˆr y N (v)) ϕ(ˆr y N (v)). Since ϕ is conversely max consistent, we have x ϕ(v). Thus, C(v) ϕ(v). So, we conclude that ϕ coincides with the core on the whole domain. 21

22 (0, 3, 0) ( 1, 2, 2) (3, 2, 2) (0, 1, 2) (2, 1, 0) (0, 0, 3) (3, 0, 0) (3, 2, 2) Figure 3: α 3. The shaded area describes the set of payoff vectors such 4 ) that (x 1, x 3 ) ϕ (ˆr x1,3 (v). 22

23 (0, 3, 0) ( 1, 2, 2) (3, 2, 2) x α y α (0, 0, 3) (3, 0, 0) (3, 2, 2) Figure 4: α 3. The shaded area describes the set of payoff vectors such 4 that (x 1, x 2 ) ϕ(ˆr 1,2 x (v)), (x 1, x 3 ) ϕ(ˆr 1,3 x (v)), and (x 2, x 3 ) ϕ(ˆr 2,3 x (v)). 23

24 Next, we prove the following claim: Claim 1. On the domain of convex games, ϕ satisfies converse max consistency. We use the following lemma in the proof. Lemma 8. Let v V N vex, x C(v), and S, T 2 N. If x(s) v(s) and x(t ) v(t ), then x(s T ) v(s T ) and x(s T ) v(s T ). Proof. Since v is convex, v(s T ) + v(s T ) v(s) + v(t ) x(s) + x(t ) x(s T ) + x(s T ). Since x C(v), we have x(s T ) v(s T ) and x(s T ) v(s T ). Thus, x(s T ) v(s T ) and x(s T ) v(s T ). Proof of Claim 1. Let N N with N 3, v Vvex, N and x R N be such that for all N N with N 2, we have ˆr N x (v) VN vex and x N ϕ (ˆr N x (v)). Since ϕ is a subsolution of the core and the core is conversely max consistent, x C(v). Suppose that there exists S S(v) such that S N and x(s) v(s). Let i S. Note that for all j N \ S, ˆr x i,j(v)(i) max [v(i, j T ) x(t )] v(s) x(s \ i) x i. T N\i,j Since (x i, x j ) ϕ (ˆr x i,j (v)) and ϕ is individually rational, x i ˆr x i,j (v)(i). This implies x j ˆr x i,j (v)(j). Thus there exists T ij N such that j T ij, i T ij, and x(t ij ) v(t ij ). Let T i j N\S T ij. Then, by Lemma 8, x(t i ) v(t i ). Note that N \ S i S T i. Again by Lemma 8, x(n \ S) v(n \ S). This implies v(s) + v(n \ S) x(s) + x(n \ S) x(n) v(n). Thus, for all y C(v), y(s) v(s), which contradicts S S(v). 24