Converse consistency and the constrained equal benefits rule in airport problems

Transcription

1 Converse consistency and the constrained equal benefits rule in airport problems Cheng-Cheng Hu Min-Hung Tsay Chun-Hsien Yeh June 3, 009 Abstract We propose a converse consistency property and study its implications, when imposed together with other properties, in airport problems. We show that the constrained equal benefits rule satisfies the property, and then offer axiomatic arguments in favor of the rule on the basis of the property. In addition, we make use of consistency and our converse consistency as guides to design an extensive form game that gives a non-cooperative interpretation for the constrained equal benefits rule. Journal of Economic Literature Classification Numbers: C71; C7; D30; D63. Keywords: Converse consistency; consistency; the constrained equal benefits rule; modified nucleolus; airport problems 1 Introduction We consider the problem of sharing the cost of a public facility among agents who have different needs for it. An example is the so-called airport problem: Department of Finance, Providence University, Tai-Chung 43301, Taiwan. cchu@pu.edu.tw Department of Finance, National Central University, Chung-Li 30, Taiwan. mhtsay@hotmail.com Corresponding author: Institute of Economics, Academia Sinica, Taipei 115, Taiwan, and Department of Economics, National Central University, Chung-Li 30, Taiwan. E- mail: chyeh@econ.sinica.edu.tw 1

2 different airlines need airstrips of different lengths. The larger the planes an airline flies, the longer the airstrip it needs. Serving a given plane implies serving all smaller planes at no extra cost. To accommodate all planes, the airstrip must be long enough for the largest plane. How should the cost of the airstrip be shared among the airlines? 1 A rule is a function that associates with each airport problem, an allocation of the cost of the airstrip, which we call a contribution vector. Our first purpose is to propose the converse of the following consistency property, and study its implications when imposed together with other properties. Suppose that for each airline, say airline i, the cost of satisfying its need is represented by the cost parameter c i. For simplicity, we assume that each airline operates one plane. Consider a problem and a contribution vector x chosen by a rule for it. Imagine now that airline i contributes x i and leaves, and reassess the situation from the viewpoint of the remaining airlines. Instead of thinking of x i as covering an abstract part of the airstrip, it is natural to think of x i as a contribution to the part of the airstrip airline i uses. For each airline j (with c j c i ), its cost parameter is revised down by the amount x i since contributing to the part of the airstrip airline i uses implies contributing to the part of the airstrip that airlines i and j use. For each airline k (with c k < c i ), how should x i be imputed to airline k s cost parameter? One can think of x i as a contribution to the part of the airstrip that airlines i and k use. If c k > x i, then agent k benefits completely from agent i s contribution. Thus, airline k s cost parameter is revised down by the amount x i ; otherwise, agent k benefits partially from agent i s contribution. The revised cost parameter of agent k is obtained by taking the maximum of zero and c k x i. Uniform-subtraction consistency (Potters and Sudhölter, 1999) of the rule says that the components of x pertaining to the remaining airlines should still be chosen by the rule for the reduced 1 For a comprehensive survey of this literature, initiated by Littlechild and Owen (1973), see Thomson (004). This property is an application of a general principle of consistency for airport problem. This general principle has been studied extensively in a great variety of models such as taxation, bargaining, exchange, matching, social choice etc. For a comprehensive survey of the literature on this idea, see Thomson (000).

3 problem just defined. 3 We say that a rule is conversely uniform-subtraction consistent if whenever for some airport problem, a contribution vector has the property that for each proper two-airline subgroups, the rule chooses the restriction of the vector to the two-airline subgroup for the reduced problem this subgroup faces, then the vector should be recommended for the initial problem by the rule. 4 We show that the so-called constrained equal benefits rule 5 (henceafter, CEB rule), which equates airlines benefits 6 subject to no one receiving a transfer, satisfies converse uniform-subtraction consistency. We then base axiomatic characterizations of the CEB rule on this property together with other desirable properties. Our second purpose is to study non-cooperative dimension of the CEB rule. It is well-known that the non-cooperative interpretation often follows from designing a specific bargaining procedure that leads to the outcome of the rule. 7 Like many others 8, we make use of uniform-subtraction consistency and converse uniform-subtraction consistency as guides in designing an extensive form game that gives a non-cooperative interpretation for the constrained equal benefits rule. We show that the outcome of any subgame perfect equilibrium of our extensive form game is the contribution vector recommended by the CEB rule. 3 Potters and Sudhölter (1999) refer to it as ψ-consistency. We adopt Thomson (004) s terminology. 4 This converse consistency is an application of a general principle of converse consistency for airport problem. This general converse consistency principle has been studied and applied to a number of models by many authors such as Chang and Hu (007); Chun (00). 5 The rule is introduced by Sudhölter (1996, 1997) under the name of modified nucleolus. The terminology we adopt here is borrowed from Thomson (004). 6 By an airline s benefit, we mean the difference between the cost of satisfying this airline s need and his contribution. 7 For this line of research, see Serrano (005), Krishna and Serrano (1990), and Serrano (1993). 8 For references, see Dagan et al. (1997); Chang and Hu (008). 3

4 Notation and definitions.1 The model Let U N be a universe of agents with at least two elements, where N is the set of natural numbers. Given a finite and nonempty set N U and i N, let c i R + be agent i s cost parameter and c = (c i ) i N R N + be the profile of cost parameters. An airport problem, or simply a problem, is a pair (N, c). Let A be the class of all problems on U. Given (N, c) A, let n denote the number of agents in N, and assume that N {1,..., n} and c 1 c n. Thus, we refer to agent 1 as the first agent and agent n as the last agent. A contribution vector for (N, c) A is a vector x R N + such that i N x i = max i N c i and for each i N, x i c i. Let X(N, c) be the set of all contribution vectors for (N, c) A. A rule is a function defined on A that associates with each problem (N, c) A a vector x X(N, c). Our generic notation for rules is ϕ. For each coalition N N, we denote (c i ) i N by c N, (ϕ i (N, c)) i N by ϕ N (N, c), and so on.. The central rules and properties We now introduce our central rules. The first one is defined only for the twoagent problem. It says that the agent with smaller cost parameter contributes half of his cost parameter, and the other contributes the remaining amount to be collected. Standard rule, S: For each i, j N and each ({i, j}, (c i, c j )) A with c i c j, c i S i ({i, j}, (c i, c j )) = S j ({i, j}, (c i, c j )) = c j S i ({i, j}, (c i, c j )). The second is the constrained equal benefits rule, which is informally introduced in Section 1. Constrained Equal Benefits rule, CEB: For each (N, c) A and each i N, 4

5 CEB i (N, c) max {c i β, 0}, where β R + is chosen such that i N CEB i(n, c) = max j N c j. Remark 1: It is known that the CEB rule, the sequential equal contributions rule, and the nucleolus coincide with the standard rule for the two-agent problem. Our central properties follow. Let (N, c) A with n, i N\{n}, and x X(N, c). The reduced problem of (N, c) with respect to N = N\ {i} and x, (N, uc x N ), is defined by (i) for each j N such that c j < c i, (uc x N ) j = max {c j x i, 0}, and (ii) for each j N such that c j c i, (uc x N ) j = c j x i. Uniform-Subtraction consistency, US consistency: ( For each (N, c) ) A with n and each i N\{n}, if x = ϕ (N, c), then N\ {i}, uc x N\{i} ( ) A and x N\{i} = ϕ N\ {i}, uc x N\{i}. Next is its converse consistency. Let (N, c) A with n, i N\{n}, and x X(N, c). The reduced problem of (N, c) with respect to N = {i, n} and x, (N, uc x N ), is defined by { (uc x N ) i = max c i } k i,n x k, 0, and (uc x N ) n = c n k i,n x k. Converse Uniform-Subtraction consistency, CUS consistency: For each (N, c) A with n > and each x X(N, c), if for each N N with N = and n N, x N = ϕ (N, uc x N ), then x = ϕ (N, c). 5

6 3 Axiomatic characterizations We make use of the following lemma to show the existences of our axiomatic characterizations of the CEB rule. Lemma 1: The CEB rules satisfies CUS consistency. Proof. Let (N, c) A. Assume that N = {1,,, n} and( c 1 c ) c n. Let x X (N, c) and for each i n, x {i,n} = CEB {i, n}, uc x {i,n}. We show that x = CEB (N, c). Let i N\ {n}. Note that ( ) and x {i,n} = CEB {i, n}, uc x {i,n}. It follows that ( ) uc x {i,n} = x i. Let β = c n x n. Then i ( ( ) uc x {i,n} n x k k N = c n = x i + x n and c i β = c i c n + x n = c i ) x k x i. k i,n If c i ( ) x k 0, then x i = uc x {i,n} = c i x k and max{c i β, 0} = k i,n i k i,n x i. If c i ( ) x k < 0, then uc x {i,n} = 0 and x i = 0. It follows that k i,n max{c i β, 0} = 0 = x i. Then x = CEB (N, c). i Q.E.D. We first establish two axiomatic characterizations of the standard rule on the basis of the following properties. The first one says that two agents with the same cost parameters should contribute equal amounts. Formally, Equal treatment of equals: For each N N, each c C N, and each pair {i, j} N, if c i = c j, then ϕ i (N, c) = ϕ j (N, c). The second one is an order property. It says that of two agents, the benefit of the agent with the larger cost parameter should be at least as much as the other does. 9 Order preservation for benefits: For each N N, each c C N, and each pair {i, j} N, if c i c j, then c i ϕ i (N, c) c j ϕ j (N, c). 9 The property is introduced by Littlechild and Thompson (1977). 6

7 Our next property has to do with possible increase of the cost parameter of the last agent. It says that if the cost parameter of the last agent increases by δ, then its contribution should increase by δ. 10 Formally, Last-agent cost additivity: For each pair {(N, c), (N, c )} of elements of A and each δ R +, if c n = c n + δ and for each j N\{n}, c j = c j, then ϕ n (N, c ) = ϕ n (N, c) + δ. The last one is a monotonicity requirement. It says that if an agent s cost parameter increases, all other agents should contribute at most as much as they did initially. 11 Cost monotonicity: For each N N, each c C N, each c C N, and i N, if c i c i and for each j N\{i}, c j = c j, then for each j N\{i}, ϕ j (N, c ) ϕ j (N, c). 1 Remark : It is known that order preservation for benefits implies equal treatment of equals. We now are ready to present two characterizations of the standard rule, which play important roles in our paper. Proposition 1 For N =. The standard rule is the only rule satisfying order preservation for benefits and cost monotonicity. Proof. It is clear that the standard rule satisfies the two properties. Conversely, let (N, c) A with N {i, j} and c i c j. Let ϕ be a rule satisfying the two properties. Let x = ϕ(n, c) and y = S(N, c). We show that x i = y i. By efficiency, we then conclude that x = y. We consider two cases. 10 It is a weaker version of the property, stronger last-agent cost additivity, appeared in Potters and Sudhölter (1999). They combine strong last-agent cost additivity and homogeneity as a property, referred to as covariance. For a related reference on covariance. See also Sudhölter (1998). 11 It appears in Thomson (004) under the name of others-oriented cost monotonicity. 1 This property is introduced by Thomson (004). The property is a complement of individual cost monotonicity (Potters and Sudhölter, 1999), which says the following. Under the same hypotheses, agent i should pay at least as much as he did initially. 7

8 Case 1: c i = c j. By Remark, order preservation implies equal treatment of equals. Thus, x i = x j = c i = y i. Case : c i < c j. Note that for the two-agent case, order preservation for benefits implies that x i c i. We show that x i = c i. Let c A with N {i, j} and c i = c i and c j = c i. Let x = ϕ(n, c ). By Case 1, x i = c i. Note that c i = c i and c j < c j. By cost monotonicity, x i x i. Thus, x i c i and we derive x i = c i. Q.E.D. Proposition For N =. The standard rule is the only rule satisfying equal treatment of equals and last-agent cost additivity. Proof. It is clear that the standard rule satisfies the two properties. Conversely, let (N, c) A with N {i, j} and c i c j. Let ϕ be a rule satisfying the two properties. Let x = ϕ(n, c) and y = S(N, c). We show that x = y. We consider two cases. Case 1: c i = c j. By equal tretament of equals, x i = x j. By efficiency, x i = x j = c i = y i = y j. Case : c i < c j. Let c A with N {i, j} and c i = c i and c j = c i. Let x = ϕ(n, c ). By Case 1, x j = c i. By last-agent cost additivity of the CEB rule, x j = x j + δ, where δ c j c i. Then, x = y. Q.E.D. We make use of the following lemma to derive our characterizations of the CEB rule. Elevator Lemma (Thomson, 000): If a rule ϕ is consistent and coincides with a conversely consistent rule ϕ in the two-agent case, then ϕ coincides with ϕ in general. It can be shown that the CEB rule satisfies equal treatment of equals, order preservation for benefits, cost monotonicity, and last-agent cost additivity. Note that the CEB rule is US consistent, and that as shown in Lemma 1, it satisfies CUS consistency. With the help of Elevator Lemma and Propositions 1 and, the next results are immediate. 8

9 Theorem 1 The CEB rule is the only US consistent rule that satisfies order preservation for benefits and cost monotonicity. Theorem The CEB rule is the only CUS consistent rule that satisfies order preservation for benefits and cost monotonicity. Theorem 3 The CEB rule is the only US consistent rule satisfying equal treatment of equals and last-agent cost additivity. Theorem 4 The CEB rule is the only CUS consistent rule satisfying equal treatment of equals and last-agent cost additivity. 4 Non-cooperative interpretation We next investigate the non-cooperative dimension of rules that satisfy US consistency and CUS consistency. Let (N, c) be an airport problem. Without loss of generality, assume that c 1 c c n. We construct the following 3-stage extensive form game Γ(N, c) to capture the concept of US consistency and CUS consistency. The game tree of Γ(N, c) is depicted in Figure 1. Stage 1 : At the beginning of the game, each agent k N\{n} proposes a number x k R + with 0 x k c k simultaneously. 13 Let x n = c n k n x k. We refer to x = (x k ) k N as the binding proposal. The game proceeds to Stage. Stage : Agent n can take either action L or action C. If L is taken, then the game ends. The binding proposal x is the outcome of the game. If C is taken, then agent n picks an agent in N\ {n}, say agent i. Each agent k N\ {i, n} contributes x k and leaves the game. The game proceeds to the last stage. 13 Dagan et al. (1997), and Chang and Hu (008) study non-cooperative dimensions of bankruptcy rules. In Dagan et al. (1997), there is one and only one proposer who must be a creditor with the largest claim. In Chang and Hu (008), the authors follow Thomson (005) to determine the unique proposer by randomization. Compared to their settings, we have n 1 proposers and each of them proposes how much he/she would like to contribute. 9

10 Each agent k N \{ n} proposes a real number x between 0 and c k, k and let x = c x. n n i i n Agent n decides to take L or C. Agent n picks an agent i N \ { n}, and each agent k N \{ i, n} contributes x and leaves the game. k C L Nature chooses an agent between agent i and agent n each with probability 1. x ( xk) k N Agent i Agent n (( uc ), x + x ( uc ), x ) x x { in, } i i n { in, } i N\{ in, } x + x max 0, c x, max 0, c x, x i n n k n k N\{ i, n} k i, n k i, n Figure 1: The game tree of Γ(N, c) 10

11 Stage 3 : Nature chooses between agent i and agent n each with probability 1. If agent i is chosen by Nature, then the game ends. Agent i { contributes z i = max 0, c i } k i,n x k and agent n contributes the residual x i + x n z i. If agent n{ is chosen by Nature, the game ends. Agent n contributes z n = max 0, c n } k i,n x k and agent i contributes the residual x i + x n z n. We make use of the following proposition to prove our next results. Proposition 3 Let (N, v) A. If the binding proposal x is a contribution vector in X (N, c), and agent n chooses agent i( in Stage, then ) the expected contribution vector of agents i and n is CEB {i, n}, uc x {i,n}. ( ) Proof. Let β = 1 uc x {i,n}. Agent n s expected contribution is and agent i s expected contribution is i 1 [ x i + x n ( ) uc x {i,n} + ( ) ] uc x i {i,n} n = x i + x n 1 ( ) uc x {i,n} i = ( ) uc x {i,n} β n ( ) = CEB n {i, n}, uc x {i,n} 1 ( ) uc x = ( ) {i,n} uc x i {i,n} β i ( ) = CEB i {i, n}, uc x {i,n}. Q.E.D. The following notations are useful to construct a subgame perfect equilibrium (henceafter, SPE) of Γ (N, c). Let x R N and i N\ {n}. Let τ i (c, x) = (τ i i (c, x), τ i n (c, x)), where { } τ i i (c, x) = 1 max{0, c i k i,n x k } + x i + x n max{0, c n k i,n x k } 11

12 and τ i n (c, x) = x i + x n τ i i (c, x). That is, τ i (c, x) is the expected contribution of agents i and n if x is the binding proposal, and agent n picks agent i in Stage of Γ (N, ( c). Note that) if x is a contribution vector in X (N, c) then τ i (c, x) = CEB {i, n}, uc x {i,n}. The next two results establish our non-cooperative interpretation of the CEB rule. The first one says that the contribution vector recommended by the CEB rule can be supported by a SPE of the game Γ(N, c). Theorem 5 There exists a subgame perfect equilibrium of Γ (N, c) with the outcome CEB (N, c). Proof. The proof is by constructing a strategy profile that is a SPE of Γ (N, c) and generates the outcome CEB (N, c). Let f be a strategy profile and be defined as follows: Stage 1: Each agent k N\{n} proposes CEB k (N, c) at the beginning of the game. Stage : Assume that x is the binding proposal. Let µ = min τ k n (c, x). k N\{n} Agent n decides to take action L or action C. If x n µ, then agent n takes action L. If x n > µ, then agent n takes action C and picks agent i N\ {n} with τ i n (c, x) = µ. Step 1: The outcome of the strategy profile f of the game Γ (N, c) is CEB (N, c). Each agent k n, by following f k, proposes CEB k (N, c) at the beginning of the game. Let x = CEB (N, c). Then x is the binding proposal. By US consistency of the CEB rule, for each k n, x n = CEB n ( {k, n}, uc x {k,n} ) = τ k n (c, x). Thus, µ = x n. By following f n, agent n then takes action L. The game ends with the outcome CEB (N, c). 1

13 Step : The strategy profile f is a subgame perfect equilibrium of the game Γ (N, c). It is easy to verify that adopting the strategy f n is the best response for agent n in Stage provided each agent k n follows f k. Let i N\ {n} and each agent k N\{i} follow f k. Suppose that agent i deviates by proposing x i = x i ε for some ε > 0 in Stage 1. Let x be the binding proposal induced by agent i deviation. Let k N\ {i, n}. Note that τ i n (c, x ) = x n, and that τ k n (c, x ) = x k + x n + ε 1 max{0, c k l k,n x l + ε}. If c k l k,n x l + ε 0, then τ k n (c, x ) x n + 1 ε. If c k l k,n x l + ε > 0, then by order preservation for benefits of the CEB rule, τ k n (c, x ) = x k + x n + ε 1 max{0, c k l k,n x l + ε} = x n + ε + 1 [ x k + l n x l c k ] = x n + ε + 1 [c n x n (c k x k )] x n + ε. Note that x X(N, c) and for each k N\{i, n}, x k = x k. Thus, x n = x n + ε. Agent n then picks agent i by following f n in Stage, and agent i contributes x i. Thus, agent i is not better off by deviation. Suppose that agent i deviates by proposing x i = x i + ε for some ε > 0 in Stage 1. If agent n takes action L, or takes action C and picks agent k N\ {i, n} in Stage, then agent i contributes x i + ε. Thus, agent i is not better off by deviation. If agent n takes action C and picks agent i, then the expected contribution of agent i is x i. Agent i is not better off by deviation. Thus, we conclude that f is a SPE of Γ(N, c). Q.E.D. The next result says that the SPE outcome of the game Γ(N, c) is unique. Moreover, it is CEB(N, c). 13

14 Theorem 6: Each subgame perfect equilibrium outcome of the game Γ (N, c) is CEB (N, c). Proof. Let g be a SPE of Γ (N, c). Assume that each agent k n proposes x k in Stage 1 by following g k. Let x n = c n k n x k and x = (x k ) k N. We consider two cases. Case 1: Agent n takes action L by following g n. Then, the game ends with the outcome x. We claim that x is a contribution vector in X (N, c). Suppose, by contradiction, that x is not a contribution vector. By the game rule, we know that for each k n, 0 x k c k and x n c n. Recall the definition of a contribution vector. We have for each k N, 0 x k c k. Thus, only x n < 0 is possible if x is not a contribution vector. Note that there exists i N\ {n} such that x i > 0. It follows that c n k i,n x k < x i. Let agent i deviate by proposing 0 in Stage 1. Let x be the binding proposal induced by agent i s deviation. If agent n takes action L, or takes action C and picks agent k i, then it follows that agent i contributes 0. Agent i is better off by deviation. It violates that g is a SPE of Γ (N, c). If agent n takes action C and picks agent i in Stage, then there are two subcases to be discussed. Subcase 1.1: c n k i,n x k < 0. Note that x n < 0 and x i > 0. The expected contribution of agent i is τ i i (c, x ) = 1 (x i + x n ) < x i. Agent i is better off by deviation. It violates that g is a SPE of Γ (N, c). Subcase 1.: c n k i,n x k 0. Note that c i c n and c n k i,n x k < x i. Thus, c i k i,n x k < x i. It follows that the expected contribution of agent i is { } τ i i (c, x ) = 1 max{0, c i x k } + (x i + x n ) max{0, c n x k } k i,n k i,n { < 1 x i + (x i + x n ) c n + } x k k i,n < x i. 14

15 Agent i is better off by deviation. It violates that g is a SPE of Γ (N, c). Thus, we conclude that x X (N, c). we next claim that x = CEB (N, c). Suppose, by contradiction, that x CEB (N, ( c). Then, by CUS consistency, ( there exists ) i n such that x {i,n} CEB {i, n}, uc x {i,n} ). If CEB n {i, n}, uc x {i,n} < x n, then agent n can deviate by taking action C and picking agent i in Stage, and ( end up with) the expected contribution CEB n ({i, n}, uc x {i,n} ). If CEB i {i, n}, uc x {i,n} < ) x i, then agent i can deviate by proposing CEB i ({i, n}, uc x {i,n} in Stage 1, ) and end up with the contribution CEB i ({i, n}, uc x {i,n}. Thus, agent i or agent n can be better off by deviation. It violates that g is a SPE of Γ (N, c). Case : Agent n takes action C by following g n. Assume that agent n takes action C and picks agent i in Stage. The outcome of the game is y = ( τ i (c, x), x N\{i,n} ). If N =, then y = CEB (N, c). We obtain the desired result. Let N 3. We claim that y = CEB (N, c). There are two subcases to be considered. Subcase.1: For each j N\ {i, n}, x j = 0. Then x X (N, c) and y {i,n} = CEB ( ) {i, n}, uc x {i,n} ( 1 = c i, c n 1 ) c i. For some j N\ {i, n}, c j > c i. If c i = 0, then by the game rule, x i = 0. Thus, for each k n, x k = 0. It follows that y n = c n. In this case, agent n can deviate by picking agent j in Stage and obtain the expected contribution c n 1c j, which is less than c n. Agent n can be better off by deviation. It violates that g is a SPE of Γ (N, c). If c i > 0, then agent i can deviate by proposing a real number 1c i ε, where { 0 < ε < min c j 1 c i, 1 } c i. Let x be the binding proposal induced by agent i s deviation. ( Clearly, ) x X (N, c), x j = 0, x n = c n c i + ε, and for k = j, n, uc x {j,n} = 15 k

16 c k 1c i + ε > 0. Note that τ j n (c, x ) = 1 [ ( ) ( ) ] x j + x n uc x {j,n} + uc x {j,n} j n ( = c n 1 ) c i 1 ( c j 1 ) c i ε = y n 1 ( c j 1 ) c i ε < y n. Thus, agent n will pick agent j, rather than agent i, in Stage. Agent i then ends up with a contribution 1c i ε, which is less than c i. Thus, agent i will be better off by deviation. It violates that g is a SPE of Γ (N, c). For each j N\ {i, n}, c j c i. Then, in this case, the β R + in the definition of the CEB rule must be equal to c i in order to solve k N CEB k(n, c) = c n. Thus, CEB (N, c) = ( 1 c i, c n 1c ) i, 0 N\{i,n}. It follows that y = CEB (N, c). Subcase.: For some j N\ {i, n}, x j > 0. By the similar arguments as for Case 1, it can be shown that x X (N, c). By subgame perfection, y n τ k n (c, x) for each k N\ {i, n}. For each k N\ {i, n}, y n < τ k n (c, x). Let ε > 0 and ε < min { τ j n (c, x) y n, x j }. Suppose that agent j deviates by proposing x j ε. Let x is the binding proposal induced by agent j s deviation. It can be shown that x X (N, c). Note that x i = x i and x n = x n + ε. It follows that { τ i n (c, x ) = x i + x n + ε 1 max 0, c i } x k + ε k i,n y n + ε < min { τ j n (c, x), x j + y n } τ j n (c, x). 16

17 Then, agent n will not pick agent j in Stage, and agent j ends up with contribution x j ε, which is less than x j. It violates that g is a SPE. For some k N\ {i, n}, y n = τ k n (c, x). If y {i,n} = x {i,n}, then y = x. By the same arguments as for case 1, it can be shown that y = CEB (N, c). Assume that y {i,n} x {i,n}. By subgame perfection, y n < x n. Note that x, y X(N, c). Thus, x i < y i. Note that 0 < y i y n. Suppose that agent i deviate by proposing x i + ε for some ε > 0 such that x i + ε < y i and x n ε > 0. Let x be the binding proposal induced by agent i s deviation. It can be shown that x X (N, c). Note that x n = x n ε and x k = x k. It follows that { τ k n (c, x ) = x k + x n ε 1 max 0, c k } x l ε < y n = τ i n (c, x) l k,n Then, agent n would not pick agent i in Stage. It follows that agent i ends up with a contribution x i + ε, which is less than y i. Thus, it violates that g is a SPE. By Subcases.1 and., we conclude that y = CEB (N, c) Q.E.D. References Chang, C., Hu, C., 007. Reduced game and converse consistency. Games and Economic Behavior 59, Chang, C., Hu, C., 008. A non-cooperative interpretation of the f-just rules of bankruptcy problems. Games and Economic Behavior 63, Chun, Y., 00. The converse consistency principle in bargaining. Games and Economic Behavior 40, Dagan, N., Serrano, R., Volij, O., A noncooperative view of consistent bankruptcy rules. Games and Economic Behavior 18, Krishna V., Serrano, R., Multilateral bargaining. Review of Economic Studies 63,

18 Littlechild S.C., Owen G., A simple expression for the Shapley value in a special case. Management Science 3, Littlechild S.C., Thompson G.F., aircraft landing fees: a game theory approach. Bell Journal of Economics 8, Peleg B., On the reduced game property and its converse. International Journal of Game Theory 15, Potters J., Sudhölter P., Airport problems and consistent allocation rules. Mathematical Social Sciences 38, Sudhölter P., The modified nucleolus as canonical representation of weighted majority games. Mathematics of Operation Research 1, Sudhölter P., The modified nucleolus: properties and axiomatizations. International Journal of Game Theory 6, Sudhölter P., Axiomatizations of game theoretical solutions for oneoutput cost sharing problems. Games and Economic Behavior 4, Thomson, W., 000. Consistent allocation rules. Mimeo, University of Rochester, Rochester, NY, USA. Thomson, W., 004. Cost allocation and airport problems. Mimeo, University of Rochester, Rochester, NY, USA. Thomson, W., 005. Divide-and-permute. Game and Economic Behavior 5,