Osaka University of Economics Working Paper Series No Hart Mas-Colell Implementation of the Discounted Shapley Value

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1 Osaka Unversty of Economcs Workng Paper Seres No Hart Mas-Colell Implementaton of the Dscounted Shapley Value Tomohko Kawamor Faculty of Economcs, Osaka Unversty of Economcs November, 2014

2 Hart Mas-Colell Implementaton of the Dscounted Shapley Value Tomohko Kawamor Faculty of Economcs, Osaka Unversty of Economcs November 13, 2014 Abstract We consder an mplementaton of the dscounted Shapley value We modfy the Hart Mas-Colell model as each player dscounts future payoffs and proposes not only an allocaton but also a coalton We show that the dscounted Shapley value s supported by any statonary subgame perfect equlbrum of the modfed game such that n any subgame, the coalton that conssts of all actve players mmedately forms We also provde condtons for such a statonary subgame perfect equlbrum to exst Keywords: Dscounted Shapley value, Coaltonal barganng, Noncooperatve support, Effcency JEL classfcaton codes: C72, C73, C78 The author s grateful to Mnoru Ktahara, Masahro Okuno-Fuwara and Kazuo Yamaguch for ther valuable comments Ths work was supported by JSPS KAKENHI Grant Number Osaka Unversty of Economcs, Osum, Hgashyodogawa-ku, Osaka , Japan Tel: Fax: none E-mal: kawamor@osaka-ueacp 1

3 1 Introducton [Joo96] ntroduced the α-dscounted Shapley value, whch s a value that takes nto account both margnalsm and egaltaransm The α-dscounted Shapley value of coaltonal game N, v s the Shapley value of coaltonal game N, w such that for any S 2 N, w S = α N v S, where α [0, 1] [Joo96] axomatzed the α-dscounted Shapley value by the Hart Mas-Colell consstency and the α- standard ness 1 If α = 1 α = 0, the dscounted Shapley value s the Shapley value egaltaran value [HMC96] gave the Shapley value a noncooperatve foundaton [HMC96] presented a noncooperatve coaltonal barganng model wth nontransferable utltes, n whch f a proposal s reected, the player that offers the proposal becomes nactve wth probablty 1 ρ [HMC96] showed that the Shapley value s mplemented as the expected payoff tuple of any equlbrum n the transferable utlty case We gve the α-dscounted Shapley value a noncooperatve foundaton We ncorporate tme dscountng nto the model of [HMC96], n whch each player s supposed to not dscount future payoffs, n the transferable utlty case The ncorporaton of tme dscountng unfes the models of [HMC96] and [Oka93], 2 n whch the equlbrum payoff tuple s the egaltaran value We show that under the dscount factor δ, the δ1 ρ 1 ρδ -dscounted Shapley value s mplemented as the expected payoff tuple of any subgame-effcent statonary subgame perfect equlbrum SSPE, whch s an SSPE n whch the full coalton the coalton that conssts of all actve players forms wthout delay n every subgame We also show that the α-dscounted Shapley value approxmately concdes wth the ex post payoff tuple 3 by any subgame-effcent SSPE when ρ and δ go to unty wth δ1 ρ 1 ρδ α convergng to [vdbf10] also gave the α-dscounted Shapley value a noncooperatve foundaton [vdbf10] ncorporated tme dscountng nto the model of [PCW01] the bddng mechansm [vdbf10] showed that under the dscout factor δ, the δ-dscounted Shapley value s mplemented as the equlbrum payoff tuple Snce δ1 ρ 1 ρδ < δ, the value by the Hart Mas-Colell mplementaton n the present paper s closer to the egaltaran value than that by the Pérez-Castrllo Wettsten nplementaton n [vdbf10] Ths dfference mght be because the proposer whose proposal s reected becomes nactve wth a certan probablty n [HMC96] and wth certanty n [PCW01] Thus, f the proposer whose proposal s reected probablstcally becomes nactve n [PCW01], both approaches may mplement the same value 1 A value ϕ s α-standard f for any two-player coaltonal game N, v and any dstnct, N, ϕ N, v = vn+αv{} αv{} 2 2 [Oka93] s an earler verson of [Oka96] In [Oka93], when a coalton forms, the game ends; n [Oka96], after a coalton forms, the players that are outsde of the formed coalton contnue barganng 3 The ex post payoff tuple s the payoff tuple that s calculated provded a proposer was selected 2

4 [Joo96] ntroduced the egaltaran Shapley value, whch also takes nto account both margnalsm and egaltaransm The egaltaran Shapley value s a convex combnaton of the Shapley value and the egaltaran value [vdbfj13] ncorporated breakdown of negotaton nto the frst round of the bddng mechansm n [PCW01] and showed that the egaltaran Shapley value s mplemented as the equlbrum payoff tuple Whle n [HMC96], each proposer proposes only an allocaton for the full coalton, n the present paper, each proposer proposes both a coalton and an allocaton for the coalton Thus, n the present paper, a coalton structure s endogenously decded and a subcoalton mght form, whch leads to the neffcency under the superaddtvty Then, ths paper provdes a necessary and suffcent condton for a subgame-effcent SSPE to exst By ths condton, t s shown that there s a subgame-effcent SSPE when ρ and δ go to unty wth δ1 ρ 1 ρδ convergng to α only f for any subgame of the underlyng coaltonal game, the α-dscounted Shapley value of the subgame s n the core of the subgame, and f for any subgame, the α-dscounted Shapley value of the subgame s n the nteror of the core of the subgame [CGL14] and the present paper have studed the Hart Mas-Colell mplementaton of the dscounted Shapley value ndependently of each other [CGL14] ntroduced tme dscountng nto the model of [HMC96] and showed the concdence of the dscounted Shapley value and the equlbrum payoff tuple of the model The man dfference between [CGL14] and the present paper s as follows In [CGL14], each proposer proposes only an allocaton for the full coalton, that s, the full coalton formaton s assumed; n the present paper, each proposer proposes a coalton as well as an allocaton for the coalton, and a condton for the full coalton formaton s derved In [CGL14], the nontransferable utlty case s also consdered; n the present paper, only the transferable utlty case s consdered The remander of ths paper s organzed as follows Secton 2 descrbes a noncooperatve coaltonal barganng model; Secton 3 shows that the α-dscounted Shapley value s supported by any subgame-effcent SSPE; Secton 4 gves condtons for a subgame-effcent SSPE to exst; Secton 5 concludes ths paper The appendx contans proofs for all propostons 2 Model A coaltonal game s a par N, v such that N s a nonempty fnte set and v s a map from 2 N to R such that v = 0 A coaltonal game N, v s superaddtve f for any dsont S, T 2 N, v S T v S + v T Defnton 1 For any coaltonal game N, v, the α-dscounted Shapley value of 3

5 N, v s x R N such that for any N, x = 1! N! N! S 2 N α N v S α N S\{} v S \ {} The α-dscounted Shapley value of N, v s the Shapley value of the coaltonal game N, w such that for any S 2 N, w S = α N v S 4 If α = 1 α = 0, the α-dscounted Shapley value s the Shapley value egaltaran value Let N, v be a superaddtve coaltonal game For any ρ, δ [0, 1] 2 \{1, 1} =: Π, defne an extensve game G ρ, δ A state s a nonempty subset of N Let S := 2 N \ { } In the game, there are nfnte rounds, whch are classfed by states The structure of the game s as follows The game begns wth a round wth state N At a round wth state S, barganng proceeds as follows: A player S s selected as a proposer wth probablty 1 Player proposes a par of a coalton ncludng and a feasble allocaton for the coalton, e, C, x such that C 2 S \ { }, C, x R C + and C x = v C Each player n C \ {} announces her acceptance or reecton of the proposal sequentally accordng to some predetermned order Then, the game proceeds as follows: If all players n C \ {} accept the proposal, the game ends If a player reects the proposal, wth probablty ρ, the game goes to the next round wth state S wth probablty 1 ρ, f > 1, t goes to the next round wth state S \ {} otherwse, the game ends If a proposal C, x such that C s accepted at the tth round, player obtans a payoff of δ t 1 x ; f a proposal C, x such that C s accepted or reecton s nfntely repeated, she obtans nothng δ s the common dscount factor In ths paper, we consder pure strateges The equlbrum concept employed n the paper s the statonary subgame perfect equlbrum SSPE, whch s the subgame perfect equlbrum such that each player takes the same actons at all rounds wth the same state For any ρ, δ Π, a strategy tuple σ of G ρ, δ s subgame effcent f at every round wth state S, every proposal n σ s mmedately accepted at the round n σ no delay and every player proposes full coalton S n σ full-coalton formaton 4 By usng ths dscounted game N, w, we can defne α-dscounted solutons correspondng to other solutons of coaltonal games 4

6 Ths defnton s based on [Oka96] Obvously, ρ < 1 or δ < 1 and the superaddtvty necesstate no delay and the full-coalton formaton for the subgame effcency, respectvely For any S S, let v S be the restrcton of v to S For any S S and any α [0, 1], let ˆϕ S α be the α-dscounted Shapley value of S, v S For any ρ, δ Π, let α ρ, δ := δ1 ρ 1 ρδ and ϕ S ρ, δ := ˆϕ S α ρ, δ Note that for any ρ, δ Π, α ρ, δ [0, 1] 3 Equlbrum payoff Theorem 1 states that the dscounted Shapley value s supported as the expected payoff tuple by any subgame-effcent SSPE Theorem 1 Let ρ, δ Π Let σ be a subgame-effcent SSPE of G ρ, δ Then, the expected payoff tuple by σ n any subgame that begns wth any state S s ϕ S ρ, δ Remark 1 For any ᾱ [0, 1], there exsts ρ, δ Π such that α ρ, δ = ᾱ For any ρ [0, 1, α ρ, 1 = 1; for any δ [0, 1, α 1, δ = 0 Theorem 1 s ntutvely explaned as follows At a round wth state S, f player s proposal s accepted, the total surplus for actve players s v S; otherwse, the expected total surplus for actve players s ρδv S + 1 ρ δv S \ {} As n [HMC96], player s SSPE payoff s determned accordng to the dfference n these total surpluses, whch s v S ρδv S + 1 ρ δv S \ {} = 1 ρδ v S α ρ, δ v S \ {} Thus, the expected SSPE payoff tuple s the α ρ, δ-shapley value Snce the future payoffs are dscounted, n the above calculaton, v S \ {} s dscounted but v S s not dscounted n the case where the proposal s accepted Thus, v S \ {} s more dscounted than v S Corollary 1 states that the dscounted Shapley value s approxmately supported as the ex post payoff tuple by any subgame-effcent SSPE Corollary 1 Let ρ n, δ n n N be a sequence n Π such that lm n ρ n = 1, lm n δ n = 1 and lm n α ρ n, δ n = ᾱ for some ᾱ [0, 1] Let σ n n N be a sequence such that for any n N, σ n s a subgame-effcent SSPE of G ρ n, δ n Let S S and S Then, the payoff tuple by σ n n any subgame that begns wth player s proposng node under state S converges to ˆϕ S ᾱ as n goes to nfnty Remark 2 For any ᾱ [0, 1], there exsts a sequence ρ n, δ n n N be a sequence n Π such that lm n ρ n = 1, lm n δ n = 1 and lm n α ρ n, δ n = ᾱ For any sequence ρ n, δ n n N n Π such that δ n = 1 for any n N, α ρ n, δ n = 1 for any n N; for any sequence ρ n, δ n n N n Π such that ρ n = 1 for any n N, α ρ n, δ n = 0 for any n N 5

7 4 Effcency Theorem 2 provdes a necessary and suffcent condton for subgame effcency n G ρ, δ Theorem 2 Let ρ, δ Π Then, there exsts a subgame-effcent SSPE of G ρ, δ f and only f for any S, T S such that S T and 2 and any T, 1 ρδ v S α ρ, δ v S \ {} + α ρ, δ T \{} ρ, δ + ρδ ϕ S ρ, δ v T T ϕ S\{} Corollary 2 provdes the condtons for subgame effcency when barganng frcton s nfntesmally small Corollary 2 Let ρ n, δ n n N be a sequence n Π such that lm n ρ n = 1, lm n δ n = 1 and lm n α ρ n, δ n = ᾱ for some ᾱ [0, 1] Then, for statements gven below, mples, and mples For some n N, for any n N such that n n, there exsts a subgameeffcent SSPE of G ρ n, δ n For any S S such that 2, ˆϕ S ᾱ s n the core of S, v S For any S S such that 2, ˆϕS ᾱ s n the nteror of the core of S, v S 1 Remark 3 Suppose that ᾱ = 1 Then, for any S S, ϕ S ᾱ s the Shapley value of S, v S Thus, f for any S S, S, v S s a strctly convex game e, for any T, U 2 S, v T U + v T U > v T + v U, then holds n Corollary 2 s ntutvely explaned as follows Suppose that there exsts a subgame-effcent SSPE σ of G ρ n, δ n n the lmt n Consder a round wth any state S Consder the lmt as n ρ n 1 and δ n 1 Then, after a reecton at the round, the game goes to the next round wth state S wthout dscountng Thus, by Theorem 1, the expected payoff tuple n the subsequent subgame s the ᾱ-dscounted Shapley value ϕ S ᾱ of S, v S Hence, any player s payoff condtonal on beng a proposer s v S S\{} ϕs ᾱ = ϕs ᾱ Suppose that ϕ S ᾱ s not n the core of S, v S Then, there exsts a coalton T S that blocks ϕ S ᾱ Thus, there exsts x R T + such that for any T, x > ϕ S ᾱ and T x = v T Hence, by the one-shot devaton to proposng T, x, whch s accepted, a player T can mprove her payoff from ϕ S ᾱ to x, whch s a contradcton 6

8 5 Concluson In ths paper, we showed that the α-dscounted Shapley value s supported by any subgame-effcent SSPE We also provded condtons for a subgame-effcent SSPE to exst Fnally, followng [MW95], we conecture that f we allow ρ, δ = 1, 1, when for any S S, the core of S, v S s nonempty, any SSPE payoff tuple n the subgame wth actve-player set S s n the core of S, v S, and any allocaton n the core of S, v S s supported as some SSPE payoff tuple n the subgame wth actve-player set S 7

9 Appendx: Proofs of propostons Lemma for proof of Theorems 1 and 2 Lemma 1 For any ρ, δ Π, any S S such that 2 and any S, ϕ S ρ, δ = 1 v S α ρ, δ v S \ {} + α ρ, δ S\{} ρ, δ ϕ S\{} Proof Snce ρ and δ are fxed, we omt ρ, δ from G ρ, δ, α ρ, δ and ϕ ρ, δ S\{} = ϕ S\{} S\{} T 2 S\{} = S\{} T 2 S\{} = = = = S\{} T 2 S \{S} T 2 S \{S} S\{} Thus, T 2 S \{S} T 2 S \{S} 1! S \ {}! S \ {}! 1! 1! 1! 1 T 1! 1! 1! 1 T 1! 1! 1! 1!! 1! 1 v S αv S \ {} + α 1! 1! 1! = 1 α v S αv S \ {} + = 1 v S αv S \ {} + = 1!!! T 2 S α S\{} v T α S\{} T \{} v T \ {} α 1 v T α T \{} 1 v T \ {} α 1 v T α T \{} 1 v T \ {} α 1 v T α T \{} 1 v T \ {} α 1 v T α T \{} 1 v T \ {} α 1 v T α T \{} 1 v T \ {} ϕ S\{} S\{} S\{} T 2 S \{S} ϕ S\{} 1!!! α v T α T \{} v T \ {} α v T α T \{} v T \ {} = ϕ S QED Proof of Theorem 1 Snce ρ and δ are fxed, we omt ρ, δ from G ρ, δ, α ρ, δ and ϕ ρ, δ For any S S, let I S := { T, T 2 S \ { } T } and 8

10 X S := R IS For any S S, any x X S, any T 2 S \ { } and any T, let x T := x T, For any S S, defne f S : X S X S as for any x X S and any T, I S, f = 1, f ST x = v T, and f 2, f ST x = 1 v T k T \{} ρδx T k \{} + 1 ρ δxt k + k T \{} 1 By the mathematcal nducton, show that for any S S, ϕ T s a T, I S unque fxed pont of f S Let S be a nonempty subset of N such that = 1 For any x X S and any T, I S, f ST x = v T = ϕ T Thus, ϕ T ρδx T + 1 ρ δx T \{k} T, I S s a unque fxed pont of f S Let n be a natural number such that 2 n N Suppose that for any S S such that S = n 1, ϕ T s a unque fxed T, I S pont of f S Let S be a nonempty subset of N such that = n Lemma 2 Let x be an element n X S such that for any T, I S wth T S, x T = ϕt Then, for any T, IS, f ST x = ρδx T + 1 ρδ ϕ T + ρδ v T x T k 2 k T Proof If = 1, the both sdes of 2 are v T, and thus, 2 holds Suppose that 2 f ST x = 1 v T k T \{} ρδx T k Note that \{} k T \{} ϕt k = v T \ {} Then, f ST x = ρδx T + 1 v T ρδ x T k k T Note that α = δ1 ρ 1 ρδ Then, f ST x = ρδx T + 1 ρδ \{} + 1 ρ δϕt k + k T \{} v T αv T \ {} + α 1 1 ρ δv T \ {} + 1 ρ δ k T \{} ϕ T \{k} + ρδ ρδx T + 1 ρ δϕ T \{k} ϕ T \{k} k T \{} v T k T x T k Thus, Lemma 1 yelds 2 QED 9

11 By Lemma 2, for any T, I S, f ST ϕ U k U,k I S = ϕ T + ρδ v T ϕ T k k T Note that ϕ k T ϕt k = v T Then, for any T, IS, f ST U k = ϕ U,k I T S Thus, ϕ T s a fxed pont of f S Let x be a fxed pont of f S Then, for T, I S x any S and any T, I S\{}, f S\{}T U k = f U,k I ST S\{} x = x T Thus, for any S, x T s a fxed pont of f S\{} Hence, by the nducton T, I S\{} T, IS\{} hypothess, for any S, x T = ϕ T T, IS\{} Therefore, for any T, I S wth T S, x T = ϕt Thus, by Lemma 2, for any S, x S = f SS x = ρδx S + 1 ρδ ϕ S + ρδ v S x S k 3 k S Sum 3 wth respect to over S Then, snce k S ϕs k = v S, k S xs k = v S Substtute ths nto 3 Then, for any S, x S = ϕ S ϕ Thus, x = T T, I S Therefore, ϕ T k s a unque fxed pont of f S Hence, by the mathematcal T,k I S nducton, for any S S, ϕ T s a unque fxed pont of f S T, I S Let σ be a subgame-effcent SSPE of G Let u be the element n X N such that for any S, I N, u S s player s expected payoff by σ at any round wth state S Then, snce σ s a subgame-effcent SSPE, for any S, I N, f = 1, u S = v S = f NS u, and f 2, u S = 1 v S = f NS u S\{} ρδu S + 1 ρ δu S\{} + Thus, u s a fxed pont of f N Therefore, snce ϕ S of f N, u = ϕ S S, I N Proof of Corollary 1 S\{} S, I N 1 ρδu S + 1 ρ δu S\{} s a unque fxed pont QED For any n N, by the subgame effcency of σ n and Theorem 1, player s payoff by σ n n any subgame that starts from player s proposng node under state S s v S k S\{} ρ n δ n ˆϕS k α ρ n, δ n + 1 ρ n δ ˆϕS\{} n k α ρ n, δ n, 10

12 whch converges to v S ˆϕ k S\{} S k ᾱ = ˆϕ S ᾱ as n goes to nfnty, and that of any player S \ {} s ρ n δ n ˆϕS α ρ n, δ n + 1 ρ n δ n ˆϕS\{} α ρ n, δ n, whch converges to ˆϕ S ᾱ as n goes to nfnty QED Proof of Theorem 2 α ρ, δ and ϕ ρ, δ Snce ρ and δ are fxed, we omt ρ, δ from G ρ, δ, Necessty Suppose that there exsts a subgame-effcent SSPE σ of G By Theorem 1, For any S S and any S, player s expected payoff by σ at any round wth state S s ϕ S For any S S such that 2 and any dstnct, S, responder s expected payoff by reectng player s proposal at any round wth state S s ρδϕ S + 1 ρ δϕ S\{} Let S and T be elements n S such that S T and 2 Let T Consder any proposng node of player at any round wth state S Snce σ s a subgame-effcent SSPE, player s payoff at the proposng node s v S S\{} ρδϕ S + 1 ρ δϕs\{} The supremum of player s payoffs by one-shot devatons to proposals wth coalton T to be accepted s v T T \{} ρδϕ S + 1 ρ δϕs\{} Snce σ s an SPE, v S S\{} ρδϕ S + 1 ρ δϕ S\{} v T T \{} ρδϕ S + 1 ρ δϕ S\{} Note that S ϕs = v S, S\{} ϕs\{} = v S \ {} and δ 1 ρ = 1 ρδ α Then, we obtan 1 Suffcency Suppose that for any S, T S such that S T and 2 and any T, 1 holds Consder strategy tuple σ such that at any round wth any state S, any player S proposes S, x such that f = 1, x = v S, and f 2, for any S \ {}, x = ρδϕ S + 1 ρ δϕs\{} and any player S accepts player s proposal f and only f her share n the proposal s greater than or equal to ρδϕ S + 1 ρ δϕs\{} Note that f 2, x = v S S\{} ρδϕ S + 1 ρ δϕ S\{} = 1 δρ v S 1 ρ δv S \ {} + ρδϕ S ρδϕ S 0 In σ, any proposal n σ s accepted by all responders In σ, any player offers a proposal wth the full coalton Thus, σ s subgame effcent σ s statonary For any S S and any S, let u S be player s expected payoff by σ at any round 11

13 wth state S Let S be a nonempty subset of N such that 2 and S Snce σ nvolves no delay, u S = 1 v S S\{} ρδϕ S + 1 ρ δϕ S\{} + Note that S ϕs = v S, S\{} ϕs\{} Then, S\{} 1 ρδϕ S + 1 ρ δϕ S\{} = v S \ {} and δ 1 ρ = α 1 ρδ u S = 1 ρδϕ S + 1 ρδ v S δ 1 ρ v S \ {} + δ 1 ρ = ρδϕ S + 1 ρδ 1 v S αv S \ {} + α ϕ S\{} S\{} ϕ S\{} S\{} Thus, by Lemma 1, u S = ϕ S Consder any round wth any state S such that 2 Frst, show the unmprovablty of respondng actons n the round Any player s payoff by reectng any other player s proposal gven other actons n σ s ρδϕ S + 1 ρ δϕ S\{} Thus, any player s respondng actons n σ are unmprovable Next, consder the unmprovablty of proposng actons of any player S at the round Player s payoff by σ at her proposng node at the round s v S S\{} ρδϕ S + 1 ρ δϕs\{} Consder any one-shot devaton to offerng any acceptable proposal wth any coalton T 2 S wth T Player s payoff by the devaton s v T T \{} ρδϕ S + 1 ρ δϕs\{} at most Note that S ϕs = v S, S\{} ϕs\{} = v S \ {} and δ 1 ρ = α 1 ρδ Then, player s gan from the devaton s at most v T ρδ ϕ S 1 ρδ v S αv S \ {} + α T ϕ S\{} T \{} By 1, ths s less than or equal to 0 Consder any one-shot devaton to offerng any unacceptable proposal Then, player s expected payoff by the devaton s ρδu S Thus, player s gan from the devaton s ρδu S v S S\{} ρδϕ S + 1 ρ δϕ S\{} = 1 δρ v S δ 1 ρ v S \ {} 0 Hence, player s proposng actons n σ s unmprovable Therefore, by the one-shot devaton prncple, σ s an SPE Thus, σ s subgame-effcent SSPE QED 12

14 Proof of Corollary 2 Suppose that holds Let S be an element n S such that 2 Let T 2 S \ { } Then, there exsts T Snce holds, by Theorem 2, for some n N, for any n N such that n n, 1 holds for ρ, δ = ρ n, δ n Thus, T ϕs ᾱ v T Hence, ϕs ᾱ s n the core of S, v S Therefore, holds Suppose that holds Then, for any S, T S such that S T and 2 and any T, T ϕs ᾱ > v T, and thus, there exsts n ST N such that for any n N such that n n ST, 1 holds for ρ, δ = ρ n, δ n Snce T S S 2 } s fnte, t has the maxmum Let n be the { n ST maxmum Let n N such that n n Then, for any S, T S such that S T and 2 and any T, snce n n n ST, 1 holds for ρ, δ = ρ n, δ n Thus, by Theorem 2, there exsts a subgame-effcent SSPE of G ρ n, δ n Therefore, holds QED 13

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