Akira Yamada

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Akira, Yamada

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ヤマダ, アキラ

Yamada, Akira

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Sapporo University Academic Repository / 札幌大学学術情報リポジトリ

99A note on alternate two-way contracts for side payments in two-player gamesA note on alternate two-way contracts forside payments in two-player games∗Akira Yamada†March 5, 2023AbstractWe studied side contracting process where two players alternatelyoffer not only schemes for side payment transfer to the other butalso schemes for side payment receipt from the other before choosingactions in the underlying game. Then, we found that there may bea kind of the second mover’s disadvantage in such side contractingprocess in contrast to Yamada (2005).1 IntroductionYamada (2005b) studied side contracting process where two players alter-nately offer schemes for side payment transfer to the other before choosingactions in the underlying game, and found that the second offerer’s equilib-rium payoff can exceed his threat point level to some extent. Then, we notedby the result that there is a kind of the second mover’s advantage in suchside contracting process.∗This note stems from a manuscript, titled ”Alternate two-way contracts for side pay-ments in two-player games”, which the author presented at the annual meeting of theJapan Law and Economics Association organized over zoom in November 2021. The au-thor is grateful to Professor Shinsuke Kambe for his helpful comments on the manuscriptin the conference, in accordance with which the author is still on the way revising themanuscript.†This work is part of achievement of 2021 Domestic Research Training Program, Sap-poro University. The author was engaged in visiting research at the Institute of InnovationResearch, Hitotsubashi University, in this program.1経済と経営　53‒1（２０23.3）〈研究ノート〉A note on alternate two-way contracts forside payments in two-player games＊  Akira Yamada100 経営と経済　53巻 1号In line with Jackson andWilkie (2005), a canonical paper in this eld, Ya-mada (2005b) focused on voluntarily offered side payments and assumed thatsuch side payments would always be accepted by transferees. This assump-tion might be thought of as arbitrary, however, since voluntarily offered sidepayments could be invalidated by spontaneous rejection to receive them.1In contrast to Yamada (2005b), we studied side contracting process wheretwo players alternately offer not only schemes for side payment transfer to theother but also schemes for side payment receipt from the other before choos-ing actions in the underlying game. Then, we found that the second offerer’sequilibrium payoff must be suppressed to his threat point level, and theremay be a kind of the second mover’s disadvantage in such side contractingprocess.In what follows we present the model in Section 2 and the analysis inSection 3. Our concluding remarks appear in Section 4.2 ModelWe consider two-player three-stage games played as follows.Stage 1: Player 1 announces a transfer function (side-payment transferscheme) and a receipt function (side-payment acceptance/rejection scheme),each of which is assumed to be binding.Stage 2: Player 2 announces a transfer function (side-payment transferscheme) and a receipt function (side-payment transfer acceptance/rejectionscheme), each of which is assumed to be binding.Stage 3: Each player chooses an action.The players alternately promises two-way schemes (transfer and receipt func-tions) on side payments in the rst and second stages. Such an alternate con-tracting process was analyzed by Yamada (2005b), where the players promisenot two-way but one-way (only transfer) schemes. Moreover, the two-wayschemes were called bilateral contracts in Yamada (2003), which studied theperformance of such two-way schemes in not alternate but simultaneous con-tracting games.1Yamada (2005a) studied another type of two-player three-stage games which are thesame as Yamada’s (2005b) except that players are allowed to offer negative side paymentsas well as nonnegative ones. In Yamada’s (2005a) setting, the second transfer-offerer takesall the payoff while the rst looses everything.2101A note on alternate two-way contracts for side payments in two-player gamesThe players are given by a set N = {1, 2}. Let i or j denote any givenone of the two players. When a player is denoted by i, let j denote the otherplayer, and vice versa.A player i’s nite pure strategy space in the third stage game is denotedby Xi, with X = ×iXi. Let ∆ (Xi) denote the set of mixed strategies for i,and let ∆ = ×i∆ (Xi). We denote by xi, x, µi, and µ generic elements ofXi, X, ∆ (Xi), and ∆ respectively. For simplicity, we sometimes use xi andx to denote µi and µ respectively that place probability one on xi and x. Aplayer i’s before-side-payment payoffs in the third stage game are given by avon Neumann-Morgenstern utility function vi : X → R.A transfer function announced by i in the rst or second stage is denotedby ti, where ti : X×Z → R+ with Z = {0, 1} represents i’s promises to j as afunction of actions chosen in the third stage and indicators 0 and 1. Indicator0 means that according to the transfer and receipt schemes announced in therst and second stages, a player rejects transfer from the other. Indicator 1means that according to the transfer and receipt schemes announced in therst and second stages, every player accepts transfer from the other. Let Tbe the set of all possible ti. Let t = (t1, t2).Note that if ti (x, z) = zτi (x) for some τi : X → R+, then the transferscheme becomes degenerate, or ti (x, z) = 0 for all x, unless every player ac-cepts transfer from the other. That is, when players are expected to promiseacceptance to each other, such transfer function can be sensitive to a player’sdeviation on the receipt scheme.A receipt function announced by player i in the rst or second stageis denoted by ri, where ri : T 2 → {0, 1} represents i’s acceptance (1) orrejection (0) of transfer from j as a function of proles of transfer functionsannounced in the rst and second stages. Let r = (r1, r2).Given a prole t of transfer functions and a prole r of receipt functionsin the rst and second stages, and a play x in the third stage game, the payoffUi to player i becomesUi (x, t, r) = vi (x) + (ri (t) tj (x, a (t, r))− rj (t) ti (x, a (t, r)))where a (t, r) = r1 (t)× r2 (t).Given a prole t of transfer functions and a prole r of receipt functionsin the rst and second stages, and a play µ in the third stage game, theexpected payoff EUi to player i becomes3102 経営と経済　53巻 1号EUi (µ, t, r) =Px×kµk (xk) (vi (x) + (ri (t) tj (x, a (t, r))− rj (t) ti (x, a (t, r))))where a (t, r) = r1 (t)× r2 (t). Let EUi (µ) =Px×kµk (xk) vi (x).Let NE (t, r) denote the set of (mixed) Nash equilibria of the third stagegame given (t, r) in the rst and second stages. Let NE represent the setof (mixed) Nash equilibria of the underlying game (the third stage gamewithout side contracts). For sake of simplicity and analyzability, we assumefor each i, there exists minµ∈NEEUi (µ).A pure strategy prole x ∈ X of the third stage game is supportable ifthere exists a subgame perfect equilibrium of the three stage game where(i) for t and r announced in the rst and second stages and x played in thethird stage on the equilibrium path, Ui (x, t, r) = ui for some u ∈ R2 suchthatPi ui =Pi vi (x), and moreover, (ii) any 1’s deviation in the rst stagecannot induce a Nash equilibrium of the following (after-deviation) subgameoff the equilibrium path so that 1 may enjoy more payoff than u1. Note (ii)guarantees a kind of stability of the supportability. This condition requiresthat there be no incentive for deviation from the beginning of the game.3 AnalysisWe study the model to obtain the next theorem.Theorem 1. If x is supportable, then 2’s equilibrium payoff is no more thanminµ∈NEEU2 (µ).Proof of Theorem 1. Since x is supportable, there exists a subgame perfectequilibrium of the three stage game where (i) for t and r announced in therst and second stages, and x played in the third stage on the equilibriumpath, Ui¡x, t, r¢= ui for some u ∈ R2 such thatPi ui =Pi vi (x), and (ii)any 1’s deviation in the rst stage cannot induce a Nash equilibrium of thefollowing (after-deviation) subgame off the equilibrium path so that 1 mayenjoy more payoff than u1.Suppose u2 >minµ∈NEEU2 (µ). Note here that there must be iµ for all i suchthat iµ ∈ NE and EUi (iµ) ≤ ui. Otherwise some i announces degenerateti and ri, or no transfer and no acceptance, and enjoy EUi (µ) > ui for any4103A note on alternate two-way contracts for side payments in two-player gamesµ ∈ NE. That is, i has an incentive to deviate from the equilibrium path.Let 2bµ =argminµ∈NEEU2 (µ) in particular.Consider τi : X → R+ such that τi (x) = 0 for all x 6= x, bui = vi (x) +τj (x)− τi (x) where bu1 = u1 +u2− minµ∈NE(EU2(µ))2and bu2 = u2 −u2− minµ∈NE(EU2(µ))2,and τi (x) > 0 implies τj (x) = 0. Note bu1 > u1 and bu2 >minµ∈NE(EU2 (µ))since u2 >minµ∈NE(EU2 (µ)). Let bt and br be as follows.bti (x, z) =½z (τi (x) + max {0, vi (x) + τj (x)− τi (x)− bui}) if x = (xi, xj)0 otherwisebri (t) =½1 if t = bt0 otherwiseSuppose 1 deviates to¡bt1, br1¢in the rst stage, considering the followingstrategy prole (µ, t2, r2) for the subgame afterwards.(1) (t2, r2) =¡bt2, br2¢;(2) if (t2, r2) =¡bt2, r2¢, wherer2¡bt¢ = 1, then µ = x;(2-1) if (t2, r2) =¡bt2, r2¢, where r2¡bt¢ = 0, then 2bµ ∈ NE.(2-2) if (t2, r2) = (t2, r2), where t2 6= bt2, then 2bµ ∈ NE.Case (1)-(2). If 2 announces¡bt2, r2¢with r2¡bt¢ = 1 in the second stage,x ∈ NE ¡bt, (br1, r2)¢and Ui¡x,bt, (br1, r2)¢= bui for each i where bu1 > u1 andbu2 >minµ∈NEEU2 (µ).Case (1)-(2-1). Consider 2 announces¡bt2, r2¢with r2¡bt¢ = 0 in the secondstage.If µ = (µ1,2 bµ2), thenEU1 (µ, t, r)=Px×kµk (xk) (v1 (x) + (r1 (t) t2 (x, a (t, r))− r2 (t) t1 (x, a (t, r))))=Px×kµk (xk) (v1 (x) + (1 · 0− 0 · 0))=Px×kµk (xk) v1 (x) = EU1 (µ1,2 bµ2) ≤ EU1 (2bµ),If µ = (2bµ1, µ2), thenEU2 (µ, t, r)=Px×kµk (xk) (v2 (x) + (r2 (t) t1 (x, a (t, r))− r1 (t) t2 (x, a (t, r))))5104 経営と経済　53巻 1号=Px×kµk (xk) (v2 (x) + (0 · 0− 1 · 0))=Px×kµk (xk) v2 (x) = EU2 (2bµ1, µ2) ≤ EU2 (2bµ) < bu2.Case (1)-(2-2). Consider 2 announces (t2, r2) with t2 6= bt2 in the secondstage.If µ = (µ1,2 bµ2), thenEU1 (µ, t, r)=Px×kµk (xk) (v1 (x) + (r1 (t) t2 (x, a (t, r))− r2 (t) t1 (x, a (t, r))))=Px×kµk (xk) (v1 (x) + (0 · t2 (x, a (t, r))− r2 (t) · 0))=Px×kµk (xk) v1 (x) = EU1 (µ1,2 bµ2) ≤ EU1 (2bµ).If µ = (2bµ1, µ2), thenEU2 (µ, t, r)=Px×kµk (xk) (v2 (x) + (r2 (t) t1 (x, a (t, r))− r1 (t) t2 (x, a (t, r))))=Px×kµk (xk) (v2 (x) + (r2 (t) · 0− 0 · t2 (x, a (t, r))))=Px×kµk (xk) v2 (x) = EU2 (2bµ1, µ2) ≤ EU2 (2bµ) < bu2.Thus, (1), (2), (2-1) and (2-2) constitute a Nash equilibrium of the sub-game of the three-stage game after¡bt1, br1¢is announced in the rst stage.In its equilibrium path (1)-(2) 1 can enjoy bu1 > u1, which implies 1 has anincentive to deviate to¡bt1, br1¢in the rst stage. The result contradicts tox’s supportability in terms of (ii), or its stability condition.Theorem 1 implies that the second offerer’s equilibrium payoff must besuppressed to his threat point level. That is, there may be a kind of thesecond mover’s disadvantage in such side contracting process.4 ConclusionWe studied side contracting process where two players alternately offer notonly schemes for side payment transfer to the other but also schemes for sidepayment receipt from the other before choosing actions in the underlyinggame. Then, we found that there may be a kind of the second mover’sdisadvantage in such side contracting process.6105A note on alternate two-way contracts for side payments in two-player gamesThe result seems in contrast to Yamada (2005b). Yamada (2005b) studiedside contracting process where two players alternately offer only schemes forside payment transfer to the other before choosing actions in the underlyinggame, and found that the second offerer’s equilibrium payoff can exceed histhreat point level to some extent. Then, we noted by the result that thereis a kind of the second mover’s advantage, not disadvantage, in such sidecontracting process as in Yamada’s (2005b) framework.In order to make it clear to what extent there really is a kind of thesecond mover’s disadvantage, we must consider in more detail the nature ofequilibrium in our setting. Next, we are to fully characterize equilibriumoutcomes in side contracting process where two players alternately offer notonly schemes for side payment transfer to the other but also schemes for sidepayment receipt from the other before choosing actions in the underlyinggame.5 ReferenceJACKSON, M. O. and WILKIE, S. (2005), ”Endogenous Games and Mech-anisms: Side Payments among Players”, Review of Economic Studies, 72,543-566.YAMADA, A. (2003), ”Efficient Equilibrium Side Contracts,” EconomicsBulletin, 3(6), 1-7.YAMADA, A. (2005a), ”Surviving Efficient Equilibria in Two-Player Gameswith Alternate Unilateral Contracts for Side Payments,” Review of the Eco-nomics and Business Administration, Sapporo University, 35(2), 81-86.YAMADA, A. (2005b), ”Alternate Contracts for Side Payments,” EconomicsBulletin, 3(46), 1-5.7

A note on alternate two-way contracts for side payments in two-player games

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A note on alternate two-way contracts for side payments in two-player games

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