Equilibria in sequential bargaining games as solutions to systems of equations

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1 Economics Letters 84 (2004) Equilibria in sequential bargaining games as solutions to systems of equations Tasos Kalandrakis* Department of Political Science, Yale University, and Wallis Institute, University of Rocester, USA Received 7 December 2003; accepted 10 Marc 2004 Available online 7 June 2004 Abstract No-delay, stationary equilibrium points of sequential bargaining games wit istory-dependent random recognition rules and general agreement rules are caracterized via a finite number of equalities and inequalities. Existence of equilibrium is establised using Brouwer s fixed point teorem. D 2004 Elsevier B.V. All rigts reserved. Keywords: Sequential bargaining; Equilibrium existence JEL classification: C72 We caracterize no-delay, stationary equilibrium points in a general class of sequential bargaining games via a finite number of equalities and inequalities. We ten adopt arguments of Nas (1951), to establis existence of equilibrium using Brouwer s teorem. Existence as been obtained for a general class of discounted games wit stationary random recognition rules by Banks and Duggan (2000), using Glicksberg s teorem. Jackson and Moselle (2002) analyze an application of a related maority rule game and establis existence using Kakutani s teorem. Te arguments we present below furter simplify te existence proof. We allow for bot discounting and fixed delay costs. We also study more general recognition rules to allow as special cases bot alternating offers models and stationary random recognition rules. Tus, wen it comes to institutions, we encompass models analyzed by Rubinstein (1982), Binmore (1987), Baron and Fereon (1989), Baron (1991), Banks and Duggan (2000), etc. Besides its simplicity, we believe te caracterization we present is enligtening as to te structure of te equilibrium set. It is used in Kalandrakis (2003) to sow generic determinacy of pure strategy equilibria for a * W. Allen Wallis Institute of Political Economy, 107 Harkness Hall, University of Rocester, Rocester, NY , USA. Tel.: ; fax: address: kalandrakis@yale.edu (T. Kalandrakis) /$ - see front matter D 2004 Elsevier B.V. All rigts reserved. doi: /.econlet

2 408 T. Kalandrakis / Economics Letters 84 (2004) subclass of tese games. It is also possible tat tis formulation may facilitate te construction of algoritms for te computation of equilibrium. We start our analysis by describing te bargaining environment. Consider a set of n z 2 players N={1,...,n}. Tey convene in periods t = 1,2,...to reac an agreement x drawn from a set X. X is a convex, compact subset of R d, d z 1. A decision requires te approval of a winning coalition, CpN. Te set of winning coalitions is determined by te underlying voting rule and is denoted by Do2 N \Ø,D p Ø. For example, if all players ave one vote and te voting rule is simple maority, D consists of all coalitions wit more tan n/2 members. As in Banks and Duggan (2000) te only restriction on te voting rule is monotonicity, so tat for any two coalitions A, B wit ApBpN, we ave AaD Z BaD. In eac period t = 1,2,..., one of te players is recognized to make a proposal zax. Having observed te proposal, players vote yes or no. If a winning coalition vote yes, ten te game ends wit z being implemented, else te game moves to te next period. Proposers in eac period are recognized probabilistically and probabilities of recognition may depend on te identity of te proposer in te last period. Tus, if te game reaces period t > 1 and player an was te proposer in period t 1, ten player ian is recognized wit probability p i z 0, P n ¼1 p ¼ 1, for eac an. Player ian derives von Neuman Morgenstern stage utility u i :X! R from te agreement x. We assume trougout u i is continuous and add furter assumptions as necessary. To complete te description of payoffs, we entertain two standard possibilities in te literature: Assumption A1 (discounting): Players discount te future by a factor d i a[0,1], ian, and u i (x) z 0, for all xax, all ian. Tus, te payoff of player i from a decision xax reaced in period t z 1 is given by d i t 1 u i (x), and it is zero in te case of perpetual disagreement. Te second possibility is: Assumption A2 (fixed delay cost): Players incur a delay cost c i >0, ian. Under tis assumption, te payoff of player i from a decision xax reaced in period t z 1 is given by u i (x) (t 1)c i, and it is l in te case of perpetual disagreement. As is standard in te literature, we focus on stationary subgame perfect (SSP) equilibria and require tat in every structurally identical subgame players beavior is ex ante (prior to any randomization) identical. A stationary proposal strategy for player ian is an element l i ap[x], were P[X] is te set of Borel probability measures over X. We say tat an SSP involves no delay if every proposal in te support of players proposal strategies is approved. Before we describe voting strategies, we define te continuation value of players as teir expected utility if te game moves in te next period. Wit no-delay proposal strategies te continuation value of player i, m i, wen te proposer in te current period is is given by m i ¼ Xn ¼1 Z p u i ðxþl *ðdxþ; for all i; an X ð1þ Tus, te space of all possible continuation values, VoR n, in no-delay equilibria is obtained as te image of a mapping v:p[x]! R n were te it coordinate is defined as m i ðlþum X u i ðxþlðdxþ. Clearly, V u v(p[x]) is convex and it is also compact as te continuous image of compact set P[X ] (Aliprantis and Border, 1999, Teorem 14.11). We specify stationary voting strategies for player i by acceptance sets A i ox, an. A i contains te proposals by player wic i approves. We restrict voting strategies in order to rule out equilibria were undesirable agreements are approved or desirable agreements are reected solely because eac of te players is not pivotal

3 T. Kalandrakis / Economics Letters 84 (2004) and ence is indifferent between er voting actions. Tus, in equilibrium we require tat A i :VOX is a correspondence tat satisfies xaa i ðv ÞZu i ðxþzd i m i ; under A1 ð2þ xaa i ðv ÞZu i ðxþzm i c i ; under A2 ð3þ for all, ian. Following Baron and Kalai (1993) we call suc voting strategies stage-undominated. Consider te subset of winning coalitions tat include player and are minimum winning in te sense tat if any player i p is removed from te coalition, te coalition ceases to be winning. Call tis set of coalitions t od, defined as t u {CaD:aC, C \{i}gd,i p }. Let te number of coalitions in t be n u At A.By non-emptiness and monotonicity of te agreement rule, D, we are guaranteed tat n z 1 for all an. We now define te agenda setting plan, f x, of proposer and coalition C x at, x = 1,...,n, as a correspondence f x :V O X, given by fx Þuarg max x u ðxþ: xa u A ðv Þ ac x : In certain modal cases, as we establis in Lemma 1 at te end of our analysis, f x is a non-empty, single-valued function for all an, x = 1,...,n. Wen all agenda setting plans, f x, are functions we are afforded te following reduction in te description of stationary proposal strategies. Instead of a measure l ap[x] we specify te proposal strategy of player as a coice among te finite number of coalitions in t. Tus, for continuation values v av, a proposal strategy for player is an element, m ad n 1, were D n 1 is te (n 1)-dimensional unit simplex in R n. Te coalition mixing probability m x denotes te probability tat coalition C x at is cosen. Implicitly, wen te cosen coalition is C x at, te proposed agreement is given by f x (v )ax. We now ave te following caracterization of no-delay SSP equilibrium for te entire class of games we consider: Teorem 1. If f x is a function for all an, x =1,...,n, ten no-delay SSP equilibria in stageundominated voting strategies under eiter A1 or A2 are caracterized by coalition mixing probabilities m *ad s 1, and continuation values v *av, an suc tat: m * i ¼ Xn ¼1 X n p c¼1 m * c u iðfc ðv* ÞÞ; for all i; an ð4þ m * x > 0Zu ðf x ðv * ÞÞzu ðf xv ðv * ÞÞ; xv ¼ 1;...; n : ð5þ Proof. Every winning coalition Cgt wit ac is a superset of some coalition CVat. Tus, wen continuation values are given by some v av, if tere exist optimal, no-delay proposals for tat are acceptable by members of C, tey are also acceptable by coalition CVat, i.e. optimal proposals must be drawn from { f x (v )} n x =1 and conditions (4) and (5) are necessary in equilibrium. Note tat Eqs. (4)

4 410 T. Kalandrakis / Economics Letters 84 (2004) and (5) ensure tere do not exist profitable one-stage deviations at te proposal stage. Also, by definition, agenda setting plans f x respect condition (2) or (3), ence tere are no profitable one-stage deviations at te voting stage eiter wit stage-undominated voting strategies. Tus, tere are no profitable finite period deviations. Now, since u i (x) z 0, xax, tere do not exist profitable infinite deviations under A1. c i >0 ensures te same under A2, and we ave establised sufficiency. 5 Te above caracterization invites a proof of existence of equilibrium using Brouwer s fixed point teorem. We provide suc a proof in te next teorem: Teorem 2. Consider a game under A1 or A2 for wic f x is a continuous function for all an, x =1,...,n. A no-delay SSP equilibrium exists. Proof. Let M = D n 1 n =1. Define a function F:V n M! V n M so tat te first n 2 coordinates (in V n ) correspond to te n vectors of continuation values, v=(v 1,...,v n ), and te P n ¼1 n remaining coordinates in M correspond to coalition mixing probabilities. Set te coordinate tat corresponds to te continuation value of te it player wen te proposer is as F ðm Þðv; mþu P n ¼1 p P n x¼1 m x u iðfx ðv ÞÞ. i To specify coordinates corresponding to te mixing probabilities m x *, we sall use Nas s (1951) analogous formulation for finite games in normal form. P For given v av and by using a particular mixing, m, among coalitions, proposer s expected utility is n x¼1 m x u ðfx ðv ÞÞ. If instead proposes only to coalition C x at er utility is u ( f x (v )). Te potential improvement in Vs payoff, u x (v,m ), obtained by proposing exclusively n to coalition C x instead of mixing among coalitions according to m, is defined as ux ðv ; m Þumax 0; u ðfx ðv ÞÞ P o n x¼1 m x u ðfx ðv ÞÞ. Tis is a continuous function by te continuity of f x, u, by te teorem of te Maximum. Ten, we define te xt coordinate of te mixing vector of proposer as F (mx) (v,m)um x +u x (v,m )/1+ P n x¼1 u x (v,m ). Importantly, F is a continuous function of (v, m)av n M as a result of te continuity of f x, u, and u x. Also, V n M V n M is convex and compact since bot M and V are convex and compact. Tus, Brouwer s fixed point teorem applies and F as a fixed point. It is trivial to see tat (v, m) is a fixed point of F if an only if it satisfies conditions (4) and (5). 5 Teorems 1 and 2 could be void if te conditions on te agenda setting plans f x are not met. In te next lemma we provide a pair of sufficient conditions: Lemma 1. Assume tat for all ian eiter: (A3) u i strictly concave, or (A4) X = D n 1,u i (x)=m i (x i ), wit m i (0) = 0, m i V>0, and m i W V 0. Ten, f x is a continuous function for all an, x =1,...,n under A1 or A2. Proof. Under eiter A3 or A4, A C :V O X defined as \ ac A (v ) for Cat is a non-empty, continuous correspondence (e.g., Banks and Duggan, 2000). Also, \ ac A (v ) is convex valued as te intersection of convex sets. Tus, A3 implies tere is a unique maximizer, and continuity of f x follows from Berge s teorem of te Maximum. Also, under A4, te inverse m 1 i exists, ence te agenda setting plan is obtained as te following continuous function: 8 0 if gc x f x ðv Þ¼ >< >: m 1 i ðmaxf0; m gþ if ac x qfg 1 X m 1 l ðmaxf0; m l gþ if ¼ lac x qfg were t = d m under A1, or t = m c under A2. 5 ð6þ

5 T. Kalandrakis / Economics Letters 84 (2004) Condition A3 covers political environments wit spatial or ideological spaces, wile A4 covers standard dividete-dollar environments. Wile in combination, tese sufficient conditions cover most applications in te literature, we note tat te same existence arguments apply if f x are correspondences tat admit continuous selections. Alternatively, if we relax A3 to u i concave, instead of strictly concave, we can use Geanakoplos (2003) satisficing principle to establis existence via Brouwer s teorem. Lastly, an easy consequence of our formulation is a bound on te number of possible agreements wic cannot be larger tan P n i¼1 n i in any equilibrium. For oligarcic rules, n i = 1, and all SSP equilibria are in pure strategies, so tat we extend Teorem 2, Part (ii) of Banks and Duggan (2000) to non-stationary recognition rules and te case of fixed delay costs. References Aliprantis, C.D., Border, K., Infinite Dimensional Analysis: a Hitciker s Guide, 2nd ed. Springer-Verlag: Berlin. Banks, J.S., Duggan, J., A bargaining model of collective coice. American Political Science Review 94 (Marc), Baron, D., A spatial bargaining teory of government formation in parliamentary systems. American Political Science Review 85 (Marc), Baron, D.P., Fereon, J.A., Bargaining in legislatures. American Political Science Review 83 (December), Baron, D.P., Kalai, E., Te simplest equilibrium of a maority rule game. Journal of Economic Teory 61, Binmore, K., Perfect Equilibria in Bargaining Models, Capter 5. In: Binmore, K., Dasgupta, P. (Eds.), Te Economics of Bargaining. Blackwell, Oxford. Geanakoplos, J., Nas and Walras equilibrium via Brouwer. Economic Teory 21 (2 3), Jackson, M., Moselle, B., Coalition and party formation in a legislative voting game. Journal of Economic Teory 103, Kalandrakis, T., Regularity of pure strategy stationary equilibria in a class of bargaining games. Mimeo, Yale University. Nas, J., Non-Cooperative games. Te Annals of Matematics, 2nd Ser. 54 (2), Rubinstein, A., Perfect equilibrium in a bargaining model. Econometrica 50,