Monotone GamesNetwork Arc hitecture, Salience and CoordinationTo be presented at Social Choice and NetworksCS 294 / Econ 207A / Math C223A / Stat 206AOct 12, 2010IntroductionThe prisoner’s dilemma game with one-shot payoffs 2 2 0 3 3 0 1 1has a unique Nash equilibrium in which each player chooses (defection),but both player are better if they choose (cooperation).If the game is played repeatedly, then ( ) accrues in every period ifeach player believes that choosing will end cooperation ( ),andsubsequent losses outweigh the immediate gain.• The Folk Theorem for infinitely repeated games demonstrates that coop-eration can be sustained in long run relationship s.• But the Folk Theorem is only partly successful as a theory of cooperativebehavior.• It guarantees the existence of a la rge class of equilibria, some of which areefficient and many m ore of which have unattractive welfare properties.• One response is to introduce more structure to guarantee efficient equilib-rium outcomes in repeated games.• A monotone game is an extensive-form game with simultaneous moves andan irreversibility structure on strategies.• It captures a variety of situations in which players make partial commit-ments.• We characterize conditions under which equilibria result in efficient out-comes.• The game has many equilibrium outcomes so the theory lacks predictivepower.• To produce stronger predictions, we restrict attention to sequential equi-libria, or Markov equilibria, or symmetric equilibria.• Whether any of these refinements is reasonable in practice is an empiricalquestion.• Multiple equilibria cannot be avoided in general and the theory cannot tellus which equilibrium is most likely to be played.• Identify the important factors in creating the “salience” of certain equilib-ria.The game• An indivisible public project with cost K and N players, each of whom hasan endowment of E tokens.• The players simultaneously make irreversible contributions to the projectat a sequence of dates t =1,...,T.• The project is carried out if and only if the sum of the contributions islarge enough to meet its cost.• If the project is completed, each player receives A tokens plus to thenumber of tokens retained from his endowment.Thegameisdefined by five parameters (positive integers except for A ≥ 0)A - value the public goodE - initial individual endowmentK - cost of the public goodN - number of playersT - number of periods.Each of these parameters influences the set of equilibria of the game in adistinct way.To avoid trivialities, we assume that— the aggregate endowment is greater than the cost of the project (com-pletion is feasible)NE > K— the aggregate value of the project is greater than the cost (completionis efficient)NA > K— the project is not completed by a single player (either it is not feasibleor it is not rational)min {A, E} <K.Information structure• To complete the description of the game, we have to specify the informa-tion available to each player.• Perfect information makes it easier for players to coordinate their actions,if they are so inclined.• In the absence of perfect information, players beliefs play a larger role insupporting (possibly inefficient) equilibria.• Asymmetry of the information structure may have an impact on the “se-lection” of equilibria.• The information structure is represented by a directed graph (or network).• Each player is located at a node of the graph and player i can observeplayer j if there is an edge leading from node i to node j.• The experiments involve three-person networks: empty, complete and allnetworks with one or two edges.• Each network has a different architecture, a different set of equilibria, anddifferent implications for the play of the game.Networks Empty Complete One-link A A A B C B C B C Line Star-out Star-in Pair A A A A B C B C B C B CThe empty networkThe game is essentially the same as the static game in which all playersmake simultaneous binding decisions.Proposition (one-shot) ( i) There exists a pure-strategy Nash equilib-rium with no completion. Conversely, there exists at least one pure-strategy equilibrium in which the project is completed with probabilityone. ( ii) The game also possesses a symmetric mixed-strategy equi-librium in which the project is completed with positive probability.The indivisibility of the public project makes each contributing player “piv-otal” (Bagnoli and Lipman (1992)).ThecompletenetworkThe sharpest result is obtained for the case of pure-strategy sequentialequilibria.Proposition (pure strategy) Suppose that A>Eand T ≥ K. Then, underthe maintained assumptions, in any pure strategy sequential equilibrium ofthe game, the public project is completed with probability one.In any pure strategy equilibrium, the probability of completion is eitherzero or one, so it is enough to show that the no-completion equilibrium isnot sequential.The logic of the proof can be illustrated by an example =31 =1 = =2Suppose, contrary to the claim, that there exists a pure sequential equilib-rium with zero provision so every player’s payoff is simply the value of hisendowment =1.If one player contributes his token at date 1, one of the remaining playerscanearnatleast1 by contributing his endowment at date 2.Thus, the good must be provided at date 2 if one player contributes atdate 1. Anticipating this response, it is clearly optimal for someone tocontribute a unit at date 1.Mixed strategies expand the set of parameters for which there exists ano-completion equilibrium.Proposition (mixed strategy) Suppose that A>Eand T ≥ K.Thenthere exists a number A∗(E,K,N, T) such that, for any E<A<A∗there exists a mixed strategy equilibrium in which the project is completedwith probability zero.The use of mixed strategies in the continuation game can discourage aninitial contribution and support an equilibrium with no completion.The example will make this clear:As long as 1 there is no pure-strategy sequential equilibrium inwhich the good is not provided.With mixed strategies, if one player contributes a token in the firstperiod, the continuation game possesses a symmetric mixed-strategyequilibrium.A necessary and sufficient condition for 0 1 to be an equilibriumstrategy is that each player
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