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on Game Theory |
| By: | Dirk Bergemann (Cowles Foundation, Yale University); Stephen Morris (Dept. of Economics, Princeton University) |
| Abstract: | We propose an incomplete information analogue of rationalizability. An action is said to be belief-free rationalizable if it survives the following iterated deletion process. At each stage, we delete actions for a type of a player that are not a best response to some conjecture that puts weight only on profiles of types of other players and states that that type thinks possible, combined with actions of those types that have survived so far. We describe a number of applications. This solution concept characterizes the implications of equilibrium when a player is known to have some private information but may have additional information. It thus answers the "informational robustness" question of what can we say about the set of outcomes that may arise in equilibrium of a Bayesian game if players may observe some additional information. |
| Keywords: | Incomplete Information, Informational Robustness, Bayes Correlated Equilibrium, Interim Correlated Rationalizability, Belief-Free Rationalizability |
| JEL: | C72 C79 D82 D83 |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:2066&r=gth |
| By: | Bó, Inácio; Hakimov, Rustamdjan |
| Abstract: | We run laboratory experiments where subjects are matched to colleges, and colleges are not strategic agents. We test the Gale-Shapley Deferred Acceptance (DA) mechanism versus the Iterative Deferred Acceptance Mechanism (IDAM), a matching mechanism based on a new family of procedures being used in the field, in which information about tentative allocations is provided while students make choices. We consider two variations of IDAM: one in which they are only informed about whether they are tentatively accepted or not (IDAM-NC) and one in which students are additionally informed at each step of the tentative cutoff values for acceptance at each school (IDAM). A significantly higher proportion of stable outcomes is reached both under IDAM and IDAM-NC than under DA. The difference can be explained by a higher proportion of subjects following an equilibrium strategy akin to truthful behavior under IDAM and IDAM-NC than truthful behavior itself under DA. Moreover, the provision of intermediate cutoff values in IDAM leads to higher rates of equilibrium behavior than in IDAM-NC and a higher robustness of stability between the rounds of experiments. Our findings provide substantial support for the rising practice of using iterative mechanisms in centralized college admissions in practice. |
| Keywords: | Market Design,Matching,Iterative Mechanisms,College Admissions,Experiments |
| JEL: | C78 C92 D63 D78 D82 |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:zbw:wzbmbh:spii2016209&r=gth |
| By: | Heinrich H. Nax; Maxwell N. Burton-Chellew; Stuart A. West; H. Peyton Young |
| Abstract: | We study behavior in repeated interactions when agents have no information about the structure of the underlying game and they cannot observe other agents’ actions or payoffs. Theory shows that even when players have no such information, there are simple payoff-based learning rules that lead to Nash equilibrium in many types of games. A key feature of these rules is that subjects search differently depending on whether their payoffs increase, stay constant or decrease. This paper analyzes learning behavior in a laboratory setting and finds strong confirmation for these asymmetric search behaviors in the context of voluntary contribution games. By varying the amount of information we show that these behaviors are also present even when subjects have full information about the game. |
| Keywords: | Learning; Information; Public goods game |
| JEL: | J1 |
| Date: | 2016–07 |
| URL: | http://d.repec.org/n?u=RePEc:ehl:lserod:68714&r=gth |
| By: | James B. T. Sanders; J. Doyne Farmer; Tobias Galla |
| Abstract: | We study adaptive learning in a typical p-player game. The payoffs of the games are randomly generated and then held fixed. The strategies of the players evolve through time as the players learn. The trajectories in the strategy space display a range of qualitatively different behaviors, with attractors that include unique fixed points, multiple fixed points, limit cycles and chaos. In the limit where the game is complicated, in the sense that the players can take many possible actions, we use a generating-functional approach to establish the parameter range in which learning dynamics converge to a stable fixed point. The size of this region goes to zero as the number of players goes to infinity, suggesting that complex non-equilibrium behavior, exemplified by chaos, may be the norm for complicated games with many players. |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1612.08111&r=gth |
| By: | Itai Arieli; H. Peyton Young |
| Abstract: | We study how long it takes for large populations of interacting agents to come close to Nash equilibrium when they adapt their behavior using a stochastic better reply dynamic. Prior work considers this question mainly for 2 × 2 games and potential games; here we characterize convergence times for general weakly acyclic games, including coordination games, dominance solvable games, games with strategic complementarities, potential games, and many others with applications in economics, biology, and distributed control. If players' better replies are governed by idiosyncratic shocks, the convergence time can grow exponentially in the population size; moreover, this is true even in games with very simple payoff structures. However, if their responses are sufficiently correlated due to aggregate shocks, the convergence time is greatly accelerated; in fact, it is bounded for all sufficiently large populations. We provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies, the length of the better reply paths, the extent to which players can influence the payoffs of others, and the desired degree of approximation to Nash equilibrium. |
| Keywords: | Population games; better reply dynamics; convergence time. |
| JEL: | J1 |
| Date: | 2016–03 |
| URL: | http://d.repec.org/n?u=RePEc:ehl:lserod:68715&r=gth |
| By: | HIRATA, Daisuke; KASUYA, Yusuke |
| Abstract: | This paper studies stable and (one-sided) strategy-proof rules in many-to-one matching markets with contracts. Not assuming any kind of substitutes condition or the law of aggregate demand, we obtain the following results. First, the number of stable and strategy-proof rules is at most one. Second, the doctor-optimal stable rule, whenever it exists, is the unique candidate for a stable and strategy-proof rule. Third, a stable and strategy-proof rule, whenever it exists, is second-best optimal for doctor welfare, in that no individually rational and strategy-proof rule can dominate it. This last result is further generalized to non-wasteful and strategy-proof rules. Due to the weak assumptions, our analysis covers a broad range of markets, including cases where a (unique) stable and strategy-proof rule is not equal to the one induced by the cumulative offer process or the deferred acceptance algorithm. |
| Keywords: | matching with contracts, stability, strategy-proofness, uniqueness, efficiency, irrelevance of rejected contracts |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:hit:econdp:2016-13&r=gth |
| By: | Nejat Anbarci; Nick Feltovich |
| Abstract: | A recurring puzzle in bargaining experiments is that individuals under–respond to changes in their bargaining position, compared to the predictions of standard bargaining theories. This result has been observed in a variety of settings, but there has been little systematic study of the factors associated with higher or lower responsiveness.We conduct a complete–information bargaining experiment using two institutions – the Nash demand game (NDG) and a related unstructured bargaining game (UBG) – and with bargaining power varied via the disagreement outcome. Importantly, in about one–fourth of bargaining pairs, one player’s disagreement payoff is more than half the cake size; in these cases, 50–50 splits are not individually rational. We find that subjects are least responsive to changes in bargaining position in the NDG with both disagreement payments below half the cake size. Responsiveness is higher in the UBG, and in the NDG when one disagreement payment is more than half the cake, but in both cases it is still less than predicted. It is only in the UBG with a disagreement payment more than half the cake that responsiveness reaches the predicted level. Our results imply that the extent to which actual bargaining corresponds to theoretical predictions will depend on (1) the institutions within which bargaining takes place, and (2) the distribution of bargaining power, in particular, whether the 50–50 norm yields a plausible focal point. We construct and analyse a simple model that characterises our main results. |
| Keywords: | Nash demand game; unstructured bargaining; real effort; equal split; experiment |
| JEL: | C78 C72 D81 |
| Date: | 2016–11 |
| URL: | http://d.repec.org/n?u=RePEc:mos:moswps:2016-21&r=gth |
| By: | Dirk Bergemann (Cowles Foundation, Yale University); Benjamin Brooks (Dept. of Economics, University of Chicago); Stephen Morris (Dept. of Economics, Princeton University) |
| Abstract: | We study auction design when bidders have a pure common value equal to the maximum of their independent signals. In the revenue maximizing mechanism, each bidder makes a payment that is independent of his signal and the allocation discriminates in favor of bidders with lower signals. We provide a necessary and sufficient condition under which the optimal mechanism reduces to a posted price under which all bidders are equally likely to get the good. This model of pure common values can equivalently be interpreted as model of resale: the bidders have independent private values at the auction stage, and the winner of the auction can make a take-it-or-leave-it-offer in the secondary market under complete information. |
| Keywords: | Optimal auction, common values, revenue maximization, revenue equivalence, rst-price auction, second-price auction, resale, posted price, maximum value game, wallet game, descending auction, local incentive constraints, global incentive constraints |
| JEL: | C72 D44 D82 D83 |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:2064&r=gth |
| By: | Satoh, Atsuhiro; Tanaka, Yasuhito |
| Abstract: | We examine maximin and minimax strategies for firms under symmetric oligopoly with differentiated goods. We consider two patterns of game; the Cournot game in which strategic variables of the firms are their outputs, and the Bertrand game in which strategic variables of the firms are the prices of their goods. We will show that the maximin strategy and the minimax strategy in the Cournot game, and the maximin strategy and the minimax strategy in the Bertrand game for the firms are all equivalent. However, the maximin strategy for the firms are not necessarily equivalent to their Nash equilibrium strategies in the Cournot game nor the Bertrand game. But in a special case, where the objective function of one firm is the opposite of the sum of the objective functions of other firms, the maximin and the minimax strategies for the firms constitute the Nash equilibrium both in the Cournot game and the Bertrand game. |
| Keywords: | maximin strategy, minimax strategy, oligopoly |
| JEL: | C72 D43 |
| Date: | 2016–12–27 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:75837&r=gth |
| By: | Dirk Bergemann (Cowles Foundation, Yale University); Benjamin Brooks (Dept. of Economics, University of Chicago); Stephen Morris (Dept. of Economics, Princeton University) |
| Abstract: | A single unit of a good is to be sold by auction to one of two buyers. The good has either a high value or a low value, with known prior probabilities. The designer of the auction knows the prior over values but is uncertain about the correct model of the buyers’ beliefs. The designer evaluates a given auction design by the lowest expected revenue that would be generated across all models of buyers’ information that are consistent with the common prior and across all Bayesian equilibria. An optimal auction for such a seller is constructed, as is a worst-case model of buyers’ information. The theory generates upper bounds on the seller’s optimal payoff for general many-player and common-value models. |
| Keywords: | Optimal auctions, common values, information structure, model uncertainty, ambiguity aversion, robustness, Bayes correlated equilibrium, revenue maximization, revenue equivalence, information rent |
| JEL: | C72 D44 D82 D83 |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:2065&r=gth |
| By: | F. Delbono; L. Lambertini |
| Abstract: | We challenge the global optimality of one-shot punishments in infi nitely repeated games with discounting. Speci fically, we show that the stick-and-carrot punishment à la Abreu (1986) may not be globally optimal. We prove our result by investigating tacit collusion in the infi nite repetition of a linear Cournot game. We illustrate the existence of the stick-and-carrot globally optimal punishment for large cartels, and fully characterise it. Then, we show that for mall cartels, global optimality may be reached only with two-period punishments. |
| JEL: | C73 L13 |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:bol:bodewp:wp1091&r=gth |
| By: | Matthew O. Jackson; Brian Rogers; Yves Zenou |
| Abstract: | We survey the literature on the economic consequences of the structure of social networks. We develop a taxonomy of ‘macro’ and ‘micro’ characteristics of social inter-action networks and discuss both the theoretical and empirical findings concerning the role of those characteristics in determining learning, diffusion, decisions, and resulting behaviors. We also discuss the challenges of accounting for the endogeneity of networks in assessing the relationship between the patterns of interactions and behaviors. |
| Keywords: | Social networks, social economics, homophily, diffusion, social learning, contagion, centrality measures, endogeneity, network formation. |
| JEL: | D85 C72 L14 Z13 |
| Date: | 2016–11 |
| URL: | http://d.repec.org/n?u=RePEc:mos:moswps:2016-45&r=gth |
| By: | HIRATA, Daisuke; KASUYA, Yusuke |
| Abstract: | This note provides three additional results that are omitted from Hirata and Kasuya (2017) but were contained in an older version (Hirata and Kasuya, 2015). |
| Keywords: | matching with contracts, stability, strategy-proofness, uniqueness, efficiency, irrelevance of rejected contracts |
| Date: | 2016–12 |
| URL: | http://d.repec.org/n?u=RePEc:hit:econdp:2016-14&r=gth |
| By: | Rothenhäusler, Dominik; Schweizer, Nikolaus; Szech, Nora |
| Abstract: | This paper analyzes how moral costs affect individual support of morally difficult group decisions. We study a threshold public good game with moral costs. Motivated by recent empirical findings, we assume that these costs are heterogeneous and consist of three parts. The first one is a standard cost term. The second, shared guilt, decreases in the number of supporters. The third hinges on the notion of being pivotal. We analyze equilibrium predictions, isolate the causal effects of guilt sharing, and compare results to standard utilitarian and non-consequentialist approaches. As interventions, we study information release, feedback, and fostering individual moral standards. |
| Keywords: | Moral Decision Making,Committee Decisions,Diffusion of Responsibility,Shared Guilt,Being Pivotal,Division of Labor,Institutions and Morals |
| JEL: | D02 D03 D23 D63 D82 |
| Date: | 2016 |
| URL: | http://d.repec.org/n?u=RePEc:zbw:kitwps:99&r=gth |
| By: | Maria Gabriella Graziano (Università di Napoli Federico II and CSEF); Claudia Meo (Università di Napoli Federico II); Nicholas C. Yannelis (University of Iowa) |
| Abstract: | We introduce the notion of stable sets with externalities and address the existence problem. The importance of this solution concept is related to the fact that the existence of core allocations for exchange economies is not in general assured in a framework with more than two traders. |
| Keywords: | Stable sets; interdependent preferences; core; types of agents. |
| JEL: | C71 D51 D70 |
| Date: | 2016–12–17 |
| URL: | http://d.repec.org/n?u=RePEc:sef:csefwp:461&r=gth |
| By: | Philip J. Grossman; Catherine Eckel; Mana Komai; Wei Zhan |
| Abstract: | This paper addresses followers’ assessment of leaders’ effectiveness in a controlled laboratory environment with salient incentives. We employ a simple game setting to examine how leaders are evaluated for the successes and failures of their groups. Followers participate in a five-person, coordination game repeated for two sets of 10 periods. Followers play each set with a different fixed group. After period 10, a leader provides (scripted) guidance on how to play the game to maximize group earnings. The gender of the leader is the only variable factor. At the end of the twentieth period, followers vote to reward (at a cost to themselves) their leader. We find that, even though leaders are all providing the same guidance, followers are more likely to heed the advice of the male leaders, followers are less likely to ascribe group success to female leaders, and followers reward male leaders more generously than female leaders. There is a premium to being male. |
| Keywords: | Leadership, Gender, Coordination Game |
| JEL: | C92 J71 J16 |
| Date: | 2016–11 |
| URL: | http://d.repec.org/n?u=RePEc:mos:moswps:2016-38&r=gth |
| By: | Benoît Julien; Sephorah Mangin |
| Abstract: | This paper generalizes the well-known Hosios (1990) efficiency condition to dynamic search and matching environments where the expected match output depends on the market tightness. Such environments give rise to a novel externality – the output externality – which may be either positive or negative. The generalized Hosios condition is simple: entry is constrained efficient when buyers’ surplus share equals the matching elasticity plus the surplus elasticity (i.e. the elasticity of the expected joint match surplus with respect to buyers). This intuitive condition captures both the standard externalities generated by the frictional matching process and the output externality. |
| Keywords: | constrained efficiency, search and matching, directed search, competitive search, Nash bargaining, Hosios condition |
| JEL: | C78 D83 E24 J64 |
| Date: | 2016–11 |
| URL: | http://d.repec.org/n?u=RePEc:mos:moswps:2016-28&r=gth |