Costly and discrete communication: an experimental investigation

Language is an imperfect and coarse means of communicating information about a complex and nuanced world. We report on an experiment designed to capture this feature of communication. The messages available to the sender imperfectly describe the state of the world; however, the sender can improve communication, at a cost, by increasing the complexity or elaborateness of the message. Here the sender learns the state of the world, then sends a message to the receiver. The receiver observes the message and provides a best guess about the state. The incentives of the players are aligned in the sense that both sender and receiver are paid an amount which is increasing in the accuracy of the receiver’s guess. We find that the size of the language endogenously emerges as a function of the costs of communication. Specifically, we find that higher communication costs are associated with a smaller language. Although the equilibrium predictions do not perform well, this divergence occurs in a manner which is consistent with the experimental communication literature: overcommunication. We find that the sender’s payoffs, relative to equilibrium payoffs, are decreasing in the cost of communication. We also find that the receiver’s payoffs, relative to equilibrium payoffs, are increasing in the cost of communication. Finally, we find imperfections in coordination on the basis of the experimental labels.     

Who gains from information asymmetry?

This article considers an asymmetric contest with incomplete information. There are two types of players: informed and uninformed. Each player has a different ability to translate effort into performance in terms of the contest success function. While one player’s type is known to both players, the other is private information and known only to the player himself. We compare the Bayesian Nash equilibrium outcome of a one-sided private information contest to the Nash equilibrium with no private information, in which both players know the type of the other player. We show conditions under which uncertainty increases the investment of the uninformed player and the rent dissipation of the contest, while decreasing the expected net payoff of the informed player. In addition, we consider conditions under which the informed player—before knowing his own type—prefers that the uninformed player knows his type. Moreover, we show conditions for the existence/non-existence of equilibrium in a two-stage contest in which the informed player declares his type (or does not declare) in the first stage and in the second stage the two players play according to the information available to them.     

Degrading network capacity may improve performance: private versus public monitoring in the Braess Paradox

The Braess Paradox (BP) is a counterintuitive finding that degrading a network that is susceptible to congestion may decrease the equilibrium travel cost for each of its users. We illustrate this paradox with two networks: a basic network with four alternative routes from a single origin to a single destination, and an augmented network with six alternative routes. We construct the equilibrium solutions to these two networks, which jointly give rise to the paradox, and subject them to experimental testing. Our purpose is to test the generality of the BP when the network is enriched as well as the effects of the information provided to the network users when they conclude their travel. To this end, we compare experimentally two information conditions when each of the two networks is iterated in time. Under public monitoring each user is accurately informed of the route choices and payoffs of all the users, whereas under private monitoring she is only informed of her own payoff. Under both information conditions, over iterations of the basic and augmented games, aggregate route choices converge to equilibrium.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

Socially Structured Games

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power. We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core. We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera.      

Communication Protocols with Belief Messages

This paper presents a communication protocol to reach consensus. In our setting, every player has asymmetric information and evaluates a fixed event on his information. According to the protocol, the sender sends non-partitional messages that he believes the event with a probability of at least his evaluation. We show that the posteriors for the event must be equal among the players after the communication. Journal of Economic Literature Classification: C62, C78.     

Perceptron versus automaton in the finitely repeated prisoner’s dilemma

We study the finitely repeated prisoner’s dilemma in which the players are restricted to choosing strategies which are implementable by a machine with a bound on its complexity. One player has to use a finite automaton while the other player has to use a finite perceptron. Some examples illustrate that the sets of strategies which are induced by these two types of machines are different and not ordered by set inclusion. Repeated game payoffs are evaluated according to the limit of means. The main result establishes that a cooperation at almost all stages of the game is an equilibrium outcome if the complexity of the machines the players may use is limited enough and if the length T of the repeated game is sufficiently large. This result persists when more than T states are allowed in the player’s automaton. We further consider a variant of the model in which the two players are restricted to choosing strategies which are implementable by perceptrons and prove that the players can cooperate at most of the stages provided that the complexity of their perceptrons is sufficiently reduced.     

Satisfaction and preferences in a legality social dilemma: Does corporate social responsibility impact consumers’ behaviour?

We investigate players’ preferences in a multiplayer prisoner's dilemma by comparing results from satisfaction-based and a choice-based approach by means of a laboratory experiment. The experimental design tests the effects of the legality rating frame on consumers’ choice between products from producers labeled by the legality rating and products by unlabeled producers. Both approaches provide strong evidence of preference heterogeneity, with players who cooperate above median being less affected in their choice by monetary payoffs vis-à-vis the public good component. The empirical findings support the hypothesis of our theoretical model that (part of the) players have, in addition to the standard self-interest component, an other-regarding preference argument that is further satisfied in the legality frame plus conformity design.A policy suggestion stemming from our experiment is based on the fact that corporate social responsibility legality frames and culture have a significant effect on an important portion of consumers. These consumers reveal that the often-declared willingness to pay for socially, environmentally and legally responsible features of products is confirmed by actual purchases of more expensive responsible products. Governments should therefore promote the creation of legality rating schemes with well-defined pre-established rules such as in the Italian case.     

A Simple Model of Secure Public Communication

Public communication is secure if a hostile third-party cannot decode the messages exchanged by the communicating parties. In Nash equilibrium, communication by computationally unbounded players cannot be secure. We assume complexity averse players, and show that a simple, secure, and costless communication protocol becomes available as the marginal complexity cost tends to zero.      

Equilibrium as compatibility of plans

This paper uses a game-theoretic framework to formalize the Hayekian notion of equilibrium as the compatibility of plans. To do so, it imposes more structure on the conventional model of strategic games. For each player, it introduces goals, goal-oriented strategies, and the goals’ probabilities of success, from which players’ payoffs are derived. The differences between the compatibility of plans and Nash equilibrium are identified and discussed. Furthermore, it is shown that the notion of compatibility of plans, in general, differs from the notion of Pareto efficiency. Since the compatibility of plans across all players can rarely be achieved in reality, a measurement is introduced to determine various degrees of plan compatibility. Several possible extensions and applications of the model are discussed.

     

A Banzhaf share function for cooperative games in coalition structure

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.     

Implementing equal division with an ultimatum threat

We modify the payment rule of the standard divide the dollar (DD) game by introducing a second stage and thereby resolve the multiplicity problem and implement equal division of the dollar in equilibrium. In the standard DD game, if the sum of players’ demands is less than or equal to a dollar, each player receives what he demanded; if the sum of demands is greater than a dollar, all players receive zero. We modify this second part, which involves a harsh punishment. In the modified game \((D\!D^{\prime })\) , if the demands are incompatible, then players have one more chance. In particular, they play an ultimatum game to avoid the excess. In the two-player version of this game, there is a unique subgame perfect Nash equilibrium in which players demand (and receive) an equal share of the dollar. We also provide an \(n\) -player extension of our mechanism. Finally, the mechanism we propose eliminates not only all pure strategy equilibria involving unequal divisions of the dollar, but also all equilibria where players mix over different demands in the first stage.     

Information transparency and equilibrium selection in coordination games: an experimental study

We experimentally investigate the role of information transparency for equilibrium selection in stag hunt coordination games. These games can be transformed from a prisoner’s dilemma game by introducing a centralized reward or punishment scheme. We aim to explore the impact of the disclosure of information on how final payoffs are derived on players’ incentive to coordinate on the payoff-dominant equilibrium. We find that such information disclosure significantly increases the tendency of players to play the payoff-dominant strategy and reduces the occurrence of coordination failure. The mechanism works directly through the positive impact of disclosure on the saliency of the payoff-dominant equilibrium, and indirectly through the positive influence of disclosure on players’ belief about the likelihood of cooperation by the opponent.     

Game-theoretic analyses of coalition behavior

Coalitions are frequently more visible than payoffs. The theory of n-person games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person game of status in which each player seeks to maximize the rank of his total score. To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by von Neumann and Morgenstern's solution of two-person cooperative games, Aumann and Maschler's bargaining set solution for cooperative n-person games, and an alternative model of coalition behavior in three-person sequential games of status.

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Effects of fixed costs in two-person sequential bargaining

Rubinstein (1982) considered the problem of dividing a given surplus between two players sequentially, and then proposed a model in which the two players alternately make and respond to each other's offers through time. He further characterized the perfect equilibrium outcomes, which depend on the players time preferences and order of moves. Using both equal and unequal bargaining cost conditions and an unlimited number of rounds, two experiments were designed to compare the perfect equilibrium model to alternative models based on norms of fairness. We report analyses of final agreements, first offers, and number of bargaining rounds, which provide limited support to the perfect equilibrium model, and then conclude by recommending a shift in focus from model testing to specification of the conditions favoring one model over another.     

Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games

A situation, in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (point-valued) solution for TU-games assigns a payoff distribution to every TU-game. In this article we discuss a class of equal surplus sharing solutions consisting of all convex combinations of the CIS-value, the ENSC-value and the equal division solution. We provide several characterizations of this class of solutions on variable and fixed player set. Specifications of several properties characterize specific solutions in this class.     

Subgame perfect equilibrium in a bargaining model with deterministic procedures

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, \(\delta _A\) and \(\delta _B\), we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if \(\delta _A+\delta _B\ge 1\), while almost no division can ever be supported in SPE if \(\delta _A+\delta _B < 1\).     

Tacit Coordination in Choice Between Certain Outcomes in Endogenously Determined Lotteries

Tacit coordination is studied in a class of games in which each of n = 20 players is required to choose between two courses of actions. The first action offers each player a fixed outcome whereas the second presents her the opportunity of participating in a lottery with probabilities that are determined endogenously. Across multiple iterations of the game and trial-to-trial changes in the composition of the lottery, we observe a remarkably good coordination on the aggregate but not individual level. We further observe systematic deviations from the Nash equilibrium solution that are accounted for quite well by a simple adaptive learning model.     

Ambiguity and decision modeling: A preference-based approach

In this article, we develop a model that permits a decision maker's preferences to depend on the decision maker's ambiguity about the probability of an event that is relevant for decision-making purposes. We deal with ambiguity through preference modeling, with ambiguity leading to modifications in the utilities of outcomes. The behavior of ambiguity premiums and probability premiums as the payoffs are varied depends on the nature of the modifications in utilities. Particular forms of the model that arise under different sets of assumptions about preferences include additive, bilinear, and ratio forms. We conclude with a brief example and some thoughts about potential generalizations and implications of the model.     

Consulting an Expert with Potentially Conflicting Preferences

We study a situation where a decision maker relies on the report of a self-interested and informed expert prior to decide whether to undertake a certain project. An important feature in this interaction is that, depending on the collected information, the two agents have potentially conflicting preferences. Information contained in the report is partially verifiable in the sense that the expert can suppress favorable information sustaining the project but he cannot exaggerate it. Our results show that this setting favors the agent which is the less eager to undertake the project in that he always succeeds to induce his most preferred action.