The role of strategic uncertainty in games: An experimental study of cheap talk and contracts in the Nash demand game

We utilise results from a human-subjects experiment to examine the connection between strategic uncertainty and outcomes in games. Our basic game is a Nash demand game where one player has an outside option available. A “chat” treatment allows bargainers to send cheap-talk messages prior to playing the basic game, and in a “contracts” treatment, they can additionally propose and accept binding contracts. We propose that strategic uncertainty comprises at least two facets: “coordination-type”, which is lower in the chat game than in the basic game, and “rationality-type”, which is lower in the contracts game than in the chat game. We find that both types of strategic uncertainty impact bargaining outcomes: moving from the basic game to the chat game, and thence to contracts, improves several aspects of outcomes, such as higher efficiency, less opting out and less under-demanding. Other results include a treatment effect on the types of agreements that are reached.     

The number of pure Nash equilibria in a random game with nondecreasing best responses

We randomly draw a game from a distribution on the set of two-player games with a given size. We compute the distribution and the expectation of the number of pure-strategy Nash equilibria of the game conditional on the game having nondecreasing best-response functions. The conditional expected number of pure-strategy Nash equilibria becomes much larger than the unconditional expected number as the size of the game grows.     

Secrecy, two-sided bias and the value of evidence

We provide necessary and sufficient conditions on players' preferences for a verifiable message game to possess a separating equilibrium and/or a pooling equilibrium, as well as sufficient conditions for every equilibrium of such a game to be pooling. We also characterize all nonseparating equilibria, and use this property to show that less information might be conveyed in any equilibrium of a verifiable message game than in some equilibrium of the cheap talk game with the same parameters.     

Cyclic Games: An Introduction and Some Examples

We introduce a model of a cyclic game. Designed to take advantage of the recurring nature of certain economic and social situations, a cyclic game differs from an extensive form game in that a cyclic game does not necessarily have an end. The same situations, although with different players, may be repeated infinitely often. We provide an example showing that, even though a cyclic game has, in a sense, perfect information, it may not have an equilibrium in pure strategies. We demonstrate existence of equilibrium and illustrate the application of our model to an oligopolistic industry. Journal of Economic Literature Classification Numbers: C72, C73, D43.     

Reputation and Reciprocity: Consequences for the Labour Relation

Recent evidence highlights the importance of social norms in many economic relations. However, many of these relationships are long term and provide repeated game incentives for performance. We experimentally investigate interaction effects of reciprocity and repeated game incentives in two treatments (one-shot and repeated) of a gift-exchange game. In both treatments we observe reciprocity, which is strengthened in the repeated game. A detailed analysis shows that, in the repeated game, some subjects imitate reciprocity. Thus, reciprocity and repeated game incentives reinforce each other. Observed behaviour is robust against experience. We conclude that long-term interaction is a "reciprocity-compatible" contract enforcement device.
JEL classification : J 30; C 91     

Cooperative games in permutational structure

Summary. By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley. Received: October 31, 1995; revised version: February 5, 1997     

Strategic complexity in repeated extensive games

This paper studies a machine (finite automaton) playing a two-player repeated game of a simple extensive-form game with perfect information. We introduce a new complexity measure called multiple complexity which incorporates a strategyʼs responsiveness to information in the stage game as well as the number of states of the machine. We completely characterize the Nash equilibrium of the machine game. In the sequential-move prisonerʼs dilemma, cooperation can be sustained as an equilibrium.     

Endogenous private information structures

We formally incorporate the option to gather information into a game and thus endogenize the information structure. We ask whether models with exogenous information structures are robust with respect to this endogenization. Any Nash equilibrium of the game with information acquisition induces a Nash equilibrium in the corresponding game with an exogenous structure. We provide sufficient conditions on the structure of the game for which this remains true when ‘Nash’ is replaced by ‘sequential’. We characterize the (sequential) Nash equilibria of games with exogenous information structures that can arise as a (sequential) Nash equilibrium of games with endogenous information acquisition.     

Cooperators and reciprocators: A within-subject analysis of pro-social behavior

We perform a within-subject analysis of pro-social behavior in the public-good and gift-exchange game. We find that participants classified as cooperators in the public-good game tend to reciprocate higher wages in the gift-exchange game with higher levels of effort. Non-cooperators do not exhibit such tendency. Both types offer similar wages.     

Equilibrium selection in similar repeated games: experimental evidence on the role of precedents

We report on an experiment examining behavior and equilibrium selection in two similar, infinitely repeated games, Stag Hunt and Prisoner’s Dilemma under anonymous random matching. We are interested in the role that historical precedents may play for equilibrium selection between these two repeated games. We find that a precedent for efficient play in the repeated Stag Hunt game does not carry over to the repeated Prisoner’s Dilemma game despite the possibility that efficient play can be sustained as an equilibrium of the indefinitely repeated game. Similarly, a precedent for inefficient play in the repeated Prisoner’s Dilemma game does not extend to the repeated Stag Hunt game. We conclude that equilibrium selection between similar repeated games may have less to do with historical precedents and might instead depend more on strategic considerations associated with the different payoffs of these similar repeated games.     

Two equivalence results for two-person strict games

A game is strict if for both players, different profiles have different payoffs. Two games are best response equivalent if their best response functions are the same. We prove that a two-person strict game has at most one pure Nash equilibrium if and only if it is best response equivalent to a strictly competitive game, and that it is best response equivalent to an ordinal potential game if and only if it is best response equivalent to a quasi-supermodular game.     

Coalition formation in general apex games under monotonic power indices

An apex game consists of one apex player and a set of minor players. We identify two key properties of apex games and use them to introduce the class of general apex games. We derive players' preferences over winning coalitions by applying strongly monotonic power indices on such a game and all its subgames and investigate whether there are core stable coalitions in the induced hedonic coalition formation game. Besides several general results, in particular, we develop conditions on the game for the Shapley–Shubik index, the Banzhaf index, and the normalized Banzhaf index.     

Characterising equilibrium selection in global games with strategic complementarities

Global games are widely used to predict behaviour in games with strategic complementarities and multiple equilibria. We establish two results on the global game selection. First, we show that, for any supermodular complete information game, the global game selection is independent of the payoff functions chosen for the game?s global game embedding. Second, we give a simple sufficient criterion to derive the selection and establish noise independence in many-action games by decomposing them into games with smaller action sets, to which we may often apply simple criteria. We also report in which small games noise independence may be established by counting the number of players or actions.     

Fictitious play property of the Nash demand game

We show that the Nash demand game has the fictitious play property. We also show that almost every fictitious play process and its associated belief path converge to a pure-strategy Nash equilibrium in the Nash demand game.     

Large newsvendor games

We consider a game, called newsvendor game, where several retailers, who face a random demand, can pool their resources and build a centralized inventory that stocks a single item on their behalf. Profits have to be allocated in a way that is advantageous to all the retailers. A game in characteristic form is obtained by assigning to each coalition its optimal expected profit. We consider newsvendor games with possibly an infinite number of newsvendors. We prove in great generality results about balancedness of the game, and we show that in a game with a continuum of players, under a nonatomic condition on the demand, the core is a singleton. For a particular class of demands we show how the core shrinks to a singleton when the number of players increases.     

Convex Games and Stable Sets

It is well known that the core of a convex coalitional game with a finite set of players is the unique von Neumann–Morgenstern stable set of the game. We extend the definition of a stable set to coalitional games with an infinite set of players and give an example of a convex simple game with a countable set of players which does not have a stable set. But if a convex game with a countable set of players is continuous at the grand coalition, we prove that its core is the unique von Neumann–Morgenstern stable set. We also show that a game with a countable (possibly finite) set of players which is inner continuous is convex iff the core of each of its subgames is a stable set.Journal of Economic LiteratureClassification Numbers: C70, C71.     

The effect of luck framing on distributional preferences

This paper experimentally investigated luck framing. Specifically, we analyzed the difference between being assigned an advantageous role in a distribution game and being assigned the same role while being explicitly told that you were lucky to be in that favorable position. We tested this difference by implementing a dictator game and a no-veto-cost ultimatum game. We observed that: a) dictators transferred larger amounts in the game when they were explicitly told they were lucky to be the dictator compared to dictators who did not receive this message, and b) responders in the no-veto-cost ultimatum game who were explicitly told they were lucky to be in that role were significantly less likely to reject a particular offer compared to responders in the game who did not receive this message. The combination of these results is consistent with the hypothesis that people are more likely to behave in a more prosocial and egalitarian manner when they are reminded that they are lucky to be in a particular position.     

Overcoming coordination failure using a mechanism based on gradualism and endogeneity

We examine three tools that can enhance coordination success in a repeated multiple-choice coordination game. Gradualism means that the game starts as an easy coordination problem and moves gradually to a more difficult one. The Endogenous Ascending mechanism implies that a gradual increase in the upper bound of coordination occurs only if coordination with the Pareto superior equilibrium in a stage game is attained. The Endogenous Descending mechanism requires that when the game’s participants fail to coordinate, the level of the next coordination game be adjusted such that the game becomes simpler. We show that gradualism may not always work, but in such instances, its effect can be reinforced by endogeneity. Our laboratory experiment provides evidence that a mechanism that combines three tools, herein termed the “Gradualism with Endogenous Ascending and Descending (GEAD)” mechanism, works well. We discuss how the GEAD mechanism can be applied to real-life situations that suffer from coordination failure.     

The museum pass game and its value

We discuss a game in which service providers (e.g., museums) team up in offering a limited time access pass allowing unlimited usage of their services. In this game, a natural way to allocate the joint income from the sale of passes among the service providers is by using the Shapley value. We show that, for the particular game considered, the Shapley value takes a very intuitive and computationally simple form.     

Moral Hazard and Bargaining Power

Abstract. We introduce bargaining power in a moral hazard framework where parties are risk-neutral and the agent is financially constrained. We show that the same contract emerges if the concept of bargaining power is analyzed in either of the following three frameworks: in a standard principal–agent (P–A) framework by varying the agent's outside opportunity, in an alternating offer game, and in a generalized Nash-bargaining game. However, for sufficiently low levels of the agent's bargaining power, increasing it marginally does affect the equilibrium in the Nash-bargaining game, but not in the P–A model and in the alternating offer game.