Persuasion with communication costs

This paper studies strategic transmission of verifiable information with reporting costs that continuously increase in the precision of the report. Contrary to previous literature, it is shown that the unraveling result first derived by Milgrom (1981) is relatively robust to costly reporting. A separating equilibrium exists even with arbitrarily high reporting costs. Intuitively, the costs work as a signaling device and a combination of disclosure of information and costly signaling accomplishes full separation. With reporting costs there are typically multiple equilibria. For example, a pooling equilibrium exists if and only if the reporting costs are high. Finally, a separating equilibrium exists when the receiver has to make a costly effort in order to access the information in a report, as long as this cost is not too high.     

Strategic knowledge sharing in Bayesian games

This paper provides a model for the study of direct, public and strategic knowledge sharing in Bayesian games. We propose an equilibrium concept which takes into account communication possibilities of exogenously certifiable statements and in which beliefs off the equilibrium path are explicitly deduced from consistent possibility correspondences, without making reference to perturbed games. Properties of such an equilibrium and of revised knowledge are examined. In particular, it is shown that our equilibrium is always a sequential equilibrium of the associated extensive form game with communication. Finally, sufficient conditions for the existence of perfectly revealing or non-revealing equilibria are characterized in some classes of games. Several examples and economic applications are investigated.     

Uncertainty and insurance in strategic market games

Summary. For perfectly competitive economies under uncertainty, there is a well-known equivalence between a formulation with contingent goods and one with state-specific securities followed by spot markets for goods. In this paper, I examine whether this equivalence carries over to a particular form of imperfect competition. Specifically, I look at three Shapley-Shubik strategic market games: one with contingent commodities, one with Arrow securities traded under imperfect competition and one with Arrow securities traded under perfect competition. First I compare the feasibility constraints of these three games. Then I compare their equilibrium sets. As in Peck and Shell (1989), the only common equilibria between the first and the second game are those which involve no transfer of income across states. However, if the securities markets are competitive, then the set of equilibria of the contingent commodities game and the securities game coincide. Received: June 16, 1997; revised version: April 30, 1998     

Revisiting games of incomplete information with analogy-based expectations

This paper studies the effects of analogy-based expectations in static two-player games of incomplete information. Players are assumed to be boundedly rational in the way they forecast their opponent's state-contingent strategy: they bundle states into analogy classes and play best-responses to their opponent's average strategy in those analogy classes. We provide general properties of analogy-based expectation equilibria and apply the model to a variety of well known games. We characterize conditions on the analogy partitions for successful coordination in coordination games under incomplete information [Rubinstein, A., 1989. The electronic mail game: Strategic behavior under ‘almost common knowledge’. Amer. Econ. Rev. 79, 385–391], we show how analogy grouping of the receiver may facilitate information transmission in Crawford and Sobel's cheap talk games [Crawford, V.P., Sobel, J., 1982. Strategic information transmission. Econometrica 50, 1431–1451], and we show how analogy grouping may give rise to betting in zero-sum betting games such as those studied to illustrate the no trade theorem.     

Learning with bounded memory in games

We study learning with bounded memory in zero-sum repeated games with one-sided incomplete information. The uninformed player has only a fixed number of memory states available. His strategy is to choose a transition rule from state to state, and an action rule, which is a map from each memory state to the set of actions. We show that the equilibrium transition rule involves randomization only in the intermediate memory states. Such randomization, or less frequent updating, is interpreted as a way of testing the opponent, which generates inertia in the player's behavior and is the main short-run bias in information processing exhibited by the bounded memory player.     

Repeated signaling games

I analyze a class of repeated signaling games in which the informed player's type is persistent and the history of actions is perfectly observable. In this context, a large class of possibly complex sequences of signals can be supported as the separating equilibrium actions of the “strong type” of the informed player. I characterize the set of such sequences. I also characterize the sequences of signals in least cost separating equilibria (LCSE) of these games. In doing this, I introduce a state variable that can be interpreted as a measure of reputation. This gives the optimization problem characterizing the LCSE a recursive structure. I show that, in general, the equilibrium path sequences of signals have a simple structure. The shapes of the optimal sequences depend critically on the relative concavities of the payoff functions of different types, which measure the relative preferences towards payoff smoothing.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

Strategic market games with a finite horizon and incomplete markets

Summary. We study a strategic market game associated to an intertemporal economy with a finite horizon and incomplete markets. We demonstrate that generically, for any finite number of players, every sequentially strictly individually rational and default-free stream of allocations can be approximated by a full subgame-perfect equilibrium. As a consequence, imperfect competition may Pareto-dominate perfect competition when markets are incomplete. Moreover - and this contrasts with the main message conveyed by the market games literature - there exists a large open set of initial endowments for which full subgame-perfect equilibria do not converge to -efficient allocations when the number of players tends to infinity. Finally, strategic speculative bubbles may survive at full subgame-perfect equilibria.Received: 24 January 2002, Revised: 21 February 2003, JEL Classification Numbers: C72, D43, D52. Correspondence to: Gaël GiraudWe thank Tim Van Zandt for his comments.     

“Very Nice” trivial equilibria in strategic market games

Following Shapley [Theory of Measurement of Economic Externalities, Academic Press, New York, 1976], we study the problem of the existence of a Nash Equilibrium (NE) in which each trading post is either active or “legitimately” inactive, and we call it a Shapley NE. We consider an example of an exchange economy, borrowed from Cordella and Gabszewicz [Games Econ. Behav. 22 (1998) 162–169], which satisfies the assumptions of Dubey and Shubik [J. Econ. Theory 17 (1978) 1–20], and we show that the trivial equilibrium, the unique NE of the associated strategic market game, is not “very nice,” in the sense that it is not “legitimately” trivial. This result has the more general implication that, under the Dubey and Shubik's assumptions, a Shapley NE may fail to exist.     

Existence of nontrivial equilibria in repeated games with imperfect private monitoring

We consider finitely repeated games with imperfect private monitoring, and provide several sufficient conditions for such a game to have an equilibrium whose outcome is different from repetition of Nash equilibria of the stage game. Surprisingly, the conditions are consistent with uniqueness of the stage game equilibrium. A class of repeated chicken is shown to satisfy the condition.     

Strategic technology choice in regulated markets with demand uncertainty

This paper presents a simple model of a non-competitive market with demand uncertainty in which firms can choose their technology of production. Technology is characterised by two parameters: capacity and flexibility. The first has a strong commitment value while flexibility is needed to face uncertainty. Lack of competition requires active regulation to ensure that the price is not set at excessive level. When choosing their technology, firms take into account not only the effects of this choice on the opponent(s) but also the effect on the regulated price. In this framework, and because of regulation, firms have an incentive to strategically manipulate their cost (cost padding). This causes monopoly regulation aiming at improving allocative efficiency to be ineffective. In fact, by “tying its hand” to a low level of capacity, the monopolistic firm is able to get round the constraint imposed by the regulator. Increasing the number of firms in the market may restore regulation effectiveness. The reason is that if demand is sufficiently volatile, then firms strategically choose flexible techniques and this effect dominates over the incentive to manipulate costs in order to escape regulation. In this case, regulation is effective precisely because cost padding is hampered by firms’ non-cooperative behaviour.

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Evidence on learning in coordination games

This paper reports an experiment designed to detect the influence of strategic uncertainty on behavior in order statistic coordination games, which arise when a player’s best response is an order statistic of the cohort’s action combination. Unlike previous experiments using order statistic coordination games, the new experiment holds the payoff function constant and only changes cohort size and order statistic.

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Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

Polling games and information revelation in the Downsian framework

We investigate the incentives faced by poll respondents when candidates use polling data to inform their selection of policy platforms. Focusing on models with a unidimensional policy space, single peaked preferences and two office-seeking candidates observing a summary statistic from polls that ask respondents their preferences, we find that for most environments honest poll response cannot occur in a perfect Bayesian equilibrium. However, simple partially-revealing equilibria exist when the poll only asks respondents which party or candidate they prefer. When the candidates learn the sample average or see all the data, there are partially revealing equilibria that mimic those of the binary message game. Interpretation of polling data requires knowledge of the equilibrium played as the meanings of poll responses are endogenously determined. The analysis suggests that naive use of polling data may be misleading.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

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.     

Transferable utility games with uncertainty

We introduce the concept of a TUU-game, a transferable utility game with uncertainty. In a TUU-game there is uncertainty regarding the payoffs of coalitions. One out of a finite number of states of nature materializes and conditional on the state, the players are involved in a particular transferable utility game. We consider the case without ex ante commitment possibilities and propose the Weak Sequential Core as a solution concept. We characterize the Weak Sequential Core and show that it is non-empty if all ex post TU-games are convex.