Garbling of signals and outcome equivalence

In a game with incomplete information players receive stochastic signals about the state of nature. The distribution of the signals given the state of nature is determined by the information structure. Different information structures may induce different equilibria.Two information structures are equivalent from the perspective of a modeler, if they induce the same equilibrium outcomes. We characterize the situations in which two information structures are equivalent in terms of natural transformations, called garblings, from one structure to another. We study the notion of ‘being equivalent to’ in relation with three equilibrium concepts: Nash equilibrium, agent normal-form correlated equilibrium and the belief invariant Bayesian solution.     

Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.     

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.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.     

Payoff continuity in incomplete information games: a comment

Kajii and Morris (J. Econ. Theory 82 (1998) 267) provide necessary and sufficient conditions for two priors to be strategically close. The restrictiveness of these conditions establishes that strategic behavior can be highly sensitive to the assumed prior. Their results thus recommend care in the use of priors in economic modelling. Unfortunately, their proof of a central proposition fails for zero probability types. This comment corrects their proof to account for these cases.     

On weighted Kalai–Samet solutions for non-transferable utility coalitional form games

We investigate the implications of the axiom of coalitional concavity for non-transferable utility coalitional form games. This axiom says that if the feasible set of some coalition is uncertain whereas the feasible sets of other coalitions are known, then all players in the coalition with the uncertain feasible set should (weakly) benefit from reaching a compromise before the uncertainty is resolved. By imposing this axiom, in addition to other minor axioms, we characterize the weighted Kalai–Samet [Econometrica 53 (1985) 307] solutions: these solutions coincide with the weighted egalitarian solutions on the domain of bargaining problems, and with the weighted Shapley values on the domain of transferable utility coalitional form games.     

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.     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

Extremism, campaigning and ambiguity

This paper studies a model of how political parties use resources for campaigning to inform voters. Each party has a predetermined ideology drawn from some distribution. Parties choose a platform and campaign to inform voters about the platform. We find that, the farther away parties are from each other (on average), the less resources are spent on campaigning (on average). Thus, if parties are extreme, less information is supplied than if parties are moderate. We also show that if a public subsidy is introduced, we have policy convergence, given some mild technical restrictions on the public subsidy.     

On redundant types and Bayesian formulation of incomplete information

A type structure is non-redundant if no two types of a player represent the same hierarchy of beliefs over the given set of basic uncertainties, and it is redundant otherwise. Under a mild necessary and sufficient condition termed separativity, we show that any redundant structure can be identified with a non-redundant structure with an extended space of basic uncertainties. The belief hierarchies induced by the latter structure, when “marginalized,” coincide with those induced by the former. We argue that redundant structures can provide different Bayesian equilibrium predictions only because they reflect a richer set of uncertainties entertained by players but unspecified by the analyst. The analyst shall make use of a non-redundant structure, unless he believes that he misspecified the players' space of basic uncertainties. We also consider bounding the extra uncertainties by the action space for Bayesian equilibrium predictions.     

Strategic alliances and internationalisation in the aircraft manufacturing industry

Based on an indicator measuring the technological level of aircraft, this paper shows that in the aircraft industry, firms are obliged to deal not only with high technological barriers, but growing financial and market barriers, too. In order to reduce these, a complex network of relationships has developed over time. This network involves both main firms belonging to the world oligopoly and firms capable of offering specialised technology and/or a potential broadening of the market. The result is a worldwide production organisation. This paper highlights the fact that the aircraft industry is undergoing a global reorganisation featuring an integration process where six groups (two in Europe and four in the United States) have come to the fore. In the future, it will be possible to imagine new forms of co-operation between the emerging European and American groups.     

A theory of sequential reciprocity

Many experimental studies indicate that people are motivated by reciprocity. Rabin [Amer. Econ. Rev. 83 (1993) 1281] develops techniques for incorporating such concerns into game theory and economics. His theory is developed for normal form games, and he abstracts from information about the sequential structure of a strategic situation. We develop a theory of reciprocity for extensive games in which the sequential structure of a strategic situation is made explicit, and propose a new solution concept—sequential reciprocity equilibrium—for which we prove an equilibrium existence result. The model is applied in several examples, and it is shown that it captures very well the intuitive meaning of reciprocity as well as certain qualitative features of experimental evidence.     

Incentives in market games with asymmetric information: the core

Summary. This paper examines the ex ante core of a pure exchange economy with asymmetric information in which state-dependent allocations are required to satisfy incentive compatibility. This restriction on players' strategies in the cooperative game can be interpreted as incomplete contracts or partial commitment. An example is provided in which the incentive compatible core with nontransferable utility is empty; the game fails to be balanced because convex combinations of incentive compatible net trades can violate incentive compatibility. However, randomization of such strategies leads to ex post allocations which satisfy incentive compatibility and are feasible on average. Hence, convexity is preserved in such a model and the resulting cooperative games are balanced. In this framework, an incentive compatible core concept is defined for NTU games derived from economies with asymmetric information. The main result is nonemptiness of the incentive compatible core. Received: December 26, 2001; revised version: June 11, 2002 RID="*" ID"*" This work was financed, in part, by contract No 26 of the programme “P?le d'attraction interuniversitaire” of the Belgian government, and, in part, by research grant SBR93-09854 from the U.S. National Science Foundation. Much of my thinking about this topic was developed during a wonderful visit to CORE for the 1991–1992 academic year (on sabbatical from the University of Pennsylvania). This paper was originally circulated in December 1991 as CARESS Working Paper #91-38, Center for Analytic Research in Economics and the Social Sciences, Department of Economics, University of Pennsylvania and in February 1992 as CORE Discussion Paper 9221, Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. RID="*" ID="*" At the very start of my research, Jean-Fran?ois Mertens was almost a co-author. Fran?ois Forges provided detailed comments at a later stage, during my visit to THEMA, Université Cergy-Pontoise, in Spring 1997. They are entitled to the customary disclaimer.     

Prudent expectations equilibrium in economies with uncertain delivery

In an economy with private information, we introduce the notion of objects of choice as lists of bundles out of which the market selects one for delivery. This leads to an extension of the model of Arrow–Debreu that is used to study trade ex ante with private state verification. Under the assumption that agents are prudent, equilibrium is characterized by the fact that agents consume bundles with the same utility in states that they do not distinguish. This is a weaker condition than the restriction of equal consumption imposed by Radner (Econometrica 36(1), 31–58, 1968), therefore, some no trade situations are avoided and the efficiency of trade increases.     

Mixed-strategy equilibria and strong purification for games with private and public information

This paper shows the existence of mixed-strategy equilibria for games with private and public information under general conditions. Under the additional assumptions of finiteness of action spaces and diffuseness and conditional independence of private information, a strong purification result is obtained for the mixed strategies in such games. As a corollary, the existence of pure-strategy equilibria follows. I am very grateful to Yeneng Sun, Nicholas C. Yannelis and M. Ali Khan for helpful discussions and suggestions. I also wish to thank an anonymous referee whose comments led to many improvements in the paper.     

Existence of Sparsely Supported Correlated Equilibria

We show that every N-player K ₁ × ... × K _N game possesses a correlated equilibrium with at least zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games. We thank an anonymous referee for most useful comments. The first author acknowledges financial support from Spanish Ministry of Science and Technology, grant SEJ2004-03619, and in form of a Ramón y Cajal fellowship. The second author acknowledges support by the PASCAL Network of Excellence under EC grant no.506778, as well as from Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324. Both authors also acknowledge financial support from BBVA grant “Aprender a jugar.”     

Computable general equilibrium with financial markets

Summary. There are a wide variety of theoretical general equilibrium models with incomplete security markets. In this paper we give a general recipe for using homotopy algorithm to compute equilibria in these models. In many models, taxes, transaction-costs or other market frictions introduce the additional difficulty that equilibrium prices or choices (but not equilibrium allocations) may be undetermined. In order to demonstrate how these difficulties can be dealt with, we develop a globally convergent algorithm to compute equilibria in a model with cash-in-advance constraints, several goods and incomplete financial markets. Furthermore we describe how to implement the algorithm using a publicly available suite of subroutines for homotopy-pathfollowing. Received: October 1, 1999; revised version: December 16, 2000     

Information acquisition in auctions: Sealed bids vs. open bids

This paper studies the incentives of a bidder to acquire information in an auction when her information acquisition decision is observed by the other bidders before bidding. Our results show that the sealed bid (second price) auction may induce more information acquisition about a common component of the value than the open (English) auction but less about the private component of the value. Moreover, we show that more information about the private value and less information about the common value may improve efficiency and revenue. Consequently, our results suggest new arguments in favor of the open auction.     

An experiment on learning with limited information: nonconvergence, experimentation cascades, and the advantage of being slow

We present the results of an experiment on learning in a continuous-time low-information setting. For a dominance solvable version of a Cournot oligopoly with differentiated products, we find little evidence of convergence to the Nash equilibrium. In an asynchronous setting, characterized by players updating their strategies at different frequencies, play tends toward the Stackelberg outcome which favors the slower player. Convergence is significantly more robust for a “serial cost sharing” game, which satisfies a stronger solution concept of overwhelmed solvability. As the number of players grows, this improved convergence tends to diminish, seemingly driven by frequent and highly structured experimentation by players leading to a cascading effect in which experimentation by one player induces experimentation by others. These results have implications both for traditional oligopoly competition and for a wide variety of strategic situations arising on the Internet.