Network games with incomplete information

We consider a network game with strategic complementarities where the individual reward or the strength of interactions is only partially known by the agents. Players receive different correlated signals and they make inferences about other players’ information. We demonstrate that there exists a unique Bayesian-Nash equilibrium. We characterize the equilibrium by disentangling the information effects from the network effects and show that the equilibrium effort of each agent is a weighted combinations of different Katz–Bonacich centralities.     

On the relationship between robustness to incomplete information and noise-independent selection in global games

This note demonstrates that a symmetric 3×3 supermodular game may fail to have any equilibrium robust to incomplete information. Since the global game solution in symmetric 3×3 supermodular games is known to be independent of the noise structure, this result implies that a noise-independent selection in global games may not be a robust equilibrium. Our proof reveals that the assumption in global games that the noise errors are independent of the state imposes a non-trivial restriction on incomplete information perturbations.     

On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom–Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619–632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205–220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.     

On the diffuseness of incomplete information game

We introduce the “relative diffuseness” assumption to characterize the differences between payoff-relevant and strategy-relevant diffuseness of information. Based on this assumption, the existence of pure strategy equilibria in games with incomplete information and general action spaces can be obtained. Moreover, we introduce a new notion of “undistinguishable purification” which strengthens the standard purification concept, and its existence follows from the relative diffuseness assumption.     

Semiparametric identification of binary decision games of incomplete information with correlated private signals

This paper studies the identification and estimation of a static binary decision game of incomplete information. We make no parametric assumptions on the joint distribution of private signals and allow them to be correlated. We show that the parameters of interest can be point-identified subject to a scale normalization under mild support requirements for the regressors (publicly observed state variables) and errors (private signals). Following Manski and Tamer (2002), we propose a maximum score type estimator for the structural parameters and establish the asymptotic properties of the estimator.     

Potential games with incomplete preferences

This paper studies potential games allowing the possibility that players have incomplete preferences and empty best-response sets. We define four notions of potential games, ordinal, generalized ordinal, best-response, and generalized best-response potential games, and characterize them using cycle conditions. We study Nash equilibria of potential games and show that the set of Nash equilibria remains the same when every player’s preferences are replaced with the smallest generalized (best-response) potential relation or a completion of it. Similar results are established about strict Nash equilibria of ordinal and best-response potential games. Lastly, we examine the relations among the four notions of potential games as well as pseudo-potential games.     

On input-output type output multipliers from incomplete information

Incomplete knowledge of data usually hinders the establishment of detailed input-output tables. It is for this reason that up-dating procedures (RAS) as well as short-cut methods have been developed. In this paper the short-cut output multiplier formula by Drake (1976) is compared with the output multiplier involved in the RAS procedures. It will turn out that both these output multiplier estimates are two special cases of a more general class of estimating procedures. It is demonstrated by an empirical example how this generalized procedure should be applied in practice. The kind of reasoning underlying this paper can be extended to other problems of input-output analysis as well.     

Subgame-perfect equilibria in stochastic timing games

We develop a notion of subgames and the related notion of subgame-perfect equilibrium – possibly in mixed strategies – for stochastic timing games. To capture all situations that can arise in continuous-time models, it is necessary to consider stopping times as the starting dates of subgames. We generalize Fudenberg and Tirole’s (Rev. Econom. Stud. 52, 383–401, 1985) mixed-strategy extensions to make them applicable to stochastic timing games and thereby provide a sound basis for subgame-perfect equilibria of preemption games. Sufficient conditions for equilibrium existence are presented, and examples illustrate their application as well as the fact that intuitive arguments can break down in the presence of stochastic processes with jumps.     

Investment timing and efficiency in incomplete contracts

This paper considers a buyer and seller capable of making observable but nonverifiable relationship-specific investments before the parties' valuations of trade (also observable but nonverifiable) are known. Our formulation allows investments to have direct external effects and the investments to occur sequentially. We characterize an incomplete contract that implements the first-best investments and trade. This optimal contract states negotiation procedures used to determine a monetary transfer made during the investment phase and the terms of trade.     

Coordinating under incomplete information

We show that, in a minimum effort game with incomplete information where player types are independently drawn, there is a largest and smallest Bayesian equilibrium, leading to the set of equilibrium payoffs (as evaluated at the interim stage) having a lattice structure. Furthermore, the range of equilibrium payoffs converges to those of the deterministic complete information version of the game, in the limit as the incomplete information vanishes. This entails that such incomplete information alone cannot explain the equilibrium selection suggested by experimental evidence. We thank Atila Abdulkadiroglu, Hans Carlsson, Ani Guerdjikova and an anonymous referee for helpful comments. Part of this work was done while Asheim was visiting Cornell University, which hospitality is gratefully acknowledged.     

Sequential mergers under incomplete information

We study sequential mergers under incomplete information where the follower is ignorant about the leader's merger synergy. When the follower's own synergy is sufficiently large, incomplete information induces both firms to merge more. These additional mergers benefit both firms and total welfare but hurt consumers. If the follower's synergy is very small, the leader is unable to take any strategic action, and most results are reversed. The analysis suggests that incomplete information strengthens the strategic complementarity between the two mergers and thereby increases the likelihood of a merger wave.     

On games without side payments arising from a general class of markets

The class of games without side payments obtainable from finite trader markets having possibly infinite dimensional commodity spaces, individual compact, convex consumption and production sets, and concave upper-semicontinuous utility functions is considered. It is shown that these market games are precisely the totally balanced games. In fact, each totally balanced game is shown to have both a finite commodity representation and an infinite commodity ‘simple’ representation.     

Switching from complete to incomplete information

We construct an elementary mechanism [Dutta, B., Sen, A., Vohra, R., 1995. Nash implementation through elementary mechanisms in economic environments. Review of Economic Design 1, 173–203] that Nash implements the constrained Walrasian correspondence. We extend it to incomplete and non-exclusive information economies by enlarging the message space of agents. In addition, measurability restrictions on allocations with respect to prices proper to constrained rational expectations equilibria are imposed in the outcome function. We show that by imposing such restrictions, the mechanism Bayesian implements the constrained rational expectations equilibrium correspondence. This result shows game-theoretic connections between these two market equilibrium concepts. However, these connections are obtained at the price of strong restrictions on the behavior of agents.     

Asymmetric sequential search under incomplete information

We study a multistage sequential search model with n agents who compete for one job. The agents arrive sequentially, each one in a different stage. The agents' abilities, which are private information, are derived from heterogeneous distribution functions. In each stage, the designer chooses an ability threshold. If an agent has a higher ability than the threshold in the stage in which he arrives, he gets the job and the search is over. The agent's ability is not revealed when he wins the job and the designer has only an estimation of this ability according to the threshold placed by him. We analyze the optimal ability thresholds imposed by the designer who wishes to maximize the ability estimation of the agent who gets the job net of the search cost. We also investigate the relation between the optimal ability thresholds as well as the optimal order of agents in all stages according to the agents' distributions of abilities.     

A profit-center game with incomplete information

Profit centers in a firm in multidivisional form agree in the ex ante stage upon a plan about their joint production and profit imputation. The plan is executed in the subsequent two periods of the interim stage: the setup period and the manufacturing period. In the setup period, each center has its private information, but a part of its information is revealed to the other centers through its action. Based on the information endogenously pooled this way, the centers take another round of actions in the manufacturing period. A core plan is defined as a Bayesian incentive-compatible plan of the grand coalition of profit centers, upon which no coalition can improve using its Bayesian incentive-compatible plan. A core plan is called full-information revealing if each center fully reveals its private information in the setup period. Three existence theorems for a full-information revealing core plan are established. The first two theorems impose alternative conditions on returns to scale: (1) the neoclassical convex technology, and (2) increasing returns to scale. In case (2), a stronger condition than Scarf's distributiveness is imposed on the total production set. The third theorem is based on a specific supplier-customer relationship among the divisions. Received: 13 October 1997 / Accepted: 26 July 1999     

On Markov games

In the paper it is demonstrated, how a dynamic programming approach may be useful for the analysis of Markov games or stochastic games. Markov games with finitely many stages are dealt with extensively. The existence of optimal Markov strategies is proven for finite stage Markov games using a shortcut of a proof by D erman for the analogous result for Markov decision processes. For Markov games with a countably infinite number of stages some results are summarized. Here again the results and the methods of proof have much in common with results and proofs for Markov decision processes. Actually the theory of Markov games is a generalisation. The paper contains short introductions into the theories of matrix games and tree games.     

Large but finite games with asymmetric information

Carmona considered an increasing sequence of finite games in each of which players are characterized by payoff functions that are restricted to vary within a uniformly equicontinuous set and choose their strategies from a common compact metric strategy set. Then Carmona proved that each finite game in an upper tail of such a sequence admits an approximate Nash equilibrium in pure strategies.