Endogenous Stackelberg equilibria with incomplete information

In this paper, the endogenous order of quantity decision is studied in a duopoly model with incomplete information. One firm knows the state of the demand curve while the other firm remains uninformed. Firms have to commit to a quantity in one out of two periods. While, a priori, simultaneous-move Cournot equilibria are possible, only Stackelberg equilibria, with either the informed or the uninformed firm moving first, emerge endogenously.     

Correlated equilibria, incomplete information and coalitional deviations

This paper proposes new concepts of strong and coalition-proof correlated equilibria where agents form coalitions at the interim stage and share information about their recommendations in a credible way. When players deviate at the interim stage, coalition-proof correlated equilibria may fail to exist for two-player games. However, coalition-proof correlated equilibria always exist in dominance-solvable games and in games with positive externalities and binary actions.     

Sequential location under one-sided demand uncertainty

In entering a new market, firms face demand uncertainty. We depart from the usual Hotelling duopoly model with sequential entry. We allow firms to locate outside of the city and assume that market conditions are common knowledge. We then introduce one-sided demand uncertainty. We find that demand uncertainty can be seen as a differentiation force when faced by the first entrant and as an agglomeration force when faced by the second entrant. Finally, the second firm’s imperfect information implies higher welfare losses.     

Belief-free equilibria in games with incomplete information: Characterization and existence

We generalize the results of Hörner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.     

Representation of constitutions under incomplete information

We model constitutions by effectivity functions. We assume that the constitution is common knowledge among the members of the society. However, the preferences of the citizens are private information. We investigate whether there exist decision schemes (i.e., functions that map profiles of (dichotomous) preferences on the set of outcomes to lotteries on the set of social states), with the following properties: (i) The distribution of power induced by the decision scheme is identical to the effectivity function under consideration; and (ii) the (incomplete information) game associated with the decision scheme has a Bayesian Nash equilibrium in pure strategies. If the effectivity function is monotonic and superadditive, then we find a class of decision schemes with the foregoing properties.     

Alternating-offer bargaining over menus under incomplete information

Summary. This paper considers bargaining with one-sided private information and alternating offers where an agreement specifies both a transfer and an additional (sorting) variable. Moreover, both sides can propose menus. We show that for a subset of parameters the alternating-offer game has a unique equilibrium where efficient contracts are implemented in the first period. This stands in sharp contrast to the benchmarks of contract theory, where typically only the uninformed side proposes, and bargaining theory, where typically the agreement only specifies a transfer. Received: September 10, 2001; revised version: March 25, 2002 RID="*" ID="*" I benefitted from discussions with Benny Moldovanu, Holger Müller, and Roland Strausz, and from comments made by an anonymous referee.     

Regulating a risk-averse firm under incomplete information

We examine the optimal regulatory policy for a risk-averse firm when the firm is imperfectly informed about its efficiency parameter for a project at the time of contracting. The firm’s risk aversion shifts the optimal regulatory policy from a fixed-price contract to a cost-plus contract. The optimal regulatory policy entails undereffort by an inefficient firm as in Laffont and Tirole (J Polit Econ 94(3):614–641, 1986) and the effort distortion increases as the firm becomes more risk-averse. Further, the regulator benefits from sequential contracting with the firm where the firm chooses contract terms gradually as it acquires information, albeit the benefit diminishes as the firm becomes more risk-averse.      

Computationally restricted unmediated talk under incomplete information

Summary. We show the role of unmediated talk with computational complexity bounds as both an information transmission and a coordination device for the class of two-player games with incomplete information and rational parameters. We prove that any communication equilibrium payoff of such games can be reached as a Bayesian-Nash equilibrium payoff of the game extended by a two phase universal mechanism of interim computationally restricted pre-play communication. The communication protocols are designed with the help of modern cryptographic tools. A familiar context in which our results could be applied is bilateral trading with incomplete information.Received: 9 September 2002, Revised: 14 March 2003, JEL Classification Numbers: C72. Correspondence to: Amparo UrbanoWe wish to thank financial aid from the Valencian Institute of Economic Research (IVIE) and partial support by DIGCYT under project PB95 - 1074. A previous version of this work appears as IVIE Working Paper WP-AD 99-07, under the title: "Unmediated talk under incomplete information".     

The evolution of conventions under incomplete information

Summary. We formulate an evolutionary learning process with trembles for static games of incomplete information. For many games, if the amount of trembling is small, play will be in accordance with the games (strict) Bayesian equilibria most of the time. This supports the notion of Bayesian equilibrium. Often the process will select a specific equilibrium. We study an extension to incomplete information of the prototype conflict known as Chicken and find that the equilibrium selection by evolutionary learning may well be in favor of inefficient Bayesian equilibria where some types of players fail to coordinate.Received: 17 March 2003, Revised: 3 December 2003, JEL Classification Numbers: C72.     

p-Best response set and the robustness of equilibria to incomplete information

In this paper, we use p-best response sets—a set-valued extension of p-dominance—in order to provide a new sufficient condition for the robustness of equilibria to incomplete information: if there exists a set S which is a p-best response set with , and there exists a unique correlated equilibrium μ^* whose support is in S then μ^* is a robust Nash equilibrium.     

Common priors under incomplete information: a unification

Summary. While the meaningfulness of the common prior assumption (CPA) under incomplete information has been established recently by various authors, its epistemic rationale has not yet been adequately clarified. To do so, we provide a characterization of the CPA in terms of a new condition called “Mutual Calibration”, and argue that it constitutes a more transparent and more primitive formalization of the Harsanyi Doctrine than the existing characterizations. Our analysis unifies the understanding of the CPA under incomplete information and clarifies the role of higher-order expectations and of the difference between situations with only two and those with at least three agents. In the concluding section, the analysis is applied to the problem of defining Bayesian consistency of the intertemporal beliefs of a single-agent with imperfect memory. The CPA yields a notion of “Bayesian updating without a prior”. Received: March 24, 2000; revised version: April 27, 2000     

Stochastic equilibria with incomplete financial markets

We demonstrate the existence of equilibria with incomplete financial markets for stochastic economies whose information structure is given by an event tree, restricting attention to purely financial securities, those paying in units of account (e.g., “dollars”). Financial markets may be incomplete: some consumption streams may be impossible to obtain by any trading strategy. Securities may be individually precluded from trade at arbitrary states and dates. Sufficient conditions for the existence of stochastic equilibria are: continuous, convex, strictly monotonic preferences and strictly positive aggregate endowments. These conditions are weakened. A corollary states that any regime of security prices precluding arbitrage can be embedded in an equilibrium.     

Incentives and incomplete information

The problem of incentives for correct revelation in a collective decision model is presented as a game with incomplete information. Two approaches to incomplete information are used, a first where the individual beliefs are not introduced and a second where they are. In the first approach it is recalled that the mechanisms for which the solution to the incentive problem is in dominant strategies lead in general to a budgetary problem for the central agency. For these mechanisms a uniqueness property is demonstrated. In the second approach it is shown that if a compatibility condition is imposed on the individual beliefs and if a Bayesian solution is given to the incentive problem, then it is possible to avoid the budgetary problem.     

Cooperative vs. non-cooperative R&D incentives under incomplete information

This paper studies incentives for cooperative research vis-à-vis non-cooperative research in an incomplete information framework. We show that with quantity competition under asymmetric information, the expected payoff from non-cooperative research goes down compared to the case of symmetric information; hence research joint venture incentives of the firms are larger under asymmetric information. In either case, however, the larger is the size of the cost-reducing innovation the lower is the incentive for cooperative research. Finally, in our model, incomplete information does not affect the consumers' welfare, but the firms become worse off.     

Bargaining under incomplete information, fairness, and the hold-up problem

In the hold-up problem incomplete contracts cause the proceeds of relationship-specific investments to be allocated by bargaining. This paper investigates the corresponding investment incentives if individuals have heterogeneous fairness preferences. Individual preferences are taken to be private information. Investments can then signal preferences and thereby influence beliefs and bargaining behavior. In consequence, individuals might choose high investments in order not to signal information that is unfavorable in the ensuing bargaining.     

Coarse matching with incomplete information

We study two-sided markets with heterogeneous, privately informed agents who gain from being matched with better partners from the other side. Our main results quantify the relative attractiveness of a coarse matching scheme consisting of two classes of agents on each side, in terms of matching surplus (output), an intermediary’s revenue, and the agents’ welfare (defined as the total surplus minus payments to the intermediary). Following Chao and Wilson (Am Econ Rev 77: 899–916, 1987) and McAfee (Econometrica 70:2025–2034, 2002), our philosophy is that, if the worst-case scenario under coarse matching is not too bad relative to what is achievable by more complex, finer schemes, a coarse matching scheme will turn out to be preferable once the various transaction costs associated with fine schemes are taken into account. Similarly, coarse matching schemes can be significantly better than random matching, while still requiring only a minimal amount of information.     

Budget-constrained sequential auctions with incomplete information

I study a budget-constrained, private-valuation, sealed-bid sequential auction with two incompletely-informed, risk-neutral bidders in which the valuations and income may be non-monotonic functions of a bidder's type. Multiple equilibrium symmetric bidding functions may exist that differ in allocation, efficiency and revenue. The sequence of sale affects the competition for a good and therefore also affects revenue and the prices of each good in a systematic way that depends on the relationship among the valuations and incomes of bidders. The sequence of sale may affect prices and revenue even when the number of bidders is large relative to the number of goods. If a particular good, say α, is allocated to a strong bidder independent of the sequence of sale, then auction revenue and the price of good α are higher when good α is sold first.