Uncertainty, co-ordination and path dependence

Our objective is to assess whether dynamics hinder or assist co-ordination in a game with strategic complementarities. We study two dynamic aspects: different agents make decisions at different points in time; and extra information about a payoff-relevant state of nature becomes available over time. We find that the dynamic resolution of information matters most for uniqueness of equilibrium. This is demonstrated by showing that the condition for uniqueness is weaker when learning occurs. We also analyse how successfully agents co-ordinate when there is a unique equilibrium. Finally, we show that path dependence occurs: the order in which signals arrive matters, as well as the total amount of information received.     

SEQUENTIAL EXPERT ADVICE: SUPERIORITY OF CLOSED‐DOOR MEETINGS

Two career‐concerned experts sequentially give advice to a Bayesian decision maker (D). We find that secrecy dominates transparency, yielding superior decisions for D. Secrecy empowers the expert moving late to be pivotal more often. Further, (i) only secrecy enables the second expert to partially communicate her information and its high precision to D and swing the decision away from first expert's recommendation; (ii) if experts have high average precision, then the second expert is effective only under secrecy. These results are obtained when experts only recommend decisions. If they also report the quality of advice, fully revealing equilibrium may exist.     

The possibility of informationally efficient markets

A rational expectations equilibrium with positive demand for financial information does exist under fully revealing asset price—contrary to a wide-held conjecture. Whereas a continuum of investors is inconsistent with fully revealing equilibrium, finitely many investors with average portfolios demand information in equilibrium if they can adjust portfolio size in an additive signal-return model. More information diminishes the expected excess return of a risky asset so that investors who only have a choice of portfolio composition or whose asset endowments strongly differ from the average portfolio are worse off. Under fully revealing price, information market equilibria both with and without information acquisition are Pareto efficient.     

Second Mover Disadvantages in a Three-Player Stackelberg Game with Private Information

I consider a three-player Stackelberg. game where each player has its private signal concerning stochastic demand. I show that at a perfectly revealing equilibrium, the second mover earns the lowest and the third mover the highest expected profit of the three. I characterize this result by the strategic substitutes or complements relationships among three firms at the equilibrium. Journal of Economic Literature Classification Numbers: C72, D82, and L13.     

Markov Perfect Equilibrium: I. Observable Actions

We define Markov strategy and Markov perfect equilibrium (MPE) for games with observable actions. Informally, a Markov strategy depends only on payoff-relevant past events. More precisely, it is measurable with respect to the coarsest partition of histories for which, if all other players use measurable strategies, each player's decision-problem is also measurable. For many games, this definition is equivalent to a simple affine invariance condition. We also show that an MPE is generically robust: if payoffs of a generic game are perturbed, there exists an almost Markovian equilibrium in the perturbed game near the initial MPE. Journal of Economic Literature Classification Numbers: C72, C73.     

Evolutionary Stability in Games of Communication

This paper identifies evolutionarily stable outcomes in games in which one player has private information and the other takes a payoff-relevant action. The informed player can communicate at little cost. Outcomes satisfying a set-valued evolutionary stability condition must exist and be efficient in common-interest games. When there is a small cost associated with using each message the outcome preferred by the informed player is stable. The paper introduces a nonequilibrium, set-valued stability notion of entry resistant sets. For games with partial common interest, the no-communication outcome is never an element of an entry resistant set. Journal of Economic Literature Classification Numbers: C72,D82.     

Informational efficiency with ambiguous information

This paper proves the existence of fully revealing rational expectations equilibria for almost all sets of beliefs when investors are ambiguity averse and have preferences that are characterized by Choquet expected utility with a convex capacity. The result implies that strong-form efficient equilibrium prices exist even when many investors in the market make use of information in a way that is substantially different from traditional models of financial markets.     

The generic existence of rational expectations equilibrium in the higher dimensional case

The study of the generic existence of rational expectations equilibrium has concentrated on three cases, according to whether the space of “states of private information” has dimension less than, equal to, or greater than the price space. Previous results have shown that in the first case fully revealing equilibria exist generically and in the second case there are open sets of environments with no equilibrium. This paper shows that in the third case there is a residual set of environments which permits equilibria which are arbitrarily close to fully revealing.     

Sequential contracting with multiple principals

This paper considers dynamic games in which multiple principals contract sequentially and non-cooperatively with the same agent. We first show that when contracting is private, i.e. when downstream principals do not observe the mechanisms offered upstream and the decisions taken in these mechanisms, all PBE outcomes can be characterized through pure-strategy profiles in which the principals offer menus of contracts and delegate to the agent the choice of the contractual terms. We then show that, in most cases of interest for applications, the characterization of the equilibrium outcomes is further facilitated by the fact that the principals can be restricted to offer incentive-compatible extended direct mechanisms in which the agent reports the endogenous payoff-relevant decisions contracted upstream in addition to his exogenous private information. Finally we show how the aforementioned results must be adjusted to accommodate alternative assumptions about the observability of upstream histories and/or the timing of contracting examined in the literature.     

Information,trade and common knowledge

In any voluntary trading process, if agents have rational expectations, then it is common knowledge among them that the equilibrium trade is feasible and individually rational. This condition is used to show that when risk-averse traders begin at a Pareto optimal allocation (relative to their prior beliefs) and then receive private information (which disturbs the marginal conditions), they can still never agree to any non-null trade. On markets, information is revealed by price changes. An equilibrium with fully revealing price changes always exists, and even at other equilibria the information revealed by price changes “swamps” each trader's private information.     

Monetary equilibrium in an overlapping generations model with productive capital

Summary We study perfect foresight competitive equilibrium in an overlapping generations model with productive capital and a fixed nominal stock of money. We obtain almost-complete characterizations of (a) the existence of a monetary equilibrium from an arbitrary initial capital stock, and (b) the existence of anefficient monetary equilibrium from an arbitrary initial capital stock. When the initial capital stock is no larger than the golden rule stock, the necessary and sufficient condition for both (a) and (b) is the dynamic inefficiency (in the sense of Malinvaud) of the autarkic (or nonmonetary) equilibrium from the same initial stock. However, this condition, though necessary, isnot sufficient for the existence of a monetary equilibrium when the initial stock exceeds the golden rule stock (and still more conditions are needed for anefficient monetary equilibrium to exist). We provide characterizations for these cases, and as corollaries obtain examples in which (a) the nonmonetary equilibrium is inefficient but no monetary equilibrium exists, and (b) monetary equilibria exist but no efficient monetary equilibrium does.We are grateful to a co-editor and an anonymous referee for comments that greatly improved the exposition in the paper.     

Bayesian–Walrasian equilibria: beyond the rational expectations equilibrium

In general rational expectations equilibrium (REE), as introduced in Radner (Econometrica 47:655–678, 1978) in an Arrow–Debreu–McKenzie setting with uncertainty, does not exist. Moreover, it fails to be fully Pareto optimal and incentive compatible and is also not implementable as a perfect Bayesian equilibrium of an extensive form game (Glycopantis et al. in Econ Theory 26:765–791, 2005). The lack of all the above properties is mainly due to the fact that the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal), which leads inevitably to the presumption that agents know all the primitives in the economy, i.e., random initial endowments, random utility functions and private information sets. To get around this problematic equilibrium notion, we introduce a new concept called Bayesian–Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market-clearing prices using Bayesian updating and evaluate their consumption in terms of Bayesian price estimates, which are different for each individual. In this framework agents maximize expected utility conditioned on their own private information about the state of nature, subject to a Bayesian estimated budget constraint. Market clearing is not an intrinsic part of the definition of BWE. However, both in the case of perfect foresight and in the case of symmetric information BWE leads to a statewise market clearing; it then becomes an ex post Walrasian equilibrium allocation. This new BWE exists under standard assumptions, in contrast to the REE. In particular, we show that our new BWE exists in the well-known example in Kreps (J Econ Theory 14:32–43, 1977), where REE fails to exist. This work was done in the Spring of 2005, when EJB was a visiting professor at the University of Illinois.     

The acquisition of information in a dynamic market

Summary This paper models the information acquisition process in an intertemporal rational expectations framework. It demonstrates that equilibria do not generally exist in intertemporal economies in which agents are assumed to know the state-contingent price path and the information acquisition process is endogenous. In addition, an example of a fully revealing equilibrium in which agents pay a strictly positive amount for information is provided. Finally, we also show that it is possible for an equilibrium to exist in which agents choose to purchase information even if all agents, including the agents who purchased the information, are made strictly worse off by the purchase.The author would like to thank Kerry Back, Gerry Feltham Rich Kihlstrom, Vasant Naik, Bryan Routledge, Harald Uhlig and Josef Zechner for their comments and suggestions. A special note of thanks is due to the (anonymous) referee.     

Menu theorems for bilateral contracting

This paper studies bilateral contracting where multiple principals negotiate contracts with multiple agents independently. It is shown that pure-strategy equilibrium allocations relative to any ad hoc set of feasible mechanisms are supported by pure-strategy perfect Bayesian equilibria relative to the set of menus. This paper also shows that all equilibrium allocations to any ad hoc set of feasible mechanisms are supported by correlated equilibria relative to the set of menus, where a state is a probability distribution function over payoff-relevant variables. Furthermore, all equilibrium allocations relative to the set of menus persist even if principals use more complex mechanisms.     

Commitment and observability in a contracting environment

This paper examines leader-follower games where a leader must purchase an essential input from a price-setting supplier in order to take an action. We show that equilibrium outcomes when the followers perfectly observe the leaders' actions cannot be approximated by mixed equilibrium outcomes of the game where followers imperfectly observe the leaders' actions, i.e. they are not accessible. Accessibility fails since in a pure strategy equilibrium, a supplier makes positive profits; however in an equilibrium where a leader randomizes, supplier profits must be zero. Our result follows from a generalized indifference principle that mixed strategies must satisfy in economic environments. While supplier profits cannot be approximated, player action profiles are accessible. Our results also apply to games with costly observation.     

Approximate equilibria in microeconomic rational expectations models

For general equilibrium models in which prices transmit information among asymmetrically informed traders, strict rational expectations approximate equilibria are defined. A state-dependent price function is an ε-equilibrium if, when agents use their own information and that conveyed by prices, aggregate excess demand (in each state of the world) does not exceed ε. For any positive ε, existence requires only very mild assumptions—continuity and compact support. Moreover, there are revealing ε-equilibria for all smooth economies satisfying a dimensionality condition. In an open neighborhood of this case, existence of maximally revealing ε-equilibria holds.     

Campaign War Chests, Entry Deterrence, and Voter Rationality

It is often claimed that the accumulation of "war chests" by incumbents deters entry by high–quality challengers in Congressional elections. This paper presents a game–theoretic analysis of the interaction between an incumbent, potential challengers, an interest group, and a representative (rational) voter, where the incumbent's "quality" (or "legislative effectiveness") is known to the interest group, but not to the voter or to potential challengers. Under certain conditions, a perfectly revealing equilibrium exists; the incumbent signals her quality by raising funds from the interest group to accumulate a war chest. The entry deterrence effect thus operates solely through the role of war chests in signaling incumbent quality.     

Existence and Uniqueness of Nash Equilibrium in Electoral Competition Games: The Hybrid Case

This paper analyzes the traditional unidimensional, two‐party electoral competition game when parties have mixed motivations, in the sense that they are interested in winning the election, but also in the policy implemented after the contest. In spite of having discontinuous payoffs, this game, referred to as the hybrid election game, is shown to be payoff secure and reciprocally upper semi‐continuous. Conditional payoffs, however, are not quasi‐concave. Hence, the existence of a pure strategy Nash equilibrium (psne ) is ensured only if parties have homogenous interests in power. In that case, an equilibrium not only exists, but it is also unique. Instead, if parties have heterogeneous motivations, depending upon the relationship between the electoral uncertainty, the aggregate opportunism, and its distribution across parties, a psne may or may not exist. The mixed extension, however, is always better reply secure. Therefore, a mixed strategy Nash equilibrium does indeed exist.     

Bargaining foundation for ratio equilibrium in public‐good economies

We provide a bargaining foundation for the concept of ratio equilibrium in public‐good economies. We define a bargaining game of alternating offers, in which players bargain to determine their cost shares of public‐good production and a level of public good. We study the stationary subgame perfect equilibrium (SSPE) without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every SSPE without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.