A number of papers have shown that a strict Nash equilibrium action profile of a game may never be played if there is a small amount of incomplete information. The authors present a general approach to analyzing the robustness of equilibria to a small amount of incomplete information. A Nash equilibrium of a complete information game is said to be robust to incomplete information if every incomplete information game with payoffs almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. The authors show that many games with strict equilibria have no robust equilibrium and examine why they get such different results from existing refinements. If a game has a unique correlated equilibrium, it is robust. A natural many-player many-action generalization of risk dominance is a sufficient condition for robustness.

Traders with short horizons and privately known loss limits interact in a market for a risky asset. Risk-averse, long horizon traders generate a downward sloping residual demand curve that faces the short-horizon traders. When the price falls close to the loss limits of the short horizon traders, selling of the risky asset by any trader increases the incentives for others to sell. Sales become mutually reinforcing among the short term traders, and payoffs analogous to a bank run are generated. A “liquidity black hole” is the analogue of the run outcome in a bank run model. Short horizon traders sell because others sell. Using global game techniques, we solve for the unique trigger point at which the liquidity black hole comes into existence. Empirical implications include the sharp V-shaped pattern in prices around the time of the liquidity black hole.

We study the informational events that trigger equilibrium shifts in coordination games with incomplete information. Assuming that the distribution of the changes in fundamentals has fat tails, we show that majority play shifts either if fundamentals reach a critical threshold or if there are large common shocks, even before the threshold is reached. The fat-tail assumption matters because it implies that large shocks make players more unsure about whether their payoffs are higher than others. This feature is crucial for large shocks to matter.

An Ascending Auction for Interdependent Values: Uniqueness and Robustness to Strategic Uncertainty by Dirk Bergemann and Stephen Morris. Published in volume 97, issue 2, pages 125-130 of American Economic Review, May 2007

What outcomes can be implemented by the choice of an information structure in binary‐action supermodular games? An outcome is partially implementable if it satisfies obedience (Bergemann and Morris (2016)). We characterize when an outcome is smallest equilibrium implementable (induced by the smallest equilibrium). Smallest equilibrium implementation requires a stronger sequential obedience condition: there is a stochastic ordering of players under which players are prepared to switch to the high action even if they think only those before them will switch. We then characterize the optimal outcome induced by an information designer who prefers the high action to be played, but anticipates that the worst (hence smallest) equilibrium will be played. In a potential game, under convexity assumptions on the potential and the designer's objective, it is optimal to choose an outcome where actions are perfectly coordinated (all players choose the same action), with the high action profile played on the largest event where that action profile maximizes the average potential.

We analyze games of incomplete information and offer equilibrium predictions that are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action–state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.

The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying mechanism design on richer type spaces. We ask when ex post implementation is equivalent to interim (or Bayesian) implementation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is implementing a social choice function (not correspondence) and (2) in a quasilinear environment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance. In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results offer new insights into the relationship between dominant strategy implementation and Bayesian implementation.

A set of players have preferences over a set of outcomes. We consider the problem of an “information designer” who can choose an information structure for the players to serve his ends, but has no ability to change the mechanism (or force the players to make particular action choices). We describe a unifying perspective for information design. We consider a simple example of Bayesian persuasion with both an uninformed and informed receiver. We extend information design to many player and relate it to the literature on incomplete information correlated equilibrium.

This paper elucidates the logic behind recent papers which show that a unique equilibrium is selected in the presence of higher order uncertainty, i.e., when players lack common knowledge.We introduce two new concepts: belief potential of the information system and p-dominance of Nash-equilibria of the game, and show that a Nash-equilibrium is uniquely selected whenever its p-dominance is below the belief potential.This criterion applies to many-action games, not merely 2 x 2 games.It also applies to games without dominant strategies, where the set of equilibria is shown to be smaller and simpler than might be initially conjectured.Finally, the new concepts help understand the circumstances under which the set of equilibria varies with the amount of common knowledge among players.