Testing the TASP: An Experimental Investigation of Learning in Games with Unstable Equilibria

We report experiments designed to test between Nash equilibria that are stable and unstable under learning. The “TASP” (Time Average of the Shapley Polygon) gives a precise prediction about what happens when there is divergence from equilibrium under fictitious play like learning processes. We use two 4 x 4 games each with a unique mixed Nash equilibrium; one is stable and one is unstable under learning. Both games are versions of Rock-Paper-Scissors with the addition of a fourth strategy, Dumb. Nash equilibrium places a weight of 1/2 on Dumb in both games, but the TASP places no weight on Dumb when the equilibrium is unstable. We also vary the level of monetary payoffs with higher payoffs predicted to increase instability. We find that the high payoff unstable treatment differs from the others. Frequency of Dumb is lower and play is further from Nash than in the other treatments. That is, we find support for the comparative statics prediction of learning theory, although the frequency of Dumb is substantially greater than zero in the unstable treatments.

Bargaining and Conflict with Incomplete Information

This paper studies bargaining and conflict under incomplete information, provides an overview and a critical account of the literature on the topic and contributes with original research. We first revise models of mechanism design and sequential bargaining that take confrontation as final. Conflict and inefficiencies are to be expected in these models whenever parties have optimistic prospects on the outcome of the all-out conflict. After examining the causes and reasons for this optimism, we move to the analysis of the recent literature that considers the existence of limited confrontations that allow bargaining to resume. In the presence of private information, these limited conflicts convey information and thus become a bargaining instrument. The paper closes with a discussion on the related empirical literature, the challenges that it faces and some potential avenues for further research.

Optimism and commitment: An elementary theory of bargaining and war

We propose an elementary theory of wars fought by fully rational contenders. Two parties play a Markov game that combines stages of bargaining with stages where one side has the ability to impose surrender on the other. Under uncertainty and incomplete information, in the unique equilibrium of the game, long confrontations occur: war arises when reality disappoints initial (rational) optimism, and it persist longer when both agents are optimists but reality proves both wrong. Bargaining proposals that are rejected initially might eventually be accepted after several periods of confrontation. We provide an explicit computation of the equilibrium, evaluating the probability of war, and its expected losses as a function of i) the costs of confrontation, ii) the asymmetry of the split imposed under surrender, and iii) the strengths of contenders at attack and defense. Changes in these parameters display non-monotonic effects.

Dominance Solvable Games with Multiple Payoff Criteria.

Two logically distinct and permissive extensions of iterative weak dominance are introduced for games with possibly vector-valued payoffs. The first, iterative partial dominance, builds on an easy-to check condition but may lead to solutions that do not include any (generalized) Nash equilibria. However, the second and intuitively more demanding extension, iterative essential dominance, is shown to be an equilibrium refinement. The latter result includes Moulin’s (1979) classic theorem as a special case when all players’ payoffs are real-valued. Therefore, essential dominance solvability can be a useful solution concept for making sharper predictions in multicriteria games that feature a plethora of equilibria.