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.

Finite Horizon Learning

Incorporating adaptive learning into macroeconomics requires assumptions about how agents incorporate their forecasts into their decision-making. We develop a theory of bounded rationality that we call finite-horizon learning. This approach generalizes the two existing benchmarks in the literature: Eulerequation learning, which assumes that consumption decisions are made to satisfy the one-step-ahead perceived Euler equation; and infinite-horizon learning, in which consumption today is determined optimally from an infinite-horizon optimization problem with given beliefs. In our approach, agents hold a finite forecasting/planning horizon. We find for the Ramsey model that the unique rational expectations equilibrium is E-stable at all horizons. However, transitional dynamics can differ significantly depending upon the horizon.

On Cobb-Douglas Preferences in Bilateral Oligopoly

Bilateral oligopoly is a simple model of exchange in which a finite set of sellers seek to exchange the goods they are endowed with for money with a finite set of buyers, and no price-taking assumptions are imposed. If trade takes place via a strategic market game bilateral oligopoly can be thought of as two linked proportional-sharing contests: in one the sellers share the aggregate bid from the buyers in proportion to their supply and in the other the buyers share the aggregate supply in proportion to their bids. The analysis can be separated into two ‘partial games’. First, fix the aggregate bid at B; in the first partial game the sellers contest this fixed prize in proportion to their supply and the aggregate supply in the equilibrium of this game is X˜ (B). Next, fix the aggregate supply at X; in the second partial game the buyers contest this fixed prize in proportion to their bids and the aggregate bid in the equilibrium of this game is ˜B (X). The analysis of these two partial games takes into account competition within each side of the market. Equilibrium in bilateral oligopoly must take into account competition between sellers and buyers and requires, for example, ˜B (X˜ (B)) = B. When all traders have Cobb-Douglas preferences ˜ X(B) does not depend on B and ˜B (X) does not depend on X: whilst there is competition within each side of the market there is no strategic interdependence between the sides of the market. The Cobb-Douglas assumption provides a tractable framework in which to explore the features of fully strategic trade but it misses perhaps the most interesting feature of bilateral oligopoly, the implications of which are investigated.

Noncooperative Oligopoly in Markets with a Continuum of Traders: A Limit Theorem µa la Cournot

In this paper, we consider an exchange economy µa la Shitovitz (1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley and known as the Shapley window model. We analyze the relationship between the set of the Cournot-Nash equilibrium allocations of the strategic market game and the Walras equilibrium allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably in¯nite, any Cournot-Nash equilibrium allocation of the game is not a Walras equilibrium of the underlying exchange economy. Accordingly, in the original spirit of Cournot (1838), we par- tially replicate the mixed exchange economy by increasing the number of atoms, without a®ecting the atomless part, and ensuring that the measure space of agents remains finite. We show that any sequence of Cournot-Nash equilibrium allocations of the strategic market games associated with the partially replicated exchange economies approximates a Walras equilibrium allocation of the original exchange economy.

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.

Information Design

There are two ways of creating incentives for interacting agents to behave in a desired way. One is by providing appropriate payoff incentives, which is the subject of mechanism design. The other is by choosing the information that agents observe, which we refer to as information design. We consider a model of symmetric information where a designer chooses and announces the information structure about a payoff relevant state. The interacting agents observe the signal realizations and take actions which affect the welfare of both the designer and the agents. We characterize the general finite approach to deriving the optimal information structure for the designer - the one that maximizes the designer's ex ante expected utility subject to agents playing a Bayes Nash equilibrium. We then apply the general approach to a symmetric two state, two agent, and two actions environment in a parameterized underlying game and fully characterize the optimal information structure: it is never strictly optimal for the designer to use conditionally independent private signals; the optimal information structure may be a public signal or may consist of correlated private signals. Finally, we examine how changes in the underlying game affect the designer's maximum payoff. This exercise provides a joint mechanism/information design perspective.