Large Strategy Spaces Research Articles

Evolution in minority games. (II). Games with variable strategy spaces

We continue our study of evolution in minority games by examining games in which agents with poorly performing strategies can trade in their strategies for new ones from a different strategy space. In the context of the games discussed in this paper, this means allowing for strategies that use information from different numbers of time lags, m. We find, in all the games we study, that after evolution, wealth per agent is high for agents with strategies drawn from small strategy spaces (small m), and low for agents with strategies drawn from large strategy spaces (large m). In the game played with N agents, wealth per agent as a function of m is very nearly a step function. The transition is at m= m t , where m t ≈ m c−1. Here m c is the critical value of m at which N agents playing the game with a fixed strategy space (fixed m) have the best emergent coordination and the best utilization of resources. We also find that overall system-wide utilization of resources is independent of N. Furthermore, although overall system-wide utilization of resources after evolution varies somewhat depending on some other aspects of the evolutionary dynamics, in the best cases, utilization of resources is on the order of the best results achieved in evolutionary games with fixed strategy spaces. Simple explanations are presented for some of our main results.

Learning Dynamics, Lock-In, and Equilibrium Selection in Experimental Coordination Games

This paper compares the leading theoretical approaches to equilibrium selection, both traditional and adaptive, in the light of recent experiments by Van Huyck, Battalio, and Beil (henceforth "VHBB") in which subjects repeatedly played coordination games, uncertain only about each other's strategy choices. The large strategy spaces of VHBB's designs and the variety of interaction patterns they considered yielded rich dynamics, with systematic differences in limiting outcomes across treatments. Explaining these differences promises to shed considerable light on equilibrium selection and coordination more generally, in the field as well as the laboratory. Following earlier analyses by Bruno Broseta and myself, I propose a model that gives a flexible characterization of individual behavior and allows for strategic uncertainty, in the form of idiosyncratic random shocks to players' adjustments. The model includes representatives of the leading approaches to equilibrium selection, which are distinguished by different values of behavioral parameters, including variances that represent the level of strategic uncertainty. The model provides a framework within which to estimate the parameters econometrically, using data from the experiments. The estimates suggest that VHBB's treatments evoked high initial levels of strategic uncertainty, declining steadily to zero as subjects learned to predict each other's responses. The resulting model has nonstationary transition probabilities and history-dependent dynamics that lock in on an equilibrium of the stage game, whose prior probability distribution is nondegenerate due to persistent effects of strategic uncertainty. The analysis shows that even when strategic uncertainty is eventually eliminated by learning, it imparts a drift to the learning dynamics, whose magnitude and direction depend on the environment and the behavioral parameters. This drift makes the distribution of the limiting outcome vary across treatments much as its empirical frequency distribution varied in the experiments. In this sense, taking the persistent effects of strategic uncertainty into account allows a simple, unified explanation of VHBB's results.

Local perfection

We investigate solution concepts for normal-form games with large strategy spaces. “Trembling hand” perfection (THP) is extended to games with compact metric strategy sets. A Nash equilibrium is THP if it is the limit of equilibria in which agents face vanishingly small uncertainty. We refine THP by imposing some structure on the kind of uncertainty that agents face: a THP equilibrium is locally perfect if the uncertainty is “predominantly local” in nature, i.e., if agents' probability assessments assign, in the limit, an overwhelmingly lage proportion of the total probability mass to events “in the vicinity of the truth.”