Ed Hopkins

Learning, information, and sorting in market entry games: theory and evidence

Previous data from experiments on market entry games, N-player games where each player faces a choice between entering a market and staying out, appear inconsistent with either mixed or pure Nash equilibria. Here we show that, in this class of game, learning theory predicts sorting, that is, in the long run, agents play a pure strategy equilibrium with some agents permanently in the market, and some permanently out. We conduct experiments with a larger number of repetitions than in previous work in order to test this prediction. We find that when subjects are given minimal information, only after close to 100 periods do subjects begin to approach equilibrium. In contrast, with full information, subjects learn to play a pure strategy equilibrium relatively quickly. However, the information which permits rapid convergence, revelation of the individual play of all opponents, is not predicted to have any effect by existing models of learning.

Learning in perturbed asymmetric games

We investigate the stability of mixed strategy equilibria in 2 person (bimatrix) games under perturbed best response dynamics. A mixed equilibrium is asymptotically stable under all such dynamics if and only if the game is linearly equivalent to a zero sum game. In this case, the mixed equilibrium is also globally asymptotically stable. Global convergence to the set of perturbed equilibria is shown also for (rescaled) partnership games (also know as games of identical interest). Some applications of these results to stochastic learning models are given.

Marriage as a Rat Race: Noisy Pre-Marital Investments with Assortative Matching

We study the efficiency of premarital investments when parents care about their child’s marriage prospects, in a large frictionless marriage market with nontransferable utility. Stochastic returns to investment ensure that equilibrium is unique. We find that, generically, investments exceed the Pareto-efficient level, unless the sexes are symmetric in all respects. Girls will invest more than boys if their quality shocks are less variable than shocks for boys or if they are the abundant sex. The unique equilibrium in our continuum agent model is the limit of the equilibria of finite models, as the number of agents tends to infinity.

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.

Cross and Double Cross: Comparative Statics in First Price and All Pay Auctions

This paper analyses comparative statics for first price auctions and all pay auctions with independent private values. In all pay auctions, bidders with low values will respond to a stochastically higher (in the sense of likelihood ratio dominance) distribution of types by playing less aggressively while high value bidders bid more. In the first price auction, a similar change results in all types playing more aggressively. Furthermore, we show that a decrease in dispersion of values, in the sense of a refinement of second order stochastic dominance, although also associated with an increase in competitiveness, may in addition result in less aggressive play by bidders with high values in both auction forms. We also find similar considerations in an oligopoly game with incomplete information: stochastically lower costs can lead to higher prices.

Inequality and Risk-Taking Behaviour

This paper investigates social in∞uences on attitudes to risk and ofiers an evolutionary explanation of risk-taking by young low-ranked males. Becker, Murphy and Werning (2005) found that individuals about to participate in a status tournament may take fair gambles even though they are risk averse in both wealth and status. Here their model is generalised by use of the insight of Hopkins and Kornienko (2010) that in a tournament or status competition one can consider equality in terms of the status or rewards available as well as in initial endowments. While Becker et al. found that risk-taking is increasing in the equality of initial endowments, it is found here that it is increasing in the inequality of rewards in the tournament. Further, it is shown that the poorest will be risk loving if the lowest level of status awarded is su‐ciently low. Thus, the disadvantaged in society rationally engage in risky behavior when social rewards are su‐ciently unequal. Finally, as greater inequality in terms of social status induces gambling, it can cause greater inequality of wealth.

Evolving Game Theory

Jorgen W. Weibull (1995) Evolutionary Game Theory. MIT Press. £18.50

Running to Keep in the Same Place: Consumer Choice as a Game of Status

30 xv+265 pp. Ken Binmore, Alan Kirman and Piero Tani (eds) (1993) Frontiers of Game Theory. MIT Press. £40.50

Two Competing Models of How People Learn in Games

47.50 350 pp.