# Reinhard Selten (2004)

## StationarityConcepts for Experimental 2 x 2 Games

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Imagine you are sitting in a separate cubicle in front of a computer screen. You are looking on a matrix with the four fields AA, AB, BA and BB. You are playing a game over 200 periods against an invisible person in another cubicle. If you are player 1, then you choose either up or down by clicking A or B, if you are player 2, you choose either left or right by clicking A or B. Your pay-off after each click depends both on your and your opponent’s decision and is clearly indicated in each of the four fields of the matrix. For you it could be for example 7, 0, 5 or 9 in AA, AB, BA or BB, while your opponent would gain 4, 11, 6 and 2 in the respective fields. For your participation you receive € 5 plus 1.6 €-Cent for each payoff point accumulated over the 200 periods.

With this setting, Reinhard Selten and his coworkers attracted 864 students of the University of Bonn to participate in an experiment to find stationary solutions (i.e. equilibria) in completely mixed 2x2 games (where two players have two options each). Their research objective was, as Selten explains in this lecture in Lindau, to compare the experimental results with four different theoretical concepts and see how the latter’s predictions match with the practical results. For this purpose, they conducted 54 sessions with 16 students each. In these sessions, they ran twelve different games, i.e. worked with twelve different pay-off matrices. The games were selected to yield a reasonably wide distribution over the parameter space. In each session, there were two independent subject groups with four participants in the role of player 1 and four participants in the role of player 2. Within each subject group players 1 and 2 were randomly chosen in each period. In contrast to experimental 2x2 games played repeatedly by the same two opponents, the experimental design used here mirrors the behavior of populations.

The four stationary concepts, which Selten compares with the data from his extensive experiment are the Nash equilibrium, the sample-7 equilibrium, the impulse balance equilibrium and the quantal response equilibrium. It turns out that the by far best performing concept is the impulse balance equilibrium. “It is related to learning direction theory”, says Selten, “the theory that I have developed on the basis of a paper together with Stöcker, long ago, in 1985, and it is better described in a paper by Selten, Abbink and Cox 2001[1]. The impulse balance theory is more a behavioral theory.”

In comparison to his vividly illustrative Lindau lecture in 1997, Reinhard Selten gives a very technical presentation here, whose understandability would certainly be facilitated if one saw its accompanying slides. In this audio file this not possible of course. One year after his lecture in Lindau, however, Selten published the results of the research he presented there in an informative and comprehensive paper, which the University of Bonn fortunately makes freely accessible [2].

Joachim Pietzsch

[1]
Selten R, Abbink K, Cox R (2001). Learning direction theory and the winner’s curse. Bonn Graduate School of Economics. Universität Bonn. Discussion Paper No. 10/2001.

[2]
Selten R and Chmura T. Stationary concepts for experimental 2x2 games. Bonn Graduate School of Economics. Universität Bonn. Discussion Paper No. 33/2005.
http://www.wiwi.uni-bonn.de/bgsepapers/bonedp/bgse33_2005.pdf

Abstract

The talk reports experimental results about 200 times repeated completely mixed 2x2 games played by randomly matched subjects with constant roles. Each of six zero-sum games was played by 12 independend groups of 8 subjects. Each of six non-zero games was played by 6 independend groups of 8 subjects. Three stationary concepts and associated learning models were examined: Nash equilibrium, sample 7 equilibrium and impulse balance. Sample 7 assumes optimization against a sample of 7 choices of the other player drawn from the stationary distribution. Impulse balance assumes choice probabilities proportional to expected payoff differences in favor of the chosen strategy. In addition to this impulse balance is based on a transformed game in which losses relative to the third highest payoff count double. The three concepts do not require parameter estimates. The comparisons are based on quadratic distances of observed choice frequencies from theoretical predicted ones or with simulation results for the learning model. An overall optimal parameter was estimated for a fourth concept, quantal response equilibrium. Impulse balance significantly outperforms all other concepts. The same result is obtained for the associated learning models.

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