Journal of Economic Behavior Organization Vol. 44 2001 221–232
Anticipatory learning in two-person games: some experimental results
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Fang-Fang Tang
∗
Division of Applied Economics, Nanyang Business School, Nanyang Technical University, Singapore 639798, Singapore
Received 12 October 1998; received in revised form 12 October 1999; accepted 19 October 1999
Abstract
Crawford [Econometrica 42 1974 885; J. Econ. Behavior Organ. 6 1985 69] has presented a striking example in which plausible adaptive learning rules fail to locate a straightforward
mixed-strategy equilibrium. However, Selten [Game Equilibrium Models I. Springer, Berlin 1991, p. 98] argued that such learning rules can be stabilized for some games if there is an anticipation
component in the learning process. This paper reports on an experiment designed to test Selten’s predictions. There is evidence in support of Selten’s stability prediction in the sense that the data
from a game predicted to be stable comes closer to Nash equilibrium than data from a game predicted to be unstable. © 2001 Elsevier Science B.V. All rights reserved.
JEL classification: C72; C73; C92; D83 Keywords: Mixed Nash equilibrium; Experimental learning; Stability
1. Introduction
Consider a model in which, each period, players from two large populations are randomly matched to play a two-person game in normal form. Between periods, players adjust their
behavior in adaptive ways. Assume the game has a unique Nash equilibrium which is completely mixed. For a natural adaptive rule — involving greater player emphasis on
better rewarded actions- Crawford 1974, 1985 proved that play will not converge to the equilibrium. This was a striking result. It showed a standard equilibrium to be unstable under
a natural adaptive dynamic. Selten 1991, however, rescued stability by hypothesizing that
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This paper is based on the first chapter of my Ph.D. dissertation, Tang 1996.
∗
Fax: +65-791-3697. E-mail address: afftangntu.edu.sg F.-F. Tang.
0167-268101 – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 0 0 0 0 1 5 5 - 4
222 F.-F. Tang J. of Economic Behavior Org. 44 2001 221–232
players anticipate and react, still adaptively, to the potential instability. When Selten’s anticipation hypothesis is added to Crawford’s dynamic, the mixed equilibrium becomes
stable for small enough adjustment speeds if and only if, a certain stability condition on payoffs is met.
In this paper, Selten’s stability condition, and thus, his anticipations hypothesis, is tested experimentally. Two games are specified. One meets Selten’s stability condition and the
other does not. The games are roughly similar otherwise. For each game, groups of exper- imental subjects are split into two populations to play the game repeatedly under random
matching. Selten’s anticipation hypothesis is supported in the sense that behavior clusters closer to the equilibrium for the game which meets Selten’s stability condition.
Selten’s development is much too long to reproduce. His stability condition can only be sketched. Let ˆ
A be the normalized payoff matrix faced by a player from the first of the two populations. Here “normalized” means that an appropriate constant is added to each column
of the original payoff matrix, say A, to make column sums equal to zero, and an appropriate constant is added to each row of A to make the row sums equal to zero. Similarly, let ˆ
B be the normalized version of the payoff matrix B, faced by a player from the second of the two
populations. Selten’s stability condition is that the non-zero eigenvalues of the matrix ˆ A ˆ
B
T
be negative where T denotes transpose. That is, Selten’s result is this: for small enough adjustment speeds, the mixed equilibrium is stable if and only if the non-zero eigenvalues
of ˆ A ˆ
B
T
are negative. A very rough intuition can be given to the stability condition on ˆ
A ˆ B
T
. Selten gives a much longer and more precise intuition. The effect of the first population’s play on the second
population’s payoff is determined in substantial part by ˆ B
T
, and the effect of the second population’s play on the first population’s payoff is in turn, determined in substantial part by
Â. Thus, the effect of the first population’s play on its own payoffs is critically influenced by the product ˆ
A ˆ B
T
. Under the adjustment dynamic, negative eigenvalues mean, very roughly speaking, that the first population’s change in mixing probabilities are negatively
related to their levels, which creates a stability-promoting effect. From the viewpoint of the second population, a similar interpretation can be given to the transpose of ˆ
A ˆ B
T
. When this stability-promoting effect is present, the instability of the Crawford dynamic is easier
to overcome. It turns out that the effect is necessary and sufficient for Selten’s anticipations mechanism to rescue stability. If the game were constant-sum, then ˆ
B = − ˆ A would hold, and
ˆ A ˆ
B
T
= − ˆ A ˆ
A
T
would automatically have negative eigenvalues also see Conlisk 1993a,b. For the two experimental games presented momentarily, the one failing Selten’s stability
condition will depart further from constant sums. Sections 2 and 3 describe the experimental design and procedures. Section 4 describes
results. Section 5 concludes the paper.
2. Game design
