یادگیری قوانین در بازی های عادی ـ فرم متقارن: تئوری و شواهد

An experiment, consisting of two 15-period runs with 5 × 5 games, was designed to test Stahl's [International Journal of Game Theory28, 111–130 (1999)] model of boundedly rational behavioral rules and rule learning for symmetric normal-form games with unique symmetric Nash equilibria. A player begins with initial propensities on a class of evidence-based behavioral rules and, given experience over time, adjusts her propensities in proportion to the past performance of the rules. The experimental data provide significant support for rule learning and heterogeneity characterized by three modes. We also strongly reject “Nash learning” and “Cournot dynamics” in favor of rule learning. Journal of Economic Literature Classification Numbers: C72, C90, C51, C52.

This paper tested a theory of boundedly rational behavioral rules, and a theory of rule learning based on past performance. The boundedly rational rules can be interpreted as weighing evidence for and against available actions based on archetypal models of the other players. An econometric model was specified and an experiment was designed to fit and test this model. Our model fits the data much better than random noise or an error-prone Nash model. We strongly reject the hypothesis of no rule learning and the hypothesis of Cournot dynamics in favor of our model. The maximum likelihood estimates of the parameters reveal substantial heterogeneity in the population of participants both in the initial log-propensities over rules and the extent of learning. A statistical test of homogeneity strongly rejects that hypothesis. After exhaustive testing of each of the nine parameters of the rule-learning model, we rejected the hypotheses that anyone of them could be dropped from the model. Herd evidence Yo and level-1 evidence Y1 receive the most initial weight on average. However, one should not conclude that level-2 evidence (Y2) and Nash evidence (Y3) are unimportant, since they can become important via learning. The average estimate of f31 is 1.53, so substantial learning can occur. Further, examining the reinforcement function s'». e, 0 1+ 1 ) , we found that the "99% best" rules in our (v, e) rule space generally included high values for V 2 and "s- Knowing that the no-rule-learning hypothesis was strongly rejected, there is no doubt that this reinforcement of level-2 and Nash evidence contributed to the success of the model. We also presented graphical evidence (Fig. 5) showing substantial changes in the weight given level-2 and Nash evidence over time, consistent with rule learning and increasing sophistication. Using a histogram approach and kernel density estimation, we were able to identify three modes in the distribution of the initial evidence weights v. This suggests that it might be possible to specify a parsimonious finite mixture model that explains the data as well (when accounting for the difference in the number of parameters).While we could not reject asymptotic stability (f3o ::s; 1) for three-fourths of the participants, it may trouble some readers that 25% appear to have explosive dynamics. However, it should be noted that the only way for the probabilistic choice function to eventually put unit mass on one rule (such as the Nash rule) is to have explosive dynamics. An apparent weakness of our learning theory is the implicit assumption that participants evaluate the whole five-dimensional space of rules Rand update their rule probabilities 'P accordingly. It might seem more realistic to assume that the players gather incomplete samples of rule performance measures depending on similarity or closeness with the rules recently used. Unfortunately, we cannot directly observe this sampling and evaluation process. On the other hand, rules that are "close" in the brains of human subjects are not necessarily those that are close in our parametric representation (and vice versa). In other words, a rule that involves a large change in the parameter space of our representation is not necessarily more distant, and hence less likely to be evaluated. Stinchcombe (1997)has shown that artificial neural networks are universal approximators even for arbitrarily small parameter sets; that is, small changes in the parameter weights of a neural network can span a large function space. Therefore, it is theoretically possible that local experimentation in the weight-space of a brain's neural network could in fact span a space of rules as large as and similar to our five-dimensional space of evidence-based rules.