Abstract
Inspired by the behavior in repeated guessing game experiments, we study adaptive play by populations containing individuals that reason with different levels of cognition. Individuals play a higher order best response to samples from the empirical data on the history of play, where the order of best response is determined by their exogenously given level of cognition. As in Young’s model of adaptive play, (unperturbed) play still converges to a minimal curb set. Random perturbations of the best response dynamic identifies the stochastically stable states. In Young’s model of adaptive play with simple best-responses, the set of stochastically stable states are sensitive to the sample size that individuals from a population can draw. In generic games with higher order best-responders in both populations, the sample size is rendered irrelevant in determination of the stochastically stable set. Perhaps counter-intuitively, higher cognition may actually be bad for both the individual with higher cognition and his parent population.
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Notes
Relatedly, Stahl (1993) develops a hierarchical model of “smartness” based on rationalizability and argues that while “smartness” may evolve over time, all levels of “smartness” would continue to be represented in the population. Our framework also allows for evolution of cognitive hierarchies.
The minimal curb set is, loosely speaking, a set which contains all its best responses and there is no proper subset contained in it with the same property. For a more precise definition, see Sect. 2.
Mohlin (2012) shows that it is possible for evolutionary learning processes to converge to a state where different cognitive types co-exist.
For the purpose of nomenclature, we retain the associated terminology of the level-\(k\) model, even though we step outside the boundaries of it. Our focus is the long-run behavior, whereas the level-\(k\) model is meant for the purpose of explaining initial play.
Although in the Nash bargaining game a higher sample size confers a benefit to the population, depending on the payoff structure it can be a bane as well—see Sect. 6 for more details.
This effect has also been shown by Sáez-Martí and Weibull (1999) when allowing for “clever” individuals (that is, of level-2) in only one population.
We refer here to the same generic class of games as referred to in Young (1998, p. 111). So, a property holds generically for a class of games if it holds for an open dense subset of that class (according to the Lebesgue measure on the Euclidean space specifying the payoffs while fixing the number of players and the number of actions they can choose from). Next, if we have a property that is generic for a class of games, we call the games in the subset for which this property holds generic.
Matros (2003) shows this for the situation where only one population has level-2 individuals and both populations have the same sample size, in which case, the presence of the level-2 individuals does not make a difference to the stochastically stable outcome. Our more general result allows for even higher levels of cognition, populations to have unequal sample sizes, and both populations to host individuals of level-2 or higher.
We do not require the probability to be selected to be equal for each individual in a population.
In cases of multiple best responses, we always assume each best response to be chosen with positive probability, not necessarily with equal chance.
In order to do so, it is necessary that the \(L2\) individual possesses knowledge of the utility function of the rival population. In Sect. 5 we are going to relax this assumption.
In Sect. 5 we replace this assumption with an alternative one.
The situation where one population contains a share of \(L2\) individuals and both populations have an equal sample size has been captured in Matros (2003).
Note that we do not explicitly require all these types actually to be contained in the rival population.
The \(L^{*}k\) individual, therefore, does not choose a best response to a distribution of types, but rather, after considering the best response to each lower cognitive type, places point mass belief on one such lower cognitive type. We remain agnostic about the process by which the particular lower cognitive type is chosen, but only require that the probability of each lower cognitive type being assigned point mass belief be strictly positive.
Examples outside this class for which the propositions below do not hold are easily constructed.
Even though we assume that an \(L^{\prime }k\) individual projects his own utility function onto the rival population, the general message of this section—that is, play converges to a minimal curb set but the set of stochastically stable states may differ—is valid even if an \(L^{\prime }k\) individual evaluates the rival’s preferences with some other cardinal utility function (under the proviso that the ordinal preferences are identical). The reason for more explicitly dealing with self-projection of utility function is that under a situation of identical ordinal preferences, it might be more reasonable to attribute one’s own preference onto another rather than to use some other arbitrary utility function to do so.
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Acknowledgments
We thank Jean-Jacques Herings and David Levine for very helpful comments and suggestions. Financial support by the Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged.
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Khan, A., Peeters, R. Cognitive hierarchies in adaptive play. Int J Game Theory 43, 903–924 (2014). https://doi.org/10.1007/s00182-014-0410-5
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DOI: https://doi.org/10.1007/s00182-014-0410-5