Case-based learning with different similarity functions

The paper applies the rule for adaptation of the aspiration level suggested by Gilboa and Schmeidler to a situation in which the similarity between acts is represented by an arbitrary similarity function [Gilboa, I., Schmeidler, D., 1996. Case-based optimization. Games Econ. Behav. 15, 1-26]. I show that the optimality result derived by Gilboa and Schmeidler in general fails. With a concave similarity function, only corner acts are chosen in the limit. The optimality result can be reestablished by introducing convex regions into the similarity function and modifying the aspiration adaptation rule. A similarity function which is "sufficiently convex" allows approximating optimal behavior with an arbitrary degree of precision.