Brock, W., Dindo, P. and Hommes, C. (2006). Adaptive rational equilibrium with forward looking agents International Journal of Economic Theory, 2(3-4):241--278.

In adaptive rational equilibrium dynamics (ARED) agents choose between a costly rational expectation forecast and a cheap naive forecast, and the fractions using each of the two strategies evolve over time and are endogenously coupled to the market equilibrium price dynamics. In this setting, agents are backward looking in the sense that strategy selection is based on experience measured by relative past realized profits. When the selection pressure to switch to the more profitable strategy is high, instability and complicated chaotic price fluctuations arise. In this paper we investigate the ARED with forward looking agents, whose strategy selection is based upon expected profits. Our findings suggest that forward looking behavior dampens the amplitude of price fluctuations, but local instability of the steady state remains. The global dynamics depends upon how sophisticated the forward looking behavior is. With perfectly forward looking agents, prices converge to a stable 2-cycle, whereas with forward looking agents who are boundedly rational concerning their estimate of expected profits, small amplitude chaotic price fluctuations might arise. We also establish an equivalence relationship between a heterogeneous agent model with switching of strategies and a representative agent framework, where the representative agent optimally chooses between the benefits of a high quality forecast and the associated information gathering costs. To an outside observer it is impossible to distinguish between the two.