Bayesian Rationality with Subjective Evaluations in Enlivened Decision Trees

A decision-making agent is usually assumed to be Bayesian rational, or to maximize subjective expected utility, in the context of a completely and correctly speci ed decision model. Following the discussion in Hammond (2007) of Schumpeter's (1911, 1934) concept of entrepreneurship, and of Shackle's (1953) concept of potential surprise, this paper considers enlivened decision trees whose growth over time cannot be accurately modelled in full detail. An enlivened decision tree involves more severe limitations than model mis-speci cation, unforeseen contingencies, or unawareness, all of which are typically modelled with reference to a universal state space large enough to encompass any decision model that an agent may consider. We consider three motivating examples based on: (i) Homer's classic tale of Odysseus and the Sirens; (ii) a two-period linear-quadratic model of portfolio choice; (iii) the game of Chess. Though our novel framework transcends standard notions of risk or uncertainty, a form of Bayesian rationality is still possible. Instead of subjective probabilities of different models of a classical finite decision tree, we show that Bayesian rationality and continuity imply subjective expected utility maximization when some terminal nodes have attached real-valued subjective evaluations instead of consequences. Moreover, subjective evaluations lie behind, for example, the kind of Monte Carlo tree search algorithm that has been used by some powerful chess-playing software packages.