Rational probabilistic deciders.

Abstract

This dissertation develops a purely probabilistic approach to the theory of rational behavior by introducing and analyzing so-called <italic> Rational Probabilistic Deciders</italic> (RPDs). RPDs are ergodic stochastic systems that choose better decisions with larger probabilities than worse ones. RPDs do not give rise to the complicated dynamics specific to other approaches such as rational automata. Thus, RPDs permit more complete investigations of their stationary and transient properties. The behavior of an RPD is characterized by its Level of Rationality (LR) and the penalty function defined by the environment. Two classes of RPDs are considered---local (L-RPD) and global (G-RPD). The former take their decisions based on the penalty in the current state while the latter consider all states. In this dissertation, we analyze the <italic>individual behavior</italic> of both L- and G-RPDs in stationary and symmetric Markov switch environments and their <italic>collective behavior</italic> in zero sum matrix games, under fractional interactions, and in the Edgeworth exchange economies. The approach of the analysis is based on Markov chain techniques. The results obtained can be summarized as follows: (1) In stationary environments, both L- and G-RPDs are able to take the least penalized decision as the LR tends to infinity. However, G-RPDs converge to steady state faster than L-RPDs. (2) In symmetric Markov switch environments, the average penalty incurred by G-RPDs is a monotonically decreasing function of the LR, while for L-RPDs, the average penalty is minimized by a nontrivial optimal LR. (3) Collectives of L- and G-RPDs are able to find their optimal strategies when playing a zero sum matrix game with a saddle in pure strategies. However, if the saddle is in mixed strategies, they are not capable of finding their optimal strategies. Also, both L- and G-RPDs are able to find their mixed optimal strategies when playing against a human that uses his mixed optimal strategy, provided that the payoff matrix is symmetric. In addition, G-RPDs, being able to converge faster, take advantage of L-RPDs by winning more in the transients of the game and in games with mixed optimal strategies. (4) A collective of all L-RPDs or all G-RPDs under homogeneous fractional interaction behaves optimally if the LR of each RPD grows at least as fast as the size of the collective. Under non-homogeneous fractional interaction, the collective behaves optimally even if the size of the collective tends to infinity as long as the LR of each individual is sufficiently large. (5) In the Edgeworth exchange economy, G-RPDs with an identical LR converge to a unique Pareto optimal point, irrespective of the initial product allocation. If their LRs are not identical, the G-RPDs converge to a Pareto equilibrium favoring the G-RPD with larger LR. These results form the theory of rational behavior for Rational Probabilistic Deciders.