Making the most of potential: potential games and genotypic convergence.

We consider genotypic convergence of populations and show that under fixed fitness asexual and haploid sexual populations attain monomorphic convergence (even under genetic linkage between loci) to basins of attraction with locally exponential convergence rates; the same convergence obtains in single locus diploid sexual reproduction but to polymorphic populations. Furthermore, we show that there is a unified theory underlying these convergences: all of them can be interpreted as instantiations of players in a potential game implementing a multiplicative weights updating algorithm to converge to equilibrium, making use of the Baum-Eagon Theorem. To analyse varying environments, we introduce the concept of 'virtual convergence', under which, even if fixation is not attained, the population nevertheless achieves the fitness growth rate it would have had under convergence to an optimal genotype. Virtual convergence is attained by asexual, haploid sexual and multi-locus diploid reproducing populations, even if environments vary arbitrarily. We also study conditions for true monomorphic convergence in asexually reproducing populations in varying environments.

A Survey on Entropy and Economic Behaviour.

Entropy plays a significant role in the study of games and economic behaviour in several ways. A decision maker faced with an n-fold repetition of a decision-making problem needs to apply strategies that become increasingly complex as n increases. When several players are involved in selecting strategies in interactive games, bounds on the memories and cognitive capacities of the players can affect possible outcomes. A player who can recall only the last k periods of history is said to have bounded recall of capacity k. We present here a brief survey of results of games played by players with different bounded recall capacities, in particular those indicating surprisingly strong relations between memory and entropy in the study of the min-max values of repeated games with bounded recall. In addition, we consider uses of entropy in measuring the value of information of noisy signal structures, also known as experiments. These are represented by stochastic matrices, with the rows representing states of the world and the columns possible signals. The classic ordering of experiments, due to David Blackwell and based on decision-making criteria, is a partial ordering, which has led to attempts to extend this ordering to a total ordering. If a decision maker has a prior distribution over the states, receipt of a signal yields a posterior. The difference between the entropy of a prior and the expected entropy of the set of possible posteriors has been proposed as a natural extension of the Blackwell ordering. We survey this alongside the theory of rational inattention, which posits that, since individuals have limited attention, they do not always follow every single piece of economic news in planning their economic behaviour. By modelling attention limits as finite channel capacity in the sense of Shannon, economists have developed a theory that explains a range of observed economic behavioural phenomena well.

Sex with no regrets: How sexual reproduction uses a no regret learning algorithm for evolutionary advantage.

The question of 'why sex' has long been a puzzle. The randomness of recombination, which potentially produces low fitness progeny, contradicts notions of fitness landscape hill climbing. We use the concept of evolution as an algorithm for learning unpredictable environments to provide a possible answer. While sex and asex both implement similar machine learning no-regret algorithms in the context of random samples that are small relative to a vast genotype space, the algorithm of sex constitutes a more efficient goal-directed walk through this space. Simulations indicate this gives sex an evolutionary advantage, even in stable, unchanging environments. Asexual populations rapidly reach a fitness plateau, but the learning aspect of the no-regret algorithm most often eventually boosts the fitness of sexual populations past the maximal viability of corresponding asexual populations. In this light, the randomness of sexual recombination is not a hindrance but a crucial component of the 'sampling for learning' algorithm of sexual reproduction.