  Strategies for sustaining cooperation and preventing exploitation by selfish
agents in repeated games have mostly been restricted to Markovian strategies
where the response of an agent depends on the actions in the previous round.
Such strategies are characterized by lack of learning. However, learning from
accumulated evidence over time and using the evidence to dynamically update our
response is a key feature of living organisms. Bayesian inference provides a
framework for such evidence-based learning mechanisms. It is therefore
imperative to understand how strategies based on Bayesian learning fare in
repeated games with Markovian strategies. Here, we consider a scenario where
the Bayesian player uses the accumulated evidence of the opponent's actions
over several rounds to continuously update her belief about the reactive
opponent's strategy. The Bayesian player can then act on her inferred belief in
different ways. By studying repeated Prisoner's dilemma games with such
Bayesian inferential strategies, both in infinite and finite populations, we
identify the conditions under which such strategies can be evolutionarily
stable. We find that a Bayesian strategy that is less altruistic than the
inferred belief about the opponent's strategy can outperform a larger set of
reactive strategies, whereas one that is more generous than the inferred belief
is more successful when the benefit-to-cost ratio of mutual cooperation is
high. Our analysis reveals how learning the opponent's strategy through
Bayesian inference, as opposed to utility maximization, can be beneficial in
the long run, in preventing exploitation and eventual invasion by reactive
strategies.
