  We study an evolutionary game of chance in which the probabilities for
different outcomes (e.g., heads or tails) depend on the amount wagered on those
outcomes. The game is perhaps the simplest possible probabilistic game in which
perception affects reality. By varying the `reality map', which relates the
amount wagered to the probability of the outcome, it is possible to move
continuously from a purely objective game in which probabilities have no
dependence on wagers, to a purely subjective game in which probabilities equal
the amount wagered. The reality map can reflect self-reinforcing strategies or
self-defeating strategies. In self-reinforcing games, rational players can
achieve increasing returns and manipulate the outcome probabilities to their
advantage; consequently, an early lead in the game, whether acquired by chance
or by strategy, typically gives a persistent advantage. We investigate the game
both in and out of equilibrium and with and without rational players. We
introduce a method of measuring the inefficiency of the game and show that in
the large time limit the inefficiency decreases slowly in its approach to
equilibrium as a power law with an exponent between zero and one, depending on
the subjectivity of the game.

Physics and Society