  We introduce an evolutionary game with feedback between perception and
reality, which we call the reality game. It is a game of chance in which the
probabilities for different objective outcomes (e.g., heads or tails in a coin
toss) depend on the amount wagered on those outcomes. 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. We study self-reinforcing games, in
which betting more on an outcome increases its odds, and self-defeating games,
in which the opposite is true. This is investigated in and out of equilibrium,
with and without rational players, and both numerically and analytically. We
introduce a method of measuring the inefficiency of the game, similar to
measuring the magnitude of the arbitrage opportunities in a financial market.
We prove that convergence to equilibrium is is a power law with an extremely
slow rate of convergence: The more subjective the game, the slower the
convergence.
