A bettor is faced with the problem of choosing optimal bets, given his (subjective) probabilities of outcomes of an experiment and the payoff odds on these outcomes. Conversely, a bookie (pari-mutuel system), faced by several bettors with different subjective probabilities, has the problem of choosing payoff odds so as to avoid the risk of loss. It is shown that, under certain reasonably broad conditions on the several bettors' utilities and subjective probabilities, such an equilibrium set of payoff odds always exists. Some examples are worked out in detail.