Betting with Budgets

    We explore betting mechanisms in which players participate by specifying a belief about an uncertain outcome (typically a probability) and a budget (an amount of money to put at risk). We identify a number of properties that seem desirable in such a mechanism, including budget balance, individual rationality, incentive compatibility, monotonicity (incentivizing agents to specify large budgets), and high stakes (transferring a large fraction of the losers’ budgets to the winners). Common betting mechanisms fall short on a number of these properties. For example, double auctions, market makers, and pari-mutuel markets are not incentive compatible for beliefs. Shared scoring rules are incentive compatible for beliefs but have no notion of budgets. Automated market makers like Hanson’s market scoring rule mechanism are not budget balanced. We propose three new betting mechanisms and evaluate them according to our desired properties. Our all-in mechanism generalizes shared scoring rules to handle budgets. Our weighted score mechanism retains every property except high/full stakes. Although our pari-mutuel score mechanism fails to satisfy incentive compatibility and full stakes, it is close in a quantifiable sense and may have good behavior in practice. We describe and examine dynamic variants of the above mechanisms and discuss the difficulty (impossibility?) in achieving all desired properties.