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We study games of chance (e.g., pokers, dices, horse races) in the form ofagents' first-order posterior beliefs about game outcomes. We ask for anyprofile of agents' posterior beliefs, is there a game that can generate thesebeliefs? We completely characterize all feasible joint posterior beliefs fromthese games. The characterization enables us to find a new variant of Border'sinequalities (Border, 1991), which we call a belief-based characterization ofBorder's inequalities. It also leads to a generalization of Aumann's AgreementTheorem. We show that the characterization results are powerful in bounding thecorrelation of agents' joint posterior beliefs.

A Belief-Based Characterization of Reduced-Form Auctions