Partial-Kelly Strategies and Expected Utility: Small-Edge Asymptotics
Decision Analysis
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In this paper we consider the problem of optimal betting on simultaneous games when the bookmaker accepts bets on the joint outcome of subsets of any combination of those events, called parlays, accumulators, or multibets. When the bookmaker's take is a fixed proportion of the wager, multibetting on all n games replicates any other betting strategy and dominates separate (simultaneous) bets on individual games. When, more typically, the bookmaker quotes multiplicative payouts, and hence takes a higher percentage on multibets than bets on single games, the optimal betting strategy depends on the bettor's utility function. We consider the special case of a Kelly (log utility) bettor, and the more general case of a λ-Kelly bettor who wagers a fixed fraction λ ≤ 1 of the full-Kelly bet. Our main result is that when the bookmaker offers multiplicative payouts, a Kelly or λ-Kelly bettor realizes the same monetary outcome by multibetting as would have been achieved by sequential betting on single games (were sequential bets possible). It follows, therefore, that even when games are sequential, a Kelly or λ-Kelly bettor should only bet sequentially when there is an expectation of the bookmaker's odds becoming more favorable nearer to game time.