Payouts in parimutuel numbers games

Session Title

Gambling Mathematics: Quantitative Finance

Presentation Type

Paper Presentation

Start Date

26-5-2026 12:00 AM

Abstract

Parimutuel numbers games, like the Massachusetts Numbers Game or California's Daily 3 and Daily 4, have payouts that may be surprising. If such a game returns 50% of its revenue in prizes, then the average bet has an expected payout of more than 50%, because gamblers do not choose their numbers uniformly. But how much more? It can be hard to estimate that. As data, it is a noisy and right-skewed time series. Analytically, there is not a good closed form for it. We address both approaches by providing a formula that better approximates the expected payout and providing a data set of gambler wagers, which enables easy computer simulations. Such simulations show how changes in rules or gambler behavior will change payouts. Finally, we explain the mystery of why early parimutuel numbers games began with very large payouts that strongly declined over time, without any changes in the rules of the game.

Author Bios

Skip Garibaldi is a mathematician who was previously Winship Distinguished Research Professor at Emory University and Associate Director of the Institute for Pure & Applied Math at UCLA. His academic work on the mathematics of lotteries has included investigations with reporters that triggered changes in state policy and jail time.

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May 26th, 12:00 AM

Payouts in parimutuel numbers games

Parimutuel numbers games, like the Massachusetts Numbers Game or California's Daily 3 and Daily 4, have payouts that may be surprising. If such a game returns 50% of its revenue in prizes, then the average bet has an expected payout of more than 50%, because gamblers do not choose their numbers uniformly. But how much more? It can be hard to estimate that. As data, it is a noisy and right-skewed time series. Analytically, there is not a good closed form for it. We address both approaches by providing a formula that better approximates the expected payout and providing a data set of gambler wagers, which enables easy computer simulations. Such simulations show how changes in rules or gambler behavior will change payouts. Finally, we explain the mystery of why early parimutuel numbers games began with very large payouts that strongly declined over time, without any changes in the rules of the game.