Alice believes that the 49ers will win the Super Bowl with probability 5/8. Bob believes that Ravens will win the Super Bowl with probability 3/4.
Assuming that Alice and Bob are both willing to accept any bet that gives them a positive expected value of winning, did you know that there’s a way to place bets with both of them so that you can make money for certain?
Here’s what you can do. Bet with Alice that you’ll pay her $2 if the 49ers win and she’ll pay you $3 otherwise. Alice agrees because her expected value on this bet is: 2 (5/8) – 3 (3/8) = $0.125, or 12.5 cents.
Bet with Bob that you’ll pay him $2 if the Ravens win, and he’ll pay you $3 otherwise. Bob agrees because his expected value on this bet is 2 (3/4) – 3 (1/4) = $0.75, or 75 cents.
Alice and Bob both believe they have positive expectation, but you will win for certain: no matter who wins the Super Bowl, you will pay out $2 and receive $3, and therefore net a dollar!
Students will be quite surprised by this result; it naturally motivates a study of the meaning of probabilities as measures of belief– a “Bayesian” view of probability.
The Math Behind the Fact:
In fact, as long as Alice and Bob have different beliefs about the probability of the outcomes of the Super Bowl, you can design a bet that will give both of them positive expectation and you positive winnings! See if you can figure out how.
Setting up these bets is a form of arbitrage that capitalizes on the difference in belief that exists between two parties, and hedges against the risk of the outcome. Hedge funds use similar reasoning all the time.
You might enjoy these other Fun Facts in game theory.
How to Cite this Page:
Su, Francis E., et al. “Risk-Free Betting on Different Beliefs.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
K. Binmore, Fun and Games: a Text on Game Theory, 1992, p.87.
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