Sure Betting on Different Beliefs

Alice believes that Gore will win the election with probability 5/8. Bob believes that Bush will win the election with probability 3/4.

Assuming that Alice and Bob are both willing to accept any bet that gives them a positive expectation 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 Gore wins and she'll pay you $3 otherwise. Alice agrees because her expectation is: $2(5/8)-$3(3/8)=$1/8.

Bet with Bob that you'll pay him $2 if Bush wins, and he'll pay you $3 otherwise. Bob agrees because his expectation is $2(3/4)-$3(1/4)=$3/4.

Alice and Bob both believe they have positive expectation, but you will win for certain: if either Bush or Gore wins, you will net a dollar!

Presentation Suggestions:
Students will be quite surprised by this result; it naturally motivates them to ponder 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 election, you can design a bet that will give both of them positive expectation and you positive winnings! See if you can figure out how.

You might enjoy these other Fun Facts in game theory.

How to Cite this Page:
Su, Francis E., et al. "Sure Betting on Different Beliefs." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References: K. Binmore, Fun and Games: a Text on Game Theory, 1992, p.87. Subjects:    probability Level:    Advanced Fun Fact suggested by:   Francis Su Suggestions? Use this form. 3.94 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! Get the Math Fun Facts iPhone App!

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