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 beliefs.

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.

How to Cite this Page:
Su, Francis E., et al. "Sure Betting on Different Beliefs." Mudd 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. 4.34 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!

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