Coherence (philosophical Gambling Strategy) - Conditional Wagers and Conditional Probabilities

Conditional Wagers and Conditional Probabilities

Now imagine a more complicated scenario. You must set the prices of three promises:

• to pay $1 if the Red Sox win tomorrow's game; the purchaser of this promise loses his bet if the Red Sox do not win regardless of whether their failure is due to their loss of a completed game or cancellation of the game, and

• to pay $1 if the Red Sox win, and to refund the price of the promise if the game is cancelled, and

• to pay $1 if the game is completed, regardless of who wins.

Three outcomes are possible: The game is cancelled; the game is played and the Red Sox lose; the game is played and the Red Sox win. You may set the prices in such a way that

(where the second price above is that of the bet that includes the refund in case of cancellation). (Note: The prices here are the dimensionless numbers obtained by dividing by $1, which is the payout in all three cases.) Your prudent opponent writes three linear inequalities in three variables. The variables are the amounts he will invest in each of the three promises; the value of one of these is negative if he will make you buy that promise and positive if he will buy it from you. Each inequality corresponds to one of the three possible outcomes. Each inequality states that your opponent's net gain is more than zero. A solution exists if and only if the determinant of the matrix is not zero. That determinant is:

Thus your prudent opponent can make you a sure loser unless you set your prices in a way that parallels the simplest conventional characterization of conditional probability.