Question

Consider the following two-person zero-sum game. Both players simultaneously call out one of the numbers f2; 3g. Player I wins if the sum of the numbers called is odd and player II wins if their sum is even. The loser pays the winner the product of the two numbers called (in dollars). Find the payoff matrix, the value of the game, and an optimal strategy for each player.

Answer 1

The pay off matrix (for numbers 1-9) is shown in the attached table.

The value of the game (expected winning per play) = $9.57

The strategy for player II is to call even numbers all the time, which guarantees a winning in every game!

The strategy for player I is to call odd numbers all the time, in case player II calls odd numbers (good luck!)

Explanation:

The pay off matrix (for numbers 1-9) is shown in the attached table.

The payoff for player II (even wins) is 775 for 9*9 = 81 possible scenarios.

Thus the value of the game is 775/81 = $9.57 (expected winnings per play)

The strategy for player II is to call even numbers all the time, which guarantees a winning in every game!

The strategy for player I is to call odd numbers all the time, in case player II calls odd numbers (good luck!)