Question

Game Theory problem

Player 1 chooses a positive integer x 〉 0 and player II chooses a positive integer y > 0. The player with the lower number pays a dollar to the player with the higher number unless the higher number is more than twice larger in which case the payments are reversed if x = y. Find the unique optimal strategy in this game

Answer 1

For simplicity we will assume that the end of each round , the loosing player plays the winning player a dollar amount equal to the winning players's score for that round.

Also, we will assume the rules require bob to chose a number other than the one already chosen by Alice.

With Alice choosing 16, bob's optimal reply is 15, but for those choices, the expected value of Alice's score minus bob's score is +0.5 , so Alice will win more points on average, than bob.

If Alice.s chooses 16, and bob chooses any number other than 15, the expected value of Alice's score minus bobs score will be greater than 0.5, so bob best choice is 15, since it minimizes bob's best choice is 15, since it minimizes bob's expected.

Optimal strategies for Bob;

If Alice chooses K with k < 16 , Bob's best reply is K - 1 , which yields a positive expectation for Bob.

Finally, If Alice chooses 16, all choices by bob yield a negative expectation for Bob, but to minimize the expected loss, bob should choose 15.