Question

5. (10 points) Alice and Barbara are playing a one-stage guessing game. Each must choose a real number between 1 and 4 (inclusive). Alice's target is to match Barbara's number. Barbara's target is to name twice Alice's number. Each receives $10 minus a dollar penalty that is equal to the absolute difference between her guess and her target. Solve this game by iteratively deleting dominated strategies. What will Alice and Barbara choose?

Answer 1

Payoff matrix is drawn according to the rules as shown below:

Barbara 10 9 10 9 10 10 9 Alice 10 9 10

For Barbara, playing "1" is strictly dominated by playing "2", which makes sense because 1 can never be double of any number which might be chosen by Alice.

So it is dominated from the payoff matrix by iterated eliminated of strictly dominated strategies.

Now in the remaining matrix,

「9, 10 8,9 7,8 10,8 9,9 8,10 9,6 10,7 9,8 8,4 9,5 10,6 For Player 1, choosing 1 is strictly dominated by choosing number 2. Hence row 1 is eliminated from payoff matrix too. [10,8 9,9 8,10 9,6 10,7 9,8 8,4 9,5 10,6 For Player 2, choosing 4 strictly dominates choosing number 2 or 3. Hence columns choosing number 2 or 3 is eliminated from payoff matrix too. 8, 10 9,8 10,6 Hence it is optimal for Plaver 1 to choose number 4. Nash equilibrium of the game: 4,.e. both players choose number 4