Question

Two people have $10 to divide between themselves. They use the following procedure to divide the money. Each person names a nonnegative integer between 0 and 10. If the sum of the amounts that the people name is at most 10 then each person receives the amount of money he names (and the remainder is destroyed). If the sum amounts of the people name exceeds 10 and the amounts named are different, then the person who names the smaller amount receives that amount and the other person receives the remaining money. If the sum of the amounts the people name exceeds 10 and the amounts named are the same then each person receives $5. Formulate this situation as a strategic game and find all pure Nash equilibria. (Hint: you need to write down the matrix of payoffs.)

Answer 1

Answer:

1. When both players choose 5 (5, 5)
2. When both players choose 6 (5, 5)
3. When Player 1 chooses 5 and Player 2 chooses 6  (5, 5)
4. And when Player 1 chooses 6 and Player 2 chooses 5 (5, 5)

Explanation:

Ok, I attached the matrix of payoffs for clarity in my answer.

As we can see, the Nash equilibria is reached when each player (in this case 2 players) dont  have another option that increases his benefit given the option chosed by the other player.

In red we can see the best posible choices for the player 1, and in blue for player 2, in green we can see the Nash equilibria so for this exercise there are four Nash equilibriums:

1. When both players choose 5 (5, 5)
2. When both players choose 6 (5, 5)
3. When Player 1 chooses 5 and Player 2 chooses 6  (5, 5)
4. And when Player 1 chooses 6 and Player 2 chooses 5 (5, 5)

Interestingly, the 4 Nash equilibria are Pareto efficient, which means that there is no other situation in which the player increases his profit without harming the benefit of the other player.