Question

1. There are two players. Each names an amount at the same time as the other. They can say $0 or say $10 or anything in between, even fractions of cents. If the amounts total $10 or less they get what they named. If they total more than $10 they get $0.

What are the Nash equilbria? To solve this try some pairs of amounts. If they aren’t equilibria – if one player or both could do better by changing the amount – then try to adjust what one or the other names in order to make them into equilibria.

Answer 1

Let the two players be A and B.

Hypothetical table:

A/B	6	5
6	(0,0)	(0,0)
5	(0,0)	($5,$5)

In the above case, if both players say more than $5, none of them get anything but if they name for any amount less than or equal to $5, they will get the amount they name as the total will be $10.

Nash equilibrium is a set of strategies, one for each player, such that no player has the incentive to change his or her strategy given what the other players are doing.

In the above table, we can see that no player can increase his payoff by changing that current amount name. If even one of them name more than $5, both of them will get zero payoffs. If they name less then both will get paid less than $5.

BR_A(5) = $5

BR_A(6) = $4

BR_B(5) = $5

BR_B(6) = $4

So, if one player names amount more than $5 then the best response of other would be saying any amount less than $5 which totals the amount $10 or less but because they have to say amount at the same time. The only point ($5,$5) should be Nash equilibrium.