I'm working on a game theory problem I can't seem to figure out.
Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These numbers do not have to be in whole dollar units. The choices are made simultaneously. Each player's payoff is equal to her own money payoff. In all the cases below, if $s_1+s_2\leq 10$, then the players get the amounts they named (and the remainder is destroyed).
(a) In the first case, if $s_1+s_2 >10$, then both players get zero and the money is destroyed.
(b) In the second case, if $s_1+s_2 >10$ and the amounts named are different, then the person who names the smaller amount gets that amount and the other person gets the remaining money. If $s_1+s_2 >10$ and $s_1=s_2$, then both players get $\$5$.
(c) In the third case, the games (a) and (b) must be played such that only integer dollar amounts can be named by both players.
Determine the pure strategy Nash Equilibria for all games (a) – (c).
I'm pretty sure I've figured out (a), noting that if player 2 chooses a strategy such that $0 \leq s_2 < 10$, the best response of player 1 is $BR_1(s_2) = 10- s_2$. If $s_2 = 10$, then $BR_1(s_2) = [0,10]$ since the payoff is $0$ regardless. The same holds for player 2. The Nash Equilibria is the intersection of the BR lines, and occur at $(10,10)$ and the line $s_1+s_2=10$.
For (b), my thought is that if player 2 chooses $s_2 \leq 5$, then $BR_1(s_2) = 10-s_2$ as in (a). However, if $s_2 > 5$, I feel that $BR_1(s_2) = s_2 -\epsilon$ for some very small $\epsilon >0$. This way, the total amount will be over $\$10$, but player 1 will have the smaller amount and thus get his money. However, I'm not sure if this is right or how to find the Nash Equilibrium in this case. Any help would be greatly appreciated.
Best Answer
This was just bumped to the homepage by Community ♦ because it had only unvoted answers. Since it turns out that these were wrong, here's a new one:
In (b), every strategy below $\$5$ is strictly dominated by $\$5$, since that always wins exactly $\$5$ independent of the other player's strategy.
On the other hand, no strategy above $\$5$ has a best response. If $s_1\gt5$, then if $s_2\ge s_1$, Player $2$ can improve by switching to $s_2=s_1-\epsilon$; whereas if $s_2\lt s_1$, Player $2$ can improve by switching to $\frac{s_1+s_2}2$.
Thus the only pure-strategy Nash equilibrium is $(5,5)$, where deviating in either direction reduces the payoff.