Consider a two-player game where Player 1 chooses a strategy $x_1=[1,3]$ and Player 2 chooses $x_2=[0,2]$ Let the payoff functions for P1 and P2 be $u_1(x_1,x_2)=\min\{x_1,x_2\}$ and $u_2(x_2,x_1)=\max\{x_1,x_2\}$.
What are the best response functions $BR_1(x_2)$ and $BR_2(x_1)$?
Are there any pure strategy Nash equilibria and what is the rationalizable set $R$?
Just by intuition, it seems like P2 would always choose $x_2=2$ to maximize their payoff and P1 should match that choice so $x_1=x_2$ so that P1's payoff is never lower than the value P2 chooses. P1 should have no reason to ever choose $x_1=3$ because his payoff is the minimum of the values and 3 is out of range for P2. I think this means there is a Nash equilibrium at (2, 2) and the rationalizable set would be $R={[2]x[2]}$ but I don't know how to actually show this mathematically. Additionally, neither player's strategy should dominate the other since their expected payoff is the same, correct?
I would really like to see how this problem should be approached and if I am thinking about it in the right way, thanks.
I’ll assume that you meant $x_1\in[1,3]$ instead of $x_1=[1,3]$, and likewise for $x_2$.
You seem to be assuming that Player $1$ wants to minimize the payoff of Player $2$. That’s not the assumption made in game theory; Player $1$ only wants to maximize her own payoff and doesn’t care about the payoff of Player $2$.
Given $x_2$, the set of best responses of Player $1$ is $\{x_1\in[1,3]\mid x_1\ge x_2\}$, because Player $1$ cannot gain more than $x_2$ but will gain less than $x_2$ if $x_1\lt x_2$.
Given $x_1$, the set of best responses of Player $2$ is $\{2\}$ if $x_1\lt2$ (because in that case Player $2$ gains $2$ only if he plays $2$ and less otherwise) and $[0,2]$ if $x_1\ge2$ (because in that case Player $2$ gains $x_1$ regardless of what he plays).
To find the rationalizable set, we need to first remove all strategies that are never best responses. But all strategies occur as best responses above, so the rationalizable set is the set of all strategies for both players, $[1,3]\times[0,2]$.
Since Player $2$ can gain at least $2$ by playing $2$, at least one player must play at least $2$ in a Nash equilibrium.
If it’s Player $2$ who plays $2$, then the set of best responses of Player $1$ is $[2,3]$, and $2$ is a best response of Player $2$ to all of these, so that yields a set $[2,3]\times\{2\}$ of Nash equilibria.
If it’s Player $1$ who plays at least $2$, then the set of best responses of Player $2$ is all of $[0,2]$, and playing more than $2$ is a best response of Player $1$ to all of these, so that yields a set $[2,3]\times[0,2]$ of Nash equilibria. Since this set contains the one above, that’s the set of all Nash equilibria.