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Suppose Player 1 chooses $x$ and Player 2 chooses $y$ where $x,y$ are chosen from $[0,1]$. Player $1$ wants to maximize the payoff function $\sin(2xy)$ and Player $2$ wants to maximize the payoff function $\sin(xy)$. Note, they choose simultaneously and without discussion. Are there ordered pairs $(x,y)$ that are equilibrium points of this game?

In general, how does one do this for payoff functions $f(x,y)$ and $g(x,y)$? I imagine that Nash Equilibrium is somehow relevant here, but I'm only familiar with discrete Nash Equilibria. Is there a continuous counterpart?

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In general, how does one do this for payoff functions $f(x,y)$ and $g(x,y)$?

  • For this example, you could determine the best response maps for each player and then find where the two best response maps intersect. In general, you could write the problem as a variational inequality (VI) and then use methods for solving VIs. The references below give algorithms for solving VIs.

Are there ordered pairs $(x,y)$ that are equilibrium points of this game?

  • The best response for Player 2 is $B_2(x) = 1$ when $x\in (0,1]$ and any value in $[0,1]$ when $x=0$. The best response for player 1 is $B_1(y) = 1$ when $y\in(0, \frac{\pi}{4}]$, $\frac{\pi}{4y}$ when $y\in(\frac{\pi}{4}, 1]$, and any value in $[0,1]$ when $y=0$. The location where the two best response maps intersect (when $x\in B_1(y)$ and $y\in B_2(x)$) is when $(x,y) = (0,0)$ and $(\frac{\pi}{4},1)$.

I imagine that Nash Equilibrium is somehow relevant here, but I'm only familiar with discrete Nash Equilibria. Is there a continuous counterpart?

  • The class of "Continuous kernel games" is the continuous counterpart. Here are some references:
  1. F. Facchinei and C. Kanzow, “Generalized Nash Equilibrium Problems”
  2. F. Facchinei and J. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems."
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