# [Economics] Showing existence of a Nash equilibrium in pure strategy

## Solution 1:

No. Consider the matching pennies game $$\begin{array}{c |c} & H & T & \\ \hline H & (1,-1) & (-1,1)\\ T & (-1,1) & (1,-1)\\ \end{array}$$ which we know no PSNE exists.

Define the function $$v_i(\cdot,\cdot)$$ by $$v_i(H,0) = 5 \\ v_i(x,y) = 0 \quad \forall (x,y) \not = (H,0)$$

At $$\delta = 0$$, we have the game $$\begin{array}{c |c} & H & T & \\ \hline H & (5,4) & (-4,1)\\ T & (-1,5) & (1,-1)\\ \end{array}$$ which has a PSNE of $$(H,H)$$.

For any other value of $$\delta$$, we're back in the original game so no PSNE exists.

Redefine $$H = 0, T = 1$$ and you have your monotone decreasing requirement.

## Solution 2:

Seems like this class of games is very general. (Or I don't get the definition.) Note that $$\theta$$ is not even used, it is just some parameter that has value $$\theta_0$$ throughout.

As long as there exist two games $$G_{\delta}$$ and $$G_{\delta_0}$$, were both have the same number of players $$J$$, each player has two strategies in both $$G_{\delta}$$ and $$G_{\delta_0}$$, and $$G_{\delta}$$ has a Nash-equilibrium but $$G_{\delta_0}$$ does not, then one can get negation by defining $$v(a; \delta_0) := u_{\delta_0}(a) - u_{\delta}(a)$$ where $$u_{\delta}$$ is the vector of payoffs in $$G_{\delta}$$, and $$u_{\delta_0}$$ is the vector of payoffs in $$G_{\delta_0}$$ given strategy profile $$a$$.

There is still the possibility that either all or no games have a Nash-equilibria, but this is not true on the class of these two strategies per player games, as shown by Walrasian Auctioneer's answer.