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I am seeking to solve for a Nash equilibrium in pure strategies $(d_2,d_2)$ involving two players, $1$ and $2$. Given that $h'(.)$ is s strictly decreasing and continuous function, $\Phi(d_1-d_2)$ denoting a convolution function, and $F(.)$ denoting a CDF, I want to prove for existence and uniqueness of equilibrium. My guess is that we we use a fixed point theorem to prove existence. The following is the first order condition for maximization.

$$g_1(d_1) \equiv h'(d_1)-\gamma\Phi(d_1-d_2)-\eta(1-F(m-d_1))=0 \\ g_2(d_2)≡ h'(d_2)-\gamma[1-\Phi(d_1-d_2)]-\eta(1-F(m-d_2))=0 $$

Note that the parameters are all positive and $d_1$ & $d_2$ are continuous and $m$ is a constant. I highly appreciate any suggestion towards the proof.

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    I have removed several tags. Questions that are about a specific mathematical problem, rather than proof writing in general, shouldn't be categorized as "proof writing", and "proof theory" is an entirely separate field of mathematics.2012-08-27
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    Showing existence of an equilibrium in pure strategies is much more demanding than proving existence using mixed strategies. To start, you will need some sort of quasi-concavity of payoffs, which is no necessary when mixing is allowed. I believe that you should try to establish existence allowing for mixed strategies and then try to see if it's possible to find one in pure strategies.2012-09-17
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    What is the game?2014-01-17

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