All-pay auction question

Two players take part in the following auction for a £1000 prize. The two players submit bids simultaneously, and the higher bid wins the prize (if bids are identical each gets £500). Both the winner and the loser have to pay the seller the amount of their bids. The players can bid any nonnegative amount.

Find Nash equilibrium in pure strategies.

Find a mixed strategy equilibrium.

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** I start with finding Nash equilibrium **

Players ={P1, P2}

Valuation= 1000

Bids= {b1, b2} $\in [0, 1000]$

Payoffs are

$U_i(b1,b2)= 1000-b_i$ if $b_i>b_j$

$U_i(b1,b2)= 500-b_i$ if $b_i=b_j$

$U_i(b1,b2)=-b_i$ if $b_i<b_j$

Case 1: b1=b2

If b1=b2=1000, then $U_i(b1,b2)= 500-1000=-500$

There is a profitable deviation for players. For example P1 may deviate to b1=0. P1 loses, P2 wins. Both players have zero payoff. So, (1000,1000) is not Nash equilibrium.

If $b1=b2=0$, then $U_i(b1,b2)= 500-0=500$

There is a profitable deviation for players. For example P1 may deviate to b1=1. P2 loses, P2 loses. $U_1(b1,b2)=999$ and $U_2=0$. So, (0,0) is not Nash equilibrium.

If $0<b1=b2=<1000$, then one of the players has an incentive to bid higher than the another’s bid in order to increase her payoff. So, this case is also not Nash equilibrium.

Case 2: $b1\not= b2$

If $b_i>1000$ then this player wins, but she gets negative payoff. So this is not logical.

If $0<b1\not= b2<1000$, then player with lower bid has an incentive to bid higher than the another’s bid in order to increase her payoff. So, this case is also not Nash equilibrium.

As a result, there is no Pure Nash equilibrium for this game.

** Next, I try to find mixed strategy Nash equilibrium. **

Assume that P1 bids $b1=b^*$

The expected payoff of P1 is as follows

$$=P(b_2<b^*)U_1(b_1,b_2)+ P(b_2=b^*)U_1(b_1,b_2)+ P(b_2>b^*)U_1(b_1,b_2)$$

$$= P(b_2<b^*)[1000-b^*]+ P(b_2=b^*)[500-b^*]+ P(b_2>b^*)[-b^*]$$

I cannot proceed the solution after this point. How can I find this mixed strategy Nash equilibrium?

And is the findings about the Pure Nash equilibrium correct?

Please share your ideas with me. Thank you.

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