# 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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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

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, 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, 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

$$=P(b_2b^*)U_1(b_1,b_2)$$

$$= P(b_2b^*)[-b^*]$$

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?

• @VARulle you are right! Thank you for your contribution. For example P1 bids 1000, then P1 wins and her utility $U_1= 0$. And P2 bids 0, then her utility $U_2$=0. But P1 have an incentive to deviate to bid $b1^*=1$ then P1 wins again, but her utility increases to $U_1=999$. So, again there is no Nash equilibrium in this case. Is this explanation enough? Are there any cases which I didn’t cover? Commented Aug 22, 2022 at 15:44