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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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    $\begingroup$ Hi! My problem with your attempt on the mixed strategy is that it lacks intention, it is just a random formulation. What theory did you read before attempting to answer this question? If this is a class you are forced to take - my sympathies! If it is not, you should read more theory and ask questions about that, you'll get a much better understanding. $\endgroup$
    – Giskard
    Commented Aug 21, 2022 at 11:13
  • $\begingroup$ Not sure why you wrote "this is a past exam question". Sorry, I don't want to write out a detailed solution; perhaps someone else. Recommend reading the textbook again though until you get a better grasp of mixed strategies over infinite strategy sets. $\endgroup$
    – Giskard
    Commented Aug 21, 2022 at 13:25
  • $\begingroup$ The two cases you consider for non-identical bids do not cover all possible cases. What happens if one bidder bids 1000 and the other 0? $\endgroup$
    – VARulle
    Commented Aug 22, 2022 at 14:36
  • $\begingroup$ @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? $\endgroup$
    – studentp
    Commented Aug 22, 2022 at 15:44
  • $\begingroup$ Dear @VARulle can you also please look at this question? I draw the tree but I am not sure about my payoff calculation. Please check it if you mind. Thank you. economics.stackexchange.com/questions/52455/… $\endgroup$
    – studentp
    Commented Aug 22, 2022 at 21:57

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