<p>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.</p>
<p>Find Nash equilibrium in pure strategies.</p>
<p>Find a mixed strategy equilibrium.</p>
<p>—-</p>
<p>** I start with finding Nash equilibrium **</p>
<p>Players ={P1, P2}</p>
<p>Valuation= 1000</p>
<p>Bids= {b1, b2} <span class="math-container">$\in [0, 1000]$</span></p>
<p>Payoffs are</p>
<p><span class="math-container">$U_i(b1,b2)= 1000-b_i$</span> if <span class="math-container">$b_i&gt;b_j$</span></p>
<p><span class="math-container">$U_i(b1,b2)= 500-b_i$</span> if <span class="math-container">$b_i=b_j$</span></p>
<p><span class="math-container">$U_i(b1,b2)=-b_i$</span> if <span class="math-container">$b_i&lt;b_j$</span></p>
<p>Case 1: b1=b2</p>
<p>If b1=b2=1000, then <span class="math-container">$U_i(b1,b2)= 500-1000=-500$</span></p>
<p>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.</p>
<p>If <span class="math-container">$b1=b2=0$</span>, then <span class="math-container">$U_i(b1,b2)= 500-0=500$</span></p>
<p>There is a profitable deviation for players. For example P1 may deviate to b1=1. P2 loses, P2 loses. <span class="math-container">$U_1(b1,b2)=999$</span> and <span class="math-container">$U_2=0$</span>. So, (0,0) is not Nash equilibrium.</p>
<p>If <span class="math-container">$0&lt;b1=b2=&lt;1000$</span>, 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.</p>
<p>Case 2: <span class="math-container">$b1\not= b2$</span></p>
<p>If <span class="math-container">$b_i&gt;1000$</span> then this player wins, but she gets negative payoff. So this is not logical.</p>
<p>If <span class="math-container">$0&lt;b1\not= b2&lt;1000$</span>, 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.</p>
<p>As a result, there is no Pure Nash equilibrium for this game.</p>
<p>** Next, I try to find mixed strategy Nash equilibrium. **</p>
<p>Assume that P1 bids <span class="math-container">$b1=b^*$</span></p>
<p>The expected payoff of P1 is as follows</p>
<p><span class="math-container">$$=P(b_2&lt;b^*)U_1(b_1,b_2)+ P(b_2=b^*)U_1(b_1,b_2)+ P(b_2&gt;b^*)U_1(b_1,b_2)$$</span></p>
<p><span class="math-container">$$= P(b_2&lt;b^*)[1000-b^*]+ P(b_2=b^*)[500-b^*]+ P(b_2&gt;b^*)[-b^*]$$</span></p>
<p>I cannot proceed the solution after this point. How can I find this mixed strategy Nash equilibrium?</p>
<p>And is the findings about the Pure Nash equilibrium correct?</p>
<p>Please share your ideas with me. Thank you.</p>

CC BY-SA 4.0
2022-08-21T11:13:18.437Z
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.
Giskard
CC BY-SA 4.0
2022-08-21T13:25:12.387Z
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.
Giskard
CC BY-SA 4.0
2022-08-22T14:36:38.443Z
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?
VARulle
CC BY-SA 4.0
2022-08-22T15:44:23.573Z
@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?
studentp
CC BY-SA 4.0
2022-08-22T21:57:02.190Z
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. https://economics.stackexchange.com/questions/52455/please-check-for-payoffs-which-i-calculated-in-the-incomplete-and-imperfect-game
studentp
CC BY-SA 4.0
2022-08-23T10:38:57.877Z
Sorry, I have no time at the moment. But concerning this question here, you can just google "all-pay auction" and easily find the answer: For two bidders, there's a unique, symmetric and completely mixed equilibrium. (You can also easily find out how to calculate it.)
VARulle
<microeconomics><game-theory><self-study><nash-equilibrium><auctions>
All-pay auction question
studentp
# All-pay auction question

(Asked by: username_0 on 2022-08-20)

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.

—-

\*\* 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.

**Tags:** microeconomics, game-theory, self-study, nash-equilibrium, auctions

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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.  
> **username_2**: 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?  
> **username_0**: @username_2 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?  
> **username_0**: Dear @username_2 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. https://economics.stackexchange.com/questions/52455/please-check-for-payoffs-which-i-calculated-in-the-incomplete-and-imperfect-game  
> **username_2**: Sorry, I have no time at the moment. But concerning this question here, you can just google "all-pay auction" and easily find the answer: For two bidders, there's a unique, symmetric and completely mixed equilibrium. (You can also easily find out how to calculate it.)  

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