<p>In response to our <a href="https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions">policy on homework questions</a>, I will work on the following abstract version of your question:</p>

<p><em>A single prize is to be allocated by an all-pay auction. Assume the value of the prize is 1 to both bidders, and this is common knowledge. Both bidders have a budget constraint <span class="math-container">$m$</span>. If there is a tie, each bidder receives the prize with <span class="math-container">$\frac{1}{2}$</span> probability. Find a Nash Equilibrium in this game.</em></p>

<p>We shall focus on the symmetric equilibrium in this game, that is, both bidders are using the same equilibrium strategy. Before we get started, let's make two observations:</p>

<p><strong>Claim 1</strong>: Whenever there is a mass point in the equilibrium strategy (i.e. there is positive probability to submit one bid), this mass should be put on the upper bound <span class="math-container">$m$</span>. For if not, one bidder can always deviate to a slightly higher bid and obtain higher surplus. </p>

<p><strong>Claim 2</strong>: Whenever a bidder randomizes, the bid should be uniformly drawn from <span class="math-container">$(0,t)$</span>, and the probability that this happens should be <span class="math-container">$t$</span>. This is because only in this case, a bidder is indifferent between any bid <span class="math-container">$b\in(0,t)$</span>: 
<span class="math-container">$$\text{Expected payoffs}=t\cdot \frac{b}{t}\cdot 1-b=0.$$</span></p>

<p>Therefore, we can guess that the equilibrium strategy takes the form
<span class="math-container">$$t\cdot \text{Uniform}(0,t)+(1-t)\cdot\delta_m.$$</span>
That is, with probability <span class="math-container">$t$</span>, a bidder submits a bid uniformly drawn from <span class="math-container">$(0,t)$</span>, and with complementary probability she submits <span class="math-container">$m$</span>. The only thing left is to pin down <span class="math-container">$t$</span>. Note that if this is an equilibrium strategy, a bidder should be indifferent between any bid <span class="math-container">$b\in(0,t)\cup\{m\}$</span>. Therefore, the payoffs from submitting <span class="math-container">$m$</span> should also be 0, i.e.
<span class="math-container">$$t\cdot 1+(1-t)\cdot\frac{1}{2}-m=0,$$</span>
which implies <span class="math-container">$t=2m-1$</span>. (Notice that for the existence of such an equilibrium, we must have <span class="math-container">$2m&gt;1$</span>.)</p>

<p>Therefore, a Nash Equilibrium in this game is that both bidders play the strategy
<span class="math-container">$$(2m-1)\cdot \text{Uniform}(0,2m-1)+(2-2m)\cdot\delta_m.$$</span></p>

<p>How does it map back to the original problem? I believe the OP can take it from here.</p>

CC BY-SA 4.0
2018-10-11T03:17:40.103Z
2018-10-11T03:17:40.103Z
Ziwei Wang
<p>I am currently struggling with this exercise. 
Professor Nash announces that he will auction off a 20 dollars bill in a competition between two students chosen at random. Each student is to privately submit a bid on a piece of paper; whoever places the highest bid wins the 20 dollars bill. In the event of a tie, each student gets 10 dollars. The auction is all-pay: each student must pay his/her bid regardless of who wins the auction.</p>

<p>HERE is the question</p>

<p>Suppose each student could borrow money from the other students in the class, so that each of them had a total of $11 to bid. Find a Nash equilibrium in this case? 
I know I have to find a mixed-strategy  nash equilibrium but I don't understand how I can't compute it.</p>

<p>Thanks</p>

CC BY-SA 4.0
2018-10-10T08:24:57.990Z
The question is incomplete. You need the probability distribution over the range of private values to calculate an exact, numerical answer.
superhulk
CC BY-SA 4.0
2018-10-10T17:17:33.430Z
You do not need the probability distribution. Question is fine as is.
Lee Sin
CC BY-SA 4.0
2018-10-10T17:20:51.920Z
Maybe start by looking at specific cases. What happens when they both are at 20 dollar bids? 10? 0? What happens when one is at 10, the other at 20?

Start mapping this out.

In the meantime: https://www.youtube.com/watch?v=1IAsV31ru4Y

Hopefully that can give you some insight.
Lee Sin
CC BY-SA 4.0
2018-10-11T01:43:52.723Z
@LeeSin Then how can one find the bidding strategy for an auction?
superhulk
CC BY-SA 4.0
2018-10-11T01:52:43.223Z
As far as I know the bidding strategy for an All Pay auction is $\beta^{AP} = \int_0^x yg(y)dy$, where $x$ is the valuation realized by the bidder, and $g(y)$ is the density function of the highest ordered statistic among $X_2 , X_3 , ... , X_n$, and all the rv's  $X_1 , X_2 , X_3 , ... , X_n$ follow the same distribution.
superhulk
CC BY-SA 4.0
2018-10-14T14:15:35.353Z
The teacher gave us the following hint : try mixing between two bids. I am wondering if it means that we expect people to play randomly.
Mathilde
CC BY-SA 4.0
2018-11-10T07:58:42.300Z
@superhulk We are auctioning a \$20 bill, so it is implicit that both players have a valuation distribution that is degenerate at \$20.
Ubiquitous
<game-theory><nash-equilibrium><auctions>
All-Pay Auction Mixed Strategy Equilibrium
Mathilde
# All-Pay Auction Mixed Strategy Equilibrium

(Asked by: username_0 on 2018-10-09)

I am currently struggling with this exercise.
Professor Nash announces that he will auction off a 20 dollars bill in a competition between two students chosen at random. Each student is to privately submit a bid on a piece of paper; whoever places the highest bid wins the 20 dollars bill. In the event of a tie, each student gets 10 dollars. The auction is all-pay: each student must pay his/her bid regardless of who wins the auction.

HERE is the question

Suppose each student could borrow money from the other students in the class, so that each of them had a total of $11 to bid. Find a Nash equilibrium in this case?
I know I have to find a mixed-strategy nash equilibrium but I don't understand how I can't compute it.

Thanks

**Tags:** game-theory, nash-equilibrium, auctions

> **Comments:**
> **username_1**: The question is incomplete. You need the probability distribution over the range of private values to calculate an exact, numerical answer.  
> **username_2**: You do not need the probability distribution. Question is fine as is.  
> **username_2**: Maybe start by looking at specific cases. What happens when they both are at 20 dollar bids? 10? 0? What happens when one is at 10, the other at 20?  Start mapping this out.  In the meantime: https://www.youtube.com/watch?v=1IAsV31ru4Y  Hopefully that can give you some insight.  
> **username_1**: @username_2 Then how can one find the bidding strategy for an auction?  
> **username_1**: As far as I know the bidding strategy for an All Pay auction is $\beta^{AP} = \int_0^x yg(y)dy$, where $x$ is the valuation realized by the bidder, and $g(y)$ is the density function of the highest ordered statistic among $X_2 , X_3 , ... , X_n$, and all the rv's  $X_1 , X_2 , X_3 , ... , X_n$ follow the same distribution.  
> **username_0**: The teacher gave us the following hint : try mixing between two bids. I am wondering if it means that we expect people to play randomly.  
> **username_3**: @username_1 We are auctioning a \$20 bill, so it is implicit that both players have a valuation distribution that is degenerate at \$20.  

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## 1 answers

### Answer by username_4 on 2018-10-11. Score: 3

In response to our [policy on homework questions](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions), I will work on the following abstract version of your question:

*A single prize is to be allocated by an all-pay auction. Assume the value of the prize is 1 to both bidders, and this is common knowledge. Both bidders have a budget constraint $m$. If there is a tie, each bidder receives the prize with $\frac{1}{2}$ probability. Find a Nash Equilibrium in this game.*

We shall focus on the symmetric equilibrium in this game, that is, both bidders are using the same equilibrium strategy. Before we get started, let's make two observations:

**Claim 1**: Whenever there is a mass point in the equilibrium strategy (i.e. there is positive probability to submit one bid), this mass should be put on the upper bound $m$. For if not, one bidder can always deviate to a slightly higher bid and obtain higher surplus.

**Claim 2**: Whenever a bidder randomizes, the bid should be uniformly drawn from $(0,t)$, and the probability that this happens should be $t$. This is because only in this case, a bidder is indifferent between any bid $b\in(0,t)$:
$$\text{Expected payoffs}=t\cdot \frac{b}{t}\cdot 1-b=0.$$

Therefore, we can guess that the equilibrium strategy takes the form
$$t\cdot \text{Uniform}(0,t)+(1-t)\cdot\delta\_m.$$
That is, with probability $t$, a bidder submits a bid uniformly drawn from $(0,t)$, and with complementary probability she submits $m$. The only thing left is to pin down $t$. Note that if this is an equilibrium strategy, a bidder should be indifferent between any bid $b\in(0,t)\cup\{m\}$. Therefore, the payoffs from submitting $m$ should also be 0, i.e.
$$t\cdot 1+(1-t)\cdot\frac{1}{2}-m=0,$$
which implies $t=2m-1$. (Notice that for the existence of such an equilibrium, we must have $2m>1$.)

Therefore, a Nash Equilibrium in this game is that both bidders play the strategy
$$(2m-1)\cdot \text{Uniform}(0,2m-1)+(2-2m)\cdot\delta\_m.$$

How does it map back to the original problem? I believe the OP can take it from here.


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