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

  • The question is incomplete. You need the probability distribution over the range of private values to calculate an exact, numerical answer. – superhulk Oct 10 at 8:24
  • You do not need the probability distribution. Question is fine as is. – Lee Sin Oct 10 at 17:17
  • 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: youtube.com/watch?v=1IAsV31ru4Y Hopefully that can give you some insight. – Lee Sin Oct 10 at 17:20
  • @LeeSin Then how can one find the bidding strategy for an auction? – superhulk Oct 11 at 1:43
  • 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 Oct 11 at 1:52

In response to our 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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