All-pay auction

Rules

The following analysis follows a few basic rules.

• Each bidder submits a bid, which only depends on their valuation.
• Bidders do not know the valuations of other bidders.
• The analysis is based on an independent private value (IPV) environment where the valuation of each bidder is drawn independently from a uniform distribution [0,1]. In the IPV environment, if my value is 0.6 then the probability that some other bidder has a lower value is also 0.6. Accordingly, the probability that two other bidders have lower value is ${0.6}^2=0.36$.

Symmetry Assumption

In IPV bidders are symmetric because valuations are from the same distribution. These make the analysis focus on symmetric and monotonic bidding strategies. This implies that two bidders with the same valuation will submit the same bid. As a result, under symmetry, the bidder with the highest value will always win.

Using revenue equivalence to predict bidding function

Consider the two-player version of the all-pay auction and ${\displaystyle v_i,v_j}$ be the private valuations independent and identically distributed on a uniform distribution from [0,1]. We wish to find a monotone increasing bidding function, ${\displaystyle b(v)}$, that forms a symmetric Nash Equilibrium.

If player ${\displaystyle i}$ bids ${\displaystyle b(x)}$, he wins the auction only if his bid is larger than player ${\displaystyle j}$'s bid ${\displaystyle b(v_j)}$. The probability for this to happen is

${\displaystyle \mathbb{P}[b(x)>b(v_j)]=\mathbb{P}[x>v_j]=x}$, since ${\displaystyle b}$ is monotone and ${\displaystyle v_j\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}[0,1]}$

Thus, the probability of allocation of good to ${\displaystyle i}$ is ${\displaystyle x}$. Thus, ${\displaystyle i}$'s expected utility when he bids as if his private value is ${\displaystyle x}$ is given by

${\displaystyle u_i(x|v_i)=v_{i}x-b(x)}$.

For ${\displaystyle b}$ to be a Bayesian-Nash Equilibrium, ${\displaystyle u_i(x_i|v_i)}$ should have its maximum at ${\displaystyle x_i=v_i}$ so that ${\displaystyle i}$ has no incentive to deviate given ${\displaystyle j}$ sticks with his bid of ${\displaystyle b(v_j)}$.

${\displaystyle ⟹u_i^{\prime }(v_i)=0⟹v_i=b^{\prime }(v_i)}$

Upon integrating, we get ${\displaystyle b(v_i)=\frac{v_i^2}{2}+c}$.

We know that if player ${\displaystyle i}$ has private valuation ${\displaystyle v_i=0}$, then they will bid 0; ${\displaystyle b(0)=0}$. We can use this to show that the constant of integration is also 0.

Thus, we get ${\displaystyle b(v_i)=\frac{v_i^2}{2}}$.

Since this function is indeed monotone increasing, this bidding strategy ${\displaystyle b}$ constitutes a Bayesian-Nash Equilibrium. The revenue from the all-pay auction in this example is

${\displaystyle R=b(v_1)+b(v_2)=\frac{v_1^2}{2}+\frac{v_2^2}{2}}$

Since ${\displaystyle v_1,v_2}$ are drawn iid from Unif[0,1], the expected revenue is

${\displaystyle \mathbb{E}[R]=\mathbb{E}[\frac{v_1^2}{2}+\frac{v_2^2}{2}]=\mathbb{E}[v^2]=\underset{0}{\overset{1}{\int }}{v}^{2}dv=\frac{1}{3}}$.

Due to the revenue equivalence theorem, all auctions with 2 players will have an expected revenue of ${\displaystyle \frac{1}{3}}$ when the private valuations are iid from Unif[0,1].

Bidding Function in the Generic Symmetric Case

Suppose the auction has ${\displaystyle n}$ risk-neutral bidders. Each bidder has a private value ${\displaystyle v_i}$ drawn i.i.d. from a common smooth distribution ${\displaystyle F}$. Given free disposal, each bidder's value is bounded below by zero. Without loss of generality, then, normalize the lowest possible value to zero.

Because the game is symmetric, the optimal bidding function must be the same for all players. Call this optimal bidding function ${\displaystyle \beta }$. Because each player's payoff is defined as their expected gain minus their bid, we can recursively define the optimal bid function as follows:

${\displaystyle \beta (v_i)\in arg\underset{b\in \mathbb{R}}{max}\left\{\mathbb{P}(\mathrm{\forall }j\ne i:\beta (v_j)\le b)v_i-b\right\}}$

Note because F is smooth the probability of a tie is zero. This means the probability of winning the auction will be equal to the CDF raised to the number of players minus 1: i.e., ${\displaystyle \mathbb{P}(\mathrm{\forall }j\ne i:\beta (v_j)\le \beta (v_i))=F(v_i{)}^{n-1}}$.

The objective now satisfies the requirements for the envelope theorem. Thus, we can write: ${\displaystyle \begin{array}{cc}{\int }_0^{v_i}F(\tau {)}^{n-1}d\tau & =(F(v_i{)}^{n-1}\cdot v_i-\beta (v_i))-(F^{n-1}(0)\cdot 0-\beta (0))\\ \beta (v_i)& =F^{n-1}(v_i)v_i-{\int }_0^{v_i}F(\tau {)}^{n-1}d\tau \\ \beta (v_i)& ={\int }_0^{v_i}\tau d{F}^{n-1}(\tau )\end{array}}$

This yields the unique symmetric Nash Equilibrium bidding function ${\displaystyle \beta (v_i)}$.

Examples

Consider a corrupt official who is dealing with campaign donors: Each wants him to do a favor that is worth somewhere between $0 and $1000 to them (uniformly distributed). Their actual valuations are $250, $500 and $750. They can only observe their own valuations. They each treat the official to an expensive present - if they spend X Dollars on the present then this is worth X dollars to the official. The official can only do one favor and will do the favor to the donor who is giving him the most expensive present.

This is a typical model for all-pay auction. To calculate the optimal bid for each donor, we need to normalize the valuations {250, 500, 750} to {0.25, 0.5, 0.75} so that IPV may apply.

According to the formula for optimal bid:

${\displaystyle b_i(v_i)=\left(\frac{n-1}{n}\right){v_i}^n}$

The optimal bids for three donors under IPV are:

${\displaystyle b_1(v_1)=\left(\frac{n-1}{n}\right){v_1}^n=\left(\frac{2}{3}\right){0.25}^3=0.0104}$

${\displaystyle b_2(v_2)=\left(\frac{n-1}{n}\right){v_2}^n=\left(\frac{2}{3}\right){0.50}^3=0.0833}$

${\displaystyle b_3(v_3)=\left(\frac{n-1}{n}\right){v_3}^n=\left(\frac{2}{3}\right){0.75}^3=0.2813}$

To get the real optimal amount that each of the three donors should give, simply multiplied the IPV values by 1000:

${\displaystyle b_{1}real(v_1=0.25)=\mathrm{\$}10.4}$

${\displaystyle b_{2}real(v_2=0.50)=\mathrm{\$}83.3}$

${\displaystyle b_{3}real(v_3=0.75)=\mathrm{\$}281.3}$

This example implies that the official will finally get $375 but only the third donor, who donated $281.3 will win the official's favor. Note that the other two donors know their valuations are not high enough (low chance of winning), so they do not donate much, thus balancing the possible huge winning profit and the low chance of winning.

A real life example of an all-pay auction can be found as part of the format of the Polish game show Awantura o kasę.