As shown by Riley and Samuelson (1981),[1] equilibrium bidding in an all pay auction with private information is revenue equivalent to bidding in a sealed high bid or open ascending price auction.
The Nash equilibrium is such that each bidder plays a mixed strategy and expected pay-offs are zero.
[2] The seller's expected revenue is equal to the value of the prize.
However, some economic experiments and studies have shown that over-bidding is common.
That is, the seller's revenue frequently exceeds that of the value of the prize, in hopes of securing the winning bid.
In repeated games even bidders that win the prize frequently will most likely take a loss in the long run.
[4] The most straightforward form of an all-pay auction is a Tullock auction, sometimes called a Tullock lottery after Gordon Tullock, in which everyone submits a bid but both the losers and the winners pay their submitted bids.
[5] This is instrumental in describing certain ideas in public choice economics.
[citation needed] The dollar auction is a two player Tullock auction, or a multiplayer game in which only the two highest bidders pay their bids.
Other forms of all-pay auctions exist, such as a war of attrition (also known as biological auctions[7]), in which the highest bidder wins, but all (or more typically, both) bidders pay only the lower bid.
The war of attrition is used by biologists to model conventional contests, or agonistic interactions resolved without recourse to physical aggression.
[8] 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.
We wish to find a monotone increasing bidding function,
, he wins the auction only if his bid is larger than player
's expected utility when he bids as if his private value is
Since this function is indeed monotone increasing, this bidding strategy
are drawn iid from Unif[0,1], the expected revenue is
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.
Because each player's payoff is defined as their expected gain minus their bid, we can recursively define the optimal bid function as follows:
This means the probability of winning the auction will be equal to the CDF raised to the number of players minus 1: i.e.,
The objective now satisfies the requirements for the envelope theorem.
This yields the unique symmetric Nash Equilibrium bidding function
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).
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.
To get the real optimal amount that each of the three donors should give, simply multiplied the IPV values by 1000: