All-Pay Auctions
Ever see a commercial where they say you can get an expensive item for EXTREMELY CHEAP by using their bidding software? Well, it turns out that those bids are in an “all-pay” format, which means that you have to pay your bid whether or not you win. Seems like a scam right? Let’s take a deeper look at these types of auctions.
Most auctions reward the item to the person who bids the highest, and only the winner either pays their bid or the second highest. The losers get their bid money back, essentially having a payoff of zero. However, an all-pay auction is a type of auction where everyone pays money regardless of whether or not they win the auction. The money they pay is either how much they bid, a fraction of their bid, or a flat fee (Gneezy & Smorodinsky). The winner is the highest bid, and they pay their bid for the item. Thus, everyone who participates in this auction tend to have a payoff less than 0 (negative payoff), except, maybe, the winner (Weisbaum). Furthermore, very few people actually make a profit when they win the auction (“Dealdash’d: A losing bet for consumers”). It begs the question: “If we were to participate in an all-pay auction at all, what strategy should we follow?”
Here is the payoff function of an all-pay auction. “u” represents the payoff, “x” represents the bid placed by the ith person, and “v” represents the ith person’s value of the item. We can see that the payoff is always negative. If someone places a losing bid, their payoff is negative “x.” If there is a tie between “m” people, then there is a one in “m” chance they will win, and their payoff will be between their value multiplied by the probability (1/m) and their bid. Finally, placing a winning bid yields a payoff of the difference between your value “v” and the bid “x.”
We know that there is no pure strategy Nash Equilibrium in this situation because there is no dominant strategy in an all-pay auction. To find the mixed strategy Nash equilibriums of this auction, we need to understand that each bidder has a cumulative distribution function (CDF), that we’ll call G, which represents the probability that they will bid less than you. This CDF represents the possible bids that given person can place, and the area under this curve represents a probability. You can input your bid (x) into the CDF, and the integral from negative infinity to x represents the probability that the person will bid less than you. Thus, the integral of the CDF from negative infinity to infinity is 1.
An auction is an exaggerated version of the basic 2-player game (that we’ve discussed in class), in which you are “playing” it against every other bidder. Thus, we can calculate the mixed strategy Nash Equilibrium from a single bidder’s perspective by setting the difference between your payoff multiplied by your winning probability and the inverse of that probability multiplied by your bid to zero. A case involving 3 people (you are person i, and the other 2 people are j and k) is written in equation form below (Baye, et al.).
While analyzing this equation, we can see that the first term involves multiplying everyone else’s CDF’s by the difference between your value and your bid because the probability that all of them bid under you is an “and” relationship. The second term is just the inverse of that probability (either they all bid under you or they don’t, so those 2 probabilities add to 1) multiplied by your bid/payment/transaction. A more general form of this equation can be seen below (written by me in LaTex).
From this, we can see that the probability of winning significantly decreases when there are more bidders. This is almost like an nth root equation, which decreases more and more as you increase the power of the root. That also means that your bid must significantly decrease as the people increase as well, simply because your payoff would very easily go extremely negative.
Now, we can look at this from the seller’s perspective. We can assume that the seller’s revenues will be very high if there are a large number of bidders in the auction, even if each of them bid a small amount. If everyone bids their value (truthfully), the revenue is just the average of everyone’s valuations multiplied by the number of bidders. In an experiment involving all-pay auctions done by professors at the University of Chicago, they found that when there were more than 8 people participating in the auction, the seller’s revenue was, on average, much more than the prize’s actual value. The independent variables of this experiment were the number of people participating in each auction: 4, 8, or 12 (Gneezy & Smorodinsky). This is because bidders all tended to over-bid when there were more people, which completely contradicts our mathematical proof for the Nash Equilibrium stated above. They also found that most of the winning bids were pretty close to the actual value, which means that the seller definitely made at least twice the value of their good most of the time.
All-pay auctions are everywhere in life, not necessarily in the auction house. We can argue that sports competitions where there is one champion at the end of the season is an all-pay auction. Every team spends money, time, and effort in creating what they believe to be a championship team. Their belief is their valuation, and most teams (that don’t tank) will bid truthfully, which means that each team is “bidding truthfully.” However, there is only one champion at the end of the season, which means that they are the winner with a positive payoff. Everyone else has already paid their “bid.” Furthermore, any kind of war or battle is an all-pay auction. The victorious side will always have leverage over the defeated, and most of the time, losing parties never receive any benefits, while the victors reap as many benefits as they can. The losers have “paid their bids” in soldiers’ lives, land, money, and more, while the winners paid their bid, but got a less negative (and maybe even positive) payoff.
In conclusion, it can be interpreted that all-pay auctions are just a fancy expression for winner-take-all competitions, where there is only one winner and they receive all the prizes. Everyone else in the competition/game expended time/money/effort in trying to win, but ultimately came out empty-handed (Baye, et al.). We haven’t really discovered too much groundbreaking information here: we just simply proved that all-pay auctions are essentially a scam that provides huge revenues for the seller.
Citations:
Baye, Michael R., et al. “The All-Pay Auction with Complete Information,” Economic Theory, vol. 8, no. 2, Springer, 1996, pp. 291–305.
Dealdash’d: A losing bet for consumers. Truth In Advertising. (2018, January 31). Retrieved September 18, 2021.
Gneezy, U., and R. Smorodinsky “All-Pay Auctions: An Experimental Study,” Journal of Economic Behavior and Organization, 61(2), pp. 255-275.
Weisbaum, H. (2017, July 4). Dealdash Penny Auction sued for running ‘perverse lotteries’. NBCNews.com. Retrieved September 18, 2021.