Penny auctions from a game theoretic point of view

Penny auction is a type of auction that started appearing on the Internet around 2005 and 2006[1]. It works by having everyone participating pay a bidding fee, usually to the auctioning website, which gives you X amount of bids. Each auction has a time limit and a starting price (usually 0$). Each bid raises the final price of the item by 0.01 cent and adds Y seconds (for example 20) to the time left timer.
Lets say a site gives 100 bids for 20$, and an item sells for 50$. Bids = 50$/0.01$ = 5000 bids; which has a total price of 1000$. So the seller earns 1050$, which is usually a lot more than the value of the item. The winner can earn a bit if he did not waste too many bids, but everyone else looses money.

All-but-winner-pays auction (abwp from here on) is a type of auction where everyone pays what they bid except the winner. It is similar to Martin Shubik’s Dollar auction paradox and the All-pay auction model, but differs in that the winner of the auction does not have to pay anything, which is meant to boost incentive to keep bidding on the item. Like the Dollar auction paradox, lets assume we are going to auction away one dollar. We have four players in this game: A, B, C and the seller S. A starts out the auction by bidding 0.05$, B proceeds to overbid A – lets say 0.10$. C joins in with 0.15$.
The three buyers continue to increase bids by 0.05$ until one of them, lets say A, reaches 1.00$. At this point A has a potential payoff of +1$ or -1$ if one of the other people overbids hem. For B and C they have a payoff of about -0.90$ / +1.0$. B may bid 1.05$ to still have a chance of winning and minimize his loss. This may keep going until the dollar sells for several times its original value.

Let’s first look at ABWP auctions from a game theoretical standpoint

We have the seller S and the players A, B and C. The players are perfectly rational and will do whatever they can to earn as much value in the short term as possible.
At any stage in the game the players are either the highest bidder, or have a chance to bid to get an item for free. Each bid increases the price by Epsilon, so the potential loss for each bid is small (Going in rounds of A, B, C, A, .. the potential loss is whatever you have already bid + 2*epsilon (The other two peoples bids) ). Epsilon is incredibly small, almost zero so players will just keep increasing the price to infinity.

So why does this happen to this type of game?
In normal auctions both seller and buyers gain between 0 and TV (True Value) value. That is all parties gain something from the trade. We can look at an auction as two groups of players – the seller group and the buyer group. The seller group have a valuation for the product of X, the buyer group have a valuation of {B1 .. Bn} >= X, where B is the highest valuation. The final price the product sells for is Z. The seller groups profit is in the space {0, Z-X} and the buyer groups profit is in the space {0, B-Z}.

Now lets look at ABWP in the same way.
The seller group still has a valuation of the product X, and the buyer group have a valuation {B1 .. Bn} >= X where B is the highest valuation. Z is the price of the money transferred which means the sum of all the bids the loser of the auctions have to pay. At the start of the game when the bid is 0 Z is also 0. Each bid increases Z. The seller groups profit is still in the space {0, Z-X}, so when Z increases the seller group as a whole earns more and more. On the other size the buyer groups profit is still in the space {0, B-Z}, so when Z increases the groups profit decreases. This means that from the buyer groups point of view – each bid they put into the auction lowers their potential earnings, even to the point where it goes into negative. This means that if the buyer group cooperated they could earn value, but if they play as individual players they will lose.

In the real world, still using rational people – this means that the only way buyers will gain >= 0 is if there is no buyers (buyer group has a gain of 0) or a single buyer, in which case the buyer group would gain the true value of the item. In other cases the price and Z will accelerate to infinity.

Penny auctions use a slightly different mechanism. It makes the prices look ridiculously low by making most of the price of the item bleed out through a side channel – the bidding fees. A rational player will enter the game and see an item with a price a lot lower than TV. He will buy some bids and bid on the item. It may not seem like much, but when the buyer bid the seller earned money permanently and buyer lost money permanently. This repeats for several rounds with multiple buyers until the price is around TV.

Putting this into the equation from ABWP from before we need to change Z a little. The new Z is the sum of the price of all the bids purchased by the buyers plus the final price. Variables would be the following:
* PB – Average price per bid
* N – number of bids
* PI – Price increase per bid

Z = N*PB + N*PI = N(PB+PI)

The trick to this auction type is that the seller only shows the buyer(s) N*PI (the current price of the item), leaving out N*PB (The price paid for all the bids so far), which is half the equation.

We have again that the seller groups profit is in the space {0,Z-X} and the buyer groups profit is in the space {0, B-Z}. With the same equation we draw the same conclusion – that this type of auction unfairly favors the seller group.

These observations are right in line with the observations from Martin Shubik’s Dollar auction paradox – that perfectly rational players with full knowledge of the game are compelled to make the irrational decision to keep bidding on items and increase the price so much that they in the end lose money / value.[4]

There are other problems with these type of auctions as well. The most common of the two – penny auctions is notorious for being involved in scamming operations where site admins go in and bid on items in order to artificially increase the prices of items, or prevent users from getting items at all [2]. One of the more commonly used CMS system for setting up a penny auction site – PHPPennyAuction had build in support for bidding bots that would keep bidding on items last second until they either reached a per-determined price or would make sure to always have the last bid (if a bot wins the site doesn’t get any complaint if they don’t send out the item on sale). The PHPPennyAuction.com site is at the time of writing shut down, possibly due to bad reputation, but a site called “ajaxphppennyauction.com” has taken it’s place[3]. Subjectively the new site does not look trustworthy.

[1] https://en.wikipedia.org/w/index.php?title=Bidding_fee_auction&dir=prev&action=history First mention on wikipedia.

[2] http://www.pennyauctionwatch.com/2010/01/pennybiddr-com-alleged-scam-with-shill-bidders/

[3] http://www.ajaxphppennyauction.com/

[4] http://www.math.toronto.edu/mpugh/Teaching/Sci199_03/dollar_auction_1.pdf

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