You and another student are called up to the front of the classroom to participate in an auction.

Here is how the auction works:

–the prize money equals the sum of money in both of your wallets

–the winner is determined by an open outcry auction (suppose the professor announces selling prices that increase by $1 increments, and the game goes until someone says “I quit”)

–the winner has to pay the final price in exchange for the prize

Let’s say you are chosen for this game, and you are allowed to peek in your wallet before the auction begins.

You see that you have $10 in your wallet. How much would you be willing to pay in the auction?

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The answer

It is apparent you should be willing to bid at least $10. At that price you will at least break-even. But you should probably be willing to spend a little more so that you can profit sometimes too, if the other person has money but backs out early.

It turns out there is a general rule that is an equilibrium. If both people use the same bidding technique, then each should be willing to spend up to double the money in his wallet, and you should be willing to bid up to $20.

Why is that the case?

Let’s analyze the bidding behavior. Suppose you know the other person will bid up to twice the amount in his wallet. What’s the maximum amount you should be willing to pay?

To begin, we set up some variables. Let’s say your wallet contains x, the other person’s wallet contains y, so that the prize money equals x + y.

If you lose the auction, you walk away with nothing so there is nothing to analyze. So let’s consider the case that you win the auction at a price of p. What’s the maximum price you want to pay?

Notice that if the other person bids up to two times their wallet, 2y, then it must be the case that you won at 2y = p.

We want to solve for the highest price p you will pay such that the prize money x + y exceeds the price you win at p. The condition we want is:

x + y = x + 0.5p > winning price of p
if and only if p < 2x

In other words, if the other person bids up to double, you should also be willing to bid up to double and win profitably.

In real life, of course, things are not so simple. You have to consider the other party may use a different bidding strategy (say bidding up to a lucky number). But still, this is a reasonable rule of thumb and at least good enough to use for a game theory class.

This type of game is called an “almost common value” auction: the prize is a common value to all parties, but each party has good information on his own wallet to glean a bit more information.

As further reading, there are some neat applications of this auction in Telecom written about in Paul Klemperer’s 1997 paper Auctions with Almost Common Values: The Wallet Game and its Applications.

My favorite part of the article was the following paragraph which highlights how ignorance of game theory can be costly:

For example, a few years ago Israel’s government added a novel twist to an auction of oil-refinery facilities. To encourage more and higher bids, the government offered a $12m prize to the second-highest bidder. It was an expensive mistake. Without the incentive, the highest bid would have been about $12m higher, an analysis showed—participants bid low because the loser would strike it rich. Combine that sum with the prize payout, and the government’s loss amounted to roughly $24m.

This is a great example of the law of unintended consequences. The government offered an incentive for people to bid higher, but it resulted in everyone competing for the second-prize and low-balling their bids.

The lesson: don’t assume people will bid like you want them to!

(I have also been trying to find the math behind the calculation to explain on this blog. My research attempts have failed, and sadly the article does not include a reference. If you’re aware of this example please let me know, thanks in advance.)

The video is short and offers great practical advice for buying a car in America. Give it a watch:

Video: how to buy a car using game theory

And to pre-empt someone saying the video “is not really about game theory,” I respond that the following game theory concepts are used in the car-buying strategy:

–changing the game: you don’t go to the dealership, you call them

–auction theory: you are turning the negotiation process into an auction amongst dealerships

–credible threats: you walk out if they mislead you

Game theory can be a wonderful way of thinking about problems.

The listing was noticed by a UC Berkeley associate professor Michael Eisen. You absolutely must read this post on his blog to get all the interesting details of how the bidding war unfolded.

I will give a short summary below with some observations of the seller strategies.

Understanding the sellers of the books

The first important detail is the listing was made by a seller in the marketplace, and not by Amazon itself. The marketplace works like eBay in that sellers can list their items, either new or used, and earn reputation points for their performance.

The book in has 2 sellers offering new copies. In such a situation, we might refer to a couple of models of duopoly. In Bertrand competition, sellers compete on a price, and repeated undercutting leads the equilibrium profits to zero. In Cournot competition, sellers compete on quantity (say, because neither seller can supply the entire market) and that leads to a profitable equilibrium price.

In this circumstance, neither of those things happened. Why not?

How did the book rise to $23.6 million?

Eisen observed the prices of the books daily and found an interesting pattern along the following lines.

Seller A was systematically listing the book for about 0.1 percent lower than seller B’s previous day price. Seller B, in turn, was updating its book listing to be about 27 percent higher than seller A.

The next day, the two would update the prices again in the same pattern.

Can you guess what might happen? Let’s imagine the starting price is $100, and here’s what will happen after 7 iterations:

Look at that: in just one week the price of the book has gone up five-fold!

A human seller may have noticed the prices were getting out of control. The fact the book got up to $23.6 million strongly suggests the two sellers were using an algorithmic pricing strategy.

Why would seller B list its book for more?

Seller A’s motivation of under-cutting price makes a lot of sense. But what was seller B doing?

Eisen came up with an interesting theory. Seller B, it turned out, had a very strong reviews. What seller B might have been doing is hoping that buyers would pay a premium price for its reputation.

As this was a rare and out of print book, there was another possibility. It was possible that seller B did not actually have a new copy of the book. If anyone ordered, then it would buy from seller A and flip the book, with a healthy profit margin!

The example suggests one should keep a healthy skepticism about sellers with good reputation. You have to wonder if you are truly paying for quality or just the illusion of quality.

I have summarized the story, but be sure to read all the details: Amazon’s $23,698,655.93 book about flies

These sites got big a couple years ago, and there are some great articles at codinghorror.com about how they are a scam and basically a lottery.

This would be fine if the sites were regulated like a lottery. But in fact, they advertise on TV as legitimate auction sites.

Here’s a video that summarizes some of the issues with penny auctions:

Video: Don’t Waste Your Money – Penny auction scams

Some of the main objections to these sites are:

• the timers are different: the auction time keeps increasing as people bid
• it costs money to bid, and often you end up with nothing
• there may be costs just to sign up

Another issue to add is:

• these sites often offer free bids for people to sign up. These free bids increase the length of the auction (as the timer resets) which forces paying bidders to have to place more bids, generating extra revenue for the site

The bottom line: avoid penny auctions unless you are looking to do something similar to gambling and playing the lottery.