How to avoid the winner’s curse using game theory

I’ve made a huge mistake–GOB and others from Arrested Development

Strangely, in many auctions it is a huge mistake to win. Winning can either be unprofitable or less profitable than expected, kind of like a Pyrrhic victory. The tendency of the winner to overpay is more commonly known as the winner’s curse.

I have earlier described how the winner’s curse affects salary negotiations, oil drilling, and broadcast rights in my article understanding the winner’s curse. The essential feature of such auctions is the winner is bidding too high based on his information. As a consequence, the bidder wins in unprofitable or less profitable situations.

How can the winner’s curse be remedied? While there is no hard and fast rule, there is a principle that can help.

The idea is to bid like a winner or a consequentialist. This idea is described in the fine book The Art of Strategy and I have recast a problem from the text to illustrate.

Buying a blog

Suppose you are in the process of buying a small blog. You wish to buy it and improve it a bit. How much should you offer to pay?

The question naturally depends on how much you think the blog is worth. You happen to know a thing or two about small blogs. After some research you conclude the small blog is worth somewhere between 4 thousand and 12 thousand. Besides this range you cannot estimate any more specifics. You think it is equally likely for the blog to be worth any of the values in the range.

You also have an advantage in the process. You happen to know a lot about websites and blogs, and you figure you can improve the website. You can apply your web savvy to increase the blog’s worth by 20 percent. You are certain of this as you have built up other blogs before.

The one thing you are not able to do is find an exact worth of the blog. The best you know is that the blog is equally likely to be worth any value from 4 thousand to 12 thousand, with an average value of 8 thousand.

It’s important to understand how the uncertainty affects the offer. Suppose you offer 5 thousand. There are various ways this could turn out.

• If the blog turns out to be actually worth 4.5 thousand, then you would be able to raise its worth to 5.4 thousand. Since you have paid 5 thousand,  you will make a decent profit.
• If the blog turns out to be actually worth only 4 thousand, then you can only raise its worth to 4.8 thousand. Since you have paid 5 thousand, you will lose a bit of money.

Your first priority in this process is to make sure you make a sensible offer. You want to be sure you are not systematically overpaying and suffering from the winner’s curse.

So you ask yourself: what is the most you should offer for the blog? In other words, at what offer will you exactly break even on average? Once you know this, you will end up shaving a little bit more so you can make an offer that will actually profit.

The wrong reasoning = winner’s curse

I’ll admit it: when I first heard this question I came to the wrong answer.

Here was my logic. I figured the company was worth 8 thousand on average. Since the buyer could raise the value of the company by 20 percent, I calculated the average buyer value as 8 x 1.2 = 9.6 thousand. Therefore, a buyer could offer 9.6 thousand and just break even on average.

Do you see what is wrong with this logic?

The issue is the answer does not take the seller into account. The seller often knows more than the buyer. The fact that the seller is willing to accept an offer is bad news. It means the seller thinks the blog is worth at most that offer.

So if a seller accepts that offer, then the correct range is the blog is worth somewhere between 4 thousand to 9.6 thousand, or an average of 6.8 thousand. The buyer will be able to raise the blog up to a value of 6.8 x 1.2 = 8.16 thousand on average. But this is not good enough–the buyer has offered 9.6 thousand and will lose on average. It’s the dreaded winner’s curse!

The right answer

The correct offer is one where in which the offer exactly equals the expected worth of the blog. You need to shave your offer to account for the information advantage of the seller.

Which offer works?

The correct answer is 6 thousand. When a seller accepts this offer, the blog is actually worth somewhere between 4 and 6 thousand, with an average of 5 thousand. You can expect to raise the blog’s value to 5 x 1.2 = 6 thousand, which is precisely the amount you have offered.

The math behind the answer

The way to solve for the offer is to set up an equation between your offer and the blog’s expected value to the buyer.

The two quantities are your offer, let’s say x, and the blog’s expected worth.

What is the blog’s expected worth? As argued above, it’s important to consider the seller. A seller accepting an offer means he thinks the blog is worth at most that amount.

So a seller accepting an offer of x means the blog is worth somewhere between 4 thousand and x, for an average of (4 + x) / 2. This is the blog’s average worth to the seller.

On top of that, you can raise the value by 20 percent. Hence, the expected blog’s worth is 1.2 multiplied by (4 + x) / 2.

The equation is therefore:

x = 1.2 (4 + x) / 2
x = 0.6 (4 +x)
x = 2.4 + 0.6 x
0.4 x = 2.4
x = 6

Now you know 6 thousand is the highest you should pay.

Getting to a profitable bid is another exercise in negotiations and game theory. I will not discuss the issue here but I will mention that in such situations patience can be a negotiating asset.

Discussion questions

1. The above discussion was about buying a blog. How could you apply the same principle in other situations where the winner’s curse applies–like signing a star athlete or bidding on broadcast rights for the Olympics?

2. Write an equation for the break-even offer if the bid is x, the valuation is equally likely to be between a and b, and the buyer’s skill leads to improving the company worth by y percent.

3. (more advanced) Rewrite the equation in 2 if the valuation is between a and b, but with a probability density function f(t).

4. Would the bidding strategy change if there were two or more buyers?

5. How would the bidding change if instead the winner paid the second highest offer?

6. Would additional bidders help the seller or the buyers?

7. How would the game change if the seller was less informed than the buyer about the worth of the seller’s company?