## Wednesday, August 18, 2010

### When does it pay to burn money?

Yesterday, a student of microeconomics brought me an excellent game theory question to discuss. The topic was credible commitments (which I previously wrote about here and here). I don't want to spoil the question for others who get the joy of working this problem, so here's the problem with context changed:

Suppose there are two firms in the smart phone industry: an incumbent and a potential entrant. The incumbent can charge a high price or a low price. The potential entrant can enter or stay out. We also know the following facts about the incumbent's profit:
1. If the potential entrant enters, the incumbent gets 10 from charging a high price, and 5 from charging a low price.
2. If the potential entrant stays out, the incumbent gets 60 from charging a high price and 55 from charging a low price.
That is, no matter what the entrant does, the incumbent wants to charge a high price. A strategy like this (best regardless of what the other player does) is called a dominant strategy. The incumbent in this setup has a dominant strategy to charge a high price.

1. If the incumbent charges a high price, the entrant gets 25 from entering (0 from not entering).
2. If the incumbent charges a low price, the entrant gets -20 from entering (0 from not entering).
This is where the problem gets interesting. If the incumbent charged a low price, he could deter the entrant from entering (ensuring a choice between 60 and 55 instead of 10 and 5). BUT, we already deduced that no matter what the entrant does, the incumbent has a dominant strategy to charge a high price. In other words, no matter how much talk there is of the incumbent charging a low price, the entrant won't believe him.

Given these payoffs, the equilibrium is that the entrant will enter, and the incumbent will charge a high price. The entrant gets 25 while the incumbent gets 5... to the dismay of the incumbent.

So, what's an incumbent to do? He knows the game and understands that he has a dominant strategy that is detrimental to profits. One possibility is to credibly change his payoffs. Here's what I would do:

Sign a contract with some advertising agency that is contingent on setting a high price. In exchange for one worthless ad, the incumbent would pay 6 (I didn't put units on my payoffs, maybe it is millions of dollars) to the advertising agency. The important feature of the contract is that it would only run if the incumbent's price is high.

This seems like a dumb contract for the incumbent to sign. After all, the incumbent would be spending money for the purpose of decreasing payoffs. But, it's not so dumb when you look at the strategy behind it. With the contract, the incumbent's payoffs are as follows:
1. If the potential entrant enters, the incumbent gets 4 from charging a high price, and 5 from charging a low price.
2. If the potential entrant stays out, the incumbent gets 54 from charging a high price and 55 from charging a low price.
Under the contract, the incumbent has a dominant strategy to charge a low price! Now, the entrant believes the incumbent when he hears rumors of a low price. The outcome of the game is that the entrant does not enter while the incumbent charges a low price. Instead of making 5 (as in the equilibrium without the contract), the incumbent makes 55, a vast improvement.

An alternative would be for the incumbent to just burn 6 units of profit if he ever sets a high price. But, notice that threatening to hold a "profit bonfire" upon setting a high price doesn't quite do the trick. That's because once the time comes to hold the bonfire, the incumbent will want to substitute monopoly money for the real profits. The entrant knows this, and for this reason, the entrant won't believe a simple threat to burn money.

This need to convince the entrant of incentive compatibility is why the incumbent must go to great lengths to take the burning of profits out of his hands. Hiring the ad agency accomplishes this task. It doesn't usually pay to burn money, but if a company needs to credibly commit to a dominated strategy, burning money the right way might be a good idea.