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Using Game Theory to Model Market Uncertainty

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Here, traps a trader can fall into by not understanding the distinction between uncertainty due to randomness and uncertainty due to actions of other people.

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In 1994, the British government wanted to keep the value of the pound sterling above 2.7780 German deutschemarks. The British treasury spent £27 billion buying sterling at that rate and had essentially unlimited borrowing ability. It also raised interest rates by 500 basis points in the middle of a recession the keep the pound attractive to investors. Yet hedge funds, led by George Soros and Stanley Druckenmiller at the Quantum Fund, forced the price down to 2.4 deutschemarks with far smaller resources.

Treasury officials were treating this like a contest in which the stronger side wins. But the hedge funds were treating it like a bet. As Druckenmiller later explained, he knew there was a chance the pound would not go down, but also there was virtually no chance it would go up. All the economic fundamentals and market pressure were downward, the only thing holding the pound at 2.7780 was buying by the British treasury. That made it a heads-I-win-tails-I-break-even bet. Not only is that irresistible to any red-blooded investor, but it makes it easy to borrow large amounts of money to increase the size of bets. If there's little downside, a lender has little risk. The more investors piled in, the more attractive the bet became.

Instead of treating the exchange rate as a Random Walk, the British treasury should have considered it as a bet. In order to win the contest, it had to be able to create a credible scenario in which speculators would have been hurt if they had failed. In the circumstances that may not have been possible, in which case the treasury could have saved £3.3 billion plus a lot of pride by bowing to the inevitable.

Modern central bankers absorbed this lesson. In No Reserve: The Limit of Absolute Power, former Argentinean central bank President Martin Redado writes:

…[A]lthough they had tolerated my exchange strategy, they didn't like it. Face to face with the president [of Argentina], I once again explained to her the need to provide the exchange rate with upward and downward movement, in order to keep the markets from thinking there was easy money to be made from it, as had happened in programs with semi-fixed exchange rates-like Martínez de Hoz's "little table" [in the late 1970s] or Cavallo's Convertibility [in 1991].

You don't have to be a central bank to make this mistake. Any trader who has ever held an illiquid position that the Street knew he had to sell knows what it's like to play a losing hand. Anyone who watched a price chart without considering who was painting the picture has paid for his oversight. Anyone who forgets that his brokers, counterparties, service providers, investors, regulators, and others are profit-maximizing entities can get a rude shock (and profit is not always measured in dollars).

A more subtle error is to adopt a simple game theory explanation without thinking it through. Going back to our airport screening example, the simplest game theory model says to search passengers completely at random because any pattern can be exploited by terrorists to reduce their probability of being caught. But there are costs to terrorists of recruiting nonstandard suicide bombers or to disguising agents. More important, there is not a single rational terrorist entity, but many potentially dangerous people with different motivations, beliefs and skills.

Thinking more deeply, the entire goal of terrorists is to provoke a response. The more you inconvenience travelers by intrusive security, the more terrorists win from each successful atrocity.

Moreover, even if you eliminated all terrorist threats to passenger airlines, you might just redirect terrorist efforts to other targets. Your goal is not to keep increasing security until you catch every bomber, it's to get to a world in which there are no bombings and no need for security. How best to get there is a tricky question; my point is only that a simple game theory model positing a single terrorist organization with predictable responses and a goal of catching as many suicide bombers as possible is inadequate for the real decisions we have to make. It may be adequate for maximizing the budget and powers of anti-terrorist professionals and the re-election prospects of politicians, but those are different games.

You hear flimsy game theory arguments all the time by people who feel the need to explain every market event after the fact. Prices went up? Buyers came into the market (but doesn't there have to be a seller for every buyer?). Prices went down? Profit-taking. These are flimsy because the actors are never identified precisely, and the commentator doesn't ask even the most basic question, "Why would those people act in a way that consistently loses money?" It takes some digging and thinking to come up with solid game theory trade ideas, and even with that effort, you get surprised much of the time (you just hope it's less than half the time).

Even simple trades motivated by game theory fail a lot, trades like:
  • Buy stocks likely to be added to the S&P500 and short stocks likely to be dropped because index funds and benchmarked managers are forced to swap those stocks on the event.
  • Estimate trades ETFs will have to do near the market close and front-run them.
  • Buy merger targets and short acquirers at time of announcement because institutions will sell targets rather than wait for the deal to go through.
These qualify technically as game theory trades because you are treating the stock price as an amount chosen by rational actors rather than a draw from a probability distribution. They are not flimsy; serious researchers have teased out actual actors and figured out their actual incentives.

But they're too simple to need any theory. Any trader has to understand these kinds of trades and to know the difference between them and flimsy game theory arguments on one hand, and probability-based arguments on the other hand.

Traders can, but don't have to, study sophisticated game theory in order to develop more complicated trades. That's a subject for another article, the use of game theory as an offensive trading weapon. For today, the lesson is only how to defend yourself against game theory mistakes, and the most basic mistake is not to understand game theory in the first place.
No positions in stocks mentioned.
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