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Using Game Theory to Model Market Uncertainty


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.


Game theory is a branch of mathematics that all traders, even traders who hate math, need to appreciate. It's one of the ways to study uncertainty. The more familiar mathematical model of uncertainty is probability theory. In that field, we model things we don't know as if they are coin flips or casino games. An application to finance is the Random Walk model of security prices. Of course, security prices are not coin flips; prices move up and down in response to market fundamentals and investor preferences and beliefs. But for some purposes it's useful to treat the prices as random.

In game theory we model uncertainty as choices made by rational entities rather than unpredictable events we cannot influence. Consider the debate about profiling for airport passenger screening. Some people argue that most suicide bombers are young men, so Transportation Security Agency agents should concentrate on them rather than, say, 80-year-old Buddhist monks or mothers with small children. Other people argue that terrorists will quickly note any discrimination by agents and enlist bombers who can be disguised as whatever category gets the least attention.

Regardless of how you stand on that issue, note that the first argument is probabilistic. If historical observation shows that people with certain characteristics are more common among bombers, screeners should devote more attention to people with those characteristics. That treats being a suicide bomber as a random event that can be correlated with observable characteristics.

The second argument is a game theory one. It assumes suicide bomber characteristics are chosen by a rational entity that observes your screening strategy and responds to it in predictable ways.

For a simpler example, consider two sports bettors. One develops a computer simulation model of sporting events and estimates the probability of the favored team covering the spread. That's treating the event like a casino game in which outcomes have calculable probabilities. His sister uses reasoning like, "Bettors tend to overbet on the teams that rewarded them last time, so I bet on teams after they've lost to the spread three times in a row and against teams that beat the spread three times in a row," or "Big-market teams have more natural bettors than small-market teams and people are more apt to bet on games they attend than games they watch on television, so I bet against the big-market team at home."

These strategies ignore the game entirely, and focus on bettor behavior, and the predictable actions of bookies to adjust the line to get the same amount bet on both sides rather than to reflect the true probability of winning. You can make money either way. (Hint: the second method is much easier, although it's wise to pair it with at least some rudimentary statistical analysis of the game itself).

It's important to remember that both views are models, simplifications of reality. It is not literally true that some lottery drawing in the sky determines who will blow himself up today, nor is it true that there is a single, predictable entity that determines who will try to smuggle explosives onto an airplane.

Sporting events are won and lost on the field by real players (unless, of course, they are fixed, in which case we have a pure game theory problem). Probability theory and game theory can be useful tools to support decisions, but outside of some artificial situations they are not exact descriptions of the world.

How does this relate to trading? I already mentioned the Random Walk model that treats security price changes as draws from a statistical distribution. A trader might use this approach by noticing that prices that go up tend to keep going up, so she buys stocks with good price momentum.

An example of game theory thinking is a trader looking for stocks with high short interest, in which the shorts have lost money and appear to have weak hands (perhaps because a popular short position just lost a lot of money, or perhaps because it's near the end of a year in which short-bias hedge funds have had poor performance) and the stock has become hard to borrow. If this stock goes up, there may be a short squeeze to send it even higher.

Note that in this example, our probability theory trader and game theory trader are in the same position, buying a stock because it has gone up. But they are in for entirely different reasons. The probabilist is looking at historical data and betting that the future will resemble the past. The gamer is looking at the situation of other traders with positions in the stock, and making a prediction based on reasoning rather than statistics. The probabilist hopes other traders are either ignorant or forced to act irrationally because she wants to profit from their bad trades. The gamer is relying on other traders to act rationally.

Up to this point, we have been abusing the word "theory." You don't need a Ph.D. in statistics to test if a momentum rule has made money in the past, and you don't need high-powered game theory mathematics to notice the signs of an impending short squeeze. This article is not about advanced quantitative trading techniques exploiting sophisticated mathematics. Rather it's about the traps a trader can fall into by not understanding the distinction between uncertainty due to randomness and uncertainty due to actions of other people.

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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