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How to Exploit Game Theory for Profit

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Game theory works best when combined with solid statistical analysis and thorough fundamental investigation.

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Last week, in Using Game Theory to Model Market Uncertainty, I covered what all traders should understand about game theory in order to protect themselves. Today I'll address what a quant trader should understand to exploit game theory for profit.

As I discussed last week, the basic idea of game theory is to model uncertainty as rational actions of other entities rather than randomness (like a coin flip) or something else. The examples I gave all concerned very simple decisions, for which little theory of any kind is needed. For a more realistic example, one where mathematical analysis is necessary to find the correct answer, consider trading a merger arbitrage strategy.

Suppose company A (the "acquirer") offers to swap one of its shares for every two shares of company T (the "target"). A's stock is selling for $50 per share and B's stock was selling for $20 per share before the offer. Therefore, T's shareholders should all accept the deal, and the value of B's stock should rise to $25. If T's shares rise only to $24, it presents a merger arbitrage opportunity. Investors can buy two shares of company T for $48, short one share of company A to get $50. They can then swap their two T shares for one A share, use that to cover their short, and have $2 riskless profit.

Well, it's not quite riskless, which is why the older name for this strategy is "risk arbitrage," which is a contradiction ("arbitrage" means riskless profit). The name recognizes that this can be a very low-risk strategy if done properly, but there are ways to lose money. The main risk is that the deal does not go through or the terms get renegotiated. If the deal fails, you would expect the price of T's shares to fall back to $20, or even below as deals often fail due to bad news about the target, giving you a loss of $8 or more. It could be even worse than this if A's shares have gone up in the interim.

The statistical approach to merger arbitrage is to analyze historical deals to determine the major factors that predict success or failure. Some important variables are the merger premium (25% in the example, since the offer was worth $25 for a $20 stock), the spread (4.17% in the example, because the arb offered $2 profit on a $48 position), the recent pre-announcement stock price movements of A and T, the predicted amount of time until closing, whether the deal is for stock (as above) or cash or something else, whether the deal is contingent on financing, whether the acquisition is important to A's business or opportunistic, whether there are regulatory issues, and whether there are other potential acquirers. This analysis would produce a set of criteria to determine which deals to pursue, how much capital to allocate to each, and when to bail out if things change.

A game theory approach starts with identifying the relevant decision makers. Some obvious candidates are A's management, the people expected to fund the deal for A, T's management, T's board, T's shareholders, other potential suitors for T, other merger arb investors and regulators. Not all of these will be relevant in all deals, but in complex situations (see, for example, the amusing and reasonably accurate account of the RJR Nabisco takeover battle, Barbarians at the Gate), you might have a lot more entities involved. For each decision maker we need to know the decisions it controls, and its "payoff function"; that is, how much it values all potential outcomes. In advanced game theory applications we might also specify what information set each decision maker has and what alliance or side-payment opportunities are available.

A common mistake is to overcomplicate things. This is the same error as data mining in statistics, where the attempt to build a model that works perfectly in the past results in a model that doesn't work at all in the future. The beauty of both statistics and game theory is that reasonably simple (but not too simple) robust models can produce useful results. We don't need to model a hundred decisions makers with a dozen decisions each and complex preferences over millions of potential outcomes, and if we try our prediction will be useless. With a lot less effort we can get a useful result limiting ourselves to a few key decision makers, decisions, and preference criteria.

Another mistake stems from the fact that elementary game theory courses and books often emphasize zero-sum games, where everything gained by one participant is lost by another. In trading, it is usually the nonzero-sum elements that drive the most interesting games. The profit you earn as a trader comes from pre-announcement holders of stock A, stock T, or some combination. To the extent this is a zero-sum game, those people would always act contrary to your interest. In order to take advantage of game theory, you should focus on the things they want other than your money.



To keep things simple, assume that benchmarked institutional investment managers held A and T shares before the announcement. When (if) the merger is completed, T will leave the benchmark. If these managers hold A and T at their benchmark weights, they will match the benchmark whether the merger goes through or not. If instead they sell you their T and use the money to buy A (and then lend you their A shares to short), there are two possible outcomes. If the merger goes through, they lose a small amount to the index. If the merger falls apart, they make a large amount relative to the index. Clearly you want to be in mergers where that is an attractive bet to the managers because in those cases they will be willing to give you a large spread relative to the probabilities of the outcomes.

You are also in competition with company A's managers, since you are short their stock and they want their stock to go up. In most cases it is the management of the acquirer that calls the shots, not the board or shareholders. However, A's managers have other interests as well. The merger will make their company larger, which could mean improved prospects for compensation and power. There is prestige to buying companies, and successful deals make it easier to do future deals, while unsuccessful deals can be embarrassing and costly and inhibit future deals (or even get the managers fired). The managers may have long-term plans that will boost the share price after you have closed out your position upon merger completion. You like mergers in which the incentives other than possible short-term stock price increase are very strong for the acquirer's management.

You have an ally in company T, since you are long that stock. It's usually the board of directors that is most important for T, although there are mergers in which you should model management or shareholders (or both) of the target as well. In this case any considerations other than short-term stock price might cause them to deviate from their natural alliance with you. They might be angling for board positions in the new company, or worried about criticism or shareholder lawsuits, or have emotional ties to T as an independent entity. They may have outside business interests that will be affected by the merger.

At this point we have enough entities to build a nontrivial game theory model; that is, a model whose conclusions are not obvious from looking at its inputs. In a model for real trading, you will almost always have more than three entities, and I cannot imagine any deal in which other merger arb players can be safely ignored.

There are a minimum of two outcomes -- the deal completes according to original terms or the deal falls apart. There may be other alternatives as well, a renegotiated deal, a delayed deal that gets done eventually, and acquisition by a third party. You can include random outcome components. For example, if the deal does fall apart, the stock prices of A and T are very important. You accommodate this by letting nature be a player with the strategy of selecting at random among available choices.

You need to estimate a payoff for each combination of entity and outcome. So that's a minimum of six (three entities, two outcomes) and more likely something like 60 (say six entities and 10 outcomes). In my experience it doesn't pay to go beyond this scale, but there are people who build elaborate computer models with thousands of outcome, or even use Monte Carlo simulation to examine millions of outcomes.

How well does all this work? I know people who claim great success, but they're usually shy about revealing exactly how they do things. From what I can personally verify, it can work surprisingly well, even when you have to guess a lot of the inputs. You often know more than think you know or can find experts to give information. One of the most useful features of this approach is that I find experts are almost always terrible at prediction, but sometimes good at answering specific questions like, "What are the most important considerations for A's management" or "How would T's board of directors rank the following potential outcomes."

I wouldn't suggest that anyone become a pure game theory trader; in my opinion, game theory is a valuable component of the quant toolbox, but not a solution to all investment issues. It works best when combined with solid statistical analysis and thorough fundamental investigation. But there are a lot more statisticians and investigators out there than game theorists, so you may find it gives you an important edge.

No positions in stocks mentioned.
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