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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 recognises 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 analyse 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 emphasise 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.