When Microsoft announced its intention to acquire Yahoo last February, the software giant knew the struggling search firm would not come easily into the fold. But Microsoft had anticipated the eventual minuet of offer and counteroffer five months before its announcement, thanks to the powers of game theory.

A mathematical method of analyzing game-playing strategies, game theory is catching on with corporate planners, enabling them to test their moves against the possible responses of their competitors. Its origins trace as far back as *The Art of War,* the unlikely management best-seller penned 2,500 years ago by the Chinese general Sun Tzu. Mathematicians John von Neumann and Oskar Morgenstern adapted the method for economics in the 1940s, and game theory entered the academic mainstream in the 1970s, when economists like Thomas Schelling and Robert Aumann used it to study adverse selection and problems of asymmetric information. (Schelling and Aumann won Nobel prizes in 2005 for their work.)

Game theory can take many forms, but most companies use a simplified version that focuses executives on the mind-set of the competition. “The formal stuff quickly becomes very technical and less useful,” says Louis Thomas, a professor at the Wharton School of Business who teaches game theory. “It’s a matter of peeling it back to its bare essentials.” One popular way to teach the theory hinges on a situation called the “prisoner’s dilemma,” where the fate of two detainees depends on whether each snitches or stays silent about an alleged crime (see “To Squeal or Not to Squeal?” at the end of this article).

Many companies are reluctant to talk about the specifics of how they use game theory, or even to admit whether they use it at all. But oil giant Chevron makes no bones about it. “Game theory is our secret strategic weapon,” says Frank Koch, a Chevron decision analyst. Koch has publicly discussed Chevron’s use of game theory to predict how foreign governments and competitors will react when the company embarks on international projects. “It reveals the win-win and gives you the ability to more easily play out where things might lead,” he says.

**Enter the Matrix**

Microsoft’s interest in game theory was piqued by the disclosure that IBM was using the method to better understand the motivations of its competitors — including Microsoft — when Linux, the open-source computer operating system, began to catch on. (Consultants note that companies often bone up on game theory when they find out that competitors are already using it.)

For its Yahoo bid, Microsoft hired Open Options, a consultancy, to model the merger and plot a possible course for the transaction. Yahoo’s trepidation became clear from the outset. “We knew that they would not be particularly interested in the acquisition,” says Ken Headrick, product and marketing director of Microsoft’s Canadian online division, MSN. And, indeed, they weren’t; the bid ultimately failed and a subsequent partial acquisition offer was abandoned in June.

Open Options wouldn’t disclose specifics of its work for Microsoft, but in client workshops it asks attendees to answer detailed questions about their goals for a project — for example, “Should we enter this market?” “Will we need to eat costs to establish market share?” “Will a price war ensue?” Then, assumptions about the motives of other players, such as competitors and government regulators, are ranked and different scenarios developed. The goals of all players are given numerical values and charted on a matrix. The exercise is intended to show that there are more outcomes to a situation than most minds can comprehend, and to get managers thinking about competition and customers differently.

“If you have four or five players, with four actions each might or might not take, that could lead to a million outcomes,” comments Tom Mitchell, CEO of Open Options. “And that’s a simple situation.” To simplify complex playing fields, Open Options uses algorithms to model what action a company should take — considering the likely actions of others — to attain its goals. The result replicates the so-called Nash equilibrium, first proposed by John Forbes Nash, the Nobel prize–winning mathematician portrayed in the movie *A Beautiful Mind.* In this optimal state, the theory goes, a player no longer has an incentive to change his position.

As a tool, game theory can be useful in many areas of finance, particularly when decisions require both economic and strategic considerations. “CFOs welcome this because it takes into account financial inputs and blends them with nonfinancial inputs,” says Mitchell.

**Rational to a Fault?**

Some experts, however, question game theory’s usefulness in the real world. They say the theory is at odds with human nature, because it assumes that all participants in a game will behave rationally. But as research in behavioral finance and economics has shown, common psychological biases can easily produce irrational decisions.

Similarly, John Horn, a consultant at McKinsey, argues that game theory gives people too much credit. “Game theory assumes rationally maximizing competitors, who understand everything that you’re doing and what they can do,” says Horn. “That’s not how people actually behave.” (Activist investor Carl Icahn said Yahoo’s board “acted irrationally” in rejecting Microsoft’s bid.) McKinsey’s latest survey on competitive behavior found that companies tend to neglect upcoming moves by competitors, relying passively on sources such as the news and annual reports. And when they learn of new threats, they tend to react in the most obvious way, focusing on near-term metrics such as earnings and market share.

Moreover, finance executives have their own sets of metrics, and when favored indicators such as net present value clash with game-theory models, choices become more complicated. “Sometimes [game theory] tells you things you don’t like,” says Koch.

Game theory is still finding its place as a tool for companies, and its ultimate usefulness may depend on how quickly it moves from novelty to accepted practice. Practice, in fact, may be key. McKinsey takes that to heart with its “war game” scenarios, in which a company’s top managers play the roles of different parties in a simulation. In effect, this boils game theory down to the schoolyard lesson that perfection comes through repetition. “Discipline is not a dirty word,” as basketball coach Pat Riley once said. Game theory is one way that companies can assess their options with more discipline.

*Alan Rappeport is a reporter for* CFO.