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In the fall of 2001, top executives at IBM Corp. realized they had a major problem on their hands. Something was happening to their industry that the technology giant could neither control nor predict. When you're IBM, this is not a feeling that goes over well. Big Blue didn't get its nickname—or, for that matter, its $105 billion worth of assets—by going with the flow (all currency in U.S. dollars).

The problem soon landed on Joel Cawley's desk. In a company packed with brains, Cawley's job as vice-president of corporate strategy is to figure out new ways for IBM to think. He's like a mechanic wandering the aisles of a hardware store, searching for gadgets to fill out his tool box—except he has a budget in the millions and his store is the entire world.

That fall, IBM was watching the development of the Linux operating system with trepidation. Unlike previous threats to the dominant Windows system, Linux was a cheap and easy-to-use library of code being collectively assembled by programmers around the world. The goal was an operating system that no single company would own but all would use, providing free digital DNA for everything from servers to cellphones. It was an ambitious idea, and just the kind of disruptive force that spells sleepless nights for anyone invested in the status quo—in this case, not just IBM but also the likes of Sun Microsystems, Intel and Microsoft.

There were just too many dangling questions. Would Linux fly? And if it did, who stood to benefit most? Would any of the big tech companies muscle their way in and hijack the development? Or would Microsoft bog down the idea in legal wrangling?

Finding the answers to those questions quickly was crucial. If the concept took off and IBM hadn't invested properly in its own Linux-compatible technology, the company could find itself drastically behind, just as it had in the 1940s, when it famously decided not to back development of the first photocopier and ended up conceding crucial ground to an upstart called Xerox. Jump in too soon, and the company risked throwing money at a flash in the pan.

At IBM's headquarters north of New York City, Cawley picked up the phone and dialled Canada. He was acting on a suggestion a colleague had made: The next time they were in a strategic jam, they should try game theory.

Cawley's call went to Open Options Corp., a tiny Waterloo, Ontario, company that specializes in the discipline. In a stout, brown building, CEO Tom Mitchell and principal scientist Niall Fraser listened as Cawley described IBM's predicament.

Ten-year-old Open Options is one of a handful of companies in the world that devote all of their energy to using game theory to solve corporate problems. The firm counts five of the Top 10 companies on the Fortune 500 as customers, including IBM, Chevron and Ford, not to mention others further down the list, such as Xerox, Caterpillar and Boeing. (Most clients prefer to stay anonymous.)

The company's roots date to the late 1960s, when Fraser, newly graduated from high school, was contemplating what to do with his life. Much of what was being taught in university didn't appeal. "I was a hippie—a proud one," Fraser says. "Part of that ethic was to abandon conventional society, look for your own interests and build your life on your own terms. So I did that."

There followed five years of travel and sundry work like tobacco picking. What Fraser really wanted was to study strategic thinking from a philosphical point of view. After all, "the heart of life is how to make good decisions," he says. But the avenues for that line of study were limited. At the time, most business schools weren't pushing the boundaries on creative problem-solving. So Fraser avoided academia until the day he noticed that an obscure course was being taught in the University of Waterloo's engineering department. Game theory? Fraser wanted in. And since his father ran the department, strings were pulled to make up for a few high school grades that would have otherwise kept him from being admitted.

The way game theory works is simple—up to a point. A list is made of all the potential players in a particular situation—a game—and the options for each player are mapped out in as much detail as possible. Since one player can have several options, each one is given a numerical value to indicate its priority. However, those calculations become complex in a hurry: A situation with eight players, where each person or company has three potential options, can lead to 16,777,236 possible outcomes.

Game theory provides a path through the maze. The goals of one player are cross-referenced against those of another, revealing that some outcomes are not sustainable. Then there are moves that may be blocked by another player. Knowing that can allow a person or company to change their own behaviour and thereby change the game. Eventually, the number of possibilities is reduced to the single best attainable outcome.

Tools like logic and reason have long been used to predict others' actions, in fields ranging from card-playing to war making. It wasn't until the 1920s that game theory joined the tool kit, thanks to John von Neumann. The Hungarian mathematician believed that if a variety of players in a game act rationally—that is, they each pursue their own best interests—their movements are predictable.

The next big name in game theory is John Nash, the Princeton math legend who, along with two others, was awarded the 1994 Nobel Prize in economics. Nash's principles, developed in the late 1940s and early '50s, have found their way into everything from auction design to strategies for contract negotiations. But it wasn't until the 2001 film A Beautiful Mind, which focused on Nash's debilitating struggle with schizophrenia, that the man and his science had their 15 minutes of fame.

There's a pivotal and wholly fictional scene that was written into the film to explain how Nash's take on game theory worked. Most game theorists cringe whenever it is discussed. In the scene, Nash—played by Russell Crowe—and four friends are sitting in the campus bar when a statuesque blonde walks in, accompanied by four friends. The young men immediately begin verbally jockeying among themselves to see who can win the blonde's attention. Wrong approach, says Nash.

Nash believed that game theory could be used not only to discern patterns in how people think but also to find the best possible outcomes for everyone in a particular game—particularly if that outcome isn't the one that most people would arrive at themselves. "If we all go for the blonde, we block each other," the fictional Nash tells his friends. "Not a single one of us is going to get her. So then we go for her friends. But they will all give us the cold shoulder, because nobody likes to be second choice. But what if no one goes for the blonde? We don't get in each other's way and we don't insult the other girls. It's the only way to win, the only way we all get laid."

During the Cold War, the U.S. government relied heavily on game theory and sank millions into researching it. The work of people like Nash helped determine how the U.S. should strategize against the Soviet Union, most notably on the nuclear front (see story below). But it was also finding applications on the ground. By the mid-'70s, Niall Fraser, now studying at the University of Waterloo, was trying to figure out why multibillion-dollar engineering projects were being stalled by unexpected circumstances. For instance, a massive plan to irrigate North Dakota farmland by diverting water from the Missouri River ground to a halt because planners failed to take account of local environmental groups and Canadian governments, which both vigorously opposed the $2-billion project. Game theory could have predicted the impasse, Fraser figured.

Classic game theory assigns numeric scores to each outcome in a game. Fraser thought the science would be faster and more nimble if it simply ranked the preferences of each player. Accordingly, he and other game theorists developed Ordinal Nonco-operative Game Theory. Thirty-two papers and one book later, Fraser was a professor of management sciences at the University of Waterloo.

As of 1987, Fraser was working on software to speed up game theory calculations. That year, a paper he'd written was presented at a conference in New Orleans. Although Fraser wasn't there in person, it turned out to be a pivotal event in his career. Shortly after the conference, he received a phone call from the CIA. Three agency staffers had attended the conference. They thought Fraser's work could come in handy.

By this time, the Soviet Union was in decline; satellite states were angling for independence, and the U.S. government was watching closely. Azerbaijan's shaky future was of particular interest. Located on the northern border of Iran, Azerbaijan was one of the oldest oil-producing regions in the world; it also had a testy relationship with one of its neighbours, Armenia. The agency wanted to know if game theory could be useful in such situations. Specifically, the CIA asked Fraser what would happen to Nagorno-Karabakh, an Armenian enclave in Azerbaijan. More broadly, it wanted to know if the region would crumble into civil war if the USSR collapsed.

"We thought it was pretty cool, working with these James Bond guys," Fraser says. "Their business cards just had 'Joe Blow, Research and Development.' No company, no address, no phone number." They did have a phone number and a mailbox, of course—but they changed every week. In the span of a few years, the CIA put millions into the development of Fraser's software. "They were very generous," Fraser says. "They paid for everything, but all they required was the rights to use it. The commercial rights remained with us." That technology would form the foundation of Open Options.

It was the complexity of the Linux puzzle that drew Joel Cawley and IBM to Open Options. If the revolutionary concept of open-source software was to become entrenched in the tech industry, IBM saw opportunities. But if Linux was destined to get bogged down in industry wrangling, there was no sense in investing in it. Cawley had to accurately predict the future.

Some companies, such as IBM and Intel, wanted Linux to succeed, while others, such as Sun Microsystems, seemed vehemently against it. Then there was the possibility that Microsoft would sue to protect its code, which cast a shadow over Linux's future even as independent programmers beavered away at extending the code.

"There were a lot of different interests at stake," says Cawley. Sometimes, smart companies can rely on instinct or gut feelings to navigate complex situations, but, in Cawley's view, this was not one of those cases. The permutations were just too extensive; the potential for human error was too apparent. "To figure out where and how it's likely to evolve, you need to look at the interaction of a whole bunch of different players and their respective interests—and if those interests diverge or converge," he says. "It really does get unwieldy, so you can't trust simple intuition."

Open Options went to work. Since founding the company, Fraser had added a handful of staff, including Tom Mitchell, an engineering physicist who had worked at IBM and Imperial Oil. Open Options charges a flat fee of $85,000 and usually spends about three weeks collecting, compiling and crunching data. Over the course of a year and a half starting in late 2001, IBM and Open Options conducted five in-depth assessments on the Linux scenario.

The first step was to get executives from IBM into a room to start mapping out the game. For the math in game theory to work effectively, all players capable of influencing the game must be identified and their potential options listed and ranked. At this stage, clients are asked to draw on a wide range of personnel, since, in the case of IBM, its marketing people would likely have a different perspective on Microsoft than would its engineers. Once the group is assembled, they are asked to determine what objectives another company is likely to pursue. "Frequently they will say to us, 'Well, we don't know about those competitors.' And our answer is, 'Yes, you do,' " Mitchell says.

IBM's top people began devising preference lists: what they believed Sun, Red Hat, Microsoft and a variety of other companies wanted to see happen with Linux. Consumer preferences were also factored in. In all, there were about eight key players in the game, with 21 feasible options among them. While Sun was against Linux, Microsoft was seen as a swing player that could go either way. However, since Microsoft was openly concerned about Linux encroaching on Windows code, the possibility of its pursuing legal action still ranked high. Red Hat, a Linux developer, naturally wanted the system to make headway. IBM was watching from the sidelines but was positively disposed toward Linux.

Once all 21 options had been assigned, in descending order, to each player—creating what are known as preference trees—the data was fed into the software, which started computing what von Neumann would have had to diagram by hand in the 1920s. Game theory algorithms perform several jobs. A "compatibility" calculation cross-references the motivations of all players and their top objectives to determine who is a potential ally and who is a potential opponent. This eliminates several theoretical outcomes. An assessment of "arrogance" uncovers which players are likely to act on their own, regardless of what other players want. This, too, cancels out a number of possible choices.

A determination of "aggressiveness," meanwhile, reveals which players are likely to act and which are likely to sit back and wait for the others to make a move. But of all the calculations, the most important may be for "power." It establishes which actions rank highly as concerns across all players' charts, regardless of whether the player is in favour of that action. The player who can wield the most influence over the action is deemed to have the most power. Companies often make the mistake of thinking they can influence a particular scenario, without realizing they don't hold the hammer, Mitchell says. "Power doesn't flow inherently to an IBM or to a Microsoft because it is big. On any specific issue, power flows to the player that controls the actions that matter."

The power equation was crucial in the IBM project. Though a number of massive tech companies were involved, the end-users of Linux—the customers—ended up holding more power than all of the others. For example, even though Microsoft wielded the threat of legal action that could stymie Linux, that concern ranked lower on most of the preference charts than satisfying the desire of consumers to have it developed.

In the end, Open Options gave IBM two key pieces of advice. First, it told IBM to hold off on Linux, even though the company was ready to start developing products. Investment would be wise in good time. "Our recommendation to IBM was: Don't lead," Mitchell says. Since a key preference of several other players was to see an operating system created independently of the major companies, any effort by IBM to climb aboard too early in the process could disrupt its evolution.

It was a difficult notion for IBM to understand. "IBM at that time and throughout its history has been about the biggest company in the computer industry," Mitchell says. "It would set standards, create standards and be the standard in many segments. So that was what it was used to, that was their normal way of behaving."

Second, Open Options deduced that not only was the threat of legal action overblown, but that a major lawsuit could actually help Linux develop. When Utah-based SCO Group took action against IBM, alleging that some of its proprietary code had been used in IBM's initial work on Linux, the suit failed to dent Linux—and instead enshrined it among independent-minded techies. Soon, the other players in the game began to align behind Linux. When the other players saw the lawsuit was not having a major impact on the development of Linux, they lost their fear. "Once a threat has been acted on, it loses its power," says Mitchell, "because it's already been acted upon."

Today, Linux is a considerable thorn in mighty Microsoft's side, since the code helps run everything from CD players to Astronomy software and supercomputers. IBM has now invested a billion dollars in Linux-compatible products. Its web servers run on Linux; so do its biggest supercomputers.

It may seem odd that IBM—whose business is solving problems for other businesses—had to run to outside help with its own decision making. But Cawley says game theory makes a crucial contribution by speeding up the process. IBM is very busy; the company can't pull together teams of top executives for long stretches at a time to mull over scenarios and debate strategy.

"What you don't realize [in many decisions] is how much time gets spent on debate that at the end of the day doesn't end up mattering that much," Cawley says. "If you can narrow the debate down to the handful of things that really matter, you can dramatically speed up decision time.

"With the Open Options guys, we've gotten to the point where we can assemble a team and, in the course of three or four days, gather the data for the workshops, feed it in, and within a couple of weeks have good outputs and good answers and good insights."

Since the late '80s, mostly thanks to Nash's influence, most top business schools in North America have started teaching introductory game theory as a strategy tool. A second Nobel prize for game theory—awarded to Thomas Schelling and Robert Aumann in 2005—has further added to its allure.

Fraser and Mitchell, who often raise their voices excitedly when talking about the principles behind game theory—the way a sports fan might recount last Sunday's football game—are aware of the images conjured up by their line of work. "We were going to wear lab coats for you today," Mitchell jokes when we meet. I in turn note that John Nash battled schizophrenia—is that a job requirement for a game theorist? "More of a job hazard," Fraser quips.

Game theory does have its detractors, who dismiss it as common-sense decision making packaged in a shiny wrapper. Martin Kihn first encountered game theory as a business student at Columbia University in the late '90s. After graduating, he spent four years as a consultant with Booz Allen Hamilton in New York, advising pharmaceutical companies, sometimes using low-level game theory on pricing strategies. After leaving the business, he penned House of Lies: How Management Consultants Steal Your Watch and Then Tell You the Time. Game theory, he argues, works well on paper, but can be ineffective outside the lab. "People don't behave rationally, particularly in business. It's all about personalities, who gets along with who," he says. "Mergers and acquisitions is probably one area where irrationality rules."

Hal Varian, a game theorist and economics professor at Berkeley, agrees that the lack of human touch is a fundamental weakness of the equations. Game theory worked in the Cold War because the Soviets, for all of Khrushchev's bluster about burying the West, always ended up thinking rationally: The USSR didn't want to see itself wiped off the map. But in situations where players are willing to harm themselves in order to minimize the gains of other players—think of suicide bombers or of a bitter divorce settlement—game theory falls apart because the behaviour is erratic, Varian says. "It works reasonably well in business situations and military situations, but maybe less well in situations where there's a lot of emotions at stake."

California Institute of Technology professor R. Preston McAfee, a leading game theorist who helped the U.S. government design auctions for broadband spectrum, says doubters ought to remember that game theory is a tool, not an answer. "Game theory is sometimes criticized because it doesn't actually completely solve the problem," McAfee says. "On the other hand, the exercise of applying game theory very often clears up things that you can dispense with—issues that aren't salient to the decision process. Sometimes just thinking it through identifies strategies that you hadn't thought available."

Martin Kihn's critique notwithstanding, Mitchell says mapping out deals is a good fit with game theory—for outfoxing other bidders in auctions by determining their threshold; and for sorting out the pretenders from the contenders. So Open Options was happy to be asked by Toronto-based broadcaster Corus Entertainment Inc. in 2004 how media consolidation might play out in Canada. Corus CEO John Cassaday was convinced that the industry was heading for another flurry of mergers, and he didn't want Corus to end up as the only wallflower left on the dance floor once everyone else hitched up. (Cassaday's public announcement that he was using game theory was received by some analysts as if he'd said he was resorting to consulting a Ouija board.)

A collection of Corus executives began brainstorming the motivations of the other players in the industry. What does Global want most? And CTV? What about radio? Which assets fit together the best? And what are the motivations of the controlling shareholders?

That game is now playing out: Alliance Atlantis Communications Inc. put itself up for sale in December, setting the stage for the merger dance that Cassaday foresaw. While he wouldn't comment specifically on Alliance Atlantis at press time in early January, Cassaday was glad to have the Open Options intelligence in his pocket.

"What we used game theory to do was to force us into a discipline of prioritizing our options in a number of dimensions, including affordability and strategic fit," he said. "And we were able to do that on the basis of understanding how others might view these opportunities. We could articulate where we thought the most heightened competition would be for a particular asset, and where we thought we might be able to slide in."

The CIA was still mulling over Fraser's report on the fate of Nagorno-Karabakh when, at 11:41 a.m. on Dec. 10, 1988, an earthquake measuring 6.9 on the Richter scale devastated Armenia. Out of a population of a few million, at least 25,000 were killed and more than 500,000 were left homeless. The threat of Armenia's sparking a civil war in Azerbaijan evaporated. As the weeks turned to months, Fraser's report seemed academic.

Yet it wasn't. Soon after the collapse of the USSR in 1991, Azerbaijan fell into a bitter war with Armenia. A ceasefire was declared three years later, but the two have yet to resolve a long-standing feud over Nagorno-Karabakh.

The brief at hand may be CIA or M&A—but luckily for Open Options, the games never end.