THE intelligence report is clear: an informer has tipped you off that a terrorist organisation is planning an imminent attack on a key military base this week. What should you do? Clearly, ignoring the tip-off could be disastrous - but the base is very well defended, making any attack pretty suicidal. Taking action is not without its problems either: it will divert resources from other vulnerable targets, and could alert the terrorists to the informer in their midst.
This is a classic problem for what mathematicians call Game Theory, a bag of tricks for finding the best strategy when faced with conflicting possibilities. In essence, you just give a score to the consequences of the various actions you might take, and then choose the course of action that gives the best score in the worst possible situation.
All this supposes, of course, that everyone is acting rationally. But what if they are not? What if the terrorists don't care about their own survival, and really are mounting a suicide mission? And what if some of your colleagues had earlier been murdered by these same terrorists - who are now giving you a golden opportunity to get even?
On the face of it, any theory that attempts to sum up something as complex as an emotionally charged conflict with a handful of numbers is asking for trouble. Certainly game theory, in which rationality decides which course is "best", isn't well-placed to deal with this kind of situation. But now a small group of mathematicians think they have found a way to boost the power of game theory to tackle such problems. They believe that classic game theory addresses just part of something much bigger - something that includes the "irrational" influences of emotion - which they call drama theory.
Attempts to make game theory applicable to real life date back to the early 1950s, when mathematicians at the RAND Corporation in California used it to advise the US Air Force on Cold War strategy. Back then, most of the advances were in "zero sum games" - simple two-player situations where what is good for one is bad for the other. In such cases, game theory recommends choosing the highest scoring tactic in the worst situation (see "A theory of choice").
But even back then it was obvious that most real-life problems aren't remotely like zero sum games: what is bad for one "player" can often be equally bad for the other. A classic case is the game of Chicken, immortalised in the 1955 film Rebel Without A Cause, starring James Dean. Two high-school kids, Jimbo and Buzz, are to race their cars toward the edge of a cliff, and the first one to "chicken out" loses.
Obviously, if Jimbo swerves first, then Buzz wins - and vice versa. But if both swerve, then both players gain - though not as much as if their opponent had chickened out. And, of course, if neither swerves then both lose big time.
As with zero-sum games, there's a rule for finding optimal strategies for these more complex games. Known as Nash's theorem, it was discovered by RAND mathematician John Nash, and ultimately won him a share of the 1994 Nobel Prize for Economics. Nash's theorem says that it is always possible for a player to choose a strategy that is best for him or her when all the other players are also following their best strategies. In this "equilibrium", no player can improve his or her prospects by choosing an alternative strategy. On the strength of Nash's insights, it seemed for a while that game theory might be able to solve realistic problems. This was an exciting prospect: swap Jimbo and Buzz for America and the Soviet Union, and you could see parallels with the Cuban missile crisis of 1962, or with the Cold War more generally.
But those hoping to solve the world's crises with pencil and paper quickly discovered a problem: Nash's rule shows that there is no single state of equilibrium for a game like Chicken. There are two: you can decide to swerve, while the other person plans to keep driving, or vice versa. In either case, neither you nor your opponent can improve your score by unilaterally changing your mind. But which strategy is "better"?
Game theorists started to look for answers, but just uncovered more problems. For example, only truly irrational players can credibly threaten to drive on no matter what - and so a rational strategy is to be completely irrational. Such "paradoxes of rationality" dogged game theorists through the 1970s and 1980s. A huge effort was made to find rules for selecting the most "rational" strategy in every game; none really worked.
As despondency began to descend, many game theorists believed that the problems of defining rationality in games such as Chicken were just distractions not worth the candle. To others, however, this smacked of sweeping difficulties under the carpet. That was the view of the group of British game theorists who met at Sheffield Hallam University in November 1991.
The meeting was called by Nigel Howard, a veteran game theorist who had advised the US government in the Strategic Arms Limitation Talks during the 1960s. Howard was well used to applying game theory in real-life situations - and well aware of its limitations. "The effects can be dire," he says, recounting a story of two economists taking a taxi to their hotel in Jerusalem.
Worried that they were going to be overcharged, they decided not to haggle about the price until they reached the hotel, when their bargaining position would be much stronger. But their entirely rational, game-theoretic strategy didn't work out too well, says Howard. "The driver was so outraged at their conduct that he locked the taxi doors, drove them back to where they'd started, and dumped them on the street." What had gone wrong? It seemed that while the taxi driver probably knew nothing of game theory, he knew when people were playing games with him, and he didn't like it. So he did something game theorists don't like: he got angry, acted against even his own preference for getting paid, and changed the game.
It was this idea of emotions playing a key role by triggering irrational responses that Howard and fellow UK-based theorists Peter Bennett, Morris Bradley and Jim Bryant kicked around at that meeting at Sheffield Hallam. "Someone pointed out that what we were really dealing with here weren't just games," recalls Howard. "They're dramas, where the beliefs and values of the characters evolve as the plot unfolds." By the end of the meeting, drama theory had been born. At its heart is the idea that games are not static, one-shot deals decided by rationality, but dynamic situations that can be utterly transformed by the emotions of the players.
But to make drama theory more than a buzz word, Howard and his colleagues had to find some way of capturing their ideas more precisely. As it happens, some of the basic ideas had been floating around in classic game theory for years. During the 1960s, Howard himself had developed "metagame theory", which focused on the role of paradoxes in determining the outcomes of games. In the game of Chicken, for example, it seems pretty rational for Jimbo to want to win. Yet to do this, he must convince Buzz that he will not swerve, no matter how much Buzz insists he won't either. But coming from a rational person, Jimbo's threat is hardly credible: no sane person would declare a determination to follow hell-bent Buzz clear off the edge of the cliff.
There's a way out of this "credibility paradox", however: Jimbo should stop acting rationally, and instead behave as if he is crazy before he goes anywhere near his car. Suddenly, his threat to keep on driving becomes all too credible. So, far from being a mere vexation, says Howard, the game of Chicken points up the failings of game theory's insistence on rationality as a guide to how to behave. Irrational behaviour sometimes pays.
The ways in which people react to such credibility paradoxes is at the very heart of drama theory, says Howard. "The basic idea is that paradoxes have an emotional effect on the characters," he says. "And the reason these emotions emerge - like anger and fear, or affection and goodwill - is that they have a drama-theoretic role. They shake the characters out of old ways of thinking, allowing them to see a new way forward."
Howard and his colleagues have now identified a pile of paradoxes which can lead to game-changing emotions. Chicken, for instance, involves an "inducement paradox", in which Jimbo must use an irrational threat to induce Buzz to swerve. Others, including the famous riddle of the Prisoner's Dilemma, involve a "cooperation paradox".
In this, two criminals are arrested by the police, and they know that if they both stay silent, they'll be held only for a short time - say a month. The police, however, have told each prisoner that they'll go free if they confess and land the other in it for years. For each prisoner as an individual, Nash's theorem gives a unique, rational solution: accept the police offer, and start talking. But for the pair as a team, both spending a month in prison is preferable to one being locked away for years. But the only way of achieving this is for both prisoners to put their trust in each other and stay silent.
So once again arguments based on rationality support two entirely different courses of action. And this creates a cooperation paradox: each must convince the other that they will act as a team despite the fact that each could do better for themselves by squealing.
According to drama theory, what actually transpires will depend both on emotions and events that took place before the prisoners ever found themselves in their predicament. For long-standing partners in crime like Butch Cassidy and the Sundance Kid, emotional bonds will come to the fore when they face the cooperation paradox. Squealing will become unthinkable - and they'll both get off. But if one of the prisoners has always been an unwilling accomplice, the cooperation paradox will trigger anger and distrust and he'll act to save his own skin.
"That games can be changed is hardly a new idea," says Bennett, now at Britain's Department of Health in London. "What is new in drama theory is the suggestion that emotion and preference change are frequently triggered in predictable ways by these paradoxes."
All this talk of emotion and preferences may seem decidedly touchy-feely, and far from the apparently solid, quantitative results of conventional game theory. But in a paper published earlier this year in the Journal of the Operational Research Society (vol 49, p 144), Howard showed that the idea of credibility paradoxes gives a firm mathematical basis for drama theory. Using set theory, he showed that it is possible to capture precisely the paradoxes and that the "gradients" they create - that is, the forces they produce - can change a game from one form to another by altering players' preferences. The resulting theorems also show that games free from the paradoxes have convincing solutions - supporting drama theory's claim that "paradox resolution" holds the key to solving games like Chicken.
For example, the theorems show how emotions can lead Jimbo to decide quite rationally that he will swerve, or won't. One possibility is that Jimbo and Buzz, responding to their own fears as the day of the contest approaches, might slowly come to respect one another's courage, and perhaps even to like one another. This emotional bond could make Jimbo come to value a life-preserving collective swerve even over his own potential victory. On the other hand, if Buzz goads Jimbo relentlessly about not being a "real man", Jimbo's blind anger could lead him to resolve his personal Chicken paradox by preferring death to dishonour - and not swerving. Understanding such actions and reactions opens the door not only to analysing potential conflicts, but also to manipulating them.
Howard has since extended his mathematics to show just how one game is transformed into another, and has linked those changes to the various paradoxes. "Now we've got a complete theory for breaking down games and showing how they go," he says. The full details will appear later this year in Howard's first book devoted to drama theory. Its publication should help dispel the suspicion of more mainstream game theorists that drama theory is too vague to be of any real use.
"It's pretty much dismissed by game theorists - to the extent that they know about it, which most don't," says Steven Brams, a leading game theorist and political scientist at New York University. "But then, most game theorists have little interest in applications - apart from maybe testing game theory models in laboratory experiments. I'm personally sympathetic with the aims of drama theory and its focus on promises, threats and the like."
Whatever its reception among game theorists, drama theory is already being taken up by researchers trying to get to grips with complex real-life conflicts. At Lancaster University, conflict resolution analyst Hugh Miall is using drama theory to study the events now unfolding in Northern Ireland. "Game theory has been useful as a very simple way of looking at human behaviour in the laboratory, and it's told us a lot about rationality," he says. "I'm excited about drama theory because it opens up the possibility of much more - linking conflicts with their political situation, for example."
The years of deadlock in Northern Ireland were, says Miall, the result of Sinn Fein and the Unionists being trapped in a game like the Prisoner's Dilemma: both sides would benefit collectively from a peace deal, but individually each preferred to carry on the struggle rather than capitulate unilaterally. "In a situation like this, where individually `rational' courses of action lead to a collectively irrational outcome, the only escape lies in changing the game," says Miall. According to drama theory, that called for the injection of some outside, emotional factor capable of building up trust.
It came when Sinn Fein's leader Gerry Adams effectively put his position on the line over the cease-fire, says Miall. "That showed he was emotionally committed to it - and helped make his claims credible with the Unionists." Miall thinks that drama theory may allow other conflicts to be analysed to reveal which peace strategies work - and which don't. For example, on Good Friday this year, seven political parties, including the Ulster Unionist Party and Sinn Fein, signed an agreement on the future of Northern Ireland. "The Good Friday agreement was reached essentially by breaking it down into lots of small steps," says Miall. "That was important for building up trust."
It is too early to say if drama theory will cast new light on the age-old problem of resolving disputes and conflict. Certainly the drama theorists are not claiming to have found the Final Answer: "Any claim to have the one, true, complete theory should be suspect," says Howard. But with conflicts growing ever more complex, all Howard and his colleagues are saying is - give drama theory a chance.
GAME THEORY is based on mathematics, but its implications are intuitive: how you respond to a situation depends on how you rate the various options. In the case of the intelligence report, you can either act on the informer's tip-off, or not, and the terrorists can carry out their attack, or not. Each of the four possible combinations has its own rating.
In a simple "zero sum" game, the ratings are captured in a "payoff matrix" in which what is good for you (a high rating) is equally bad for the other party. For example, you may think that acting on the tip-off will at least give your team some useful practice, and that the payoff matrix is something like that shown. In this matrix, the two highest scores lie in the upper row. So by acting, you will be guaranteed a higher payoff than by not acting. Of course, if the terrorists act rationally too, they will foresee your choice and will not attack, giving you only 1 rather than 5.
But if you are worried that acting may jeopardise your informant, you may reassess the payoffs. The decision might still be simple. But if the two highest values end up in opposing corners, the more sophisticated analysis of game theory will tell you to adopt a "mixed strategy", and to act with a frequency calculated from the matrix.
Things become more complicated in non-zero sum games, in which what is good for one player can also be good for the other. In Chicken, for example, the payoff matrix might be written with Jimbo's score given first, followed by that for Buzz.
Clearly, swerving can be good for both - and driving on disastrous. Sorting out the most "sensible" course of action in cases like Chicken is far less obvious - and that is where drama theory comes in.