快猫短视频

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鈥攂ut 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鈥檛 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鈥攚ho 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 鈥渂est鈥�,
isn鈥檛 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鈥攕omething that includes the 鈥渋rrational鈥�
influences of emotion鈥攚hich 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 鈥渮ero sum games鈥濃�攕imple 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 鈥淎 theory of choice鈥�).

But even back then it was obvious that most real-life problems aren鈥檛
remotely like zero sum games: what is bad for one 鈥減layer鈥� 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 鈥渃hicken out鈥� loses.

Obviously, if Jimbo swerves first, then Buzz wins鈥攁nd vice versa. But
if both swerve, then both players gain鈥攖hough 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鈥檚 a rule for finding optimal strategies for
these more complex games. Known as Nash鈥檚 theorem, it was discovered by RAND
mathematician John Nash, and ultimately won him a share of the 1994 Nobel Prize
for Economics. Nash鈥檚 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 鈥渆quilibrium鈥�, no player can
improve his or her prospects by choosing an alternative strategy. On the
strength of Nash鈥檚 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鈥檚 crises with pencil and paper quickly
discovered a problem: Nash鈥檚 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 (see Diagram). But which strategy is 鈥渂etter鈥�?

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鈥攁nd so a rational strategy is to be completely irrational.
Such 鈥減aradoxes of rationality鈥� dogged game theorists through the 1970s and
1980s. A huge effort was made to find rules for selecting the most 鈥渞ational鈥�
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鈥攁nd well aware of its limitations. 鈥淭he effects can be dire,鈥�
he says, recounting a story of two economists taking a taxi to their hotel in
Jerusalem.

All change

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鈥檛 work out too well, says Howard. 鈥淭he driver was so outraged at their
conduct that he locked the taxi doors, drove them back to where they鈥檇 started,
and dumped them on the street.鈥� 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鈥檛 like it. So he did something game
theorists don鈥檛 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.
鈥淪omeone pointed out that what we were really dealing with here weren鈥檛 just
games,鈥� recalls Howard. 鈥淭hey鈥檙e dramas, where the beliefs and values of the
characters evolve as the plot unfolds.鈥� 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 鈥渕etagame theory鈥�, 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鈥檛 either. But coming from a rational person, Jimbo鈥檚 threat is
hardly credible: no sane person would declare a determination to follow
hell-bent Buzz clear off the edge of the cliff.

There鈥檚 a way out of this 鈥渃redibility paradox鈥�, 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鈥檚 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. 鈥淭he basic idea is that paradoxes have an
emotional effect on the characters,鈥� he says. 鈥淎nd the reason these emotions
emerge鈥攍ike anger and fear, or affection and goodwill鈥攊s 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.鈥�

Howard and his colleagues have now identified a pile of paradoxes which can
lead to game-changing emotions. Chicken, for instance, involves an 鈥渋nducement
paradox鈥�, in which Jimbo must use an irrational threat to induce Buzz to swerve.
Others, including the famous riddle of the Prisoner鈥檚 Dilemma, involve a
鈥渃ooperation paradox鈥�.

Testing trust

In this, two criminals are arrested by the police, and they know that if they
both stay silent, they鈥檒l be held only for a short time鈥攕ay a month. The
police, however, have told each prisoner that they鈥檒l go free if they confess
and land the other in it for years. For each prisoner as an individual, Nash鈥檚
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鈥攁nd they鈥檒l 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鈥檒l act to save
his own skin.

鈥淭hat games can be changed is hardly a new idea,鈥� says Bennett, now at
Britain鈥檚 Department of Health in London. 鈥淲hat is new in drama theory is the
suggestion that emotion and preference change are frequently triggered in
predictable ways by these paradoxes.鈥�

Chicken solution

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
鈥済radients鈥� they create鈥攖hat is, the forces they produce鈥攃an change
a game from one form to another by altering players鈥� preferences. The resulting
theorems also show that games free from the paradoxes have convincing
solutions鈥攕upporting drama theory鈥檚 claim that 鈥減aradox resolution鈥� 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鈥檛
(see Diagram). 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鈥檚 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 鈥渞eal man鈥�,
Jimbo鈥檚 blind anger could lead him to resolve his personal Chicken paradox by
preferring death to dishonour鈥攁nd 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.
鈥淣ow we鈥檝e got a complete theory for breaking down games and showing how they
go,鈥� he says. The full details will appear later this year in Howard鈥檚 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.

鈥淚t鈥檚 pretty much dismissed by game theorists鈥攖o the extent that they
know about it, which most don鈥檛,鈥� says Steven Brams, a leading game theorist and
political scientist at New York University. 鈥淏ut then, most game theorists have
little interest in applications鈥攁part from maybe testing game theory
models in laboratory experiments. I鈥檓 personally sympathetic with the aims of
drama theory and its focus on promises, threats and the like.鈥�

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. 鈥淕ame theory has
been useful as a very simple way of looking at human behaviour in the
laboratory, and it鈥檚 told us a lot about rationality,鈥� he says. 鈥淚鈥檓 excited
about drama theory because it opens up the possibility of much
more鈥攍inking conflicts with their political situation, for example.鈥�

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鈥檚 Dilemma:
both sides would benefit collectively from a peace deal, but individually each
preferred to carry on the struggle rather than capitulate unilaterally. 鈥淚n a
situation like this, where individually `rational鈥� courses of action lead to a
collectively irrational outcome, the only escape lies in changing the game,鈥�
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鈥檚 leader Gerry Adams effectively put his position on
the line over the cease-fire, says Miall. 鈥淭hat showed he was emotionally
committed to it鈥攁nd helped make his claims credible with the Unionists.鈥�
Miall thinks that drama theory may allow other conflicts to be analysed to
reveal which peace strategies work鈥攁nd which don鈥檛. 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. 鈥淭he Good
Friday agreement was reached essentially by breaking it down into lots of small
steps,鈥� says Miall. 鈥淭hat was important for building up trust.鈥�

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: 鈥淎ny claim to have the one, true,
complete theory should be suspect,鈥� says Howard. But with conflicts growing ever
more complex, all Howard and his colleagues are saying is鈥攇ive 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鈥檚 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 鈥渮ero sum鈥� game, the ratings are captured in a 鈥減ayoff matrix鈥� 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 (see diagram).
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 鈥渕ixed strategy鈥�, 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鈥檚 score given first, followed by that
for Buzz (see diagram).FIG-mg21555302.JPG

Clearly, swerving can be good for both鈥攁nd driving on disastrous.
Sorting out the most 鈥渟ensible鈥� course of action in cases like Chicken is far
less obvious鈥攁nd that is where drama theory comes in.