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The name of the game
CONTRARY to popular belief, there is no Nobel prize for mathematics. The reason apparently is that mathematical theories cannot be tested, as they are essentially tautological in nature. For that matter, can you imagine Begin’s peace theories being tested? And, how would one even begin to formulate criterion for good literature?
However, it is not as if mathematicians do not get a look in. For instance, the mathematician John Nash — about whom the film A Beautiful Mind was made —received a Nobel Prize for Economics for his work in the theory of games. I am not joking: a mathematician actually got the Nobel Prize for Economics for his work in game theory.
How can there be a serious theory of games and why should it get a Nobel prize? Let me try to explain.
When one thinks of games it is usually between two people, or teams, one of whom wins and the other loses. It may also be tied, though.
Even if there are several players, the number of wins and losses balances out. The same applies to games involving more players, like poker. Such games are called “zero-sum games” because the sum of the pay-off for the various players is zero.
It is not necessary, however, that all games be of the zero-sum type. Consider the game of “Passing the parcel.” The people who are left holding the parcel when the music stops suffer penalties while the person who is never caught gets a prize.
Again, consider Olympics. Nobody loses but three people “win” in each game. The wins and losses do not balance out. Such games are “non-zero sum games.”
There is yet another way of looking at games, which has relevance for us. The games can be competitive, as we have been considering, but they could also be cooperative.
Consider mountaineering as a game. Mountaineers may be competing among themselves to be the first to the top but they may also be cooperating with each other to help scale the difficult parts. This game is partly competitive and partly cooperative.
A non-zero sum and non-cooperative game, known as “The Prisoners’ dilemma,” may be constructed as follows. Two thieves are picked up by the police, but there is no evidence against them. To get a conviction, the police need a confession.
The two are locked up in separate cells, so they cannot confer. Then each is offered the option of becoming the state’s witness, thereby becoming eligible for a lighter sentence as compared to the other person.
“However,” they are both told, “if your partner in crime confesses first he will get the lighter sentence and you will get the full sentence.” Each thief knows that if neither confesses, neither gets convicted, but as each fears that the other may confess before he does, each is likely to confess sooner than later to beat the other to it.
A cooperative game is when two people are trapped in a room on fire, of which the door needs to be broken down. Neither can break down the door alone but by cooperating they can both escape. In such a case, they will cooperate.
Where then, in all of this, is mathematics and where is the Nobel prize? Suppose Aliya and Ali are playing a zero-sum game in which there are two strategies for each player. Both want to get the maximum pay-off, regardless of what the other gets.
Assume that both know the pay-off for each strategy available to a player. Suppose that for the first strategy of both, Aliya gets 4; for her first and Ali’s second strategy she gets 1; for her second and Ali’s first she loses 1; and for the second strategy of both they break even.
She will look for the strategy that gives the maximum pay-off to her when Ali chooses the best strategy for himself. So she chooses the minimum of her maximum pay-off for all the strategies: her first strategy, knowing that Ali will choose his second.
In this case, it does not matter whether she chooses the strategy first or he does. The answer comes out the same. However, there is no guarantee that it will not matter in general.
Consider the possibility that the pay-offs in the above pairs of strategies are a gain of 6, a loss of 4, a loss of 2 and a gain of 5, respectively. In this case, if she chooses the strategy first she loses 2 by choosing the first strategy, but if he chooses the strategy he will choose his least loss of 5 by taking the second strategy and she will gain 5.
This is best seen by putting the strategies in a table with her strategy choices along rows and his choices along columns. If the row minimum and the column maximum are the same that pair of strategies will be chosen that gives that particular pay-off. Instead, if the row minimum and the column maximum are not same there will be no final choice of strategies possible.
If we can put in probabilities for the other player to choose each strategy, we can arrive at a definite answer. This is where mathematics comes in. One needs a similar (but more complicated) analysis for non-zero sum games.
The calculations involve a lot of probability theory and linear algebra as well as the optimization theory. If the choices involved vary continuously, like setting some dials or turning joysticks, the games are called “differential games” and differential calculus has to be used.
In the above example, since it is a zero-sum game it is competitive and cannot be cooperative. Real interest comes when one considers non-zero sum games that are cooperative in that the players negotiate in advance. In general, the same analysis gave that these games are non-deterministic, by which is meant that there is no answer obtainable about what strategy will be adopted.
It was taken for granted that when this theory is applied to economic activity the conclusion would be that only competitive games can occur. This should have been seen as a suspect conclusion because societies evolved through some cooperation in economic activity and not just because of competition.
As Nash realized, it should be possible to model partly cooperative and partly competitive economic behaviour. He found that this could be done by providing a ‘threat’ value, which is the penalty that is paid for reneging on the agreement for cooperation between “players.” This sets a “lower limit” for what each player can expect from the other.
Cooperation is made stable by this minimum level. That is, people may get much greater benefits than the minimum when they cooperate.
What has happened is that people are now able to trust each other more because they know that the other will not go back on the agreement. The benefits of cooperation can be attained because the supposed “inevitable human greed” of the “players” has been curbed by the threat of penalty for non-cooperation.
We now come to the reason why John Nash was awarded a Nobel prize. Since the time of Adam Smith — who formulated the dictum which said “a reasonable man will maximize his profits” — it had been thought that “one would sell one’s own grandmother for profit.”
This belief had distorted the development of societies. After Nash’s theories, one arrived at the view that reasonable people do not pursue profits at the expense of all else.
People do not have to be philanthropists to cooperate in economic activity. Enlightened self-interest is enough. The Nobel prize was conferred on him to show, for the first time in the framework of standard economic theory, that cooperative economic behaviour is reasonable.
The writer, an HEC-distinguished national professor, is director of the Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Rawalpindi. He can be contacted at
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