A scientific formula for playing games in the dark

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As John and Linda try to rendezvous at a convenient restaurant, a little game theory might suggest some strategies.

Two star-crossed lovers – let us call them Linda and John – have a date this evening. But, once again, John has let the battery on his mobile phone go flat. They cannot talk to each other. What should they do?

John Nash, whose biography has just been published, is an American mathematician who devised a framework for thinking about such problems – the theory of non-co-operative games. Two or more agents make independent decisions and the outcome depends, for better or worse, on the interaction between them. That is the situation faced by Linda and John. Everyone engaged in business, politics, or everyday life plays non-co-operative games again and again.

John sits there thinking “what will Linda do”? And what she will do will depend, in turn, on what she thinks John will do. But what John will do depends, in turn, on what John thinks Linda thinks that John will do. And so on, for ever.

But suppose Linda and John share a favourite restaurant – the Elizabeth in Oxford, perhaps. Then Linda might think: suppose John were to be at the Elizabeth, what would be the best course of action for me? To go to the Elizabeth, of course. And the same is true for Linda. So even without a chat on the mobile phone, it would seem to make good sense for each to turn up at the Elizabeth at eight o’clock.

That happy event is a Nash equilibrium. It cuts through endless rounds of speculation about the motives of others. Is there an outcome in which everyone is doing what is best for them, given the choices which have been made by everyone else? That is the definition of a Nash equilibrium. If the concept seems rather obvious, but demonstrates that Nash’s idea has one of the hallmarks of real scientific originality. It is obvious, once it has been pointed out to you. It wasn’t before.

Still, there are problems to which the Nash solution is less trite than the recommendation to go to the Elizabeth at eight. Take the familiar business problem of how much capacity to build. The price you realise for your output depends on the total amount of capacity which everyone installs. This has a precise Nash equilibrium solution. The total amount of capacity which will be built increases with the number of firms which enter – but at a diminishing rate – and the amount of excess capacity will rise with the ratio of fixed to variable costs. We can write out the mathematics of it. And understand better why industries with higher fixed costs are prone to cyclical instability, why they experience periods of acute price competition, and tend to be subject to cartels and to regular phases of rationalisation and consolidation.

But a Nash equilibrium is a funny kind of solution. It need not be the outcome. Linda might get fed up and go home. John might have misunderstood what Linda said about the Elizabeth. And Linda and John could have found each other in many different Nash equilibria. If they want to be together sufficiently then meeting in Joe’s greasy spoon café is another solution.

If there are many Nash equilibria, and there are often are, then better ones should be preferred to worse. Still, it can happen the other way round. The QWERTY layout for keyboards is grossly inefficient. It was invented to slow users down in the days when typewriter keys were liable to jam. Other systems, like the Dvorak layout, are simpler and quicker to use. But if everyone else is using QWERTY, if most machines are made in QWERTY, the sensible thing to do is to learn and use QWERTY. And that is true for everyone else. Which is why we are stuck, for ever, with an inefficient Nash equilibrium. It is a demonstration of the power of the concept. Having reached a Nash equilibrium, we cannot leave it, even for a better one.

More confusing still are the cases where several Nash equilibria are equally good. Suppose John likes the Elizabeth but Linda prefers the Petit Blanc. If each follows their own preference, each will dine alone. But if Linda goes to the Elizabeth, she takes the risk that John, with equivalent altruism, will be at Le Petit Blanc – leaving them with the worse of all outcomes, in which they dine separately at their least preferred restaurants. The game is called the Battle of the Sexes by mathematicians, because it displays familiar features of personal relationships – lack of communication, confusion, and scope for the best of motives to be misunderstood.

But the Battle of the Sexes is actually the universal problem of co-ordination in business. It does not matter much what the outcome is, only that we should all pursue the same actions. The general answer to this issue is hierarchy. If John’s wishes are paramount, the outcome is likely to be better – not just for John, but for Linda. It does not really matter which side of the road we drive on, but it matters a great deal that everyone should drive on the same side of the road, and that is why it is necessary that someone should tell us what to do and that we should obey them.

Not an idea that goes down well in Oxford, or at Princeton, where Nash spent his career. That career – characterised by phases of outstanding originality broken by long bouts of paranoid schizophrenia – is a telling reminder that the originality and sensitivity associated with great insight may actually be disabling, not helpful, in everyday life. But we should not conclude that we cannot learn from these insights. Just as we need not admire Mozart’s character to enjoy his music, or applaud Clinton’s habits to benefit from his economic policies, we need not envy Nash’s life to benefit from his ideas.

*Sylvia Nasar A Beautiful Mind Faber & Faber

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