# What exactly is 'game theory'?

The Greek Finance Minister Yanis Varoufakis - currently negotiating over the fate of his country's debt - is a student of "game theory". But what is it, asks Chris Stokel-Walker.

"Some commentators rushed to presume that as Greece's new finance minister I was busily devising bluffs, stratagems and outside options, struggling to improve upon a weak hand," wrote Varoufakis in the New York Times this week. "If anything, my game-theory background convinced me that it would be pure folly to think of the current deliberations between Greece and our partners as a bargaining game to be won or lost via bluffs and tactical subterfuge."

Game theory can be described as the mathematical study of decision-making, of conflict and strategy in social situations.

It helps explain how we interact in key decision-making processes.

Imagine you're buying a car from a dealership.

The dealer wants to sell the car, and has a fixed price beyond which he cannot drop without making a loss. You want to buy the car, but get the best price. The car dealer bases his negotiating tactics on the fact that you want the car, that he could sell the car to another customer, that he'll win praise for selling the car, and the break-even point beyond which he cannot negotiate down.

You base your negotiating on the fact that the dealer wants to sell the car, that there is a profit margin into which you can haggle to bring down the car's price, and are mindful that the dealer knows your presence in his office means you want to buy a car.

Every time we interact with another human being, whether it's eyeing up the last muffin at the coffee shop counter, or doing a favour for a work colleague that we hope will be repaid in kind, we're using logic that could be described using the rules of game theory.

These "games" are vital even to animals, says Antonio Cabrales, a professor of economics at University College London.

"I take an action, you take an action," explains Cabrales. "Something happens. The something that happens is going to depend on what both you and I do."

The game is a kind of mathematical model to understand decision making and the interaction between decision makers. The best known game is the Prisoner's Dilemma.

Two people are arrested, imprisoned and given a date for a trial.

The prosecutor of the case approaches each prisoner in turn and presents them with an offer - if you confess against your partner in crime, all charges will be dropped against you and used as evidence to convict the other. He would get 20 years.

If you stay silent and your partner confesses, you'll be convicted and get 20 years and he'll be freed. If both of you confess, you'll be convicted but only get five years in prison. If you both stay silent, you'll both be convicted and get one year in prison.

The prisoner's dilemma is that each prisoner's fate relies on the other's actions. Individually, confessing would be the better option - but if both confess, the punishment is worse than it would be if they both held their tongue.

"These people make choices, and these choices then have an impact on other people," says Paul Schweinzer, a senior lecturer at the University of York's department of economics. "Game theory is when I take these impacts my choices have on other people into account when making my decision."

The "game" is the interaction between two or more parties, and relies on people acting rationally, knowing the boundaries of the "game", and knowing that the other party is equally cognisant of the rules. These strategic interactions form the crux of game theory. "Sometimes we use it knowingly, sometimes we do it intuitively," notes Cabrales.

Even if people - and certainly animals - do not reason consciously about what strategies to take, other forces, such as evolution, or learning from past mistakes, often make them behave in the same way as if they were cool rational game players.

The theory has entered popular culture, perhaps most notably through Russell Crowe's portrayal of Nobel laureate John Nash in the film A Beautiful Mind.

A pioneer of the theory, John D Williams, wrote a 1954 study - The Compleat Strategyst - which attempted to bring it to the masses. He said society "would benefit from having more persons informed regarding its nature; and that the knowledge would benefit the persons".

Now it's used by many different people across a broad spectrum of interests. "The major reason for the success was that in a variety of settings people began to realise they had to think formally and systematically about strategic interactions," explains Rakesh Vohra, an economics professor at the University of Pennsylvania and senior member of the Game Theory Society. "Game theory revolutionised the study of economics."

Auctions for infrastructure projects or Premier League TV rights deploy game theory. Dating applications and services rely on it. Companies selling consumer goods use game theory to predict how their competitors - and customers - would react in a price war.

Away from the decisions made by retailers, commercial negotiators and so on, quiz shows often use game theory. The Weakest Link is a notable example, where partnerships are formed and strategic decisions are made about who to cut from the team.

Some of the first codified uses of game theory were in war. Both the British and American military used early computers to run models that would utilise game theory to help commanders decide whether, where and when to attack the enemy.

Since then, the concept has evolved. As Vohra explains, "when game theory was first born, there was a group of people who thought that if we build a model large and complex enough and we crank that handle we will know what to do."

That was seen as too ambitious, and the theory has changed. "Instead what we're trying to do is inform judgement," says Vohra. "We cannot tell you what to do, except in very limited circumstances. But what we can do is tell you the important things you have to make a judgement about. In a complex world where there are many things you have to pay attention to, this is still enormously useful in terms of focusing your attention."

The concept is not solely based in conflict and combat, though - it can also help co-operation. "In a zero-sum game, you can think about chess, where one person's win means immediately another player loses," argues Schweinzer. "There are other games like joint production. If the two of us write something together, we can both gain from that. There's no winner and no loser, but the act of playing together generates something we can both benefit from - a win-win game."

Ethicist Carissa Veliz, of the Oxford Uehiro Centre for Practical Ethics, agrees. "While game theory may appear to be essentially about situations involving opposed self-interests, it needn't be.

"Selfishness need not be among the assumptions of game theory. In the classical Prisoner's Dilemma, it is often assumed that each prisoner only values his or her own well-being. But we can imagine the case where both prisoners are genuinely disinterested activists, and each wants to leave prison as soon as possible because they sincerely believe they will do the most good in the world by furthering the respective cause they are committed to."

The negotiations currently taking place between Greece and its creditors are a prime place to deploy game theory - not least because of Varoufakis's past as a theorist in the field. "If you think about the renegotiation of the loans to Greece, this is a very good example of where you have a tension between competition and co-operation," says Vohra.

Not everybody agrees. Sean Hargreaves Heap, a professor of political economy at King's College London, who co-authored a critical introduction to game theory in the 1990s, believes that game theory is of little use to Greece's financial negotiations.

"If you think the talks are like a game of chicken, it's a case of who's going to blink first - the Greek government or the Germans? That's useful, but game theory merely tells you there are three different things you should expect in such a game of chicken," he says.

"One is the Germans blink, the other is the Greeks blink, the third is that both sides toss a coin as to whether they blink or not. Game theory is a useful way of characterising the problem, but in terms of telling you what someone is going to do in a game of chicken, it's completely hopeless."

Hargreaves Heap wrote his book on game theory with a young academic born in Athens and educated in Essex and Birmingham. His co-author's name? Yanis Varoufakis.

## Thomas Schelling

• One of the first to apply game theory to international relations in his book The Strategy of Conflict which analysed the nuclear arms race
• Argued the capability to retaliate was more useful than the ability to resist an attack and that uncertain retaliation was more credible than certain retaliation
• Both the US and the former Soviet Union adopted such a strategy - known as mutually assured destruction - during the Cold War

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