What is game theory? The math of strategic choices

Game theory is the study of strategic decisions: choices where what you get depends not only on what you do, but on what others do, and where everyone knows it. A "game" is any such situation, from two shops setting prices to countries arming or bidders in an auction. Game theory writes down the players, their options and their payoffs, then asks what each player would sensibly choose, given that the others are choosing too.

In short

• A game has players, the strategies open to each, and payoffs for every combination of choices, usually drawn as a table.
• A Nash equilibrium is a set of choices where no player could do better by changing only their own. It predicts where play settles, not that the result is good.
• Games differ: in a zero-sum game one side's gain is the other's loss; in most real games both can win, or both can lose.

What counts as a game?

A game in this sense has three parts. The players are the people, firms, countries or even animals making choices. The strategies are the moves open to each. The payoffs say how much each player values every possible combination of moves, whether in money, votes, years in prison or offspring. The Stanford Encyclopedia of Philosophy defines the subject as "the study of the ways in which interacting choices of economic agents produce outcomes", outcomes that none of the players may have intended.

The key word is interacting. Choosing an umbrella is not a game: the weather doesn't care what you do. Choosing a price when a rival across the street is choosing one too is a game, because your best price depends on theirs, and theirs on yours. Game theory assumes, as a starting point, that players are rational, meaning they pick what serves them best given what they expect the others to do, and that each knows the others are doing the same.

How do you find the answer to a game?

Take two cafés on the same street, Big Bean and Little Cup, each choosing a price for its coffee: high or low. Big Bean has more seats and loyal customers; Little Cup lives on passers-by. Their daily profits for each combination are in the table, Big Bean's first:

	Little Cup charges high	Little Cup charges low
Big Bean charges high	$600, $300	$380, $400
Big Bean charges low	$550, $200	$400, $250

Work through it as each café would.

1. Little Cup's view. If Big Bean charges high, Little Cup earns $300 by charging high and $400 by charging low: low is better. If Big Bean charges low, Little Cup earns $200 high or $250 low: low is better again. Charging low is Little Cup's best move whatever Big Bean does, which game theorists call a dominant strategy.
2. Big Bean's view. Big Bean has no dominant strategy. If Little Cup charged high, Big Bean would do best charging high, $600 against $550. If Little Cup charges low, Big Bean does best charging low, $400 against $380. So Big Bean has to predict its rival.
3. Putting them together. Big Bean can see from step 1 that Little Cup will charge low. Its best reply to that is to charge low too. The game settles at low, low: $400 for Big Bean and $250 for Little Cup.
4. The check. At low, low, could either café do better by changing only its own price? Big Bean switching to high would get $380, less than $400. Little Cup switching would get $200, less than $250. Neither wants to move alone.

That last check is the definition of a Nash equilibrium: a set of choices in which no player can do better by changing only their own choice, given what the others are doing. The mathematician John Nash showed in 1950 that every game with a finite number of players and strategies has at least one, once players are allowed to choose at random.

Notice what the equilibrium is not. Both cafés would earn more at high, high, $600 and $300, than at the equilibrium. Getting there would need an agreement that each would be tempted to break, and in many countries price-fixing agreements are illegal. When each player's sensible choice leaves everyone worse off, the game has the shape of the prisoner's dilemma, the most studied game of all.

Can a game have more than one answer, or none?

Yes to both. Which side of the road to drive on is a game with two equilibria: everyone on the left, or everyone on the right. Either is stable, since nobody gains by switching alone, and which one a country ends up with is a matter of history, not logic. Games like this, called coordination games, explain why standards and conventions stick once they're set.

Rock-paper-scissors has no equilibrium in fixed choices: whatever you always play, your opponent can beat. Its equilibrium is to play each of the three at random, a third of the time each, so there is nothing to exploit. That is a mixed strategy, the reason penalty takers in soccer vary which side they kick to, and the reason probability runs through game theory.

What's the difference between zero-sum and non-zero-sum games?

	Zero-sum game	Non-zero-sum game
What it means	The payoffs always add to the same total, so one side's gain is the other's loss	The total can grow or shrink with the choices made
Examples	Chess, poker between two players, rock-paper-scissors	Trade, price competition, arms races, climate agreements
Is cooperation useful?	No: there is nothing to share	Often: both can gain, or both can lose
What theory says	John von Neumann's 1928 minimax theorem: each side can guarantee itself a best worst case	Nash equilibrium, which may leave everyone worse off than they could be

The most common mistake about game theory is to treat every contest as zero-sum. Most of economic life is not: a trade happens because both sides expect to gain, and a monopoly or a price war shows how the total itself can change with the players' choices.

Where is game theory used?

• Auctions. In July 1994 the US Federal Communications Commission sold radio frequencies for mobile phones with a new format, the simultaneous multiple round auction, designed by the economists Paul Milgrom and Robert Wilson with Preston McAfee. The first sale took 47 rounds of bidding for 10 licenses and raised $617 million for spectrum the government had previously handed out practically for free. By 2014 the FCC's auctions in this format had raised more than $120 billion, according to the Nobel committee's account, which gave Milgrom and Wilson the 2020 prize.
• Deterrence. In the nuclear arms race of the Cold War, the economist Thomas Schelling argued that the capability to retaliate can be more useful than the ability to resist an attack, and that a side can strengthen its position by visibly limiting its own options, as the 2005 prize announcement put it.
• Shared resources. Overfishing and overgrazing are many-player games over a scarce resource, the tragedy of the commons, and so are climate treaties.
• Evolution. In the 1970s the biologists John Maynard Smith and George Price used games to explain animal behavior, such as why most fights between animals of the same species are ritual displays rather than fights to the death. Here the payoffs are offspring and the strategies are inherited, so no one needs to be rational for the theory to apply.

Where did game theory come from?

Year	Step
1838	Antoine Augustin Cournot analyzes two firms competing on output, an equilibrium idea in all but name
1928	John von Neumann proves the minimax theorem for two-player zero-sum games
1944	Von Neumann and Oskar Morgenstern publish Theory of Games and Economic Behavior, founding the field
1950	John Nash defines the equilibrium that bears his name; at the RAND Corporation, researchers run the first prisoner's dilemma experiment
1960	Thomas Schelling's The Strategy of Conflict applies game theory to bargaining and deterrence
1973	Maynard Smith and Price bring game theory into biology
1994	Nash, John Harsanyi and Reinhard Selten share the Nobel prize in economics for equilibrium analysis
2020	Milgrom and Wilson win the prize for auction design

The 1994 prize announcement said that 50 years after von Neumann and Morgenstern, game theory had "become a dominant tool for analyzing economic issues".

Still argued

Game theory's predictions rest on players being rational and expecting each other to be. In laboratory experiments people often aren't: they share more than the theory predicts and punish unfairness at a cost to themselves. Some economists see this as a flaw in the theory's assumptions; others reply that payoffs should include what people care about, such as fairness, and that play moves toward equilibrium as people learn a game. Behavioral game theory tries to combine the two, and the Stanford Encyclopedia of Philosophy sets out both readings of experiments run in the lab and in small-scale societies on several continents.

Check yourself

• Without looking back, explain why Big Bean has to predict Little Cup's price but Little Cup doesn't have to predict Big Bean's.
• What does a Nash equilibrium guarantee, and what doesn't it guarantee?
• Why is trade a non-zero-sum game?

Sources

• Game Theory — Don Ross, Stanford Encyclopedia of Philosophy, revised 2023
• The Prize in Economic Sciences 1994, press release — The Royal Swedish Academy of Sciences, 1994
• The Prize in Economic Sciences 2005, press release — The Royal Swedish Academy of Sciences, 2005
• The quest for the perfect auction, popular science background — The Royal Swedish Academy of Sciences, 2020
• Nash equilibrium — Wikipedia