A two-page paper published by John Nash in 1950 is a seminal contribution to the field of Game Theory and of our general understanding of strategic decision-making. That paper, “Equilibrium points in N-person games”, introduced a cornerstone concept which came to be known as Nash equilibrium.
Game theory is concerned with situations where decisions interact – where the “payoff” or reward for a decision maker depends not only on his or her own decision but also on the decisions of others.
Such situations are pervasive in real life. The payoff for a buyer in an auction, for example, depends not only on the amount he bids but also on the bids of the other buyers. If the buyer’s bid is not the highest, then he loses the auction. Likewise, the profit realised by a firm depends not only on the price it sets for its product but also on the prices set by its competitors. In a tennis match, the likelihood the server will win a point depends on whether she delivers the serve to the receiver’s left or right and whether the receiver correctly anticipates it.
Auctions, price setting and tennis are all examples of “non-cooperative” strategic interactions that mathematicians and economists refer to as “games”. They are non-cooperative because decision makers take their actions independently and are unable to enter into binding agreements with others regarding their actions, either because such agreements are illegal (when setting prices) or because they have no incentive to do so (as in tennis).
The notion of Nash equilibrium, developed in Nash’s 1950 paper, is the basis of how economists predict the outcome of strategic interactions.
Informally, a Nash equilibrium is a list of actions, one for each decision maker, such that each decision maker’s action is best for him, given the actions of the others. Such a list of actions is an equilibrium (or stable point), since no decision maker has an incentive to change his action.
Consider a driver approaching an intersection. She stops when she approaches a red light and she continues without concern when she approaches a green light. It is a Nash equilibrium when all drivers behave this way. When approaching a red light it is best to stop since the crossing traffic has a green light and will continue. When approaching a green light it is best to continue since the crossing traffic has a red light and will stop. Thus it is in each driver’s own interest to play her part in the equilibrium, given that everyone else does. No traffic cop is required.
Nash equilibrium also allows for the possibility that decision makers follow randomised strategies. Allowing for randomisation is important for the mathematics of game theory because it guarantees that every (finite) game has a Nash equilibrium.
Randomisation is also important in practice in commonly played games such as Two-up, Rock-Paper-Scissors, poker and tennis. We all know from our own experience how to play Rock-Paper-Scissors against a sophisticated opponent: play each action with equal probability, independently of the actions and outcomes in past plays. Indeed, this is exactly what Nash equilibrium predicts. Nash’s theory applies to any game with any number of decision makers, whereas John von Neumann’s 1928 Minimax Theorem applies only to “zero-sum” games with two players.
Interestingly, data collected from championship tennis matches has shown that the serve-and-return behaviour of professional players is consistent with both von Neumann’s Minimax Theory and Nash’s generalisation.
Nash’s work dealt with games in which each decision maker takes his or her action without knowing the actions taken by others, and in which no decision-maker has private information. John Harsanyi extended the notion of Nash equilibrium to deal with strategic interactions, such as in auctions, in which decision-makers have private information. (In an auction, buyers know the value they place on the item being sold, but they don’t know how other buyers value the item.)
Reinhardt Selten extended the notion of Nash equilibrium to deal with dynamic interactions, in which one or more decision-makers observe the action of another before taking their own action. In 1994, Nash shared the Nobel Prize in Economics with Harsanyi and Selten for these contributions.
While Nash is best known for his contribution to non-cooperative game theory, he also made a seminal contribution to cooperative game theory with the development of the Nash bargaining solution.
Nash’s work has had a profound impact in economics. Knowledge of game theory is essential training for every professional economist and it is a common – and popular – subject for undergraduate students as well. Nash’s work not only revolutionised modern economics, it has also had an important impact in fields as diverse as computer science, political science, sociology and biology.
His work on game theory won him a Nobel Prize for economics in 1994 and he just recently received Norway’s Abel prize for mathematics.
John Nash remained active at scientific conferences around the world. He was happy to talk with students, many of whom wanted a picture with him too. He and his wife, Alicia were both killed in a car accident on the New Jersey Turnpike on Saturday, May 24 on their way home from receiving the prize.
Both were kind people and they will be missed.