Numeracy #10: Nash Equilibrium

Imagine two criminals, Bob and Sam, who were arrested today for selling drugs. It’s an open and shut case, but the district attorney has a hunch that Bob and Sam were involved in a murder that took place a week prior. So the district attorney puts Bob and Sam into separate rooms and offers them both the same deal.

  • If you both keep quite, you both get 1 year in prison for selling drugs.
  • If one of you rats out the other for the murder and the other stays quite, the confessor will go free and the other will get life imprisonment.
  • If you both rat each other out, you will both get 10 years in prison.

This scenario is called the prisoner’s dilemma and though it may seem like the obvious solution is for both Bob and Sam to stay quite, in reality both of them acting rationally will lead to a sub-optimal outcome. But we will get back to this.

John Nash

In 1948, John Nash arrived a Princeton with a PhD recommendation that simply read: “He is a mathematical genius”. If you watched the movie A Beautiful Mind, he was the character played by Russell Crowe who took issue with Adam Smith’s idea that individual ambition and competition always serves the common good.

In 1949, Nash sent a note to the Proceedings of the National Academy of Sciences laying out the concept of what would come to be known as the Nash equilibrium. That one page note would end up winning him a Nobel prize in economics for his contribution to game theory.

Let’s get some definitions out of the way. Game theory is the study of strategies involved in complex games between intelligent and rational decision-makers. Basically, you are in a game if your fate is affected by the actions of others. For example, two companies competing for market share can be modeled as a game or two countries threatening global nuclear warfare.

A Nash equilibrium describes the stable strategic outcome that will be produced when players consider what other players will do. In Nash’s words:

an equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player’s strategy is optimal against those of the others.

Put more simply, a group of players will be in a Nash equilibrium if no player can make themselves better off by acting differently, given what other players are doing. Put even simpler, no player can make themselves better off by changing their strategy. In that global nuclear war example, each side knows that launching first will cause the other player to launch. Mutual assured destruction sets the equilibrium at not starting a war. Neither player will be better off by moving away from this equilibrium.

Nash went a bit further and proved that any game with a finite number of players and strategies, would have at least one Nash equilibrium.

The Prisoner’s Dilemma

Though the Nash equilibrium states that each player is doing as well as they can, it does not mean the group as whole will do as best as they could. The classic example of this is the prisoner’s dilemma that we opened with. Consider the following payoff matrix that Bob and Sam face:

Payoff Matrix.png

Put yourself in Sam’s shoes. If Bob confesses, you have the option of either 10 years if you confess or life in prison if you keep quite. Confess wins out. If Bob stays quite, you have the option of either 0 years if you confess or 1 year if you stay quite. Confess wins out again. Bob will use the same rational and also confess. Sam and Bob both end up with 10 years in prison, even though both would have been better off if they worked together and kept quiet. This is a powerful idea: what is best for the individual can be disastrous for the group.

The Dismal Science

By expanding on this idea of incentives, economics now had a tool for modeling how self-improving individuals could lead to self-harming crowds. Over-fishing and the tragedy of the commons is the most cited example. But there are also some less known examples.

Consider Braess’s Paradox, the idea that an improvement to a road network will actually make traffic worse. The reason is that the new Nash equilibrium will be sub-optimal. To help understand this, imagine that we are trying to get from city A to city B. Currently there are two paths, one starts north and one starts south. Let’s say that there are 5000 drivers and about half use each path, otherwise one path would become less attractive. Now a city planner comes along and adds a new path that branches out of the north path. Self-interested drivers will now opt for the new shortcut and end up jamming the north road with all 5000 drivers. Even stranger, it has been shown that closing a major road can actually improve traffic flow.

Consider the story of spectrum auctions. Before 1994, the FCC gave away spectrum for free in a lottery and court hearing system. This led to some horrible incentives, as people who had no interest in using the spectrum would enter the lottery and then sell the rights in secondary markets. By the 1990s, facing the threat of a spectrum shortage, the FCC knew the system needed to change. They brought in two game theorists who devised the world’s first simultaneous and ascending auction:

To see how this might work in practice, imagine a very simple example. Two companies are bidding on cell phone licenses in San Francisco and Chicago. Company A is interested in the Bay Area market, while Company B is only willing to pay a premium for Chicago if it can land the San Francisco license too.

In each round, the two companies will submit bids on the licenses that they are interested in acquiring. In the above example, Company A might place $500,000 on San Francisco, while Company B might submit $510,000 on both. When the round ends, both companies will be able to view the highest bid in each market and, now having some understanding of the preferences of their competitor and the likely trajectory of prices in each market, they can adjust their strategies accordingly before making their bids in the second round.

The auction finally ends when a round passes in which no bids are made on any of the licenses.

Lastly, consider the American hospital system in the 1940s. The war had made medical students a rare commodity and hospitals were in fierce competition for them. A sub-optimal Nash equilibrium formed where hospitals were incentivized to send out job offers to candidates¬†before they even graduated. As offers went out earlier and earlier, hospitals didn’t know what quality they would get and candidates had no chance to consider competing prospects. This was solved in the 1990s by implementing a system where candidates submitted a list of the hospitals they wanted to work at and an algorithm matched them with hospitals that would be happy to take them.

The solution to the problem is often to set the incentives in a way that the group as a whole isn’t harmed. If the system still fails to work properly, economists assumed people just aren’t informed enough to make rational decisions. But even with full information, people can still be irrational.

Refining the Idea

In experiments, people faced with the prisoner’s dilemma only chose to confess about half the time. The rationality of the incentive structure isn’t always enough to predict actions, since real people tend to be a wild card. Nonetheless, when the experiments are repeated, people learn that it is their interest to confess.

By accounting for this and other real world limitations, the Nash equilibrium retains a central role in modern microeconomics.

As it relates to investing

Consider the run up in mortgage lending before the housing collapse of 2007. Each individual lender had an incentive to lend more, even though the rise in housing prices made the whole system more prone to a decline. It was a sub-optimal Nash equilibrium that most financial manias exhibit: profit seeking rational individuals leading to the madness of crowds.

The Nash equilibrium also plays a role in competition. Consider the fight between Uber and Didi in China. Both were trapped in a sub-optimal Nash equilibrium where they would spend billions of dollars to edge out tiny portions of market share. If any one of them stopped spending, the other would win by default and they would disappear. They could have agreed to stop fighting and form a duopoly cartel, but then the Nash equilibrium would be to cheat. In the end, Uber gave up and sold its China business to Didi.


In 2015, John Nash died in a car crash. He had a complicated life, but his ability to pair mathematical models with real world applications in a simplistic manner made him a national icon. Nash taught us all that the interactions of a group, rather than the individual, is where the magic happens.



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