What is the Nash Equilibrium?

A Nash equilibrium is like feeding a stray cat...

Imagine a stray cat crying for food at your door. You feel bad, so you give it a can of tuna. But now the cat knows where to get food. The next day, it comes back. Even though you don’t want to keep feeding it, you give it another can of tuna to make it stop crying out. Now you basically have a pet. A Nash equilibrium works the same way. People react to each other’s behaviors until they reach an outcome that persists.

A Nash equilibrium (NE) is a game theory concept with applications in many disciplines, especially in the social sciences. Biology, economics, finance, and sociology are examples of fields of study that benefit from game theory and use the NE concept.

In the context of game theory, almost any interaction qualifies as a game. The participants are called players, their choices are called strategies, and the results are called payoffs. Payoffs don't have to be money. They include any positive outcome — translated into a value called utility (a metric that quantifies how happy a person is with an outcome). To game theorists, the goal of any game is to get the greatest payoff you can. But, unlike the way we usually think about games, you don’t win by out-scoring your opponent.

A real-world example would be to look at two drivers on the same road, driving in opposite directions, as players in a game. They each have a set of strategies to choose from — driving on the left, or driving on the right. Their health is the payoff measure. If they both pick left or right, they'll pass each other safely. If they choose opposite strategies (one chooses to drive on their left, the other on their right), they'll crash into each other.

So, there's an incentive to change strategies if they initially picked opposite options. And there's no incentive to change course if they both opt to drive on the right or left side of the road. That means that everyone driving on the right side of the road is a NE. So is everyone driving on the left side of the road.

The idea of a Nash equilibrium is named after John Nash, an American mathematician who won a Nobel prize in economics in 1994. Nash earned his PhD from Princeton at age 22. His battle with mental illness later became famous in the 2001 movie A Beautiful Mind. Nash's work extended the ideas put forth by John Von Neumann and Oskar Morgenstern in their book Theory of Games and Economic Behavior.

Finding a Nash equilibrium (NE) is simplest using a matrix. A payoff matrix describes the results of a game where people interact. It lists the available strategies, along with what each player receives for each combination of strategies. Player 1’s strategies are located on the left column, and their payoffs are listed on the left of each payoff box. Player 2’s strategy profile is on the top row, and their payoffs are on the right side of each payoff box.

A combination of strategies is a NE if neither player can improve their payoff by choosing different strategies.

For example, consider the following hypothetical pricing game. Both fictitious companies Skip’s Chips and Captain Shark are popular fish and chips fast food restaurants that compete with each other. If they both charge the same price on their best selling fish and chips plate, they split the market. If either undercuts the competition by lowering the price, it will capture a much larger market share.

When both restaurants charge $12, they both enjoy higher economic profits than if they both charged $10. But there is always the incentive to lower the price to capture a larger market share.

If Captain Shark lowers its price to $10 and Skip’s Chips keeps its price at $12, Captain Shark benefits from higher profits, and Skip’s Chips loses market share. Similarly, if Skip’s Chips lowers its price to $10 and Captain Shark keeps its $12 price, Skip’s Chips wins and Captain Shark loses.

It stands to reason that the best thing either company can do — whether the other keeps its price at $12 or lowers it to $10 — is to lower its price to $10. Charging $10 is the dominant strategy for both players.

The easiest way to see this is by finding the best response to each of the strategies of the other player. First, imagine Skip’s Chips charges $12. The best response for Captain Shark’s is to charge $10. So, underline that payoff. Next, look at the best response for Captain Shark if Skip’s Chips charges $10. Again, the best response for Captain Shark is also to charge $10. So, underline that payoff.

Now, go across each row and underline the best response for Skip’s Chips to each of Captain Shark’s pricing strategies. If any combination of strategies have both payoffs underlined, that is a NE. In this game, that only happens when both companies charge $10. Even though both companies would earn more profits if they charged $12, they are stuck at $10 with no incentive to do anything different. That’s the NE.

There might not always be a pure strategy that is a Nash Equilibrium. But John Nash (an American mathematician) won a Nobel prize in economics for proving that every noncooperative game has at least one mixed strategy solution (which makes choices by probability).

For example, think about playing the simple game “rock, paper, scissors.” There's no single strategy equilibrium — a stable solution to which playing the game naturally leads. If your friend always picks rock, you just need to pick paper to beat them. Realizing that they keep losing, they have an incentive to change their own strategy — So, they start picking scissors. Now, you keep losing and want to change your approach — So, you switch to playing rock.

This series of reactions will never lead to a point where both players in this two-player game are satisfied with their choices. That means there's not a pure strategy Nash equilibrium. But there is a mixed strategy solution. If you randomly pick each choice with equal weight, there's no way for your friend to get an edge against you. And if they randomly pick each choice with equal weight, no response gives you a leg up. Each player's strategy of assigning a uniform probability distribution to the mix of options is the Nash Equilibrium for rock, paper, scissors.

A dominant strategy solution is always a Nash equilibrium; but every Nash equilibrium isn't necessarily a dominant strategy solution. A dominant strategy is one that is the best choice, regardless of what strategy the other players choose. If a player has a dominant strategy, the opponent’s best response to that option is a Nash equilibrium. But not every game has a dominant strategy.

Imagine a company is considering moving into a new market. They have the strategy options of expanding or not expanding. The only reason this company is hesitating is that it's not sure how the incumbent company will respond. The incumbent company will have the strategy options of dropping its prices or leaving them the same.

First, consider the situation in which predatory pricing by the incumbent would reduce the new venture’s profit potential from $5M to $2M. While the new company’s executive team might lose sleep over the $3M difference, entering the market is a dominant strategy. Since it’s a profitable venture, either way, it doesn’t really matter if the incumbent retaliates or not. Knowing that retaliation won’t prevent the entry, the incumbent probably won’t do it. So, the combination of entering and not retaliating is a unique Nash equilibrium and a dominant strategy solution.

Situations that have multiple Nash equilibria aren’t dominant-strategy solutions. For instance, everyone driving on the right side of the road is a Nash equilibrium. That’s because it doesn’t make sense for anyone to switch lanes. The same would be true if everyone drove on the left. Driving on the same side as everyone else is a Nash equilibrium, but it doesn't follow the dominant strategy solution concept.

Understanding the Nash equilibrium (NE) is essential to good decision-making. Economic theory assumes that people are rational. So outlining the players, strategies, and payoffs allows an entrepreneur or investor to anticipate how a scenario will unfold. If the current situation doesn’t lead to a desirable outcome, they can position themselves to change the result.

For instance, giving an employee the skills, connections, and experience they need to be successful also increases the threat that they will eventually use that knowledge against the company. Seeing that risk in advance, the company might require a non-compete agreement as part of the hiring and training process. Other contract terms, such as performance bonuses and stock options, also help to align the interests of the employee and the corporation when the natural NE doesn't result in a favorable outcome.

The Prisoner’s Dilemma is a classic example of game theory and Nash equilibria. It shows how individual self-interest can result in worse outcomes than would occur with cooperation or collusion. In some cases, it reveals how specific market failures can exist. In other cases, it shows how competition drives prices downward.

In the original version of the Prisoner’s Dilemma, two suspects are detained by the police in separate interrogation rooms. The police don’t have enough evidence to convict either of them without a confession, but they have enough to put them in jail for a lesser charge.

If they both stay quiet, they will go to jail for one year. The police officer gives the suspects a deal. If they both confess, they will get a reduced sentence of three years. But, if one friend pins the blame on the other one who doesn’t cooperate, the friend will walk free while the other suspect will go to jail for 10 years.

The reason the Prisoner’s Dilemma is interesting is that cooperation would lead to the best possible outcome. If they both stay quiet, they will both serve a shorter sentence. But the individual incentives make cooperation unlikely. The reward for ratting out the friend is too tempting, and the risk of that friend flipping is too significant. So, the only strategy that makes sense to a rational player is to confess. The choices that are driven by self-interest end up with both confessing and spending more time in prison. That’s the Nash equilibrium.

Other situations outside the interrogation room are also Prisoner’s Dilemmas. For instance, choosing whether to run a marketing campaign for a well-known brand in an established market falls into this category. While companies would be better off keeping their marketing budget for other purposes, failing to market the brand, while the competition does, is too great a risk.

So, the companies end up spending more money than they would like. Competition in pricing sometimes leads to the same result. One company thinks it can win more business by cutting prices. That individual incentive leads them to act. But such a price reduction forces competitors to match the lower price. Consequently, all of the firms end up charging less.

Breaking free from a Prisoner’s Dilemma requires changing the payoffs. In the case of actual criminals, if the suspect knows that punishment awaits a snitch on the streets, going free is not preferable to spending a year in jail. Companies colluding to keep prices higher or to not advertise can escape the Prisoner’s Dilemma. But antitrust laws often prevent that kind of cooperation from occurring — under the theory that it can hurt consumers.

But other means of achieving the same result are perfectly legal. For instance, a “price match guarantee” is a signal to competitors not to bother trying to steal their customers with lower prices. It’s a preemptive commitment to reduce the price automatically, often preventing the price reduction from happening in the first place.