The Nash equilibrium

According to this concept - named after the mathematician John F. Nash -, a game-theoretic strategy is in an equilibrium when none of the players gets an advantage by unilaterally changing the strategy, as long as all the other players do not change their strategies.

In other words, the players are in a Nash-equilibrium, when a unilateral decision change of one player would diminish his pay-off. No player could improve his performance by unilaterally changing his decisions.

A strategy denotes a strict Nash-equilibrium, if every decision of the strategy represents the best response (i.e. the response yielding the highest pay-off) to the decisions of others.

In European road traffic, for example, driving on the right side of the street is a strategy that yields a (very simple) Nash-equilibrium. No car driver would benefit from changing this strategy as long as the others do not change their strategy too. Driving on the right side is a best response to the behavior of others as long as all others drive on the right side too.

Since best responses often are found in self-organization, Nash-equilibria can be attractors for certain interactions.

Example: battle of the sexes

Alice and Bob like each other and love to spend their spare time together. However, Alice prefers the theatre, whereas Bob would rather go to the cinema. Nevertheless, both prefer going out together rather than insisting on their preferences and going to the theatre or cinema alone. A pay-off matrix for this situation can look as follows:

1  /  2
0  /  0
0  /  0
2  /  1

For both Alice and Bob, an evening spent together yields more pay-off than an evening spent alone. Both want to coordinate themselves, but they want to do so in different ways. This means that there are two strategies (1 / 2 and 2 / 1), in which none of the actors has an advantage of unilaterally deviating from his/her decision, as long as the other player does not deviate, too. External factors, directing the development in favor of one of the Nash-equilibria, often affect such situations. Using the example of road users driving on the left or the right side of the street, this could be a random inequality in the initial distribution, for instance.

Example Nash-equilibrium: the commuter game

A group of commuters has the choice between going to work with their own car or taking a public bus. Driving by car yields a higher individual well-being pay-off, but leads to traffic jams which in turn might lower the overall pay-off. Taking a public bus yields a lower well-being pay-off, but does not jam the streets.

commuter game

In the figure above, the horizontal black line between 0 and 1 indicates the share of car drivers. The vertical lines represent the pay-offs; the green (higher) line depicts the pay-off of car drivers, while the red line describes the pay-off of bus riders. In this scenario, car drivers always have a higher payoff, irrespectively of the size of their share. Driving by car is a dominant strategy in this scenario, and a dominant strategy-equilibrium entails that everyone is driving by car. Of course, if too many commuters decide to go by car, their pay-off is negative, whereas the pay-off would be positive if everyone decided to take the bus.

Hence, if everyone decides for the strategy he expects to have the highest pay-off with, they actually deteriorate their pay-off.

McCain's, Roger (2010) Game Theory: A Nontechnical Introduction to the Analysis of Strategy. World Scientific.

A slightly varied scenario assumes that taking the bus is less prone to traffic jams than driving by car. For example, there might be bus lanes in some districts, granting that buses get ahead even if the cars are stuck in a traffic jam.

commuter game 2

Hence, in this scenario the bus traffic is less obstructed by the cars. The pay-off of people taking the bus decreases more slowly than the pay-off of car drivers. At the point q, the decrease of car drivers' pay-off overtakes the decrease of the bus riders' pay-off. For higher shares of car drivers (at the right side of q), car drivers obtain a significantly lower pay-off than bus riders.

Hence, the game does no longer exhibit a dominant strategy-equilibrium. But there is still a Nash-equilibrium at q: if a bus rider switches to his car at this point, he will shift the ratio into the region where car drivers perform more poorly, and therewith will decrease his own pay-off. On the other hand, a car driver switching to the bus at q would also decrease his own pay-off by shifting the ratio to the left. Thus, nobody can improve by unilaterally switching the strategy - it is a Nash-equilibrium.

A Nash-equilibrium neither needs to be ideal nor fair! In the commuter game, everyone would be better off if everyone was riding the bus (the red line in the zero point is located significantly higher than the intersection point of green and red at the point q). However, if everyone acts with regard to his pay-off, this situation will not occur without coordination. The individual pay-off for driving a car stays higher until q is reached. Furthermore, also in a Nash-equilibrium, a single player can yield a very high pay-off and many others might yield nothing or only as little as needed to keep the sum of payoffs higher than the sum of a "fair" (equally low) payoff.

Questions about fairness and the distribution in game-theoretical contexts are answered in the Pareto-improvement and the Pareto-optimum, inter alia.

The Pareto-optimal strategy

Named after Vilfredo Pareto (1848-1923), a strategy is considered Pareto-optimal, if there is no other strategy making any single individual better off, without making at least one other individual worse off at the same time.

To better understand this Pareto-optimum, it is helpful to have a look at the Pareto-improvement first. The Pareto-improvement is given, when an option within a strategy is changed in a way that increases the payoff of the player making the change, but at the same time does not decrease the pay-off of the other players.

If no further Pareto-improvement can be achieved in a strategy, the strategy is Pareto-optimal.

In the social situation in such a case, no player has the possibility to complain that he could be better off, without harming anyone else. In this sense, the strategy is fair. In the given situation it is not possible to make any better decisions (for anyone) without making anyone else worse off. In this regard, in economy the term pareto-efficiency is used.

Example: pig and piglet

Two pigs are stabled together and can get themselves food with the help of a lever device. The lever is located at the one end of the stable and once pressed dispenses food into a trough located at the other end of the stable. One of the pigs is big and stolid, the other pig is young and agile.

If the big pig pushes the lever, the small pig has the chance to already wait at the trough and eat part of the food before the other pig manages to turn around, reach the food and edge the small pig aside. In contrast, if the small pig pushes the lever, it will not have this chance. The big pig will prevent the small pig from eating. If both pigs push the lever, the small pig will reach the trough more quickly and manage to eat a bit before it is edged aside by the big one.

A pay-off matrix for this game could look as follows:

Big pig
Do nothing
Push lever
Young pig
Do nothing
0 / 0
5 / 1
Push lever
-1 / 6
1 / 5

Dominant option for the piglet: "do nothing"

Common goods solutions: "do nothing"/"push lever" and "push lever"/"push lever" (5/1 and 1/5, together 6)

Pareto-optimal solution: "do nothing"/"push lever" and "push lever"/"push lever", and "push lever"/"do nothing" (5/1, and 1/5, and -1/6)
None of the pigs can improve based on these strategies without harming the other pig at the same time.

Nash-equilibrium: "do nothing"/"push lever" (5/1)
As long as the other pig does not change its behavior, no pig has a benefit from taking another decision.

In the long run, the pigs' behavior results in a Nash-equilibrium. Experiments with real animals confirmed this result.