Nash equilibrium is a very crucial concept of game theory. It helps to determine an optimal solution in a non-cooperative game where all players do not have any incentive to deviate from their initial move. In other words, this is the situation where everyone in the game is putting in their best, assuming and understanding clearly what the other players would be supposed to be doing.
We can also say that in a Nash equilibrium, a player knows the strategy of the other players and even then does not change their initial strategy because they have no incentive to do so. Once there is an equilibrium, players will be worse off by deviating from their initial strategy. However, we need to appreciate that in a given game situation, the Nash Equilibrium can be there, can not be there at all, or there can be multiple Nash Equilibrium points.
This equilibrium concept takes into account the behavior and interactions of the players to come up with the outcome. Moreover, it also helps predict the moves that a player would make, provided all players are aware of the likely decision (decisions) of the other players, and they all are making the decision at the same time.
Examples of Nash Equilibrium
Let us consider an example to better understand the Nash Equilibrium. Suppose there are two companies, A and B, and both are planning to advertise to attract new customers. Following are the outcomes – if only one of the two advertises, the one advertises gains 200 new customers. And if both companies advertise then each will get 100 new customers. Further, if none of the two advertise, neither would get any new customers.
Given the above situation, it is preferable and seems to be the best option that both A and B should go ahead with advertising their product and increase their customers. Because by following this strategy, both will get additional customers and more sales rather than not doing any advertising at all. So, the outcome of both companies’ advertising is the Nash equilibrium.
Now, let us study the example where there are possibilities of multiple equilibria. Suppose there are two students, A and B, and both are friends. Both have to register for one foreign language, either Hindi or Chinese. If both A and B register for the same language, then they have an option to study together. But, if they select different languages, then they will not be able to study together. Given the situation, it is difficult for them to speak to each other before the registration.
In such a scenario, there are two desirable outcomes, i.e., both A and B register for the same foreign language, either Hindi or English. In this strategy, both the students can have the opportunity to do the study together. So, the outcome Hindi/Hindi or Chinese/Chinese will be Nash Equilibrium in this case.
Prisoner’s Dilemma & Nash Equilibrium
We can also apply Nash Equilibrium to the popular prisoner’s dilemma. In this, police arrest two criminals – A and B – and put them in two separate cells. As both are kept in different cells, they have no way to communicate with each other and jointly decide the action plan. Police have no proof against them, so it offers them options to either testify that the other was part of the crime or do not speak anything.
Following are the likely outcomes – both testify against each other, and each serves a five-year prison. If A speaks up against B, but B chooses not to say anything, then A gets no prison, and B gets a 10-year prison or vice versa. Lastly, if both of them do not speak or remain silent, they will get a jail term of one year.
Since both prisoners knew that the other also got the same options, the Nash equilibrium would be both A and B testifying against each other out of fear. A point to note is that it is not the best outcome. Since fear, emotion, and psychological pressures come into play, the best option gets ignored. As in this situation, it would have been better and would have been in the best interest of both A and B that both of them keep mum and serve one year in prison. Rather than both speaking and both getting prison for a longer term.
Applications of Nash Equilibrium
Nash equilibrium owes its name to American Economist John Nash (1928-2015). In 1994, Nash got the Nobel Prize in Economics for his contribution to game theory. He also got the prestigious Abel Prize for Mathematics.
Nash’s concept is regarded as among the most crucial concept of game theory. And the primary reason for this is its wide applicability across various disciplines, be it social sciences, economics, and more.
Following are some of the applications of Nash equilibrium:
There is a general belief that more roads would reduce road congestion. This, however, is not true because more roads induce demand, and the new roads get to fill up quickly. And, this again results in congestion of roads.
The equilibrium for the commuters is to drive on new roads because it is faster. Since everyone thinks so, it eventually leads to congestion on roads. This is not the optimal solution. The optimal solution would be less traffic on roads, but it is not possible without government intervention.
All athletes are better if no one consumes performance-enhancing drugs. Since every player knows that no other player is taking drugs, they feel tempted to take the drug to perform better than others. This results in everyone taking drugs, and anyone not taking them is at a comparative disadvantage.
Every country knows that the world needs to cut on CO2 emissions to keep a check on climate change. However, each country has an economic interest in emitting CO2. Such thinking by all countries leads to global inaction on CO2 emissions.
In an oligopoly, both firms are better if they limit their output and use a monopoly price. Since the two are rivals, they manufacture more number of units, and this reduces the payoff for the two.
Two firms engage in a legal conflict would be collectively better off if they didn’t hire a lawyer. However, each firm eventually hires a lawyer out of fear that if they don’t, and others hire a lawyer, then they would lose.