By Clemence Vidal Economics is all about the consumer and producer behavior as it is about finance or the allocation of resources. This short article will explain to you one of the most fundamental tools economists use to frame competitive decision-making. Game theory is a “tool” used to study strategic decision-making. |

So what is Game theory?

Game theory is a type of methodology. So like all methodologies, it attempts to predict outcomes. While other methodologies use tools like case studies or statistics, game theory involves the use of “games” not only to display all possible but also to determine the likelihood of certain outcomes. The essential feature of Game theory is that it provides a formal modelling approach while looking at strategic interaction between agents.

Now let’s focus on one specific concept: the Nash equilibrium. This concept is named after its inventor John Nash, an American mathematician and is used in multiple disciplines (ranging from behavioral ecology to economics).

In order to try to understand this concept, look at this game:

Game theory is a type of methodology. So like all methodologies, it attempts to predict outcomes. While other methodologies use tools like case studies or statistics, game theory involves the use of “games” not only to display all possible but also to determine the likelihood of certain outcomes. The essential feature of Game theory is that it provides a formal modelling approach while looking at strategic interaction between agents.

Now let’s focus on one specific concept: the Nash equilibrium. This concept is named after its inventor John Nash, an American mathematician and is used in multiple disciplines (ranging from behavioral ecology to economics).

In order to try to understand this concept, look at this game:

Let’s say player A makes choice i and player B makes choice j. (i,j) is a Nash equilibrium if neither player can improve their payoff by changing their choice.

It seems quite evident what the players should do in this game (as both of them are rational and human…). Both players should make choice 1 as they both win 10 points.

Besides, this pair of choices (i,j)=(1,1) is a Nash equilibrium. In other words: neither player can improve their payoff by unilaterally changing their choice! Suppose player A switches to choice 2, his payoff drops to -1 points. And if player B switches to choice 2, his payoff drops to -1 points. Therefore, both players won’t change their strategy.

So let’s give a definition of Nash equilibrium before moving on:

Remember, player A's choices are i=1,2,…,m

while player B's choices are j=1,2,…,n

Condition

1) Player A can't improve their payoff by switching their choice from i to any other choice i′.

2) Player B can't improve their payoff by unilaterally switching their choice from j to any other choice j′.

Let’s look at a trickier example: the prisoner’s dilemma. It’s a classic example typically used to teach game theory.

The prisoner’s dilemma is probably the most widely used game in game theory. Its use has completely changed Economics, but is also used in other fields such as biology or psychology. It was named by Albert Tucker in 1950 who developed it from earlier works.

So the prisoner’s dilemma describes a situation where two prisoners, suspected of burglary, are taken into custody.

The problem is that policemen do not have enough evidence to convict them of that crime. And the situation is the following: if none of them confesses (they cooperate with each other), the two of them will be charged the lesser sentence (a year of prison). The fact is that police will question them on separate interrogation rooms, so the two prisoners cannot communicate. The police will try to convince each prisoner to confess by offering them a “no jail card”, while the other prisoner will be sentenced to a ten years’ term. If both prisoners confess, each prisoner will be sentenced to eight years. The two prisoners are offered the same deal and know the consequences of their actions. Besides, they are completely aware that the other prisoner has been offered the exact same deal.

The situation is here modelled by the following table:

It seems quite evident what the players should do in this game (as both of them are rational and human…). Both players should make choice 1 as they both win 10 points.

Besides, this pair of choices (i,j)=(1,1) is a Nash equilibrium. In other words: neither player can improve their payoff by unilaterally changing their choice! Suppose player A switches to choice 2, his payoff drops to -1 points. And if player B switches to choice 2, his payoff drops to -1 points. Therefore, both players won’t change their strategy.

So let’s give a definition of Nash equilibrium before moving on:

Remember, player A's choices are i=1,2,…,m

while player B's choices are j=1,2,…,n

- Definition: Given a 2 player normal game, the pair of choices (i,j) is a Nash equilibrium if:
- 1)For all 1≤i′≤m, Ai′j≤Aij
- 2) For all 1≤j′≤n, Bij′≤Bij

Condition

1) Player A can't improve their payoff by switching their choice from i to any other choice i′.

2) Player B can't improve their payoff by unilaterally switching their choice from j to any other choice j′.

Let’s look at a trickier example: the prisoner’s dilemma. It’s a classic example typically used to teach game theory.

The prisoner’s dilemma is probably the most widely used game in game theory. Its use has completely changed Economics, but is also used in other fields such as biology or psychology. It was named by Albert Tucker in 1950 who developed it from earlier works.

So the prisoner’s dilemma describes a situation where two prisoners, suspected of burglary, are taken into custody.

The problem is that policemen do not have enough evidence to convict them of that crime. And the situation is the following: if none of them confesses (they cooperate with each other), the two of them will be charged the lesser sentence (a year of prison). The fact is that police will question them on separate interrogation rooms, so the two prisoners cannot communicate. The police will try to convince each prisoner to confess by offering them a “no jail card”, while the other prisoner will be sentenced to a ten years’ term. If both prisoners confess, each prisoner will be sentenced to eight years. The two prisoners are offered the same deal and know the consequences of their actions. Besides, they are completely aware that the other prisoner has been offered the exact same deal.

The situation is here modelled by the following table:

Since prisoners cannot communicate and will (supposedly) make their decision at the same time, it is considered to be a simultaneous game, and can be therefore analysed using the strategic form. As described before, if both prisoners confess the crime they will be charged an eight years sentence each. If neither confesses, they will be charged one year each. If only one confesses, that prisoner will go free, while the other will be charged a ten years sentence. So we see clearly that for each set of strategies, in total there are four payoffs.

In order to solve this game, we need to eliminate all dominated strategies, in order to get the dominant strategy. Let’s call prisoner 1 P1 and prisoner 2 P2.

P1 has to build a belief about what choice P2 is going to make, in order to choose the best strategy. So, if P2 confesses, he will get either -8 or 0, and if he lies he will get either -10 or -1. It can be seen that P2 will choose to confess, as he will be better off. So given that P2 will choose to confess, P1 must choose the best strategy: P1 can either confess which pays -8 or lie which pays -10. The rational thing to do for P1 is to confess. Proceeding inversely, by analising the beliefs of P2 about P1’s strategies, gets us to the same point. So the rational thing to do for P2 is to confess. Hence “to confess” is the dominant strategy. But in this case “to confess” is not only the dominant strategy, it’s also the Nash equilibrium in this game since it is the set of strategies that maximise each prisoner’s utility given the other prisoner’s strategy.

In the prisoner's dilemma, the Nash equilibrium is not the optimal decision. The optimal decision is to both stay quiet, but if the two prisoners do so, they both risk being ratted out by the other person and end up spending a lot more time in prison. Nash equilibrium is not to necessarily find the best outcome; it's about finding the outcome where no players have the incentive to change their behavior.

At that point you might be wondering: what is the use of Nash equilibrium?

The answer is simple; thinking through behavioral decisions in a game theory for instance as competitive situation by identifying both the optimal and Nash equilibrium can help you optimise decisions. Moreover, Game theory and Nash equilibrium are really useful as they can not only be used in economy but in many other fields. For instance, some politicians try to explain the Soviet missile installation in Cuba, and the resulting American attempts to remove the weapons by relating it to Game theory.

For those who might be interested a film A beautiful mind (2001) chronicles John Forbes Nash’s life in which there’s a special scene explaining Nash Equilibrium, though there is just one problem: the Nash equilibrium is explained incorrectly… By trying to simplify simplify Nash's discovery director Ron Howard made an enormous mistake.

In order to solve this game, we need to eliminate all dominated strategies, in order to get the dominant strategy. Let’s call prisoner 1 P1 and prisoner 2 P2.

P1 has to build a belief about what choice P2 is going to make, in order to choose the best strategy. So, if P2 confesses, he will get either -8 or 0, and if he lies he will get either -10 or -1. It can be seen that P2 will choose to confess, as he will be better off. So given that P2 will choose to confess, P1 must choose the best strategy: P1 can either confess which pays -8 or lie which pays -10. The rational thing to do for P1 is to confess. Proceeding inversely, by analising the beliefs of P2 about P1’s strategies, gets us to the same point. So the rational thing to do for P2 is to confess. Hence “to confess” is the dominant strategy. But in this case “to confess” is not only the dominant strategy, it’s also the Nash equilibrium in this game since it is the set of strategies that maximise each prisoner’s utility given the other prisoner’s strategy.

In the prisoner's dilemma, the Nash equilibrium is not the optimal decision. The optimal decision is to both stay quiet, but if the two prisoners do so, they both risk being ratted out by the other person and end up spending a lot more time in prison. Nash equilibrium is not to necessarily find the best outcome; it's about finding the outcome where no players have the incentive to change their behavior.

At that point you might be wondering: what is the use of Nash equilibrium?

The answer is simple; thinking through behavioral decisions in a game theory for instance as competitive situation by identifying both the optimal and Nash equilibrium can help you optimise decisions. Moreover, Game theory and Nash equilibrium are really useful as they can not only be used in economy but in many other fields. For instance, some politicians try to explain the Soviet missile installation in Cuba, and the resulting American attempts to remove the weapons by relating it to Game theory.

For those who might be interested a film A beautiful mind (2001) chronicles John Forbes Nash’s life in which there’s a special scene explaining Nash Equilibrium, though there is just one problem: the Nash equilibrium is explained incorrectly… By trying to simplify simplify Nash's discovery director Ron Howard made an enormous mistake.