Nash Equilibrium

Machine learning (ML) and game theory, are basically the concept of Nash Equilibrium. Machine Learning is known for its ability to find patterns and make predictions from data, while Nash equilibrium is a fundamental concept in game theory used to analyze strategic interactions. In this article, we will explore more about the Nash equilibrium.

Nash equilibrium

Nash equilibrium is an important concept in game theory that provides the optimal outcome in case the player doesn’t deviate from their initial strategy. This is done in response to no incentive provided to the players for such deviation. This was named after the Mathematician, John Nash, defining the solution of a non-cooperative game involving two or more players.

Because the strategy remains optimal and the user doesn’t receive any incentive also denotes that the players are aware of each other’s strategy and so will not deviate at all.

Considering the other player remains constant in their strategy, an individual can receive no incremental benefit from such a deviation. A game can have multiple or none Nash equilibrium.

Key Points

1. Nash equilibrium provides an optimal solution to encounter the required outcome by not deviating from their initial strategy.
2. Since individuals are already aware of each other’s strategy, both the players win as everyone gets the outcome that they thought for.
3. One such example is the Prisoner’s dilemma.
4. Since the players quest to win, they would be performing that strategy that would lead them to such a state. Thus the chosen strategy is the best and the optimal solution that they can use. This is also in conjunction with the dominant strategy.
5. Further, as stated there cannot be any Nash equilibrium in a game. So, it is not always true that the strategy chosen is the optimal one.

Nash equilibrium vs Dominant strategy

Nash equilibrium reveals the optimal solution to be one when the player doesn’t deviate from their initial strategy and when he is aware of other players’ strategies as well. None of the players will be changing their strategy and will be keeping it the same.

Dominant strategy, on the other hand, makes sure that the chosen strategy will lead to better results out of all the possible strategies available. It doesn’t conform to the strategy the opponent is supposed to use.

Though both seem quite similar, there is a grave difference between them.

Nash equilibrium connotes that none of the players will win if one of the users deviates from its strategy, keeping the other players constant, though deviating from its initial strategy, as per dominant strategy, will lead to win of the player, as this being his best solution.

Example of Nash equilibrium

In the given game, we have two players with strategies S1 and S2.

S1 being the optimal strategy considering both the players know each other’s approach or they are made aware of it and both will be moving their initial strategy. They will opt for it because they know deviating from it will not be in the interest of the match.

Hence in case they opt for S1, both will win. In other cases, deviating to S2 will result in defeat of one of them.

Prisoner’s dilemma

The Prisoner’s dilemma is considered as a classic example of Nash equilibrium.

Consider two prisoners are arrested on a severe crime for which the sentence is 7 years imprisonment. They were party to that crime but the prosecutors don’t have any sort of evidence to support their claim.

To make sure they arrive at some conclusion, they kept both of them in different prisons, completely isolated and with no means of communication.

They have some choices before them.

1. They can either betray the other, the sentence will then be revoked for the one who had provided the evidence.
2. They can remain silent, in which they would have to serve 1 year in imprisonment.
3. Both of them can betray the other, and thus both now serve for 7 years in prison.

In this case, the Nash equilibrium is when both betrays each other which will make each of them to serve 7 years in prison. This is because, in case one had remained silent and the other betrayed, then one would be facing the worst consequence.

How to find Nash Equilibrium

To find Nash equilibrium, one has to first postulate all the possible scenarios in a game and then find the optimal one out of it.

In a game of two-players, they will have to surpass all the ways one can play. In case, none of them had deviated from their strategy, then we can say, Nash equilibrium has arrived.

Games with multiple Nash Equilibria

Consider a scenario where two students are to enroll to a particular course out of two provided courses.

From the above chart, it becomes quite clear that both the students when not had enough time to decide can opt for any of the two courses. In case both opted for the same course, they would be having an advantage of studying for the exams together (a situation of win-win), else would not be able to learn together.

Thus, it represents multiple Nash equilibriums where the students can opt for either C1 or C2 to attain perfect equilibrium.

Games without a Nash equilibrium

It is not always necessary for the games to have a Nash equilibrium. Games with randomized strategies between the pure strategies can be considered an example for the same.

Below mentions some of them.

1. Rock-Paper-Scissors game with discontinuous no randomized strategy
2. Games where players have opposite preferences

Mixed Strategies

Consider the below game of tennis where we have the server and the receiver with target strategies at forehands or backhands.

There is no pure strategy involved in this game. For playing the same, the server and the receiver are unpredictable about each other’s strategy. Here comes mixed strategy Nash equilibrium.

The potential strategies for the server will be –

• In case the server targets forehands and the same is anticipated by the receiver, the probable payoff will be 90 and 10 for receiver and the server, respectively.
• In case the server targets backhands and the same is anticipated by the receiver, the payoff then would be 60 and 40 for the receiver and the server, respectively.

To increase the payoff, we can assume the server mixes both the forehands and the backhands.

For example, let’s say the server targets the forehands and the backhands with a 50:50 probability. Then the receiver’s payoff will be

• 0.5 * 90 + 0.5 * 20 = 55, in case of forehands
• 0.5 * 30 + 0.5 * 60 = 45, in case of backhands

Thus, the receiver will be opting for forehands as the payoff offered will be 55 and the server will be increasing his payoff to 45 with mixing 50:50 probability.

Generalization of Mixed Strategy Nash Equilibrium

Let us now try to generalize the above expression for increasing the server’s payoff.

Suppose, the server targets forehands with a probability q and backhands with a probability 1-q. The receiver’s payoff in this case would be

• q * 90 + (1-q) * 20 = 20 + 70q, in case of forehands
• q * 30 + (1-6) * 60 = 60 – 30q, in case of backhands

To increase the payoff, the receiver can move either forehand or backhand based on the following interpretations.

• if 20 + 70q > 60 – 30q, he will be opting for forehands
• if 20 + 70q < 60 – 30q, he will be opting for backhands
• if 20 + 70q = 60 – 30q, he can opt for any of them, as both can fetch the same payoff

The server, on the other hand, to maximize his payoff should have to minimize the payoff of the receiver. He can do so by setting 20 + 70q and 60 – 30q equal, giving q = 0.4.

That means the server should mix forehands and the backhands in the ratio of 40:60 to maximize his payoff.

Limitations of Nash equilibrium

The following are the limitations in Nash equilibrium.

1. This is based on the fact, the player knows the strategies of its opponent. This can be rarely seen in practical situations where we are not aware of it. In a war torn situation, it is quite impossible to know the opponent’s steps.
2. Unlike dominant strategy, Nash equilibrium doesn’t always lead to an optimal solution.
3. In case of multiple games played with the same player, it doesn’t take into account his/her past decisions. In many of the cases, even the past decisions are helpful in identifying the future strategy.

Conclusion

Nash equilibrium relies on the fact that the player will continue to remain on its initial strategy knowing the strategy of his opponent. They won’t be changing their due course as no incentive is associated with it. Based on the knowledge of your opponent’s plan of action, it can provide the best pay off to apply and can be applied in different real life situations.