Nash Equilibrium
What is the Nash Equilibrium? Have you ever thought about how our emotional feelings can affect your decisions? Sometimes, it is better to choose mathematics. Well, Nash Equilibrium is something you should learn in order to make less emotional decisions in particular cases. Keep reading!
In the game theory, the Nash equilibrium is about a solution to the games where two people know each other’s strategy. This solution method can be applied to many different situations in real life such as economics or even to the game “rock-paper-scissors”.
Mixed Strategies Nash Equilibrium
Mixed Strategy Nash Equilibrium is used when two players shouldn’t randomize because the payoffs are not the same. There are two players and two different strategies in this game, A and B. Let’s say, strategy A is to help somebody and B is to refuse to help somebody. The numbers inside the boxes show how much money they are going to have according to their choices. The first number shows the money player 1 will get and the second number shows the money player 2 will get. Columns refer to player 1’s choices and rows refer to player 2’s choices. Both players know how much each player get according to their choices. For example, they both know that if they both choose A to play, player 1 will get 3 dollars and player two will get only 1 dollar. Our purpose is to find out how much probability should a player choose strategy A or strategy B in order to get as much money as possible.
STEP 1: Figure out player 1’s the payoffs for each situation.
Both players are clever thus they consider each other's payoffs before they choose which strategy to play.
Let’s say x = P( player 2, A). Then 1-x = P( player 2, B)
Let us define player 1’s payoff from each strategy.
If player 1 chooses A to play, the payoff will be 3.x +(-2).(1-x) = 5x-2 dollars.
If player 1 chooses B to play, the payoff will be 0.x +2.(1-x) = 2–2x dollars.
Having defined the payoffs for player 1, player 2 can decide the best possible x value now.
STEP 2: Make the graphic to see easier.
x is the probability of player 2 choosing A, as we have defined. Player 2 is clever thus he defines the payoffs for each situation and makes the graphic. What we understand from the graphic is if x =0, Player 1 will choose the strategy B. Likewise, If x ≤ 4/7, player 1 will choose the strategy B. If x ≥ 4/7. The junction point is the point which gives the lowest payoff for player 1. This is why player 2 should define x according to the junction point. Thus the probability of player 2 choosing strategy A is
5x-2=2–2x ⇒ 7x = 4 ⇒ x=4/7.
STEP 3: Figure out player 2’s the payoffs for each situation.
Let’s say y = P( player 1, A) so 1-y = P( player 1, B)
Let us define player 2’s payoff from each strategy.
If player 2 chooses A to play, the payoff will be 1.y + 4.(1-y) = 4–3y
If player 2 chooses B to play, the payoff will be 1.y. + 1(1-y) = 1
Having defined the payoff for player 2, player 1 can decide the best possible y value now.
STEP 4: Make the graphic to see easier.
y is the probability of player 1 choosing A, as we have defined. Player 1 is clever as well as player 2 thus player 1 also calculates player 2’s payoff for each situation and makes the graphic. In order to make player 2 get less money player 1 should define y according to the payoffs. If y<1 player 2 will choose strategy A to play in order to gain more money.
The junction point would be the best point to consider when player 1 is deciding the probability. Thus, 4–3y=1⇒y=1. This means player 1 had better choose strategy A to play.
PRISONERS DILEMMA
The most popular example given in the game theory field is prisoners dilemma both because of its mathematical view and psychological view. It was originally framed by Merril Flood and Melvin Dresher working at RAND during 1948 and 1950. The jail punishment and the name “Prisoners Dilemma” was given to the problem by Canadian mathematician Albert W. Tucker.
From Wikipedia,
Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:
If A and B each betray the other, each of them serves two years in prison
If A betrays B but B remains silent, A will be set free and B will serve three years in prison (and vice versa)
If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).
Let’s figure out the diagram.
Now, let us define the payoff for A. As you see from the diagram, the payoffs from every single situation is symmetric thus it is expected to have the same payoffs for A and B. It will be adequate to just define the payoff for A.
Let’s say x = P( B, stays silent) so 1-x = P( B, betrays)
If A stays silent, the payoff will be (-1).x + (-3)(1-x) = 2x-3.
If A betrays, the payoff will be 0.x + (-2).(1-x) = 2x-2.
Now, let us make the graphic.
If B is a person who wants his accomplice to suffering, then B should not stay silent which means x = 0. If A is also that kind of person, A should also betray rather than staying silent. We can approach the problem from a different viewpoint.
If B cooperates, A should defect because going free is better than serving 1 year. If B defects, A should also defect, because serving 2 years is better than serving 3. So either way, A should defect. Parallel reasoning will show that B should defect.
You can easily figure out the diagrams and the graphic for the game “rock-paper-scissors” which will be pretty funny and enjoyable. Also, adjusting these games with more emotional cases might give some surprising strategies that might contradict with your feelings. :)
Resources,
1 Prisoner’s Dilemma. Wikipedia. Web. 08.04.2020
2 Nash Equilibrium. Wikipedia. Web. 01.04.2020
3 Nash Equilibrium. Colombia University. Web. 02.04.2020
4 Oyunlar Kuramı 11 (Ali Nesin). Youtube. Web. 30.03.2020
