Game Theory: Nash Equilibrium

An introduction to solving for Nash Equilibrium in non-cooperative games

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Welcome back! Last time we looked at games involving Prisoner Dilemma and how to solve them. I introduced both sequential and simultaneous games with my examples focusing on the later. We also discussed games of complete information and perfect rationality. Now we will look at the Nash Equilibrium solution for solving games.

Nash equilibrium was developed by the late John Nash (pictured). A mathematician who in 1994 won the Nobel Memorial Prize in Economic Sciences for his contribution to non-cooperative games (the games we are currently analyzing). Nash Equilibrium has been widely used in the fields included but not limited to economics, political science, and evolutionary biology. In 2001, the film A Beautiful Mind was released which won the Academy Award for Best Picture (one of my favorite films).

Nash

A Nash Equilibrium is defined as a profile of strategies such that each player’s strategy is an optimal response to the other players’ strategies (What?). Let’s look back at the example of Prisoner’s Dilemma:

1\2 Silent Betray
Silent -1,-1 -6,0
Betray 0,-6 -4,-4

Remember, a major premise of game theory is each player will look to optimize (or maximize) their utility (or payoff). With this in mind, we know that both players 1 and 2 will choose to betray each other. This results in a payoff of -4 for both players. You may be wondering why the players would end up with payoff of -4 when better payoffs exist. If player 1 chose the strategy of silent, player 2 could choose the strategy of betray resulting in payoff of -6 which is not optimal. So player 1’s best interest is to choose the strategy of betray.

1\2 Silent Betray
Silent -1,-1 -6,0
Betray 0,-6 -4,-4

Now let’s look at player 2. Following the premise of optimization and perfect rationality, player 2 would also choose the strategy of betray. By definition the strategy choices of (Betray, Betray) with payoff (-4,-4) is a Nash Equilibrium! It would not be optimal for Player 1 to choose silent since Player 2 would optimize by choosing betray and vice versa.

1\2 Silent Betray
Silent -1,-1 -6,0
Betray 0,-6 -4,-4

Let’s now take a look at an example from a previous blog and solve for Nash Equilibrium.

1\2 Talk Avoid
Talk 4,4 -2,6
Avoid 6,-2 1,1

In this scenario you frequent the night scene with your friends and are at a club (been there done that). You see someone from across the way and take in interest in that person. You have the strategy of either talking to that person of interest or avoiding them (so does the individual). If you both choose to talk to each other you both get a payoff of 4 (barring an awkward conversation). If you both choose to not talk to each other then you get a payoff of 1 (avoid embarrassment). But you can face the ultimate humiliation if you choose to talk and the person chooses to avoid or vice versa (people can be ruthless). The Nash Equilibrium is (Avoid, Avoid) with a payoff of (1,1).

1\2 Talk Avoid
Talk 4,4 -2,6
Avoid 6,-2 1,1

Going forward we will be solving games using Nash Equilibrium. Feel free to follow my blog and leave comments below as your feedback is appreciated.

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