Games, Strategies, and Decision Making By Joseph Harrington, Jr. First Edition Chapter 4: Stable Play: Nash Equilibria in Discrete Games with Two or Three Players Prepared by Debashis Pal Copyright © 2009 by Worth Publishers Lecture Notes on

Game of chicken: Two cars move towards each other. As the cars come hurtling towards each other, each driver must decide whether to swerve to avoid a collision or to hang tough. If at least one driver swerves, a collision is avoided. If no one swerves, the cars get into a serious collision. The goal is to avoid being the first to swerve. What are the outcomes of the game of chicken? What are the “real world” examples of the game?

Game of chicken in movies:

The game of chicken:

How to solve the game of chicken: Neither player has a strictly dominated strategy. The Iterative Deletion of Strictly Dominated Strategies (IDSDS) won’t help us solve the game. We use the concept of Nash equilibrium to solve the game. Nash equilibrium is credited to the brilliant mathematician John Nash, who earned the Nobel Prize in Economics for his contribution. Have you seen the movie “A Beautiful Mind”?

Nash equilibrium A strategy profile is a Nash equilibrium if each player’s strategy maximizes his or her payoff, given the strategies used by other players. At a Nash equilibrium, no player has an incentive to unilaterally change his or her strategy. Nash equilibrium identifies strategy profiles that are stable in the sense that each player is content to do what she is doing, given what everybody else is doing. Behavior generated by a Nash equilibrium is expected to persist over time.

Finding Nash equilibria in the game of chicken: If Driver 2 chooses “hang tough” then Driver 1 is better off choosing “swerve.”

Finding Nash equilibria in the game of chicken : If Driver 1 chooses “swerve,” then Driver 2 is better off choosing “hang tough.”

Finding Nash equilibria in the game of chicken : There are two Nash equilibria. At each, one driver chooses “swerve” and the other driver chooses “hang tough.” For example, if Driver 1 chooses “swerve,” then it is best for Driver 2 to choose “hang tough,” and if Driver 2 chooses “hang tough,” then it is best for Driver 1 to choose “swerve.” Each driver’s strategy maximizes his or her payoff, given the strategy of the other driver.

Which driver will swerve? Nash equilibrium does not predict which driver will swerve and which driver will hang tough. Other specification(s) are needed to make that prediction. For example, if it is common knowledge that Driver 1 is crazy and always hangs tough, then at a Nash equilibrium, Driver 2 will swerve and Driver 1 will hang tough.

Prisoner’s dilemma game: Most widely examined game in game theory. Two members of a criminal gang have been arrested and placed in separate rooms for interrogation. Each is told that if one testifies against the other and the other does not testify, the former will go free and the latter will get three years of jail time. If each testifies against the other, they will both be sentenced to two years. If neither testifies against the other, each gets one year.

Prisoner’s dilemma game: There is a unique Nash equilibrium, where each testifies against the other. (The numbers in the box are not years in jail. They represent preference rankings.)

Driving conventions game: Two Nash equilibria: either both drivers drive on the left-hand side of the road, or both drivers drive on the right-hand side of the road.

A game of coordination and conflict: telephone Colleen is chatting with Winnie and suddenly they are disconnected. Should Colleen call Winnie, or should Winnie call Colleen? There are two Nash equilibria: (Call, Wait) and (Wait, Call).

Rock-paper-scissors game: Can you find a Nash equilibrium?

Try It: find all the Nash equilibria of the following game

Solution: There are three Nash equilibria: (a, z), (d, z), and (b, x).

Conflict and mutual interest in games: constant sum game If for every strategy pair, the sum of their payoffs is a constant number, the game is known as a constant sum game. When the number happens to be zero, the game is known as a zero sum game. The rock–paper–scissor game is a zero sum game. In a constant sum game, the players are in a conflict. If one player wins, the other player loses.

Conflict and mutual interest in games: mutual interest game Consider the driving conventions game. Both drivers prefer (Left, Left) and (Right, Right) to (Left, Right) and (Right, Left). If the rankings of the strategy pairs by their payoffs coincide for the players, the game is known as a mutual interest game. The driving conventions game is a mutual interest game.

Conflict and mutual interest in games: A game may lie in between a constant sum game and a mutual interest game. The strategic settings provide room for both conflict and mutual interest. In the game of chicken, both drivers want to avoid (hang tough, hang tough). They have a mutual interest. There is also room for conflict; the drivers disagree as to how they rank (hang tough, swerve) and (swerve, hang tough).

A useful technique to find the Nash equilibria: the best reply method Let us find out the Nash equilibria for the following game between Jack and Diane:

A useful technique to find the Nash equilibria: the best reply method For each of Diane’s strategies, let us circle the best outcome(s) for Jack.

A useful technique to find the Nash equilibria: the best reply method Next, for each of Jack’s strategies, let us circle the best outcome(s) for Diane.

A useful technique to find the Nash equilibria: the best reply method Any box where both numbers are circled corresponds to a Nash equilibrium outcome. The corresponding pair of strategies gives rise to a Nash equilibrium. In this game, there are two Nash equilibria: (a, y) and (b, x).

A useful technique to find the Nash equilibria: the best reply method Use the technique to see that the rock–paper–scissors game does not have a Nash equilibrium.

Three-player games Alicia, Kaitlyn, and Lauren will attend this week’s episode of American Idol. They are trying to decide what to wear for the occasion. They all support one of the contestants, Ace Young. They have planned to coordinate their T-shirts to spell “ACE.” Lauren will wear the “A,” Kaitlyn will wear the “C,” and Alicia will wear the “E.” However, each girl prefers to wear a Bebe shirt, and must independently decide whether to wear a lettered T-shirt or a Bebe shirt.

Three-player American Idol game: Each player gets a payoff of 2 if they all wear their lettered T-shirts. If at least one player does not wear the lettered T-shirt, then a player get 1 if she wears her Bebe top, and she gets 0 if she wears her lettered T-shirt.

Nash equilibria for the three-player American Idol game: Use the circling technique that we discussed earlier. There are two Nash equilibria (two strategy profiles in which all three payoffs are circled).

Promotion and sabotage Three candidates are running in U.S. presidential primaries. Each candidate has one unit of effort to spend. Each candidate may spend effort to enhance her own performance (positive effort) or to denigrate one of the two competing candidates (negative effort). Candidate i’s existing performance is V_i. If a candidate exerts positive effort, then her performance goes up by 1. If exactly one candidate exerts negative effort against i, then i’s performance goes down by 1. If two candidates exert negative efforts against i, then i’s performance goes down by = 4.

Promotion and sabotage Performance of Candidate i:

Promotion and sabotage If a candidate’s performance is better than the performances of the other two candidates, then she wins with probability 1, and her payoff is 1. If two candidates tie for the highest performance, then each has a 50% chance of winning and each candidate’s payoff is ½. If all three candidates have equal performances, then each has a 1/3 chance of winning and each candidate’s payoff is 1/3. A candidate has zero payoff if her chance of winning is 0.

Promotion and sabotage Suppose V_1 = 2 and V_2 = V_3 = 0. That is, candidate 1 is the front-runner. What are the Nash equilibria of this game? There are two Nash equilibria: Nash equilibrium 1: All candidates exert positive efforts. Nash equilibrium 2: Candidates 2 and 3 exert negative efforts against candidate 1. Candidate 1 exerts positive effort.

Find all the Nash equilibria for the following game:

Relationship between Nash equilibria and the strategies that survive IDSDS :