Defining Nash Equilibrium
Defining Nash Equilibrium DreamWorks initially threw down the gauntlet in the clash of the ’toon titans way back in June 2002, claiming a release date of November 5, 2004 for Sharkslayer. . . . The studio’s staking out of the November date was seen as a slap at Disney, which has traditionally released its Pixar pictures that month. Disney . . . kicked up the brinkmanship factor, announcing that Sharkslayer or no, The Incredibles would also open on November 5. . . . DreamWorks [then] blinked [as it] moved the release date for its film . . . [to] October 1.1
IN SPITE OF ITS REFERENCE to a nonlethal passive fowl, Chicken is a dangerous game. In its classic form, it begins with two cars facing each other in duel-like fashion (and typically occupied by male teenagers in pursuit of testosterone- inspired adventures). As the cars come hurtling towards one another, each driver is frantically deciding whether to swerve to avoid a collision or to hang tough (hoping that the other will swerve). The goal is to avoid being the first to swerve, although if both hang tough, then the result is a mangled mess of metal and flesh. Chicken has been played in many contexts, including contests between movie executives (with release dates) and between the leaders of the United States and the former Soviet Union (with nuclear weapons). TABLE 4.1 lists a few other games of Chicken that have arisen in fact and fiction.
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Stable Play: Nash Equilibria in Discrete Games with Two or Three Players
TABLE 4.1 CHICKEN IN ACTION
Mode Description
Tractors Footloose (1984, movie)
Bulldozers Buster and Gob in Arrested Development (2004, TV)
Wheelchairs Two old ladies with motorized wheelchairs in Banzai (2003, TV)
Snowmobiles “[Two adult males] died in a head-on collision, earning a tie in the game of chicken they were playing with their snowmobiles.” <www.seriouslyinternet.com/278.0.html>
Film release dates Dreamworks and Disney–Pixar (2004)
Nuclear weapons Cuban Missile Crisis (1963): “Since the nuclear stalemate became apparent, the Governments of East and West have adopted the policy which Mr. Dulles calls ‘brinksmanship.’ This is a policy adapted from a sport which, I am told, is practised by some youthful degenerates. This sport is called ‘Chicken!’ ” (Bertrand Russell, Common Sense and Nuclear Warfare, 1959)
90 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
FIGURE 4.1 provides a strategic form representation of Chicken.* Because neither player has a strictly dom- inated strategy, the iterative deletion of strictly domi- nated strategies (IDSDS, Section 3.4.2) won’t help us solve this game. But don’t forsake hope, as game the- ory has many more game-solving tricks to offer.
If you’ve either read the book or seen the movie A Beautiful Mind, then you know about the brilliant schizophrenic mathematician John Nash. In his
doctoral thesis at Princeton University, Dr. Nash made two striking game- theoretic advances—one of which became known as Nash equilibrium—that resulted in his winning the Nobel Prize in Economics more than 40 years later.
To understand what Nash equilibrium is and why it is an appropriate method for solving a game, let us return to the discussion of the previous chapter. In the context of a game, a player is rational when he chooses a strat- egy to maximize his payoff, given his beliefs about what other players will do. The tricky part is figuring out what is reasonable for a player to believe about the strategy another player will select. Chapter 3 used the assumption that ra- tionality is common knowledge among the players to derive those beliefs. For example, if player 2 has a dominant strategy and player 1 believes that player 2 is rational, then player 1 believes that player 2 will use her dominant strat- egy. In this manner, we derived player 1’s beliefs regarding player 2’s strategy.
The approach of Nash equilibrium maintains the assumption that players are rational, but takes a different approach to nailing down beliefs. What Nash equilibrium does is require that each player’s beliefs about other players’ strate- gies be correct. For example, the strategy that player 1 conjectures that player 2 will use is exactly what player 2 actually does use. The definition of Nash equi- librium is then made up of two components:
1. Players are rational: Each player’s strategy maximizes his payoff, given his beliefs about the strategies used by the other players.
2. Beliefs are accurate: Each player’s beliefs about the strategies used by the other players are true.
Condition (1) is innocent enough; it’s condition (2) that is tougher to swallow. It requires that players be effective prognosticators of the behavior of others. In some settings, that may be a reasonable assumption; in others, it may not. Combining the assumptions about behavior—that it is always rational—and be- liefs—that they are always true—gives us the definition of Nash equilibrium.
✚ DEFINITION 4.1 A strategy profile is a Nash equilibrium if each player’s strategy maximizes his or her payoff, given the strategies used by the other players.
With n players, there are, then, n conditions that must be satisfied in order for a strategy profile to be a Nash equilibrium—one condition for each player
FIGURE 4.1 The Game of Chicken
2,2
3,1
1,3
0,0 Driver 1
Driver 2
Swerve
Hang tough
Swerve Hang tough
*You might be tempted to put large negative numbers for the strategy pair in which both participants choose hang tough, since this means certain injury and possible death. You can do so, but it’ll make no dif- ference as regards the solution. As long as the payoffs when both hang tough are less than all the other payoffs in the matrix, our conclusions regarding behavior will be the same. This condition reflects the property that what matters is the ranking of the payoffs, not their actual values.