ECO290E: Game Theory Lecture 3 Why and How is Nash Equilibrium Reached?

Three Reasons for NE 1.By rational reasoning 2.A result of discussion 3.A limit of some adjustment process  Which factor serves as a main reason to achieve Nash equilibrium depends on situations.

1. Rationality Players can reach Nash equilibrium only by rational reasoning in some games, e.g., Prisoners’ dilemma. However, rationality alone is often insufficient to lead to NE. (see Battle of the sexes, Hawk-Dove game, etc.) A common (and correct) belief about future actions combined with rationality is enough to achieve NE.  2 and 3 help players to share a correct belief.

Focal Point A correct belief may be shared by players only from individual guess.  Class room experiments, i.e., Choose one city in Japan! (you will win if you can choose the most popular answer)  Most of the students named “Tokyo.” Like this experiment, there may exist a Nash equilibrium which stands out from the other equilibria by some reason.  Focal Point (by Thomas Schelling)

2. Self-Enforcing Agreement Without any prize or punishment, verbal promise achieves NE while non equilibrium play cannot be enforced.  NE=Self-Enforcing Agreement Example: Prisoners’ Dilemma Even if both players promise to choose “Silent,” it will not be enforced since (S,S) is not a NE.

3. Repeated Play Through repeated play of games, experience can generate a common belief among players. Example: Escalator Either standing right or left can be a NE. Example: Keyboard “Qwerty” vs. “Dvorak”  History of adjustment processes determine which equilibrium is realized: Economic history has an important role.  “Path Dependence” (by Paul David)

Roles of Social Science Analyze the frequently observed phenomena and explain the reason.  NE serves as a powerful tool. Predict what will happen in the future.  Although it is usually difficult to make a one-shot prediction, NE may succeed to predict the stable situation after some (long) history of adjustment processes.

What is Rationality? A player is rational if she chooses the strategy which maximizes her payoff given other players’ strategies. The Definition implies that a rational player takes a dominant strategy whenever it is available. never takes (strictly) dominated strategies.

Dominance Let x and y be feasible strategies for player i. Then strategy x is strictly dominated by y if the following is satisfied: That is, x is strictly dominated by y when y gives i strictly higher payoffs than x does irrespective of other players’ strategies.

Iterated Elimination of Strictly Dominated Strategies Player 2 Player 1 LeftMiddleRight Up Down

Rational Solution Step 1: “Right” is strictly dominated by “Middle,” so player 2 never takes “Right.” Step 2: Given the belief that Player 2 never takes “Right,” “Down” is strictly dominated by “Up.” Therefore, Player 1 will not take “Down.” Step 3: Given the belief that Player 1 will not take “Down,” “Left” is strictly dominated by “Middle.” Therefore, Player 1 will not take “Left.” Step 4: Only (Up, Middle) is survived after the iterated elimination process!  This reasonable solution coincides with NE.