ECO290E: Game Theory

Lecture 3Why and How is Nash Equilibrium Reached?

Three Reasons for NE1. By rational reasoning2. A result of discussion3. 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 onl

y from individual guess. Class room experiments, i.e., Choose one cit

y 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.

),(),( iiii

ii

syusxuSs

Iterated Elimination of Strictly Dominated Strategies

Player 2Player 1

Left Middle Right

Up 0 1

2 1

1 0

Down 3 0

1 0

0 2

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.