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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:
si  S i
u i ( x, s  i )  u i ( y , s  i )
• 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
Left
2
Player 1
Up
Middle
0
1
Down
2
1
3
0
Right
1
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.