eco290e: game theory lecture 3 why and how is nash equilibrium reached?
Post on 19-Jan-2018
212 Views
Preview:
DESCRIPTION
TRANSCRIPT
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.
top related