Lecture 5: Mixed Strategies
Advanced Microeconomics I
Yosuke YASUDA
National Graduate Institute for Policy Studies
December 24, 2013
Why and How is Nash Equilibrium Reached?
There are at least four reasons why we can expect Nash equilibrium (NE) would realize:
1. By rational “reasoning” 2. Being an “outstanding choice” 3. A result of “discussion”
4. A limit of some “adjustment process”
Which factor serves as a main reason to achieve NE 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, Chicken game, etc.) A correct belief about players’ future strategies combined with rationality is enough to achieve NE. 2 - 4 help players to share a correct belief.
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Ex Prisoners’ Dilemma✆
Each player’s best response is “confess” independent of other player’s strategy.
2. Focal Point:
A correct belief may be shared by players only from individual guess.
Example : Choose one number from {1, 2, 3, 4, 100}. (you will win if you can choose the most popular answer)
⇒ Most of the players choose “100,” because 100 looks distinct from other 4 numbers.
Like this example, there may exist a Nash equilibrium which stands out from the other equilibria by some reason. Such a outstanding Nash equilibrium is called a focal point (discovered by Thomas Schelling).
3. Self-Enforcing Agreement
Without any prize or punishment, verbal promise achieves NE whereas non-equilibrium play cannot be enforced, that is, Nash equilibrium = self-enforcing agreement.
4. Repeated Play
Through repeated play of games, experience can generate a common belief among players.
Examples
◮ Escalator: Either standing right or left can be a NE.
◮ Keyboard: “Qwerty” vs. “Dvorak”
History of adjustment processes determines which equilibrium would be realized, which is sometimes mentioned as path dependence(proposed by Paul David).
Indifference Property (1)
Indifference Property Since expected utilities are linear in the probabilities, if a player uses a (non-degenerate) mixed strategy in a Nash equilibrium, she must be indifferent between all pure strategies to which she assigns positive probability.
That is, for any two pure strategies si and s′i chosen with positive probabilities in equilibrium, i.e., σi∗(si), σ∗i(s′i) > 0, we must have
ui(si, σ−i∗ ) = ui(s′i, σ∗−i) = ui(σi∗, σ−i∗ ). Application Asymmetric Matching Pennies
Consider a slightly modified version of the matching pennies (assume x > 0):
1 2 Heads Tails Heads −x, x 1, −1
Indifference Property (2)
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Q Calculate a Nash equilibrium. For what values of x, does✆ player 2 choose “Heads” with higher probability than “Tails”?
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A Let p✆ i be the probability for player i to choose “Heads.” Then, by indifference property, player 1 must be indifferent between the two strategies,
u1H = p2(−x) + (1 − p2) = p2− (1 − p2) = u1T.
⇒ p2 = 2 3 + x.
Similarly, the indifference property for player 2 gives, u2H = p1x− (1 − p1) = −p1+ (1 − p1) = u2T.
⇒ p1 = 2 3 + x.
