Download a > -r

Section 8: Redux
Outline
• Overview of equilibrium concepts so far
• Example of perfect Bayesian equilibrium
• Example of monetary policy (see handout
from last section)
– Theories of delegation
– Time-consistency
Overview
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Nash Equilibrium
Mixed-strategy Nash Equilibrium
Subgame Perfect Equilibrium
Bayesian Equilibrium
Perfect Bayesian Equilibrium
Key Terms
• Equilibrium path: a node to node path observed in
equilibrium
– A strategy off the equilibrium path is never observed
– Any branch on which there is positive support (probability) for
play forms part of the equilibrium path
– There may be multiple equilibrium paths if there are multiple
equilibria
• In Nash equilibrium, the credibility of best responses judged along
equilibrium path; off the path, NE may include incredible strategies
• In subgame perfect NE, best replies are judged in every subgame
• In perfect Bayesian equilibrium, best replies judged at each
information set
• Beliefs: a probability distribution for a particular
information set
– In mixed-strategy NE, a belief is a probability distribution over
the nodes in the other player’s information set
– In Bayes Nash equilibrium, a belief is a probability distribution
over the nodes in nature’s information set and the corresponding
type of other player
– In perfect Bayesian equilibrium, a belief is a probability
distribution over the nodes in one’s own information set.
Nash Equilibrium
• A strategy profile S is a Nash equilibrium if and
only if each player is a playing a best response
to the strategies of the other players
• A strategy profile S is a strict Nash equilibrium if
and only if each player’s strategy is the single
best-response to the strategies of the other
players
• Note that there is no strategic uncertainty
– Each player’s belief about another’s strategy is
concentrated on actual strategy the other player uses
Mixed-Strategy Nash Equilibrium
• A mixed-strategy Nash equilibrium is a mixedstrategy profile whereby each player is playing a
best response to the others’ strategies
– Consists of a probability distribution over the set of
strategies, for each player
• Of course, probability density may be 0 for some strategies
– To solve, look for a mixed strategy for one player that
makes the other players indifferent between a
subset of their pure strategies
• If a player mixes over a set of strategies, it must be the case
that each of those strategies yields the same expected payoff
• Thus the player is indifferent about which strategy is played
• This indifference will occur when other players are mixing
over their own strategies in the appropriate way
• Useful for simultaneous games
Subgame Perfect Equilibrium
• A strategy profile is called a subgame perfect
Nash equilibrium if it specifies a Nash
equilibrium in every subgame
• The SPNE is the “no bluffing” equilibrium.
– Here, all strategies are credible
• Think of the pirate game: (99,0,1) is SPE, but there are many
not-credible NE of the form (100-x, 0, x)
• Grim trigger strategies are supported as
subgame perfect equilibria
– Player 1 cooperates, given that player 2 will punish
– Cooperation is a best-response if the future matters
– Grim trigger is credible: NE in every subgame
• Useful for sequential games
Bayesian Equilibrium
• Bayesian equilibrium consists of
– Each player’s strategy, which is a best response,
given the strategies of the other players,
– and given players’ beliefs (prior probabilities) about
the probability distribution over moves by nature
• Nature’s moves determine the ‘type’ of player, where ‘type’
corresponds with the payoffs associated with that player
• Useful for simultaneous games of incomplete
information
– Typically, transform such games into games of
imperfect information and different ‘types’, modeled
as a simultaneous game
– Helpful to make a normal form (given beliefs about
nature) and solve
Perfect Bayesian Equilibrium
• A perfect Bayesian equilibrium is:
– a belief-strategy pairing, over the nodes at all information sets,
such that
• each player’s strategy specifies optimal actions, given everyone’s
strategies and beliefs, and
• beliefs are consistent with Bayes’ rule whenever possible
• Beliefs mean something new here!
– Probability distribution over location in the information set.
• Strategies corresponding to events off the equilibrium
path may be paired with any beliefs, because these
strategies are not consistent with Bayes’ Rule (and occur
with probability zero.)
• Two major types of equilibria:
– Pooling: the types behave the same—updated beliefs=old beliefs
– Separating: the types of behave differently
• Useful for sequential games of incomplete information
Perfect Bayesian Equilibrium
Nuclear Deterrence Example
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2 players, each contemplating nuclear war
Neither knows if the other is about to strike
Each can attack (A, a) or delay (D, d)
If war, there’s a 1st strike advantage
Outcomes:
– No war results in (0, 0)
– The outcome of being the first-striker in war is –a
– The outcome of NOT being the first-striker in war is –r
• Assume 0 > -a > -r
Perfect Bayesian Equilibrium
Nuclear Deterrence Example
1
½
A
(-a, -r)
2
D
N
½
2
Such that
0 > -a > -r
(-r, -a)
a
d
(0, 0)
(-a, -r)
(-a, -r)
A
a
d
1
D
(0, 0)
Perfect Bayesian Equilibrium
Nuclear Deterrence Example
1
½
A
(-a, -r)
2
D
N
½
2
Such that
0 > -a > -r
P(
(-r, -a)
a
d
and
Player 2 forms beliefs over
and
Equilibrium will look like:
p = Prob(Player 1 plays D), m = Prob( )
d
(0, 0)
(-a, -r)
(-a, -r)
A
a
Player1 forms beliefs over
q = Prob(Player 2 plays d), n = Prob( )
1 Play D: m*[q*0+(1-q)*(-r)]+(1-m)*0
= m*(1-q)*(-r)
1
D
Bayes Rule
|at info set) = P(at info set | )*P(
(0, 0)
) / P(at info set)
1 Play A: m*(-a)+ (1-m)*(-a)
= -a
Always play A if m = 1, q = 0, else play D
PBE: (A, a, 1, 1)…though Nature goes (½, ½).
Another: (D, d, ½, ½)… again, by symmetry, and because beliefs are consistent w/ play.
PBE outcomes are consistent w/ players’ PBE beliefs
Only 1 more PBE: When m*(1-q)*(-r) = -a and m = (½) / (½+½*q) -----This is Bayes’ rule!
-a = -r*(1-q) / (1+q), q* = (r–a)/(r+a) = p*, m* = (r–a)/2*(r+a) = n*: PBE: (p*, q*, m*, n*)