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Lecture 4 Game Plan
February Madness
Games with multiple Nash equilibria
… which equilibrium does evolution select?
… mixed strategies or pure strategies?
Sequential move games
… escaping from Annoying Servant Game
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Nash Equilibria Sequential Move Games-Game Theory For Managers-Lecture Slides, Slides of Game Theory
Content of Game Theory for Managers course includes ratiinality, prisoner dilemma, Loyal servant, assurance, credibility, stratigic subtitutes, entry, brinkmanship, negotiation, auctions, signaling, reputation etc. This file contains: Equilibrium, Sequential, Game, Nash, Move, Player, Assurance, Coordinate, High
Typology: Slides
2011/2012
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Lecture 4 Game Plan^ ^ February Madness^ ^ Games with multiple Nash equilibria^ ^ … which equilibrium does evolution select?^ ^ … mixed strategies or pure strategies?^ ^ Sequential move games^ ^ … escaping from Annoying Servant Game
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Mixed Strategies
“Ni bhionn an rath ach mar ambionn an smacht” “There is no luck except where
there is discipline”
- old Irish proverb
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Analysis of Bluffing Game ^ You get Good Card 15/48, about 1/3 ^ What do you do with Bad Card?^ ^ If you
never raise
, player B will always Fold when you have a Good Card.^ • get +100 when Good, -100 when Bad^ • average payoff about
-
^ If you
always raise
, player B will always Call you on it (even worse!)^ • get + 200 when Good, -200 when Bad^ • average payoff about
-
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How Often to Raise in Eqm? ^ Need to Raise enough for Player B tobe indifferent between Fold and Call^ ^ B gets –100 if Folds^ ^ B gets either –200 or +200 if Calls^ • By Call, B “risks 100 to gain 300” relative to Fold^ • So we need Prob(Bluff | Raise) = 25%^ ^ 15 Good Cards so we Bluff on 5 Bad Cards^ • So, Raise with 5/33 Bad Cards^ • When 1/3 chance of Good Card, Bluff with prob. 1/
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Payoffs in Equilibrium Player B Folds 33% of time^ ^ Good Card
Æ^ 33%(+100)+67%(+200),
so get
167 when Good Card
^ … & Player A indifferent to Raise orNot given a Bad Card^ ^ –100 when Bad Card ^ Overall payoff is about
–^
for A
^ much better than always/never bluffing
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Best responses in bluffing^ 100% Call^ 67% Fold^ 0%^
100%
0%^
17% Don’t Raise w/Bad Card
Raise w/ BadCard
^ Suppose other Raises & Folds^ ^ What’s your best response?
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Bluffing on a boundary
0%^
100% 17% 100%^ Zone I^ 67% Zone 0%
II
Zone IVZoneIII
Call^ Fold^ Don’t Raise w/
Bad Card
Raise w/ BadCard
^ Who will you beat if you choose onboundary of Zone I and Zone IV?
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Bluffing by equilibrium^ 100%
Zone I ZoneII
Zone IVZoneIII
Call^ 67%^ Fold^ 0%
100%
0%^
17% Don’t Raise w/Bad Card
Raise w/ BadCard
^ Who will you beat if you choose theequilibrium strategy?
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Which Side of the RoadShould We Drive On? ^ Map of the world showing which countriesdrive on the right (the majority) and whichdrive on the left. ^ Photograph of a 1967 pamphlet explainingSweden’s change to driving on the right. ^ Photograph of a car with a sign “Keep to theRight” on its dashboard reminding the driverof the new law. From Nova Scotia, 1923.
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Driving Game
Me
Left^
Right
Left You Right What are the Nash equilibria in this game?
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Mixed Strategies in theDriving Game
Me
Left^
Right^
Random
Left Right You Random
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Assurance Game
Column Player
Low High
High Row Player
Low
^ Key features:• Each wants to do the same thing as the other• Both better off if both choose High
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Mixed Strategies in theAssurance Game ^ Any player who mixes between twoactions
must be indifferent
between
those actions^ ^ This requires that
the other player
mix
with just the right probabilities to createthis indifference If Prob(High)=1/4 & Prob(Low)=3/4 is my^ mixture, then
you^ get payoff 3/4 no
matter what you do.^ ^ Each player adopts the better action
less
often (!!)
in the mixed-strategy eqm
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Reaction Curves inAssurance Game^ 100%
Row’s prob. of High Col’s prob.of High
25%^ 0%0%^ 25%
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