Transcript Game Theory and the Nash Equilibrium

Eponine Lupo
 Game
Theory is a mathematical theory that
deals with models of conflict and
cooperation.

It is a precise and logical description of a
strategic setting
 It
can be applied to many social sciences,
evolutionary biology, and has many
applications in economics.
 Game Theory is often used in more complex
situations where chance and a player’s
choice are not the only factors that are
contributing to the outcome.

Ex. Oil deposits
 Games—situations
where the outcome is
determined by the strategy of each player
 Strategy—a
complete contingent plan
outlining all the actions a player will do
under all possible circumstances
 Key
assumption: players are rational with
complete information and want to maximize
their payoffs
 Classic
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Games
Matching Pennies
Coordination
Battle of the Sexes
 Prisoner’s
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Dilemma
Normal Form
Extensive Form
Strategies—pure strategy set
Solution
Nash Equilibrium (D,D)
A
probability distribution over the pure
strategies for a player
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Must add up to 1 or 100%
Infinite number of mixed strategies
Choose a mixed strategy to keep opponents
guessing
Use a mixed strategy if the game is not solvable
using pure strategies (no cominant or efficient
strategies)

Dominance—Prisoner’s Dilemma
S1 is dominated by S11 if S11 gives Player 1 better
payoffs than S1, no matter what the other players do.
 Compares 1 strategy to another of a single player
 Iterated Dominance—Pigs
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Efficiency—Pareto Coordination
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S is more efficient than S1 if everyone prefers S to S1
Compares 2 strategy combinations involving all
players
S is efficient if there is nothing that’s more efficient
than S.
Best Response
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S1 is a Best Response to S2 if S1 gives player 1 the
highest payoff given player 2 is playing S2
 Named
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after John Nash
American mathematician
Subject of A Beautiful Mind
 Definition:
A strategy profile is a Nash
equilibrium if and only if each player’s
prescribed strategy is a best response to the
strategies of the others.
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No player can do better by unilaterally changing
his or her strategy
Equilibrium that is reached even if it is not the
best joint outcome
 Pure
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Some games do not have a pure strategy N.E.
One always exists in a mixed form
 All
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and Mixed Strategy N.E.
finite games have at least one N.E.
A N.E. will/must be played in the last stage
 In
a Mixed N.E., each player chooses his
probability mixture to maximize his value
conditional on the other player’s selected
probability mixture.
 Matching
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Pennies—mixed strategy only
(.5,.5)X(.5,.5)
 Coordination
 Prisoner’s
Dilemma
 Find
the Dominant strategies
 Find the Best Responses for each player
 Find the pure strategy N.E.
 Find
the mixed strategy N.E. for 2X2 games
 Find more than 1 mixed strategy NE
 2 player games with more than 2 strategies
 3 player games