Game Theory and the Nash Equilibrium
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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
Games
Matching Pennies
Coordination
Battle of the Sexes
Prisoner’s
Dilemma
Normal Form
Extensive Form
Strategies—pure strategy set
Solution
Nash Equilibrium (D,D)
A
probability distribution over the pure
strategies for a player
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
Efficiency—Pareto Coordination
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
S1 is a Best Response to S2 if S1 gives player 1 the
highest payoff given player 2 is playing S2
Named
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.
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
Some games do not have a pure strategy N.E.
One always exists in a mixed form
All
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
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