Eponine Lupo

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Game Theory is a mathematical theory that deals with models of conflict and cooperation.
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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.
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Ex. Oil deposits

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Games —situations where the outcome is determined by the strategy of each player
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Strategy —a complete contingent plan outlining all the actions a player will do under all possible circumstances
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Key assumption : players are rational with complete information and want to maximize their payoffs

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Classic Games
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Matching Pennies
Coordination
Battle of the Sexes
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Prisoner’s Dilemma
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Normal Form
Extensive Form
Strategies—pure strategy set
Solution
Nash Equilibrium (D,D)

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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)

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Dominance —Prisoner’s Dilemma
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S
1 is dominated by S payoffs than S
1
1
1 if S
1
1 gives Player 1 better
, no matter what the other players do.
Compares 1 strategy to another of a single player
Iterated Dominance—Pigs
Efficiency —Pareto Coordination
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S is more efficient than S 1 if everyone prefers S to S 1
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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S
1 is a Best Response to S
2 if S
1 gives player 1 the highest payoff given player 2 is playing S
2

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Named after John Nash
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American mathematician
Subject of A Beautiful Mind
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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

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Pure and Mixed Strategy N.E.
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Some games do not have a pure strategy N.E.
One always exists in a mixed form
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All finite games have at least one N.E.
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A N.E. will/must be played in the last stage
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In a Mixed N.E., each player chooses his probability mixture to maximize his value conditional on the other player’s selected probability mixture.

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Matching Pennies—mixed strategy only
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(.5,.5)X(.5,.5)
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Coordination
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Prisoner’s Dilemma

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Find the Dominant strategies
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Find the Best Responses for each player
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Find the pure strategy N.E.

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Find the mixed strategy N.E. for 2X2 games
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Find more than 1 mixed strategy NE
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2 player games with more than 2 strategies
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3 player games