Game Theory
Model
Do not take other agents’ actions as given – contributes consumer theory and gen eq
Agents are conscious of interdependence and that actions have consequences ¿
Information
Complete information: every player knows everything there is to know about the game,
including the characteristics eg cost functions of other players eg Prisoner’s Dilemma,
Stackelberg
Perfect information: when a player has to choose an action, they know what actions, if any,
have already been chosen up to that point by other players eg Stackelberg
Common knowledge assumption: every player knows that every other player knows the
characteristics of the game, knows they know they know, etc
Dominant Actions
Unique best response, regardless of what the other agent does
Payoffs and Best Responses
Payoff function is ui ( s i , s−i ), best response is si in response to s−i that maximises i’s payoff ie
ui ( s i , s−i ) ≥ ui ( s'i , s−i ), may be multiple best responses
Strictly dominant action: strict best response to every action profile s−i
o ⇒ ui ( si , s−i ) ≥ui ( s'i , s−i ) ∀ s−i ≠ si
Nash Equilibrium
Action profile s= ( s 1 , … , s N ) s.t. each player’s action is a best response to the actions of other
players – no profitable deviation
No player wants to change their action given what other players are doing
Stronger requirement than requiring rational behaviour alone, conjectures must be correct
Battle of Sexes
Players do not have dominant strategies – best response depends on actions of others
Y X
Y 1,1 0,0
X 0,0 1,1
Playing Chicken
Swerve Straight
Swerve 0,0 −1,1
Straight 1 ,−1 −10 ,−10
Could limit options ie put steering lock on, so B knows A will go straight, so B will swerve
A must ensure B knows this limitation, and that the threat is credible
Matching Pennies
H T
H −1,1 1 ,−1
T 1 ,−1 −1,1
B wants to match, A does not – so no pure strategy NE
Mixed Strategies
B chooses H w.p. p, and T w.p. 1− p
A gets payoff from H−1 ( p )+ 1 ( 1−p )=1−2 p, payoff of T 1 ( p )−1 ( 1− p )=2 p−1
In eqbm expected payoffs equal so 2 p−1=1−2 p ⇒ p=1/2
Repeat for A w.p. q ⇒ q=1/2
Best Responses
1
H a best response for A iff p ≤1/ 2, T iff p ≥1/2, p= ⇒ any q s . t . 0 ≤ q ≤1 is BR
2
So ( p , q ) =( 12 , 12 ) is NE in mixed strategies
Model
Do not take other agents’ actions as given – contributes consumer theory and gen eq
Agents are conscious of interdependence and that actions have consequences ¿
Information
Complete information: every player knows everything there is to know about the game,
including the characteristics eg cost functions of other players eg Prisoner’s Dilemma,
Stackelberg
Perfect information: when a player has to choose an action, they know what actions, if any,
have already been chosen up to that point by other players eg Stackelberg
Common knowledge assumption: every player knows that every other player knows the
characteristics of the game, knows they know they know, etc
Dominant Actions
Unique best response, regardless of what the other agent does
Payoffs and Best Responses
Payoff function is ui ( s i , s−i ), best response is si in response to s−i that maximises i’s payoff ie
ui ( s i , s−i ) ≥ ui ( s'i , s−i ), may be multiple best responses
Strictly dominant action: strict best response to every action profile s−i
o ⇒ ui ( si , s−i ) ≥ui ( s'i , s−i ) ∀ s−i ≠ si
Nash Equilibrium
Action profile s= ( s 1 , … , s N ) s.t. each player’s action is a best response to the actions of other
players – no profitable deviation
No player wants to change their action given what other players are doing
Stronger requirement than requiring rational behaviour alone, conjectures must be correct
Battle of Sexes
Players do not have dominant strategies – best response depends on actions of others
Y X
Y 1,1 0,0
X 0,0 1,1
Playing Chicken
Swerve Straight
Swerve 0,0 −1,1
Straight 1 ,−1 −10 ,−10
Could limit options ie put steering lock on, so B knows A will go straight, so B will swerve
A must ensure B knows this limitation, and that the threat is credible
Matching Pennies
H T
H −1,1 1 ,−1
T 1 ,−1 −1,1
B wants to match, A does not – so no pure strategy NE
Mixed Strategies
B chooses H w.p. p, and T w.p. 1− p
A gets payoff from H−1 ( p )+ 1 ( 1−p )=1−2 p, payoff of T 1 ( p )−1 ( 1− p )=2 p−1
In eqbm expected payoffs equal so 2 p−1=1−2 p ⇒ p=1/2
Repeat for A w.p. q ⇒ q=1/2
Best Responses
1
H a best response for A iff p ≤1/ 2, T iff p ≥1/2, p= ⇒ any q s . t . 0 ≤ q ≤1 is BR
2
So ( p , q ) =( 12 , 12 ) is NE in mixed strategies
