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Game Theory and the Environment
Game Theory and the Environment
November 8, 2011
Game Theory and the Environment
Game Theory
Static Games of Complete Information
Game theory attempts to mathematically capture behavior instrategic situations
Normal Form Game: Each Player simultaneously choose astrategy, for instance,
Eliminate externalities: Noise, Garbage, emissions, etc..
Game Theory and the Environment
Game Theory
Normal Form Game
Classical Example: Prisoners�Dilemma "Two suspects arearrested and charged with a crime"
Prisoner 2Not Confess Confess
Prisoner 1 Not Confess �1,�1 �6, 0Confess 0,�6 �3,�3
Game Theory and the Environment
Game Theory
Normal Form Game
The Normal Form representation of a game speci�es:
1 The players in the game2 The strategies available to each player3 The payo¤ received by each player for each combination ofstrategies
�Chicken Game �: Assumptions:
1 Two Countries: Country X and Country Y2 Countries must choose whether or not to abate pollution.3 Abatement cost $74 Bene�ts $5 (To both countries)5 Doing nothing exposes both countries to serious pollutiondamage. Cost $4
Game Theory and the Environment
Game Theory
Chicken Game
PolluteAbate
Pollute (4, 4) (5, 2)
Abate (2, 5) (3, 3)
YX
Country X’sStrategies
Country Y’s Strategies
Country X’sPayoff
Country Y’sPayoff
Game Theory and the Environment
Game Theory
How do we obtain the Nash equilibrium solution?We have to keep �x the strategy of the other player (orcountry) and to analyze what is the best response for theplayer who does not have its strategy �xed. For instance:
Let us assume that country Y always pollutes, so what iscountry X�s best response in this speci�c case?
UX (AbatenPollute) > UX (PollutenPollute),since � 2 > �4
Pollute Abate
Pollute (4, 4) (5, 2)
Abate (2, 5) (3, 3)
YX
Game Theory and the Environment
Game Theory
Chicken Game
Now, let us assume that country Y always abates, so what iscountry X�s best response in this speci�c case?
UX (AbatenAbate) < UX (PollutenAbate),since 3 < 5
Pollute Abate
Pollute (4, 4) (5, 2)
Abate (2, 5) (3, 3)
YX
Game Theory and the Environment
Game Theory
Chicken Game
The second step: Analyze what happens when country Xalways abates.So the question will be: what is country Y �s best response inthis speci�c?
UY (AbatenAbate) < UY (PollutenAbate),since 3 < 5
Pollute Abate
Pollute (4, 4) (5, 2)
Abate (2, 5) (3, 3)
YX
Game Theory and the Environment
Game Theory
Chicken Game
Finally, let us assume that country X always pollutes, sowhat is country Y �s best response in this speci�c?
UY (AbatenPollute) < UY (PollutenPollute),since � 2 > �4
Pollute Abate
Pollute (4, 4) (5, 2)
Abate (2, 5) (3, 3)
YX
Therefore, we have can say that the Nash equilibrium of theChicken game is {(5,�2) and (�2, 5)} or {(abate, pollute)and (pollute, abate)}.
Game Theory and the Environment
Game Theory
Sequential game
Extensive form
Sequential game:
Country X chooses �rst (First Mover of the game)Country Y then observes country X�s choice and decides on itsown action.
Game Theory and the Environment
Game Theory
Sequential game
Extensive form
The solution of this game can be found by the method ofbackward induction
We will star analyzing the bottom of the tree, it means,country Y �s best response
If country Y observes that country X has chosen to pollutethen country Y �s best response is to abate (since �4 < �2)If country Y observes that country X has chosen to abatethen country Y �s best response is to pollute (since 3 < 5)
Country X
Country Y
Country Y
Pollute
Pollute
Pollute
Abate
Abate
Abate
(4, 4)
(3,3)
(5, 2)
(2, 5)
Game Theory and the Environment
Game Theory
Sequential game
Extensive form
Now, lets us analyze country X�s best response (assumingthat country X knows country Y �s best response)
Country X knows that when it chooses to pollute country Ywill abateand when it chooses to abate country Y will decide to pollutethen country X best response is to abate (since 5 > �2).
Country X
Country Y
Country Y
Pollute
Pollute
Pollute
Abate
Abate
Abate
(4, 4)
(3,3)
(5, 2)
(2, 5)
Therefore the Nash equilibrium of this game is (5,�2) wherecountry X pollutes and country Y abates
First Mover Advantage