Georges Zaccour Chair in Game Theory and Management, GERAD, HEC Montréal, Canada 1Georges Zaccour...

Georges Zaccour Universidad de Valladolid 1

International Environmental Agreements: Game Theoretic Approaches

Georges Zaccour Chair in Game Theory and Management, GERAD, HEC Montréal, Canada


Outline

Some generalities on IEAs

Game theory 101

IEA as a non-cooperative game

IEA as a cooperative game


IEA: Ingredients

n countries (transboundary context) Interdependent payoffs Examples: tropical forest, biodiversity, fisheries,

emissions reduction, etc. Asymmetry

Benefits Costs Views on means to be deployed

Long-term problem (inertia in technology and behavior)


IEA: Examples

Some treaties: Climate change (Kyoto Protocol, COP 15)

Ozone layer depletion (Montreal Protocol)

Acid rain (Sulphur Emissions Reduction Protocol)

Biodiversity loss (Biodiversity Convention)

http://www.cbd.int/

http://www.undp.org/chemicals/montrealprotocol.htm

http://en.wikipedia.org/wiki/Acid_rain

http://en.wikipedia.org/wiki/Acid_rain

http://www.cbd.int/

http://www.cbd.int/


IEA: Why they are needed?

Typical features of many environmental problems: Public good, Externalities, Free riding

Excludable Non-excludable

Rivalrous Pure private good Open-access resource (common good) Ocean fishery

Non-rivalrous Club goodWilderness Area

Public good Air, Pollution abatement


IEA: Why they are needed

Externalities: one agent’s decision has an impact on utility of other agents in an unintended way and when no compensation/payment is made by the generator.

Free riding

Provision of a public good usually leads to Market Failure

No international institution to correct this; voluntary international cooperation efforts to provide the public good.


Game theory 101

Branch of mathematics

Applications in many areas: Economics, politics, engineering, biology, ecology, computer science, etc.

Strategic Interactions between players (firms, countries, automata, etc.)

Payoff of one depends on what the others do

Optimization problem vs. game problem


Game theory 101

Cooperative vs. non-cooperative games

Description of a game Normal or strategic form Extensive form Characteristic function form


Game theory 101

Game in strategic form:

Set of players

Strategic players, dummy players (e.g., nature)

Set of strategies of each player

Payoffs

Function of selected strategies by all players


Game theory 101

Further assumptions:

Each player is rational

Common knowledge that each player is rational

Information: Perfect / imperfect; Complete / incomplete


Game theory 101

Non-cooperative game: Nash equilibrium No player has an interest in deviating unilaterally Best response (BR) to other players’ strategies Existence: Strategy set is compact and convex;

payoff is continuous and quasi-concave in own strategy; proof relies on a fixed-point argument of BR

Uniqueness: BR is a contraction


Game theory and IEAs

Non-cooperative approach Voluntary participation Club idea Mechanisms to increase participation

Cooperative approach Collective optimum Sharing of benefits (and costs)


Games which depict an IEA

Prisoners’ Dilemma

-1, -1

-9, 0

0, -9

-6, -6

Prisoner 2

Pri

son

er

1

Not Confess Confess

Not Confess

Confess

• Normal form representation: agents choose their strategy simultaneously

Prisoners’ Dilemma


Games which depict an IEA

Pollution’s Dilemma

-1, -1

-9, 0

0, -9 -6, -6

Prisoner 2

Pri

son

er

1

Abate

Pollute

Abate

Pollute

• Pollute is the dominant strategy for each player

Prisoners


Chicken game (Hawk-Dove)

Two drivers drive towards each other on a collision course: one must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a "chicken," meaning a coward;

Hawk-Dove refers to a situation in which there is a competition for a shared resource and the contestants can choose either conciliation or conflict.


Chicken Game

0, 0 -100, 100

100, -100

-1000, -1000

Swerve Do not swerve

Swerve

Do not swerve


Chicken Game

0, 0 -100, 100

100, -100

-1000, -1000

Abate Pollute

Abate

Pollute


Sequential play in Chicken game

Country 1

Country 2

pollute

abate

pollute

abate

abate

pollute

(-4, -4)

(5, -2)

(-2, 5)

(3, 3)

•Which equilibrium will be played?

•Extensive form game: sequential choice leads to a unique Nash Equilibrium

• Backward induction

•„First-Mover advantage“


First-mover advantage:

„The lady who pushes her child‘s stroller across the intersection in front of a car that has already come to a dead stop is in no particular danger as long as she sees the driver watching her: even if the driver prefers not to give her the right of way she has the winning tactic“

[Schelling (1966). Arms and Influence. New haven: Yale University Press, pp: 117-118]


Insurance game

0, 0 0, -8

-8, 0

4, 4

Country 2

Countr

y 1

Do not contribute Contribute

Do n

ot

contr

ibute

C

ontr

ibut

e

•Cost of contribution

•Benefit of contribution (only if both countries contribute)

•Two Nash equilibria

•Cooperative solution is stable


Multiple players

Cartel (club) problem Entry test Exit test

Self enforcing: no participant has an incentive to deviate and no non-participant has an incentive to accede to the agreement

Size of the agreement: how many countries will join?


Preliminary conclusions

Different payoff structures lead to different equilibria.

Signatories and non-signatories would both do better if all cooperate (prisoners’ dilemma)

Non signatories do better than the signatories, because they free-ride (chicken game)

Full cooperation is not usually stable (it is not self enforcing)



How can international treaties be structured, such that the mutually preferred outcome is an equilibrium?

Repeated games: cooperative equilibria become reachable

Fraction of members decreases when there are many countries affected

Breadth versus depths of an agreements Modest target?



Literature: predictions for self-enforcing agreements are rather pessimistic

Since treaties must be self-enforcing, they must do more than simply telling countries what to do. Treaties must manipulate the incentive structure of countries

How can the incentive structure be manipulated?



Existence of an external institution which coordinates the process

Leadership role by one important nation

Define minimum participation threshold (e.g., Kyoto)



Side payments to induce cooperation of the non contributors

Establish more agreements than only one. For instance for each group of countries which has particular characteristics; Kyoto protocol and developing countries

Linkage of negotiations and linked benefits.


Cooperative approach

Starting point: No obstacles to cooperation (economic, sociological, psychological, political, etc.)

Two-step algorithm

1. Determine the best collective outcome

2. Share this outcome


Tools of cooperative games

Characteristic function measure of strategic force of coalitions

Set of Imputations individual and collective rationality

Solutions Value-type solutions (unique imputation) Set of imputations


Most used solutions

Core: stable allocation (but core can be empty)

Shapley Value (linearity, Pareto-optimality, fairness) Each player gets a weighted average of her

marginal contributions


On fairness

Horizontal equity: One man, one vote Vertical equity: Altruism Market justice: Efficient allocation of

resources Sovereignty: No invasion of a player’s right Consensus : Diplomacy Compensation : Extension of Pareto... Principle of Rawls : Horizontal + vertical Shapley value: Symmetry, strategic force


A third step: sustainability

Dynamic problem: long-term agreement

How to guarantee sustainability? Binding agreement Time-consistent agreement Cooperative equilibrium

Mechanisms: Side payments, punishment, threat, etc.


Incentive equilibrium

Incentive Strategies I behave as a gentleman if you do the same (tooth for

tooth, eye for eye)

Incentive equilibrium Credibility

Two-player context


Trigger strategies

Agreement to punish a deviator

Based on past behavior

Random events

Discontinuities


Time consistency

Design an agreement

cooperative payoff-to-go dominates non-cooperative payoff-to-go

Not an equilibrium: a minimal requirement


General conclusion

Shed a light on a series of questions:

Will countries behave selfishly and continue to pollute?

Does mutually beneficial cooperation take place between independent states?

What can be done to increase the chances of cooperative behavior?


Yet more Difficulties

Modeling payoff functions

Measure of damages

Techno-economic models and large-scale optimization models

Learning

Non linearities (almost in everything!)

Correlations between pollutants


Advertising Spot

Jorgensen, Martín-Herrán, Zaccour (2010)

Zaccour (2008)

Breton, Sbragia, Zaccour (2010)

Other papers in forest mangement, climate-change negotiation, etc.

Georges Zaccour Chair in Game Theory and Management, GERAD, HEC Montréal, Canada 1Georges Zaccour...

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