Dynamic Games with Applications to Climate Change
Treaties
Prajit K. Dutta
Roy Radner
MSRI, Berkeley, May 2009
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
1 / 27
Motivation: Dynamic Commons
Principal features of climate change problem
Global Common - sources of carbon buildup are local but global
stock determines warming
Near-irreversibility - stock of GHGs depletes slowly so e¤ect of
current emissions is felt into distant future
Asymmetry - some regions will su¤er more than others
Nonlinearity & Uncertainty - costs can be very nonlinear and driven
by "unknown" & uncertain processes
Models need to accommodate these features
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
2 / 27
Motivation: Games & Treaties
Focus of current research is on treaty-formation
Main Question - what agreements or treaties will sovereign nations
sign & then carry out?
That requires a
Strategic Model - Although the players (countries) are relatively
numerous, there are some very large players and blocks of like-minded
countries
"Philosophical Perspective" - Climate Change is an International Issue
and hence a Treaty cannot be mandated but has to be
Incentive-Compatible. The Treaty needs to be an equilibrium of the
game
Additional requirement
Simplicity in Solution - Desirable for Political Reasons
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
3 / 27
Overview
Talk in Two Parts
Part I - Dynamic (Commons) Games: Set-Up, Strategies,
Equilibria, Applicability to Climate Change Treaties
Part II - Economics of Climate Change: Recent Research & Results
Part I builds on literature going back to Shapley (1953) and Levhari &
Mirman (1973). Recent work of relevance includes
Abreu-Pearce-Stachetti (1990) and Dutta & Sundaram (2005).
Part II builds on recent work by Dutta & Radner (2004, 2006, 2008),
Dockner et al (1999, 2006), Barrett (2006) and others.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
4 / 27
Dynamic Games: In Words
Introduced as "Stochastic Games" by Lloyd Shapley (1953), generalizing
Dynamic Programming
Multiple Periods - Each period players interact by picking an action.
Action interaction take place at a given state which changes as a
consequence. Payo¤ received by each player in each period is based on
action vector played as well as the state in that period.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
5 / 27
Dynamic Games: Notation
t = time period (0, 1, 2, ...T ).
i
s (t )
ai ( t )
a (t )
π i (t )
q (t )
δ
=
=
=
=
=
=
=
players (1, ..., I ).
state at beginning of period t, s (t ) 2 S
action taken by player i in period , ai (t ) 2 Ai
(a1 (t ), a2 (t ), ... aI (t )) vector of actions taken in period t.
π i (s (t ), a(t )) payo¤ of player i in period t.
q (s (t + 1) j s (t ), a(t )) conditional distribution of state at period
discount factor, δ 2 [0, 1).
Exogenous Variables - Initial value of state, s (0), discount factor δ and
game horizon, T (…nite or in…nite)
Endogenous Variables - Everything else
Special Cases - Dynamic Programming (I = 1) & Repeated Games
(#S = 1)
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
6 / 27
Connection to Climate Change Treaties
Players = Nations/Blocs; potential signatories
State = stock of GHGs - Common State
State = technology (coal, oil, nuclear ..), capital stock, R&D level,
etc. - Private States
Action (Common State relevant) = Emissions of GHGs
Action (Private State relevant) = switching technologies, investment,
R&D expenditures, etc.
Remark - Actions "implemented" through national policies, cap &
trade, carbon tax, R&D subsidy, contributions to international
organizations, foreign aid etc.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
7 / 27
Histories & Strategies
History at time t, h(t ) - list of prior states & action vectors
h(t ) = s (0), a(0), s (1), a(1), ......s (t )
Assumption - past actions and states including current state
observable
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
8 / 27
Histories & Strategies
History at time t, h(t ) - list of prior states & action vectors
h(t ) = s (0), a(0), s (1), a(1), ......s (t )
Assumption - past actions and states including current state
observable
Strategy for player i at time t, σi (t ) - a complete conditional action
plan for every history
σi (t ) : fh(t )g ! P (Ai )
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
8 / 27
Histories & Strategies
History at time t, h(t ) - list of prior states & action vectors
h(t ) = s (0), a(0), s (1), a(1), ......s (t )
Assumption - past actions and states including current state
observable
Strategy for player i at time t, σi (t ) - a complete conditional action
plan for every history
σi (t ) : fh(t )g ! P (Ai )
Strategy for entire game, σi - a list of strategies, one for every
period:
σi = σi (0), σi (1), ...σi (t ), ...
Let σ = (σ1 , σ2 , ...σI ) denote a vector of strategies, one for each player
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
8 / 27
Examples of Strategies
Pure Strategy σi - where σi (t ) is a deterministic choice (from Ai ).
May however be conditional on history
Example - Emission of country i is high if country j had high
emissions in previous period but low if j had low previous emissions
("Tit for Tat")
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
9 / 27
Examples of Strategies
Pure Strategy σi - where σi (t ) is a deterministic choice (from Ai ).
May however be conditional on history
Example - Emission of country i is high if country j had high
emissions in previous period but low if j had low previous emissions
("Tit for Tat")
(Stationary) Markovian Strategy fi - When the action map is
independendent of history (Markovian) and time (stationary)
fi : S
! P (Ai )
Example - Switch to new technology happens only when capital stock
is su¢ ciently "large" (because capital & energy are complements).
Switching strategy is non-stationary if it depends on the number of
periods left
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
9 / 27
Outcomes & Payo¤s
Consider a pure strategy vector σ & suppose that state transition q is also
deterministic. Then there is a unique history generated by σ:
h(t; σ ) = s (0), a(0; σ), s (1; σ), a(1; σ), ......s (t; σ)
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
10 / 27
Outcomes & Payo¤s
Consider a pure strategy vector σ & suppose that state transition q is also
deterministic. Then there is a unique history generated by σ:
h(t; σ ) = s (0), a(0; σ), s (1; σ), a(1; σ), ......s (t; σ)
where
a(τ; σ) = σ (τ; h(τ; σ))
and
s (τ + 1; σ) = q (s (τ + 1) j s (τ; σ), a(τ; σ))˙
Above called outcome path for σ.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
10 / 27
Outcomes & Payo¤s
Consider a pure strategy vector σ & suppose that state transition q is also
deterministic. Then there is a unique history generated by σ:
h(t; σ ) = s (0), a(0; σ), s (1; σ), a(1; σ), ......s (t; σ)
where
a(τ; σ) = σ (τ; h(τ; σ))
and
s (τ + 1; σ) = q (s (τ + 1) j s (τ; σ), a(τ; σ))˙
Above called outcome path for σ. Associated lifetime payo¤ is
T
Ri (σ) =
∑ δt πi (s (τ; σ), a(τ; σ))
t =0
Generalize when σ not a pure strategy or q not deterministic
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
10 / 27
Equilibria
Subgame - the game that remains after every history h(t )
Restriction of σ to the subgame is denoted σ j h(t )
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
11 / 27
Equilibria
Subgame - the game that remains after every history h(t )
Restriction of σ to the subgame is denoted σ j h(t )
Nash Equilibrium (NE) - Strategy vector σ is a NE if
Ri (σ )
Prajit K. Dutta, Roy Radner ()
Ri (σi , σ i ), for all i, σi
Games & Climate Change
MSRI, Berkeley, May 2009
11 / 27
Equilibria
Subgame - the game that remains after every history h(t )
Restriction of σ to the subgame is denoted σ j h(t )
Nash Equilibrium (NE) - Strategy vector σ is a NE if
Ri (σ )
Ri (σi , σ i ), for all i, σi
Subgame Perfect (Nash) Equilibrium (SPE) - σ is a SPE of the
game if it is a NE for every subgame
Ri (σ j h(t ))
Ri (σi , σ
i
j h(t )), for all i, σi , h(t )
Remark - Not all NE are SPE since NE only requires incentive inequality to
be satis…ed on outcome path generated by σ . But o¤-equilibrium
credible?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
11 / 27
Equilibria
Subgame - the game that remains after every history h(t )
Restriction of σ to the subgame is denoted σ j h(t )
Nash Equilibrium (NE) - Strategy vector σ is a NE if
Ri (σ )
Ri (σi , σ i ), for all i, σi
Subgame Perfect (Nash) Equilibrium (SPE) - σ is a SPE of the
game if it is a NE for every subgame
Ri (σ j h(t ))
Ri (σi , σ
i
j h(t )), for all i, σi , h(t )
Remark - Not all NE are SPE since NE only requires incentive inequality to
be satis…ed on outcome path generated by σ . But o¤-equilibrium
credible?
Markov Perfect Equilibrium (MPE) - A stationary Markov strategy
vector f is a MPE if
Ri (f )
Prajit K. Dutta, Roy Radner ()
Ri (fi , f i ), for all i, fi
Games & Climate Change
MSRI, Berkeley, May 2009
11 / 27
Climate Change - Transition Equation
Greenhouse gases form a global common - hence studied by dynamic
commons game (DCG)
Aggregate E¤ect of Emissions - (Part of) state space S is a
single-dimensional variable with a "commons" structure a¤ected by
aggregate emissions
s (t + 1) = q (s (t ), A(t ))
A(t ) denote the global (total) emission during period t;
I
A(t ) =
∑ ai ( t )
i =1
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
12 / 27
Climate Change - Transition Equation
Greenhouse gases form a global common - hence studied by dynamic
commons game (DCG)
Aggregate E¤ect of Emissions - (Part of) state space S is a
single-dimensional variable with a "commons" structure a¤ected by
aggregate emissions
s (t + 1) = q (s (t ), A(t ))
A(t ) denote the global (total) emission during period t;
I
A(t ) =
∑ ai ( t )
i =1
Global stock of greenhouse gases (GHGs) at the beginning of period t
denoted g (t ).
Linear Transition g (t + 1) = A(t ) + σg (t )
where 0 < σ < 1
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
12 / 27
Climate Change - Payo¤s
Separable E¤ect of Emissions - Bene…ts from emissions & costs
from GHG stock
hi (ai (t )) ci (g (t ))
hi may be thought of as country’s GDP. Linked to emissions via energy
use. Standard assumptions - concavity, continuity - made.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
13 / 27
Climate Change - Payo¤s
Separable E¤ect of Emissions - Bene…ts from emissions & costs
from GHG stock
hi (ai (t )) ci (g (t ))
hi may be thought of as country’s GDP. Linked to emissions via energy
use. Standard assumptions - concavity, continuity - made.
In Dutta & Radner it is assumed that the marginal cost of GHG is constant
Linear Cost of GHG
ci (g (t )) = ci g (t )
where ci is country speci…c
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
13 / 27
Climate Change - Payo¤s
Separable E¤ect of Emissions - Bene…ts from emissions & costs
from GHG stock
hi (ai (t )) ci (g (t ))
hi may be thought of as country’s GDP. Linked to emissions via energy
use. Standard assumptions - concavity, continuity - made.
In Dutta & Radner it is assumed that the marginal cost of GHG is constant
Linear Cost of GHG
ci (g (t )) = ci g (t )
where ci is country speci…c
Discussion of Linearity Assumption - Strong assumption that rules
out catastrophes. Made for three reasons:
- Simple Conclusions
- Full Characterization possible, not just qualitative features. Hence
calibration of solutions possible
- What is the right non-linear
cost function?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
13 / 27
More General State Variables
Recall, State Variable multi-dimensional including "private"
components such as each country’s capital stock, technology, R&D
level, population etc.
Associated action vectors - respectively, investment, technology
switching, R&D expenditures, etc.
Relevant transition equations - e.g. capital stock grows through
savings less destruction through depreciation
Relevant payo¤ functions - e.g. switching costs paid when technology
is switched
Histories, strategies, outcomes and payo¤s de…ned as in general
model of Dynamic Games
MPE and SPE studied
If the actions are unobservable, then strategies and notion of
equilibrium needs modi…cation. Similarly if there are sources of
private information.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
14 / 27
Literature
Dynamic Games
- Shapley (1953), Parthasarathy (1973), Mertens & Parthasarathy (1987)
Du¢ e et al (1994), Abreu, Pearce & Stachetti (1990), Dutta (1995)
Commons Games
- Levhari & Mirman (1973), Benhabib & Radner (1993), Dutta &
Sundaram (1993, 1994), Dockner, Long & Sorger (1996), Tornell &
Velasco (1992)
Climate Change Treaties via Dynamic Games
- Dutta and Radner (2004, 2006, 2008), Dockner & Nishimura (1999),
Long & Sorger (2006)
Climate Change Treaties via Repeated Games
- Barrett (2003), Finus (2007)
Climate Change Agreements Non-Strategic
- IPCC Report (1973), Stern Report (2006), Nordhaus & Yang (1996),
Nordhaus & Boyer (2000)
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
15 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
16 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
State evolves according to
g (t + 1) = A(t ) + σg (t )
where A(t ) is total emissions
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
16 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
State evolves according to
g (t + 1) = A(t ) + σg (t )
where A(t ) is total emissions
Country i 0 s period t payo¤, vi (t ) given by
vi (t ) = hi [ai (t )]
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
ci g ( t )
MSRI, Berkeley, May 2009
16 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
State evolves according to
g (t + 1) = A(t ) + σg (t )
where A(t ) is total emissions
Country i 0 s period t payo¤, vi (t ) given by
vi (t ) = hi [ai (t )]
ci g ( t )
Payo¤ over the game horizon given by
∞
vi =
∑ δt vi (t )
t =0
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
16 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
State evolves according to
g (t + 1) = A(t ) + σg (t )
where A(t ) is total emissions
Country i 0 s period t payo¤, vi (t ) given by
vi (t ) = hi [ai (t )]
ci g ( t )
Payo¤ over the game horizon given by
∞
vi =
∑ δt vi (t )
t =0
Solution concept - SPE
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
16 / 27
Simple Model
Only state variable stock of GHGs, g (t ) - constant capital,
technology, labor
State evolves according to
g (t + 1) = A(t ) + σg (t )
where A(t ) is total emissions
Country i 0 s period t payo¤, vi (t ) given by
vi (t ) = hi [ai (t )]
ci g ( t )
Payo¤ over the game horizon given by
∞
vi =
∑ δt vi (t )
t =0
Solution concept - SPE
Example studied for calibration purposes
γ
1 γi
hi [ai (t )] = φi Ki i Li
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
βi
ai ( t )
fi
βi
MSRI, Berkeley, May 2009
16 / 27
A Simple MPE - "Business as Usual"
There is a simple MPE - dubbed BAU equilibrium - that involves
constant emissions ai
The level of emissions determined by
hi0 (ai *) = δwi
where wi is the lifetime marginal cost wi =
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
ci
1 δσ
MSRI, Berkeley, May 2009
17 / 27
A Simple MPE - "Business as Usual"
There is a simple MPE - dubbed BAU equilibrium - that involves
constant emissions ai
The level of emissions determined by
hi0 (ai *) = δwi
where wi is the lifetime marginal cost wi =
ci
1 δσ
Same result holds when there are additional state variables such as
capital, technology, labor, ... (say zi ) There is a "BAU equilibrium"
characterized by
hi 1 (ai *,zi ) = δwi
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
17 / 27
A Simple MPE - "Business as Usual"
There is a simple MPE - dubbed BAU equilibrium - that involves
constant emissions ai
The level of emissions determined by
hi0 (ai *) = δwi
where wi is the lifetime marginal cost wi =
ci
1 δσ
Same result holds when there are additional state variables such as
capital, technology, labor, ... (say zi ) There is a "BAU equilibrium"
characterized by
hi 1 (ai *,zi ) = δwi
Result follows directly from the fact that a unit of emission in period t
is of size σ in period t + 1, σ2 in period t + 2, etc. On account of
linearity in cost, these surviving units add ci
δσ in period t + 1, ci
(δσ)2 in period t + 2, etc.; or over lifetime
ci
1 δσ
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
17 / 27
Global Pareto Optima
Let x = (xi ) be a vector of positive numbers, one for each country. A
Global Pareto Optimum (GPO) is a solution to
Maxσ ∑ xi vi
i
For every x there is a GPO that involves constant emissions b
ai
The level of emissions determined by
xi hi0 (âi ) = δw
where w is the lifetime social marginal cost
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
1
1 δσ
∑i xi ci
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18 / 27
Global Pareto Optima
Let x = (xi ) be a vector of positive numbers, one for each country. A
Global Pareto Optimum (GPO) is a solution to
Maxσ ∑ xi vi
i
For every x there is a GPO that involves constant emissions b
ai
The level of emissions determined by
xi hi0 (âi ) = δw
where w is the lifetime social marginal cost
1
1 δσ
∑i xi ci
Same result holds when there are additional state variables such as
capital, technology, labor, ... (say zi ) There is a GPO characterized by
xi hi 1 (ai *,zi ) = δwi
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
18 / 27
Global Pareto Optima
Let x = (xi ) be a vector of positive numbers, one for each country. A
Global Pareto Optimum (GPO) is a solution to
Maxσ ∑ xi vi
i
For every x there is a GPO that involves constant emissions b
ai
The level of emissions determined by
xi hi0 (âi ) = δw
where w is the lifetime social marginal cost
1
1 δσ
∑i xi ci
Same result holds when there are additional state variables such as
capital, technology, labor, ... (say zi ) There is a GPO characterized by
xi hi 1 (ai *,zi ) = δwi
Tragedy of the Common - Since xi ci < ∑j xj cj it follows from the
concavity of hi that
ai * > âi
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
18 / 27
Numerical Results on GPO & BAU
Table 1 - Benchmark Case (δ = 0.97, cost = Fankhauser ci , Year =
1998)
Region/Emissions
BAU (Gtc ) GPO (Gtc ) % Di¤erence ( BAUBAUGPO )
United States
1.50
1.36
9%
Western Europe
0.86
0.79
8%
Other High Income
0.59
0.53
9%
Eastern Europe
0.74
0.45
39%
Middle Income (MI )
0.41
0.36
12%
Lower MI
0.58
0.41
30%
China
0.85
0.56
34%
Lower Income
0.66
0.48
28%
Total
6.18
4.93
20%
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
19 / 27
Numerical Results on GPO & BAU 2
Table 2 - Benchmark Case (δ = 0.995, cost = Fankhauser ci , Year =
1998)
Region/Emissions
BAU (Gtc ) GPO (Gtc ) % Di¤erence ( BAUBAUGPO )
United States
1.50
1.15
23%
Western Europe
0.86
0.69
20%
Other High Income
0.59
0.46
22%
Eastern Europe
0.74
0.27
64%
Middle Income (MI )
0.41
0.29
28%
Lower MI
0.58
0.27
54%
China
0.85
0.32
62%
Lower Income
0.66
0.32
52%
Total
6.18
3.76
39%
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
20 / 27
Other Equilibria
Could there be other - better - equilibria than the BAU? Are they
simple?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
21 / 27
Other Equilibria
Could there be other - better - equilibria than the BAU? Are they
simple?
Are there treaties - SPE - sustained by the BAU itself? "Countries pledge
to cut emissions z% from the BAU and if someone doesn’t then we revert
to the BAU"
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
21 / 27
Other Equilibria
Could there be other - better - equilibria than the BAU? Are they
simple?
Are there treaties - SPE - sustained by the BAU itself? "Countries pledge
to cut emissions z% from the BAU and if someone doesn’t then we revert
to the BAU"
Proposition. There is a continuum of emission cuts that are possible as
SPE. There is a (strictly positive) maximum emission aggregate cut
sustained by the BAU threat
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
21 / 27
Other Equilibria
Could there be other - better - equilibria than the BAU? Are they
simple?
Are there treaties - SPE - sustained by the BAU itself? "Countries pledge
to cut emissions z% from the BAU and if someone doesn’t then we revert
to the BAU"
Proposition. There is a continuum of emission cuts that are possible as
SPE. There is a (strictly positive) maximum emission aggregate cut
sustained by the BAU threat
The "best" SPE sustainable by the BAU threat need not be the one
that implements the maximal cuts
E¤ect of Asymmetry in Costs - can be shown that the greater the
cost asymmetry the lower are aggregate payo¤s in the "best" SPE
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
21 / 27
Maximal Emission Cuts
Table 3 - Minimum Sustainable Emissions in Benchmark Case
(δ = 0.97, cost = Fankhauser ci , Year = 1998)
Country /Emissions
UnitedStates
Brazil
China
India
Korea DPR
Poland
Russia
Ukraine
Prajit K. Dutta, Roy Radner ()
MIN GPO
(%)
BAU
Games & Climate Change
9%
31%
21%
9%
+22%
+5%
+17%
+21%
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22 / 27
All SPE
Can there be better threats than the BAU? Are they simple? What
is the best such equilibrium?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
23 / 27
All SPE
Can there be better threats than the BAU? Are they simple? What
is the best such equilibrium?
Can one characterize the set of all SPE?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
23 / 27
All SPE
Can there be better threats than the BAU? Are they simple? What
is the best such equilibrium?
Can one characterize the set of all SPE?
Answer - The SPE payo¤ correspondence has a surprising simplicity; the
set of equilibrium payo¤s at a level g is a simple linear translate of the set
of equilibrium payo¤s from some benchmark level, say, g = 0.
Consequently, the set of emission levels that can arise in equilibrium is
state-independent. Though in a particular equilibrium, emission levels
may vary with g .
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
23 / 27
All SPE
Can there be better threats than the BAU? Are they simple? What
is the best such equilibrium?
Can one characterize the set of all SPE?
Answer - The SPE payo¤ correspondence has a surprising simplicity; the
set of equilibrium payo¤s at a level g is a simple linear translate of the set
of equilibrium payo¤s from some benchmark level, say, g = 0.
Consequently, the set of emission levels that can arise in equilibrium is
state-independent. Though in a particular equilibrium, emission levels
may vary with g .
Proposition. The equilibrium payo¤ correspondence Ξ is linear; there is a
compact set U
<I such that for every initial state g
Ξ (g ) = U
Prajit K. Dutta, Roy Radner ()
fw1 g , w2 g , ...wI g g
Games & Climate Change
MSRI, Berkeley, May 2009
23 / 27
All SPE
Can there be better threats than the BAU? Are they simple? What
is the best such equilibrium?
Can one characterize the set of all SPE?
Answer - The SPE payo¤ correspondence has a surprising simplicity; the
set of equilibrium payo¤s at a level g is a simple linear translate of the set
of equilibrium payo¤s from some benchmark level, say, g = 0.
Consequently, the set of emission levels that can arise in equilibrium is
state-independent. Though in a particular equilibrium, emission levels
may vary with g .
Proposition. The equilibrium payo¤ correspondence Ξ is linear; there is a
compact set U
<I such that for every initial state g
Ξ (g ) = U
fw1 g , w2 g , ...wI g g
Bootstrapping ideas - the Abreu-Pearce-Stachetti operator - employed
in proof
Computing the equilibrium value set U not an easy task - unlike
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
23 / 27
Second-Best Equilibrium
Second-Best Problem - To maximize a weightem sum of equilibrium
payo¤s
I
max
∑
i =1
xi Vi (g ), V (g ) 2 Ξ(g )
Second-Best Equilibrium
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
24 / 27
Second-Best Equilibrium
Second-Best Problem - To maximize a weightem sum of equilibrium
payo¤s
I
max
∑
i =1
xi Vi (g ), V (g ) 2 Ξ(g )
Second-Best Equilibrium
Proposition. There exists a constant emission level a a1 , a2 , ....aI such that no matter what the initial level of GHG, the second-best policy
is to emit at the constant rate a. In the event of a deviation from this
constant emissions policy by country i, play proceeds to i 0 s worst
equilibrium.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
24 / 27
Second-Best Equilibrium
Second-Best Problem - To maximize a weightem sum of equilibrium
payo¤s
I
max
∑
i =1
xi Vi (g ), V (g ) 2 Ξ(g )
Second-Best Equilibrium
Proposition. There exists a constant emission level a a1 , a2 , ....aI such that no matter what the initial level of GHG, the second-best policy
is to emit at the constant rate a. In the event of a deviation from this
constant emissions policy by country i, play proceeds to i 0 s worst
equilibrium. Furthermore, the second-best emission rate is always strictly
lower than the BAU rate, i.e., a < a .
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
24 / 27
Second-Best Equilibrium
Second-Best Problem - To maximize a weightem sum of equilibrium
payo¤s
I
max
∑
i =1
xi Vi (g ), V (g ) 2 Ξ(g )
Second-Best Equilibrium
Proposition. There exists a constant emission level a a1 , a2 , ....aI such that no matter what the initial level of GHG, the second-best policy
is to emit at the constant rate a. In the event of a deviation from this
constant emissions policy by country i, play proceeds to i 0 s worst
equilibrium. Furthermore, the second-best emission rate is always strictly
lower than the BAU rate, i.e., a < a . Above a critical discount factor
(less than 1), the second-best rate coincides with the GPO emission rate b
a.
Second-Best Simple, Implementable by an across the board cut
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
24 / 27
Worst Equilibrium
Consider second-best solution in which xi = 0. Denote that emission level
a(x i ), the i less second-best
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
25 / 27
Worst Equilibrium
Consider second-best solution in which xi = 0. Denote that emission level
a(x i ), the i less second-best
Worst Equilibrium for Country i Is in two parts:
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
25 / 27
Worst Equilibrium
Consider second-best solution in which xi = 0. Denote that emission level
a(x i ), the i less second-best
Worst Equilibrium for Country i Is in two parts:
1. There exists a ”high” emission level a(i ) (with ∑j 6=i aj (i ) > ∑j 6=i aj )
s.t. each country emits at rate aj (i ) for one period (no matter what g is),
j = 1, ..I .
2. From the second period onwards, each country emits at the constant
rate aj (x i ), j = 1, ..I .
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
25 / 27
Worst Equilibrium
Consider second-best solution in which xi = 0. Denote that emission level
a(x i ), the i less second-best
Worst Equilibrium for Country i Is in two parts:
1. There exists a ”high” emission level a(i ) (with ∑j 6=i aj (i ) > ∑j 6=i aj )
s.t. each country emits at rate aj (i ) for one period (no matter what g is),
j = 1, ..I .
2. From the second period onwards, each country emits at the constant
rate aj (x i ), j = 1, ..I .
So a sanction is made up of two emission rates, a(i ) and a(x i ). The
…rst imposes immediate costs on country i by increasing the emission
levels of countries j 6= i. For the punishing countries, however, the
resultant cost increase is o¤set by a subsequent permanent change,
the switch to the emission vector a(x i ), which permanently increases
their quota at the expense of country i 0 s.
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
25 / 27
Generalizations & Extensions
Technology Choice - Switching to cleaner technologies
With linear costs to switching, the solution is "bang-bang" - Tragedy
obtains
Capital Growth - does reducing emissions inhibit growth?
Ongoing research suggesting that treaty formation incentives can be
very di¤erent for countries with varying capital stocks
Foreign Aid -can aid be Pareto improving for donor & recipient?
Yes in some circumstances
Other Issues - Treaty negotiation issues. Who joins …rst? How
many countries are enough? Should climate change treaties be
"linked" to trade treaties?
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
26 / 27
Conclusions
Climate change treaties can be modeled via dynamic commons games
Tragedy of the Commons obtains in the simple BAU equilibrium - on
emissions, technology switching etc.
BAU can be strictly improved by using it as a threat to reduce emissions
Best equilibria may be relatively simple to implement
Asymmetry makes treaties harder to implement but foreign aid can help
Prajit K. Dutta, Roy Radner ()
Games & Climate Change
MSRI, Berkeley, May 2009
27 / 27