IntroductionThe GameThe Model

General ResultsNumerical Simulations

The Thin Green Line:Transboundary Pollution Problems in Coupled Lake

Systems

W. Davis DechertSharon O’Donnell and Stuart McDonald

2nd Workshop Game Theory in Energy, Resources and theEnvironment 2008

29 June 2008

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

I The Game

I Non Cooperative Solution

I Dynamic Programming

I Numerical Results

I Stochastic Elements & Droughts

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

The Single Lake Game

I Players: Communities that share the lake

I Pollution: Phosphorus applied to land (controls)Accumulation in the lake (state)

I Benefits: Increased crop production −→ profits

I Costs: Phosphorus −→ nutrients for weeds and algæ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

The Single Lake Game

I Players: Communities that share the lake

I Pollution: Phosphorus applied to land (controls)Accumulation in the lake (state)

I Benefits: Increased crop production −→ profits

I Costs: Phosphorus −→ nutrients for weeds and algæ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

The Single Lake Game

I Players: Communities that share the lake

I Pollution: Phosphorus applied to land (controls)Accumulation in the lake (state)

I Benefits: Increased crop production −→ profits

I Costs: Phosphorus −→ nutrients for weeds and algæ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

Upstream/Downstream Game

I Players: Upstream and downstream communities

I Pollution: Upstream and downstream application of P

I Interconnection: Some upstream pollution flows downstream

I Benefits: Increased crop production

I Costs: Accumulation of phosphorus in the lakes

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

Upstream/Downstream Game

I Players: Upstream and downstream communities

I Pollution: Upstream and downstream application of P

I Interconnection: Some upstream pollution flows downstream

I Benefits: Increased crop production

I Costs: Accumulation of phosphorus in the lakes

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

Upstream/Downstream Game

I Players: Upstream and downstream communities

I Pollution: Upstream and downstream application of P

I Interconnection: Some upstream pollution flows downstream

I Benefits: Increased crop production

I Costs: Accumulation of phosphorus in the lakes

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Single Lake ProblemCoupled Lake Problem

Upstream/Downstream Game

I Players: Upstream and downstream communities

I Pollution: Upstream and downstream application of P

I Interconnection: Some upstream pollution flows downstream

I Benefits: Increased crop production

I Costs: Accumulation of phosphorus in the lakes

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Assumptions

I Separable functional form for costs and benefits

I Agreement on functional form of costs

I Player’s benefits are a function of his/her own loading

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Assumptions

I Separable functional form for costs and benefits

I Agreement on functional form of costs

I Player’s benefits are a function of his/her own loading

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Assumptions

I Separable functional form for costs and benefits

I Agreement on functional form of costs

I Player’s benefits are a function of his/her own loading

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Assumptions

I Separable functional form for costs and benefits

I Agreement on functional form of costs

I Player’s benefits are a function of his/her own loading

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

State Equations

I Upper lake

xt+1 = bxxt +x2t

1 + x2t

+∑

i

ai ,t

I Lower lake

yt+1 = byyt +y2t

1 + y2t

+∑

i

ai ,t + µxt

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

State Equations

I Upper lake

xt+1 = bxxt +x2t

1 + x2t

+∑

i

ai ,t

I Lower lake

yt+1 = byyt +y2t

1 + y2t

+∑

i

ai ,t + µxt

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Weighted Social Welfare Criteria

I The Planner’s Problem

max{ai,t}

∞∑t=0

βt(W1(a1,t , · · · , an,t , xt)− λW2(a1,t , · · · , am,t , yt)

)

xt+1 = bxt +x2t

1 + x2t

+∑

i

ai ,t

yt+1 = byt +y2t

1 + y2t

+∑

i

ai ,t + µxt

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Objectives and the Game Potential

I Each upper lake player solves

max{ai,t}

∞∑t=0

βt(ui (ai ,t)− kic(xt)

)

xt+1 = bxt +x2t

1 + x2t

+∑

i

ai ,t

I Potential function

max{a1,t ...an,t}

∞∑t=0

βt

(∑i

ui (ai ,t)/ki − c(xt)

)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Objectives and the Game Potential

I Each upper lake player solves

max{ai,t}

∞∑t=0

βt(ui (ai ,t)− kic(xt)

)

xt+1 = bxt +x2t

1 + x2t

+∑

i

ai ,t

I Potential function

max{a1,t ...an,t}

∞∑t=0

βt

(∑i

ui (ai ,t)/ki − c(xt)

)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Objectives and the Game Potential

I Each lower lake player solves

max{aj,t}

∞∑t=0

βt(uj(aj ,t)− kjc(yt)

)

yt+1 = byt +y2t

1 + y2t

+∑

j

aj ,t + µxt

I Potential function

max{a1,t ...am,t}

∞∑t=0

βt

∑j

uj(aj ,t)/kj − c(yt)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

CharacterizationCooperative GameNoncooperative Game

Objectives and the Game Potential

I Each lower lake player solves

max{aj,t}

∞∑t=0

βt(uj(aj ,t)− kjc(yt)

)

yt+1 = byt +y2t

1 + y2t

+∑

j

aj ,t + µxt

I Potential function

max{a1,t ...am,t}

∞∑t=0

βt

∑j

uj(aj ,t)/kj − c(yt)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Properties of Solutions

I Two possible optimal steady states:

Low P oligotrophic low in nutrientsHigh P eutrophic high in nutrients

I Upper lake converges monotonically to SS

I Skiba points (both steady states are optimal)

I Lower lake very sensitive to µ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Properties of Solutions

I Two possible optimal steady states:

Low P oligotrophic low in nutrientsHigh P eutrophic high in nutrients

I Upper lake converges monotonically to SS

I Skiba points (both steady states are optimal)

I Lower lake very sensitive to µ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Properties of Solutions

I Two possible optimal steady states:

Low P oligotrophic low in nutrientsHigh P eutrophic high in nutrients

I Upper lake converges monotonically to SS

I Skiba points (both steady states are optimal)

I Lower lake very sensitive to µ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Properties of Solutions

I Two possible optimal steady states:

Low P oligotrophic low in nutrientsHigh P eutrophic high in nutrients

I Upper lake converges monotonically to SS

I Skiba points (both steady states are optimal)

I Lower lake very sensitive to µ

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60

Initi

al a

ccum

ulat

ion

of p

hosp

horu

s -

dow

nstr

eam

lake

(y 0

)

Periods

Convergent path for downstream lake as upstream lake converges to its steady state

Steady state- downstream lake (dashed line)

Path when upstream lake is at steady state (solid black line)

Path when µx0,j < µx* (red)

Path when µx0,j > µx* (blue)

Figure

2:C

onvergentpaths

4

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5

Tot

al lo

adin

g pe

r pe

riod

(con

trol

)

Accumulated phosphorus in the lake (state)

yt+1=yt function and policy functions for downstream lake, µ=0.01, upstream steady state xt=0.3347

yt+1=yt when upstream is at steady state

Downstream policy when upstream lake is at steady state, µx0=0.003347

Downstream policy - 35 periods prior to steady state, µx0=0.014424

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Upst

ream

= m

u=0.

0

Downstream

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake -- Reversible

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake -- Irreversible

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake

c = 0.6x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake

c = 0.4x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Upstream Lake

c = 0.25x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.00

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.01

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.02

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.03

x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.00

c = 0.6x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.01

c = 0.6x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.02

c = 0.6x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.03

c = 0.6x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.00

c = 0.4x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.01

c = 0.4x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.02

c = 0.4x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.00

c = 0.25x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.01

c = 0.25x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line

IntroductionThe GameThe Model

General ResultsNumerical Simulations

Upper LakeLower Lake

-0.1

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

Tota

l load

ing

per p

erio

d (c

ontro

l)

Accumulated phosphorus in the lake (state)

Downstream Lake mu = 0.02

c = 0.25x(t+1) = x(t)

W. Davis Dechert Sharon O’Donnell and Stuart McDonald 2nd Workshop Game Theory in Energy, Resources and the Environment 2008The Thin Green Line