Game Theory and Applications to Fisheriescermics.enpc.fr/~delara/MABIES/Pereau-gametheory_MABIES2013.pdfBailey-Sumaila-Lindroos (2010) Application of game theory to ﬁsheries over

,

Steady state approachDynamic approach

Game Theory and Applications to FisheriesMathematics of Bio-Economics, Institut Henri Poincare, Paris

Jean-Christophe PEREAU, Luc DOYEN

March 11, 2013

Pereau, Doyen Game Theory MABIES

,


Tragedy of the commons


,


Fishermen dilemma in open-access

Two users exploiting a common fish stock: Cooperation failure

user 1\2 Conserve Deplete

Conserve 3 , 3 1 , 4

Deplete 4 , 1 2 , 2

Management of transboundary stocks between two or morecoastal states (G. Munro (1979) Canadian Journal ofEconomics)

Management of straddling stocks between coastal states andDWFNs (Distant Water Fishing Nations)

Myopic player: Whatever I do not harvest may be harvestedby others and therefore I do not have any incentives to savethe resource for tomorrow


,


Main modeling features

Bailey-Sumaila-Lindroos (2010) Application of game theory tofisheries over three decades Fisheries Research

Technical points outlined

Static game and dynamic game

Discrete time model

Identical players and heterogenous players

Single stocks and multi-species stocks


,


The basic game

Stock dynamics

x(t + 1) = F (x(t))− h(t)

Catches

h(t) =

n∑

i=1

hi (t) =

n∑

i=1

qiei (t)x(t)

Rent of a player i with xoai is the zero-rent level

πi (x(t), ei (t)) = phi (t)− ciei (t)

= pqi

(x(t)− ci

pqi

)ei (t)

= pqi (x(t)− xoai ) ei (t)


,


Outline of the presentation

1 Steady state approach

2 Dynamic approach


,


Outline of the presentation

1 Steady state approach

2 Dynamic approach


,


References

Basic game : Mesterton-Gibbons (1993), Game-theoreticResource modeling, Natural Resource Modeling 7, 93-146.

Stage games : Ruseski (1998) Fleet and vessels:International Fish Wars: The strategic roles for fleet licensingand effort subsidies, J. of Economics and EnvironmentalManagement 36:70-88.

Repeated game : Hannesson (1997) Fishing as a supergame,J. of Economics and Environmental Management

Coalitionnal game : Pintassilgo & Lindroos (2008) Coalitionformation in straddling stock fisheries: a partition functionapproach, International Game Theory Review, 10(3):303-317


,


At the equilibrium

Assume Logistic

F (x) = x + rx(1−

x

K

)

Steady state: sustainable yield

F (x)− x = h ⇔ x =K

r

(r −

∑

i

qiei

)= g(e)

with

h =

n∑

i=1

hi =

n∑

i=1

qieix

Assume asymmetric users

xoa1 ≤ xoa2 ≤ .... ≤ xoan

Rent of a player i with xoa is the zero-rent levelPereau, Doyen Game Theory MABIES

,


Non cooperative outcome

Each player maximizes its profit wrt ei

maxei≥0, x=g(e)

πi (x , ei ) = maxei≥0

πi (g(e), ei ) = maxei≥0

Πi (e)

Nash equilibrium: e∗ = (e∗1 , e∗2 , . . . , e

∗n )

Πi (e∗i , e

∗−i ) ≥ Πi (ei , e

∗−i ), ∀i , ∀ei ,

with e−i = e1, .., ei−1, ei+1, .., en


,


Non cooperative outcome

Assume all players are active e∗i > 0

FOCs

ei =xoai − g(e)

g ′(e)

reaction function

2qiei = r

(1−

xoaiK

)−

n∑

j 6=i

qjej

Aggregate effort

n∑

i=1

qiei =r

1 + n

(n−

∑ni=1 x

oai

K

)


,


Non cooperative equilibrium

Optimal individual fishing mortality rate

F ∗nci = qie

∗nci =

r

(n + 1)

1 +

n∑

j 6=i

xoajK

− nxoaiK

Non cooperative stock

x∗nc =1

n + 1

(K +

n∑

i=1

xoai

)


,


Active users

Number of active players ?

n⋆ = max (i , e∗nci > 0)

Assume xoa1 ≤ xoa2 ≤ .... ≤ xoan : n⋆ = max (i , xoai < x∗nc )

Optimal Fishing mortality rate

qie∗nci =

{r

(n∗+1)

(1 +

∑n∗

j 6=i

xoajK

− n∗xoai

K

)if 1 ≤ i ≤ n∗

0 if n∗ + 1 ≤ i ≤ n

Stock

x∗nc =1

n∗ + 1

(K +

n∗∑

i=1

xoai

)


,


Application

n = 7 players such thatxoa1

K= 0.4 <

xoa2

K= 0.45 <

xoa3

K= 0.5 <

xoa4

K= 0.55 <

xoa5

K=

xoa6

K=

xoa7

K= 0.6

Only the fourth first players are active:

n⋆ = 4


,


Cooperative outcome

Optimal cooperative effort

maxei≥0, x=g(e)

∑

i

πi(x , ei )

Who will be the harvester?

A solution: only the most efficient user

e∗ci =

{r

2q1

(1−

xoa1K

)if i = 1

0 otherwise

Cooperative stock

x∗c =1

2(K + xoa1 )


,


The tragedy of the commons

Assume all players are identical :qi = q, ci = c and xoai = xoa ≤ K .

Nash equilibrium

e∗nc =r

q (n + 1)

(

1−xoa

K

)

x∗nc =n

n + 1

(

K

n+ xoa

)

π∗nci =

pqKr

(n + 1)2

(

1−xoa

K

)2

Cooperative solution

e∗c =r

2nq

(

1−xoa

K

)

x∗c =1

2(K + xoa)

π∗ci =

pqKr

4n

(

1−xoa

K

)2


,



Assume all players are identical :qi = q, ci = c and xoai = xoa ≤ K .

Nash equilibrium

e∗nc =r

q (n + 1)

(

1−xoa

K

)

x∗nc =n

n + 1

(

K

n+ xoa

)

π∗nci =

pqKr

(n + 1)2

(

1−xoa

K

)2

<?? >

≥

≤

≤

Cooperative solution

e∗c =r

2nq

(

1−xoa

K

)

x∗c =1

2(K + xoa)

π∗ci =

pqKr

4n

(

1−xoa

K

)2


,



When n → ∞ we have

π∗nci = 0, x∗nc = xoa

Open access equilibrium where each agent gets a null payoff

The open-access tragedy is reduced with asymmetric users


,


Comparison with reference points

0 50000 100000 150000 200000 250000 300000 350000 4000000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Abundance (blue whale unit)

catc

hes

(blu

e w

hale

uni

t)


,


Cooperation failure and incentives for cheating

users i/j Conserve Deplete

Conserve πi(eci , e

cj ), πj(e

ci , e

cj ) πi(e

ci , e

dj ), πj (e

ci , e

dj )

Deplete πi(edi , e

cj ), πj(e

di , e

cj ) πi(e

nci , encj ), πj (e

nci , encj )

Optimal defection effort: best reply when all others choosethe cooperative strategy

edi =r

4q

(1 +

1

n

)(1−

xoa

K

)


,


Repeated game to sustain cooperation

Benefits from cheating

πdi = pq

(xd − xoa

)edi

xd = K(1−

q

R

(edi + (n − 1)eci

))

Condition for cooperation: Punishment strategy

∞∑

t=0

(1

1 + δ

)t

πci > πd

i +

∞∑

t=1

(1

1 + δ

)t

πnci

A high discount factor sustains the cooperation equilibrium


,


The basic game with dynamic externalityTwo-species dynamic gameCoalition formation gameViability and Coalition formation game

Problem statement

The problem faced by agent i , i = 1, .., n is to maximize

maxhi (t0),...hi (T−1)

(T−1∑

t=t0

ρtiUi(hi (t)) + ρTi U(x(T ))

)

Dynamics and constraint

x(t + 1) = F

(x(t)−

n∑

i=1

hi(t)

)

0 ≤n∑

i=1

hi (t) ≤ x(t)

Utility function with U ′ > 0 and U ′′ < 0


,



References

Basic game : Levahri and Mirman (1980) The great fish war:an example using a dynamic Cournot-Nash solution. Bell J. ofEconomics 11:322-344

Two species game : Fischer and Mirman (1996) Thecompleat Fish Wars: Biological and Dynamics Interactions, J.of Economics and Environmental Management, 30:34-42,Fischer and Mirman (1992) Strategic dynamic interactions:Fish Wars, J. of Economic Dynamics and Control, 16:267-287.

Coalition formation game : Kwon (2006) PartialInternational Coordination in the Great Fish War,Environmental & Resource Economics, 33:463-483

Viability and coalition : Doyen and Pereau (2012)Sustainable coalitions in the commons, Mathematical SocialScience, 63:57-64


,



The Fish War

The problem faced by agent i is to maximize

maxhi (t0),...hi (T−1)

(T−1∑

t=t0

ρtiU(hi (t)) + ρTi U(x(T ))

)

with

Logarithmic utility function U(hi ) = log hi

Dynamic F (x) = bxα with 0 < α < 1 ≤ b

K = b1/(1−α) normalized to 1


,



Dynamic programming: Bellman method

Sequential resolution

Start at the final time horizon and apply backward inductionmechanism at each step

See Chapter 5 of De Lara-Doyen (2008)


,



Bellman cooperative

Problem

maxu(.)

T−1∑

t=0

n∑

i=1

U(x(t), ui (t), t) +

n∑

i=1

Mi(T , x(T ))

x(t + 1) = F (x(t), u(t), t)

x(0) = x0

Value function for t = 0...T − 1

V (T , x) =n∑

i=1

Mi(T , x)

V (t, x) = maxu

(n∑

i=1

Ui (t, x , ui) + V (t + 1,F (x , t, u))

)

u∗(x , t) = argmaxu

V (t, x)


,



Bellman non cooperative

Value functions for t = 0...T − 1

Vi(T , x) = Mi(T , x(T ))

Vi(t, x) = maxui

(U(t, x , ui ) + Vi (t + 1,F (x , t, u)))

u∗i (x , t) = argmaxui

Vi (t, x)


,



2 players and T = 2 periods

ρ1 = ρ2 = ρ

Vi(2, x) = Mi (2, x(2)) = ρ2 log(x(2))

Vi(1, x) = maxhi

(ρ log(hi) + Vi(2, (x − hi − hj)α))

Vi(1, x) = maxhi

(ρ log(hi ) + αρ log(x − hi − hj))

FOCs for i = 1, 2 and i 6= j :

hi =x − hj1 + αρ

Linear optimal feedback:

h∗i (1, x) =1

2 + αρx


,



Value function

Vi (1, x) = ρ log(h∗i ) + ρ2α log (x − H∗)

= ρ log

(1

2 + αρx

)+ ρ2α log

(αρ

2 + αρx

)

= C1 + D1 log(x)

with constants

C1 = ρ log(

12+αρ

)+ ρ2α log

(αρ

2+αρ

)

D1 = ρ(1 + αρ)


,



One-period backward value function

Vi(0, x) = maxhi

{log(hi ) + Vi(1, (x − H)α)}

= maxhi

{log(hi ) + C1 + D1 log (x − H)α}

= maxhi

{log(hi ) + C1 + αD1 log (x − H)}

Focs for i = 1, 2 and i 6= j

hi (x , hj ) =x − hj1 + αD1

Linear optimal feedback

h∗i (0, x) =1

2 + αD1x


,



One-period backward value function

Vi(0,B) = C0 + D0 log(B)

whereD0 = αD1 + 1

and

C0 = f (C1,D1) = log

(1

2 + αD1

)+C1+αD1 log

(αD1

2 + αD1

)


,



Linear feedback

Period 0

h∗i (0, x) =1

2 + αρ(1 + αρ)x

Period 1

h∗i (1, x) =1

2 + αρx


,



Non-cooperative outcome for T -period horizon

Optimal Nash feedback and value function

h∗nci (t, x) =1

2 + αDt+1ρ−tx , V ∗nc

i (t, x) = Ct + Dt log(x)

where Dt = αDt+1 + ρt and Ct = f (Ct+1,Dt+1) with finalconditions DT = ρT and CT = 0.

We deduce

h∗nci (t, x) =

(2 +

T−t∑

s=1

(αρ)s)−1

x


,



Non-cooperative steady-state (long term)

Fishing mortality rate:

F ∗nci =

1− αρ

2− αρ

Harvest:h∗nci = F ∗nc

i x∗nc

Stock level:

x∗nc =

(αρ

2− αρ

) α

1−α


,



Non cooperative game with n identical players

Optimal catches

h∗nci (t, x) =

(n − 1 +

T−t∑

s=0

(αρ)s

)−1

x

Steady state fishing mortality rate

F ∗nci =

(1− αρ

n− αρ (n − 1)

)

Steady state stock level

x∗nc =

(αρ

n − αρ (n − 1)

) α

1−α

Sustainable catches h∗nc = nh∗nci = nF ∗nci x∗nc


,



Cooperative game with n identical players

Optimal catches

h∗ci (t, x) =1

n∑T−t

s=0 (αρ)sx

Fishing mortality rate

F ∗ci =

1

n(1− αρ)

Cooperative steady state

x∗c = (αρ)α

1−α

Cooperative sustainable catches h∗c = nh∗ci = nF ∗ci x∗c


,



Tragedy of open-access

Cooperation performs better than non cooperation

x∗nc < x∗c , F c∗i < F nc∗

i , hnc∗i < hc∗i

Stock can collapse in the non cooperative case

limn→+∞

x∗nc = 0


,



K = 1, x0 = K/4, ρ = 0.98, α = 0.6, n = 5biomass

50454035302520151050

0.25

0.20

0.15

0.10

0.05

0.00

periodes

Non-Coop

Hnc(t)

Bnc(t)

biomass

50454035302520151050

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Coop

Bc(t)

Hc(t)

biomass

50454035302520151050

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Coop/Non Coop

Bc(t)

Bnc(t)


,



K = 1, x0 = K/2, ρ = 0.98, α = 0.6, n = 5

periodes

50454035302520151050

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

biomass

Non-Coop

Bnc(t)

Hnc(t)

biomass

50454035302520151050

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

periodes

Coop

Bc(t)

Hc(t)

biomass

50454035302520151050

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Coop/Non Coop

Bc(t)

Bnc(t)


,



Higher ρ ⇔ Higher xbiomass

50454035302520151050

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Non-Coop

Bnc(t)

Hnc(t)

Bnc(t)

Hnc(t)

biomass

50454035302520151050

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

periodes

Coop

Hc(t)Hc(t)

Bc(t)Bc(t)

biomass

50454035302520151050

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Coop/Non Coop

Bc(t)

Bnc(t)

Bc(t)

Bnc(t)


,



Higher α ⇔ Lower xbiomass

50454035302520151050

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

periodes

Non-Coop

Bnc(t)

Hnc(t)

Bnc(t)

Hnc(t)

biomass

50454035302520151050

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

periodes

Coop

Hc(t)Hc(t)

Bc(t)Bc(t)

biomass

50454035302520151050

0.6

0.5

0.4

0.3

0.2

0.1

0.0

periodes

Coop/Non Coop

Bc(t)

Bnc(t)

Bc(t)

Bnc(t)


,



A technical note on feedback strategies

Feedback strategies can be difficult to compute analytically

Transform the optimization problem in a linear problem withrespect to the transformed state variable

The FOC with respect to the control variable is independentof the state variable


,



Each user chooses a linear decision rule hi (t) = Fi(t)x(t)

Transition equation z(t) = log x(t)

z(t + 1) = αz(t) + α log(1− Fi(t)− F−i(t))

Objective function of player i

maxFi (t)

∞∑

t=0

ρt(z(t) + log Fi (t))


,



Bellman equation

Vi (z(t)) = maxFi (t)

{z(t) + log(Fi (t)) + ρVi(α(z(t) + log(1− Fi (t)− F−i (t))))

i thinks that −i uses the constant strategy F−i(t) = F−i

Foc1

Fi(t)= ρV ′

i (z(t + 1))α

1 − Fi(t)− F−i

Conjecture V (z) = C + Dz such that V ′(z) = D with D aconstant

1

Fi=

ραD

1− Fi − F−i

⇒ Fi =1− F−i

1 + ραD


,



With symmetry Fi = F ,∀i

F = (n + ραD)−1

By substitution

C + Dz(t) = z(t) + log F + ρC + ρD(αz(t) + α log(1− nF ))

By identification D = (1− ρα)−1


F ∗nc =1− ρα

n − ρα(n − 1)


,



2 countries and two species

In each country people consume both type of fish.

The two species x and y interact according three cases (i)symbiotic, (ii) negative and (iii) predator-prey.

The dynamic without harvest is

x(t + 1) = f (x(t), y(t)) = xα1(t)yβ1(t)

y(t + 1) = g (x(t), y(t)) = xβ2(t)yα2(t)

3 cases

Symbiotic relation βi > 0 i = 1, 2.

Mutual predators βi < 0 i = 1, 2

Predator-prey relation : βi > 0 and βj < 0, i 6= j = 1, 2

In all cases: αi > βi .


,



Two countries case

The general problem of country i is to

sup{hi1(.),hi2(.)}

(∞∑

t=t0

ρtiU(hi1(t), hi2(t))

)

under the dynamic constraints

x(t + 1) = (x(t)− h1(t))α1 (y(t)− h2(t))

β1

y(t + 1) = (x(t)− h1(t))β2 (y(t)− h2)

α2

and the scarcity constraints

0 ≤ h1(t) = hi1(t) + hj1(t) ≤ x(t)

0 ≤ h2(t) = hi2(t) + hj2(t) ≤ y(t)

Logarithmic utility function

U(hi1, hi2) = a1 log hi1 + a2 log hi2

ρ1 = ρ2 = ρ Pereau, Doyen Game Theory MABIES

,



Non cooperative outcome for 2 countries

Linear feedback rules a1 = a2 = 1

F nc1 =

(1− ρα1) (1− ρα2)− ρ2β2β1(2− ρα1) (1− ρα2) + ρβ2 (1− ρβ1)

F nc2 =

(1− ρα1) (1− ρα2)− ρ2β2β1(2− ρα2) (1− ρα1) + ρβ1 (1− ρβ2)


,



Cooperative outcome

Linear feedback rules a1 = a2 = 1

F c1 =

(1− ρα1) (1− ρα2)− ρ2β1β2(1− ρα2) + ρβ2

F c2 =

(1− ρα1) (1− ρα2)− ρ2β1β2(1− ρα1) + ρβ1

Under non cooperation, there is always overfishing ascompared to cooperation.

Overfishing is reduced (increased) when species are mutualpredators (symbiotic)

Always over-fishing for the predator but indeterminate sign forthe prey.


,



Symbiotic relationship

K1 = K2 = 1, ρ = 0.98, α1 = α2 = 0.5, β1 = β2 = 0.2

years

50454035302520151050

0.40

0.35

0.30

0.25

0.20

biomass

years

50454035302520151050

0.40

0.35

0.30

0.25

0.20

biomass

ncc


,



Negative relationship

K1 = K2 = 1, ρ = 0.98, α1 = α2 = 0.5, β1 = β2 = −0.2

years

50454035302520151050

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

biomass

years

50454035302520151050

0.60

0.55

0.50

0.45

0.40

0.35

0.30

0.25

biomass

ncc


,



Predator-prey relationship

K1 = K2 = 1, ρ = 0.98, α1 = α2 = 0.5, β1 = 0.2,β2 = −0.2

biomass

50454035302520151050

0.25

0.20

0.15

0.10

0.05

0.00

years

Predator

years

50454035302520151050

7

6

5

4

3

2

1

0

biomass

Prey

cnc


,



Coalition formation: Kwon (2006)

N identical countries

Nash equilibrium between a coalition of size J and N − Joutsiders (singletons)

Size of the coalition ?

• J = N Grand Coalition

• J = 0 No cooperation

• 2 < J < N Partial cooperation

Objective function of player i

maxhi (t)

∞∑

t=0

ρt log hi(t)


,



Programs

Program of a member i of the coalition J

V pci (t, J, x) = max

hpci(t)

{J log hpci (t) + V pc

i (t + 1, x)}

s.t x(t + 1) = f(x(t)− Jhpci (t)− (N − J)hok (t)

)

Program of a outsider k

V ok (t, J, x) = max

hok(t)

{log hok (t) + V ok (t + 1, x)}

s.t x(t + 1) = f(x(t)− Jhpci (t)− (N − J − 1)hoj (t)− hok(t)

)


,



Optimal harvest

Reaction function

hpci =x − (N − J)hok

(αρ(1− ρα)−1 + 1) J

hok =x − Jhpci

(αρ(1 − ρα)−1 + N − J)


F pci (j) = =

1

J (N − J + (1− ρα)−1)

F ok (j) = =

1

N − J + (1− ρα)−1= JF pc

i (J)

Total Fishing mortality is F = JF pci (J) + (N − J)F o

k (J).


,



Value functions

V pci (J, x) = Cpc(J) +Dpc log x

V ok (J, x) = C o(J) + Do log x

with

Do = Dpc =1

1− ρα

and

C pc(J) =1

1− ρ

[lnF pc(J) +

(ρα

1− ρα

)log (1− F )

]

C o(J) =1

1− ρ

[lnF o(J) +

(ρα

1− ρα

)log (1− F )

]


,



Particular cases

Grand Coalition

V ci (x) =

1

1− ρ

[log F c +

(ρα

1− ρα

)log (1− NF c)

]+

log x

1− ρα

No-cooperation

V nci (x) =

1

1− ρ

[log F nc +

(ρα

1− ρα

)log (1− NF nc )

]+

log x

1− ρα


,



Profitability, Internal and external stability conditions

Profitability condition: the value function of an insider ishigher than his non-cooperative payoff

V pci (J) ≥ V o

i (1) ⇔ Cpc(J) ≥ C o(1)

Internal stability condition:the value function of an insider ishigher than his payoff if he decides to withdraw the initialcoalition and which become of size J − 1.

V pci (J) ≥ V o

i (J − 1) ⇔ Cpc(J) ≥ C o(J − 1)

External stability condition: the value function of an outsideris higher than his payoff if he decides to join the coalition ofsize J which become J + 1.

V ok (J) ≥ V pc

k (J + 1) ⇔ C o(J) ≥ Cpc(J + 1)Pereau, Doyen Game Theory MABIES

,



Internal and external stability conditions

Internal stability

S(J) = Cpc(J)− C o(J − 1) ≥ 0

External stability:

S(J + 1) = Cpc(J + 1)− C o(J) ≤ 0


,



Results

For any N ≥ 3 and J ∈ [2,N − 1], an outsider has noincentive to join the coalition

For any N ≥ 3, if the size of the coalition is greater than 2,then an insider will always want to withdraw from the coalition

for any N, α and ρ, the maximum size of a sustainablecoalition is two. For any given N, there exists an α∗ρ∗ suchthat the coalition of size 2 is sustainable for anyαρ ∈ [α∗ρ∗, 1[. If αρ < α∗ρ∗, then there does not exist anysustainable coalition

For large N, it means that the growth stock has to be almostlinear α → 1 and almost no discounting, ρ → 1.


,



Viability approach: Doyen-Pereau (2012)

How to avoid bio-economic collapses?

Which balance between resource dynamics and strategicinteractions?

Formation of coalition using viability control

Shape of viable coalitions?

Minimum number of users in a viable coalition?

Marginal contribution of agents to maintain safe the resource?


,



The exploited renewable resource

Dynamics of the exploited resource x(t)

x(t + 1) = f

(x(t)− h(t)

)

with f ′ > 0, f (0) = 0, f (K ) = K .

0 xoa1 xoa2 .. .. xoan K

Catches by n agents with ei effort of agent i and qicatchability

h(t) =

n∑

i=1

qiei (t)x(t)


,



The agents exploiting the resource

n heterogeneous agents: ci cost of effort

c1 < c2 < .... < cn

Open-access levels

xOAi =

cipqi

We assume

xOA1 < xOA

2 < .... < xOAn < K

Rent of agent i with p price

Πi (x(t), ei (t)) = pqi (x(t)− xOAi ) ei (t)


,



Game issue

Coalition S : global sustainable rent

∑

i∈S

Πi (x(t), ei (t)) > 0

An ecological viability condition

x(t) > xS = mini∈S

xOAi

Outsiders j /∈ S :

Πj(x(t), ei (t)) ≥ 0


,



Viability kernel for a coalition S

Viability kernels Viab(S , t) for a coalition S ???

Dynamic programming:

At the terminal date T

Viab(S ,T ) ={x | x > xS

}

For any time t = 0, 1, ...,T − 1, there exists efforts ei ∈ Ssuch that ∀ outsiders j /∈ S

Viab(S , t) =

x > xS | f

x

1−

∑

i∈S

qiei −∑

j /∈S

qjej

∈ Viab(S , t + 1)


,



Viability kernels in the general case

Theorem

The viability kernels at time t are

Viab(S , t) = ∅ if 1 /∈ S

Viab({1, ..., n} , t) = ]xOA

1 ,+∞[

Viab({1, ..., j} , t) =]xOA

1 , xOA

j+1

[for j < n

Viab(S , t) = Viab(S , t) for S = ∪i ({1...i} ⊂ S)


,



Example: n = 3: The tragedy of open-access revisited

Coalition singletons:the smallest viability

Viab({1}) = ]xOA1 , xOA

2 [

Empty coalition

Viab({2}) = Viab({3}) = ∅

Partial Coalition

Viab({1, 2}) = ]xOA1 , xOA

3 [

Grand Coalition:the largest viability

Viab({1, 2, 3}) = ]xOA1 ,+∞[


,



A focus on the grand coalition S = {1, 2, 3}

Largest viability

Viab(N) = ]xOA1 ,+∞[

MEY viable

xMEY > xOA1 ∈ Viab(S)

0 10 20 30 40 50 60 70 80 90 100

xOA1

t

x(t) Stockx(t)

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

2000

2500

3000

Effortof agentsei (t)

0 10 20 30 40 50 60 70 80 90 100

0

20000

40000

60000

80000

100000

Rentof agentsΠi (t)


,



The grand coalition S = N

At t = T :Viab(N,T ) = {x > xN = xOA

1 }

At T − 1

Viab(N,T−1) =

{x > xOA

1 , ∃e

∣∣∣∣∑n

i=1Πi (x , ei ) > 0,f (x(1−

∑ni=1 ei )) ∈ Viab(N,T )

}

Viability kernel

Viab(N,T − 1) =

{x > xOA

1 ,maxe

f

(x(1−

n∑

i=1

ei )

)> xOA

1

}

Viab(N,T − 1) ={x > xOA

1 , x > f −1 (xOA1 )}

Viab(N,T − 1) =]xOA1 ,+∞[


,



A focus on the partial coalition S = {1, 2}

Reduced viability

Viab({1, 2}) = ]xOA1 , xOA

3 [

Agent 3 is neutralizedby 1 and 2

0 10 20 30 40 50 60 70 80 90 100

xOA1

xOA3

x(t) Stockx(t)

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

2000

2500

3000

3500

t

e(t)


0 10 20 30 40 50 60 70 80 90 100

0

5000

10000

15000



,



Viability kernel for a coalition S

At t = T : Viab(S ,T ) = {x > x{1,...,j} = xOA1 }

At t = T − 1:

Viab(S ,T−1) =

{x > xOA

1

∣∣∣∣∃eS ∀(el)l /∈S ,

f(x(1−

∑i∈S ei −

∑l /∈S el)

)> xOA

1

}

If x ∈]xOA1 , xOA

j+1[ then players j + 1, . . . , n become passive:

]xOA1 , xOA

j+1[⊂ Viab(S ,T − 1)

If x ≥ xOAj+1, the stock of the resource is too high to ensure a

positive rent for the coalition S whatever the effort played byplayer j + 1, .., n

[xOAj+1,∞[

⋂Viab(S ,T − 1) = ∅


,



Viable efforts for the viable coalitions

The viable feedbacks e∗S(t, x) = (e∗i (t, x))i∈S for a coalition S

and any stock x in Viab{1...j}(t) =]xOA1 , xOA

j+1

[are solutions of

the linear constraints∑

i∈S e∗i (px − ci ) > 0

∑i∈S e

∗i (t, x) =

α

(1−

f −1(xOA1 )

x

)+ (1− α)max

(0, 1 −

f −1(xOAj+1)

x

)

with 0 < α < 1.

Flexible decisions:

Ecological and economic performances

Passive outsiders


,



A focus on the singleton S = {1}

Smallest viability


2 [

0 10 20 30 40 50 60 70 80 90 1000

5000

10000

15000

20000

25000

30000

time t

Stockx(t)

0 10 20 30 40 50 60 70 80 90 100

0

20

40

60

80

100

120

140

160

t


0 10 20 30 40 50 60 70 80 90 100

0

1000

2000

3000

4000

5000

6000

t



,



A focus on the singleton S = {1}

Smallest viability


2 [

0 10 20 30 40 50 60 70 80 90 1000

5000

10000

15000

20000

25000

30000

time t

Stockx(t)

0 10 20 30 40 50 60 70 80 90 100

0

20

40

60

80

100

120

140

160

t


0 10 20 30 40 50 60 70 80 90 100

0

1000

2000

3000

4000

5000

6000

t



,



Minimum number of agents in a viable coalition

Extension of Sandal-Steinshamn (2004) (equilibrium):

n∗(x) = min (|S | | x ∈ Viab(S , 0))

where |S | the cardinal of coalition S

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

8

9

Stock x

Numb

er of pl

ayers n

*(x,c)

The number of viable agents increases with the stock


,



A ”simple” game formulation

Indicator function

1Viab(S,t)(x) =

{1 if x ∈ Viab(S , t)0 if x 6∈ Viab(S , t).

Maxmin formulation

V (T , x) = 1{x>xS}(x)V (t, x) = sup

ei , i ∈ S

ei ≥ 0,∑

i∈S

Πi (x , ei ) > 0

inf{ej , j /∈ S,ej ≥ 0Πj (x , ej ) ≥ 0

1{x>xS}V (t + 1, f (.))

t = 0, 1, ..,T − 1

with f (.) = f(x(1−

∑i∈S ei −

∑j /∈S ej

)).


,



Marginal contribution to viability

Shapley value

Shi(x) =∑

i∈S⊆N

(|S | − 1)!(n − |S |)!

n!

(1Viab(S,t)

(x)− 1Viab(S\{i},t)(x))

Application n = 3

Agents i\ Stock x 0 xOA1 xOA

2 xOA3

agent 1 | 0 | 1 | 1/2 | 1/3

agent 2 | 0 | 0 | 1/2 | 1/3

agent 3 | 0 | 0 | 0 | 1/3

Agent 1 always veto agentAgent 1 dictator if resource lowEqual contributions for active agents but nothing for passiveusers


,



Conclusion

To be continued...for the next 3 decades...


Documents

Game Theory and Applications to Fisheriescermics.enpc.fr/~delara/MABIES/Pereau-gametheory_MABIES2013.pdfBailey-Sumaila-Lindroos (2010) Application of game theory to ﬁsheries over