Non-cooperative game theory: Three fisheries games

Marko Lindroos

JSS

This lecture is about

Non-cooperative games classification Nash equilibrium

Applications in fisheries economics basic game (Mesterton-Gibbons NRM 1993) stage games (Ruseski JEEM 1998) repeated games (Hannesson JEEM 1997)

Non-cooperative games

Individual strategies for the players Reaction functions, best reply Nash equilibrium definition

Stages games at different levels

Repeated games, folk theorems, sustaining cooperative

behaviour as equilibria

Dynamic games

Why non-cooperative

Classification: strategic (static), extensive (dynamic),

coalition

Important in fisheries non-cooperation (competition) vs

cooperation

Division not clear, almost all games have both non-

cooperative and cooperative elements

Typically in economics non-cooperative game theory

dominates

What are non-cooperative games about

How fisher’s decisions interact with other fishers’

decisions

What is the best strategy for the fishers

What is exected to happen is the fishery? Depends on

rules of the game, number of players, biological factors

Why fishers behave as they do?

Assume rational choice

International fisheries negotiations

Nature of negotiations Countries attempt to sign and ratify agreements to

maximise their own economic benefits

Negotiations typically time-consuming

Agreements not binding self-enforcing or voluntary

agreements

Explaining the tragedy of the commons

Can we explain the seemingly irrational behaviour in the

world’s fisheries, overexploitation, overcapitalisation,

bycatch…

Non-cooperative game theory explains this behaviour

Non-cooperative games vs open access (freedom of the

seas)

Nash equilibrium

Each player chooses the best available decision

It is not optimal for any single player to unilaterally change

his strategy

There can be a unique equilibrium, multiple equilibria or

no equilibria

Fisher’s dilemma

Modified prisoner’s dilemma

Non-cooperation vs cooperation

Example 1: Two countries exploiting a common fish stock

Country 2

Deplete Conserve

Country 1 Deplete 3, 2 40, -5

Conserve -5, 40 30, 20

Fisher’s dilemma explanation

Deplete: Corresponds to non-cooperation. The country is

only interested in short-run maximisation of economic

benefits. No regulation. Conserve: Optimal management of the fishery.

Cooperative case. The cooperative solution (Conserve, Conserve)

maximises the joint payoffs to the countries, equal to 50.

However, neither of the countries is satisfied with the

cooperative strategy. Both would gain by changing their

strategy to Deplete (free-riding). This is the game-

theoretic interpretation of tragedy of the commons. In the Nash equilibrium (Deplete, Deplete) unilateral

deviation is not optimal for the countries.

Reaction (best response) functions

Gives the best decisions a player can make as a function

of other players’ decisions If a decision is not a best response it can not be a Nash

equilibrium Typically best response functions are derived from a set

of optimisation problems for the players. In an n player

game there are n best response functions. Nash equilibrium is found at the intersection of the best

response functions (solution to the system of equations) Strategy is best response if it is not strictly dominated

Repeated games

deterring short-term advantages by a threat or

punishment in the fisher’s dilemma escaping the non-

cooperative Nash equilibrium

folk theorems (understood not published)

credibility of threats

Numerical repeated game

Assume that the game in example 1 is repeated infinite

number of times. If one player deviates from the

cooperative strategy Conserve to the non-cooperative

strategy Deplete, it will also trigger the other player to

choose Deplete forever after the deviation. This means

that both countries punish severely deviations from the

common agreement. Cooperation can be sustainable if the present value of

choosing Conserve is higher than deviating once from

cooperation. Present value of cooperation to player 1 when discount

rate is 5%:

Cooperation vs. deviation

This infinite sum of the geometric progression and can be

solved as follows: = 600

Next we calculate the present value of deviation. Country 1

first receives 40 and thereafter only 3 since country 2 uses its

trigger strategy, according to which it never again signs an

agreement. Hence, the present value of deviation is:

= 37 +3/(1-0.95) = 97

nCPV05.1

30...

05.1

30

05.1

3030

2

95.01

30

CPV

nDPV05.1

3...

05.1

3

05.1

340

2

n05.1

3...

05.1

3

05.1

33340

2

Tragedy of the commons solved

We see that the present value of deviating is clearly

smaller and thus, cooperation (Conserve, Conserve) is

now the equilibrium of the repeated game.

Note that the discount rate is critical in repeated games.

As discount rate approaches infinity the present value of

cooperation approaches 30 and the present value of

deviation approaches 40. The critical discount rate, over

which deviation is profitable, is therefore finite.

The first non-cooperative fisheries game

Assume there are n players (fishers, fishing firms,

countries, groups of countries) harvesting a common fish

resource x Each player maximises her own economic gains from the

resource by choosing a fishing effort Ei

This means that each player chooses her optimal e.g.

number of fishing vessels taking into account how many

the other players choose As a result this game will end up in a Nash equilibrium

where all individual fishing efforts are optimal

Building objective functions of the players

Assume a steady state:

By assuming logistic growth

the steady state stock is then

0)(

1

n

i

ihxFdt

dx

)1( 1

R

Eq

Kx

n

i

i

Stock biomass depends on all fishing efforts

hi=qEix

Objective function

Players maximise their net revenues (revenues – costs)

from the fishery max phi –ciEi

Here p is the price per kg, hi is harvest of player i, ci is unit

cost of effort of player i

ii

n

ii

ii EcR

EqKpqE

)1(max 1

Deriving reaction curves of the players

The first order condition for

player i is

The reaction curve of player i

is then

0

21

22

i

n

ijji

i

i cR

KEpqKEpq

pqKE

)1(22

1

i

n

ij

ji b

q

REE

bi=ci/pqK

Equilibrium fishing efforts

Derive by using the n reaction curves

The equilibrium fishing efforts depend on the efficiency of

all players and the number of players

1

)1()1(

)1()1(

n

ij

jii bqn

Rb

qn

nRE

Illustration

Nash-Cournot equilibrium

Symmetric case

Schäfer-Gordon model

Exercises

Compute the symmetric 2-player and n player equilibrium.

First solve 2-player game, then extend to n players.

A two-stage game (Ruseski JEEM 1998)

Assume two countries with a fishing fleet of size n1 and n2

In the first stage countries choose their optimal fleet

licensing policy, i.e., the number of fishing vessels. In the second stage the fishermen compete, knowing how

many fishermen to compete against

The model is solved backwards, first solving the second

stage equilibrium fishing efforts Second, the equilibrium fleet licensing policies are solved

Objective function of the fishermen

The previous steady state stock is then

The individual domestic fishing firm v maximises

))(

1( 21

R

EEqKx

vvv cexpqe 111max

1

111

1

wheren

vw

wv eeE

Reaction functions

In this model the domestic fishermen compete against

domestic vessels and foreign vessels

The reaction between the two fleets is derived from the first-

order condition by applying symmetry of the vessels

02

1

22

12

12

1

1

1

cR

KEpqKepqKepqpqK

e

n

vwwv

v

v

venEbq

R

n

nE 112

1

11 ])1([

1

Equilibrium fishing efforts

Analogously in the other country

By solving the system of two equations yields the equilibrium

])1([1 1

2

22 Eb

q

R

n

nE

)1

1(

21

22 nn

b

q

RnE

)1

1(

21

11 nn

b

q

RnE

Equilibrium stock

Insert equilibrium efforts into

steady state stock

expression

The stock now depends

explicitly on the number of

the total fishing fleet

21

21

1

])(1[

nn

bnnKx

Equilibrium rent

Insert equilibrium efforts and

stock into objective function

to yield

221

21

1)1(

)1(

nn

bnRpKP

First stage

The countries maximise their welfare, that is, fishing fleet

rents less management costs

The optimal fleet size can be calculated from the FOC

(implicit reaction function)

FnPW 111max

0)1(

)1)(1(3

21

221

1

1

Fnn

bnnRpK

n

W

Results

Aplying symmetry and changing variable m = 1+2n

With F=0 open access

0)21(

)1(3

2

F

n

bRpK

1

)1(

2

13/12

1 F

bRpKn

Discussion

Subsidies

Quinn & Ruseski: asymmetric fishermen

entry deterring strategies: Choose large enough fleet so

that the rival fleet is not able make profits from the fishery

Kronbak and Lindroos ERE 2006 4 stage coalition game

Repeated games – a step towards cooperation

When cooperation is sustained as an equilibrium in the

game

The game is repeated many times (infinitely)

The players use trigger strategies as punishment if one of

the players defects from the cooperative strategy Trigger here means that defection triggers non-

cooperative behaviour for the rest of the game

Cooperation means higher fish stock than non-

cooperation, in the defection period the stock is between

cooperative and non-cooperative levels

Cooperative strategies

Cooperative effort from

SG-model

Cooperative fish stock

Cooperative benefits

)1(2

bnq

RECi

1

Ci

CCiC

icExpqE

)1(2

bK

xC

Optimal defection effort

Best response when all

others choose the

cooperative strategy

Optimal defection effort

)1(4

)1()1(

2b

nq

Rnb

q

REDi

)1

1(4

)1(

nq

bREDi

Non-cooperative strategies

Effort

Stock

)1()1(

bqn

RENi

)1

1(

n

nbKxN

Cooperation vs. cheating

Benefits from cheating

Condition for cooperative

equilibrium

Di

Di

Di cE

nq

bnRE

R

qKKpqE

)

2

)1)(1(((

1

)( Ni

NNiD

iDD

iCi

cExpqEcExpqE

Discussion

Hannesson (JEEM 1997) similar results

Higher costs and lower discount rate enable a higher

number of countries in the cooperative equilibrium

Self-enforcing agreements

On Species Preservation and Non-Cooperative Exploiters

Outline

Motivation

Model

Results

Conclusion

Discussion

Motivation

Combining two-species models with the game theory

What are the driving force for species extinction in a two-

species model with biological dependency?

Does ‘Comedy of the Commons’ occur in two-species

fisheries?

What are the ecosystem consequences of economic

competition?

Modelling approach

Two-species

n symmetric competitive exploiters with non-selective

harvesting technology

Fish stocks may be biologically independent or dependent

What is the critical number of exploiters?

Analytical independent species model

S-G model

Derive first E* as the optimal effort, it depends on the

relevant economic and biological parameters

An n-player equilibrium is then derived as a function of

E*and n.

Relate then the equilibrium to the weakest stock’s size to

compute critical n*, over which ecosystem is not

sustained.

Dependent vs independent species

Driving force of extinction:

Independent species Biotechnical productivity Economic parameters

Dependent species Biological parameters must be considered Gives rise to a complex set of conditions For example:

Natural equilibrium does not exist‘The Comedy of the Commons’

Numerical dependent species model

Cases illustrated: Biological competition, symbiosis and

predator-prey

Case 1: Both stocks having low intrinsic growth rate

Case 2: Both stocks having a high intrinsic growth rate

Case 3: Low valued stock has a low intrinsic growth rate,

high value stock has a high intrinsic growth rate.

Case 4: Low valued stock has a high intrinsic growth rate,

high value stock has a low intrinsic growth rate.

Parameter values applied for simulationp1 p2 Rlow Rhigh K1=

K2

c q OA MS θ1 θ2

1 2 0.3 0.9 50 7 0.5 60 60 [-0.2;0.2] [-0.2;0.2]

Case 1: low intrinsic growth rate

-0.2

-0.1

0

0.1

0.2 -0.2

-0.1

0

0.1

0.2

0

20

40

60

theta2(beta)theta1(alpha)

ncrit

Case 2: High growth

-0.2

-0.1

0

0.1

0.2 -0.2-0.1

00.1

0.2

0

10

20

30

40

50

60

theta2(beta)theta1(alpha)

ncrit

Case 4: Low valued stock has a high intrinsic growth rate, high value stock has a low intrinsic growth rate.

-0.2

-0.1

0

0.1

0.2 -0.2

-0.1

0

0.1

0.2

0

20

40

60

theta2(beta)theta1(alpha)

ncrit

Opposite case 3

Conclusion

‘Tragedy of the Commons’ does not always apply

A small change in the interdependency can lead to big

changes in the critical number of non-cooperative players

With competition among species a higher intrinsic growth

rate tend to extend the range of parameters for which

restricted open access is sustained

Discussion

From single-species models to ecosystem models

Ecosystem approach vs. socio-economic approach

Agreements and multi-species