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 noncooperative 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
Country 1
Deplete
Conserve
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 gametheoretic 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 noncooperative 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
PVC  30 
30
30
30

 ... 
1.05 1.052
1.05n

 This infinite sum of the geometric progression and can be
solved as follows:
30
 PVC  1  0.95 = 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:
PVD  40 
3
3
3

 ... 
1.05 1.052
1.05n
= 37 +3/(1-0.95) = 97
 40  3  3 
3
3
3

 ... 
1.05 1.052
1.05n
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:
dx
 F ( x) 
dt
n
 hi  0
i 1
 By assuming logistic growth
the steady state stock is then
n
q
hi=qEix
 Ei
x  K (1  i 1
R
)
Stock biomass
depends on all
fishing efforts
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
n
max  i  pqEi K (1 
q  Ei
i 1
R
)  ci Ei
Deriving reaction curves of the players
 The first order condition for
player i is
n 1
2 pq Ei K   pq 2 E j K
2
 i
 pqK 
Ei
j i
R
 ci  0
bi=ci/pqK
 The reaction curve of player i
is then
n 1
Ei  
2
j i
Ej

R
(1  bi )
2q
Equilibrium fishing efforts
 Derive by using the n reaction curves
Ei 
nR
(1  bi ) 
(n  1)q
n 1

j i
R
(1  b j )
(n  1)q
 The equilibrium fishing efforts depend on the efficiency of
all players and the number of players
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
q( E1  E2 )
x  K (1 
)
R
n1 1
where E1  e1v 
 e1w
w v
 The individual domestic fishing firm v maximises
max  1v  pqe1v x  ce1v
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 firstorder condition by applying symmetry of the vessels
n1 1
 1v
 pqK 
e1v
2 pq 2 e1v K   pq 2 e1w K  pq 2 E2 K
w v
R
n1 R
E1 
[ (1  b)  E2 ]  n1e1v
n1  1 q
c  0
Equilibrium fishing efforts
 Analogously in the other country
E2 
n2 R
[ (1  b)  E1 ]
n2  1 q
 By solving the system of two equations yields the equilibrium
E2 
E1 
Rn 2
1 b
(
)
q 1  n1  n2
Rn1
1 b
(
)
q 1  n1  n2
Equilibrium stock
 Insert equilibrium efforts into
steady state stock
expression
 The stock now depends
explicitly on the number of
the total fishing fleet
x
K [1  (n1  n2 )b]
1  n1  n2
Equilibrium rent
 Insert equilibrium efforts and
stock into objective function
to yield
P1  RpK
n1 (1  b) 2
(1  n1  n2 ) 2
First stage
 The countries maximise their welfare, that is, fishing
fleet rents less management costs
max W1  P1  n1F
 The optimal fleet size can be calculated from the FOC
(implicit reaction function)
W1
(1  n1  n2 )(1  b) 2
 RpK
F 0
3
n1
(1  n1  n2 )
Results
 Aplying symmetry and changing variable m = 1+2n
RpK (1  b) 2
F 0
3
(1  2n)


2 1 / 3
1  RpK (1  b)

n1  
  1
2 
F




 With F=0  open access
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 noncooperative behaviour for the rest of the game
 Cooperation means higher fish stock than noncooperation, in the defection period the stock is between
cooperative and non-cooperative levels
Cooperative strategies
 Cooperative effort from
EiC
SG-model
 Cooperative fish stock
 Cooperative benefits
R

(1  b)
2nq
K
x  (1  b)
2
C
 iC
pqEiC xC  cEiC

1 
Optimal defection effort
 Best response when all
others choose the
cooperative strategy
 Optimal defection effort
EiD
R
(n  1) R

(1  b) 
(1  b)
2q
4nq
EiD
R(1  b)
1

(1  )
4q
n
Non-cooperative strategies
 Effort
 Stock
EiN
x
R

(1  b)
(n  1)q
N
1  nb
 K(
)
n 1
Cooperation vs. cheating
 Benefits from cheating
pqEiD ( K
q D R(n  1)(1  b)
 K ( Ei 
)  cEiD
R
2nq
 Condition for cooperative
equilibrium
N N
N

(
pqE
x

cE
i
i )
 iC  pqEiD x D  cEiD 
1 
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 NonCooperative 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 twospecies 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 simulation
p1
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
60
ncrit
40
20
0
-0.2
-0.1
0.2
0.1
0
0
0.1
theta1(alpha)
-0.1
0.2
-0.2
theta2(beta)
Case 2: High growth
60
50
ncrit
40
30
20
10
0
-0.2
-0.1
0.2
0.1
0
0
0.1
-0.1
0.2
theta1(alpha)
-0.2
theta2(beta)
Case 4: Low valued stock has a high intrinsic
growth rate, high value stock has a low
intrinsic growth rate.
60
ncrit
40
20
0
-0.2
0.2
-0.1
0.1
0
0
0.1
theta1(alpha)
-0.1
0.2
-0.2
theta2(beta)
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