Farsightedness in a coalitional Great Fish War
Michèle Breton
GERAD and HEC Montréal
michele.breton@hec.ca
Michel Y. Keoula
GERAD and HEC Montréal
michel-yevenunye.keoula@hec.ca
Preliminary version. Please do not cite without permission
Abstract
We explore the implications of the farsightedness assumption on the conjectures of
players in a coalitional Great Fish War model with symmetric players, derived from the
seminal model of Levhari and Mirman (1980). The farsightedness assumption for players
in a coalitional game acknowledges the fact that a deviation from a single player will lead
to the formation of a new coalition structure as the result of possibly successive moves
of his rivals in order to improve their payo¤s. It departs from mainstream game theory
in that it relies on the so-called rational conjectures, as opposed to the traditional Nash
conjectures formed by players on the behavior of their rivals.
For values of the biological parameter and the discount factor more plausible than the
ones used in the current literature, the farsightedness assumption predicts a wide scope for
cooperation in non trivial coalitions, sustained by credible threats of successive deviations
that ruin the shortsighted payo¤ of any prospective deviator. Compliance or deterrence
of deviations may also be addressed by acknowledging that information on the …sh stock
or on the catch policies actually implemented may not be available in real time (dynamic
farsightedness). In that case, the requirements are stronger and the sizes and number of
possible farsighted stable coalitions appear to be di¤erent.
In the sequential move version, which could mimic some characteristics of …shery models, the results are not less appealing, even if the dominant player or dominant coalition
with …rst move advantage assumption provides a case for cooperation with the traditional
Nash conjectures.
Key Words: Coalition stability; …sheries; farsightedness.
1
Introduction
In recent years, many …shery economists have called for a merger between the cooperative
approach and the non-cooperative one in …shery game models (see for example Lindroos et
al. (2007) and Kaitala and Lindroos (2007)).
The cooperative approach has focussed on the assessment of the relative strengths of various coalitions of participating countries. Recently, the partition function approach, designed
to account for externalities, has superseded the characteristic function approach (Pintassilgo
This research was supported by FQRSC (Québec) and NSERC (Canada).
1
(2003)). The subsequent issue of sharing the bene…ts of cooperation has also yielded a large
literature.
On the other hand, the non-cooperative approach, especially when applied to high seas
…sheries, yields a prisonner’s dilemma type of result, thus heightening the necessity of at least
a partial coordination of international …sh wars (see for example Kwon (2006)). In providing
non-cooperative foundations for cooperative arguments, the apparatus of non-cooperative
game theory seems to be promising for the uni…cation of the two strands of the game theory
literature.
In this vein, the farsightedness assumption in a coalitional game acknowledges the fact
that a deviation of a single player will lead to the formation of another coalition structure, as
the result of possibly successive moves of his rivals in order to improve their payo¤s. Thus,
the new coalition structure may not correspond to the extreme schemas of either the collapse
of all cooperative agreements or a status quo in the previous coalition stripped of its deviant
member. Hence, the farsightedness assumption departs from mainstream game theory in that
it relies on the so-called rational conjectures, as opposed to the traditional Nash conjectures
formed by players on the behavior of their rivals.
This paper aims at exploring farsightedness in the context of international …shery games,
more speci…cally in the Great Fish War model popularized by Levhari and Mirman (1980).
It mainly investigates the possibility of extending, by a rational conjectures behavioral assumption, the results obtained for Nash conjectures in the coalitional Great Fish War model
presented in Kwon (2006). In addition, this paper investigates the concept of dynamic farsighted stability in that context, along the same lines as in de Zeeuw (2008), which investigates
the issue of coalition stability in the context of international agreements for pollution abatement.
The paper is organized as follows. In Section 2, we recall the bioeconomic model of the
Great Fish War and introduce most of the de…nitions and assumptions in our coalitional game.
This framework is used in Section 3 to derive the main results of the literature on the stability
of coalitions in the Great Fish War. Section 4 is a detailed exposition of the implications of
rational conjectures on the stability of coalitions, using both static farsightedness and dynamic
farsightedness assumptions. Before concluding in Section 6, a similar analysis is carried on
in Section 5 using a Stackelberg-type information structure, where the coalition acts as a
dominant player and has the …rst move advantage, in order to investigate how our results are
modi…ed in the case of sequential moves.
2
2.1
Model and assumptions
The bioeconomic model
The bioeconomic model underlying the …shery coalition game consists in de…ning a welfare
function for the …shery owner, and a growth function for the …sh stock. In this model, the
market side of the exploitation of the …shery (prices, demand curves) is not modelled, and
the one-period welfare function of the …shery owner is assumed logarithmic:
u(x) = log x;
where x it the catch.
The one-period growth function of the …sh stock is given by the relation
st+1 = bst ; 0 <
2
<1
b;
where s is the level of the …sh stock,
is a biological parameter and b a constant. This
growth function is well known in the …shery games litterature, especially since the seminal
paper of Levhari and Mirman (1980) on Fish Wars. It displays the essential features of the
continuous-time Schaefer growth function used in basic bioeconomic models of …sheries.
One can easily verify the Schae¤er growth function properties by rewriting the previous
relation as a di¤erence equation:
st+1
st = bst
st :
The analog of the Schae¤er growth function that emerges is F (s) = bs
s. It is indeed
1
1
1
1
concave such that F (0) = F (b
) = 0 and F (s) > 0 for s 2 (0; b
). The positive constant
1
1
b
is the saturation level, or the natural equilibrium level, of the biomass. In the sequel, b
is normalized to one.
The Bellman equation de…ning the optimal exploitation of the …shery in discrete time over
an in…nite horizon by a sole owner is thus:
V (s) = max flog x + V (s)g
x
where
2.2
is the one-period discount factor, 0
< 1:
The coalition game model
The coalition formation game is based on two key assumptions:
Ai all countries are identical,
Aii only one coalition forms, that is, all the outsiders to the coalition act as single players.
The symmetry assumption is made for analytical convenience. As a consequence, one
obtains an equal sharing rule of the payo¤ of the coalition. Some of the shortcomings of this
assumption in the context of the coalition game model are discussed at the end of Section 4.2.
The second assumption is in line with the 1995 UN Fish Stock Agreement, which admonishes countries to cooperate in the management of the high seas …sheries. Nevertheless, the
agreement does not preclude any interested country to have access to the …sheries, con…rming
the open access nature of the high seas.
The coalition approach is based on the comparison of the possible outcomes of the game
according to each of the 2n possible partitioning of a set N of n players. The current favorite
tool of analysis in …shery economics is the partition function. Considered to be the most
advanced achievement of cooperative game theory applied to …shery games, the partition
function approach is presented in Pintassilgo (2003).
More formally, …rst de…ne a coalition structure as in Carraro and Marchiori (2003):
De…nition 1 A coalition structure ! = fC1 ; C2 ; :::; Cl g is a partition of the player set N ,
that is, Ci \ Cj = ? for i 6= j and [li=1 Ci = N .
Denote by the set of all coalition structures of the player set N . The total number of
elements in is 2n .
3
De…nition 2 A partition function P : ! R is a mapping which associates to each coalition
structure, a vector in Rj!j1 , representing the worth of all coalitions in !: A per-member partition function p : ! Rn is a mapping which associate to each coalition structure a vector
of individual payo¤ s in <n .
Under the assumptions of symmetric players, and unique coalition, the size of the coalition is su¢ cient to characterize the coalition structure. We denote ! m a coalition structure
consisting in one coalition of m members and n m outsiders. The per-member partition
function’s value corresponding to a coalition member is denoted by pcm ; while the per-member
partition function’s value corresponding to an outsider is denoted pom 2 . When no coalition
forms, the “individual” partition function is denoted po0 , and can be termed the no-coalition
partition function.
Finally, de…ne a pro…table coalition, as in Carraro and Marchiori (2003, p.164) :
De…nition 3 A coalition is pro…table if each cooperating player gets a payo¤ larger than the
one she would get when no coalition is formed. Formally, a coalition of size m is pro…table if
(m)
where
pcm
po0 > 0;
is called the pro…tability function.
As pointed by the two authors, pro…tability is a necessary condition for a coalition to
form.
As far as membership rules are concerned, the economic coalition theory analyzes coalition
games according to one of the three following rules: open membership, exclusive membership
and coalition unanimity membership. Open membership means that members can freely
access or leave a coalition. Exclusive membership requires the consensus of members to accept
a new member in a coalition, but members can leave it freely. Finally, coalition unanimity
membership means that no player can enter or leave a coalition without the consensus of all
the members. Exclusive membership seems to be the most appealing choice in the context of
…sheries, since adhesion of new members in a …shery management organization requires the
agreement of existing members, while members’withdrawal is free and of no direct consequence
on the existence of the organization. However, we shall address this …sh war question under
the open membership rule, because it allows to focus on the very issue of this paper, that
is, stability of cooperation agreements. Furthermore, in the case under consideration, we will
show that the per-member partition function for members of the coalition is increasing in
the size of the coalition; in that case, enlargement of membership is bene…cial for all existing
members of a coalition, and the two membership rules coincide.
3
Nash conjectures
3.1
The partition function
For n countries participating in the …shery, the …sh stock evolves as
n
X
st+1 = (st
xi ) ;
i=1
j!j is the cardinality of !. R is a set product of the <j!j :
Notice the abuse since the above de…nition of a per-member coalition function assign a payo¤ to both
coalition members and non-members.
1
2
4
and the Bellman equation corresponding to the optimization problem of Country i is
! !)
(
n
X
Vi (s) = max log xi + Vi
s
xk
:
xi
(1)
k=1
The essential feature of this relation is the negative externalities imposed on each country by
the others.
Since countries are identical, assume that the value function of a given country has the
following logarithmic form:
Vi (s) = A + B log s; i = 1; : : : n:
The …rst order condition for Country i is then
1
=
xi
s
which yields the optimal catch
xi =
s
B
Pn
k=1 xk
P
k6=i xk
:
(2)
B+1
Simultaneoulsy solving (2) for the n countries yields the feedback Nash equilibrium …shing
strategies of the n players:
s
xi =
; i = 1; : : : n:
n+B
Replacing in (1), one obtains by identi…cation
B=
A = (1
1
)
1
1
1
log
+
log
;
(3)
n
(n 1) 1
n
(n 1)
which veri…es the logarithmic functional form assumption. The corresponding feedback Nash
equilibrium …shing strategies are linear in the stock variable:
xi (s) =
1
n
(n
1)
s
i = 1; :::; n:
Notice that Equation (1) de…nes the no-coalition partition function as a function of s, yielding
po0 (s) = A + B log s:
Now suppose that under the aegis of a Regional Fishery Management Organisation (RFMO),
m countries decide to coordinate their …shing strategies in a coalition. This game has been
fully studied by Kwon (2006); we brie‡y recall the results, which are obtained in a similar
fashion to above, where the Nash equilibrium involves n m + 1 players, that is, the coalition
and the remaining n m countries.
The m-member coalition optimizes the total welfare of its members, taking into account
the externalities of the outsiders. The per-member partition function, denoted pcm (s); veri…es
the Bellman equation:
(
! !!)
n
X
pcm (s) = max log x +
pcm
(s mx
xk
;
x
k=m+1
5
where x is the catch of each individual member of the coalition. Assuming that the perc log s yields the …rst order condition
member partition function has the form pcm (s) = Acm +Bm
1
=m
x
s
c
Bm
Pn
mx
:
k=m+1 xk
(4)
Similarly, the non-member partition function pom (s) veri…es
(
! !!)
n
X
pom (s) = max log xi +
pom
s
xk
;
xi
k=1
o log s yields the …rst order conditions
assuming pom (s) = Aom + Bm
Bo
Pnm
1
= (
xi
s
), i = m + 1; :::n:
k=1 xk
(5)
Simultaneously solving (4) and (5) yields the Nash equilibrium feedback strategies of
members and non-members, and routine computations yield the following expressions for the
parameters of the partition functions:
1
o
c
Bm
= Bm
=B=
Acm = (1
Aom = (1
)
1
log
)
1
1
m((1
) (n
m) + 1)
log
1
) (n
m) + 1
(1
1
+
+
;
(6)
log
1
log
1
(1
) (n
m) + 1
(1
) (n
m) + 1
;
(7)
:
(8)
The equilibrium …shing policies are linear functions of the stock:
xc (s; m) =
1
m((1
xo (s; m) =
) (n
1
) (n
(1
m) + 1)
m) + 1
s;
(9)
s:
(10)
Notice that each coalition member harvests m times less than an outsider, so that the harvest
of the coalition equals that of a single individual outsider.
For the full cooperative solution (m = n), the coe¢ cient is
Acn = (1
)
1
log
1
n
+
1
log
;
while for m = 1 either in (7) or (8), one recovers the Nash equilibrium solution (3)
Ac1 = Ao1 = A:
c = B; the pro…tability function is independent of the level of the …sh
Notice that, since Bm
stock:
(m) = pcm (s)
po0 (s) = Acm
A
6
= (1
)
1
1
1
log
n
(1
(n 1)
) (n m) + 1
log m :
It will be convenient to extend the domain of the pro…tability function to the interval
[1; n] R; similarly, we will consider continuous extended partition functions de…ned by
c
P c (m; s) = Acm + Bm
log s : [1; n]
o
P o (m; s) = Aom + Bm
log s : [1; n]
R+ ! R
R+ ! R
where the per-member partition function pcm (s) coincides with P c (m; s) for integer values of
m, and similarly for the non-member partition function.
It is straightforward to verify that the pro…tability function is convex, vanishes at m = 1,
)
admits a minimum at m = 1+n(1
> 1 and is strictly increasing for m > m : In addition,
2
m ! 1 and the pro…tability function becomes arbitrarily large when
! 1: Properties of
the extended partition functions are also obtained in a straightforward way.
Some results and their interpretation are summarized as follows:
Ri The no-coalition partition function po0 (s) is equal to the per-member partition function
pc1 (s) and to the non-member coalition function po1 (s) for m = 1.
Rii If
is su¢ ciently large, there exists m1 , 1 < m < 2
coalition is pro…table, that is, (m) > 0 for m m1 .
m1
n
1, such that the
This result indicates that the necessary condition for forming a coalition is satis…ed for
su¢ ciently large values of
. Appendix 8.1.2 reports a table of threshold values
above
which there exists a non trivial coalition (i.e. di¤erent from the grand coalition) that is
pro…table, for various values of the number n of players, and of threshold values c above
which the pro…tability function is always positive, that is m1 = 2:
Furthermore, the open membership rule appears to be a natural one since the pro…tability
function is increasing whenever it is positive.
Riii At any s, the non-member partition function is strictly increasing in m and the permember partition function is strictly increasing in m for m m :
This result means that for m
coalition.
m , coalition members see an interest in broadening the
Riv At any s, the value of the per-member partition function pcm (s) is smaller than the value
of the non-member partition function pom (s), for all m > 1:
This result indicates that, when the coalition forms, the welfare of the coalition members
is smaller than that of the outsiders. There is indeed an opportunity cost of being a member
of the coalition.
Rv At any s, the marginal impact of an increase in m on the per-member partition function
is smaller than its marginal impact on the non-member partition function.
This result means that, when the coalition broadens, outsiders to the new coalition gain
more than previous coalition members, so that the opportunity cost of a coalition member
rises.
7
3.2
Myopic stability
Pro…tability is only a necessary condition for a coalition to form. For a full characterization
of a coalition, one needs su¢ cient conditions. Following d’Aspremont et al. (1983), many
authors have used what is termed in de Zeeuw (2008) the myopic stability concept. Myopic
stability refers to stability for Nash conjectures, whereby a coalition member contemplating
a deviation forms his anticipation on the basis that the strategies of the other players will
remain constant – namely, that his deviation would not change how the other players are
partitioned into the coalition structure.
Myopic stability consists in two conditions: internal stability, which requires no defection
temptation from individual coalition members, and external stability, which requires that
outsiders not be tempted to join the coalition. In the game under consideration here, this
amounts to:
IS pcm (s)
pom
1 (s),
or equivalently Acm
Aom
1
for internal stability; and
ES pom (s) > pcm+1 (s); or equivalently Aom > Acm+1 for external stability.
Internal stability in this setting is usually referred to in the literature as stand-alone
stability, which conveys the idea that the rest of the coalition remains constant in the event
of a defection whereby the defector do not integrate any other coalition. For m = 1, it is
assumed that the singletons are internally and externally stable.
It is worthwhile to de…ne the stability function S : [2; n] ! R
S(m) = P c (s; m)
=
S(m) = (1
)
1
1
1
Acm
log
P o (s; m
1)
Aom 1 ;
(1
(1
) (n m + 1) + 1
) (n m) + 1
log m :
The stability function S is obviously continuous and di¤erentiable with respect to m:
Now, internal and external stability conditions are equivalent to S(m) 0 and S(m + 1) < 0.
Furthermore, one can establish that S(m) < 0 for 3
m
n (see Appendix 8.1.1). Thus,
two (2) is the only candidate for the size of a non trivial stable coalition. Furthermore, the
stability function for a size-two coalition, which has the same expression as the pro…tability
function for a size-two coalition. Using observation Riii, the pro…tability condition is then
su¢ cient.
To wrap up, a full characterization of the myopic stable coalition is summarized in Proposition 1.
Proposition 1 If it exists, the largest myopic stable coalition of the coalitional Great Fish
War game is of size two and the pro…tability condition is necessary and su¢ cient.
A similar result is obtained in Kwon (2006).
For a wide range of values of the parameters and , it is not possible to …nd a non trivial
coalition of the smallest size (m = 2) that is myopically stable. In Appendix 8.1.2, we provide
a table of the thresholds c above which it is possible to coordinate a coalition of size 2, for
di¤erent values of the the total number n of countries participating in the …shery (see also
8
Kwon, 2006, p. 374). Those thresholds are the roots in
of (m) with m = 2. A choice of
above c ensures that the pro…tability function for m = 2 is positive.
When the total number of countries n = 8, this threshold is 0:9318. For example, Table 1
and Figure 1 illustrate the case = 0:99 and = 0:95 ( = 0:9405), where a myopic stable
coalition of size 2 is obtained.
m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8
c
Am
-964.33 -961.53 -926.72 -876.59 -816.12 -747.29 -670.87 -587.09
Aom
-964.33 -892.21 -816.86 -737.96 -655.18 -568.11 -476.28
(m)
2.81
37.62
87.74
148.21 217.05 293.46 377.25
S(m)
2.81
-34.51
-59.74
-78.16
-92.11
-102.76 -110.81
Table 1: Outsider and insider payo¤s, pro…tability and stability functions,
= 0:99; = 0:95 and n = 8:
400
Profitability
200
Payoff
0
Stability
-200
-400
Outsiders
-600
-800
Insiders
-1000
1
2
3
4
5
6
7
8
Size of the coalition
Figure 1: Outsider and insider payo¤s, pro…tability and stability functions in
the simultaneous moves model, = 0:99; = 0:95 and n = 8.
4
4.1
Rational conjectures
Farsighted coalitional stability
From Table 1, the following inequalities can be observed:
Ac2 > Ao1
Ac4 > Ao2
Ac7 > Ao4
or equivalently
or equivalently
or equivalently
pc2 (s) > po1 (s)
pc4 (s) > po2 (s)
pc7 (s) > po4 (s):
This example illustrates observations Riii and Rv above. It also stresses the fact that although
it is desirable to broaden coalitions (by the monotonicity of the per-member partition function), no single outsider can implement an improvement for herself by her individual move
into a coalition of at least 2 players. Similar patterns are found in de Zeeuw (2008) for an
international environmental game.
Speci…cally, when seeking for better prospects of cooperation than the size-two coalition,
one can notice the Pareto-improvement of being a member of a size-four coalition, rather than
an outsider to a size-two coalition. A similar observation applies to the size-seven coalition
9
member against the size-four coalition outsider. Thus, it is possible that partial coordination
may bring together up to seven out of the eight countries participating in the …shery. In the
example of the above table, it appears that, in some sense which will be made clear below,
the size-four and the size-seven coalition can be appealing to outsiders of smaller coalitions.
Conversely, suppose that a member of the size-four coalition is contemplating a deviation.
She can observe that her defection will result in a size-three coalition, which in turn is bound
to su¤er at least one defection that improves the payo¤ of one of the three members. The
end result is the size-two coalition, internally and externally stable according to the myopic
stability concept presented in the previous section, for which it is not Pareto-improving for her
to be an outsider. This reasoning, which relates to the ability of the player to look arbitrarily
far ahead, brings about another kind of internal stability, provided it is better for her to be a
member of the size-four coalition than an outsider to the size-two coalition.
Similarly, a prospective deviator of the size-seven coalition will come up with the conclusion
that the size-six coalition resulting from her defection is unstable, as well as the size-…ve
coalition. In fact, the prospective deviator of the size-six coalition would notice that the end
result of her own defection is not the payo¤ of a size-…ve coalition outsider, but rather that of
a size-four coalition outsider, which is a Pareto-improvement for her. Eventually, the initial
prospective deviator of the size-seven coalition will end up an outsider to a size-four coalition,
which is immune from defection, as already argued.
A kind of external stability can also be investigated in that context. The prospect of
broadening a coalition is obviously not amenable to a notion of external stability with respect
to the size-two and the size-four coalitions, which can be respectively enlarged to a size-four
and a size-seven coalition. Only the size-seven coalition is in this sense externally stable when
players look far ahead, because its unique outsider has no incentive to merge since the end
result of the move would not be Pareto-improving for her.
This assumption that players can look arbitrarily far ahead is termed rational conjecture.
It has been sketched by many authors, but one of the rigourous formulations can be credited
to Greenberg (1990). Chew (1994) proposed the stability concept alluded to in the previous
lines and coined the term farsighted coalitional stability in reference to it. The mechanism by
which players come together under rational conjectures is not speci…ed. Chew (1994) pointed
out that the subgame perfect Nash equilibrium concept embodies a similar assumption.
In the settings of Greenberg (1990) and Chew (1994), some stable sets of outcomes can be
implemented by players in environments where coalitions do not make binding agreements.
Back to our example of n = 8 players, these stable sets are the sets of outcomes corresponding
to the coalition structures ! N C = f18 g 3 , ! 2 = f2; 16 g ; ! 4 = f4; 14 g ; ! 7 = f7; 11 g. The
stability concept “does not determine what will happen but what can possibly happen”4 , in the
sense that although some moves may be desirable to improve the fate of some coalitions, the
concept does not garantee their inevitability. On the other hand, under rational conjectures,
there is no consistent story in which the outcomes corresponding to ! 3 = f3; 15 g, ! 5 = f5; 13 g,
! 6 = f6; 12 g, and to the grand coalition ! 8 = N can be reached. This is why Chew (1994)
refers to the outcomes of ! N C , ! 2 , ! 4 and ! 7 as consistent stable sets of outcomes.
The largest consistent stable set of outcomes is that of the coalition structure ! 7 = f7; 11 g :
It is not only internally stable but also externally stable under rational conjectures.
3
4
As with Nash conjectures, we assume that the no coalition structure (m = 1) is farsighted stable
Chwe (1994), p.300
10
4.2
Static farsightedness
Internal stability as introduced in the previous section is termed static farsighted stability in
de Zeeuw (2008). Two cases of general results on static farsightedness in the coalitional Great
Fish War can be ‡edged out, based on conditions on the pro…tability function. The de…nition
of the pro…tability function is the same as the one given under Nash conjectures.
4.2.1
Case 1:
<
is the threshold under which no non trivial coalition is pro…table (see Appendix 8.1.2). In
Case 1, the pro…tability function is negative for all m, 2 m n 1. It is only positive for
m = n, the grand coalition.
Proposition 2 If
farsighted stable.
<
; only the grand coalition and the no-coalition structures are static
Proof. The no-coalition structure is static farsighted stable by assumption. The grand
coalition is obviously static farsighted stable since any prospective deviator sees that his
deviation, which will lead to the collapse of the coalition, is not Pareto-improving.
The importance of this result lies in the fact that under Nash conjectures, there is no room
to argue for the stability of the grand coalition.
4.2.2
Case 2:
<1
In Case 2, there exists m 2 [2; n
m
1] such that the pro…tability function is positive for m
Proposition 3 Denote m1 = dme the …rst integer for which the pro…tability function is
positive. A sequence of internally farsighted stable coalitions of sizes 1 < m1 < m2 < m3 <
::: < mi < ::: < n can be constructed according to the rule:
1. m0 = 1;
2. for i 1, mi is the smallest integer larger than mi
equivalently, Acmi Aomi 1 :
1
such that pcmi (s)
pomi 1 (s), or
Proof. Notice …rst that the …rst inequality in the sequence, namely Acm1
Ao1 , is the
pro…tability condition.
Suppose that a sequence of internally farsighted stable coalitions has been constructed up
to the size mi 1 . We shall prove by mathematical induction that the size-mi coalition is also
internally stable.
Denote ki = mi mi 1 :By de…nition of mi ,
Acmi
Acmi
j
Aomi
1
< Aomi
1
for j = 1; : : : ; ki
1:
(11)
Now, monotonicity of P o with respect to m (Riii) implies Aok < Aom for k < m. Combining
(11) with the monotonicity of Ao yields
Acmi
j
< Aomi
j 1
for j = 1; : : : ; ki
11
1:
This implies that sucessive myopic defections from players would lead to a coalition of size
mi 1 , which we assumed internally farsighted stable. Since Acmi Aomi 1 , farsighted players
have no incentive to deviate, and the size-mi coalition is internally stable.
-40
-60
Ao(18)
-80
Payoff
-100
-120
Ao(11)
Ac(18)
-140
-160
Outsiders
-180
Ac(11)
-200
-220
Insiders
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Size of the coalition
Figure 2: Illustration of the recursive procedure for the construction of consistent stable sets for static farsightedness with n = 20, = 0:82, = 0:95.
Figure 2 illustrates the recursive procedure for the construction of farsighted stable sets
for n = 20,
= 0:82, and = 0:95: Taking one year as a time step, a discount factor of
= 0:95 or lower, and a biological paramater value = 0:82 are more realistic than the
values used in Figure 1 ( = 0:99 and = 0:5), which correspond to an almost linear growth
function and a very high relative valuation of future gains. The choice of
= 0:82 stems
from an estimation of Spence (1975), …tting blue whales annual data to the growth function
under consideration.
For that example, the stable sets are of size 1, 11 and 18. Under myopic stability, the
only stable sets are of size 1, that is, the Nash equilibrium. The reason is that
= 0:779 is
less than the threshold c (20) = 0:9761, above which the pro…tability function is positive for
m 2. To our knowledge, up to now, this is the only result on cooperation in the coalitional
Great Fish War model with simultaneous move of players.
Hence, under rational conjectures, the concept of farsightedness makes it possible to extend
the results of the existing literature and to prove that the prospects for cooperation in the
coalitional Great Fish War are not as grim as suggested under Nash conjectures. It su¢ ces
to choose
greater than the threshold
to exhibit a non trivial farsighted stable coalition.
For comparison, (20) = 0:2693. Appendix 8.1.4 provides the level of the steady-state …sh
stock in the steady state as a function of the size of the coalition.
Figure 3 summarizes those results.
12
myopic stability zone
1
alphadelta-hat
alpha*delta
0.8
static farsightedness with non trivial
coalitions
0.6
0.4
alphadelta-bar
0.2
0
static farsightedness with trivial
coalitions
4
6
8
10
12
14
number of players
16
18
20
Figure 3: Possibilities of coalition according to the value of
and the number
of players.
Before closing this section, a remark is in order. As pointed out by Finus (2001), when
restricted to symmetric countries, the concept of stability lacks the equilibrium property5 .
Since it is better to be an outsider to the size-seven coalition, what country would be that
outsider? Every country would be eager to be the outsider and membership in the size-seven
coalition circulates. That is the reason why the notion of a “steady state” can seem more
convenient to qualify the outcome. Also, the threat of a size-four coalition considered by a
player in the size-seven coalition is less operational, since the defector knows that it is in
the interest of his farsighted rivals to coordinate in another coalition that would include the
former outsider. In the words of Barrett (2003), the coalition formation process is "neither
envy-free, nor role-neutral"6 .
However, we think that the full force of those arguments does not apply to farsighted
players who would see an interest to coordinate the game, even if it is obvious that outsiders
are gaining more as the coalition broadens. Therefore, the assumption of symmetric players
remains theoretically useful to explore the essential features of the farsighted coalitional stability of the …shing game and to avoid fat modelling. Evidently, there is a need to take up
asymmetry in further investigations.
Finally, notice that time is not taken into account in the process of successive defections
from the coalitions, in the sense of making pro…ts before detection. In fact, in the proof of
Proposition 3, the successive deviations of players is only a reasoning scheme. In the event of a
defection in a size-mi coalition, all players rationally conjecture that the next stable coalition
is of size mi 1 and deviate together.
When this feature is altered, especially when it takes time to detect the deviation of a
player, the stability concept becomes dynamic farsighted stability, on which we elaborate in
the next section.
5
6
Finus (2001, p. 242)
Barrett (2003, Chap 13)
13
4.3
Dynamic farsightedness
In this section, following De Zeeuw (2008), we allow for a one-period delay in the detection
of deviations in the partially coordinated game. The motivation is straightforward, in that
the monitoring technology may not allow a real-time observation of the level of the biomass,
or the harvest of the players. One can also think of a stickiness in the dynamics of the
adjustment of the strategies, as suggested by the author to motivate the use of the term
dynamic farsightedness. Speci…cally, it is assumed that a member contemplating a deviation,
say Player i, anticipates that she may free-ride for one period. The corresponding optimization
problem is:
8
0
0
1 19
n
<
=
X
o
@
A
A
max log xi + @Aom k + Bm
log
s
(12)
x
j
k
xi :
;
j=1
n
X
s.t.
xj = (m
1) xc (s; m) + (m
n)xo (s; m) + xi
j=1
where
Pn k is the total number of deviations, including the …rst deviator’s move. In the sum
j=1 xj , it is assumed that, for the current period, the other players stick to their feedback
Nash equilibrium strategies either as members or as outsiders, as given by (9) and (10). Notice
o
that the parameters of the value function for the next period are Aom k and Bm
k , meaning
that the size of the remaining coalition is not necesarily m 1 but can be any m k with
1 k < m.
The …rst order condition for an interior solution yields the optimal deviation strategy:
(1
)(1 + (m 1) )
s
m((1
) (n m) + 1)
= xc (s; m) (1 +
(m 1)) :
xd (s; m) =
Replacing in (12) yields the value for the deviating player, given by
Adm + B d log s
where
Bd =
Adm =
1
1
Aom
= Bo = Bc
k
+ Fm
and
Fm = log
(1
m ((1
)(1 + (m 1) )
+
) (n m) + 1) 1
log
(1 + (m 1) )
m ((1
) (n m) + 1)
:
A size-m coalition is dynamically farsighted stable if deviations, detected one period later,
can be deterred, that is:
Adm = Fm + Aom
k
< Acm for all k; 1
k<m
n:
In dynamic farsightedness, the pro…tability condition is no more the only necessary condition. For example, members of the lowest pro…table coalition, of size m1 , must also care
14
about deterrence of deviations, so the pro…tability condition Acm1 Ao1 must be supplemented
by Acm1
Ao1 + Fm1 . This is the …rst inequality of the recursive procedure in the analog for
Proposition 3 that follows.
Proposition 4 A sequence of dynamically farsighted stable coalitions of sizes 1 < m1 < m2 <
m3 < ::: < mi < ::: n can be constructed according to the rule:
1. m0 = 1 ,
2. m1 is the smallest integer larger than 1 such that Acm1
3. mi (i
2) is the smallest integer larger than mi
Aomi 1 + Fmi :
1 such
Ao1 and Acm1
that Acmi
Adm1 ;
Admi with Admi =
According to the relative e¤ects of and Fmi , the sizes of the stable sets and their number
may be di¤erent from those of the consistent stable sets under static farsightedness. Table 2
presents a numerical illustration for various values of the number n of players, the biological
parameter and the preference parameter .
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
n = 8; = 0:95; = 0:99
n = 10; = 0:97; = 0:99
n = 10; = 0:82; = 0:95
n = 10; = 0:82; = 0:90
n = 20; = 0:82; = 0:95
n = 10; = 0:70; = 0:70
n = 20; = 0:70; = 0:70
n = 10; = 0:50; = 0:50
n = 20; = 0:50; = 0:50
n = 30; = 0:50; = 0:50
n = 50; = 0:50; = 0:50
n = 20; = 0:99; = 0:99
n = 30; = 0:82; = 0:95
n = 50; = 0:82; = 0:95
n = 20; = 0:82; = 0:90
n = 30; = 0:82; = 0:90
n = 50; = 0:82; = 0:90
n = 30; = 0:70; = 0:70
n = 50; = 0:70; = 0:70
myopic
1, 2
1, 2
1
1
1
1
1
1
1
1
1
1, 2
1
1
1
1
1
1
1
0.9405
0.9603
0.7790
0.7380
0.7790
0.4900
0.4900
0.2500
0.2500
0.2500
0.2500
0.9801
0.7790
0.7790
0.7380
0.7380
0.7380
0.4900
0.4900
static farsightedness
1, 2, 4, 7
1, 2, 4, 7, 10
1, 6, 10
1, 6, 10
1, 11, 18
1, 9
1, 18
1, 10
1, 20
1, 29
1, 49
1, 2, 4, 7, 10, 13, 17
1, 17, 26
1, 30, 44
1, 12, 19
1, 19, 28
1, 33, 47
1, 27
1, 45
dynamic farsightedness
1, 2, 4, 7
1, 2, 4, 7, 10
1, 6, 10
1, 6, 10
1, 11, 18
1, 10
1, 19
1
1
1
1
1, 2, 4, 7, 10, 14, 18
1, 17, 27
1, 30, 45
1, 13, 20
1, 20, 29
1, 35, 48
1, 29
1, 48
*: above the threshold
for a pro…tability function positive for all integer values m 2:
**: between c and
, c being the threshold for a pro…tability function positive for an integer
m 2 [2; n 1].
Table 2 : Sizes of stable coalitions in myopic stability, static farsightedness and
dynamic farsightedness in the simultaneous move model
Some of the results in Table 2 have already been announced: when
coalitions are the largest ones under myopic stability.
15
>
, size-2
On the other hand, some new insights emerge; For examples 1-5, all the static farsighted
stable coalitions are also dynamically stable. In our illustration, this is atributable to the high
value of the discount factor . For examples 6-7, where the discount factor is relatively low,
some of the static farsighted stable coalitions are not dynamically stable: coalitions of sizes 9
or 18 are replaced by coalitions of sizes 10 or 19: the impact of Fm is dominant.
Similar patterns emerge for large n;even when
and are relatively high (examples 1219). When the discount factor is relatively small (examples 8-11), the dynamic farsightedness
criterion can even rule out any non trivial coalition.
Notice that the plurality of non trivial farsighted stable coalitions stems from the relaxation
of the external stability criterion, in order to capture, under rational conjectures, all that can
possibly occur in terms of coalitions. Otherwise, one should restrict to the largest non trivial
set representing the largest consistent set of outcomes.
5
Farsightedness in a sequential move model
In this section, it is assumed that the (single) coalition in the Great Fish War model acts as a
leader, who announces its harvest strategy to the outsiders, and has the …rst-move advantage
in every time-period. Myopic stability in this setting has been explored in Kwon (2006) where
it is suggested that the …rst-move advantage can be attributed to a coordination advantage
that outsiders do not have. Unlike the simultaneous move model, the sequential, or dominantplayer, Great Fish War model exhibits an incentive structure that is much more favorable to
the formation of large coalitions, even more so when the parameter
is relatively low.
This is expected since the …rst-move advantage is one of preemption on the biomass, leaving
smaller shares to the outsiders, or for future harvest, as the discount factor or the biological
parameter decreases.
Formally, a quota X is allocated to each coalition member and announced. Outsiders to
a coalition of m, who move after coalition members, derive their value functions as Vmo~ (s) =
~ + B o~ log s, satisfying the Bellman equation:
Aom
m
(
! )
n
X
o~
o~
o~
o~
xk mX
Am + Bm log s = max log xi + (Am + Bm log s
:
xi
k=m+1
An interior solution satis…es the relation:
Pn
s
k=m+1 xk
xi =
o~
Bm
mX
; i = m + 1; :::; n;
which yields by symmetry for each of the outsiders
s mX
o~ + m
Bm
x s; X; m =
n
:
(13)
Cognizant of (13), the sophisticated coalition has in fact announced the appropriate X(s; m)
that maximizes its value function according to the Bellman equation:
~
c~
~
c~
Acm
+ Bm
log s = max log(X) + (Acm
+ Bm
log(s
X
(n
m)x(s; X; m)
mX) )
(14)
~ and B c~ are the coe¢ cients of the value function of each member of the coalition.
where Acm
m
16
Rewrite (14) as:
~
c~
~
c~
+ Bm
log(s
Acm
+ Bm
log s = max log(X) + (Acm
(n
X
m)
s mX
o~ + m
Bm
n
mX) ) :
The …rst order condition for an interior solution yields
X (s; m) =
s
:
c~ )
m (1 + Bm
Routine computations yield the following expressions for the parameters of the value functions:
1
c~
o~
Bm
= Bm
=
1
~
Acm
= (1
)
1
~
Aom
= (1
)
1
log
1
+
m
log
1
(1
)
+ (n m)(1
(
log
+ (n
)
+
)2
m)(1
log
1
)
(
+ (n
)2
m)(1
)
:
The feedback Stackelberg equilibrium …shing policies are again linear functions of the
stock:
1
X(s; m) =
s
(15)
m
x
~ s; X; m
=
s
1
mX(s; m)
(1
x
~ (s; m) = s
+ (1
)
) (n
+ (1
m)
) (n
m)
:
(16)
A coalition member catches more than an outsider if and only if m < n
5.1
(n
1).
Myopic stability
We proceed, as in the simultaneous move case, by studying the per-member partition function
~
~
pcm
(s) = Acm
+
1
log s;
1
the non-member partition function
~
~
pom
(s) = Aom
+
the pro…tability function for m
~
~ (m) = pcm
(s)
log s;
1
2
~
pom
(s)
~
= Acm
~
Aom
= (1
)
1
log
and the stability function for m
~
~
S(m)
= pcm
(s)
1
~
pom
+ (n
1)(1
)
m
2
1 (s)
17
+
1
log
+ (n
+ (n
1)(1
m)(1
)
)
;
~
= Acm
~
Aom
= (1
)
1
1
log
+ (n
m + 1)(1
m
)
+
1
log
+ (n m + 1)(1
+ (n m)(1
)
)
Figure 3 provides a graphical representation of these fuction for an illustrative case where
n = 20 and = = 0:7: Again, using appropriate extensions of these functions, it is easy to
verify the following properties.
Si For m = 1, the per-member partition function pc1~(s) is greater than the non-member
partition function po1~(s).
Sii The pro…tability function is always positive.
This observation indicates that the necessary condition for forming a coalition is always
satis…ed.
Siii The non-member partition function is strictly increasing in m. The per-member partition function and the pro…tability function strictly are increasing for m
m
e =
n
(n 1).
This property and Sii imply that for m < m
e ;the pro…tability function is positive and
decreasing. Hence, the open membership assumption is no more a natural one. It must be
assumed exogenously.
Siv The per-member partition function value is larger than the non-member partition function value for all m < m
e and is smaller for all m > m
e :
Notice that, unlike in the simultaneous move case, the non-member partition function
does not always dominate the per-member partition function. There is indeed an opportunity
cost of being out of the coalition for small coalition sizes. For a given n 2, the threshold
m
e is a decreasing function of , implying that the lower is ; the greater are the prospects
to expand the coalition.
Sv The stability function is strictly decreasing for all m, m
2 (see Appendix 8.2.1).
A similar reasoning to the one carried out for the simultaneous move model allows to
establish the following proposition, which is the sequential move analog of Proposition 2.
Proposition 5 The size m
e of the largest myopic stable coalition of the sequential move coalitional Great Fish War game is obtained by the following necessary and su¢ cient conditions:
1. m
e is greater than m
e =n
(n
2. m
e is the largest integer satisfying m
e
1)
~
m
~ 0 where m
~ 0 is the unique solution of S(m)
= 0:
Proof. See Appendix 8.2.2
18
:
Figure 4: Outsider and insider payo¤s, pro…tability and stability functions in
the sequential moves model, = = 0:70 and n = 20.
5.2
Static and dynamic farsightedness
For static farsightedness, the recursive procedure to construct stable coalitions presented in
~ is greater than Ao~ (and thus greater than Ao~
Section 4.2 is still valid. However, since Acm
m
m 1)
c
~
o
~
for all m < m
e , the range ki in the recursion Ami Ami 1 is equal to 1 when mi m
e . All
the sets smaller than the largest myopic stable coalition are picked by the recursive procedure,
but above m,
e patterns are similar to what happens in the simultaneous move case, in that
the range mi mi 1 can be larger than 1.
~
~
Likewise, the recursion Acm
Aom
+ Fmi for dynamic farsightedness holds with
i
i 1
F~m = log
+
1
2 2 (n
1
m
log
1) + (1 2 ) (n
(1
) (n m) +
2 2 (n
m
m) +
1) + (1 2 ) (n
(1
) (n m) +
m) +
:
(for a proof, see Appendix 8.2.3).
Table 3 presents a numerical illustration for various values of the number n of players,
the biological parameter and the preference parameter . Notice that very large sets can
be obtained, even for the myopic stability concept. However, the stable sets are even larger
when the rational conjectures (static and dynamic farsightedness assumptions) come into play.
Again, the plurality of non trivial stable coalitions is the result of the innocuous relaxation of
the external stability criterion.
Appendix 8.2.4 derives the steady-state stock corresponding to a given coalition size.
19
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
n = 8; = 0:95; = 0:99
n = 10; = 0:97; = 0:99
n = 10; = 0:82; = 0:95
n = 10; = 0:82; = 0:90
n = 10; = 0:70; = 0:70
n = 10; = 0:50; = 0:50
n = 20; = 0:99; = 0:99
n = 20; = 0:82; = 0:95
n = 20; = 0:82; = 0:90
n = 20; = 0:70; = 0:70
n = 20; = 0:50; = 0:50
n = 30; = 0:82; = 0:95
n = 30; = 0:82; = 0:90
n = 30; = 0:70; = 0:70
n = 30; = 0:50; = 0:50
n = 50; = 0:82; = 0:95
n = 50; = 0:82; = 0:90
n = 50; = 0:70; = 0:70
n = 50; = 0:50; = 0:50
0.9405
0.9603
0.7790
0.7380
0.4900
0.2500
0.9801
0.7790
0.7380
0.4900
0.2500
0.7790
0.7380
0.4900
0.2500
0.7790
0.7380
0.4900
0.2500
myopic
1, 2
1, 2
1, 4
1, 4
1, 6
1, 8
1, 2
1, 6
1, 7
1, 11
1, 16
1, 8
1, 9
1, 16
1, 23
1, 12
1, 14
1, 27
1, 38
static farsightedness
1, 2, 4, 6
1, 2, 4, 6, 9
1, 2, 3, 4, 6, 9
1, 2, 3, 4, 6, 9
1 to 6, 8
1 to 8, 10
1, 2, 4, 6, 9, 12, 15, 18
1 to 6, 9, 14, 18
1 to 7, 10, 15, 19
1 to 11, 14, 19
1 to 16, 19
1 to 8, 11, 17, 24, 29
1 to 9, 12, 18, 25, 30
1 to 16, 19, 26
1 to 23, 26, 30
1 to 12, 14, 20, 30, 40, 47
1 to 14, 17, 25, 36, 45
1 to 27, 34, 45
1 to 38, 42, 50
dynamic farsightedness
1, 2, 4, 6
1, 2, 4, 6, 9
1, 2, 3, 4, 6, 9
1, 2, 3, 4, 6, 9
1 to 6, 9
1 to 8
1, 2, 4, 6, 9, 12, 15, 19
1 to 6, 9, 14, 19
1 to 7, 11, 16, 20
1 to 11, 15
1 to 16
1 to 8, 11, 17, 24, 29
1 to 9, 12, 18, 25, 30
1 to 16, 20, 29
1 to 23
1 to 12, 14, 21, 31, 41, 48
1 to 14, 17, 25, 36, 46
1 to 27, 36, 49
1 to 38
Table 3: Sizes of stable coalitions in myopic stability, static farsightedness and
dynamic farsightedness for the sequential move model
6
Conclusion
This study has shown the contribution of alternative assumptions about the conjectures of
players in the debate on the coordination of policies in the management of high seas …sheries.
Rational conjectures, less myopic than the traditional Nash conjectures, make a strong case for
cooperation in the coalitional Great Fish War model with symmetric players. For plausible
values of the biological parameter of the growth function and of the discount factor, the
farsightedness assumption predicts a wide scope for cooperation, sustained by credible threats
of subsequent deviations that would ruin the shortsighted payo¤ of any prospective deviator.
Compliance or deterrence of deviations has been also adressed by ackowledging that a
real time information may not be available on the …sh stock or on the catch policies actually
implemented. The dynamic farsightedness concept is at the core of this aspect of the investigation. The requirements are stronger, and the sizes and number of possible farsighted stable
coalitions di¤er.
Finally, in the sequential move model, which could mimic in some ways some realities of
…shery models, the results are no less appealing, even if the dominant player or dominant
coalition with …rst move advantage assumption has provided an argument for cooperation
with the traditional Nash conjectures.
By providing non-cooperative arguments to derive plausible cooperative results, the farsightedness assumption appears as a promising avenue for the merge of cooperative game and
non cooperative game theory applied to the management of high seas …sheries.
20
7
References
1. d’Aspremont, C., Jacquemin, A., Gabszewicz, J. and Weymark, J. (1983) “On the
Stability of Collusive Price Leadership”, Canadian Journal of Economics 16:17-25.
2. Barrett, S.(2003) Environment and Statecraft: The Strategy of Environmental TreatyMaking. Oxford, Oxford University Press.
3. Carraro, C. and Marchiori, C. (2003)“Stable Coalitions”in The Endogenous Formation
of Economic Coalitions, Carraro, C. (editor), Cheltenham, Edward Elgar, 156-193.
4. Chwe, M. (1994) “Farsighted Coalitional Stability”, Journal of Economic Theory 63:299325.
5. Finus, M. (2001) Game Theory and International Environmental Cooperation. Cheltenham, Edward Elgar.
6. Greenberg, J. (1990) The Theory of Social Situations: an Alternative Game-Theoretic
Approach. Cambridge, Cambridge University Press.
7. Kwon, O. S. (2006) “Partial Coordination in the Great Fish War”, Environmental and
Resource Economics 33:463-483.
8. Kaitala, V.T. and Lindroos, M. (2007) “Game Theoretic Applications to Fisheries” in
Handbook of Operations Research in Natural Resources, Weintraub, A. et al.(editors),
Springer, 201-215.
9. Levhari, D. and Mirman, L. J. (1980) “The Great Fish War: an Example Using a
Dynamic Courant-Nash Solution”, Bell Journal of Economics 11: 649-661.
10. Lindroos, M., Kaitala, V. and Kronbak, L.G. (2007) “Coalition Games in Fishery Economics”in Advances in Fishery Economics, Bjorndal, T. et al.(editors), Blackwell Publishing, 184-195.
11. Pintassilgo, P. (2003) “A Coalition Approach to the Management of High Seas Fisheries
in the Presence of Externalities”, Natural Resource Modeling 4-16(2): 403-26.
12. Spence, A. M. (1975) “Blue Whales and Applied Control Theory”, in System Approaches and Environmental Problems, H.W. Gottinger (ed.), Gottingen, Vandenhoeck
and Ruprecht .
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Agreements”, Journal of Environmental Economics and Management 55: 163-174.
21
8
Appendix
8.1
The simultaneous move model
8.1.1
The stability function is negative for 3
(1
)S(m) =
=
=
<
1
1
1
1
1
1
(1
m
n
) (n m + 1) + 1
log m
) (n m) + 1
) (n m) + 1 + 1
log
log m
(1
) (n m) + 1
1
log m
log 1 +
(1
) (n m) + 1
1
log m
) (n m) + 1
log
(1
(1
(1
(17)
by the inequality log(1 + y) < y for all y > 0: For m n, the …rst term in (17) is bounded by
1, so that
(1
)S(m) 1 log m;
which is negative for m > e = 2: 718 3 > 2: Hence S(m) is negative for all m
8.1.2
3:
Thresholds of the pro…tability function in the simultaneous move model
Threshold
for which the pro…tability function is positive for some m 2 [2; n 1]
(computations based on m = n 1)
n
n
n
11 0.339
25
0.2495
12 0.3269 30
0.2353
3
0.7015 13 0.3165 35
0.2244
4
0.5732 14 0.3073 40
0.2156
5
0.5001 15 0.2992 45
0.2084
6
0.4519 16 0.2920 50
0.2023
7
0.4173 17 0.2855 60
0.1926
8
0.3910 18 0.2796 70
0.1850
9
0.3702 19 0.2742 80
0.1788
10 0.3532 20 0.2693 100 0.1694
Threshold c for which the pro…tability is positive for all m 2 [2; n 1] (computations based on m = 2). See also Kwon (2006, p. 474)
n
3
4
5
6
7
8
9
10
c
0.7015
0.8222
0.8733
0.9015
0.9194
0.9318
0.9409
0.9479
n
11
12
13
14
15
16
17
18
19
20
c
0.9534
0.9579
0.9615
0.9646
0.9673
0.9695
0.9715
0.9732
0.9748
0.9761
n
25
30
35
40
45
50
60
70
80
100
c
0.9812
0.9844
0.9868
0.9885
0.9899
0.9909
0.9924
0.9935
0.9944
0.9955
22
8.1.3
Existence and uniqueness of
Proof of existence and uniqueness of c is provided in Kwon (2006). It is based on the
determination of the zeroes in
of S(2) = (2), that is, the stability function for a size-two
coalition, which has the same expression as the pro…tability function for a size-two coalition.
It is also shown that c is a strictly increasing function of n.
Likewise, the values of the threshold
are the roots in
of (n 1), that is, the
pro…tability function for a size-n 1 coalition,
(n
1) = (1
)
1
1
1
One can readily check that :
i) lim (n 1) = (1
) 1 log 2(nn
!0
ii) lim (n
1) = (1
)
1 (n
2
!1
Furthermore, di¤erentiating (1
1
2
(1
)
log
Using ln(1 + y) >
log
(1
(1
y
1+y
) (n
(2
2)
)
2)
+1
(n
1)
log (n
2
log (n
)
for y >
n
1) :
<0
1)
) (n
) (n
(2
log
1)) > 0
1) with respect to
yields
n
+1
(1
) (2
2
) (n
(n
1))
:
(18)
1 and y 6= 0;
>
(n
n
2) (1
(n
)
>
1)
(2
(n
2) (1
) (n
)
(n
1))
which implies that (n 1) is a strictly increasing function of . Since this function takes
on negative values before taking on positive values for
2 ]0; 1[, by continuity, (n 1) has
a unique zero on (0; 1) :
8.1.4
Level of the …sh stock in steady state
From the …shing policies
xc (s; m) =
1
m((1
xo (s; m) =
) (n
1
) (n
(1
m) + 1)
m) + 1
s;
s:
one derives the steady state stock s according to
(s
(1
1
) (n
m) + 1
s
(n
(1
m) (1
)
s ) =s
) (n m) + 1
that is
1
s =
(m
n) + n
m+1
:
It can be readily checked that s increases in m and : The impact of
23
is ambiguous.
8.2
8.2.1
The sequential move model
The stability function is decreasing for m 2 [2; n]
For m 2 [2; n] ;
~
S(m)
= (1
)
1
+ (n
log
m + 1)(1
m
)
+
+ (n m + 1)(1
+ (n m)(1
log
1
)
)
:
Di¤erentiating with respect to m yields
1
m (1
)
(1
) (n
(n 1) + 1) (n m) +
( + (1
) (n m)) ((1
) (n m) + 1)
which is clearly negative for
8.2.2
2 (0; 1) and 0 < m
n:
Proof of Proposition 5
The necessary and su¢ cient conditions for a myopic stable coalition are:
1. the pro…tability function is positive,
2. the external stability condition and the internal stability condition are met.
~
Condition 1 is satis…ed for all coalition sizes. Condition 2 is equivalent to to S(m)
~
and S(m + 1) < 0. For m = n
(n 1) ;
~
S(n
where
(n
(1
(n
)(n 1)+1
(n 1))
1)) = (1
)
1
(1
)
1
> 1 if
~
< 1, so that S(n
~
S(n)
=
(1
)
1
ln
1
ln
(1
(n
(n
+ log n
) (n 1) + 1
(n 1))
0
:
1)) > 0: For m = n;
< 0:
~
Since S(m)
is strictly decreasing for given n and
, there exist a unique m
~ 0, n
0
0
~m
(n 1) < m
~ < n; such that S(
~ ) = 0:This implies, using again the fact that S~ is
0
~ m)
~m
strictly decreasing, that m
e = m
~ satis…es S(
e
0 and S(
e + 1) < 0.
8.2.3
Dynamic farsightedness condition
In that case, contemplating a deviation, say Player i, anticipates that she may free-ride for
one period. The corresponding optimization problem is:
8
0
0
1 19
n
<
=
X
~o
@
A
A
max log xi + @A~om k + B
log
s
x
(19)
j
m k
xi :
;
j=1
s.t.
n
X
xj = (m
1) X(s; m) + (n
m)~
x(s; m) + xi
j=1
where
Pn k is the total number of deviations, including the …rst deviator’s move. In the sum
j=1 xj , it is assumed that, for the current period, the other players stick to their feedback
Nash equilibrium strategies either as members or as outsiders, as given by (15) and (16).
24
The …rst order condition for an interior solution yields the optimal deviation strategy
0
1
X
x
~d (s; m) = (1
) @s
xj A
j6=i
= s
(1
) (1
2
m
= X(s; m)
(1
2
) (n m) +
( (n 1) + 1)
(
1) (m n) +
) (n m) +
( (n 1) + 1)
:
(
1) (m n) +
Replacing in (19) yields the value for the deviating player, given by
~ d log s
A~dm + B
where
~d =
B
A~dm =
1
~o
=B
m
1
A~o
m k
k
+ F~m
and
F~m = log
+
8.2.4
2 2 (n
1
m
log
1
1) + (1 2 ) (n
(1
) (n m) +
2 2 (n
m) +
1) + (1 2 ) (n
(1
) (n m) +
m
m) +
Level of the …sh stock in steady state
From the …shing policies
X(s; m) =
x (s; m) = s
1
m
(1
+ (1
s
)
) (n
m)
:
one derives the steady state stock s~ according to
(~
s
(1
)~
s
(n
(1
m)
+ (1
)
) (n
m)
s~ ) = s~ ;
that is,
2 2
se =
(1
) (n
1
m) +
The steady-state stock increases in m and : The impact of
25
:
is ambiguous.
: