Download Strategic Choice of Sharing Rules in Collective Rent

Strategic Choice of Sharing Rules in Collective
Rent-Seeking and the Group Size Paradox
Pau Balart∗
Sabine Flamand†
Orestis Troumpounis‡
February 2, 2014
Abstract
We provide new insights regarding the occurrence of the group size paradox
—a situation in which a small group outperforms a large one— when two groups
of different size compete for a mixed public-private good by considering that (i)
the sharing rules determining how the private part of the prize is allocated among
group members are public information and thus their choice is strategic, and (ii)
one of the two groups may set its sharing rule faster than the other. We show
that regardless of the timing of the interaction, the large group uses its sharing
rule as a mean to address free-riding among its members, thereby making the small
group inactive for most degrees of privateness of the prize. In turn, the group size
paradox occurs only when groups choose their sharing rules simultaneously and the
good is sufficiently private. If the choice of sharing rules is sequential the group
size paradox always vanishes regardless of the leader’s size. This results from an
increase (decrease) in between-group competition when the large (small) group is
the leader.
Keywords: collective rent-seeking, group size paradox, strategic choice of sharing rules, strategic complements, strategic substitutes
∗
Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe, Spain,
e-mail: [email protected]
†
Nova School of Business and Economics, Campus de Campolide, P-1099-032, Lisbon, Portugal,
e-mail: [email protected]
‡
Department of Economics, Lancaster University, Bailrigg, Lancaster, LA1 4YX, United Kingdom,
e-mail: [email protected]
1
1
Introduction
Collective contests where group members organize and compete for a prize are a common phenomenon: research and development races, procurement contests, funds to be
allocated among different departments within a university, regions within a country or
countries in the EU, prizes and projects allocated among different divisions of a firm,
or party members that participate in pre-electoral campaigns. Furthermore, one can interpret several prizes sought by competing groups as a mixture between a public and a
private good. Regional governments competing for a budget typically use the latter to
provide both monetary transfers and local public goods. Similarly, prizes in research and
development races involve both reputational and monetary benefits for the winning team.
More generally, any contest between groups for a monetary reward can be considered as
involving a public good component as long as the winners enjoy some benefits in terms
of status, reputation, or satisfaction related to the victory.
The sharing rules determining the way the private part of the prize is allocated among
the group’s members, the composition of the prize and the size of the competing groups
are known to affect individual incentives to contribute to such contests. In turn, those
incentives ultimately determine the occurrence of the group size paradox (GSP henceforth), a situation where a small group outperforms a large one. The notion of the GSP
dates back to the seminal work by Olson (1965), who stressed the severity of the free-rider
problem within large groups. Esteban and Ray (2001) and Nitzan and Ueda (2011) have
recently studied the occurrence of the GSP for mixed public-private goods. In this paper,
we consider two additional factors that may further affect the occurrence of the GSP,
namely (i) the strategic choice of sharing rules when these are public information and (ii)
the non-simultaneity of the choice of sharing rules. Let us further motivate both.1
1
Our analysis relates to the collective rent-seeking literature, which focuses on the ways to allocate a
prize among group members, starting after Nitzan (1991a). This literature considers both exogenously
given sharing rules under different specifications as well as the endogenous choice of such rules. This
literature is large and we do not aim to review it. For exogenous sharing rules see for example Nitzan
(1991a,b); Davis and Reilly (1999); Ueda (2002), and for endogenous sharing rules see for instance Lee
(1995); Baik and Lee (1997); Lee and Hyeong Kang (1998); Baik and Lee (2001); Noh (2002); Baik and
Lee (2007, 2012a); Nitzan and Ueda (2011).
2
Our model aims at filling the gap of the strategic choice of sharing rules within competing groups. While Nitzan and Ueda (2011) extend the analysis of Esteban and Ray
(2001) by considering the possibility of endogenous sharing rules, they do so in a setup
of imperfect information where sharing rules are private information, hence their choice
does not involve a strategic interaction between groups. While it is often reasonable
to think of groups’ internal organization (sharing rules) as private information, many
other cases are better described by a situation of perfect information where the decisions regarding sharing rules are potentially strategic.2 Think for instance of university
departments competing for funds, with the departments allocating part of these funds
among their members according to a publicly known incentive scheme. More generally,
any competition taking place between groups belonging to a common organization is often likely to involve common knowledge of the individual incentive schemes within each
group. Likewise, it is not unreasonable to consider that in some instances, firms have
some knowledge regarding the incentives schemes of their competitors.
At the same time, we consider that a sequential choice of sharing rules is of further
interest as it is often likely that one of the competing groups has the opportunity of
establishing its sharing rule faster than its competitor. This may result from the organizational skills of the groups, their relative size, their access to information regarding
the contest itself, etc.3 Interestingly, and as we mention in the discussion, if the two
groups were able to endogenously decide the timing of the strategic interaction, in most
instances it is the sequential version of the game that would prevail in equilibrium. Thus,
the very fact that the equilibrium timing of the game may be a sequential one provides
2
In fact, in the literature on collective rent-seeking and sharing rules the assumption of perfect information as in this paper is the usual one. To the best of our knowledge, the only papers analyzing the
case of private information are Baik and Lee (2007, 2012a); Nitzan and Ueda (2011).
3
The sequential choice of sharing rules in collective rent-seeking for the case of a pure private good has
been studied by Gürtler (2005). He shows that if the more numerous group determines its sharing rule
prior to the smaller group, rent dissipation in the group contest is higher than in an individual contest.
The introduction of a mixed public-private good allows us to stress the strategic use of sharing rules by
the large group to keep the small group inactive for most intermediate degrees of privateness. See also
Noh (1999), who analyzes the interactions between group sharing rules, the competition between two
groups over the common output, and the allocation of resources between productive and appropriate
activities. He shows that the sequential choice of sharing rules and resource allocations results in the
adoption of fully egalitarian sharing rules in both groups.
3
further justification for departing from the assumption of simultaneity. In turn, and as
our results point out, such a sequential choice of sharing rules provides a new explanation
for the complete elimination of the GSP.
These new features are introduced in a standard model of collective rent-seeking, in
which two groups of different size compete for a mixed public-private good. All individuals face the same linear cost of effort, and they have the same valuation of the prize.
Individuals belong to one of the two competing groups, which have to select the sharing
rule according to which the private part of the good will be divided among the group
members.
The dynamics of collective action in group contests for pure private and pure public
goods have been widely investigated and are well understood. Esteban and Ray (2001)
and Nitzan and Ueda (2011) have considered the possibility of mixed public-private goods,
and find that no group can be inactive in equilibrium. In stark contrast with those
findings, we show that regardless of the particular timing of the interaction between the
two groups, the small group is inactive for most degrees of privateness of the contested
prize. If such prize is a pure public good, the choice of sharing rules is irrelevant and
both groups face the same winning probability. However, as long as the good exhibits
an arbitrarily small degree of privateness, the large group is able to use its sharing rule
as a coordination device so as to address free-riding among its members, thereby leading
to the inactivity of the small group. In turn, the large group is able to provoke such an
outcome because the sharing rules allow for net transfers from low to high performers
within the group (i.e., the individual reward from the private component of the prize may
be more than proportional to individual effort). This characteristic of the group sharing
rules constitutes the fundamental difference between our analysis and the one of Nitzan
and Ueda (2011) regarding the possibility of one group being inactive in equilibrium,
as they exclude the possibility of such transfers. As it turns out, this extreme outcome
occurs for most intermediate degrees of privateness of the prize, except the ones very
close to full privateness. Conversely, if the good is mostly private, both groups are active
4
in equilibrium.
For a given level of privateness, the most direct effect of group size is that it yields
better odds at winning the competition between groups, since there is a larger number of
potential contributors. Clearly, this effect goes against the occurrence of the GSP. Then,
the degree of privateness of the contested good determines the relative meritocracy of
the selected sharing rules and thus the level of individual effort in each group. More
specifically, the returns to effort vary depending on the private-public nature of the good,
and are smaller for the large group the higher the degree of privateness. Indeed, if the
good is (at least partially) private, a larger group size also implies a lower individual
share of the prize in case of victory. This is the group-size deterrence effect (Nitzan,
1991a), which is the driving force of the GSP in Esteban and Ray (2001). Furthermore,
in the presence of (at least partially) meritocratic group sharing rules (i.e., individual
reward depends on relative performance to some extent), the degree of privateness of the
prize has yet another dimension that influences the occurrence of the GSP. As with such
sharing rules, individuals receive more of the private good the more they contribute, it
follows that the higher the degree of privateness of the good, the larger the incentives to
compete with the other group members. We show that the endogenous choice of sharing
rules, regardless of its timing, is such that the higher the degree of privateness, the more
likely that the small group adopts a more meritocratic rule than the large one. This, in
turn, reinforces the effect of privateness on the GSP.4
When the prize is sufficiently private and the choice of sharing rules is simultaneous, the size deterrence effect prevails, which decreases the attractiveness of a potential
victory from the perspective of the large group’s members. As a result, the large group
implements a less meritocratic rule than the small group, inducing low levels of effort that
4
Kolmar and Wagener (2013) analyze the adoption of efficient incentive schemes within competing
groups, separate from the group sharing rule, and offer an explanation as to why the GSP may not
hold: larger groups may end up with a better organizational structure precisely because their larger
free-rider problem builds up higher pressure to establish efficient incentive schemes. Nitzan and Ueda
(2009) consider the case where the intra-group division of the prize is determined anarchically and show
that it strengthens the possibility of the GSP. For more on the relationship between group size and group
success, see Kolmar (2013) and Konrad (2009).
5
ultimately sustain the GSP. Conversely, for a sufficient degree of publicness of the contested good, the size deterrence effect is relatively unimportant, which in turn eliminates
the GSP.5
Our results under a simultaneous strategic choice of sharing rules substantially differ
from the two closest works in the literature. Esteban and Ray (2001) assume that the
private part of the prize is shared among group members on an egalitarian basis. In
that case, for a linear cost of effort, the GSP takes place for any strictly positive degree
of privateness of the good. On the other hand, Nitzan and Ueda (2011) show that the
simultaneous non-strategic choice of sharing rules eliminates the GSP unless the prize is
purely private. Our results for the simultaneous case are less extreme than these two. We
show that the GSP may arise as long as the contested good is sufficiently private, but not
necessarily a pure private good. While the two previous works provide further results on
the influence of a convex cost of effort on the elimination of the GSP, the introduction of
a strategic choice of sharing rules obliges us to restrict ourselves to the linear cost case.6
Introducing a non-simultaneous choice of sharing rules alters the previous result by
fully eliminating the GSP regardless of the leader’s size. With sequential sharing rules, in
fact, the two groups face the same probability of winning the prize whenever the latter is
either a purely public or a purely private good. For all intermediate values of privateness,
the large group outperforms the small one. If the large group is the leader, the intuition is
similar to the standard Stackelberg duopoly model. That is, the large group commits to
a more meritocratic sharing rule than in the simultaneous game (hence effort increases),
while the small group selects a less meritocratic rule than in the simultaneous game
(hence effort decreases), which prevents the occurrence of the GSP. The small group
5
Again, in the case of most intermediary degrees of privateness of the prize, the aggregate effort effect
intensively dominates the size deterrence one, leading to situations where the small group is inactive in
equilibrium independently of the timing of the strategic interaction between groups.
6
A convex cost of effort penalizes higher levels of individual effort, hence it works against the occurrence of the GSP. Corchón (2007) provides an intuition for the latter:“Intuitively, it is clear that Olson’s
conjecture cannot hold if costs rise very quickly with effort: for instance if costs are zero up to a point,
say Ḡ where they jump to infinity, all agents will make effort Ḡ and smaller groups will exert less effort
than large ones.” Therefore, the elimination of the GSP that we find in the sequential case should also
hold with convex costs. Likewise, the occurrence of the GSP in the simultaneous case should be less
likely under the assumption of convex costs.
6
acting as the leader follows a very different strategy. It commits to a less meritocratic
rule than in the simultaneous game, thereby lowering the level of competition between
groups. In turn, the large group also responds with a less meritocratic rule than in the
simultaneous game, but the reduction is relatively lower than the one implemented by the
small group. As a result, total effort decreases relatively less in the large group, which
again eliminates the GSP. Even tough the GSP vanishes independently of the leading
group’s size, it does so for two very different reasons: the sharing rules are strategic
substitutes from the perspective of the small group, while they are strategic complements
from the perspective of the large group.
The paper is structured as follows: In Section 2, we present the model. In Section 3,
we characterize the equilibrium of the effort stage. In Section 4 and Section 5, we analyze
the simultaneous and sequential choice of sharing rules, respectively. Section 6 concludes
with a discussion. All proofs can be found in the Appendix.
2
The Model
There are two groups i = A, B of size ni ∈ N+ and without loss of generality A is the
large group (i.e., nA > nB ). All individuals within a group are identical and the cost
of exerting effort is linear.7 The valuation of the contested (divisible) good is the same
for all individuals, and is denoted by V . Individuals’ choice of efforts within each group
takes place simultaneously given the sharing rules. Individuals are risk neutral, and the
expected utility of individual k = 1, ..., ni in group i is given by
7
We abstract from the possibility of intra-group heterogeneity regarding lobbying effectivity, which
can be reflected in different valuations of the prize by the members. One can ask whether a group whose
members have highly unequal interests in the collective action will be more or less active. The literature
has provided contrasted answers to the latter question, due to the difference in the assumed form of
the effort cost function (see the discussion in Nitzan and Ueda (2013) and the references they cite).
Assuming very weak and plausible restrictions on the form of the effort cost functions, Nitzan and Ueda
(2013) show that if a group competes for a prize with sufficiently many rivals or with a very superior
rival, unequal stakes among the members can enhance its performance.
7
(
EUki =
where Ei =
Pni
j=1 eji
)
eki
1
Ei
αi
+ (1 − αi )
p + (1 − p)
V − eki
Ei
ni
EA + EB
denotes the aggregate effort of group i and p ∈ [0, 1] measures
the degree of privateness of the contested good (p = 0 corresponds to the case of a pure
public good while p = 1 corresponds to the case of a pure private good). The sharing rule
αi ∈ [0, ∞) is the relative weight given to meritocracy as opposed to egalitarianism in the
division of the prize within the corresponding group. If αi = 1, the private part of the
prize is shared according to relative effort, while if αi = 0, it is shared equally among the
group members regardless of their relative contribution. Therefore, as long as αi > 0, high
performers in group i receive a larger share of the prize than low performers. A value of
αi larger than one implies that the prize is allocated more than proportionally to relative
i
V
effort within group i. More precisely, when αi > 1, group i collects −(1 − αi ) npi (EAE+E
B)
from all of its members and allocates it according to relative effort within the group (Baik
and Lee, 1997; Lee and Hyeong Kang, 1998). Consequently, for values of αi larger than
one, an individual exerting no effort pays a transfer to the group members who exert
positive effort. Thus, if the efforts of the active group members are sufficiently high,
expected utility may be negative for an individual exerting no effort.
Our game consists in the strategic choice of the groups’ sharing rules, followed by the
simultaneous choice of individuals’ effort levels.8 We analyze both the simultaneous and
sequential choice of sharing rules by the two groups, hence the game has two or three
stages respectively. The equilibrium concept is subgame perfection in pure strategies. In
the effort stage we focus on within-team symmetric equilibria.
8
As individuals are identical within groups, we assume that in each group, the choice regarding the
sharing rule is made by the representative individual by maximizing his expected utility function.
8
3
Effort Stage
Before analyzing the choice of sharing rules, we first solve the last stage of the game,
which essentially extends the results of Ueda (2002) and Davis and Reilly (1999) by
allowing for a mixed public-private prize.
Proposition 1. Let χi (αA , αB ) = (1−αi )p(ni nj −nj )−(1−αj )p(ni nj −ni )−ni nj (1−p)−
pni with i = A, B and j 6= i. There exists a unique within-team symmetric equilibrium in
pure strategies in the effort subgame characterized as follows:
• When χi (αA , αB ) < 0 for all i = A, B:
eˆi =
Λi Φi
V
ni [nj p+ni (2nj (1−p)+p)]2
• When χi (αA , αB ) ≥ 0 for group i = A, B:
ẽ∗j =
ẽi = 0
αj p(nj −1)
V
n2j
with Λi = [nj (1 − p) + p] [ni (1 − p) + p + (ni − 1)pαi ] + (nj − 1) [ni (1 − p) + p] pαj and
Φi = nj p(1 − αi ) + ni [nj (1 − p(1 − αi + αj )) + pαj ].
The condition χi (αA , αB ) < 0 determines whether group i is active in equilibrium.
As can be seen from the analytical expression of χi (αA , αB ), the more egalitarian is the
sharing rule within a group, the more likely that its members are inactive. Indeed, as
free-riding incentives increase, individual effort eventually drops to zero. Furthermore,
the more meritocratic the rule of its opponent, the more likely that a group is inactive.
When both groups are active (i.e., χi (αA , αB ) < 0 for all i = A, B), equilibrium effort
in each group is increasing in the relative meritocracy of its sharing rule and in the value
of the prize. When only one group is active, its members compete in a standard Tullock
contest with equilibrium effort increasing in the relative meritocracy of the sharing rule
and decreasing in the size of the group.
Definition 1. The GSP arises if and only if EA < EB .
9
Given the contest success function used for the competition between groups, the
previous definition is equivalent to the small group facing a higher probability of winning
than the large group. Using the above equilibrium for exogenous sharing rules, we obtain
our first result regarding the GSP:
Proposition 2. With exogenous sharing rules,
• For p = 0, the GSP never arises.
• For p ∈ (0, 1], the GSP arises if and only if
αA <
nA − nB + 2αB nA (nB − 1)
2nB (nA − 1)
Esteban and Ray (2001) assume that the prize is shared on an egalitarian basis among
the group members regardless of their individual contribution, and find that with a linear
cost of effort, the GSP arises for all values of p except for the extreme case of p = 0 (i.e.,
when the good is purely public).9 In fact, and as our results confirm, when the prize is
a pure public good the two groups face the same probability of winning.10 Introducing
the possibility of (exogenous) group-specific sharing rules leads to a more nuanced result.
In particular, for any degree of privateness of the good, the less (more) meritocratic
the sharing rule of the large (small) group, the more likely that the GSP arises. As
with sharing rules, individual reward depends on relative performance within the group
to some extent, the degree of privateness of the prize has another dimension. Here,
individuals receive more of the private good the more they contribute. Since they do not
coordinate, there is an additional expenditure of effort besides the one arising from the
9
Pecorino and Temimi (2008) modify the model of Esteban and Ray (2001) to a standard public good
setting, and show that their results are robust to the presence of small fixed costs of participation in the
case of a pure public good, but not in the case of a fully rival good. Furthermore, when the degree of
rivalry is sufficiently high, the introduction of small fixed costs of participation implies that collective
action must break down in large groups.
10
On pure public goods, see also Baik (2008); Riaz et al. (1995), Ursprung (1990) and Katz et al.
(1990), among others. They have shown that with a pure public good, a group with larger membership
attains a winning probability larger than or equal to that of a smaller group. Hence the GSP is not valid
in such cases.
10
classical effect on the group winning probability: the lower the degree of privateness of
the good, the smaller the incentives to compete with the other members of your group.
In the limit, when the prize is purely public, individuals do not get any reward linked
to personal effort. The effect of relative publicness on the GSP (i.e., the size deterrence
effect) is driven by this factor, which is absent in Esteban and Ray (2001).11
4
Simultaneous Choice of Sharing Rules
We now consider the full two-stage game: In stage one the two groups A and B choose
S
S
their sharing rules (αA
, αB
) simultaneously, and in stage two they choose their effort levels
(eSA , eSB ), also simultaneously. As the sharing rules are irrelevant in the case of a pure
public good, we assume that p > 0. While the equilibrium concept is subgame perfection,
in our formal results we present the equilibrium choice of sharing rules, as effort levels
chosen in stage two can be derived directly from Proposition 1.
Proposition 3. Let c. If the two groups choose their sharing rules simultaneously:
• If 0 < p ≤ p1 , only group A is active and the continuum of sharing rules in the
subgame perfect equilibrium is given by
S
– αA
=
nA
nA −1
h
1−p
p
+
(nB −1)αS
B +1
nB
i
S
– αB
∈ [ nB (1−p)+p
, nB [nA (1−p)+3p−2]−2p
]
p
(nB −1)p
• If p > p1 , both groups are active and the sharing rule for i = A, B in the unique
subgame perfect equilibrium is given by
ni
2ni nj (ni − 1) + Ai p + Bi p2
αi =
(ni − 1) p [2ni nj − p(ni (2nj − 1) − nj )]
where Ai = ni nj (9 − 4ni ) − nj (nj + 2) − 2ni and Bi = ni nj (2ni − 7) + nj (nj + 3) + 3ni − 2.
11
This additional effect of privateness on the GSP is also present in Nitzan and Ueda (2011), as they
also consider partially meritocratic group sharing rules.
11
For any positive degree of privateness smaller than the threshold p1 , the large group
implements a sufficiently meritocratic sharing rule so as to keep the small group inactive.
This is in stark contrast with the case of a pure public good where both groups are
active and face the same probability of winning the prize (Baik, 2008). Remarkably, an
arbitrarily small and negligible degree of privateness of the prize is sufficient to allow the
large group to prevent any activity by the small group. Indeed, the large group uses its
sharing rule as a coordination device among its members to prevent the small group from
exerting any effort, thereby securing its own victory. Observe, furthermore, that such
preemptive behaviour by the large group occurs for most degrees of privateness, as the
threshold p1 is in fact very close to one.12 Such a coordination mechanism among the
members of the large group is possible to the extent that we allow the sharing rule αi to
exceed one, in which case an inactive group member has to pay a transfer to the other
(active) group members. As a result, selecting a very meritocratic sharing rule allows the
large group to address free-riding among its members even for low degrees of privateness,
thereby leading to high levels of individual effort and ultimately to the inactivity of the
small group. The fact that one group may be inactive in a subgame perfect equilibrium
constitutes a fundamental difference with Ueda (2002), who assumes both a pure private
good and sharing rules such that there may be no such transfers of resources between
individuals within a group.
If the good is sufficiently private (p > p1 ), we are at an interior solution and both
groups are active in equilibrium. The small group always implements a sharing rule such
that the expected share of the private prize is more than proportional to individual effort
S
(αB
> 1 for all p > p1 ). The relative meritocracy of the large group’s sharing rule is
strictly decreasing in the degree of privateness of the prize and larger than one whenever
12
Notice that the numerical value of p1 is large and very close to one as the size of the groups increases,
and/or when there is a large difference between group sizes. For instance, if nA = 15 and nB = 7 then
p1 = 0.98. However, this critical value of the degree of privateness can be rescaled by assuming that
the public good is congestible. This could be captured by a parameter c ∈ [0, 1) such that the utility
attributed to the public good by any member of group i is (1 − p)V /nci . In that case the critical value
n [ncA (1+nB )−nA ncB ]
p1 is given by pc1 = nc nBB(1+n
, which for the previous example rescales p1 to pc1 = 0.76
c
c
B )−nB (nA +nA nB )
A
for c = 0.6. In order to avoid unnecessary complications in the main text, we consider the case of c = 0.
12
the good is sufficiently public.
Interior (eA>0,eB>0)
(eA>0,eB=0)
0
1
P2
P1
αA< αB
αA>αB
Figure 1: Simultaneous choice of sharing rules
Figure 1 summarizes our comparisons: when the degree of privateness is larger than
p1 , both groups are active, although the large group implements a more meritocratic rule
S
S
than the small one (αA
> αB
) as long as p ≤ p2 .13 As the good gets relatively more
private (p > p2 ), it is the small group that implements the most meritocratic sharing rule
S
S
(αA
< αB
), which may lead to the emergence of the GSP for a sufficiently high degree of
privateness, as the following proposition establishes:
Proposition 4. With a simultaneous choice of sharing rules, the GSP arises if and only
if p > pGSP =
2nA nB 14
.
1+2nA nB
!"#%'(!$)%
%&!"&!$
!"#!$#%
%
./
.1
*+(,-.
.0
/
.,-.
,-.
Figure 2: Simultaneous choice of sharing rules and the GSP
13
To avoid introducing more notation, we do not define formally the thresholds p2 and p3 presented in
the interpretation of our results. They can be easily calculated and one can show that p1 < p2 < p3 <
pGSP .
14
The numerical value of pGSP is large and very close to one as the size of the groups increases,
and/or when there is a large difference between group sizes. For instance, if nA = 15 and nB = 7
then pGSP = 0.99. Again, this value can be rescaled if we assume that the public good is congestible.
nA nB [ncB +2nA ncB −ncA (1+2nB )]
, which for the previous example yields
In that case pcGSP = nc nc (n −n
1+c
A
B )+nA (1+2nA )nB ]−nA nB (1+2nB )
B[ A
c
pGSP = 0.95 for c = 0.6.
13
The GSP arises above a high degree of privateness of the contested good (p > pGSP ).
When the good is purely private, therefore, the GSP always occurs. As can be seen in
Figure 2, the members of the large group exert more effort than the ones of the small
group (eSA > eSB ) as long as p 6 p3 . Clearly, the GSP can only arise for degrees of
privateness of the prize that guarantee eSA < eSB , hence it must exceed p3 . In order for the
GSP to hold, individual effort by the members of the small group must be large enough
so as to compensate for the smaller number of members in the group.
One can identify two opposite effects of group size on returns to effort driving the
previous results. On the one hand, there is an aggregate effort effect. As groups’ performance is additive in the contributions of their members, a larger group size implies
better odds at winning the competition between groups. On the other hand, there is a
group-size deterrence effect (Nitzan, 1991a). If the good is (at least partially) private, a
larger size also translates into a lower share of the prize for each group member in case
of victory.15 Clearly, this latter effect disappears when the good is purely public, while
it exacerbates when the degree of privateness increases.
For degrees of privateness close to p1 the size deterrence effect is not sufficient to
compensate for the aggregate effort effect. This means, in turn, that effort is more
valuable for the members of the large group than it is for the members of the small
group. As a result, the large group implements a relatively more meritocratic sharing rule,
yielding higher levels of effort by its members. When the degree of privateness exceeds
p2 , the size deterrence effect increases accordingly, and effort progressively becomes less
valuable for the large group members. Therefore, from p2 on, the large group selects a
sharing rule that is more egalitarian than the one of the small group. However, as the
two effects are at play, returns to effort are still higher for the members of the large
group, so that they exert higher effort until a yet larger degree of privateness is reached.
As the prize becomes more private, the large group progressively decreases the degree of
15
As we explained before, this effect of relative privateness on the GSP is also driven by the existence
of (partially) meritocratic group sharing rules.
14
meritocracy of its sharing rule such that from p3 on, returns to effort are smaller for its
members than for the members of the small group (from p > p3 it holds that eSB > eSA ).
Before analyzing the sequential choice of sharing rules by the two groups, the following
lemma will be useful in order to understand its implications for the GSP.
Lemma 1. If both groups are active, the sharing rules are strategic complements from the
perspective of the large group, whereas they are strategic substitutes from the perspective
of the small group.
Given its size advantage, the large group reacts to an increase in αB by increasing
αA . Conversely, the small group chooses to decrease αB following an increase in αA .
Intuitively, an increase in αB would not be worthwhile in terms of its effect on the group’s
winning probability, given the size disadvantage of the small group.
5
Sequential Choice of Sharing Rules
We consider the following three-stage game: In stage one the leader (group i ∈ {A, B})
chooses its sharing rule αiL . In stage two the follower (group j 6= i) chooses its sharing
rule αjF . In stage three both groups simultaneously choose their effort levels (eLi , eFj ). As
for the simultaneous case, we assume that p > 0 and we present the equilibrium choices
of sharing rules (αiL , αjF ) in our formal results, as effort levels (eLi , eFj ) chosen in stage
three can be derived directly from Proposition 1.
5.1
Large group A is the leader
Proposition 5. Let p01 =
nA (nA −nB −1)
.
1+nA (nA −nB −1)
If group A is the leader:
• If 0 < p ≤ p01 , only group A is active and the continuum of sharing rules in the
subgame perfect equilibrium is given by
L
– αA
=
nA nB (1−p)+p
(nA −1)
p
15
F
– αB
]
∈ [0, nB (1−p)+p
p
• If p > p01 , both groups are active and the sharing rules in the unique subgame perfect
equilibrium are given by
L
– αA
= 1 + nA 1−p
p
F
– αB
=
(nA −1)(nA −nB )p[nA (1−p)+p]−nA [nB (1−p)+p][nA (2−2nB (1−p)−3p)−(nB −2)p]
2nA (nB −1)[nA (1−p)+p]p
Observe that when the large group is the leader it commits to a more meritocratic
sharing rule than in the simultaneous case, inducing the small group to be inactive for
an even larger range of p (i.e., p01 > p1 ). In other words, the large group acting as the
leader is capable of tying the small group’s hands and oblige the latter to react with a
sharing rule leading to its own inactivity, and this for a larger range of p. The remaining
intuitions regarding the relative levels of meritocracy and individual efforts within each
group are similar to the ones of the simultaneous case.
The following lemma will be useful in order to understand the implications of the
sequential interaction when the large group chooses first.
L
S
F
S
Lemma 2. If p > p01 then αA
> αA
and αB
< αB
.
When the large group has the leading advantage, we are in a situation that is very
similar to a standard Stackelberg duopoly model. Once the large group commits to a
L
S
more meritocratic rule than in the simultaneous case (αA
> αA
), the follower reacts by
F
S
selecting a less meritocratic rule than in the simultaneous case (αB
< αB
). Indeed, recall
that the sharing rules are strategic substitutes from the perspective of the small group
(Lemma 1).
For low values of privateness (i.e., p < p02 ), the large group ties more the prize to
relative effort than the small group and each member of the large group exerts higher
effort than each member of the small group.16 Once the prize is private enough these
Again, to avoid introducing more notation, we do not define formally the thresholds p02 and p03
presented in the interpretation of our results. They can be easily calculated and one can show that
p01 < p02 < p03 .
16
16
relationships alter. First the small group selects a more competitive sharing rule than
the large group (from p > p02 ) and from a point on each member of the small group exerts
higher effort than each member of the large group (from p > p03 it holds that eFB > eLA ).
Proposition 6. If the large group is the leader and for all values of p including full
privateness, the GSP never arises.
Contrary to the simultaneous case, the GSP never takes place regardless of the degree
of privateness of the prize, which is a direct consequence of Lemma 2. Indeed, as the
large group acting as the leader selects a more meritocratic sharing rule than in the
simultaneous case, and as the small group reacts with a less meritocratic one, it follows
that aggregate effort increases in the large group, while it decreases in the small group.
As it turns out, these effort variations are large enough to eliminate the GSP.17
5.2
Small group B is the leader
Proposition 7. If group B is the leader:
• If 0 < p ≤ p1 , only group A is active and the continuum of sharing rules in the
subgame perfect equilibrium is given by
–
F
αA
=
nA
nA −1
h
1−p
p
+
(nB −1)αL
B +1
nB
i
L
∈ [0, nB [nA (1−p)+3p−2]−2p
– αB
]
(nB −1)p
• If p > p1 , both groups are active and the sharing rules in the unique subgame perfect
equilibrium are given by
F
– αA
=
nB [nA (1−p)+p][nB (2−2nA (1−p)−3p)−(nA −2)p]−(nB −1)(nA −nB )p[nB (1−p)+p]
2(nA −1)nB [nB (p−1)−p]p
L
– αB
= 1 + nB (1−p)
p
17
In fact, for both the cases of full publicness (p = 0) and full privateness (p = 1) of the prize, the two
groups face the same probability of winning.
17
For intermediate degrees of privateness covering most of the interval of p (0 < p 6 p1 ),
the small group is not able to take advantage of its leadership. Regardless of the choice
of the small group in stage one, in stage two the large group selects a very meritocratic
sharing rule such that the small group remains inactive in stage three. Observe that
the threshold level of privateness that determines whether the small groups is active or
inactive is identical to the one obtained in the simultaneous case (Proposition 3), that
is, the small group cannot take advantage of its leadership by becoming active for any
p 6 p1 .
F
S
L
S
Lemma 3. If p > p1 then αA
< αA
and αB
< αB
.
For all degrees of privateness of the prize yielding an interior equilibrium, and in stark
contrast with the case of the large group being the leader, it turns out that when the
small group has the leadership advantage it strategically chooses to lower competition
L
S 18
by selecting a less meritocratic rule than in the simultaneous game (αB
< αB
).
In
turn, the large group also reacts with a less meritocratic rule than the one it selects when
S
F
). Indeed, recall that the sharing rules are strategic
< αA
the game is simultaneous (αA
complements from the perspective of the large group (Lemma 1). Therefore, both groups
adopt relatively more egalitarian rules, and their members exert a lower level of effort
than in the simultaneous game ((eFA , eLB ) < (eSA , eSB )).
Proposition 8. If the small group is the leader and for all values of p including full
privateness, the GSP never arises.
Observe that here, the GSP does not arise for a very different reason than when the
large group is the leader. The small group selects a sharing rule such that its members are
inactive when the prize is public enough (but not purely public), which clearly prevents
the occurrence of the GSP as in the previous cases. Then, if p > p1 the small group acting
18
This is broadly consistent with previous findings on the effects of precommitment in contests with
asymmetric players. Dixit (1987) establishes that with two asymmetric players, the “favorite” commits
effort at a higher level than in a Nash equilibrium without commitment, and the “underdog” at a lower
level.
18
as the leader strategically chooses to lower competition by selecting a less meritocratic
rule than in the simultaneous game, which in turn induces the large group to adopt
a similar behaviour. However, the reduction in meritocracy implemented by the small
group is relatively larger than the corresponding reduction in the large group, and is such
that the GSP vanishes.
The fact that the leading advantage of the small group always eliminates the GSP is
somewhat surprising. Indeed, one could expect that the small group could exploit such
a leadership advantage by adopting more meritocratic rules, thereby inducing high levels
of effort by its members, potentially giving rise to the GSP for a larger set of p0 s. As it
turns out, the small group chooses instead to lower competition between the two groups,
which in turn eliminates the GSP for all values of p.19 Hence, despite being the leader
—and thus obtaining a higher payoff than in the simultaneous game— the small group
chooses to eliminate the GSP. In turn, this suggests that the GSP is not always valuable
from the perspective of the small group.
6
Discussion
We have shown that the strategic choice of sharing rules has important implications
regarding the occurrence of the GSP. In particular, the timing of the strategic interaction
is important. While the GSP may occur for different degrees of privateness of the prize
in the case of a simultaneous interaction, when there is a first mover, the GSP vanishes
independently of the leader’s size. Furthermore, regardless of the order of moves, the
large group selects a sharing rule leading to the inactivity of the small group for most
values of privateness of the prize, even arbitrarily small ones.
We have analyzed the relationship between the size of a group and its performance in
terms of its probability of winning the prize. However, another relevant notion of group
effectiveness relates its size to per-capita payoffs. Even though the large group always
19
As in the case where the large group is the leader, for both the extreme cases of full publicness
(p = 0) and full privateness (p = 1) of the prize, the two groups face the same probability of winning.
19
outperforms (or ties) the small one in terms of aggregate effort (and thus in terms of
winning probability) whenever the order of moves is sequential, the members of the small
group achieve strictly higher expected utility than the members of the large group when
the prize is sufficiently private. This is also the case when the choice of sharing rules is
simultaneous, from a degree of privateness strictly smaller than the one required for the
GSP to arise.20 In fact, irrespective of the order of moves, when the prize is sufficiently
private so that both groups are active, individual utility is decreasing in privateness in
the large group, while it is increasing in the small group. With this alternative notion
of group effectiveness, therefore, the members of the small group always outperform the
members of the large group in a competition over a pure private good regardless of the
timing of the interaction. As a larger group size decreases the per-capita value of the
private component of the prize, the large group has some intrinsic disadvantage in terms
of payoffs.
Although the size of the leader does not matter in terms of the GSP, it does have
some implications regarding the expected utility of groups’ members and rent dissipation.
Consider the case where both groups are active for any timing of the interaction. When
the large group is the leader, all its members clearly have higher expected utility than in
the simultaneous version of the game, while all the members of the small group (the follower) are worse off. Therefore, a transition from simultaneous to sequential competition
where the large group is the leader never consists in a Pareto improvement, and increases
rent dissipation. Conversely, when the small group is the leader, the members of both
groups exert less effort than in the simultaneous game, and they have higher expected
utility. Thus, going from simultaneous to sequential competition where the small group
is the leader consists in a Pareto improvement and decreases rent dissipation.
Given these implications, a natural extension of our model consists in endogenizing
√
−1
√
The exact value of the threshold is given by p = 1 − 1 + ni ( nA + nB )2 − 1
when group
i is the leader (i = A, B). Although the corresponding threshold for the simultaneous case cannot be
derived analytically, one can show that it is unique given the strict monotonicity of expected utility with
respect to privateness in both groups.
20
20
the timing of the strategic interaction between the two groups. More specifically, let us
assume that prior to the choice of sharing rules, groups are able to declare their intention
to be the leader or the follower of the game.21 If both groups choose the same order
the simultaneous game is played, while if they agree on the order the sequential game
is played. It is easy to verify that when both groups are active, the unique equilibrium
of the game with such an endogenous choice of the timing structure gives the leadership
to the small group.22 Therefore, with an endogenous timing selection previous to the
rent-seeking activity, the GSP should never take place.23
21
In the context of contests, Baik and Shogren (1992) analyze a standard Tullock contest with two
individuals, and find that the unique equilibrium with endogenous timing is a sequential game where the
weak player is the leader (see also Leininger (1993) and Morgan (2003)). Gürtler (2005) analyzes endogenous timing in the context of collective rent-seeking, and our results do not differ qualitatively for the
case of p = 1. Baik and Lee (2012b) study the endogenous timing of contests with delegation, and show
that the delegate of the higher-valuation player chooses his effort level after observing his counterpart’s.
The industrial organization literature has extensively considered endogenous timing extensions. See for
instance Amir and Grilo (1999), Hamilton and Slutsky (1990) and Van Damme and Hurkens (1999) in
the context of a duopoly model.
22
Since this equilibrium is unique it is worth mentioning that it yields lower aggregate effort than in
the simultaneous game. Thus, if from the designer’s point of view effort is valuable, he should possibly
oblige groups to choose their sharing rules simultaneously, or even force the large group to move first. If
effort is instead considered as wasteful, the designer should opt for the design where the small group is
the leader and thus should not intervene in the game.
23
If the small group is inactive, the simultaneous game may arise as an equilibrium in some particular
cases, which clearly also prevents the occurrence of the GSP.
21
7
Appendix
Proof of Proposition 1. This proof relies on an extension of Ueda (2002). In order to
facilitate comparison, let us transform αi = 1 − α̂i .
The expected utility of individual k = 1, ..., ni in group i = A, B is
i
[(1 − α̂i ) Pneiki e p + α̂i n1i p + (1 − p)] (EAE+E
V − eki
B)
j=1 ji
By the Kuhn-Tucker Theorem, the necessary and sufficient conditions to characterize
the unique symmetric pure-strategy Nash equilibrium are the following: (i) eki = Ei /ni
for each member of group i = A, B and (ii)
V
{(1
E
− α̂i )p +
α̂i
p
ni
+ (1 − p) −
Ei
[1
E
− p(1 −
1
)]}
ni
− 1 ≤ 0 for all i
Similarly to Ueda (2002) we can define:
γ̂i = (1 − α̂i )p +
We also define n̂i =
1
,
1−p(1− n1 )
α̂i
p + (1 − p)
ni
which is always positive and larger than one. Then the
i
above necessary and sufficient conditions for the pure strategy Nash equilibrium can be
written as
V
{γ̂i
E
−
Ei n̂i
}
E
− 1 ≤ 0 for i = A, B.
This expression is analogous to expression (2) in Ueda (2002, p. 616), which guarantees that Proposition 1 and Corollary 1 in Ueda (2002) hold in our case. Then, in our
two groups case, from Corollary 1(b), members of group i are inactive if and only if
γ̂i ≤ γ̂j −
1
, ∀i 6= j
n̂i
Rewriting this condition in terms of αA , αB yields χi (αA , αB ) ≥ 0 in Proposition 1.
Solving the system of equations arising from the first-order conditions in the interior and
corner solutions, we find the equilibrium effort levels.
22
Proof of Proposition 2. By definition the GSP takes place if and only if EA < EB . Consider the equilibrium effort levels when both groups are active. Then ẼA = nA êA < ẼB =
nB êB can be written as
pA(αA , αB )B(αA , αB )
V >0
C
where:
A(αA , αB ) = − [nB (1 − p) + p] [nA (1 − p) + p + (nA − 1)pαA ] − (nB − 1) [nA (1 − p) + p] pαB < 0,
B(αA , αB ) = nB (1 − 2αA ) + nA [2nB (αA − αB ) + 2αB − 1]
2
and C = [nA (2nB (p − 1) − p) − nB p] > 0
The above expression is equal to zero for p = 0, hence the GSP does not arise in that case.
For other values of p we need to consider the sign of each expression. Since C is always
positive and A(αA , αB ) is always negative, the GSP arises if and only if B(αA , αB ) < 0,
which can be rearranged as
αA <
nA − nB + 2αB nA (nB − 1)
2nB (nA − 1)
Proof of Proposition 3.
From Proposition 1 and the equilibrium effort levels if both groups are active, the
expected utility for group i = A, B is given by
ˆ i (αA , αB ) =
EU
[ni nj +((αi −1)(ni −1)nj +αj ni (1−nj ))p]V
ni [nj p+ni (2nj (1−p)+p)]2
[ni (2ni − 1)nj − Ap + B(ni − 1)p2 ]
where
A = ni (1 − αj − ni ) + nj (1 − αi ) + ni (4ni + αi + αj − 5)nj
B = αi (nj − 1) + αj (nj − 1) + (ni − 1)(2nj − 1)
From Proposition 1 and the equilibrium effort levels if only group i is active, the
expected utility for group i is given by
23
˜ i (αi ) = V − (ni − 1)(ni + αi ) pV
EU
n2i
If group i is inactive then from Proposition 1 it must be that χi (αi , αj ) > 0. Notice
that χi (αi , αj ) + χj (αi , αj ) = −nA [2nB (1 − p) + p] − nB p < 0. Hence if χi (αi , αj ) > 0
then χj (αi , αj ) < 0, meaning that if i is inactive j must be active. Therefore if i is
inactive it holds that EUi (αi ) = 0.
Solving χj (αi2 (αj ), αj ) = 0 we define αi2 (αj ) =
ni
ni −1
h
1−p
p
+
(nj −1)αj +1
nj
i
. Given that
χj (αi , αj ) is strictly increasing with respect to αi it is true that χj (αi , αj ) > 0 for all
values αi > αi2 (αj ) and hence αi2 (αj ) is the minimum value of αi that guarantees that
members of group j are inactive in equilibrium given any αj .
Solving χi (αi3 (αj ), αj ) = 0 we define αi3 (αj ) =
nj p+ni [nj +αj p−(αj +1)nj p]
.
(1−ni )nj p
Given that
χi (αi , αj ) is strictly decreasing with respect to αi then for all α̇i ∈ [0, αi3 (αj )] it is true
that χi (α̇i , αj ) > 0 and hence group i is inactive.
Comparing the expressions of αi2 (αj ) and αi3 (αj ) notice that αi2 (αj ) > αi3 (αj ) and
therefore we can write the expected utility as follows:



0
if 0 ≤ αi ≤ αi3 (αj )



ˆ i (αA , αB ) if αi3 (αj ) < αi < αi2 (αj )
EUi (αi ) =
EU




 EU
˜ i (αi )
if αi ≥ αi2 (αj )
Notice that EUi (αi ) is a continuous function since
ˆ i (αA , αB ) = 0 and limα →α (α ) EU
ˆ i (αA , αB ) = EU
˜ i (αi )
limαi →αi3 (αj ) EU
i
i2
j
ˆ i (αA , αB ) is
Moreover, the second derivative of EU
2nB (nA −1)2 [nB (p−1)−p]p2 V
nA [nB p+nA (2nB (1−p)+p)]2
< 0. Thus,
ˆ i (αA , αB ) is a strictly concave function in the unrestricted domain αi ∈ (−∞, +∞)
EU
and obtains a global maximum at:
αi1 (αj ) =
nj [ni (1−p)+p][nj (2−2ni (1−p)−3p)−(ni −2)p]−(nj −1)(ni −nj )p2 αj
2(ni −1)nj [nj (p−1)−p]p
˜ i (αi ) is strictly decreasing with respect to αi .
Finally, notice that EU
24
If αi1 (αj ) ≤ αi3 (αj ) then for all αi > αi3 (αj ) the expected utility EUi (αi ) is strictly
decreasing and hence negative. Therefore, EUi (αi ) takes its maximal value of zero for all
values α̇i ∈ [0, αi3 (αj )]. Comparing the analytical expressions of αi1 (αj ) and αi3 (αj ) it is
true that αi1 (αj ) ≤ αi3 (αj ) if and only if
αj ≥
nj ni (1 − p) + p
= α̂j
nj − 1
p
˜ i (αi ) is strictly decreasing with respect to
If αi1 (αj ) ≥ αi2 (αj ) and given that EU
αi then EUi (αi ) has a unique global maximum at αi2 (αj ). Comparing the analytical
expressions of αi1 (αj ) and αi2 (αj ) it is true that αi1 (αj ) ≥ αi2 (αj ) if and only if
αj ≤
nj [ni (1 − p) + 3p − 2] − 2p
= α̃j
(nj − 1)p
If αi3 (αj ) < αi1 (αj ) < αi2 (αj ) then EUi (αi ) has a unique global maximum at αi1 (αj ).
Summing up, the best response of group i can be written as:



αi2 (αj )
for αj ≤ α̃j



αi (αj ) =
αi1 (αj )
for α̃j < αj < α̂j




 α̇i ∈ [0, αi3 (αj )] for αj ≥ α̂j
We now consider the nine possible combinations of the two groups’ best responses:
Case 1: Both groups play the interior
If i = A, B plays αi1 (αj ), it yields the following sharing rule:
ni
2ni nj (ni − 1) + Ai p + Bi p2
αi =
(ni − 1) p [2ni nj − p(ni (2nj − 1) − nj )]
where Ai = ni nj (9 − 4ni ) − nj (nj + 2) − 2ni and Bi = ni nj (2ni − 7) + nj (nj + 3) + 3ni − 2.
From the best responses, this is an equilibrium if and only if α̃i < αi < α̂i for i = A, B,
which is true as long as
25
p>
nB (nA − nB − 1)
= p1
1 + nB (nA − nB − 1)
Hence, both groups being active is an equilibrium if and only if p > p1 .
Case 2: Both groups play the corner
If i = A, B plays αi2 (αj ), there is no solution, as the best responses are two parallel
lines with positive slope [(nj − 1)ni ] / [nj (ni − 1)].
Case 3: A plays the interior and B plays the corner
If A plays αA1 (αB ) and B plays αB2 (αA ), it yields the following sharing rules:
α A = 1 + nA
αB =
1
p
−1
nB [nA (1−p)+p]
(nB −1)p
From the best responses, this is an equilibrium if and only if αA 6 α̃A and α̃B <
αB < α̂B . As the sharing rules above are such that αB > α̂B , the latter condition is never
satisfied. Hence we do not reach an equilibrium.
Case 4: A plays the corner and B plays the interior
If A plays αA2 (αB ) and B plays αB1 (αA ), it yields the following sharing rules:
αA =
nA [nB (1−p)+p]
(nA −1)p
αB = 1 + nB
1
p
−1
From the best responses, this is an equilibrium if and only if αB 6 α̃B and α̃A <
αA < α̂A . As the sharing rules above are such that αA > α̂A , the latter condition is never
satisfied. Hence we do not reach an equilibrium.
Case 5: A plays the interior and B plays inactivity
If A plays αA1 (αB ) and B plays α̇B ∈ [0, αB3 (αA )], we are at an equilibrium if and
only if αA > α̂A and α̃B < αB < α̂B , which is never satisfied. Let αA = αA1 (αB ). Then,
the condition αB < αB3 (αA ) reduces to αB < α̃B , hence we reach a contradiction.
26
Case 6: A plays inactivity and B plays the interior
If A plays α̇A ∈ [0, αA3 (αB )] and B plays αB1 (αA ), we are at an equilibrium if and
only if α̃A < αA < α̂A and αB > α̂B , which is never satisfied. Let αB = αB1 (αA ). Then,
the condition αA < αA3 (αB ) reduces to αA < α̃A , hence we reach a contradiction.
Case 7: Both groups play inactivity
If i = A, B plays α̇i ∈ [0, αi3 (αj )], we immediately reach a contradiction, as αA 6
αA3 (αB ) and αB 6 αB3 (αA ) cannot hold simultaneously. The condition αA 6 αA3 (αB ) is
equivalent to
αB >
nB [nA − (1 − αA )(nA − 1)p]
nA (nB − 1)p
Then, we have that
αB3 (αA ) >
nB [nA −(1−αA )(nA −1)p]
nA (nB −1)p
if and only if p >
2nA nB
2nA nB −nA −nB
>1
Hence we reach a contradiction.
Case 8: A plays the corner and B plays inactivity
If A plays αA2 (αB ) and B plays α̇B ∈ [0, αB3 (αA )], we are at an equilibrium if and
only if αA > α̂A and αB 6 α̃B , which holds if and only if αB ∈ [ nB (1−p)+p
, α̃B ] and p 6 p1 .
p
Case 9: A plays inactivity and B plays the corner
If A plays α̇A ∈ [0, αA3 (αB )] and B plays αB2 (αA ), we are at an equilibrium if and
only if αA 6 α̃A and αB > α̂B , which is never satisfied. Let αB = αB2 (αA ). Then,
αB > α̂B reduces to αA > [nA (1 − p) + p] /p. As α̃A < [nA (1 − p) + p] /p, the conditions
for an equilibrium are never satisfied.
Proof of Proposition 4. Clearly, if the small group is inactive, the GSP cannot occur. If
p > p1 , from Proposition 2, the GSP arises if and only if
27
S
αA
<
S
nA − nB + 2αB
nA (nB − 1)
2nB (nA − 1)
Substituting for the equilibrium value of αiS (i = A, B) from Proposition 3, the above
condition reduces to
(nA − nB ) [2nA nB (1 − p) − p]
<0
2(nA − 1)nB p
which holds if and only if p >
2nA nB
1+2nA nB
= pGSP .
Proof of Lemma 1. Taking derivatives of αA1 (αB ) and αB1 (αB ) we directly obtain the
result:
(nB −1)(nA −nB )p
2(nA −1)nB [nB (1−p)+p]
∂αA1 (αB )
∂αB
=
∂αB1 (αA )
∂αA
A −nB )p
= − 2nA(n(nAB−1)(n
< 0, hence αA is a strategic substitute for group B.
−1)[nA (1−p)+p]
> 0, hence αB is a strategic complement for group A.
Proof of Proposition 5. The best response of group B can be obtained from the simultaneous case:



αB2 (αA )
for αA ≤ α̃A



αB (αA ) =
αB1 (αA )
for α̃A < αA < α̂A





α̇B ∈ [0, αB3 (αA )] for αA ≥ α̂A
The expected utility of group A being the leader is given by:



0
if αA ≤ α̃A



ˆ A (αA , αB1 (αA )) if α̃A < αA < α̂A
EUA (αA ) =
EU




 EU
˜ A (αA )
if αA ≥ α̂A
Notice that EUA (αA ) is a continuous function since
ˆ A (αA , αB1 (αA )) = 0 and limα →α̂ EU
ˆ A (αA , αB1 (αA )) = EU
˜ A (α̂A )
limαA →α̃A EU
A
A
28
ˆ A (αA , αB1 (αA )) is
Moreover, the second derivative of EU
[p(nA −1)]2 V
2n2A [nA (p−1)−p]
< 0. Thus,
ˆ A (αA , αB1 (αA )) is a strictly concave function in the unrestricted domain αA ∈ (−∞, +∞),
EU
L
L
and obtains a global maximum at αA
, where αA
= 1 + nA 1−p
is the solution of the firstp
order condition
ˆ A (αA ,αB1 (αA ))
∂ EU
∂αA
˜ A (αA ) is strictly decreasing with
= 0. Given that EU
L
respect to αA , α̂A strictly dominates any αA > α̂A . As αA
> α̃A , it follows that EUA (αA )
L
is maximized at min{αA
, α̂A } and thus the corresponding expected utility is strictly posL
itive. Observe then that αA
< α̂A if and only if p >
nA (nA −nB −1)
1+nA (nA −nB −1)
= p01 . Therefore,
• If p 6 p01 , group A selects α̂A and group B selects αB ∈ [0, nB (1−p)+p
], with the
p
upper bound being the solution of αB = αB3 (α̂A ).
L
L
• If p > p01 , group A selects αA
and group B selects αB1 (αA
), given by
L
αB1 (αA
)=
(nA −1)(nA −nB )p[nA (1−p)+p]−nA [nB (1−p)+p][nA (2−2nB (1−p)−3p)−(nB −2)p]
2nA (nB −1)[nA (1−p)+p]p
F
= αB
L
S
Proof of Lemma 2. Using the equilibrium levels of (αA
, αA
) we can see that
L
S
αA
≥ αA
⇐⇒
(nA −nB )[nA (1+nA −nB )(p−1)−p]
(nA −1)[nA (2nB (p−1)−p)−nB p]
≥0
The numerator of this expression is strictly positive if and only if p > 1+ nA (1+nA1−nB )−1 >
1, whereas the denominator is positive if and only if p >
2nA nB
2nA nB −nA −nB
> 1. Therefore, the
L
S
previous expression is strictly positive for any p ∈ [p1 , 1], which guarantees that αA
≥ αA
when both groups are active.
F
S
Using the equilibrium levels of (αB
, αB
) we can see that
F
S
αB
≤ αB
⇐⇒
(nA −nB )2 [nA (1+nA −nB )(p−1)−p]p
2nA (nB −1)[nA (p−1)−p][nA (2nB (p−1)−p)−nB p]
The numerator is strictly negative for any p ∈ [0, 1 +
≤0
1
],
nA (1+nA −nB )−1
which includes
the set [0, 1]. The demominator is only negative for values of p larger than one. Therefore,
F
S
the previous expression is negative for any p ∈ [0, 1], which guarantees that αB
≤ αB
.
29
Proof of Proposition 6. Clearly, if the small group is inactive, the GSP cannot occur. If
p > p01 , from Proposition 2, the GSP arises if and only if
L
<
αA
F
nA − nB + 2αB
nA (nB − 1)
2nB (nA − 1)
L
F
Substituting for the equilibrium value of αA
and αB
from Proposition 5, the above
condition reduces to
nA (nA − nB )(p − 1) [nA (2nB (p − 1) − p) − nB p]
<0
2(nA − 1)nB p [nA (1 − p) + p]
As the denominator is positive, the condition holds if and only if the numerator is
negative, which requires p > 1. Hence we reach a contradiction.
Proof of Proposition 7. The best response of group A can be obtained from the simultaneous case:



αA2 (αB )
for αB ≤ α̃B



αA (αB ) =
αA1 (αB )
for α̃B < αB < α̂B




 α̇A ∈ [0, αA3 (αB )] for αB ≥ α̂B
The expected utility of group B being the leader is given by:



0
if αB ≤ α̃B



ˆ B (αA1 (αB )), αB ) if α̃B < αB < α̂B
EUB (αB ) =
EU




 EU
˜ B (αB )
if αB ≥ α̂B
Notice that EUB (αB ) is a continuous function since
ˆ B (αA1 (αB )), αB ) = 0 and limα →α̂ EU
ˆ B (αA1 (αB )), αB ) = EU
˜ B (α̂B )
limαB →α̃B EU
B
B
ˆ B (αA1 (αB )), αB ) is
Moreover, the second derivative of EU
[p(nB −1)]2 V
2n2B [nB (p−1)−p]
< 0. Thus,
ˆ B (αA1 (αB )), αB ) is a strictly concave function in the unrestricted domain αB ∈
EU
30
L
L
(−∞, +∞), and obtains a global maximum at αB
, where αB
= 1 + nB 1−p
is the solution
p
of the first-order condition
ˆ B (αA1 (αB )),αB )
∂ EU
∂αB
˜ B (αB ) is strictly de= 0. Given that EU
L
creasing with respect to αB , α̂B strictly dominates any αB > α̂B . Observe that αB
< α̂B
L
always holds, while αB
6 α̃B if and only if p 6 p1 . Therefore,
• If p 6 p1 , group B selects αB ∈ [0, α̃B ] and group A selects αA2 (αB ).
L
L
• If p > p1 , group B selects αB
and group A selects αA1 (αB
), given by
L
αA1 (αB
)=
nB [nA (1−p)+p][nB (2−2nA (1−p)−3p)−(nA −2)p]−(nB −1)(nA −nB )p[nB (1−p)+p]
2(nA −1)nB [nB (p−1)−p]p
F
= αA
S
L
) we can see that
, αB
Proof of Lemma 3. Using the equilibrium levels of (αB
L
S
αB
≤ αB
⇐⇒
(nA −nB )[(nA −nB −1)nB (p−1)+p]
(nB −1)[nA (2nB (p−1)−p)−nB p]
≤0
The numerator of the above expression is positive if and only if p ≥ p1 . The denominator is negative if and only if p <
2nA nB
,
2nA nB −nA −nB
hence it is negative for any p ∈ (0, 1].
Therefore, the above expression is negative for any p ∈ [p1 , 1], which guarantees that
L
S
αB
< αB
when both groups are active.
L
S
Using the equilibrium levels of (αA
, αA
) we can see that
F
S
αA
≤ αA
⇐⇒
(nA −nB )2 p[(nA −nB −1)nB (p−1)+p]
2(nA −1)nB [nB (p−1)−p][nA (2nB (p−1)−p)−nB p]
≥0
The numerator of the above expression is non-negative if and only if p ≥ p1 . The
F
S
denominator is positive for any p ≤ 1, which guarantees that αA
< αA
when both groups
are active (i.e., p ∈ [p1 , 1]).
Proof of Proposition 8. Clearly, if the small group is inactive, the GSP cannot occur. If
p > p1 , from Proposition 2, the GSP arises if and only if
F
αA
<
L
nA − nB + 2αB
nA (nB − 1)
2nB (nA − 1)
31
F
L
Substituting for the equilibrium value of αA
and αB
from Proposition 7, the above
condition reduces to
(nA − nB )(p − 1) [nA (2nB (p − 1) − p) − nB p]
<0
2(nA − 1)p [nB (1 − p) + p]
As the denominator is positive, the condition holds if and only if the numerator is
negative, which requires p > 1. Hence we reach a contradiction.
32
References
Amir, R. and I. Grilo (1999): “Stackelberg versus Cournot equilibrium,” Games and
Economic Behavior, 26, 1–21.
Baik, K. H. (2008): “Contests with group-specific public-good prizes,” Social Choice
and Welfare, 30, 103–117.
Baik, K. H. and D. Lee (2012a): “Do rent-seeking groups announce their sharing
rules?” Economic Inquiry, 50, 348–363.
Baik, K. H. and J. H. Lee (2012b): “Endogenous Timing in Contests with Delegation,” Economic Inquiry, 51, 2044–2055.
Baik, K. H. and S. Lee (1997): “Collective rent seeking with endogenous group sizes,”
European Journal of Political Economy, 13, 121–130.
——— (2001): “Strategic groups and rent dissipation,” Economic Inquiry, 39, 672–684.
——— (2007): “Collective rent seeking when sharing rules are private information,”
European Journal of Political Economy, 23, 768–776.
Baik, K. H. and J. F. Shogren (1992): “Strategic behavior in contests: comment,”
The American Economic Review, 82, 359–362.
Corchón, L. C. (2007): “The theory of contests: a survey,” Review of Economic Design,
11, 69–100.
Davis, D. D. and R. J. Reilly (1999): “Rent-seeking with non-identical sharing rules:
An equilibrium rescued,” Public Choice, 100, 31–38.
Dixit, A. (1987): “Strategic behavior in contests,” The American Economic Review, 77,
891–898.
Esteban, J.-M. and D. Ray (2001): “Collective action and the group size paradox,”
American Political Science Review, 95, 663–672.
Gürtler, O. (2005): “Rent seeking in sequential group contests,” Bonn Economic
Discussion Papers #47.
Hamilton, J. H. and S. M. Slutsky (1990): “Endogenous timing in duopoly games:
Stackelberg or Cournot equilibria,” Games and Economic Behavior, 2, 29–46.
Katz, E., S. Nitzan, and J. Rosenberg (1990): “Rent-seeking for pure public
goods,” Public Choice, 65, 49–60.
Kolmar, M. (2013): “Group Conflicts. Where do we stand?” Tech. rep., University of
St. Gallen, School of Economics and Political Science.
Kolmar, M. and A. Wagener (2013): “Inefficiency as a strategic device in group
contests against dominant opponents,” Economic Inquiry, 51, 2083–2095.
33
Konrad, K. A. (2009): “Strategy and dynamics in contests,” Oxford University Press.
Lee, S. (1995): “Endogenous sharing rules in collective-group rent-seeking,” Public
Choice, 85, 31–44.
Lee, S. and J. Hyeong Kang (1998): “Collective contests with externalities,” European Journal of Political Economy, 14, 727–738.
Leininger, W. (1993): “More efficient rent-seeking: a Münchhausen solution,” Public
Choice, 75, 43–62.
Morgan, J. (2003): “Sequential contests,” Public choice, 116, 1–18.
Nitzan, S. (1991a): “Collective rent dissipation,” Economic Journal, 101, 1522–1534.
——— (1991b): “Rent-seeking with non-identical sharing rules,” Public Choice, 71, 43–
50.
Nitzan, S. and K. Ueda (2009): “Collective contests for commons and club goods,”
Journal of Public Economics, 93, 48–55.
——— (2011): “Prize sharing in collective contests,” European Economic Review, 55,
678–687.
——— (2013): “Intra-group heterogeneity in collective contests,” Social Choice and Welfare, forthcoming.
Noh, S. J. (1999): “A general equilibrium model of two group conflict with endogenous
intra-group sharing rules,” Public Choice, 98, 251–267.
——— (2002): “Resource distribution and stable alliances with endogenous sharing
rules,” European Journal of Political Economy, 18, 129–151.
Olson, M. (1965): The logic of collective action, Harvard University Press.
Pecorino, P. and A. Temimi (2008): “The group size paradox revisited,” Journal of
Public Economic Theory, 10, 785–799.
Riaz, K., J. F. Shogren, and S. R. Johnson (1995): “A general model of rent
seeking for public goods,” Public Choice, 82, 243–259.
Ueda, K. (2002): “Oligopolization in collective rent-seeking,” Social Choice and Welfare,
19, 613–626.
Ursprung, H. W. (1990): “Public goods, rent dissipation, and candidate competition,”
Economics & Politics, 2, 115–132.
Van Damme, E. and S. Hurkens (1999): “Endogenous Stackelberg leadership,”
Games and Economic Behavior, 28, 105–129.
34