Advantageous multiple rent seeking

Marhl Compul. Modelling, Vol. 12. No. 415, pp. 51 lL518, 1989 Printed in Great Britain. All rights reserved

0895-7177/X9 $3.00 + 0.00 Copyright :i? 1989 Pergamon Press plc

ADVANTAGEOUS MULTIPLE RENT SEEKING

MARK GRADSTEIN

Department of Economics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel and Department of Economics, University of Toronto, Toronto, Ontario, Canada

SHMUEL NITZAN~

Department of Economics, Bar-Han, University, Ramat-Gan 52100, Israel

Abstract-This paper analyzes a special rent seeking game which has not been studied in the literature on pressure groups competition, rent seeking, lobbying or auctions. We consider a multiple rent contest where n identical allocators (players) with constrained resources compete under full information over a fixed supply of M (2 < 1)2 < n) advantageous rents (prizes).

Pure strategy Nash equilibria always exist in this game. Symmetric mixed strategy Nash equilibria (SMSEs), however, may not exist. A general existence condition for SMSEs is provided. When n is sufficiently large, SMSEs always exist. Assuming that the rents are identical we derive a necessary and sufficient condition (in terms of the relationship between n and m) for the existence of a unique SMSE. We also provide the characterization of the equilibrium allocation of the constrained resources of the players among the homogeneous or heterogeneous rents. It turns out that, in equilibrium, interest group leaders always concentrate their resources in attempting to secure just one of the political benefits. The distribution of allocators over the political favors (prizes) they wish to win almost reveals the relative values of the prizes.

I. INTRODUCTION

Political scientists (Salisbury 1984) have come to view pressure politics in a new way. Interest group leaders are seen as allocators of scarce resources among projects. Given that political activity must be purchased with funds diverted from alternative organizational purposes a primary question is how to allocate organizational resources between nonpolitical programs and political activities. Since interest group politics is typically competitive (Schlozman and Tierney 1986; Berry 1984), this allocation problem can be quite complicated when the interest group leader becomes aware of the interdependence between his decision and the allocation decisions of other rival interest groups. A preliminary analysis of this allocation problem, assuming that government policy depends on the activities of just two groups, has been recently presented by Johnson (1988) [for a related setting, see Becker (1983)]. Although Johnson studies the allocation problem between political and other projects, he does not attempt to pursue the (general) equilibrium analysis for the simultaneous two-group political competition. In fact, a rigorous representation of the way that groups compete has not been provided by the political scientists studying pressure group politics. In particular, no one has studied the problem of allocating political effort among a number of issues in the presence of opposing (competitive) action. In the current paper we model and analyze a stylized game of competitive allocation of fixed resources among a number of political activities. This game is henceforth referred to as the advantageous multiple rent seeking game. An informal description of this game follows.

In the advantageous multiple rent seeking game a number of interest group leaders (more generally, allocators of resources, like state governors) decide how to divide their political funds among a number of projects yielding relatively high indivisible benefits (favors). These benefits are the advantageous rents. The identical interest group leaders do not allocate their fixed resources between political and other projects but spend them on political activity only. They are competing on the indivisible rents and are aware of this competition. We suppose (with some loss of generality)

tTo whom all correspondence should be addressed.

512 MAKK GRADSTEIN and SHMUEL NITZAN

that the agent spending the most in the effort to obtain a particular prize is the winner.? In other words, the indivisible rents are distributed among the rent seekers using an auction type mechanism.1 Within our setting of advantageous rents, the value of the resources available to the allocators (the rent seekers) is lower than the value of the prizes. Consequently, even though in equilibrium the contenders’ resources are necessarily exhausted, the standard result of rent dissipation (Hillman and Samet 1986; Tollison 1982), obviously cannot hold. Our main concern, therefore, is not with the identification of the link between the value of the contestable prizes and the value of the resources attracted into the contest, but rather with the existence and characterization of the equilibrium allocation of the allocators’ resources among the different rents.

The story which accompanies our model can be that of competing interest group leaders allocating political effort among a number of projects. But the model is also applicable to other political or economic contexts such as firms’ competition for certain licenses or contracts, competition of art museums over art pieces belonging to potential donors or competition of constrained R&D units for a number of fixed indivisible grants each awarded to a single winner. To further illustrate the nature of our game consider the competition of states for a number of projects allocated by the Federal Government. These projects are indivisible and they may include defense sites, science labs, a supercollider etc. The final decisions of the government are presumably based on the merit of the proposals submitted by the competing states. The quality of the proposals certainly depends on the resources spent in their preparation. Now these resources are limited as, by law, states have to balance their budget and their revenue base is of course limited, especially in the short run. (Note that most state constitutions forbid capital raising to cover operating expenses but allow it to fund capital projects.) Consequently, then the resources available for the preparation of proposals are limited and are often considerably lower than the benefits associated with the all or nothing winning (states entirely lose or win) of the competitions. The state governors face, therefore, a typical economic problem, namely, how to allocate their limited resources across independent competitions for the various indivisible “prizes”. Strategic considerations in this context are transparent. A game theoretic approach seems to be a natural tool for analyzing the composition of the resources expended by the rival contenders on the various potential “prizes”.

The players in the advantageous multiple rent seeking game on which we focus are identical:

they are informationally identical, they share common tastes and are endowed with identical resources to be spent in the quest over the prizes. Section 2 presents the fully symmetric version of the model, whereby the identical players face homogeneous rents (prizes). Section 3 contains the equilibrium analysis for this symmetric situation. In our game pure strategy equilibria (PSEs) always exist. Proposition 1 provides the characterization of these equilibria. Specifically, in equilibrium each player concentrates on a single prize, and the rent seekers’ efforts are almost evenly distributed over the identical prizes. It turns out that the existence of symmetric mixed strategy Nash equilibria (SMSEs) is not guaranteed. Proposition 2 provides the condition ensuring the existence of SMSEs. It implies in particular that such equilibria do exist when the number of prizes is sufficiently small relative to the number of players. Section 4 deals with the extended version of the game where the players are identical but the prizes are no longer homogeneous. Proposition 3 characterizes the PSEs in this more general case. It turns out that, as before, each player concentrates his efforts on a single prize. However, the rent seekers’ efforts are not evenly distributed over the different prizes. The equilibrium ratio between the number of players concentrating on any two prizes approximates the ratio between their respective values. Finally, Proposition 4 provides a general condition for the existence of SMSEs. When the number of players is sufficiently large, existence is guaranteed. Such equilibria are mixtures of the pure strategy equilibria of Proposition 3. A brief summary of the results appears in Section 5.

tin contrast, in the rent seeking literature the common assumption is that the rent seeking expenditures determine the winning probabilities of the players; see, for example, Tullock (1980).

IWhereas in standard auctions there exists a single prize and only the winning contender pays a price equal to his bid (Riley and Samuelson 1981; Milgrom and Weber 1982), here there are several prizes and all rent seeking outlays are nonrefundable. Also note that, since we are abstracting from problems of incomplete information, our setting is basically different from that of the standard auction models.

Advantageous multiple rent seeking 513

2. THE MODEL

Consider n contenders (players), n 3 3, competing over m indivisible advantageous rents (prizes), 2 < m < n. Assume that the players as well as the prizes are identical. Homogeneity of the players implies that they are informationally identical, they are all endowed with the same amount of (rent seeking) resources, and they share identical valuations of the m prizes. With no loss of generality, let the amount of resources to be spent by each player in the quest over the prizes be equal to 1. The players have to decide how to allocate their constrained resources among the prizes realizing that each prize is awarded to that contender who makes the largest effort to win it (expends the largest amount of resources in his seeking to gain that particular rent). In case of a tie-when some players spend the same amount of resources in their attempt to win a particular prize-the prize is awarded to each of these contenders with an equal probability. In particular, when no resources are allocated to a particular prize, each player wins that prize with probability l/n. Let V denote the common value of a prize. I’ is assumed to be large relative to the resources of each contender, so that each player exhausts his available resources in the rent seeking competition. Finally, assume that the payoff of any player is given by the expected value of the prizes he wins.

Formally,letN={l,..., n}, n 3 3, denote the set of players. The strategy space of player i is the set X7 consisting of all vectors of the form x, = (x,,, . . . , x,,) satisfying

j=i

and Vj, x0 2 0. Let Lf = {I: I # i and x0 = xii} and nJ = # LJ, where # A denotes the cardinality of the set A.

That is, nJ is the number of players who tie with player i on prize j. Player i’s payoff is given by

j=l

where hj is player i’s expected payoff in the contest over the rent j,

V

hi= nj+ 1 xv 2 XI,, l#i

I 0 otherwise.

With the specifications of the set of players, their sets of strategies and their payoff functions, the definition of the symmetric advantageous multiple rent seeking game in strategic form is complete.? The following section is concerned with the existence and characterization of equilibrium in this game.

3. EQUILIBRIUM ANALYSIS

In this paper we focus on two commonly used Nash equilibrium concepts: pure strategy Nash equilibrium (PSE) and symmetric mixed strategy equilibrium (SMSE). x* = (XT, . . ,x,*) is called a pure strategy Nash equilibrium if, for each i and xi, H,(x*) > H,(x:, . . . , xi, . . . , x,*). The following proposition deals with the existence and the characterization of a PSE:

Proposition 1. The symmetric advantageous multiple rent seeking game possesses PSEs. Let M be the largest integer smaller than n that can be divided by m. That is,

M E max km. (!ck<(n/rn))

In a PSE the players are divided into m subsets, and the k* = M/m players in each subset concentrate their efforts on prize j, j = 1, . . , m; while the remaining n - k *m

players concentrate their resources on different prizes.

tNote that since the prizes are always distributed among the players, all game situations are Pareto optimal, hence the game defined above is a zero-sum game.

514 MARK GRADSTEIN and SHMUEL NITZAN

Proof. Under the suggested strategies, the payoffs of (n - M) (k* + 1) players are equal to V/(k * + l), the payoff of each of the remaining players is equal to V/k*, and no player can advantageously deviate from his strategy.

Notice that there are multiple PSEs in our game. Furthermore, in any PSE no subset of players can benefit by deviating from the equilibrium strategy. Hence, the set of PSEs constitutes the core of the cooperative counterpart of the advantageous multiple rent seeking game.

Let Fj(xi) denote a probability distribution function over Xr. (F*(x,), . . . , F*(x,)) is called a symmetric mixed strategy equilibrium if, for each i and F(x,),

EH,(F*(x,), . . * 2 F*(xn)) 2 E ffi(F*(x,)t . . . ,F*(xi- I), F(xi), F*(xi+ I), . . . ,F*(x,)),

where E denotes the expectation operator. Despite the fact that our game is symmetric, the existence of SMSEs is not secured.? In

particular, Theorem 6 in Dasgupta and Maskin (1986), which states sufficient conditions for the existence of SMSEs in symmetric games, cannot be applied since their required continuity property of the payoff functions is not satisfied in our case. The following proposition characterizes the SMSEs in our game and provides a necessary and sufficient condition for their existence.

Proposition 2. A unique SMSE exists whenever

This equilibrium is characterized by each player concentrating all his resources on one prize where each prize is chosen in equal probability. In other words, each player’s equilibrium mixed strategy is given by the following distribution function F:

over XT:

1

F,+(xi) = m

I-

when x, satisfies: x,/ = 1 for some Z, I=],... ,m, and x,=0 ‘dj, j#I

0 otherwise.

Proof. Since the value of each prize is large enough relative to the value of each player’s resources, in equilibrium the players’ resources will be exhausted. It is easy to see that our game does not possess symmetric PSEs.$ Hence we may narrow our search for equilibria to mixed strategy equilibria in which resources are entirely consumed. Furthermore, there is a positive probability p, that each player will concentrate all his available resources in attempting to win prize j,

j=l,..., m. [Otherwise, each player can guarantee himself a payoff V, V > V(m/n), by concentrating on one prize with probability 1.1

We now proceed by proving that these probabilities are identical, that is, for each j, F*

(0,. ’ * . ,, 3 . . . 3 0) = p, = p. For suppose that these probabilities are not identical and let us consider the following possible deviation of any player (say, player n): player n plays the same strategy as before, however, instead of concentrating his resources on prize j with probability p,, he is doing so with probability 1. Denote his expected payoff in this case by H’,. Clearly, H’, # H$ for j Zj’.

Note that

max {Hi,} > Vm/n i

(otherwise, player n’s payoff, when playing the previous strategy, would be < Vm/n, which is impossible). Thus, player n can increase his payoff, which implies that the players’ probabilities of concentrating their resources on the different prizes must be identical.

tThe search for SMSEs is common, and usually also fruitful, in the literature on auctions, see Riley and Samuelson (1981). $The proof is as follows: in a symmetric equilibrium the payoff of each player must be the same and (since the number

of players exceeds the number of prizes) it is < V. Clearly, then, in a symmetric PSE it is necessary that there exists a prize on which each player concentrates less than all his resources. But in this case each player can guarantee himself a payoff of V by concentrating all his resources on that prize and winning it. Therefore, no symmetric PSE exists.


To complete the proof, let us examine the following two possible cases. Case 1: pm < 1. Without loss of generality, players’ strategies in this case can be written as

follows: for i = 1, . . ,n,

F,(O, . . ,‘, . . . ,O) =p, F,(x,,, . . . ,x,,) = 1 -pm, (1)

where xii > 0 for each j. Consider, however, the following deviation of player n:

F,(O, . . ,;, . . . ,O) = pj = p, Fi(xi, + 6, xi2 + t, . . . ,xim - (m - 1)~) = 1 -pm,

where t is small. Clearly, this deviation increases player n’s payoff; therefore, equations (1) cannot constitute an

equilibrium. Case 2: pm = 1. Consider player it’s possible advantageous deviations from the symmetric

mixed strategy where he concentrates his resources on prize j; j = 1, . . . , m, with probability l/m. First, it can be easily verified that by using the pure strategy of concentrating his resources on some single prize, he cannot secure a payoff > Vm/n. (More precisely, in this case his payoff is exactly Vm/n.)

This implies that in an advantageous deviating strategy player n has to split his resources among the prizes. Let us assume then that he chooses some vector x, = (x,,, . . . ,x,,,,,), such that

m

and for every j, 0 < xd < 1 .t His expected payoff in this case is

n-l H,=Vm(l-p)“-‘=Vm 1-i .

( 1

If this payoff is less than the payoff Vm/n resulting in the symmetric mixed strategy case, then (and only then) the symmetric mixed strategy case where every player concentrates his resources on prize j, j=l,... ,m, with probability 1 constitutes an equilibrium. In other words, the SMSE exists whenever

( > l-1

n-l 1 <-. (2)

m n

This inequality is the necessary and sufficient condition for the existence of a unique SMSE.

Proposition 2 implies that to secure the existence of the SMSE, the ratio players/prizes must be sufficiently high. It follows, for example, that when m = 2, the SMSE always exists. When m = 3, the SMSE exists provided that n >/ 5. If n is sufficiently large, then the existence of the SMSE is secured.

4. AN EXTENSION: HETEROGENEOUS OBJECTS

Let us dispense with the restrictive homogeneous rents assumption of the fully symmetric model and suppose that the identical players assign different values to the m prizes. The value of prize j,j=l,... ,m, is now denoted V,. For convenience it is assumed that for every j, V, z V,+, and 3 j: V, > V,,,. In this case we obtain the following generalization of Proposition 1:

Proposition 3. There exist PSEs in the extended game. These equilibria are characterized by the partition of the set of players into m subsets each consisting of

tit should be noted that, when splitting his resources, the largest payoff is attained by allocating some part of the resource to each prize. This is due to the fact that when resources are split he can only win a prize when all other individuals happen to direct their resources to the other prizes. In such a case then any positive amount of resources spent on a prize has the same probability of winning it. It is clear therefore that splitting the resources among more prizes can only increase the probability of winning them.


n,,j=l,..., m, players. A player belonging to a subset with nj players concentrates his efforts on object j. The numbers {nj}J’!!, satisfy the following conditions:

d nj = n, z 2 J& for every I, j, 1 fj. I

Proof. The reader should verify first that if there exists a partition of the set of players into m subsets with n,, j = 1, . . . ,m, players [the numbers {nj),“=, satisfying conditions (3)], then this partition indeed characterizes a PSE. We now present a constructive proof for the existence of such a partition. Let us consider the game in extensive form in which the players decide sequentially on their allocations. That is, player 1 chooses his strategy, then comes player 2’s turn, and so forth. Our argument is based on the backward reasoning. After n - 1 players have made their allocations, the best response of player n is to concentrate his resources on prize j for which

V v, l>- iij+ 1 n,+ 1

31 is the number of players+xcluding player n-who have concentrated their resources on prize I,

zfi,=n-1.

Player n - 1, knowing what form the strategy of player n would take, cannot do better than follow the same kind of strategy. Proceeding backward, one can see that no player can increase his payoff by deviating from the outlined strategy: thus, player 1 will concentrate his resources on prize 1; player 2 will either follow him, or will concentrate his effort on prize 2, according to whether V,/2 > V2 or not, and so on. This completes our (constructive) proof.

We conclude with the investigation of SMSEs under the more general version of the rent seeking game. Our search for SMSEs is confined to equilibria of the form:

for everyj, F*(O, . ..j ,..., O)=p,, 2 pj= 1. (4) j=l

If a player (say player n) concentrates his resources on some prize j with probability 1, while other players play according to conditions (4) his payoff is given by

c vk(l -pk)n-’

= v

I

1 -Cl -pjr+kfj . (5)

Pj n

(H{ denotes the expected payoff of player i when he concentrates his resources on prize j with probability 1.)

In an SMSE (if it exists) the following equalities must be simultaneously satisfied:

for every I and j, H!, = H’,. (6)

The values {pj}J’!!, can be determined from equations (5) and (6). (In what follows we assume

for simplicity that for every j, pj > 0.) For example, when n + co,

p,‘J. (7)

g If/

The only possible way for any player (say n) to benefit by a deviation from playing according to conditions (4) [with {p,}?!, determined by equations (5) and (6)] is by splitting his resources among


the prizes. His payoff in this case is equal to

f v,<l -P,Y, j=l

while if he plays according to equilibria (4) his payoff is

Therefore, an SMSE exists whenever

Clearly, when n + co, condition (8) is satisfied. To summarize,

Proposition 4. In the extended game an SMSE exists if there exist

m

{P,L c p,= 13 ,=I

satisfying conditions (7) and (8). An SMSE always exist when n + co.

Remark. Comparison of Propositions 3 and 4 reveals that, when n -+ cc the SMSE is obtained by mixing all the PSEs of Proposition 3.

5. SUMMARY

Our results suggest that in equilibrium of advantageous multiple rent seeking games contenders will concentrate their resources on single objects. In the context of interest group leaders competing for various indivisible advantageous projects or of firms’ competition over different licenses or contracts we may thus anticipate extreme “specialization” in the rent seeking activity.

This conclusion is obtained regardless of whether the prizes are equally or differently valued by the identical contenders. The indivisibility of the rents, the special auction mechanism determining the winners, the complete information assumption and the restricted competition in our model play a central role in producing the extreme form of each player’s “rent seeking portfolio”. Competition

in our case is restricted since although our symmetric players face free access to participation in the quest for each of the m prizes where the largest outlay on a rent designates its successful contender, their resources are limited relative to the value of each of the contestable rents. Allocation strategies characterized by diversification of the rent seeking portfolio are possible of course when the rents are divisible, the rents are not advantageous, the rents are allocated by some probabilistic mechanism that is positively related to the contenders’ efforts, or when there is no competition over the potential rents (at least the allocators are unaware of such a competition).

We have tried to extend the model to the more general case where contenders are different either in terms of their tastes, or in terms of the amount of resources they possess. Unfortunately, in these cases we could not even prove the existence of equilibrium. For the partial extension of the game presented in Section 4 where identical contenders face heterogeneous prizes, PSEs exist. Further- more, in this case we were able to expose the relationship between the players’ valuations of the prizes and the distribution of their concentrated resources over the prizes. If the relative values of the prizes are unknown then they can be at least approximately determined by the information on the number of contenders concentrating on the various prizes [see equations (4)]. Put differently, in our special advantageous multiple rent seeking game privately held information on the relative values of the prizes can be effectively revealed by the known equilibrium distribution of players over the prizes on which they focus. In our game SMSEs do not necessarily exist. We have derived, however, the necessary and sufficient conditions for their existence.

Acknowledgements-We are indebted to D. Samet, S. Slutsky, J. Hamilton, an anonymous referee and especially P. Johnson for their valuable advice.


REFERENCES

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Studies 53: l-26. Hillman, A. L. and D. Samet. 1987. Dissipation of Rents and Revenues in Small-numbers Contests. Pub/it Choice 54: 63-82. Johnson, Paul E. 1988. On The Theory of Political Competition: Comparative Statics from a General Allocative Perspective.

Public Choice 58: 217-236. Milgrom, P. R. and R. J. Weber. 1982. A Theory of Auctions and Competitive Bidding. Econometrica 50: 1089%1122. Riley, J. G. and W. F. Samuelson. 1981. Optimal Auctions. American Economic Review 76: 381-392. Salisbury, Robert H. 1984. Interest Representation: The Dominance of Institutions. American Political Science Review 78:

64-76. Schlozman, Kay L. and John Tierney. 1986. Organized Interests and American Democracy, New York: Harper & Row. Tollison, R. D. 1982. Rent Seeking: A Survey. Kylos 35: 5755602. Tullock, Gordon. 1980. Efficient Rent Seeking. In Toward a Theory of the Rent Seeking Society, Eds James M. Buchanan

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Advantageous multiple rent seeking