Rent-seeking with Non-Identical Sharing Rules: An Equilibrium Rescued

Transcription

1 Rent-seeking with Non-Identical Sharing Rules: An Equilibrium Rescued by Douglas D. Davis and Robert J. Reilly * September 1997 Abstract Nitzan s (1991) analysis of differential sharing rules in a collective rent-seeking setting is reconsidered. Two groups, each with more than one member, are presumed to use different linear combinations of two sharing rules, one based on an equal-division of the prize, and the other on each member s relative effort. We show that an equilibrium always exists for this type of game, and then characterize the equilibrium. Our result is contrary to Nitzan s claims that (a) in the general case an equilibrium often does not exist, and (b) and equilibrium never exists when the groups use the polar extreme rules. * Middlebury College and Virginia Commonwealth University, and Virginia Commonwealth University, respectively. Financial support for this research was provided by the VCU Faculty Excellence Fund, and by the National Science Foundation (grant SBR ).

2 Rent-Seeking with Non-Identical Sharing Rules: An Equilibrium Rescued by Douglas D. Davis and Robert J. Reilly 1. Introduction In a 1991 paper appearing in this journal, Shmuel Nitzan extends the theory of the collective rent-seeking game in which two groups of risk-neutral bidders compete for a fixed prize. Each group has agreed upon a sharing rule to divide the prize among group members either (i) equally, or (ii) in proportion to each member s individual contribution, or (iii) using an intermediate rule that is a weighted average of rules (i) and (ii). Nitzan s paper focuses on those cases in which the two groups employ different rules. 1 He first argues that under the polar case where one group uses rule (i) and the other uses rule (ii) equilibrium never exists, provided that there is more than one bidder in each group. Second, he examines the general case in which the two groups employ different rules, including ones that specify different weights for rule (iii). In this more general setting Nitzan finds that no equilibrium exists under a wide variety of conditions. Nitzan s results, combined with previous work, such as that by Allard (1988), where a multiplicity of equilibrium arise when bidders are non-identical, suggest the rather pessimistic conclusion that the extension of Tullock s rent-seeking game to more general contexts creates severe problems of indeterminacy. The present paper removes part of the foundation of this concern by demonstrating that, contrary to Professor Nitzan s assertions, an equilibrium always 1 As Nitzan points out, if both groups adopt the same rule, an equilibrium always exists.

3 exists in this type of collective rent-seeking game. We also characterize the equilibrium for each of the cases considered by Nitzan. In retrospect, the existence of equilibrium should have been anticipated. Noting that each individual s strategy set (individual bid domain) in the game that Nitzan models may be taken as the closed, bounded (and hence compact) interval between zero and the value of the prize, and that his specified individual reward functions are continuous and quasi-concave, the existence of an equilibrium is assured by Nash s theorem (Luenberger, 1995). The source of the confusion is Nitzan s reliance on classical programming to identify the equilibrium. Taking the classical programming approach, he confines his attention to interior solutions for all bidders and correctly determines that an equilibrium with positive bids by the members of both groups often will fail to exist. However, his assumption that this finding implies absence of an equilibrium in such cases is incorrect. A reexamination the collective rent-seeking game with differential sharing rules using a nonlinear programming approach precipitates the equilibria that Nitzan overlooked. Below, the new equilibria are developed and characterized in section 3, following a brief overview statement of the model in section. Some concluding comments are presented in section 4. Hillman and Riley (1989) demonstrate equilibrium existence in a related game where private prize values differ across players. The game analyzed by Hillman and Riley differs critically from the present analysis in that here free-riding incentives are present: For one of the groups analyzed above an individual wins if and only if his group wins. Members of this group are thus motivated to reduce their individual contributions and base winning expectations on the overall group bid.

4 . The Model Suppose two groups i, i=1, compete for a fixed prize S. The probability Πi that group i wins the prize is determined by i, the aggregate bid of the group relative to the total bid, or Πi(1, ) = i/ (1) where = 1 +. Each group i consists of n(i) members. The expected prize for any member k of group i, is given by Vki = Πi (Sfki(1i,...,ki,... n(i)i)- ki) - (1 - Πi )ki () where fki(1i,...,ki,... n(i)i) is member k s share, as determined by the group s division rule. Sharing rules are assumed to be a linear combination of two possibilities, a division based on relative effort, ki/i, and an equal-division rule, 1/n(i). Thus, the sharing rule may be rewritten as f a ki (1i,...,n(i)i) = [(1 - a) ki / i + a/n(i)] (3) where 0 a 1. Since submission of zero bids by both groups is not an equilibrium, we restrict our attention to cases where >0 (as did Nitzan). Substituting (1) and (3) into (), the expected value to player k of group i when submitting bid ki is: Vki = i ( S[(1 - a) k i i + a n(i) ] - ki ) - ( - i ) ki, if ki>0 (4) 3

5 = 0, if ki=0 and i=0. Each bidder solves the nonlinear programming problem described by the maximization of (4) subject to the nonnegativity constraints on the individual bids. 3 The Lagrangian function for member k of group i may be written as L = S(1 - a) ki + Sa i n(i) ki + λiki (5) where ki 0, and λi 0. First order conditions for member k of group i are L ki = S (1 a) [ ] + Sa ki n(i) [ i ] 1 + λi = 0, (6) ki 0, λi 0, and λiki = 0 (i.e., the complementary slackness condition or csc). Attention is restricted to symmetric equilibria. Denoting ki, the bid of each member k of group i, by xi, equation (6) may be rewritten as: n(i)(1 a)( xi) + a( i) = n ()( i 1 λi) S (7) with i = n(i)xi and = 1 + = n(1)x1 + n()x. 3 As Nitzan points out in his note, the second partial derivative of each bidder s expected value function with respect to his or her own bid is strictly negative, making these functions concave. However, his supporting expression is in error. It should read Vki ki = - S(1-a)(-ki) 3 - Sa(-i) n(i) 3 < 0. Strict negativity holds as long as k is not the only bidder submitting a positive bid. 4

6 3. Non-Identical Sharing Rules Revisited Suppose that the groups employ sharing rules fk1 a1 and fk a. Without loss of generality assume, that a1>a, i.e., that group 1 places more weight on sharing than group. Applying equation (7) to the two groups yields the following first order conditions: Group 1: (1-a1)(-x1) + a 1 = n(1) S (1-λ1) (8) Group : (1-a)(-x) + a 1 = n() S (1-λ) (9) where, as before, 1, 0, = 1 + > 0, and λ1, λ 0. Lemma 1: λ1, λ < 1. Proof: Suppose λ1 1. Then the right hand side of (8) 0. But by the csc x1 = 0, and so =>0. Since 0 a1 1 the left hand side of (8)>0, a contradiction. Thus λ1<1. An identical argument establishes λ<1. Given Lemma 1, we may divide (8) by (9). Rewriting 1 =n(1)x1 and =n()x this quotient may be written as: a1 ( 1 a1)[ n( 1) x1 + n( ) x x1] + n( ) x n() 1 a ( 1 a)[ n( 1) x1 + n( ) x x] + n() 1 x n( ) 1 = (1 λ1) (1 λ) (10) 5

7 Collecting terms in x on the left hand side, and terms in x1 on the right hand side, we obtain: ( 1 λ1 ) n()x [n(1)n() (a - a ( 1 λ ) 1 ) + n()a1 + ( 1 λ ) 1 ( 1 λ ) ( 1 λ ) ( 1 λ ) n(1)(1-a)] = n(1)x1[n(1)n() (a1- ( 1 λ ) 1 a + ( 1 λ ) 1 ( 1 λ ) ( 1 λ ) - 1) + ( 1 λ ) 1 n(1)a + n()(1-a1)] (11) 4 ( 1 λ ) Denoting the bracketed expressions on the left hand side and right hand side by A1 and A respectively, (11) may be rewritten as n()xa1 = n(1)x1a. (1) Lemma : A1=0 if and only if x1=0. Proof: (i) Suppose A1=0. Then A = A1+A. It is readily shown that A1+A = n() + (1 λ1) n(1), which must be positive by Lemma 1. Thus A>0. (1 λ) Hence (1) implies that x1=0. (ii) Suppose x1=0. Then >0 implies that >0, and hence by (1) that A1=0. Lemma 3: Given a1>a, x1>0 implies that x>0. Proof: Suppose x1>0 and x=0. λ1=0 by the csc. We may rewrite (10) as ( 1 a1)([ n( 1) 1] x1) a ( 1 a)[ n( 1) x1] + n() 1 x n( ) 1 = 1 (1 λ ), 4 Equations (10) and (11) are equivalent to Nitzan equations (1) and () when λ1=λ=0, as when x1, x>0, for example. Notably, λ1=λ>0 would imply x1=x=0, a contradiction to >0. 6

8 [n(1) -1] (1 - a 1 ) 1 i.e., = >1, by Lemma 1. n(1) a (1 - a ) + (1 λ ) n() If n(1)=1 then the left hand side equals zero, a contradiction. If n(1) then the above expression implies that (1-a1) > n(1) [n(1) - 1] [(1-a)+ a n() ] > (1-a)+ a n() 1-a. But this contradicts a1>a. Hence x1>0 implies x>0. by B. In what follows, denote the expression a1[n(1)n()-n()] - a[n(1)n()-n(1)] - n(1) Lemma 4: (a) x1>0 if and only if B<0. (b) x1=0 and λ1>0 if and only if B>0. 5 (c) x1=0 and λ1=0 if and only if B=0. Proof: >0 and Lemma 3 imply that x>0: For any x1>0, x>0, if x1=0, x>0 to insure >0. Hence λ=0 by the csc, and the expression for A1 may be rewritten as: A1 = n(1)n()[(1-λ1)a-a1+λ1] + n()a1 + (1-λ1)n(1)(1-a). (13) Collecting terms in λ1 we have: A1 = [n(1)n()-n(1)](1-a)λ1 - B (14) 5 The case where B>0 was the one considered by Nitzan. As seen here, it implies that the equilibrium cannot have x1>0, not that an equilibrium fails to exist. As shown, an equilibrium does exist with x1=0 and x>0. 7

9 Part (a) (i) Suppose x1>0. Then λ1=0 by the csc and thus A1 = -B. Also A1 0 by Lemma. Thus (3) implies that A1 and A have the same sign. We may write the expression for A as: A = n(1)n()(a1-a) + n(1)a + n()(1-a1) which is positive, since a1>a. Thus A1>0 and B<0. Hence x1>0 implies B<0. (ii) Now suppose B<0 and x1=0. Then Lemma 3 implies that A1=0. Thus (14) implies that [n(1)n()-n(1)](1-a)λ1 < 0. But λ1 0 and 0 a<a1 1 contradicts this. Thus B<0 implies x1>0. This establishes part (a). Comment: With n()=1, B may be written as B = a1[n(1)-1] - n(1) = - (1-a1)n(1) - 1 which is negative. Thus by part (a), x1>0. For parts (b) and (c), below this implies that n(). Part (b) (i) Suppose x1=0 and λ1>0. Then A1=0 by Lemma, and given n(), (14) implies that B>0. (ii) Suppose B>0. Part (a) and the nonnegativity of bids implies x1=0. By Lemma A1=0. Hence (14) implies that λ1>0. This establishes part (b). Part (c) (i) Suppose x1=0 and λ1=0. Lemma implies A1=0 and then (14) implies that B=0. (ii) Suppose B=0. Part (a) implies x1=0 and then A1=0 by Lemma. Then (14) implies that λ1=0. This establishes part (c). 8

10 Finally, we observe that, as in the polar case, when x1=0 (i.e., when B 0) the equilibrium bid x is independent of n(1). In particular, with x1=0 and x>0, ==n()x and λ=0 by the csc. Thus (9) becomes (1-a)[n()x-x] = Solving for x>0: [n()x ] S. x = [ 1 a ]( ( ) ) n 1 S, which is independent of n(1). [ n( )] The foregoing results on the existence of equilibrium are summarized in the following proposition. Proposition 1 : Let groups 1 and use non-identical sharing rules fk1 a1 and fk a where 0 a < a1 1, and define B = (a1[n(1)n()-n()] - a[n(1)n()-n(1)] - n(1)). If B<0 then a Nash equilibrium exists with x1>0 and x>0. If B 0 then a Nash equilibrium exists with x1=0 and x= [ 1 a ]( ( ) ) n 1 S, independent of n(1). [ n( )] 4. Conclusion As Nitzan correctly observed an equilibrium always exists when bidders use the same sharing rule. However, contrary to Nitzan s assertion of the opposite, this paper establishes the existence of equilibrium in cases where the groups use different sharing rules. Thus, an equilibrium always exists for this class of games. 9

11 In closing, we draw attention to the special case of polar sharing rules analyzed by Professor Nitzan in his section. Letting a = 0 and a1 = 1, B = n(1)n() - n() - n(1). B<0 implies n(1) + n() > n(1)n(), the existence condition cited by Professor Nitzan in his proposition 1. Note that this condition is never satisfied when group sizes are nontrivial (e.g,. n(1) >1 and n()>1). But, as is evident from our above proposition, this condition is necessary for positive equilibrium bids by members of both groups. When B 0, free-riding incentives for members of group 1 are overwhelming, and bids for group 1 members drop to zero. Nevertheless, an equilibrium still exists in this case, with group 1 and group bids equal to x1 = 0 and x= [ n ( ) 1 ] S, respectively. [ n( )] 10

12 References Allard, R. J. (1988) Rent-seeking with non-identical players. Public Choice 57: Hillman, A. L, and J.G. Riley (1989) Politically Contestable Rents and Transfers. Economics and Politics 1, Luenberger, D. (1995), Microeconomic Theory, McGraw-Hill, Inc., Nitzan, S. (1991), Rent-seeking with non-identical sharing rules, 71 Public Choice,