~~.~ Discussion ~~~ paper o ' Research I~NOII I N N~~ N~I~e~;IpA~l~l Center for Economic Research No. 9368 THE SOLUTION TO THE TULLOCK RENT-SEEKING GAME 6JHEN R~ 2: MIXED-STRATEGY EQUILIBRIA AND MEAN DISSIPATION RATES by Michael R. Baye, Dan Kovenock and Casper G. de Vries October 1993 ISSN 0924-7815 BIBLIOTHEEK TILBURG The Solution to the Tullock Rent-Seeking Game When R~ 2: Mixed-Strategy Equilibria and Mean Dissipation Rates Michael R. Baye, The Pennsylvania State University Dan Kovenack, Purdue University Casper G. de Vries, Etasmus Universiteit RotterdamlTinbergen Institute March 1993 Abstract The original rent seeking game devised by Tullock, whereby the probability of winning a rentseeking contest is a function of the rent seeking expenditures raiseá to the power R, is solved for any value of R~ 0. In particular, we show that a mixed-strategy Nash equilibrium exists when R 1 2. The possibility of over dissipation of rents - which was conjectured in the early literature for the case where R~ 2-- dces not arise in any Nash equilibrium. We provide a tight bound on the amount of under dissipation of rents that arises in a symmetric equilibrium. This bound explains earlier experimental work which could not be rationalized before. General representations of symmetric Nash equilibrium mixed strategies are provided, as well as numerical examples based on values of R 1 2 used in some of the recent experimental literature. Correspondent: Casper G. de Vries Erasmus University Rotterdam Tinbergen Institute Oostmaaslaan 950 3063 DM Rotterdam, The Netherlands 'We are gtateful to Dave Furth and Ftans van Winden for stimulating conversations, and for comments provided by workshop participants from the CORE-ULB-KUL IUAP project, Purdue University, Pennsylvania State University, Rijksuniversiteit Limburg, and Washington State University. We also thank Max van de Sande Bakhuyzen and Ben Heijdra for useful discussions, and Geert Gielens for computational assistance. An earlier version of the paper was presented at the FSEM 1992 in Brussels and the Mid-West Mathematical Economics Confet~ertce in Pittsburgh. All three authors would like to thank CentER for its hospitality during the formative stages of the paper. The sewnd author has also benefited from the financial support of the Katholieke Universiteit Leuven and the ]ay N. Ross Young Faculty Scholar Award at Purdue University. The third author benefitted from visiting IGIER where part of the paper was written. The third author also benefitted from grant IUAP 26 of the Belgian Government. z 1. INTRODUCTION ]n Tullock (1980) the following interesting rent seeking game is described. Consider two players who bid for a poGtical favor commonly known to be worth Q dollars (Q ~ 0 and finite). Their bids influence the probability of receiving the favor. Let x and y denote the bids of agents 1 and 2 respectively, and let ar(x,y) denote the probability the first agent is awarded the political favor. The payoff to agent 1 from bidding z when the other agent bids y is U~(~ly) z rc~,y)Q - x , (i) while that of player two is symmetrically defined: U:(Y i x) -[I - x(x,Y))Q - Y. Because the politician awarding the prize may have other considerations, or because he can only imperfectly discriminate between the bids (if bids are not made in the money metric), the high bidder is not guaranteed the prize. This is a common assumption in (I) the principalagent literature (Lazear and Rosen, 1981; Nalebuff and Stiglitz, 1983; Bull, Schotter and Weigelt, 1987), (2) the political campaign expenditure literature (Snyder, 1990); and (3) the literature on rationing by waiting in line (Holt and Sherman, 1982). Presumably, given y, the probability of winning is an increasing function of x. Tullock suggested the specification 1 2 z" xR , yR ij.z~y~0 (2) otherwue (x Z 0, y Z 0) 3 where R 1 0. This specification has become standard in the rent seeking literature and other fields, see e.g. Snyder (1990). The case where R- 1 is studied most (Ellingsen, 1991; Nitzan, 1991a; Paul and Wilhite, 1991), but it is of interest to consider other values of R, as in Applebaum and Katz (1986) and Millner and Pratt (1989). Loosely speaking, the case 0 e R c 1 represents decreasing returns, while R~ 1 repn.saits increasing retums to aggn~sive bidding. While the two agent pure strategy symmetric Nash equilibrium is straightforward to calculate from the first order conditions when 0 c R 5 2, this is not the case when R~ 2. Consequently Tullock (1980) devoted a large part of his discussion to these latter cases. To date, there are only conjectures concerning the ezistence of a Nash equilibrium for R~ 2 but finite. Rowley (1991), in his review of Tullock's work, lists this as one of the three important theoretical problems for a research program in the area of rent seeking. The problem is not so much that the first-order condition for a maximum cannot be calculated; the problem is that the symmetric (x - y) solution to the two player's first-order wnditions does not necessarily yield a global maximum (if R 1 2 the symmetric solution to the first order conditions implies a negative ezpected payoff, which is dominated by a zero bid). In such a case the sum of the solutions to the first-order conditions exceed the value of the prize Q; there is the false appearance of an over dissipation of rents. Tullock (1980, 1984, 1985, 1987, 1989) devoted considerable attention to the case of over dissipation because of the induced excess social waste; see Dougan (1991) for a critical comment, and Laband and Sophocleus (1992) for estimates of the resource expenditures. In Tullock (1984) it was acknowledged that over 4 dissipation may be due to a failure of the second order conditions.' In the vernacular of game theory, over dissipation is not part of a Nash equilibrium. This notwithstanding, the possibility of over dissipation is a recutTertt theme in the rent sesking literature. In particular, Millner and Pratt (1989) examined the rent seeláng model experimentally for the cases where R- 1 and R- 3. Due to the use of laboratory dollars, the strategy space used in their experimcnt is discrete. For a prize worth 8 U.S. dollars they formulate two hypolheses concerning the mean of the individual expenditures and the mean dissipation rates. These hypotheses are stated in Table 1, together with the'u experimental results.' Both hypotheses are rejected for either value of R, but at markedly different p-values. The p-value for the R- 1 case is at least .015, while the p-value for R- 3 is at the most ]0~. Thus, Ho is only rejected marginally for the case R- 1, while I-io is strongly rejected for the case R3. Baik and Shogren (1991) point out, however, that Millner-Pratt's null hypothesis for the c~se R- 3 is not the correct one. The problem, however, is that the equilibrium to the game is not known when R 1 2. Our paper resolves this issue. Briefly considering the n-player variant, n z 2, the second order conditions fail if R~ nl(n-2), cf. Tullock (1984) (where the reverse condition is reported erroneously). Note that for the case n- 2 the second order conditions are always satisfied. But it is easily checked that for R~ 2 the symmetric solution to the first-order conditions yields U,(. ~.) c 0, and hence is not a global maximum. Thus the two agent case is the most interesting case to consider, because with n~ 2 the posited solutions obviously do not make sense if R 1 nl(n-2). z The null hypotheses should be interpreted with caution because the experimental setup of Millner and Pratt (1989) is not entirely congruent with the simultaneous move n~uirement (neither dces it fit the altemating move version studied in Leininger 1990 a,b). 5 Table 1: Millner and Pratt (1989) Hypotheses and Experimental Results Iio Exponent Experiment Iia R- 1 Mean Individual 2 Expenditures Mean Dissipation Rates Number of Observations 2.24 R- 3 6 (2.42) 5096 5696 Experiment 3.34 (-24.28) 15096 849b (2.3T) (-13.37) 146 100 More specifically, for R- 1, the symmetric Nash equilibrium is known, and the associated expenditure and dissipation rates are readily verified to correspond with the hypothesiud values in Table 1. This is further wrroborated by a recent experiment by Millner and Pratt (1991) which shows that risk avenion can explain the discrepancies between the hypothesized and realiud values in Table 1 for the case when R- I. A major benefit of the results presented below is that we will be able to explain the discrepancy between the hypothesiud values and experimental results for the case when R~ 3. The punch-line is that the formula based on the first-order equations (which yields a rent dissipation of 15096) is incorrect. In fact, there is not a symmetric pure-strategy equilibrium when R- 3. We characteriu the "correct" Nash equilibrium, and show that the results of the Millner-Pratt experiments are in line with the theoretically correct Nash equilibrium mixed strategies. To this end we mainly focus on the two agent case in discrete strategy space. consider a continuous strategy space by taláng limits of the finite game. In the last section we 6 Before we embark on this, we briefly review the approaches others have used to deal with the R ~ 2 case. The approach in the existing literature is to modi~ the original gome to remove the apparent over dissipation of rents. In his original contribution Tullock (1980) suggested three modifications. The fu-st is to let R be infinite, which tums the game into an allpay auction. Within the rent seeking literature this version has been studied by Hillman and Samet (198Tj. The complete charactetization of all equilibrium strategies has ber.n obtained by Baye et al. (1990), and the equilibrium level of r~ent dissipation is derived in Baye et al. (1993). The second type of modification is to change the one shot game into a dynamic game. Tullock (1980) discusses the case of altemating bids, and this has been formalized recently by Leininger (1990a, b). In Corcordn (1984), Corcoran and Karels (1985), and Higgins et al. (198T) the game is changed into a two stage game. In the first stage the number of participants is selected such that, when the rent seeking game is played in stage two, the number of participants is consistent with (almost) complete rent dissipation. Similarly, Michaels (1988) devises a setting within which the politician has the incentive to adjust the exponent such that the first and second order conditions are met. The third modification deals with asymmetries between the players. This was briefly dealt with in Tullock (1980) and has been further investigated by Allard (1988). Finally Nitzan (1991b) introduces coalition behavior on the part of the contestants. None oj these contributions, though, offers o solution to the original simultaneous moue rent seeking game when R~ 2. The next secdon provides this solution and relates it to the experimental and theoretical literatures. 2. SOLVIIVG THE RENT SEEKIIVG GA11~ Consider the two agent rent seelàng game with conditional payoffs and winning probabilities as given in equations ( 1) and (2). The exponent satisfies R~ 0. Suppose a pure strategy equilibrium exists. Given y~ 0, the first and second order conditions for an unconstrained (local) maximum of U~(x ~ y) are readily calculated as Y .`R R xR-' (x 4 Y~ - 1` ~~ (3) and "!R XR-7 Q(xR 4 y~~ I(R-I)(XR t Y~ - 2RxRJ c 0. (4) Assuming a symmetric solution, condition (3) yields x- y- QRl4, for which condition (4) is readily seen to hold locally for any R 1 0. Substituting back into equation (1) yields (S) U,(x - Y- QR) ~ Q(1 - 2); i- 1, 2. Note that in this case U;(. ~.) is non-negative as long as R 5 2. Moreover, for any z, y~ 0 the factor (R-1)(xR f yR) - 2R x" in the second order condition (4) is unambiguously negative if R S 1, while it is positive over some interval to right of x- 0 if R~ 1 and becomes negative thereafter. In particular, (4) is satisfied when x - y. Thus for R 5 2, the symmetric solution x - y- QRl4 constitutes a Nash Equilibrium. For R ~ 2, U(QRl4 ~ QRI4) in (S) becomes negative and hence the first order conditions do not yield a symmetric Nash equilibrium point ( because one can choose x- 0 given that y- QRl4; and earn a higher payoff. But if x- 0 is chosen, player two has an incentive to lower y to small e~ 0). 8 Generally, the fust and second order conditions (3) and (4) fail to characterize the global maximum when R ~ 2.~ In order to find a solution for the cxse R~ 2, we focus on the game with a discrete strategy space. This yields a version of the game similar to that used in the laboratory experiments by Millner and Pratt (1989, 1991).` Due to the use of laboratory dollars, the bids are necessarily discrr,te, and thus the game is a so-called finite game.' Nash's (1951) theorem guarantees that every finite game has a mixed-strategy equilibrium.~ It follows immediately that the Tullock rent-seeking game in discrete strategy spact has a Nash equilibrium, possibly in nondegenerate mixed strategies, for any R~ 2. While it is in general difficult to characterize the equilibria, we may be more specific in this case. Note that for any strategy pair (x,y), the payoff to the second agent is the same as the payoff to the first agent if the strategies played by the two agents are interchanged; the game is symmetric. Recalling that an equilibrium is defined to be a symmetric equilibrium if all players choose the same strategy, we may apply Dasgupta and Maskin's (1986) L.emma 6; a finite symmetric game has a symmetric mixed-strategy equilibrium. Baye, Tian, and Zhou (1993) show that one cannot generally blame the non-existence of a pure-strategy equilibrium on the failure of payoff functions to be quasiconcave or upper semi-continuous. Although Millner and Pratt claim to be testing the Tullock model, the experiment actually allows the rent-seekers to expend resources continuously over a small time interval. Hence, the experiment does not formally test the original one-shot simultaneous-move Tullock game. This problem is corrected in the experiments of Shogren and Baik (1991), who do not rtject the theoretical prediction when r- 1. s The continuous strategy space ( infinite game) is dealt with below. 6 The mixed-strategies may be degenerate, i.e., in the c~se of a pure strategy equilibrium. 9 In summary, the Tullock rent seeking game with a discn~e strategy space certainly has a symmetric Nash equilibrium, even when R 1 2. following questions: These results immediately raise the (i) Can we characterize the equilibria for R 1 2, even though previous authors have been unable to do so? In particular, is it possible to provide an ezplicit soludon for the symmetric equilibria that arise for differcnt values of R? (ii) Can the equilibria of the finite game be used to shed light on infinite game (o~ntinuous strategy space) equilibria? A derivative question is: (iii) How do the answers to these ques6ons relate to the experimental work reported by Millner and Pratt for the c~se R- 3? We answer question (i) by employing a device which was first used by Shilony (1985). The payoffs to the game will be written in matrix format. We then show this yields a matrix equation which can be manipulated to yield the symmetric mixed strategy solution. Some numerical examples and a special case of this procedure are provided. To answer the derivative question (iii) we manipulate the matrix equation to obtain tight bounds on the equilibrium dissipation rate. Question (ii) is answered by letting the mesh of the strategy space become small relative to the value of the prize. Recall equation (1) which gives the conditional payoffs for agent l. unconditional or expected payoffs from playing x, To obtain the EU~(x), the conditional payoffs are premultiplied by the (mixed-strategy) probability py that a particular y value is being played by player one's opponent, and subsequently these are summed over y. Thus ~ lo Q EU,GY) ~ ~ P, ~(x.Y)Q - X. Y-o Denote the expected payoffs to agents 1 and 2 in an arbitriry Nash equilibrium by v, and v, respectively. In the case of a symmetric Nash equilibrium note that the players' expected payoffs are identical, v, - vz - v(however, v need not be unique). The manipulations below make repeated use of the following general result. Tbeorem 1. In any equilibrium: (i) EU,(x) 5 v,, (ii) EU,(x) - v, when p, ~ 0, while (iii) pj - 0 if EU,(x) c v,. Similar results hold for player 2. A proof of this theorem can be found in Vorob'ev (1977, sec. 3.2.2., 3.4.2. and 3.4.3.). For a symmetric equiGbrium -- which we Irnow exists by L.emma 6 in Dasgupta and Maskin (1986) -- we can use equations (6) and (2) to restate the condition EU,(x) 5 v as Q R (7) QPr xR f YR 5 vQx Conditions (ii) and (iii) in Theorem 1 imply a complementary slaclmess-type condition for a symmetric equilibrium of the form V x: P~ x ~ Ps RsR R- v; Q Y.p X 4Y ~ 0. ~~l Now note that EU,(x - Q) 5 0, and in fact EU,(x - Q) c 0 if pr.a c 1(and R is finite). Thus in a symmetric equilibrium no mass will be placed at Q, i.e. p,.Q - pr.Q - 0. Suppose 11 (without loss of generality but for ease of notation) that Q e N, and that x and y can only take on the integer values 0, 1, ..., Q. Note that there are exactly Q oonditions ('7) for x s 0, 1, ..., Q-1. These can be conveniently expn~sod in matrix format: 1 2 0 0 0 1 1 2 1 1.2R 1 1}(Q'1)R ZR 1 1 1 2Ra1 (Q-lI (Q-1)R'1 2 ... (Q-1)R ... (Q-1~~2R v Po ZR 2Rt(Q-]I 1 2 Q v.l P, Pz PQ-, Q 5 v;i (8) v.Q-1 Q In addition to this Q x Q matrix condition, the following constraints must be imposed: Q-1 ~ pr-1; prz0, y~0, 1,..,Q. y-o (9) Condition (8), together with the constraints (9) and the complementary slackness condition (7') provide a complete, but implicit characterization of the symmetric equilibrium, which we know exists by Dasgupta and Masldn's Lemma 6. These conditions form a linear programming problem which, at least in principle, can be solved for (po, ..., po-„ v). We have thus proved Tóeorem 2. Suppose the strategy space is discrete. Then for any R~ 2, the Tullock rentseeking game has a symmetric mixed-strategy Nash equilibrium, defined implicitly by the solution to conditions C1'), ( 8) and (9). 12 In order to illustrate the practical utility of Theorem 2, we will investigate two special cases: R L oo and R~ 3. The latter case is that examined in Millner and Pratt's experiments, while the former is the discrete atrategy spaoe version of the all pay auction examined in Baye, Kovenock, and de Vries (1990; 1993). We begin with the case when the exponent R~ oo and assume Q~ 1 for simplicity. In this case the matrix expression in (8) becomes I 2 0 0 ... 0 Po 1 l 2 0 0 P, 1 1 1 ... 0 2 P, I 1 1 ... l 2 5 (10) PQ., It is straightforward to find symmetric e~uilibria if it is assumed that all p; 1 0. In this case the matrix inequality (10) becomes an equality by Theorem 1. The lower triangular matrix equation can then be solved through recursive substitution. This yields po - p~ -... - po-~ 2vIQ and p, - pj -... - pQ, - 2(1-v)IQ. In addition to (8), conditions (9) and (7') have to hold. For even walues of Q this restricts v E[0, 1], while for odd values of Q, we necessarily have v- ll2 (see Bouckaert et al. for a proof of this claim). Note that we may make the grid in the formulation of the game (7) finer and finer and normaliu the value of the prize to be one by dividing all dolLv units by Q and letting Q tend to infinity. The e~uilibrium distributions in this discrete game with r- oo then converge 13 uniformly to the continuous uniform distribution, and the expected payoff vIQ converges to uro; therc is full rcnt dissipation. Also note that equations (1) and (2) can be expressed as U~(xly) - Q- x ~f x 1 y 2Q - x 4lx - y -x jfx cy which is precisely tht defuution of the all-pay aucáion (ef. Baye, Kovenock, and de Vries, 1993). It follows that the symmetric equilibria of the discrcte all-pay suction converge to the unique (see Baye et al., 1990) equilibrium of the continuous strategy space rwo player all-pay auction. Next, consider the case of finite exponents. When 0 c R 5 2, the game has a symmetric purc strategy equilibrium (x - y- QRl4) as discussed earlier. Because R- 3 is used in Millner and Pratt's experimental work on the game, and as pointed out by Shogrcn and Baik ( 1991) the 'solution" examined by Millner and Pratt is not really a Nash equilibrium, we will focus on this case.' For R~ 2 and finite, the solutions to the game cannot be given in the same compact form as the solution for R - m, although conditions (7'),(8) and (9) still provide a complete but implicit description of the game and its soludon. For any specific values of R and Q, it can be solved explicitly through linear progrdmming. We list some examples. ' Shogren and Baik (1991) state that the behavioral inconsistency Pratt "... is due to the nonexistence of a Nash equilibrium. predictable behavioral benchmarks to measurc the experimental Theorem 2, however, provides such a benchmark. Shogren and non-existence of a symmetric pure strategy Nash equilibrium. reported in Millner and In this case there is no evidence against.' t~ur Baik are rcferring to the 14 (i) R- 3, Q- 1. Therc is one pure strategy solution: both agents bid zero and receive v- 112. lnter alia, this result holds for any fuute value of R. (ii) R- 3, Q 3 2. There exist multiple pure strategy solutions: (1) both bid zero and receive v~ 1, (2) one agent bids uro and the other bids one with respective payoffs v~ - 0 and v~ ~ 1, and (3) both agents bid one and rrceive v~ 0. Mixed strategies whereby agents randomiu over (some) of the pure strategy solutions exist as well. (iii) R- 3, Q- 3. 1'his case is still solvable by hand. In particular, condidon (8) becomes 1 2 0 0 Po 1 1 2 1 9 P, 9 2 P, 1 8 l v 3 5 v;l 3 (12) v}2 3 It is readily verified that (po, p~, p~ ~(-;-, ~, ~r) and v- 14 satisfy condition ( 12) and the other conditions of Theorem 2, and hence constitute an equilibrium to the game. (iv) R- 3, Q- 4. This case is alrcady too cumbersome to solve by hand, so we relied on the analydcal computer program 'Derive" to solve this game. It can be checked that there are two symmetric solutions: (i) (po, pi, p2, P~ -( 14, 0, 14, Ol with v--3f-, and 15 (ii) (Ao~ P~, Pi~ P~) - f O1 with v' 38~ 0' 38' 19 ~ For R~ 3 and Q~ 4, one genetally finds that all probability mass is loaded on the first few probabilities q, with most mass lo~aded on the higher q's, and 0 t v c 1. For Q~ 15 the computational burden incre~ases rapidly and exact solutions take an excessive amount of computer time. This is a bit unfortunate berause the experiment conducted by Millner and Pratt (1989) used R- 3 and a grid of Q- 80 (at the end of the experiment the laboratory dollars were converted into U. S. dollars at an exchange rate of 10. But subject payments were also rounded to the nearest 25 cents, generating a grid of 32 with unequal grid sizes). Their hypotheses and tests, however, all concern mean individual expenditures and mean dissipation rates. The question therefore is whether we have something to offer concerning these quantities, without explicitly calculating the solutions.' 71ie expected individual expenditures and the expected dissipation rates can be calculated from equation (6). Note that premultiplication of EU,(x) by p, and summation over x gives the expected equilibrium payoff to player 1 in a symmetric equilibrium: ' In future work it may be of interest to repeat the experiment for R- 3 and Q small such that all the properties of the symmetric equilibrium can be evaluated, i.e. the values of the p,'s. 16 Q Q EU, -~ P, EU~4r) -~ P. x-0 x-0 Q ~ P, tGY,y)Q - x y-0 (13) Q -~ P~v ` v, x~0 berzuse player one only loads mass on those x's which generate the same (highesy expected payoff equal to v (see Theorem 1 above). In order to dispel the claim by Millner and Pratt that over dissipation of rents is expeated when R~ 2, first note that if agent 1 chooses x~ 0 with probability 1, then Q EU, -~ P, ~(o,Y) Q' Po 2 Q Z 0. y~o Hence each player can guarantee a non-negative expected payoff. (14) Secondly, the expected dissipation rate is easily calculated from EU, t EUT. Note that v, ~ EU, - Prob{agtnt 1 wins} Q- x, where x -~p~r is the average individual expenditure. Adding up yields v, . v2 -[Prob{agent 1 wins} . Prob{agent Z wtns}] Q- x- y. But since the prize is always awarded, there is always a winning agent and hence by (14) o s v, ~vi-Q-x-y, (is~ so that x~ y 5 Q. The expected rate of rent dissipation, D, is defined as D -(i .~IQ. Thus 17 D~1-Y'~v'S1. Q We have thus proved: Theor~ 3. The two player finite rent sceláng game devised by Tullock never involves over dissipation in any (possibly mixed-strategy) Nash oquilibrium for any R~ 0. That is, D 5 1 always. The dissipadon rate is also bounded from below. But in contrast with the upper bound, the lower bound depends on the value of the exponent R. investigating the two limiting cases R- 0 and R - oo. This can be easily seen by In the former case there is no dissipation, while in the latter case dissipation can be complete. Therefore, we will investigate specific values of R. To explain the Millner-Pratt ezperimental results for the case R- 3, one requires precise information about the size of D, and hence the tighter the lower bound on D the better. ]t is not too difficult to show for Q~ 2, R~ 2, that in any equilibrium the dissipation rate is at least 5096. With more effort, for Q 1 3 a sharper lower bound for the symmetric equilibria is obtained in Theorem 4. Theorem 4. In any symmetric Nash oquiGbrium of the two player Tullock rent-seeking game with ~ ~ R~ 2 and ~ 1 Q~ 2, the dissipation rate is bounded from below by 1-~-. ts ProoG The proof comes in two parts. In part 1 we auume that p~ ~ 0, and show that this implies v 5 1. Hertct D 2 1- 2IQ. In part 2 we show that pb ~ 0 necessarily. Some of the computations for part 2 arc relegated to the Appendix. Part 1. Suppose that p, 1 0. Then (by 7lieorem 1) for x~ 0 condition (~ ne~ly beromes an e~uality: po ~ 2vIQ, so that v ~ Qpol2. Because po is bounded above by 1, v is bounded above by Q12. This implies D Z 0. To improve the upper bound on v, i.e. to lower it fmm Ql2 to 1, we continue the presumption pp 1 0. From condition (~, for x- 1 we have po ~ a 5 yQ 1; 0 5 a G~. To sce this note that all the probabilities r(l,y) except the first in the second row of matrix condition (8) are less than or oqual to 112. Combine the prrsumption pa - 2vIQ with the above inequality to get as Q-Q. (1~ Hemce 1- v 2 aQ z 0. Theroforo 1 2 v. Pa~t 2. We now show po ~ 0 necessarily. Let x be the first row for which p, ~ 0, x ;c 0, i.e. pb ~... z PI-~ - 0. Then condidon ('n holds as an equality for this row, i.e. 19 xR 1 xR 2 p,. xR}(Z~l~p,,, }... ~ xR~(Q-I~pQ-, - V t X Q~ (18) We will show that v~ 1 and pb ~ 0 are incompatible. For x f 1, condition (~ rads as follows: (xtl~ ~ 1 (z~lj' r xR p' a. 2 p'~' t (x~l~ 5 v}1 tX Q (z;11R .(Q-1~ pQ-~ (19) Compute p~ from the equality ( 18), and substitute this into the weak inequality (19). This yields the following weak inequality: 1 - 2 (x~l)R 2 (x;l)R ~ xR L xR xA ~ (x~l~, G~'1)R {(Q-1)" - 2(Z}1~ } XR ~;1)R ~~t~ 5 Q {v~l {x - 2(vtx) (x P,.~ ~ ...t xR 4(Q-1~ pQ.~ 5 xR J r ~l~x }. ~ R In the Appendix we manipulate the two sides of inequality (20) to show that the left-hand-side is non-negative while the right-hand side is strictly negative. (Note that the proof would be particularly simple if R- oo, since then (20) reduces to 0 5 -~- p,,, 5(1-v-x)IQ). This yields a contradiction so that the supposition po - 0 and v Z 1 are incompatible.) Q.E.D. 3. 11SII.LNER AND P'RATT REVISITED How do the above theoretical results compare with the experimental evidence reported by Millner and Pratt (1989)? Note that for Q large Theorems 3 and 4 provide tight bounds. In particular, given the values of R~ 3 and Q a 80 used in the Millner and Pratt experiments, Zo the symmevic ( mixod-strategy) equióbrium expected outlays are z~ y- 3.9 (after catversion to U.S. dollars) and the corraponding interval for the expecKed rent dissipation is D E[97.596, 10096] -- it is not the 150 percent dissipation rate used as the null hypothesis by Millner and Pratt. Using the experimental evidence reported by Millner and Pratt we find the following tstatistícs for the null hypotheses: -5.11 and -2.73 respectively' Compare these to the values reported by Millner and Pratt and reproduced in Table 1 above. (If the rounding to the nearest 25 cents in the actual payout is taken into acoount, the mean dissipation rate is reducod to approximately 93.75, which does not differ significantly from the experimental result at the 596 level.) Note that these t-statistics are of the same order of magnitude as those for the case R1. Also rocall the rectnt experimental work by Millner and Pratt ( 1991) which relates the relatively small discrepancy for the case R- 1 to the ezistence of risk aversion.'o Our cx]njecture is that the remaining discrepancy for the case R - 3 can be explained in a similar way. Importantly, though, the above shows that when the correct symmetric (mixed-strategy) Nash equilibrium is used as the theonr.tical benchmark to form the null hypothesis, Millner and Pratt's empirical results for the case R- 3 and Q- 80 accord well with state-of-the art rentsetking theory. Individuals seem to behave quite efficiently after all. ' Calculations are based on (3.34 - 3.9)Is, --5.11 and (84 - 97.5)Is2 - -2.73, where s, and si were calculated from Millner and Pratt ( 1989) using (3.34 - 6)Is, --24.28 and (84 - 150)Is, - -13.37. 'o Sce also Shogren and Baik, who run a related experiment for R- 1 and find that the Nash equilibrium dissipation hypothesis cannot be rejected at the 90 percent level. 21 4. S[JNIII~IARY AND RESULTS FOR THE CONTIIVUOUS STRATEGY SPACE CASE In this paper we have solved the original rent seeking game devised by Tullock for the case where the rent-seeking exponent (R) exceads two. A oonstructive method was used to find the explicit solution for the finite game (i.e, the TuUock game in discrete strateSY space). Our theoretical n~ults, which establish that rents are under dissipated when R 1 2, accord well with the existing experimental evidence. We also provide tight bounds on the rate of dissipation as the mesh of the strategy space decreases. Up to this point we have not addressed the solution to the infinite rent seeking game, i.e. when the strategy space is continuous and R 1 2. It tums out the payoff functions in oquation (1) satisfy the conditions of Theorem 6 in Dasgupta and Maskin (1986), guaranteeing the existence of a symmetric mixed strategy oquilibrium for the rent seeking game with a continuous stntegy space. The proof of their thoorem relies on finite approximation of the game and then letting the grid size become finer and finer, as we did in our example with an infinite R. Thus the construction of the equilibrium to the finite game in the previous section is driven to the limit. Under sufficient regularity conditions this method indeed yields a solution to the infinite game. The application of Dasgupta and Maskin's Theorem 6 requires four conditions, each of which is satisfied for the Tullock game with a continuous strategy space. In particular, this thoorem re~uires: ( i) The sum of the payoffs must be upper semi~ontinuous. From equations (1) and (2) we easily see that U,(x ~ y) t Uz(y ~ x) ~ Q- x- y, which is continuous and therefore upper semi~ontinuous as well. ( ii) The subset of discontinuities in the payoffs must be of a dimension lower than 2, and one must be able to express the elements of this subset as functions ~ 22 which relatc the strategy of one player to the strategy of the olher. For the Tullock game with R c oo, this oondition is simple to check, as x a y~ 0 cx)nstitutes the only point of discontinuity. Th~ condition guarantees that the discontinuities are relatively unimportant (have measure zero). (iii) The payoff U,(x ~ y) must be boundod. This holds evidently as -Q 5 U,(x ~ y) 5 Q on [0, Q]. (iv) Finally, U,(x ~ y) must be weakly lower semi-continuous. The only point where there could arise a problem is at the point of disoontinuity, but as U,(x ~ y- 0) is lower semi~ontinuous, it is oertainly weakly lower semi-oontinuous. This last condition guarantees that, loosely spe.alvng, a player daes not want to put weight on the discontinuity point even if the other player does, bccause payoffs may jump down but do not jump up. Thus we conclude that a symmetric mixed strategy oquilibrium exists for the continuous strategy space rent sceking game for all R~ 2 as well. An explicit closed form solution remains for future invatigation. For the special case R- oo, a full characteriration of all the equilibria is available even when there are more than two players; sce Baye et al. (1990, 1993). Other interesting questions include the explicit solurion to asymmetric versions of the game, as well as further ezperimental work along the lines suggestod above. These remain the focus of our future research. 23 APPENDIX In this Append'u we show that the Ieft-hand-side of inequality (20) is posidve, while the right-hand-side is negative. Manipulate the right-hand-side as follows: v.l ~ x - 2(v.x) ~~1~ ~ l~41Vt 1 MN xR ~ 0 i (v.l ;xlr a j (v- l .x)(~r.l ~ a 1 a 2 ~ ViX-1 C (1 . ~ ~. X Note that the left-hand-side of this last inequality is decreasing in v. Hence, to show that the right-hand-side of (20) is negative, it is sufficient to show that such is the case for v a 1. Assuming that v~ 1, we can further manipulate the last inoquality: 1 t l;X j(1,~~-~ a 1; 1~ x ~(1 { z)(1 a X~ 2. Evidently, for any R Z 2 and x 2 1 1 4 1 ~ 1 ~ X 1~ 1. X Thus the right-hand-side is strictly negative for any v~ 1. To obtain the left-hand-side result we need to show that for any t such that Q-12t2xf1, 24 ~R ~a]~ ~~1~ 2 Z 4~41~` ; tR 4~~1~` ~ TR xR ~ tx~ Manipulation yields [~~1~ t ~~MR ~ t~ ~ ~~~~1~ 4 tR~ FI xR~~lyr ; tR~tlYr ' ~u ; ~rtR j ~R ~al~ t ~Rte M [(x~l~ - xx][t" - sxj j 0. Because t Z x t l~ x, the left-hand-side of this lau inequality is unequivocally positive, and hcnce the left-hand-side of (20) is non-negative. 25 REFERENCFS Allard, R. J., "Rent-seeking with tan-identical players," Public Choice, 1988, 3-14. Applebaum, E. and E. Katz, "Ttansfer seeking and avoidanca: On the full social cosu of rent seeking," Public Choice, 1986, 175-181. Baye, M., D. Kovenock and C. G. De Vries, "Rigging the Lobbing Process: An Application of the All-Pay Auction,' ',(March 1993), pp. 289-294. Baye, M., D. Kovenock and C. G. De Vries, "The all-pay suction with complete information," CentER discussion paper 9051, 1990. Baye, M., G. Tian, and J. 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Pischke Individual Income, Incomplete Information, and Aggregate Consumption . 9239 H. Blcemen A Model of Labour Supply with Job Offer Restrictions 9240 F. Drost and Th. Nijman Temporal Aggregation of GARCH Processes 9241 R. Gilles, P. Ruys and J. Shou Coalition Fortnation in Large Network Economies 9242 P. Kort The Effects of Maricetable Pollution Permits on the Firm's Optimal Investment Policies 9243 A.L. Bovenberg and F. van der Plceg Environmental Policy, Public Finance and the Labour Marlcet in a Second-Best World 9244 W.G. Gale and 1.K. Scholz IRAs and Household Saving 9245 A. Bera and P. Ng Robust Tests for Heteroskedasticity and Autocorrelation Using Scon: Function 9246 R.T. Baillie, C.F. Chung and M.A. Tieslau The Long Memory and Variability of Inflation: A Reappraisal of the Friedman Hypothesis 9247 M.A. Tieslau, P. Schmidt and R.T. Baillie A Generalized Method of Moments Estimator for Memory Processes Long- No. Aethor(a) Titk 9248 K. WBmeryd Partisanship as Information 9249 H. Huizinga The Welfare Effects of Individual Retirement Accounts 9250 H.G. Blcemen Job Search Theory, Labour Supply and Unemployment Duration 9251 S. Eijffinger and Central Bank Independence: Searching for the Philosophers' Stone 9252 A.L. Bovenberg and R.A. de Mooij Environmental Taxation and Labor-Market Distortions 9253 A. Lusardi Permanent Income, Current Income and Consumption: Evidence from Panel Data 9254 R. Beetsma Imperfect Coedibility of the Band and Risk Premia in the European Monetary System 9301 N. Kahana and S. Nitran Cn~dibility and Duration of Political Contests and the Extent of Rent Dissipation 9302 W. GOth and S. Nitzan Are Moral Objections to Free Riding Evolutionarily Stable7 9303 D. Karotkin and S. Nitzan Some Peculiarities of Group Decision Making in Teams 9304 A. Lusardi Euler Equations in Micro Data: Merging Data from Two Samples 9305 W. GOth A Simple Justification of Quantity Competition and the CoumotOligopoly Solution 9106 R S. G. A. T'hr t'onsisirncy Principle For Gmnes 9307 E. Schaling Pclrt; and I íjs Imbens and Lancaster in Strnte},ic Fnrm Case Control Studies with Contaminated Controls 9308 l'. Ellingsen and K. W3meryd Foreign Direct Investment and the Political Economy of Protection 9309 H. Bester Price Commitment in Search Markets 9310 T. Callan and A. van Scest Female Labour Supply in Farm Households: Farm and Off-Farm Participation 931 I M. Pr~dhan and A. van Scest Formal and Informal Sector Employment in Urban Areas of Bolivia 9312 Th. Nijman and E. Sentana Marginalization and Contempotaneous Aggregation in Multivariate GARCH Processes 9313 K. WBmeryd Communication, Complexity, and Evolutionary Stability No. Aatóor(s) Titk 9314 O.P.Attanasio and M. Browning Consumption over the Life Cycle and over the Business Cycle 9315 F. C. Drost and B. J. M. Werlcer A Note on Robinson's Test of Independence 9316 I I. I Iamer:, P. Bomi and S. Tijs On Games Corresponding to Sequencing Situations with Ready 7'imcs 9317 W. GUth On Ultimatum Bargaining Experiments - A Personal Review 9318 M.J.G. van Eijs On the Determination of the Control Parameters of the Optimal Can-order Policy 9319 S. Hurkens Multi-sided Pre-play Communication by Burning Money 9320 J.J.G. Lemmen and S.C.W. Eijffinger The Quantity Approach to Financial Integration: The Feldstein-Horioka Criterion Revisited 9321 A.L. Bovenberg and S. Smulders Environmental Quality and Pollution-saving Technological Change in a Two-sector Endogenous Growth Model 9322 K.-E. Wifineryd The Will to Save Money: an Essay on Economic Psychology 9323 D. Talman, The (2"~" - 2}Ray Algorithm: A New Variable Dimension Simplicial Algorithm For Computing Economic F.quilibria on S" x R" 9324 H. Huizinga The Financing Abroad 9325 S.C.W. Eijffinger and E. Schaling Central Bank Independence: Theory and Evidence Y. Yamamoto and '!.. Yang and Taxation of U.S. Direct Investment 9326 T.C. To Infant Industry Protection with Learning-by-Doing 9327 J.P.J.F. Scheepens Bankruptcy Litigation and Optimal Debt Contracts 9328 T.C. To Tariffs, Rent Extraction and Manipulation of Competition 9329 F. de Jong, T. Nijman and A. RtSell A Comparison of the Cost of Trading French Shares on the Paris Bourse and on SEAQ Intemational 9330 H. Huizinga The Welfare Effecis of Individual Retirement Accounts 9i31 IL Fluizinga Time Prcfercnce and Intemational "fax Competition 9332 V. Feltkamp, A. Koster, A. van den Nouweland, P. Borrn and S. Tijs Linear Production with Transport of Products, Resources and Technology No. Autóor(s) Titk 9333 B. Lauterbach and U. Ben-Zion Panic Behavior and the Performance of Circuit Breakers: Empirical Evidence 9334 B. Melenberg and A. van Soest Semi-panunetric Estimation of the Sample Selection Model 9335 A.L. Bovenberg and F. van der Plceg Grcen Policies and Public Finance in a Small Open Economy 9336 E. Schaling On the Economic Independence of the Central Bank and the Persistence of Inflation 9337 G.-1.Otten Characterizations of a Game Theoretical Cost Allocation Method 9338 M. Gradstein Provision of Public Goods With Incomplete Decentralization vs. Central Planning Information: 9339 W. GOth and H. Kliemt Competition or Co-operation 9340 T.C. To Export Subsidies and Oligopoly with Switching Costs 9341 A. DemirgOg-Kunt and H. Huizinga Barriers to Portfolio Investments in Emerging Stock Markets 9342 G.J. Almekinders Theories on the Scope for Foreign Exchange Market Intervention 9343 E.R. van Dam and W.H. Haemers Eigenvalues and the Diameter of Graphs 9344 H. Carlsson and S. Dasgupta Noise-Proof Equilibria in Signaling Games 9345 F. van der Plceg and A.L. Bovenberg Environmental Policy, Public Goods and the Marginal Cost of Public Funds 9346 J.P.C. Blanc and R.D. van der Mei The Power-series Algorithm Applied to Polling Systems with a Dortnant Server 9347 J.P.C. Blanc Perfortnance Analysis and series Algorithm Optimization with the Power- 9348 R.M.W.J. Beetsma and F. van der Plceg Intramarginal Interventions, Bands and the Pattem of EMS Exchange Rate Distributions 9349 A. Simonovits Intercohort Heterogeneity and Optimal Social Insurance Systems 9350 R.C. Douven and J.C. Engwerda Is There Room for Convergence in the E.C.? 9351 F. Vella and M. Verbeek Estimating and Interpreting Models with Endogenous Treatment Effects: The Relationship Between Competing Estimators of the Union lmpact on Wages No. Aatóor{s) Titk 9352 C. Meghir and G. Weber Intertemporal Non-separability or Borrowing Restrictions? A Disaggregate Analysis Using the US CEX Panel 9353 V. Feltkamp Altemative Axiomatic Characterizations of the Shapley and Ban7haf Values 9354 R.J. de Groof and M.A. van Tuijl Aspects of Goods Maricet Integration. A Two-Country-Two -Sector Analysis 9355 Z. Yang A Simplicial Algorithm for Computing Robust Stationary Points of a Continuous Function on the Unit Simplex 9356 E. van Damme and S. Hurkens Commitment Robust Equilibria and Endogenous Timing 9357 W. GUth and B. Peleg On Ring Formation In Auctions 9358 V. Bhaskar Neutral Stability ln Asymmetric Evolutionary Games 9359 F. Vella and M. Verbeek Estimating and Testing Simultaneous Equation Panel Models with Censored Endogenous Variables Data 93tí0 W.B. vsn den Hout and J.P.C. Blanc The Power-Series Algorithm Extended to the BMAP~PHII Queue 9361 R. Heuts and 1. de Klein An (s,q) Inventory Model with Stochastic and Interrelated Lead Times 9362 K.-E. Wámeryd A Closer Look at Economic Psychology 9363 P.J: J. Herings On the Connectedness of the Set of Constrained Equilibria 9364 P.J- J. Herings A Note on "Macroeconomic Policy in a Two-Party System as a Repeated Game" 9365 F. van der Plceg and A. L. Bovenberg Direct Crowding Out, Optimal Taxation and Pollution Abatement 93tí6 M. Pradhan Sector PaRicipation in Labour Supply Models: Preferences or Rationing7 9367 H.G. Bloemen and A. Kapteyn The Estimation of Utility Consistent Labor Supply Models by Means of Simulated Scores 9368 M.R. Baye, D. Kovenock and C.G. de Vries The Solution to the Tullock Rent-Seeking Game When R~ 2: Mixed-Strategy Equilibria and Mean Dissipation Rates pn Qnv nn~~~ ~nnn i G Tii RI IR(, Bibliotheek K. U. Brabant TNF t~FTNFRLAND; II IIII VInItlI MININII1AIW l I ~ 7 000 01 1 33579 2