some results on rent-seeking contests with shortlisting

Some Results on Rent-Seeking Contests with ShortlistingAuthor(s): J. Atsu AmegashieSource: Public Choice, Vol. 105, No. 3/4 (2000), pp. 245-253Published by: SpringerStable URL: http://www.jstor.org/stable/30026394 .

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Public Choice 105: 245-253, 2000. © 2000 KluwerAcademic Publishers. Printed in the Netherlands.

245

Some results on rent-seeking contests with shortlisting *

J. ATSU AMEGASHIE Department of Economics, Simon Fraser University, Burnaby, British Columbia, Canada

V5A 1S6, E-mail: [email protected]

Accepted 19 April 1999

Abstract. We show that rent-seeking contests with shortlisting have comparative static properties which depend on the stage of the contest. By generalising the argument in Amegashie (1997), we show that well-known comparative static results in single-stage rent-seeking contests do not necessarily carry over to contests with shortlisting. The analysis suggests that policy prescriptions for reducing rent-seeking expenditures in contests with shortlisting cannot be given without taking into account the stage of the contest. We also extend the model in Amegashie (1997) to the case of unequal class sizes. Finally, the shortlisting method in Amegashie (1997) is compared with that in Clark and Riis (1996).

1. Introduction

Since Tullock (1980) presented his seminal rent-seeking game, there has been an enormous amount of research into the determinants of rent-seeking expenditures. It is now known, among others, that the number of winners (Amegashie, 1997; Clark and Riis, 1996), the attitude of rent-seekers to risk (Hillman and Katz, 1984; Konrad and Schlesinger, 1997), the number of rent-seekers and the rent-giver's sensitivity to seeking expenditures (Tullock, 1980), the number of rent-givers (Congleton, 1984; Amegashie, 1997) affect rent-seeking expenditures.1

In general, we have a fair idea of how changes in these variables affect rent-seeking expenditures. For example, we know, subject to some excep- tions, that an increase in the degree of risk aversion of the seekers or the number of winners will decrease aggregate rent-seeking expenditures, and an increase in the number of rent-seekers, and the sensitivity of the rent-givers or the returns to rent-seeking will increase aggregate rent-seeking expenditures. I shall refer to these results as comparative statics results. There are several of such comparative statics results in the literature. In this paper, I argue that even if these comparative statics results are correct, they may not hold

* Sections 3 and 4 of this paper are based on my MDE thesis at Dalhousie University, Canada. My thanks are due to the members of my thesis committee, Melvin Cross, Iraj Fooladi, and Kuan Xu for comments.



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in contests with shortlisting.2 Given that contests with shortlisting are very common,3 this implies that we may know very little about the determinants of rent-seeking expenditures. Indeed, whether or not, these comparative statics results hold in contests with shortlisting depends on the stage of the contest. This was shown in Amegashie (1997). In this paper, I argue that this result is more general than the discussion in Amegashie (1997) may suggest.

In the next section, I present the model in Amegashie (1997) and show that contests with shortlisting have indeterminate comparative statics results depending on the stage of the contest. Section 3 examines the model in Amegashie (1997) with unequal group sizes and Section 4 compares the model with the shortlisting method in Clark and Riis (1996). Section 5 concludes the paper.

2. A rent-seeking contest with shortlisting

In Amegashie (1997), the following model is presented. F > 1 contestants are to be shortlisted from N(> F) risk-neutral contestants to compete in a final rent-seeking contest. The contestants are divided into F groups each with N/F players,4 where there is a preliminary competition for the members of each group. A contestant is then shortlisted from each group to compete in a final contest. Using Tullock's (1980) probability function, let a > 0 and B > 0 be the parameters which capture the sensitivity of the admin- istrators of the preliminary and final competitions respectively. Let xil and

xi2 be the expenditure of the i-th seeker in the preliminary stage and in the final stage (given that he is shortlisted), respectively. The probability of being shortlisted is given by pil(a, = (xil)a/[(xil) + (xjl)], where xjl is the expenditure of j-th contestant among the (N/F-1) contestants in the i-th player's group in the preliminary contest. The probability of winning the prize given that the i-th contestant is shortlisted is given by Pi2(B,xi2,xj2) = (Xi2) + 1j= where Xj2 is the expenditure of the j-th contestant among the other (F-l) contestants in the final contest. Let V be each seeker's common valuation of the prize. We look for a subgame perfect pure-strategy (Nash) equilibrium of this game by backwards induc- tion. Given F contestants in the final stage and complete information, the expenditure by each contestant, in a symmetric Cournot-Nash equilibrium, is x2 = [f(F which implies that aggregate rent-seeking expenditure in the final stage is T2 = Fx2 = [fl(F-1)/F]V The equilibrium expected payoff is r2 = (1/F)V-x2 = [F-B(F-1)]/F2 where we assume that /3 < F/(F-1) to ensure that i2 > 0.

In the first stage, the i-th contestant chooses xil, given the outlay of the other (N/F - 1) contestants in his group, to maximise



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it = Pi()n2 (1)

In a symmetric Cournot-Nash equilibrium, this gives xl = a[(N/F 1)/(N/F)2]r2. Aggregate rent-seeking expenditure in the first stage is given by

T1 = Nx1 = a[F(N-F)/N]n2 (2)

The expected payoff in this subgame perfect equilibrium is n1 =

(F/N)r2 - xl, which is non-negative if a _ N/(N-F). The grand rent- seeking expenditure is T = T1 + T2. It follows that TlI/8a > 0 and 8T2/8/ < 0. To quote from Amegashie (1997), this implies that "...contrary to the standard result in the rent-seeking literature, an increase in the rent-giver's sensitivity to rent-seeking expenditures leads to a fall in aggregate rent- seeking expenditures. This result is due to the dynamic nature of this game. While an increase in B leads to increase in rent-seeking expenditures in the final competition, this conversely leads to a fall in rent-seeking expenditures in the preliminary stage (holding a fixed), because the increase in expenditures in the final stage reduces the expected payoff, T2, for any given number of shortlisted contestants."5 Having stated the intuition, it is obvious that in another model with a different method of shortlisting the relative magnitude of the opposing effects on rent-seeking expenditures in the preliminary and final contests, due to a change in B, may be indeterminate.

It must be noted that it is not only the returns parameter which may have an indeterminate or different comparative static effect in a contest with shortlisting, given that it has a determinate effect in a single-stage contest. This result is more general. Suppose we know that an increase in say D, where D may be the number of winners, results in a decrease in the aggregate rent-seeking expenditures in a single-stage contest. Then in a contest with shortlisting, an increase in D in the final stage will reduce aggregate rent- seeking expenditures in that stage and hence increase, r2, the contestants' expected payoff in that stage6 (which is their valuation, in the preliminary stage, of being shortlisted). This increase in r2 is likely to increase aggregate rent-seeking expenditures in the preliminary stage leading to an indeterminate effect of the increase in D on aggregate rent-seeking expenditures. However, if the increase in D were to occur only in the first stage, then the determinate comparative static result in the single-stage contest will carry over to the contest with shortlisting. This is because what happens in the first stage has no dynamic effects, since the dynamic effect of any change in a given stage is felt in previous stages (but not in subsequent stages). It follows that any policy prescription, based on comparative static results known in single-stage



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contests, may be lead to different results if applied at any stage, other than the first stage, of a contest with shortlisting.

Some more examples of this result are (i) increasing the number of winners may have no effect on aggregate rent-seeking expenditures in a contest with shortlisting, although it reduces aggregate rent-seeking expenditures in a contest with no shortlisting (see Amegashie, 1997) and (ii) reducing the number of contestants in a contest with no shortlisting leads to a reduction in aggregate rent-seeking expenditures but may not necessarily be the case in a contest with shortlisting. For example, reducing the number of contestants in the second stage has an indeterminate effect. That is, 8T/8F = -[tF2(1 + fN(a-1)]V/NF2 has an indeterminate sign (Amegashie, 1997).

It is easy to formalize this argument. We do this by noting that

8T/D = (8T1//2r)( + aT2/aD. (3)

We can rewrite (3) as

BT/8D = (8T2/8D)[(BT1/8T2) + 1], (4)

where (aT1/ar2)(ar2/aT2) = (aT1/aT2). Thus 8T1/BT2 < 0, given aT1/a82 > 0 and 8ar2/aT2 < 0. From (4), it follows that the sign of 8T/8D is not the same as the sign of 8T2/aD, if 8T1/8T2 + 1 < 0. That is, the direction of the change in rent-seeking expenditures in stage 2 will not be the same as the direction of the change in the overall aggregate rent-seeking expenditures, if the magnitude of the change in the stage 2 rent-seeking expenditures is not greater than the magnitude of the (opposite) change in stage 1 rent-seeking expenditures.

Note that our argument does not depend on the specifics of the game; for example, a Tullock probability function, identical contestants, a Cournot conjectural variation, complete information, and the method of shortlisting the contestants. What drives the argument is the dynamic nature of contests with shortlisting which accounts for the fact that what happens in one stage may have different effects in different stages.

3. The shortlisting method in Amegashie (1997) with unequal group sizes

In Amegashie (1997), the equilibrium probability of being shortlisted from each group, F/N, is the same across the groups. This is because all the groups are equal in size (i.e., N/F). In this section, I examine the case of unequal group sizes resulting in unequal equilibrium probabilities of being shortlisted.



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Specifically, I shall show that this design may result in lower rent-seeking expenditures, given identical contestants.' I need to show that the unequal- group size design gives lower rent-seeking expenditures in stage 1 than the equal-group size design for any number of shortlisted contestants. Note that in stage 2, rent-seeking expenditures would be the same for both designs since we assume that the aggregate number of shortlisted contestants, F, is the same under both designs. Suppose there are F exogenous groups with ng contestants and a contestant is shortlisted from the g-th group, where g = 1, 2, ..., F and N = CFg=1ng.,

As before, the i-th contestant in group g in stage 1, will choose xf1, given the outlay of the other (ng - 1) contestants in his group to maximise

ri = Plrt2 - Xgl, (5)

where pi is the probability that in stage 1, the i-th contestant in the g-th group will be shortlisted. As before, using Tullock's probability function, the optimal value of xl =

~x2. Aggregate rent-seeking expenditures in stage 1 is

TI' =

ngxg

= an2 (ng

(6)

Note that T1 is a special case of TI' with p -- 1/ng = F/N. In this equilibrium, the expected profit is

g = p - xg = pg[1 (7)

where 8rpg/apg > 0. Equation (6) can be simplified to obtain

T1' = a[F-J]n2, (8) where J - Fg It follows that Ti' is maximised when J is minimised.

It is easy to show that J is minimised when the group sizes are equal, given N = ng.8 Thus any unequal group size design results in lower stage 1 rent-seeking expenditures than rent-seeking expenditures when the group sizes are all equal.

The above result is consistent with the finding by Hillman and Riley (1989) and Katz and Tokatlidu (1996) that asymmetry reduces rent-seeking expenditures (see also Nitzan, 1994). However, in our model, this result is a necessary but not a sufficient condition (for lower rent-seeking expenditures) because the unequal-group size design results in an additional rent-seeking competition. The various contestants would lobby in order to be in groups where the probability of being shortlisted is higher. This is because the higher



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is the probability of being shortlisted from a particular group, the higher is the expected profit in stage 1 (i.e., a8rr/apf > 0). So the unequal-group size design results in a three-stage game. Thus, the unequal-group size design will result in lower aggregate rent-seeking expenditures, if the aggregate rent-seeking expenditures in this additional stage is sufficiently low.

An implication of the result in this section, is that the assumption that N/F be an integer may not be crucial. This is because if N/F is not an integer, then we would have to resort to unequal group sizes. But this may improve rather than worsen the efficiency properties of this method of shortlisting, since it may result in lower aggregate rent-seeking expenditures.

4. Comparing the shortlisting method in Amegashie (1997) with the sequential shortlisting method in Clark and Riis (1996)

Clark and Riis (1996) examine the following multi-winner rent-seeking contest: k > 1 winners are to be chosen from N (> k) identical players in a rent-seeking contest.9 The players make one rent-seeking contribution. The winners are then determined sequentially. The first winner is determined by using the probability distribution which arises after the rent-seeking outlays of all N players are collected. The first winner is then eliminated and the second winner is determined by using the probability distribution which arises when the outlay of the first winner is excluded. This process continues until all k winners are determined. As noted in Amegashie (1997), any method for selecting multiple winners could be used to shortlist.

Using the Tullock probability function with the returns parameter equal to 1, Clark and Riis (1996) show that aggregate rent-seeking expenditures, when N (> k) risk-neutral and identical contestants with valuation V compete for k identical prizes (which are awarded in the sequential manner described above) are given by

k(N-

V. (9)

It follows that if F > 1 contestants are shortlisted sequentially from N (> F) identical contestants, each with valuation, r2, of being shortlisted, then aggregate rent-seeking expenditures in stage 1 are

Tl1 =

F(N-1)/N-

r2 (10)



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Since the design of the contest is identical in stage 2 under both shortlisting methods, we only need to focus on the stage 1 rent-seeking expenditures to determine the relative magnitude of rent-seeking expenditures under the two shortlisting methods.

From Equations (2) and (10), we get

T11 - TI =

(F/N)(F- r2 (11)

where a = 1. Equation (11) can be rewritten as

T11 - T1 =

Nf-1j=1 r2. (12)

Note that F/N > (F-j)/()N-j) for all j > 1. It follows that T11 - Ti > 0, since the term in square brackets is positive for all j > 1 and K2 > 0.

Note that in this "adapted" Clark and Riis (1996) method, a contestant competes with all other contestants, while in the method in Amegashie (1997), each contestant competes with only those in his group. Examples of this method of shortlisting were given in Amegashie (1997). Some other examples are the following: university students who compete for university (not departmental) scholarships are nominated to the graduate school by different academic departments. Different members of a hiring committee may go to different regions to interview candidates who they may shortlist. Of course, there is less likely to be complete information in these situations than has been assumed in this paper.

Given that the 'adapted' shortlisting method in Clark and Riis (1997) is more thorough, it is not surprising that it results in higher aggregate rent- seeking expenditures. It may, however, have the advantage of picking the most efficient contestant(s) with higher probabilities than the method in Amegashie (1997) in a model with non-identical contestants.

5. Conclusion

The paper has presented some results on rent-seeking contests with shortlisting. We may know a lot about the comparative statics of rent-seeking expenditures in single-stage contests. However, we may know very little in the case of contests with shortlisting. We can, however, reduce the cost of this ignorance by limiting our policy prescriptions to the first stage of such contests. If we wish to go beyond that stage, then we need to move our research from single-stage contests to multi-stage shortlisting contests in order



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to understand how changes in the exogenous variables of such contests affect rent-seeking expenditures in such contests. We cannot depend on the known results in single-stage contests. Indeed, the result in this paper must be taken seriously, if one believes that shortlisting is a common feature of the design of rent-seeking contests.

It has also been shown that the assumption that N/F should be an integer may not be crucial to the efficiency properties of the shortlisting method in Amegashie (1997), since its violation may lead to lower aggregate rent- seeking expenditures. Finally, we show that the sequential shortlisting method in Clark and Riis (1997) may result in higher rent-seeking expenditures than the method in Amegashie (1997).

Notes

1. See also the survey by Nitzan (1994). 2. We may define shortlisting as the practice whereby a group of players in a contest are

chosen in a preliminary competition or in successive preliminary competitions to compete in a final contest, from which the winner(s) is (are) chosen.

3. Amegashie (1997) gives reasons why such contests are used. See also Baye, Kovenock, and de Vries (1993).

4. For simplicity, it is assumed that N/F is an integer. If N/F is not integer, then we have to resort to unequal group sizes. In section 3, It is shown that this may lead to lower rent-seeking expenditures.

5. The quote is essentially taken from Amegashie (1997). However, the word "finalists" has been replaced with "shortlisted contestants".

6. To see this, note that r2 = (V-T2)/F Thus a82/aT2 < 0. We obtain the same result in the case of non-identical contestants, where the i-th contestant has valuation, Vi and / = 1. Then r2i = (Vi - T2')2/Vi (see Ellingsen, 1991). Hence a policy (i.e., a change in any of the model's exogenous variables) which reduces aggregate rent-seeking expenditures, T2', in stage 2 will increase the i-th contestant's expected payoff.

7. Some results in this section have been independently proven in Gradstein and Konrad (1998). They focus on a similar contest design but with more than two stages and an identical value of the returns parameter in all stages.

8. The proof is available on request. 9. Note that while all k winners receive a prize in a multi-winner contest, not all the

shortlisted contestants may receive a prize.

References

Amegashie, J.A. (1997). The design of rent-seeking competitions: Committees, preliminary, and final contests. Typescript. Public Choice forthcoming.

Baye, M., Kovenock, D. and de Vries, C.G. (1993). Rigging the lobbying process: An application of the all-pay auction. American Economic Review 83: 289-294.



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Clark, D.J. and Riis, C. (1996). A multi-winner nested rent-seeking contest. Public Choice 87: 177-184.

Congleton, R.D. (1984). Committees and rent-seeking effort. Journal of Public Economics 25: 197-209.

Ellingsen, T. (1991). Strategic buyers and the social cost of monopoly. American Economic Review 81: 648-657.

Gradstein, M. and Konrad K.A. (1998). Orchestrating rent seeking contests. Unpublished manuscript.

Hillman, A.L. and Katz, E. (1984). Risk-averse rent seekers and the social cost of monopoly power. Economic Journal 94: 104-110.

Hillman, A.L. and Riley, J.G. (1989). Politically contestable rents and transfers. Economics and Politics Spring: 17-39.

Katz, E. and Tokatlidu, J. (1996). Group competition for rents. European Journal of Political Economy 12: 599-607.

Konrad, K.A, and Schlesinger, H. (1997). Risk aversion in rent-seeking and rent-augmenting games. Economic Journal 107: 1671-1683.

Nitzan, S. (1994). Modelling rent-seeking contests. European Journal of Political Economy 10: 41-60.

Tullock, G. (1980). Efficient rent seeking. In J.M. Buchanan, R.D. Tollison and G. Tullock (Eds.), Toward a theory of the rent-seeking society. Texas A&M University Press.



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