Complete rent dissipation when the number of rent seekers is uncertain

Economics Letters 141 (2016) 8–10

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Complete rent dissipation when the number of rent seekers is uncertain Nava Kahana a,b , Doron Klunover a,∗ a

Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel

b

IZA, Bonn, Germany

highlights • A Tullock contest with population uncertainty is studied. • The conditions for symmetric pure strategy equilibrium are presented. • When the existence of the contest is certain, complete dissipation arises.

article

info

Article history: Received 7 October 2015 Received in revised form 19 January 2016 Accepted 29 January 2016

JEL classification: C72 D72 D82

abstract The focal question in the theory of rent seeking is the extent of rent dissipation as the indicator of social loss. In general, rent-seeking models assume that the number of contenders in a rent-seeking contest is known with certainty but, given the hidden nature of rent seeking, participants in rent seeking more usually can be expected to face an uncertain number of rivals. We show that, in a Tullock contest with a stochastic number of participants, complete ex-ante rent dissipation is consistent with a pure-strategy Nash equilibrium. This contrasts with but complements previous results of under-dissipation when the number of rent seekers is uncertain. Our complete-dissipation result expands the range of circumstances consistent with association of an observed rent with social loss. © 2016 Elsevier B.V. All rights reserved.

Keywords: Contests Population uncertainty Rent dissipation

1. Introduction Rent seeking results in social loss because of unproductive use of resources (Tullock, 1967, 1980). The focal question in the theory of rent seeking is the magnitude of rent dissipation as indicator of social loss (Hillman, 2013).1 Complete rent dissipation arises for a general contest-success function (CSF) under risk neutrality with free entry into contests (Hillman and Katz, 1984). There is also complete rent dissipation for any number of contenders when the CSF is of the all-pay auction or ‘‘discriminating’’ type (Hillman and Samet, 1987). With rent seeking usually not directly observed, the theoretical complete-dissipation results provide a foundation

∗

Corresponding author. E-mail address: [email protected] (D. Klunover).

1 Rent dissipation is the ratio between the resources expended in contesting the rent and the value of the prize sought. It is complete when it equals one. http://dx.doi.org/10.1016/j.econlet.2016.01.025 0165-1765/© 2016 Elsevier B.V. All rights reserved.

for associating the value of an observed rent with social loss. The models used to establish conditions of complete rent dissipation have assumed that the number of participants in a contest is known with certainty. More usually, however, we can expect that the number of rent seekers in a contest to be uncertain, in particular given the hidden nature of rent seeking. An uncertain or stochastic number of rent seekers has been shown to result in underdissipation (Myerson and Wärneryd, 2006; Lim and Matros, 2009; Kahana and Klunover, 2015), thereby adding uncertainty about the number of participants (known as ‘‘population uncertainty’’ in the literature) to the various reasons for under-dissipation (Konrad, 2009). We show that uncertainty regarding the number of participants in a contest does not necessarily result in underdissipation. We apply the model of Myerson and Wärneryd (2006) to describe a contest in which the number of contenders is a random variable with a commonly known prior probability distribution. When the presence of a contest is certain, i.e., with certainty there will be at least two contenders, and with a Tullock

N. Kahana, D. Klunover / Economics Letters 141 (2016) 8–10

CSF, we show that, at the maximum value of the return-to-effort parameter r for which a pure Nash equilibrium exists, ex-ante (i.e., before the realization of the number of participants), rent dissipation is complete. It is well known that, in a Tullock contest with the number of contenders known with certainty, complete dissipation is the maximum amount of (ex-ante) dissipation (Baye et al., 1994).2 On the other hand, it has been shown that, in each case for a different distribution for the stochastic number of contenders that, with at least one certain contender (Myerson and Wärneryd, 2006), a binominal distribution of contenders (Lim and Matros, 2009) and a Poisson distribution of contenders (Kahana and Klunover, 2015), in a contest with an uncertain number of contenders, exante dissipation is smaller than dissipation in a contest with a known number of contenders equal to the expected number of contenders in the uncertain contest.3 Therefore, an explanation that separates our complete-dissipation result from these previous results is required. We show that the set of scale parameters r in the Tullock CSF for which there exists a pure-strategy Nash equilibrium when the number of contenders is certain, is a sub-set of the set of r for which there exists a pure-strategy Nash equilibrium when the number of contenders is stochastic with the same expectation. Uncertainty regarding the number of contenders increases a contender’s expected utility and thus leads to a higher value of r for which the expected net-payoff equals zero. In addition, rent dissipation monotonically increases with r regardless of information about the number of contenders.4

9

We relax the assumption that the set of players is common knowledge according to the commonly used population uncertainty notion of Myerson (1998a,b). That is, we assume that contenders cannot be distinguished or labeled and therefore denote a contender’s effort by x and the common effort of his opponents by y. Considering (2), a contender’s expected utility is: EU (x, y) = v

∞ 

π˜ (n)

n=2

f (x) f (x) + (n − 1)f (y)

− x.

The first-order condition for the contender is: ∞  (n − 1)f ′ (x)f (y) ∂ EU =v π˜ (n) − 1 = 0. ∂x (f (x) + (n − 1)f (y))2 n =2

Following Myerson and Wärneryd (2006), we consider a contest with n identical risk neutral contenders who compete for a prize with a common value v , where n ∈ {0, 1, 2, . . .} is a random variable with a commonly known prior probability distribution π (n) ∞ and a finite expected number of contenders µ = π ( n ) n, if it n =0 exists, with π (0) = π (1) = 0 and π (n) ̸= 1 (i.e., the number of contenders is uncertain but known to be at least two). Contenders are drawn from a pool of m potential rent seekers. Given that a contender has been selected to participate in the contest, the Bayesian updated probability distribution for the number of participants in the contest is:

π( ˜ n) =

π (n)n . µ

(1)

Denoting by x = (x1 , . . . , xn ) the rent-seeking effort chosen simultaneously by the n selected contenders, the probability that contender i wins is:

pi (x , n) =

1  ,   n

f (xi ) , n      f (xj ) j =1

 if x = 0,    

otherwise ,

(2)

  

(4)

In equilibrium, (4) must hold for all contenders. With x∗ the common effort, we have: ∞ v f ′ (x∗ )  n−1 ∂ EUi = π˜ (n) 2 − 1 = 0. ∗ ∂ xi f ( x ) n =2 n

(5)

From (4), the second-order condition for a symmetric Nash equilibrium is:

∂ E 2 Ui ∂ x2i = v f (x∗ )

∞ 

π˜ (n)

n=2

(n − 1)nf (x∗ )f ′ ′ (x∗ ) − 2(n − 1)(f ′ (x∗ ))2 n3 f 3 (x∗ )

< 0. 2. The model

(3)

(6)

In a case in which the number of contenders is commonly known to be µ and f (x) = xr , rent dissipation cannot exceed the value of the prize v , where, in a pure Nash equilibrium, only for µ r = µ−1 the rent is completely dissipated (Pérez-Castrillo and Verdier, 1992). Lim and Matros (2009) have shown for f (x) = xr and a binomial distribution of rent seekers that, in a pure Nash equilibrium, for a given number of expected contenders, uncertainty regarding the actual number of contenders results in lower ex-ante rent dissipation. This is consistent with Myerson and Wärneryd’s (2006) finding for a CSF as described in (2) (but with f strictly concave, i.e., limited to non-increasing return to effort), and a distribution with infinite potential contenders (or less) when it is known with certainty that there will be at least one contender. Nevertheless, in the following, we show that when it is known with certainty that there will be a contest (that is, the number of participants is with certainty at least two), for the Tullock CSF, i.e., f (xi ) = xri , even with an uncertain number of contenders, exante complete-dissipation can occur. Proposition. In a contest in which f (xi ) = xri : (i) When the number of contenders is stochastic and known to be at least two, a symmetric unique pure Nash equilibrium exists iff 0 < r ≤ 1 1 , while, when the number of contenders is known 1 −E n

to be µ ≥ 2, a symmetric unique pure Nash equilibrium exists iff µ µ 0 < r ≤ µ−1 , where µ−1 < 1 1 .5 1−E n

where f (0) = 0and f (xk ) > 0 for all xk ≥ 0 and n is the number of contenders who participate in the contest. ′

(ii) With a stochastic number of contenders, when n ≥ 2 and r = 1 1 1 = n−1 , the rent is ex-ante completely-dissipated. 1 −E n

E n

Proof of (i). With f (x) = xr and considering (1), the solution to (5) is: 2 There are exceptions of over-dissipation under behavioral assumptions and over-dissipation has also been observed in experiments. See the surveys in Sheremeta (2013, 2015) and Dechenaux et al. (2014). For a particular case, see Baharad and Nitzan (2008). 3 Münster (2006) has studied other aspects of contests with ‘‘population uncertainty’’. 4 Myerson and Wärneryd (2006) were the first to observe that the expected payoff under population uncertainty is strictly greater.

x∗ =

rv n − 1 E . µ n

(7)

5 For a complete characterization of the pure strategy equilibria under certainty, see Pérez-Castrillo and Verdier (1992).

10

N. Kahana, D. Klunover / Economics Letters 141 (2016) 8–10

Substituting f (x∗ ) = (x∗ )r and (1) into (6) and rearranging terms, the second-order condition (6) becomes:

∂ E 2 Ui vr = µ(x∗ )2 ∂ x2i 1 1−E 1n

For r =

=

  1 − 3E

r 1 1 E n− n

1 n

+ 2E

1

 −1+E

n2

1 n



< 0.

(8)

and n ≥ 2 we have:

  1 − 3E 1n + 2E n12 ∂ E 2 Ui 1 Sign − 1−E = Sign n ∂ x2i 1 − E 1n   2 E 2−2n − E 1n = Sign n < 0. 1 − E 1n 1 1−E 1n

That is, for r ≤

∞  π( ˜ n)

n

n =2

− x∗ =

Acknowledgments

and n ≥ 2 the second-order condition for

v rv n − 1 − E ≥ 0. µ µ n

(10)

Alternatively, the condition for a solution to be a global maximum is: r ≤

1 1 − E 1n

.7

(11) µ

Notice that from (11) it follows that µ−1 is the highest r for which a pure symmetric Nash equilibrium exists, when there is common 1 knowledge that the number of contenders is µ. Since g (n) = n− is n strictly concave in n, by Jensen’s inequality it follows that 1 E n−1

>

n

µ . µ−1

1 E (1− 1n )

=



Proof of (ii). Substituting r =

1 1 E n− n

into (7) and multiplying by

µ results in expected aggregate expenditure that equals to the prize,v , i.e., the rent is ex-ante completely-dissipated.   ∂ E2U  may not be negNotice that with π (1) ̸= 0, Sign 2 i  ∂x 1 i

r=

1−E 1 n

ative and thus the assumption that n ≥ 2 is necessary for the above proposition. For example, consider a distribution for which

π(1) = π (ˆn) = 0.5. Then, from (9) it follows that Sign ∂∂Ex2Ui = 2

i

n 1+ 2−ˆ −0.5(1+ 1nˆ )2 nˆ 2 Sign 1− 1 nˆ

which is positive for nˆ > 3.

6 Notice that Sign ∂ E 2 Ui = Sign B, where B = r (1 − 3E 1 + 2E 1 ) − 1 + E 1 and 2 n n n2 ∂ xi

∂B ∂r

= (1 − 3E 1n + 2E n12 ). Thus, if (1 − 3E 1n + 2E n12 ) > 0 then, given that B is negative for r = 1 1 it is also negative for r < 1 1 . While if, (1 − 3E 1n + 2E n12 ) < 0 since 1−E n 1−E n   2 1 ∀n ≥ 2 E n − 1 < 0, it follows that B < 0 ∀ r > 0. Hence if Sign ∂ ∂Ex2Ui  <0 i r= 1 1 1 −E n   ∂ E2 U then, Sign 2 i  < 0. ∂x i

7 If r >

1 1−E 1n

0
We have expanded the range of circumstances in which the social loss of rent seeking can be associated with the value of a contested rent by showing that uncertainty regarding the number of rent seekers is consistent with complete rent dissipation. We have also shown that ex-ante complete rent dissipation can occur with common knowledge regarding the existence of a contest, which is a weaker assumption than common knowledge regarding the number of contenders in a contest.

(9)

maximization is fulfilled.6 Moreover, for x∗ to be a global maximum, the following condition, which assures a non-negative expected payoff, must hold:

v

3. Conclusions

1 1 −E 1 n

, the symmetric solution to (5) yields a negative expected payoff

and thus, is dominated by a non-bidding strategy. Thus, a symmetric equilibrium in pure strategies does not exist and rather involves mix-strategies (Baye et al., 1994).

We thank Arye Hillman and an anonymous referee for helpful comments. References Baharad, E., Nitzan, S., 2008. Contest efforts in light of behavioural considerations. Econ. J. 118, 2047–2059. Baye, M.R., Kovenock, D., De Vries, C.G., 1994. The solution to the Tullock rentseeking game when R > 2: Mixed strategy equilibria and mean dissipation rates. Public Choice 81, 363–380. Dechenaux, E., Kovenock, D., Sheremeta, R.M., 2014. A survey of experimental research on contests, all-pay auctions and tournaments. Exp. Econ. 1–61. Hillman, A.L., 2013. Rent seeking. In: Reksulak, M., Razzolini, L., Shughart II, W.F. (Eds.), The Elgar Companion to Public Nhoice, secondnd ed.. Edward Elgar, Cheltenham U.K, pp. 307–330. Hillman, A.L., Katz, E., 1984. Risk-averse rent seekers and the social costs of monopoly power. Econ. J. 94, 104–110. Hillman, A.L., Samet, D., 1987. Dissipation of contestable rents by small numbers of contenders. Public Choice 54, 63–82. Kahana, N., Klunover, D., 2015. A note on Poisson contests. Public Choice 165, 97–102. Konrad, K.A., 2009. Strategy and Dynamics in Contests. Oxford University Press, Oxford U.K. Lim, W., Matros, A., 2009. Contests with a stochastic number of players. Games Econom. Behav. 67, 584–597. Münster, J., 2006. Contests with an unknown number of contestants. Public Choice 129, 353–368. Myerson, R.B., 1998a. Extended Poisson games and the Condorcet Jury Theorem. Games Econom. Behav. 25, 111–131. Myerson, R.B., 1998b. Population uncertainty and Poisson games. Internat. J. Game Theory 27, 375–392. Myerson, R.B., Wärneryd, K., 2006. Population uncertainty in contests. Econom. Theory 27, 469–474. Pérez-Castrillo, J.D., Verdier, T., 1992. A general analysis of rent-seeking games. Public Choice 73, 335–350. Sheremeta, R.M., 2013. Overbidding and heterogeneous behavior in contest experiments. J. Econ. Surv. 27, 491–514. Sheremeta, R.M., 2015. Behavioral dimensions of contests. In: Congleton, R.D., Hillman, A.L. (Eds.), A Companion to the Political Economy of Rent Seeking. Edward Elgar, Cheltenham UK, pp. 150–164. Tullock, G., 1967. The welfare costs of tariffs, monopolies, and theft. West. Econ. J. 5, 224–232. Tullock, G., 1980. Efficient rent seeking. In: Buchanan, J.M., Tollison, R.D., Tullock, G. (Eds.), Towards a Theory of the Rent-seeking Society. Texas A&M University Press, College Station, TX, pp. 97–112.