Rent seeking and rent dissipation: A neutrality result

Journal of Public Economics 94 (2010) 1–7

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Journal of Public Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j p u b e

Rent seeking and rent dissipation: A neutrality result José Alcalde a,⁎, Matthias Dahm b a b

IUDESP and Dep. of Economics, University of Alicante, Ctra. San Vicente s/n, E-03071, Alicante, Spain Dep. of Economics University Rovira i Virgili of Tarragona Av. de la Universitat, 1 43204 Reus (Tarragona), Spain

a r t i c l e

i n f o

Article history: Received 16 September 2008 Received in revised form 3 November 2009 Accepted 12 November 2009 Available online 16 December 2009 JEL classification: C72 (Noncooperative Games) D72 (Economic Models of Political Processes: Rent Seeking, Elections) D44 (Auctions)

a b s t r a c t We consider rent seeking contests between at least two agents who might value the prize differently. We capture a wide range of institutional aspects of contests by analyzing a class of contest success functions fulfilling several properties. The main properties are anonymity and a condition on the elasticity of a rent seeker's win probability with respect to her effort. We show the existence of a mixed-strategy equilibrium and establish equilibrium payoffs. In this equilibrium complete rent dissipation holds. Our results imply a partial robustness result for the all-pay auction. © 2009 Elsevier B.V. All rights reserved.

Keywords: (Non-)deterministic contest All-pay auction Contest success functions

1. Introduction Many economic allocations are determined by contests for a prize. For instance, monopoly rights may be awarded to the interested party that makes the greatest effort in lobbying. Disputed property rights may be assigned through the legal system in response to expenditure in litigation. The outcome of elections may depend on the amount of resources devoted to electoral campaigns. Regulations such as protectionist international trade policies may take into account the interests of the affected industries which lobbied actively prior to the political decision. In such contests the effort exerted is usually irreversible and losers are not generally compensated. Non-market allocations in which resources are dedicated to contest rents rather than to productive activities are commonly referred to as being subject to rent seeking behavior (see Tullock, 1967; Krueger, 1974; Posner, 1975; Tullock, 1975; Bhagwati, 1982; Congleton, 1986). Social losses associated with rent seeking activities are large. Krueger (1974) analyzes import licenses in India and Turkey. She estimates social losses to be 7% of the gross national product for India and 15% of the gross national product for Turkey. Mohammad and Whalley (1984) provide much higher estimates for India. Posner (1975) investigates losses from monopolies in U.S. industries and estimates that for some industries they constitute between 10 ⁎ Corresponding author. E-mail addresses: [email protected] (J. Alcalde), [email protected] (M. Dahm). 0047-2727/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2009.11.005

and 30% of sales.1 Recent estimates by Angelopoulos et al. (2009) indicate that in Europe the social cost of rent seeking for privileges provided by governments, including income transfers, subsidies, and preferential tax treatment, account for 7% of gross national product. In the literature, there are two prominent ways to represent the institutional aspects of rent seeking contests. Both are based on the assumption that the probability of obtaining the prize is an increasing function of rent seeking outlays. This relationship between each contestant's probability of winning the prize and the rent seeking efforts of all rent seekers is called a contest success function (CSF, henceforth).2 In the first formulation, following Tullock (1980), expenditures on rent seeking activities are like buying lottery tickets. This can be interpreted as situations in which rent seeking outlays are not the only determinant of the winner of the prize, and consequently a contender with little effort might actually win the contest. In the second formulation, following Hillman and Samet (1987), rent seeking outlays are the only determinant, so that the contestant with the largest outlays wins. Tullock (1980) also included a scale parameter in his lottery model of rent seeking, which determines the return from rent seeking efforts.3 In the sequel, we will refer to this 1 Cowling and Mueller (1978) and Littlechild (1981) provide further estimates for the U.S. and the U.K. 2 We use the words “rent seeker”, “contestant”, “contender”, and “bidder” interchangeably. 3 Subsequently generalizations of the Tullock contest success function have been analyzed, for example in Szidarovszky and Okuguchi (1997). Alternative mathematical structures, such as difference-forms have also been proposed, for example in Che and Gale (2000).

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model as the Tullock Rent Seeking Game (TRSG). The limiting case of the TRSG, when the return from effort approaches infinity, is the formulation considered by Hillman and Samet (1987). This case is also frequently called all-pay auction (APA).4 The core question of the rent seeking literature was to investigate to what extent resources are attracted to these activities. Following Krueger (1974), the studies mentioned above base their measurement on the assumption that the rent is dissipated completely; the resources unproductively used in contesting the rent equal its value. In the TRSG, this presumption tends to be satisfied in the limiting case of the APA. For small returns from rent seeking efforts, this is not true.5 The analysis, however, is not complete. When the scale parameter reaches a threshold value, equilibria in pure strategies no longer exist and the equilibrium is in mixed-strategies. The gap in the analysis of the TRSG between the pure strategy equilibria for low values of the scale parameter and the mixedstrategy equilibria of the limiting case of the APA has been partially closed by Baye et al. (1994). They analyze the intermediate cases with two rent seekers who value the prize equally and show that the complete dissipation presumption holds. For these intermediate cases, however, with more than two rent seekers or when two rent seekers value the prize differently, there is still no prediction.6 Furthermore, most of the existing predictions are narrow in the sense that they assume that the institutional aspects of rent seeking contests are captured by a variation of the Tullock contest success function, which has a basic additive or proportional structure.7 To estimate the extent to which resources are diverted to rent seeking rather than productive activities, it is critical to have theoretical results indicating when the assumption of complete rent dissipation is justified. This paper develops a model of rent seeking contests which is not based on a specific functional form for the contest success function and makes no restriction on the number of rent seekers or on the valuations they have for the prize. Our analysis applies to a wide range of CSFs that can have a mathematical structure which might be very different from Tullock's CSF. Our model also applies, however, to the TRSG for intermediate values of the scale parameter. We identify sufficient conditions on the CSF of a rent seeking contest to guarantee the existence of an equilibrium with properties similar to that of the APA.8 There are two main conditions. The first is anonymity, used to construct an equilibrium for general situations by building on an equilibrium of the symmetric contest with two rent seekers. The second condition applies to contests with two rent seekers, in which we approximate the continuous strategy space by means of a finite grid. The condition says that the elasticity of a rent seeker's win probability with respect to her effort must be sufficiently high when the two rent seekers tie, and lower than the corresponding elasticity when the rent seeker is facing an opponent who exerts a higher effort. We interpret this as relaxing the extreme sensitivity of the APA with respect to effort which implies that the rent seeker exerting the highest effort wins the contest for sure. This paper thus establishes that the complete dissipation presumption holds even with only two rent seekers for a wide range of 4 Given that, in rent seeking contests, the effort of all contenders is usually irreversible; it is more exact to use the qualification all-pay auction without noise (or deterministic all-pay auction) in order to distinguish this case from the one with noise. We use the term all-pay auction for brevity. 5 For instance, under constant returns, exactly half of the rent is dissipated when there are two contenders who value the prize equally. Increases in the number of contestants or in the scale parameter lead to greater rent seeking losses. 6 A consequence of the lack of a unified model and the desire of robustness is that some studies, such as Ellingsen (1991) and Glazer and Konrad (1999), analyze specific questions concerning rent seeking in both the lottery model and the APA. 7 An exception is the study by Che and Gale (2000). 8 Although in many instances the APA admits a unique equilibrium, this is not always the case (see Baye et al., 1996). When there is a multiplicity, there is one equilibrium analogous to the one we focus on. See Section 3 for more details.

rent seeking contests capturing a variety of institutional aspects. This result links the complete dissipation of rents to contests governed by anonymous CSFs which are sufficiently elastic with regard to effort. The application of this result to the TRSG shows that rents might be completely dissipated in rent seeking situations in which the extreme elasticity of the APA might be relaxed to as much as one. This paper also provides equilibrium payoffs for a wide range of contests for which no characterization of equilibrium was previously available. The third contribution is to establish a partial robustness result for the APA. The literature of rent seeking was recently reviewed by Congleton et al. (2008). Konrad (2009) reviews the more general literature on contests. The APA has been analyzed by Hillman and Samet (1987), Hillman and Riley (1989), Baye et al. (1993, 1996), Che and Gale (1998) and Siegel (2009), among others. Studies of the TRSG have been produced by Tullock (1980), Pérez-Castrillo and Verdier (1992), Skaperdas (1996) and Cornes and Hartley (2005), among others. This paper complements Baye et al. (1994) by helping to close the gap in the analysis of both strands of literature. We also follow the approach of Baye et al. (1994), by approximating the continuous strategy space by means of a finite grid. Studies in the contest literature usually specify a particular CSF and analyze equilibrium. Consequently, there are few papers dealing with a general class of CSFs and we are not aware of any carrying out an analysis at our level of generality.9 This paper is organized as follows. The next section introduces the class of contests analyzed in the present paper and defines an auxiliary contest with a finite grid on the bidding space. Section 3 establishes our main result, which in Section 4 we apply to the TRSG. The last section offers some concluding remarks. 2. Contests 2.1. Preliminaries There are n N 1 rent seekers or bidders denoted by B = fB1 ; …; Bi ; …; Bn g. Each contestant has a valuation for the prize, denoted by Vi, and submits a bid bi ∈ ℝ+. Outlays are irreversible. Bidders are risk-neutral, and they bid simultaneously. The valuations are common knowledge and without loss of generality are ordered such that V1 ≥V2 ≥ … ≥Vn N 0. It is assumed that the contest administrator commits to determine the winner through a CSF. This function associates a lottery to each vector of bids b = (b1, …, bn), specifying a probability of getting the prize for each rent seeker. Definition 2.1. [CSF] A contest success function is a mapping n

n

Ψ : ℝþ →Δ

such that for each b ∈ ℝn+, Ψ(b) is in the n − 1 dimensional simplex, i.e. Ψ(b) is such that, for each i, Ψi(b) ≥ 0, and ∑ni = 1Ψi(b) = 1. Throughout this paper we assume that CSFs satisfy the following Monotonicity Property. Definition 2.2. [MP] We state that CSF Ψ satisfies the Monotonicity Property if, for each bidder Bi, and other rent seekers' bids b− i ∈ ℝn+− 1 ∖ {0}, ′

′

Ψi ðbi ; b−i Þ≥Ψi ðbi ; b−i Þ whenever bi ≥bi ; ′

ð2:1Þ ′

Ψj ðbi ; b−i Þ≤Ψj ðbi ; b−i Þ for all j≠i; whenever bi ≥bi ; and

ð2:2Þ

Ψi ðbi ; b−i ÞN 0 only if bi N 0:

ð2:3Þ

9 For example, Szidarovszky and Okuguchi (1997) focus on logit formulations of the CSF with twice differentiable, strictly increasing, and concave effectivity functions. Malueg and Yates (2006) study homogeneous CSFs.

J. Alcalde, M. Dahm / Journal of Public Economics 94 (2010) 1–7

3

Let us observe that [MP] is a natural condition that is satisfied by all the (homogeneous) CSFs studied in the literature. In particular, Eq. (2.1) specifies a weak Monotonicity Property, of each bidder's winning probability, in their own bid; Eq. (2.2) establishes that when a bidder's winning probability increases (resp. decreases), then the winning probability of any other bidder decreases (resp. increases); and Eq. (2.3) states that no bidder has a positive winning probability unless her bid is positive (or all bidders bid zero). Given the CSF Ψ, each rent seeker's expected utility from participating in the contest, when the vector of bids is b, is

(G − 1) / G, m + 1} for all m ∈ ℕ+ are feasible. We refer to this game as the finite contest and indicate an arbitrary element of the grid by x / G, where x ∈ ℕ+. As a starting point for our analysis, we follow Baye et al. (1994) and apply results by Dasgupta and Maskin (1986) to our model. Consider a contest with n bidders and a common value. Let − μ G = ð− μ G1 ; ::; − μ Gn Þ denote an equilibrium to the contest with finite grid G. The next lemma establishes existence of a symmetric mixedstrategy equilibrium to both the continuous and finite contest and relates these equilibria.11

EΠi ðbÞ = Ψi ðbÞVi −bi ;

Lemma 2.3. Consider a contest with common value V and contest success function satisfying (A) and (DS). This contest possesses a symmetric mixed-strategy Nash equilibrium, when the strategy space is finite as well as when it is continuous. Furthermore, the profile − μ = limG→∞ − μ G exists and is a mixed-strategy Nash equilibrium to the continuous contest.

∀Bi ∈B:

ð2:4Þ

We denote a mixed-strategy for player Bi by μi and indicate the associated strategy profile by μ. 2.2. A class of contest success functions Here we describe properties of the class of CSFs analyzed in this paper. The first axiom is Anonymity, a property which establishes that each rent seeker's win probability is independent of her label and depends only on the vector of bids. (A) Anonymity. For any permutation function π of B (i.e., a bijection π : B→B) we have Ψ(π(b)) = π(Ψ(b)) for all b. Note that this axiom also implies that all bidders submitting identical bids must obtain equal probabilities of winning. This paper makes use of the continuity properties of rent seekers' payoff functions by applying the results of Dasgupta and Maskin (1986) to contests. Therefore, a natural requirement is CSF continuity. However, the most commonly used CSFs, such as the APA or the TRSG, are not continuous everywhere.10 To avoid excluding these CSFs, we allow for a weaker form of continuity of the CSF. Given a vector of bids b = (bi, b− i) ∈ ℝn+, let bimax denote the highest bid of a rent seeker other than Bi, i.e. i

bmax = max bj : j≠i

The following property assures that the set of discontinuities of the CSF is ‘small’ and that it ‘pays’ to increase outlays slightly at these points. (DS) Discontinuity Set: Given Bi ∈B, if Ψi is discontinuous at b, then: bi = bimax;

Proof. See Appendix A. □ 2.4. The condition on the elasticity of the CSF We now introduce an important condition on CSFs. For the sake of illustration, consider the example of the TRSG. Given a vector of efforts b and R, a positive parameter measuring returns to scale from effort, in Tullock's specification the probability that bidder Bi wins the contest is given by12 T

Ψi ðbÞ =

R bi n ∑j = 1 bRj

:

ð2:5Þ

It is well known that the APA is a limiting case of the TRSG. It corresponds to the case in which R → ∞ and the CSF is highly sensitive to effort. When several bidders tie at the highest bid, an arbitrarily small additional effort will definitively yield the prize. As the returns to scale parameter decreases, the CSF becomes less and less sensitive to effort. This can be formalized by considering the elasticity of a rent seeker's win probability with regard to her effort, ηi ðbÞ =

∂ΨTi ðbÞ bi = ∂bi ΨTi ðbÞ

(a) and (b) Ψi(b′i, b− i) = 1 for all b′i N bi. For instance, in the popular TRSG the CSF is continuous everywhere except at the degenerate bid vector, when all rent seekers bid zero. Thus, bimax = 0. In such a case, the second part of (DS) simply requires that in order to obtain a strictly positive win probability, contestant Bj who is not Bi, has to participate actively in the contest. In other words, zero outlays by a rent seeker imply that this contestant has no chance of winning the contest. But note that (DS) is general enough to accommodate the APA in which the CSF is continuous everywhere except when two or more contestants tie for the highest bid. 2.3. The continuous and the finite contest This paper follows the approach of Baye et al. (1994) by relating the original contest with continuous strategy space to another one in which there is a finite grid on the bidding space. Note that the latter is realistic when there is a smallest monetary unit, as in experimental settings. Given some G ∈ ℕ+, the contest is finite with grid G and smallest monetary unit 1 / G if the strategy space is discrete, such that only bids that coincide with the grid {m, m + 1 / G, m + 2 / G, …, m + 10 In fact, it is easy to see that homogeneity of degree zero of the CSF implies that if Ψi(b) is continuous at the degenerate bid vector (all rent seekers bid zero), then Ψi(b) must be constant. This fact was pointed out by Corchón (2000).

R

:

R

bi ∑j≠i bRj

+1

When this rent seeker ties with the other bidders at the highest bid, ηi(b) is strictly increasing in R. Hence, we might think of R as specifying to what degree the extreme sensitivity of the APA is relaxed by chance in the assignment of the prize. Consider now a finite rent seeking contest with two active contestants.13 Remember that, in the discrete setting bidding, (x + 1) /G represents a marginal increase of the bid x /G. For the sake of simplicity, let x̄ denote the integer such that  x≤GV1 b x + 1. Consider the elasticity of rent seeker 1's win probability, defined by
x y Ψ1 = η ; G G




x+1 y ; G G


 x y −Ψ1 ;

x

G G

1 G

Ψ1


G

;

x y ; G G

11 From Dasgupta and Maskin's results it also follows that the equilibrium strategy has no atom at zero effort, when this is a point of CSF discontinuity. 12 Eq. (2.5) supposes that there is at least one strictly positive bid. Otherwise, Anonymity and the definition of a CSF imply that the CSF assigns win probability 1 / n to all contestants, as e.g. in Baye et al. (1994). 13 Since we are dealing with anonymous CSFs, there is no loss of generality by assuming that the active rent seekers are B1 and B2. Thus, we abuse notation and denote by (b1, b2) a vector in ℝn such that bi = 0 for all i ≠ 1, 2.

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if Ψ1(x / G, y / G) N 0 and η(x / G, y / G) = ∞ otherwise. The following condition puts bounds on this elasticity when rent seeker 1 ties with the other rent seeker. On one hand, it must be higher than a lower bound and, on the other, lower than the corresponding elasticity when rent seeker 1 is facing an opponent who exerts a greater effort. (E) Elasticity: ∀x∈f0; 1; …; − x g and ∀y∈fx + 1; …; − x + 1g,
x y
x x  x η ; ≥η ; ≥ : G G G G x+1

ðE1; E2Þ

In the sequel we will refer to the first inequality as (E1) and the second as (E2). Notice that in the case of the APA, (E) holds.14 The fact that (E) relaxes the extreme properties of the CSF of the APA is very visible by rewriting (E2) as Ψ1

  x+1 x x+2 ; ≥ : G G 2ðx + 1Þ

ð2:6Þ

Note that the right-hand side (RHS) is a strictly decreasing function with values in (1/2, 1]. Thus, (E2) specifies a minimum win probability that outbidding the opponent by the minimum amount must yield, implying that the CSF must be sufficiently discriminating in favor of the higher bidder. This specifies a lower bound on how much the extreme case of the APA, in which the higher bidder definitely wins the contest, can be relaxed. 3. The main result Our main result identifies a wide range of rent seeking contests with non-deterministic CSFs allowing an equilibrium that exhibits the same properties as an equilibrium of the first-price all-pay auction for the given set of rent seekers and values. We must first define what we mean by an all-pay auction equilibrium (see Hillman and Riley, 1989; Baye et al., 1996). Definition 3.1. In an all-pay auction equilibrium μ⁎, the expected bid of rent seeker B1 is E(μ1⁎) = V2 / 2, and the bid of contestant B2 is E(μ ⁎2) = (V2)2 / (2V1). All other rent seekers abstain from the contest (by bidding zero). Contestant B1's expected equilibrium payoff is EΠ1(μ⁎) = V1 −V2, while for all other contestants, EΠi ≠ 1(μ⁎) = 0. The expected revenue is ER(μ⁎) =V2(V1 + V2) / (2 V1). This is a natural equilibrium to focus on, because it is the unique equilibrium if V2 N V3. When there is a multiplicity of equilibria, there is one equilibrium fulfilling these prescriptions, and in no equilibrium is there a rent seeker whose expected payoff exceeds the one specified in the statement. Moreover, the only case in which there is no revenue equivalence among equilibria is when more than one contestant have the second highest valuation, which is strictly lower than the highest one. See Baye et al. (1996) for more details. We are now in a position to present our main result. Theorem 3.2. Suppose the contest success function satisfies (A), (DS) and (E). Then the contest possesses an all-pay auction equilibrium. We prove this result in Appendix B using several lemmata. In order to illustrate the steps of the proof, consider the following example with particularly simple equilibrium strategies. Example 3.3. Consider the TRSG with R = 2. To show that there is an all-pay auction equilibrium, we proceed in four steps. Firstly, suppose that there are two contestants who have a common value for the prize and notice that the first order conditions

14 Note also that in the ‘continuous’ formulation, (E1) is fulfilled in the above example of the TRSG, where in the case of two bidders, ηi = R / (bRi / bRj + 1).

characterize maximizers of expected utility for each rent seeker.15 This implies the existence of a symmetric equilibrium in the example. Lemma 2.3 established the existence of a symmetric equilibrium to common value contests for more general situations. Secondly, in the example the analysis of the first order conditions also implies that both rent seekers bid half of their common valuation. Therefore the rent is completely dissipated and contestants obtain zero payoffs. Using the auxiliary finite contest, Lemma B.1 shows that the same is true under (A) and (E) in symmetric equilibria of the original contest. Thirdly, building on this symmetric equilibrium, we construct an equilibrium to two-player contests without common value when, say, V1 ≥ V2. Notice that in the example the increase in B1's valuation w.r.t. the symmetric situation does not change the problem of contestant B2. Hence, given b1⁎ = V2 / 2, her best reply is still to bid b ⁎2 = V2 / 2, on one hand, or on the other, to bid zero. So she is also willing to mix between the two. If she mixes with the right frequency, then the maximization problem of contestant B1 allows for the same solution as in the symmetric game. What follows is therefore an equilibrium to the asymmetrical contest. Rent seeker B1 bids the optimal strategy of the symmetric game b1⁎ = V2/2 and contestant B2 abstains with probability (1 − V2 / V1), and bids b2⁎ = V2 / 2 whenever she participates. Lemma B.2 establishes such a result for any symmetric equilibrium in which the rent is completely dissipated. The last step is to observe that further rent seekers with valuations lower than V2 cannot do better than B2. Given the specified bids of the first two contestants, they prefer to abstain from the contest. Thus, the described strategies constitute an all-pay auction equilibrium in mixed-strategies to the n-player TRSG with R = 2 and asymmetric valuations. 4. Tullock's rent seeking game In this section we apply Theorem 3.2 to the TRSG, mainly by checking conditions (E1) and (E2). This shows the practical applicability of these conditions.16 Condition (E1) is automatically fulfilled, while (E2) puts a bound on how much one can move away from the APA by lowering R. This bound is reached when the elasticity of the win probability in a twobidder contest at a tie reaches one. Tullock's CSF is defined as in Eq. (2.5). Using homogeneity of the CSF, conditions (E1) and (E2) simplify. The former becomes R

R

R

R

R

R

R

ðx + y Þððx + 1Þ + x Þ≥ððx + 1Þ + y Þ2x ;

ð4:1Þ

which is true for all R ≥ 0. The latter condition (E2) can be written as x+2 ðx + 1ÞR R R ≤ ⇔ðx + 2Þx ≤xðx + 1Þ ; 2ðx + 1Þ ðx + 1ÞR + xR

ð4:2Þ

which is fulfilled for x = 0. For x N 0, (following Baye et al., 1994, p. 379) we obtain      x+2 x + 1 R−1 1 1 1 R−2 ≤ ≤ 1+ 1+ ⇔1 + : x+1 x x+1 x x This holds for R ≥ 2. We have proved the following result. Proposition 4.1. For R ≥ 2, Tullock's Rent Seeking Game has an allpay auction equilibrium.

15 See e.g. Pérez-Castrillo and Verdier (1992) or Nti (1999) for a formal analysis of this maximization problem. 16 Further applications to other CSFs, including the Serial CSF (Alcalde and Dahm, 2007), which has a very different mathematical structure from the TRSG, can be found in Alcalde and Dahm (2008).

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Notice that R ≥ 2 is equivalent to the requirement that in a twobidder contest the CSF is not inelastic at a tie, that is, ηi(b1, b2) ≥ 1 as ηi(b1, b2) = R / 2 (see Section 2.4). 5. Discussion The present paper analyzed non-deterministic rent seeking contests and derived sufficient conditions to guarantee the existence of an equilibrium exhibiting the same equilibrium payoffs and total efforts as an equilibrium of the first-price all-pay auction, given a set of rent seekers and valuations for the prize. This result applies to any configuration of valuations and to any number of contestants. The principal conditions are anonymity of the contest success function and a requirement on the elasticity of the win probability with regard to a rent seeker's bid when there is a tie in contests with two rent seekers. This can be interpreted as relaxing the extreme sensitivity of the contest success function of the deterministic all-pay auction, in which the highest bidder always wins with a probability of one. Our result applies to a wide range of contests reflecting a variety of institutional aspects of rent seeking. We thus establish a neutrality result concerning rent dissipation, which is crucial for empirical estimates: In many rent seeking contests in which the assignment of the prize is responsive enough to rent seeking outlays, an equilibrium exists in which rent dissipation is complete. A special case of the class of contests considered is Tullock's Rent Seeking Game for exponents greater than two. For this popular rent seeking game and parameter values it was previously not known what the equilibrium set is, when there are more than two contestants or different valuations. Our result helps to unify models of contests. A common modelling approach is to consider the two polar cases of Tullock's rent seeking game with exponent equal to one and the limiting case of the all-pay auction in which the exponent goes to infinity. For example, Ellingsen (1991) analyzes rent seeking for a monopoly position in both models and asks whether it is socially beneficial when buyers can engage in rent defending activities aimed at regulating the monopoly.17 Ellingsen shows that buyer lobbying is always beneficial in the deterministic allpay auction, whereas in Tullock's rent seeking game with an exponent equal to one, the effect is ambiguous. Broadly speaking, the result in this paper implies that when the rent seeking game is governed by a sufficiently elastic anonymous contest success function, then there is an equilibrium in which buyer lobbying is always beneficial. Our result also establishes a partial robustness result for the all-pay auction. Robustness is important for this model because the deterministic case has important properties — some of which are known to be not fulfilled when the contest is non-deterministic enough, such as in Tullock's Rent Seeking Game with an exponent equal to one (see Che and Gale, 2000 or Fang, 2002). For instance, Baye et al. (1993) use the all-pay auction to analyze the lobbying process and discover the exclusion principle: a politician interested in maximizing lobbying outlays might have an incentive to preclude the rent seekers most valuing the prize from participating in the contest. Since the proof of this result relies on an expression for total lobbying outlays that coincides with the one in our equilibrium, it is immediate that the result holds through in our setting. The robustness result, however, is partial because we do not determine the complete equilibrium set.18 We leave this challenging question for future research. 17 Notice that for the intermediate cases the results in Baye et al. (1994) cannot be applied, as the buyers' valuation differs by the deadweight loss from the one of the sellers. 18 See Che and Gale (2000) for a related result in the context of a linear differenceform CSF, which is not a special case of the class of CSFs considered here. Note that our analysis implies that the class of contests analyzed in the present paper admits multiple equilibria. Lemma 2.3 implies that there is a symmetric equilibrium for three rent seekers, for example, with common value. However, Theorem 3.2 establishes an asymmetric equilibrium. It seems reasonable, therefore, that results similar to those found in Baye et al. (1996) could be derived.

5

Acknowledgements A previous version of this paper circulated under the title “All-Pay Auction Equilibria in Contests.” We wish to thank Amihai Glazer, Carolina Manzano and Stergios Skaperdas as well as seminar participants at Barcelona Jocs and the University of California, Irvine for useful comments. The detailed suggestions by the editor Kai A. Konrad and two anonymous referees substantially improved the paper. Any remaining errors are ours. This work is partially supported by the Institut Valencià d'Investigacions Econòmiques. Alcalde acknowledges support by FEDER and the Spanish Ministerio de Educación y Ciencia under project SEJ2007-62656/ECON. Dahm acknowledges the support of the Barcelona GSE, the Government of Catalonia, the Departament d'Universitats, Recerca i Societat de la Informació (Generalitat de Catalunya), project 2005SGR00949, and the Spanish Ministerio de Educación y Ciencia, project SEJ200767580-C02-01. A. Appendix A. Proof of Lemma 2.3 Note that the existence of a common value and (A) imply that both the finite and the continuous contest are symmetric games. Therefore, the existence of a symmetric equilibrium for the contest with finite grid G follows from Lemma 6 in Dasgupta and Maskin (1986). We show that the conditions of their Theorem 6 are also satisfied. This theorem guarantees the existence of a symmetric mixed-strategy equilibrium when the strategy space is continuous. In addition, the proof of Dasgupta and Maskin's Theorem 6 shows that the limiting equilibrium of a finite approximation to the strategy space, as the grid size goes to zero, is indeed an equilibrium to the continuous game. The application of their theorem requires some conditions to be fulfilled. First, the sum of payoffs must be upper semi-continuous. Since ∑ni = 1EΠi(b) = V − ∑ni = 1bi is continuous, it is also upper semicontinuous. Secondly, EΠi(b) must be bounded, which holds as −V ≤ EΠi(b) ≤ V for bi ∈ [0, V] and i = 1, 2, …, n. This completes the proof when the CSF is continuous. For discontinuous CSFs fulfilling (DS) two further properties must be fulfilled. Thirdly, one must be able to express a set of points that includes the discontinuities as a function relating the strategies of pairs of rent seekers. Given (DS), the identity function can be used to define this set. Fourthly, a socalled property α must hold. Let t ≥ 1 denote the cardinality of the bid i bmax in b− i. Property α is fulfilled, since lim

inf

bi ↦þ bimax

i

EΠi ðbi ; b−i Þ = V−bmax N

V i i −bmax ≥EΠi ðbmax ; b−i Þ; t+1

holds. Thus, Theorem 6 in Dasgupta and Maskin (1986) can be applied. □ B. Appendix B. Proof of Theorem 3.2 The next lemma, and its proof, follows some of the reasoning in Baye et al. (1994) and generalizes it to a broader class of contests. Lemma B.1. Consider a two-bidder contest with finite grid G and common value V, in which the contest success function satisfies (A) and (E). In any symmetric Nash equilibrium − μ G = ð− μ G1 ; − μ G2 Þ, it is true that for i = 1, 2: 1 μ G Þ≤ and ð1Þ 0≤EΠi ð− G V G ð2Þ Eð− μ Gi Þ = −EΠi ð− μ Þ: 2 Proof. First of all, let us introduce some additional notation. Given G, and rent seeker Bi's strategy μGi , let μGik denote the probability that

6

J. Alcalde, M. Dahm / Journal of Public Economics 94 (2010) 1–7

contestant Bi assigns to bidding k / G. It is easy to see that, at equilibrium, − μ Gik = 0 for any rent seeker Bi and bid such that k N − x. To prove Lemma B.1, we will concentrate on rent seeker B1. A similar reasoning applies to contestant B2. (1) (a) For the lower bound: The expected payoff from bidding x / G when the opponent follows the equilibrium strategy − μ G2 is EΠ1


x G

;− μ2

G



−
x y x x G =V ∑− μ 2y Ψ1 ; − : G G G y=0

ðB:1Þ

Choosing x = 0, contestant B1 can secure herself EΠ1 ð0; − μ G2 Þ≥0. Thus, EΠ1 ð− μ G Þ≥0. (b) For the upper bound, since − μ G is an equilibrium, rent seeker B1 must react optimally to B 2 's strategy. Thus, for all x / G: (i) EΠ1 ðx = G; − μ G2 Þ≤EΠ1 ð− μ G Þ, (ii) EΠ1 ðx = G; − μ G2 Þ = EΠ1 ð− μ G Þ if − μ G1x N 0 G G G − − − and (iii) μ 1x = 0 if EΠ1 ðx = G; μ 2 ÞbEΠ1 ð μ Þ. Using Eq. (B.1), condition (i) can be rewritten as − x

μ 2y Ψ1 V ∑− G

y=0


x y x G ; ≤EΠ1 ð− μ Þ+ : G G G

ðB:2Þ

Let x / G ≥ 0 be the lowest bid that is part of the symmetric mixedstrategy equilibrium.19 By (ii) condition (B.2) holds with equality. Using (A), we have 1 G μ + 2 2x

− x


x y 1h xi G G − = EΠ1 ð− : μ Þ+ μ 2y Ψ1 ; G G V G y=x + 1 ∑

ðB:3Þ

For x + 1 condition (B.2) becomes Ψ1

  x + 1 x −G 1 G ; μ 2x + − μ G G 2 2ðx + G EΠ1 ð− μ Þ x+1 + : ≤ V VG

  x+1 y G − ðB:4Þ ; μ 2y Ψ1 G G y=x + 2 − x

1Þ

+

∑

Computing − μ G2x from Eq. (B.3) and substituting in inequality (B.4) yields      1 x+1 x x x + 1 −G Ψ1 −2Ψ1 ; ; μ 2ðx + 1Þ + 2 G G G G        x+1 x+2 x+1 x x x + 2 −G + Ψ1 ; −2Ψ1 ; Ψ1 ; μ 2ðx + 2Þ + G G G G G G !    − ! x+1 x x+1 x x x − + … + Ψ1 ; ; Ψ1 ; μ G2 − −2Ψ1 x≤ G G G G G G   
1 x+1 x x+1 x G G ≤ μ Þ+ μ Þ+ EΠ1 ð− −2 EΠ1 ð− Ψ1 ; = V G G G G         1 1 x+1 x x+1 x G = μ Þ 2Ψ1 x + 1−2xΨ1 ; −EΠ1 ð− ; −1 : V G G G G G

ðB:5Þ Note that (E1) implies that every term on the left-hand side (LHS) of condition (B.5) is non-negative. Suppose, by way of contradiction, μ G ÞN 1 = G. The RHS of condition (B.5) is strictly smaller than that EΠ1 ð−       1 x+1 x x+1 x x + 1−2xΨ1 ; −2Ψ1 ; +1 : VG G G G G Under (E2), this expression is smaller than zero, which is a contradiction. (2) We have that in a symmetric equilibrium EΠi ð− μ GÞ = VPrfBi winsg−Eð− μ Gi Þ. Summing up for both rent seekers gives 19

G G I.e., − μ 1x N 0, and − μ 1y = 0 for all y b x.

2EΠ1 ð− μ G Þ = V½PrfB1 winsg + PrfB2 winsg−2Eð− μ G1 Þ and rearranging yields the statement. □ Lemma B.2. Let CS be a (continuous) two-bidder contest with common value Ṽ. Let CA be the same contest with asymmetric valuations V1 ≥ V2 = Ṽ. If μ⁎ = (μ1⁎, μ2⁎) is a symmetric (possibly mixed) Nash equilibrium strategy profile to CS, in which the rent is completely dissipated (in expectation), then the following strategy profile υ⁎ = (υ1⁎, υ2⁎) is a Nash equilibrium to CA: • Rent seeker B1 bids υ1⁎ = μ1⁎ and • Rent seeker B2's strategy υ2⁎ is such that she abstains from the contest with probability (1 − V2 / V1) and bids μ2⁎ whenever she participates. Proof. Note first that in CS the complete dissipation of rents implies that Ṽ = E(μ1⁎) + E(μ2⁎). Since the equilibrium is symmetric, we have that E(μ⁎i ) = Ṽ / 2, i ∈ {1, 2}. The symmetry of the game assures that, on average, each player wins half of the times and, thus, in CS we have EΠi(μ⁎) = 0, i ∈ {1, 2}. To see that in CA rent seeker B2 has no profitable deviation from υ2⁎, note that, since υ1⁎ = μ1⁎ and V2 = Ṽ is the same in CS and CA, any pure strategy in CA yields the same as in CS and B2 obtains EΠ2(υ⁎) = 0. She is, hence, willing to abstain with probability (1 − V2 / V1). For B1, note that in CS, given the mixed-strategy μ2⁎ by B2, all pure strategies b1 in the support of μ1⁎ maximize ˜ ⁎ EΠ1 ðb1 ; μ2⁎ Þ = VE½PrfB 1 wins jb1 ; Ψ; μ2 g−b1 ;

ðB:6Þ

where E[Pr{B1 wins |b1, Ψ, μ2⁎}] is B1's expected win probability from the pure strategy b1 when the CSF is Ψ and B2 mixes according to the equilibrium strategy μ2⁎. Note that, although we do not know whether μ2⁎ is a continuous, discrete, or partially continuous and discrete distribution, the following must be true. For any constant A, any b1 which is a maximizer of Eq. (B.6) is also a maximizer of ˜ ⁎ A + VE½PrfB 1 wins jb1 ; Ψ; μ2 g−b1 :

ðB:7Þ

The proof is completed by noticing that Eq. (B.7) with A = (1 − V2 / V1)V1 is the payoff of the pure strategy b1 in CA, with υ2⁎ = μ2⁎ conditional on entry.20 □ We are now in a position to prove Theorem 3.2 Proof of Theorem 3.2. Suppose V1 ≥ V2 ≥ ··· ≥ Vn, and that the CSF satisfies (A), (DS) and (E). We show the existence of an all-pay auction equilibrium by construction. Suppose there were two rent seekers with common value V2. Lemma 2.3 and then Lemma B.1 can be applied. This establishes the existence of a symmetric mixed-strategy equilibrium in which the rent (in expectation) is completely dissipated because both contestants bid (in expectation) V2 / 2. By applying Lemma B.2, it can be concluded that in any two-bidder contest without common value, such as V1 ≥ V2, equilibrium strategies exist for B1 and B2 with the properties specified in Definition 3.1. Assume there are further bidders with valuations lower or equal to V2. These contestants Bj with j N 2 can do no better than to bid zero and obtain expected payoffs of zero. To see this, take any pure strategy b′. Given μ1⁎, rent seeker B2 obtains EΠ2(μ1⁎, b′) ≤ 0 in the two-contestant game. By (A) and V2 ≥ Vj, we have that EΠj(μ1⁎, μ2⁎, b′) ≤ EΠ2(μ1⁎, b′, μ2⁎); and by Condition (2.2), EΠ2(μ1⁎, b′, μ2⁎) ≤ EΠ2(μ1⁎, b′). The expected bids imply the expressions for expected equilibrium payoffs and revenue in the statement of Theorem 3.2. □ 20

This is because Ṽ = V2, and V1 is cancelled out.

J. Alcalde, M. Dahm / Journal of Public Economics 94 (2010) 1–7

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