Mixed Equilibria in Tullock Contests

Any symmetric mixed-strategy equilibrium in a Tullock contest with intermediate values of the decisiveness parameter ("2 < R < ∞") has countably infinitely many mass points. All probability weight is concentrated on those mass points, which have the zero bid as their sole point of accumulation. With contestants randomizing over a non-convex set, there is a cost of being "halfhearted," which is absent from both the lottery contest and the all-pay auction. Numerical bid distributions are generally negatively skewed, and exhibit, for some parameter values, a higher probability of ex-post overdissipation than the all-pay auction.


Introduction
Even after several decades, the game-theoretic analysis of Tullock's (1980) model of a political contest is still incomplete. Indeed, Nash equilibria in either pure or mixed strategies have been described explicitly only for a range of lower values of the decisiveness parameter (Pérez-Castrillo and Verdier, 1992;Nti, 1999), and for the limit case of the all-pay auction (Hillman and Samet, 1987;Hillman and Riley, 1989;Baye et al., 1996), but not so for intermediate values. As has been widely acknowledged, this lack of a game-theoretic prediction is undesirable, in particular because the resulting constraints on the decisiveness parameter do not have a proper economic motivation (Che and Gale, 2000;Konrad and Kovenock, 2009;Schweinzer and Segev, 2012;Szymanski and Valetti, 2003).
For the case of intermediate values of the decisiveness parameter ("2 < R < 1"), in which a pure-strategy Nash equilibrium does not exist, Baye et al. (1994) proved the existence of a symmetric mixed-strategy Nash equilibrium with complete rent dissipation, and subsequently approximated the limit distribution by calculating equilibria of rent-seeking games with …nite strategy spaces. Building on those results, Alcade and Dahm (2010) showed that many contests of intermediate decisiveness allow a mixed-strategy equilibrium that shares important statistics with the equilibrium of the corresponding all-pay auction. 1 However, a more structural understanding of the limit distribution remained elusive. For example, it was not known if the limit distribution is continuous like in the case of the all-pay auction, or a …nite col-lection of mass points as in Che and Gale's (2000) analysis of di¤erence-form contests, or something completely di¤erent. Moreover, numerical calculations based on contests with …nite strategy spaces have tended to o¤er only rather low-resolution images of the limit distribution. 2 The present paper addresses these issues by deriving new structural properties of mixed-strategy Nash equilibria in the rent-seeking game. Speci…cally, it is shown that any symmetric mixed-strategy equilibrium in the Tullock contest of intermediate decisiveness entails countably in…nitely many mass points. Moreover, all probability weight is concentrated on these mass points. Finally, the mass points form a discrete set in the strategy space, and accumulate only at the zero bid, which itself is played with probability zero.
These …ndings are potentially important because they imply that the equilibrium prediction for intermediate values of the decisiveness parameter di¤ers structurally (even though not necessarily statistically) from the tractable cases that have been studied more frequently in the literature.
Indeed, contrasting both the lottery contest and the all-pay auction, the equilibrium bid distribution for intermediate values of the decisiveness parameter has a non-convex support, which is implied by a cost of bidding a strict convex combination of any two distinct positive optimal bids. 3 Figure 1. Equilibrium payo¤ functions in the two-player Tullock contest The main tools for proving the results of this paper are taken from the realm of complex analysis. More speci…cally, one considers a contestant's equilibrium payo¤ function, i.e., the expected payo¤ of a contestant as a function of her own expenditure, assuming that the other contestants adhere to the equilibrium strategy. See Figure 1 for illustration. It is then shown that the equilibrium payo¤ function allows a complex-analytic extension to an open connected subset of C that encompasses the real interval (0; 1). 4 But any non-constant function that is analytic over an open connected subset of the complex numbers has a discrete set of zeros. Since all expenditure levels in the support of a mixed equilibrium strategy necessarily yield the same expected payo¤, this allows deducing that all positive bids used in an equilibrium strategy must be isolated points of the support, which is the key ingredient of the equilibrium analysis. 5 The remainder of the paper is structured as follows. The necessary material from complex analysis is reviewed in Section 2. Section 3 states and proves the discreteness result. The equilibrium characterization can be found in Section 4. Section 5 o¤ers numerical illustrations. Concluding remarks are collected in Section 6. Appendix A describes the numerical approach that has been used to calculate examples of bid distributions.

Background on analytic functions
This section recalls some concepts and results from complex analysis. For further details and proofs, the reader is referred to any textbook on the topic, such as Conway (1978).
Here is the de…nition of a complex-analytic function.
De…nition 2.1. A complex-valued function f is complex-analytic in an open set U C if, at any point z 0 2 U , there is a power series P 1 m=0 m (z z 0 ) m in z around z 0 , with coe¢ cients m 2 C for m = 0; 1; 2; :::, that converges to The following result says that the zeros of a non-constant complex-analytic function on a connected open set necessarily form a discrete set.

Lemma 2.2. If f is a complex-analytic function on an open connected
set U C and if there is a sequence of distinct points z 1 ; z 2 ; ::: in U with z 0 = lim n!1 z n 2 U and such that f (z n ) = 0 for n = 1; 2; :::, then f (z) = 0 for all z 2 U .
The following two standard results in complex analysis serve as the main technical tools to show that the equilibrium payo¤ function in the rent-seeking game allows an analytic extension to some complex domain that contains the interval (0; 1).

Lemma 2.3 (Cauchy' s Theorem) Let f be a complex-analytic function
on the open set U C. Then for every closed recti…able curve that is homotopic to zero in U .

The support of mixed equilibria in the Tullock contest
In the sequel, I will discuss only the simple example of a two-player rentseeking game. 6 Each player i = 1; 2 chooses a level of expenditure x i 0.
For a …xed value of the parameter R 0, player i's payo¤ in the rent-seeking game is given by where j 6 = i, and the ratio is interpreted as 1 2 if the denominator vanishes. Recall the following facts about the equilibrium set of this game. For 0 R 2, there exists a symmetric pure-strategy Nash equilibrium in which each agent invests R 4 . For 2 < R < 1, however, there does not exist a pure-strategy equilibrium. Instead, there exists a symmetric mixed-strategy equilibrium with complete rent dissipation. 7 The following lemma collects those properties of Tullock's "impact function"that are relevant for the discreteness result stated further below.
Proof. Denoting by ln z i the principal value of the complex logarithm, the This proves the …rst claim. The second claim is obvious.
Fix now some parameter value R, and take some symmetric mixedstrategy equilibrium of the rent-seeking game with decisiveness R. The following result provides some information about the support S of .
Theorem 3.2. S \ (0; 1) is discrete, and allows only the zero bid as a potential accumulation point.
7 To de…ne mixed-strategy equilibria, I follow the usual approach according to which mixed strategies correspond to probability distributions on the respective player's space of pure strategies (Dasgupta and Maskin, 1986;Baye et al., 1994;Yang, 1994).
Proof. For " 2 (0; 1) and > 0 small, consider the rectangular domain that is illustrated in Figure 2. It is claimed …rst that, for any " 2 (0; 1), there is some = (") > 0 such that the complex-valued function is well-de…ned and bounded on U ("; ) R + . To prove this claim, take some " 2 (0; 1). On the non-empty compact interval I " = ["; 1 " ], the continuous function h assumes a minimum value M > 0. Moreover, by Lemma 3.1, h allows a complex-analytic extension b h to an open neighborhood H of (0; 1) Exploiting the compactness of I " another time, there are …nitely many x 1 i ; :::; x i 2 I " such that for any x i 2 I " , there is some 0 2 f1; :::; g with jx i x 0 i j < 1 (x 0 i ).
Hence (see again Figure 2), there exists a > 0 such that Re( b h(z i )) > M 2 for any z i 2 U ("; ). For this value of , the function e i is indeed well-de…ned and bounded on U ("; ) R + . To prove the theorem, consider now the complex-valued equilibrium payo¤ function Because the integrand is bounded on U ("; ) R + as well as continuous in z i over U ("; ) for any x j 0, and because has a compact support, Lebesgue's Dominated Convergence Theorem implies that i is continuous on U ("; ). To show that i is even complex-analytic in U ("; ), consider an arbitrary triangular path in U ("; ). Then, for any x j 0, since U ("; ) is contractible and e i ( ; x j ) is complex analytic in U ("; ), Cauchy's Theorem Integrating over yields Since has a compact support, and e i is bounded on U ("; ) R + , one may exchange the order of integration in equation (7), so that But was arbitrary, so that Morera's Theorem implies that i is indeed analytic in the complex domain U ("; ). On the other hand, i is nonconstant in U ("; ) for " su¢ ciently small because i (x i ) 1 x i for any is an open neighborhood of x i in the strategy set R + such that x i is the only best response to in e V .

Equilibrium characterization
By Theorem 3.2, any symmetric Nash equilibrium of the rent-seeking game has the property that the intersection of its support S with (0; 1) is discrete and allows only the zero bid as a potential accumulation point.
Thus, either S is …nite, or the zero bid is an accumulation point of S .
For R > 2, however, it will be shown below that the origin necessarily is an accumulation point of S . 8 It is also shown that for R 2, there are no symmetric mixed-strategy equilibria in addition to the well-known purestrategy equilibrium.
As before, I will restrict attention to the simplest of all cases, and leave any discussion to Section 6.
Theorem 4.1. In any symmetric equilibrium of the two-player rent-seeking game with 2 < R < 1, the support of the distribution of expenditure levels has the zero bid as an accumulation point. Thus, the equilibrium is characterized by a sequence of mass points y 1 > y 2 > ::: > 0, chosen with respective positive probabilities q 1 ; q 2 ; :::, so that lim k!1 y k = 0 and P 1 k=1 q k = 1. Moreover, for any integer K 1. Finally, there are no non-degenerate mixed-strategy equilibria for R 2.
Proof. Suppose that the zero bid is not an accumulation point of S . Then, using Theorem 3.2, S is discrete and compact, hence …nite. Let y 1 > y 2 > ::: > y L be the mass points of the equilibrium bid distribution, used with respective probabilities q 1 ; :::; q L , where P L k=1 q k = 1. From the …rst-order condition at y L , one obtains But since y k y L for k = 1; :::; L, it follows that Thus, for R > 2, bidding y L yields a negative expected payo¤ in equilibrium, which is impossible. The contradiction shows that the origin is necessarily an accumulation point of S . To prove equation (10), one notes that for any index K 1, where is the expected equilibrium payo¤, as before. Taking the limit K ! 1, and subsequently exchanging the sum and the limit via Lebesgue's Dominated Convergence Theorem implies then that = 0. Finally, it is shown that there are no non-degenerate mixed-strategy equilibria for R < 2. Indeed, by Theorem 3.2, any symmetric equilibrium bid distribution consists of discretely located mass points fy k g L k=1 that are chosen with probabilities fq k g L k=1 , where L 1. Consider now the …rst-order condition at y 1 , i.e., Arguing as above, this implies where the inequality is strict if L > 1. Hence, for R 2, rent dissipation would be imperfect in any non-degenerate mixed-strategy equilibrium. As pointed out above, however, > 0 is feasible only if the zero bid is not an accumulation point of S . Thus, L is …nite. Comparing now (17) with (14) shows that, indeed, L = 1. Thus, any symmetric equilibrium in the two-player Tullock contest with R 2 is necessarily in pure strategies.
The equilibrium description provided by Theorem 4.1 contrasts with both the unique (pure-strategy) Nash equilibrium in the lottery contest and the unique (mixed-strategy) equilibrium in the all-pay auction. As already explained in the Introduction, the peculiar nature of the mixed-strategy equilibria in the Tullock contest is caused by the nonconvexity of the relevant best-response set, which is illustrated by Figure 1. Intuitively, the suboptimality of bids placed, e.g., strictly between bids y 1 and y 2 , captures a cost of being "halfhearted"in the sense that such positive bids are too low to be e¤ective against a decisive action by the opponent, but at the same time too high as a measured defense against speculative underbidding.

Solving the in…nite system
While Theorem 4.1 clari…es the structure of the mixed-strategy equilibrium in the Tullock contest, it is also desirable to learn more about the speci…c values of the parameters y k and q k . Since an explicit solution of equations (10-11) is not readily available, I truncated the in…nite system and solved the resulting …nite system numerically. 9 Parameter values obtained along these lines, rounded to the fourth digit, are shown in Table I. As can be seen, for R kept …xed, the probability weight q k is generally declining in k. Moreover, an increase in R pushes the mass points more tightly together and to higher levels, at the same time lowering q 1 . In sum, this may be seen as re…ning somewhat an earlier description given by Baye et al. (1994). 10 Table I. Numerical bid distributions As the right-upper panel illustrates, the skewness of the bid distribution is generally negative for R > 2, in contrast to the corresponding case of the all-pay auction. Moreover, with variance increasing and skewness vanishing for higher R, a higher degree of decisiveness seems to foster speculative underbidding.  is not hard to see that the proof extends to settings with more than two players, heterogeneous valuations, and alternative impact functions. 11

Higher moments of the bid distributions
Similarly, variants of Theorem 4.1 can be derived for other classes of contests. In particular, the arguments made above extend to the case of 11 For example, it su¢ ces to assume that the impact function h(x i ) is real-analytic on the interval (0; 1), which is the case for many functional forms considered in the literature. probabilities fq k g 1 k=1 , player 2 randomizes over f0g [ fy k v 2 g 1 k=1 with probabilities 1 v 2 =v 1 and fq k v 2 =v 1 g 1 k=1 , while players 3; :::; N remain inactive.
The characterization of the symmetric two-player equilibrium thereby sheds light also on the structure of equilibria in more general contests.
where K max = 14, and by stipulating that y k = 0 for k > K max . This approach led to a system of equations for K = 1; :::; K max , with "error terms" Ignoring all error terms generated useful initial values for approximate solution vectors fy k g K max k=1 and fq k g K max k=1 . In explicit terms, these initial values were y K = R(R 2) K 1 (R+2) K and q K = 4(R 2) K 1 (R+2) K , for K = 1; :::; K max . Approximate solutions could then be improved locally at the K-th mass point, for any K = 1; :::; K max , by solving (18-19) numerically for "updated" values e y K and e q K of y K and q K . To ensure a cumulative probability of one, any updated probability vector (q 1 ; :::q K 1 ; e q K ; q K+1 ; :::; q K max ; 1 P K max k=1 q k ) was multiplied through with 1=(1 q K + e q K ). A Visual Basic macro executed about 40 round-robin iterations of such updating. For any considered value of R > 2, the procedure always converged to the same distribution, regardless of changes made to initial conditions.