Structural Analysis of Tullock Contests with an Application to U.S. House of Representatives Elections

Transcription

1 Structural Analysis of Tullock Contests with an Application to U.S. House of Representatives Elections Ming He Yangguang Huang March 22, 27 Abstract We study the econometrics of an asymmetric Tullock contest model with incomplete information and establish nonparametric identification of the distributions of players private cost signals. We propose estimators for the quantile functions, establish their consistency, and develop an asymptotic inference procedure. Monte Carlo experiments demonstrate the good finite sample performance of the estimators and confidence intervals. We employ this newly developed method to analyze data from U.S. House of Representatives elections. The structural estimation results provide insights into the technological advances of the past four decades and quantify the incumbency advantage and the partisan advantage. Keywords: Tullock contest, nonparametric identification, structural econometrics, U.S. House of Representatives elections JEL codes: C57, D72 We thank Yanqin Fan, Fahad Khalil, Yasutora Watanabe, Quan Wen, and seminar participants at University of Washington for valuable comments and suggestions. We are very grateful to Professor Gary Jacobson for sharing his data with us. Economics Discipline Group, University of Technology Sydney, Box 23, Ultimo, NSW, 27, AU. Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.

2 Introduction Contest games, including all-pay auctions as a special case, formalize strategic interactions among players and have wide applicability in studies of competitive environments. Beginning with the seminal works of Tullock (967, 98), contest theory has been developed in recent decades and has provided profound insights into a wide class of applications. For instance, in a rent-seeking process, lobbyists exert effort to compete for political resources (Tullock, 967, 98; Krueger, 974; Becker, 983; Baye et al., 993); in a lawsuit, both the plaintiff and the defendant hire attorneys in order to obtain a favorable judgment (Baye et al., 25; Skaperdas and Vaidya, 22); and in research, time and money are spent to file patents or compete for research award (Taylor, 995; Che and Gale, 23). Other examples include sports (Szymanski, 23), military conflicts (Hwang, 22), labor market promotions (Connelly et al., 24), college admissions (Bodoh-Creed and Hickman, 25), and Internet marketplaces (Wang and Xu, 26), among others. Congleton et al. (28) and Vojnović (26) provide comprehensive surveys of contest theory and its applications. In a standard contest model, each player exerts costly effort to compete for a prize that she may not be awarded. The efforts of all players are mapped onto winning probabilities by a contest success function (CSF), which is a probabilistic allocation rule reflecting the nature of the competitive environment. On the one hand, if the player who exerts the highest effort wins the prize for certain, the contest reduces to an all-pay auction and is referred to as perfectly discriminating (Hillman and Riley, 989). On the other hand, most contests are imperfectly discriminating: a player s effort positively influences her winning probability but does not guarantee a win, even she exerts the most effort (Skaperdas, 996). For example, the army that incurs most casualties may not win the battle; the company that spends the most resources might not win a patent race; and the candidate who spends the most money on a campaign may not win the election. Depending on whether a player knows her competitors private information, there are contest models with complete or incomplete information. For contests with complete information, there is a large body of theoretical literature, for example, Ellingsen (99), Che and Gale (998), and Siegel (29), among many others. There are also a few structural empirical studies that focus on the estimation of parametric CSFs. For example, Hwang (22) uses data on spending in military conflicts to estimate the CSF. Kang (26) studies lobbying activities in the U.S. energy sector and shows how spending affects voting outcomes. In this paper, we focus on contest models with incomplete information, in which the In addition, there are also some reduced-form empirical studies of contests. For example, Jia (28) and Malueg and Yates (2) study data from sports, and Jia and Skaperdas (22) study data from military conflicts. 2

3 Bayesian Nash equilibrium (BNE) quantifies how each player chooses spending (effort) based on her realized type and knowledge about other players. 2 According to Fearon (995), given the opportunity to negotiate and settle, incomplete information is the main reason why a conflict situation turns into a war. In the theoretical literature on incomplete-information contests, Hurley and Shogren (998b), Malueg and Yates (24), and Sui (29) consider contests in which players have discrete types. Harstad (995), Hurley and Shogren (998a), and Schoonbeek and Winkel (26) analyze contests with one-sided private information. Wärneryd (23) studies a two-player contest for a prize with common but uncertain value. Ewerhart (2) finds a closed-form equilibrium in a two-sided private information contest for a particular family of continuous type distributions. Fey (28) establishes the existence of equilibrium in a two-player contest with uniformly distributed types. Ryvkin (2) extends the result in Fey (28) to cases with multiple players and general distribution functions. Ewerhart (24) establishes the uniqueness of the equilibrium in both symmetric and asymmetric environments. We study the identification, estimation, and inference of model primitives in a Tullock contest with asymmetric and privately-informed players. To the best of our knowledge, there is no previous study on identification and structural estimation of incomplete-information contest models, despite their importance and wide applicability. Our modeling framework uses contest data to unveil the distributions of players types as the model primitives, 3 which might reflect the state of contest-relevant technologies and factors that affect players ex ante positions. 4 Learning the model primitives of a competitive environment by using structural econometric analysis is crucial for policy intervention and contest design. Our contributions to the literature are in threefold. First, we establish nonparametric identification of the quantile functions (or equivalently the distribution functions) of players private cost signals. Our identification analysis is related to that in the structural auction literature. For example, Guerre et al. (2) consider the first-price sealed-bid auction and identify the private value distribution by the distribution of bids. 5 2 Baik and Shogren (995) and Kovenock et al. (25) note that information acquisition and sharing have a significant impact on the equilibrium effort. 3 This private type can be treated as either a private cost signal or a private prize value. See additional discussion in Section 2. As a first step, we assume a known CSF and focus on the distributions of players types. More generally, considering an unknown CSF under the incomplete information framework is very interesting and challenging. We leave this for future research. 4 For example, in political campaigns, the distributions can reflect campaign-relevant technologies such as printing and advertisement. They can also reflect candidates ex ante positions such as incumbency and partisan advantages. 5 A Tullock contest game differs from a standard auction in the allocation rule and the payment rule. In a Tullock contest, the allocation rule is probabilistic, and the payment rule has the all-pay structure, while in 3

4 Second, we develop estimators for the quantile functions of private cost signals, establish their consistency, root-n asymptotic normality, and propose estimators for the asymptotic variances. Through Monte Carlo simulations, we find that our estimators and confidence intervals (CIs) perform reasonably well in finite sample. Third, we apply our method to study several questions using data from U.S. House of Representatives elections. All previous studies, such as Jacobson (978) and Levitt (994), focus on the reduced-form relationship of how campaign spending affects the election outcome. We analyze two sub-samples corresponding to incumbent-versus-challenger elections and open-seat elections. By taking into account the game theoretical structure explicitly and nonparametrically estimating the distributions of cost signals, our results shed light on the following issues. First, the estimates suggest that the distributions of private cost signals have evolved in a roughly decreasing manner in terms of first-order stochastic dominance over the past four decades, which may be attributed to advances in technologies related to campaign costs. Second, based on the estimated quantile functions, we quantify the incumbency advantage as the gap between candidates equilibrium winning probabilities given either their cost signals or spending. For example, in the 2s, a median-type incumbent wins with a probability of 6.8%, while a median-type challenger wins only with a probability of 22.5%. Third, we similarly quantify the partisan advantage. We find that a candidate from one party has an advantage over one from the other party within a decade, and this advantage switches across decades. This pattern is roughly in line with the political atmosphere in each period. For example, in the 97s, Republicans were at a serious disadvantage because of the Watergate scandal, the collapse of the Nixon administration, and the unpopular pardon of the former president by Ford. During this period, for two candidates who spent an average amount of money, the Democratic candidate s winning probability was 3.2% higher than that of the Republican candidate. The rest of this paper is organized as follows. Section 2 presents the asymmetric Tullock contest model with incomplete information. Section 3 provides the key identification result for the model primitives. In Section 4, we propose nonparametric estimators for the quantile functions and establish their asymptotic properties. In Section 5, we conduct simulations to evaluate the finite sample performance of the estimators and CIs. Empirical analysis of the U.S. House election campaign is conducted in Section 6. Concluding remarks are provided in Section 7 and all proofs are relegated to the Appendix. standard auctions, such as first-price or second-price sealed-bid auctions, the allocation rule is deterministic and the payment rule has a winner-pay structure. 4

5 2 The Model There are L risk-neutral players and their private cost signals, C = (C,, C L ), are independent but not identically distributed. C l has CDF F l ( ) and Lebesgue density function f l ( ) on the support C l = [c l, c l ]. 6 F l (or its quantile function Q l ), l =,, L, reflect the state of contest-relevant technologies and factors affecting players ex ante positions. Denote the spending vector for all players as b = (b,, b L ). Given that player l s realized private cost is C l = c and the spending vector is b, her ex post expected payoff is π l (c, b) = p l (b) cφ(b l ), (2.) where p l (b) = b l / L j= b j is the well-known Tullock CSF. The prize is normalized to be one, and φ is a known baseline cost function with φ >, φ and can take different forms such as linear or quadratic. Note that, in the above formulation, we adopt a private cost signal framework instead of a private prize value one due to the equivalence of the two setups. Therefore, the prize of the contest game is normalized to be one without loss of generality, and the spending should be interpreted as spending per unit of prize. 7 Given φ and p = (p,, p L ), the model primitives are F = (F,, F L ), the private cost signal distributions. Let the BNE spending functions be β( ) = (β ( ),, β L ( )). Given that C l = c and other players follow their BNE strategies, player l s interim expected payoff is [ b l πl (b l, c; β) = E b l + j l β j(c j ) cφ(b l) C l = c [ ] = E b l + j l β b l cφ(b l ). (2.2) j(c j ) 6 c l is allowed to be, in which case C l = [c l, ). 7 To see this, we can divide the ex post expected payoff function by c and define v = /c. We obtain vp l (b) φ(b l ). Moreover, by a monotone transformation of the spending: e l = φ(b l ), the payoff function is strategically equivalent to vφ (e l )/ L j= φ (e j ) e l. See Cornes and Hartley (25), Ryvkin (2), and Wasser (23) for details. Skaperdas (996) shows that, the relevant CSF, φ (e l )/ L j= φ (e j ), is equivalent to assuming five basic axioms of a contest, for example, p l (b) = e r l / L j= er j, r >. ] 5

6 By the definition of BNE, β must satisfy [ ] πl (β l (c), c; β) = E b l [β l (c) + j l β β j(c j )] 2 l (c) [ ] + E β l (c) + j l β cφ (β l (c)) = (c), (2.3) j(c j ) where ( ) is the zero function. The existence and uniqueness of β are established in Ryvkin (2) and Ewerhart (24). Example. Let L = 2, φ(x) = x, and the CDFs of private costs be F l (c), l =, 2, with quantile functions Q (τ) = a log + a τ τ + a τ, Q a( τ) + 2(τ) = log, τ (, ), a( τ) a( τ) + where a > is a parameter. In this case, (2.3) becomes [ˆ c2 (β (c) + β 2 (c 2 )) 2 df 2(c 2 ) c 2 [ˆ c c (β (c ) + β 2 (c)) df (c 2 ) ] ] β (c) + β 2 (c) + ˆ c2 β (c) + β 2 (c 2 ) df 2(c 2 ) c = (c), c 2 ˆ c c β (c ) + β 2 (c) df (c ) c = (c), where c = log(a+) a +a, c 2 = log a+ a a+, and c = c 2 =. It can be shown that the unique solution is β (c) = F (c), c [c, ), β 2 (c) = a( F 2 (c)), c [c 2, ). Figure illustrates the quantile functions and equilibrium spending functions for two players when a = 2. Note that β l ( ) < because a higher type corresponds to a weaker player.8 The equilibrium spending functions are determined by the quantile functions of cost signals, so are the distributions of equilibrium spending. In the next section, we establish nonparametric identification of the primitive distributions using the observed distributions. 8 Alternatively, if we adopt a private prize value framework, the equilibrium spending function will be increasing in a player s valuation. 6

7 Figure : Illustration of Example 3 Identification Given n independent and identical plays of the contest game, we obtain a random sample {B i,, B il } n i= of equilibrium spending, where B i,, B il are independent but not identically distributed, with B il = β l (C il ), l =,, L. Then the CDFs G l (b) and PDFs g l (b) of B il are directly identified from the data. For τ (, ), t l (, ) L, let q l (τ) = G l (τ), Q l (τ) = F l (τ), ω l (τ, t l ) = q l (τ) + k l q k (t k ), and δ lj (τ) = ˆ ω j l (τ, t l) dt l, for j =, 2, 3, where dt l = dt dt l dt l+ dt L. Assumption. For l =,, L, F l (c) is continuous and strictly increasing on C l = [c l, c l ]. In the equilibrium, one player s spending is monotonically decreasing in her cost signal. The probability of observing a spending higher than b is G l (b) = P(B l b) = P(β l (B l ) β l (b)) = F l (β l (b)). A change of variables gives q l ( F l (c)) = β l (c). (3.) Equation (3.) is the key to identifying F l from the observed bid distributions. Theorem. For l =,, L, under Assumption, the quantile functions of private costs 7

8 are nonparametrically identified as Q l (τ) = Proof. See the Appendix. [ ] δ φ l ( τ) q l ( τ)δ l2 ( τ), τ (, ). (3.2) (q l ( τ)) Learning the cost signal distributions is very meaningful for policy reference. For example, the contest designer of a patent race can induce the desired distribution of equilibrium spending by adjusting the prize; likewise, an international organization can discourage military conflict between two countries by imposing the proper amount of economic sanctions. Example (Continued). By (3.) and the fact that β (c) = F (c) and β 2 (c) = a( F 2 (c)), we obtain q (τ) = τ, q 2 (τ) = aτ for τ (, ). Then ω (τ, t ) = τ + at 2 for (τ, t 2 ) (, ) 2, ω 2 (τ, t 2 ) = aτ + t for (τ, t ) (, ) 2, and δ (τ) = a log τ + a, δ 2 (τ) = τ Therefore, by (3.2), we obtain τ(τ + a), δ 2(τ) = log aτ +, δ 22 (τ) = aτ Q (τ) = δ ( τ) q ( τ)δ 2 ( τ) = a log + a τ τ a( τ) + Q 2 (τ) = δ 2 ( τ) q 2 ( τ)δ 22 ( τ) = log a( τ) + a τ, aτ(aτ + ). a( τ) +. 4 Estimation and Inference In this section, we propose estimators for the quantile functions of private cost signals, study their asymptotic properties, namely, consistency and asymptotic normality, and propose estimators for the asymptotic variances. Let B (k:n) l denote the k-th lowest order statistic out of the n i.i.d. spending from player l and denote the ceiling function. Let q l (τ) be the empirical quantile estimator of q l (τ) using {B il } n i=, that is, q l (τ) = B ( nτ :n) l. Assumption 2. For l =,, L, (a) G l ( ) is strictly increasing and twice differentiable with compact support [b L l, bu l ], where bl l and sup g l (q l(τ)) <. τ (,) ; (b) inf τ (,) g l(q l (τ)) >, sup g l (q l(τ)) <, τ (,) Assumption 3. For l =,, L and τ (, ), the diagonal elements of Σ l (τ) are finite, with Σ l (τ) defined in (A.2). 8

9 Assumption 4. For l =,, L, (a) g l has an r-th continuous bounded derivative on [b L l, bu l ], r =, 2; (b) K is a second-order symmetric kernel function with support [, ] and a twice continuous and bounded derivative; (c) the bandwidth h l satisfies < h l < (b U l bl l )/2, h l, nh l / log n as n ; (d) h l = O(h l ), / nh 3 l = O(h l), h l = O(h l ), α l > /3, where h l, h l, h l, and α l are defined in Lemma A.2. Assumption 2 imposes standard conditions from the statistical literature to ensure uniform convergence of the empirical quantile function on (, ). Assumption 3 is the standard regularity assumption to invoke the multivariate central limit theorem for the i.i.d. sequence. Assumption 4 is adapted from Li and Liu (25) to make use of their boundary-corrected kernel estimator for the density functions of equilibrium spending, so that we obtain uniform convergence for the associated quantile density estimators. Let ω l (τ, t l ) = q l (τ) + k l q k (t k ), δ lj (τ) = ˆ ω j l (τ, t l) dt l, j =, 2, 3, and Q l (τ) = [ δl ( τ) q φ l ( τ) δ l2 ( τ)]. (4.) ( q l ( τ)) Theorem 2. For l =,, L and τ (, ), under Assumption 2, Q l (τ) Q l (τ) = o P (). Proof. See the Appendix. Next, we establish the asymptotic distribution related to Q l (τ). Theorem 3. Under Assumptions 2 and 3, for l =,, L and τ (, ), ( ) ) d n Ql (τ) Q l (τ) N(, Ω l ( τ) where Ω l (τ) = A T l (τ)σ l(τ)a l (τ), and A l (τ), Σ l (τ) are defined in (A.) and (A.2). Proof. See the Appendix. Next, we show that the asymptotic covariance function can be consistently estimated. 9

10 Theorem 4. Under Assumptions 2 and 4, for l =,, L and τ (, ), Ω l (τ) Ω l (τ) = o P (), where Ω l (τ) = ÂT l (τ) Σ l (τ)âl(τ), and Âl(τ), Σ l (τ) are defined in (A.3) and (A.4). Proof. See the Appendix. 5 Simulation In this section, we conduct Monte Carlo simulations to evaluate the finite sample performance of the estimators and the CIs. We use the same setup as in Example. We set L = 2, φ(x) = x. The private costs are independent and with distribution functions F ( ) and F 2 ( ) associated with quantile functions Q (τ) = a log + a τ τ + a τ and Q a( τ) + 2(τ) = log a( τ) a( τ) +. Note that Q () = log(a+) a +a, Q 2() = log a+ a a+, and Q () = Q 2 () =. Let a {, 2, 4}, n {, 5, }, and the number of repetitions be 2 for each design. By (4.), we have Q l (τ) = δ l ( τ) q l ( τ) δ l2 ( τ), l =, 2, where δ lj ( τ) = [ q l ( τ)+ k l q k(t k dt )] j l for j =, 2, 3. In estimating the asymptotic covariance matrix, we use kernel function K(u) = 2 ( u ), and set h l = ( log(n) n ) 5, h l = n 5, h l = 3.87h l, and α l = In Tables, 2, and 3, for the quantile functions evaluated at levels τ =.,.3,,.9, we report the estimated bias and root mean squared error (RMSE) of the estimators, and the empirical coverage probability (ECP) and the empirical interval length (EIL) of the 95% point-wise confidence intervals (CIs). The results are similar, and we focus on Table in the discussion. For the estimators, first, the bias and RMSE at a given quantile level decrease as the sample size increases. For example, when τ =.5, the biases and RMSEs of Q (τ) are.46,.3,.4 and.344,.53,.6 when n =, 5,, while those of Q 2 (τ) are 9 It is easy to see that in this case, c K = 3.87, where c K is defined in (A.5).

11 .54,.7,.2 and.277,.9,.79 when n =, 5,. Second, in our design, the estimation is more precise for the lower quantile level than for the upper quantile level. For example, when n = 5, the RMSEs of Q (τ) are.45,.89,.53,.27,.67 for quantile levels τ =.,.3,,.9, and those of Q 2 (τ) have a similar pattern. For the CIs, first, the ECPs are very close to the nominal levels in most cases, except for a few under coverage when n =. We attribute this to finite sample approximation error. In fact, the ECPs improve as the sample size increases. Second, the EILs increase with the quantile level due to a larger estimation error at the higher quantile level. In Figure 2, we graph the estimated quantile functions and the 95% point-wise CIs for τ {.,.2,,.99}, each curve is averaged across all the repetitions. First, the estimated curves are very close to the true quantile function in all cases. Second, the confidence bands tend to be wide at upper quantiles when n =, but they become increasingly tight as the sample size increases. In sum, our estimators and CIs perform reasonably well. Table : Performance of Estimators and CIs (a = ) ˆQ (τ) ˆQ2 (τ) n τ Bias RMSE ECP EIL Bias RMSE ECP EIL This is because, for our design, it can be shown analytically that Ω l ( τ) increases as τ increases.

12 Table 2: Performance of Estimators and CIs (a = 2) ˆQ (τ) ˆQ2 (τ) n τ Bias RMSE ECP EIL Bias RMSE ECP EIL Table 3: Performance of Estimators and CIs (a = 4) ˆQ (τ) ˆQ2 (τ) n τ Bias RMSE ECP EIL Bias RMSE ECP EIL

13 Figure 2: Mean Estimates and Point-wise CIs for Quantile Functions 6 Empirical Analysis of U.S. House Elections Every two years, elections are held to elect representatives from 435 congressional districts across all states in the U.S. The outcomes are determined by popular vote and are highly influenced by candidates campaign spending. Political campaigns have been modeled as contests since the beginning of contest theory (Baron, 994; Skaperdas and Grofman, 995). In this section, we use the structural estimation method to study election campaigns for the 3

14 U.S. House of Representatives. Our data set covers twenty House election cycles from the 93rd Congress (972 election) to the 2th Congress (2 election). The following questions are of great interest: First, for a generic candidate, the distribution function of her private cost signal reflects the state of technologies related to political campaigns: therefore, how has these technologies changed in the past decades? Second, while it has been argued that competition is desired in political campaigns, the incumbent usually has an advantage over the challenger in terms of fundraising ability and reputation. This incumbency advantage can be learned by comparing the distribution functions of candidates cost signals. How great is the incumbency advantage? Does the amount of spending affect the equilibrium vote shares or winning probabilities differently for an incumbent and a challenger? Third, the political atmosphere may change across election cycles. Is there any partisan advantage for a candidate, say a Democrat, during a certain period? Does the amount of spending affect the equilibrium vote shares or winning probabilities differently for Democratic and Republican candidates? In the following, we analyze the data and uncover the primitives of the election contests, which allows us to quantify technological changes and to assess the incumbency and partisan advantages. To answer the above questions, we focus on two sub-samples. To quantify the incumbency advantage, we use an incumbent-challenger sub-sample of 5776 elections. To quantify the partisan advantage, we use an open-seat sub-sample of 853 elections. To quantify the potential changes in the model primitives, we further divide each sub-sample by decades in accordance with the timing of electoral redistricting Incumbent-versus-Challenger Elections For the incumbent-challenger sub-sample, descriptive statistics are provided in Table 4 and Figure 3. All spending data are in thousands of 29 dollars for rest of this paper. Figure 3(A) depicts the trend in spending, averaged across all the contests in a given election cycle. It shows that levels of campaign spending increase over time. Given that spending is a decreasing function of the private cost signal, this pattern indicates that the private cost has tended to decrease in recent decades. Figure 3(B) depicts the proportion of incumbents spending and their winning probability. On average, incumbents spend more than twice what challengers do. Incumbents receive, on average, 64.3% of votes, and more than 9% of incumbents are reelected. It is clear that an incumbent has certain advantage over a U.S. House election data have been well documented by the Federal Election Commission since 972 ( 2 Electoral district boundaries are redrawn at the beginning of each decade. The 97s sub-sample contains five election cycles: 972, 974, 976, 978, and 98. The same occurs for other decades. For details, see redistricting.lls.edu. 4

15 challenger. We expect that this advantage will be reflected in the difference between the distributions of their private cost signals. In Table 5, we report two sets of reduced-form regression results. In Panel A, we report the results of linear regressions of the incumbent s vote share on her spending and the challenger s spending. In Panel B, we use election outcomes as the binary dependent variable and report the results of logistic regressions. On the one hand, the reduced-form results suggest a negative and statistically significant effect of the challenger s spending on the incumbent s vote share or winning probability, which is consistent with economic intuition. On the other hand, these results suggest either a small but statistically insignificant positive effect or a moderate and significant negative effect of the incumbent s own spending on her vote share or winning probability. Table 4: Summary Statistics (Incumbent-challenger) Obs Mean SD Min Max Incumbent spending Challenger spending Incumbent vote share (%)* Incumbent wins dummy * There are 5 missing values for the incumbent vote share. Figure 3: Descriptive Statistics (Incumbent-challenger) 5

16 Table 5: Reduced-form Results (Incumbent-challenger) All 97s 98s 99s 2s Panel A: OLS Dependent variable: vote share of Incumbent Incumbent spending -.8** ** -.64** -.779** (.2) (.) (.547) (.365) (.227) Challenger spending -.72** -.26** ** ** -.443** (.2) (.) (.688) (.474) (.256) R Obs Panel B: Logit Dependent variable: = Incumbent wins Incumbent spending..* ** (.) (.) (.342) (.96) (.3) Challenger spending -.3** -.4** ** -.65** -.69** (.) (.) (.339) (.22) (.) McFadden pseudo R Obs Note: * p <.5, ** p <.. Figure 4: Money-vote Relation (Incumbent-challenger) This counter-intuitive result is in line with Jacobson (978, 99), Abramowitz (988), and Figure 4. One explanation is that the candidates campaign spending varies with the competitiveness of an election. On the one hand, a weak challenger does not spend much and would not provide a strong incentive for an incumbent to spend. On the other hand, a 6

17 strong challenger would push an incumbent to fight for her incumbency and still convert a fair proportion of the incumbent s vote. Reduced-form regressions do not take into account the endogeneity issue caused by the strategic interactions, as argued in Erikson and Palfrey (2). We emphasize that levels of spending are simultaneously determined by the equilibrium of a contest game, which further determines the equilibrium winning probabilities. Next, we structurally estimate the quantile functions of cost signals and use the results to quantify how a candidate s realized cost signal or campaign spending determines her equilibrium winning probability. Figure 6 reports the estimated quantile functions and CIs. First, from the two upper panels, the incumbent and the challenger s estimated quantile functions have both shifted outwards in the past four decades. There is a first-order stochastic dominance relationship: the private cost signals become more and more concentrated at lower values. Advances in technologies related to political campaigns are a possible explanation. For example, laser printers have been introduced to replace the mimeograph machines, and e- mail serves as a substitute for paper mail in campaign advertisements. Second, from the four lower panels, there is a large gap between the two estimated quantile functions, which clearly reflects the incumbency advantage: the incumbent s cost signal is first-order stochastically dominated by the challenger s cost signal. This summarizes the overall advantage of the incumbent in terms of fundraising ability and the effect of reputation, etc. Next, define the equilibrium [ winning probabilities ] [ given the cost ] signal or spending β to be P(l wins c) = E l (C l ) β l (C l )+ C q j l β j(c j ) l = c = E l ( F l (c)) q l ( F l (c))+ or P(l wins b) = j l B j [ ] [ ] E = E, respectively. These two quantities are different from the b b+ j l β j(c j ) b b+ j l B j CSF. Instead of ex post winning probability, they take into account the strategic interactions and the equilibrium concept, and provide players with expected winning probabilities given their private signals or spending levels. We estimate them by P(l wins c) = n P(l wins b) = n n q l ( F l (c)) q l ( F l (c)) + j l B, ij (6.) n b b + j l B. ij (6.2) i= i= The estimation results are summarized in Figures 5 and 7. Figure 5 shows that the equilibrium winning probability is decreasing and convex in a candidate s own cost signal, which suggests increasing return to a lower cost signal. 3 It also 3 Given the state of contest-relevant technologies and other factors that affect players ex ante positions, a candidate s realized cost signal can be interpreted as her ability. 7

18 quantifies the incumbency advantage. If the incumbent and the challenger have the same cost signal, their chances of winning can be very different. For example, in the 2s, for an incumbent and a challenger with the median type (c =.54), the incumbent wins with a probability of 6.8% and the challenger wins with a probability of only 22.5%. Figure 7 shows that equilibrium winning probability is increasing and concave in a candidate s own spending, and also quantifies the incumbency advantage. For example, in the 2s, by spending the average amount of money (b = 962), an incumbent wins with a probability of 77.6% and a challenger wins with a probability of only 45.9%. Therefore, even if the challenger has no disadvantage in campaign funding, her winning probability is 3.7% lower than that of the incumbent. This might be attributed to the effect of reputation. Figure 5: Equilibrium Winning Probability Given Cost Signal (Incumbent-challenger) 8

19 Figure 6: Quantile Estimates and CIs by Period (Incumbent-challenger) 9

20 Figure 7: Equilibrium Winning Probability Given Spending (Incumbent-challenger) 6.2 Open-seat Elections For the open-seat sub-sample, descriptive statistics are provided in Table 6 and Figure 8. Figure 8(A) depicts the trends in spending, averaged across all the contests in a given election cycle. It shows that the levels of campaign spending increase over time, which again suggests that the private cost tended to decrease in recent decades. It also shows that the candidates from the two parties spend similar amounts of money on average. Figure 8(B) depicts the proportion of spending by Democratic candidates and that of elections they win: these two proportions are similar. This observation supports the ratio-form CSF of 2

21 Tullock (98), 4 which implies that a contestant s proportion of total spending determines her winning probability. Table 6: Summary Statistics (Open-seat) Obs Mean SD Min Max Democrat spending Republican spending Democrat vote share (%) Democrat wins dummy * There is missing value for the Democrat vote share. Figure 8: Descriptive Statistics (Open-seat) In Table 7, we report two sets of reduced-form regression results. In Panel A, we regress the vote share received by the Democrat candidate on her spending and the Republican candidate s spending. In Panel B, we use the election outcomes as the dependent variable and perform logistic regressions. The results from the two panels lead to similar conclusions: First, as suggested by the signs of the estimates, a candidate s own spending has a positive effect on her campaign outcome and a negative effect on the opponent s outcome. Therefore, the counter-intuitive phenomenon in Table 5 disappears after putting aside the incumbency advantage. Second, across decades, the magnitude of marginal effect of one extra unit of spending decreases. This is also consistent with the ratio-form CSF of Tullock (98), which implies that the winning probability is determined by the spending ratio. In later decades, 4 Another popular CSF is the difference form, which means that the difference in spending determines the winning probability. See Jia et al. (23). 2

22 both parties spend more; hence, an extra unit of spending by one side has a smaller effect on the ratio. Table 7: Reduced-form Results (Open-seat) All 97s 98s 99s 2s Panel A: OLS Dependent variable: vote share of Incumbent Democrat spending.**.463**.38**.3775**.592** (.) (.224) (.4) (.8) (.73) Republic spending -.** ** ** -.639** -.44** (.) (.25) (.655) (.879) (.788) R Obs Panel B: Logit Dependent variable: = Democrat wins Democrat spending.7**.2998**.355**.223**.345** (.2) (.583) (.62) (.324) (.239) Republic spending -.23** -.535** -.392** -.87** -.4** (.2) (.796) (.679) (.332) (.282) McFadden pseudo R Obs Note: * p <.5, ** p <.. Figure 9 summarizes and compares the estimated quantile functions for all periods and for candidates from both parties. First, from the two upper panels, there is a rough first-order stochastic dominance relationship across decades for candidates from both parties, which again might be explained by technological advances. 5 Second, from the four lower panels, we see an insightful pattern that the partisan advantage switches every ten years. This pattern roughly corresponds to the political atmosphere and the president s partisanship during each period. For example, in the 97s, the Republicans were at a serious disadvantage because of the Watergate scandal, the collapse of the Nixon administration, and the unpopular pardon of the former president by Ford. In the 98s, the advantage switched back to the Republicans when Ronald Reagan was in office. The Democrats had the advantage when Bill Clinton was in office during the 99s, while the Republicans had it when George Bush was in office during most of the 2s. Next, similar to the incumbent-challenger sub-sample, we use the estimated quantile 5 However, it is both interesting and puzzling to see that this stochastic dominance relationship does not hold for Democrat candidates between the 99s and 2s, and Republican candidates between the 98s and 99s. 22

23 functions to compute the relationships between either the private cost signal or spending and the equilibrium winning probability by (6.) and (6.2), respectively. Figure shows that the equilibrium winning probability is decreasing and convex in a candidate s own cost signal, and it quantifies the partisan advantage. Moreover, the partisan advantage is higher among high-cost-signal candidates, which suggests that for candidates with lower ability, partisanship plays a more important role in determining the election outcome. In Figure, the equilibrium winning probability is increasing and concave in spending. 6 It also quantifies the partisan advantage. For example, in the 97s, if Democratic and Republican candidates spent the average amount of money (b = 425) on their campaigns, they would win with probabilities of 57.4% and 54.2%, respectively. The 3.2% difference in the winning probabilities reflects the political atmosphere during that era. 6 For the 99s, although the gap is small, the equilibrium winning probability curve for the Democrats is strictly above that of the Republicans at all spending levels. 23

24 Figure 9: Quantile Estimates and CIs by Period (Open-seat) 24

25 Figure : Equilibrium Winning Probability Given Cost Signal (Open-seat) 25

26 Figure : Equilibrium Winning Probability Given Spending (Open-seat) 7 Conclusion In this paper, we study the econometrics of an asymmetric Tullock contest model with incomplete information. We establish nonparametric identification of the distributions of players private cost signals. We propose estimators for the quantile functions, establish their consistency, and develop an asymptotic inference procedure. Monte Carlo experiments are conducted to show the good finite sample performance of our estimators and CIs. We apply our method to data from U.S. House of Representatives elections. While all previous studies focus on the reduced-form relationship between campaign spending and election outcomes, we treat candidates campaign spending as equilibrium outcomes of the 26

27 underlying contest game with incomplete information. Our empirical results offer the following insights: First, the structural estimates of quantile functions across periods suggest that the distributions of private cost signals have evolved roughly in a decreasing manner in terms of first-order stochastic dominance over the past four decades. This phenomenon may be attributed to advances in technologies related to campaign costs. Second, the estimated quantile functions shed light on the incumbency advantage, which might be attributed to incumbents fundraising ability and reputation. We further quantify this advantage by the gap between equilibrium winning probabilities for incumbents and challengers. Third, we find that candidates from one party tend to have an advantage over those from the other party within a decade, and this advantage switches across decades. This pattern is roughly in line with the political atmosphere of each period. This paper yields several avenues for future research. First, we focused on the case with independent private cost signals, but in reality, private values or costs could be correlated, which highlights the need to study a contest model with correlated private types. The existence and uniqueness of equilibrium, identification, estimation, and inference in this situation is of great interest to study. Second, from an empirical perspective, it would be useful to employ our method to answer questions in other contest games, such as how research investments affect patent race outcomes, and how litigation spending affects the results of lawsuits. 27

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33 A Appendix Proof. (Theorem ) For l =,, L, let the operator T l be [ˆ ] c T l (c, β ; F, φ) = c [β l (c) + j l β j (c j)] df l(c l ) β 2 l (c) ˆ [ ] c + β l (c) + j l β j (c j) df l(c l ) cφ (β l (c)), c where β = (β,, β L ), with β l : C l R + to be the candidate functions, and c df c l(c l ) = c c l cl+ c L df c c l c l+ c (c ) df l (c l )df l+ (c l+ ) df L (c L ). The vector of equilibrium spending functions β = (β,, β L ) satisfies T l (c, β; F, φ) = (c), l =,, L L. Due to the facts that B il = β l (C il ) and β l ( ) is strictly decreasing, it is easy to show that G l (t) = F l (β l (t)), which implies that β l (s) = q l ( F l (s)). Plugging this equation into (2.3), we obtain [ˆ ] c c [q l ( F l (c)) + j l q j( F j (c j ))] df l(c 2 l ) q l ( F l (c)) ˆ [ ] c + q l ( F l (c)) + j l q j( F j (c j )) df l(c l ) cφ (q l ( F l (c))) = (c). c Under Assumption, we apply changes of variables F l (c) = τ (and thus c = Q l ( τ)), and F j (c j ) = t j, j l. Solving for Q l ( τ) yields Q l ( τ) = φ (q l (τ)) φ (q l (τ)) ˆ [ q l (τ) + j l q j(t j ) dt l [ δ l (τ) q l (τ)δ l2 (τ) ] ], τ (, ), where dt = dt dt l dt l+ dt L. [ˆ ] q l(τ) φ (q l (τ)) [q l (τ) + j l q j(t j )] dt 2 l Proof. (Theorem 2) Given any l =,, L and τ (, ), by the continuous mapping theorem, it suffices to show that (i) q l (τ) q l (τ) = o P (); (ii) δ lj (τ) δ lj (τ) = o P (), j =, 2, 3. Step (i): Under Assumption 2, sup τ (,) q l (τ) q l (τ) = o P () by Corollary.4. in Csörgo (983), which implies the point-wise convergence result. 33

34 Step (ii): for τ (, ), δ lj (τ) δ lj (τ) ˆ ω j l (τ, t l) ω j l (τ, t l) dt l ˆ () j ω j+ l (τ, t l ) ω l(τ, t l ) ω l (τ, t l ) dt l (2) [ j q l (τ) q l (τ) + ] ˆ sup q k (t k ) q k (t k ) k l t k (,) [ j q l (τ) q l (τ) + ][ sup q k (t k ) q k (t k ) k l t k (,) (3) = o P ()O P () = o P (), j =, 2, 3, ω j+ l (τ, t l ) dt l inf ω l (τ, t l ) t l (,) L where () is by Taylor expansion with ω l (τ, t l ) between ω l (τ, t l ) and ω l (τ, t l ), (2) is by the Triangle inequality, and (3) is by Corollary.4. in Csörgo (983) and the fact that /X nl = O P () under Assumption 2 (a), where X nl inf t l (,) L ω l(τ, t l ). To show that /X nl = O P (), by the Triangle inequality and the fact that ω l (τ, t l ) is between ω l (τ, t l ) and ω l (τ, t l ), we obtain ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) ω l (τ, t l ) X nl inf ω l (τ, t l ) sup ω l (τ, t l ) ω l (τ, t l ) t l (,) L t l (,) L q l (τ) + j l b L j q l (τ) q l (τ) j l p sup q j (t j ) q j (t j ) Y nl yl, t j (,) ] j+ where y l q l (τ) + j l bl j that y l ɛ >, > by Assumption 2 (a) and that τ (, ). For ɛ > such P(X nl > ɛ) P(Y nl > ɛ) P( y l + ɛ < Y nl y l < y l ɛ) = P( Y nl y l < y l ɛ), implying that P ( X nl ɛ ) since Xnl >, which in turn implies that /X nl = O P (). Proof. (Theorem 3) Given any l =,, L and τ (, ), let θ l (τ) = (q l (τ), δ l (τ), δ l2 (τ)) T, θ l (τ) = ( q l (τ), δ l (τ), δ l2 (τ)) T. Define m : R 3 R as m(θ l (τ)) = φ (q l (τ)) [δ l(τ) q l (τ)δ l2 (τ)]. For notational simplicity, we equivalently study Q l ( τ) Q l ( τ) instead of Q l (τ) Q l (τ). 34