Optimal Favoritism in All-Pay Auctions and Lottery Contests

Transcription

1 Optimal Favoritism in All-Pay Auctions and Lottery Contests Jörg Franke 1, Wolfgang Leininger 1, and Cédric Wasser 2 1 University of Dortmund (TU) Department of Economics Vogelpothsweg Dortmund Germany joerg.franke@tu-dortmund.de wolfgang.leininger@tu-dortmund.de 2 University of Bonn Department of Economics Lennéstr Bonn Germany cedric.wasser@uni-bonn.de March, 2016 Abstract We analyze the revenue-enhancing potential of favoring specific contestants in completeinformation all-pay auctions and lottery contests with many heterogeneous contestants. Two instruments of favoritism are considered: Head starts, that are added to the bids of specific contestants, and/or multiplicative biases, that give idiosyncratic weights to the bids. We show that head starts are highly effective in all-pay auctions and less effective in lottery contests. Using biases as additional instrument yields maximal revenue in the all-pay auction, while head starts are revenue-dominated by appropriately designed biases in the lottery contest. We characterize optimal head starts, biases, and the combination of both and show that they can be revenue-ranked in a complete and unambiguous way which holds for any degree of heterogeneity among contestants. Key Words: All-pay auction, lottery contest, favoritism, head start, revenue dominance. JEL classification: C72; D72 This paper supersedes Ruhr Economic Paper No. 524: Revenue Maximizing Head Starts in Contests. The authors would like to thank audiences at various seminars, workshops and conferences for helpful comments. Jörg Franke is grateful to Dan Kovenock for helpful discussions and his hospitality during a research visit at the Economic Science Institute at Chapman University.

2 1 Introduction Contests are frequently and increasingly used to allocate scarce resources among competing agents when other allocation mechanism like markets, matching, or bargaining protocols are not feasible, impractical, or not desired. A characteristic feature of contests is that participating agents exert effort or pay non-refundable bids to win an indivisible prize such that all agents incur their respective costs of effort exertion irrespectively of winning the prize or not. Examples range from promotion tournaments within firms to lobbying, from public procurement to rent-seeking, from high school admission to crowdsourcing, and from the allocation of research grants to innovation contests; see Konrad (2009),[24], and Vojnović (2016), [36], for excellent textbooks on contest theory and their applications, as well as Frank and Cook (2010), [15], and English (2005), [10], for popular approaches regarding the related phenomenon of winner-take-all-markets and the ubiquity of contests in arts and culture. As contests are typically organized by contest organizers, who have substantial discretionary power in designing the contest rules, there is also some tendency for explicit or implicit favoritism with respect to specific agents. Consider, for instance, the preferential treatment of internal or external candidates in hiring decisions or for domestic or small business firms in public procurement, handicap systems in sports, affirmative action in high school admission, or simply discrimination in the sense that the contest organizer favors specific contestants which is manifested by tailoring the conditions of the contest to the advantage of the preferred contestants. In all these cases agents are treated asymmetrically, which might have profound implications for the underlying incentive structure of the contest. Often, these forms of asymmetric treatment can be interpreted as either granting head starts or as biasing the contest rule in favor of specific agents. Agents that enjoy head starts benefit from their advantageous position in the sense that their rivals must first pass the head start to be able to compete on equal footing. Agents that are favored by a biased contest rule enjoy a higher weight on their effort in the process of determining the winner of the prize. 1 Naturally, the extent of head starts and bias has strategic implications for favored agents, their respective rivals, and therefore also on the revenue that is generated in the contest. A contest organizer who has the option to fit bias and/or head starts to the underlying heterogeneity of the contestants can 1 The University of Michigan, for example, added a head start of 20 (out of 150) points to the score of minority applicants for their undergraduate program, while small business firms are granted a 5 percent bid discount in California highway procurement auctions, see Krasnokutskaya and Seim (2011), [25]. There are also instances where both instruments are applied simultaneously: Kirkegaard (2012), [21], reports on the Canadian research promotion program, where researchers with excellent past performance receive a head start while the research proposals of junior scientists get a higher weight in the evaluation process. 1

3 therefore influence the generated contest revenue to some degree. In this paper we focus on these two instruments of favoritism (bias and head starts) and analyze their potential to generate additional revenue in contest games with several heterogeneous players. We concentrate on two frameworks that are predominantly used in the contest literature; that is, all-pay auctions with complete information, see Hillman and Riley (1989), [20], for an early analysis, and lottery contests introduced by Tullock (1980), [35]. Both models are sufficiently tractable and have been extensively applied in various contexts, see the two mentioned text books and Corchón (2007), [6] for a survey. However, our model deviates from the standard setup by allowing the contest organizer to specify idiosyncratic head starts and/or biases; that is, granting additive lump-sum boni (head starts) or as weighting the respective bids or effort levels with different factors (bias). Hence, applying both instruments simultaneously amounts to an affine transformation of the bid or effort. From the perspective of a contest organizer who is interested in maximizing aggregated equilibrium effort or bids (the expected revenue of the contest), specifying the framework-specifc optimal bias and/or head start becomes then the crucial instrument to increase contest revenue. 2 Our analysis proceeds by characterizing the equilibrium for a given bias-head-start combination for each of the two frameworks. For the lottery contest we modify the share-function approach by Wasser (2013), [37], to obtain an indirect characterization of equilibrium revenue. We then consider an unbiased lottery framework with head starts to show that head starts in general are not very effective in inducing additional revenue in a lottery contest: Basically, favored players use their head start to substitute for own effort while other players are not affected, which tends to decrease equilibrium revenue. Moreover, for each feasible head start vector there exists an appropriate bias that induces higher revenue. Hence, in the lottery contest framework the optimal bias revenue-dominates the optimal head start and the optimal combination of bias and head starts involves zero head starts. For the all-pay auction framework we concentrate on a restricted class of bias-head-start combinations under which exactly two players are active, which allows us to construct equilibrium bid functions in closed form. We derive the optimal head start in the unbiased all-pay auction and show that it levels the playing field among those two active players who endogenously decide to participate under the respective head start. We then identify the optimal set of active players and determine the respective revenue, which is a lower bound for the unrestricted class where more 2 Arguably, in some of the applications mentioned above, asymmetric treatment of agents is not (at least officially) implemented to increase contest revenue but rather for some normative reasons. However, even if normatively derived deviations from symmetric treatment are applied, the forgone revenue should be an important evaluation criterion for these policies. Our paper provides a benchmark of comparison by deriving the maximal revenue that can be obtained through optimally designing asymmetric treatment of agents. 2

4 than two players could be active. If both instruments are applied simultaneously, then the optimal bias-head-start combination can be designed in such a way that the strongest contestants faces a head start which is close to her valuation. Hence, the strongest player is forced to bid close to her valuation to overcome the high head start which implies that the entire surplus is extracted from the players and transformed one-to-one into additional revenue. Thus, in the all-pay auction framework head starts are highly effective instruments for revenue extraction and their efficiency is even higher if combined with appropriately designed biases. Finally, we compare the generated contest revenue with-in and between the two frameworks under the optimally designed instruments. We obtain a complete and unconditional revenueranking among all frameworks and instrument combinations, which holds for any degree of heterogeneity among the players. The resulting ranking implies, for instance, that revenue in the all-pay auction under optimal head starts is higher than under the optimal bias. Hence, head starts are specifically effective to induce additional revenue if the contest is highly competitive as in the all-pay auction, where strong players have to surpass the head start of their rivals to maintain their chance of winning the prize. Multiplicative biases, in contrast, are more effective in the lottery contest because they can be designed to induce additional entry by weak players which tends to increases revenue, comp. Franke et al. (2013), [17]. In the lottery contest this mechanism does not work with additive head starts because favored players use head starts as a substitute for own effort exertion. Nevertheless, the all-pay auction revenue-dominates the lottery contest independently of the fact whether the respective optimal head start, or the optimal bias, or both are used in the two frameworks, generalizing the revenue dominance result of Franke et al. (2014), [18], to any affine bid transformation. Hence, if revenue maximization is the objective of the contest organizer, then she should resort to the all-pay auction framework with optimal bias-head-start combination. Our study contributes to the literature on revenue maximization with biased contest success functions and heterogeneous players, comp. Nti (2004), [30], Fu (2006), [19], Franke (2012), [16], and Epstein et al. (2011 and 2013), [11] and [12]; see also Mealem and Nitzan (forthcoming), [29], for a recent survey on discrimination in contest games. 3 Moreover, our study contributes to the recent interest in the analysis of head starts in different competitive situations, comp. Kirkegaard (2012), [21], Seel and Wasser (2014), [32], Li and Yu (2012), [26], Segev and Sela (2014), [33], and Konrad (2002), [23]. However, all of the mentioned studies (as most of the literature) are either restricted to the two-player case or focus exclusively on only one of the two instruments. In this sense our characterization of the optimal head start in the multi-player 3 Similar issues have also been analyzed experimentally, comp. Calsamiglia et al. (2013), [5], and Balafoutas and Sutter (2012), [2]. 3

5 all-pay auction framework constitutes, for instance, a direct extension of Li and Yu (2012), [26], to the multi-player case, while a similar relation holds for the lottery contest analyzed in Nti (2004), [30]. To our knowledge there only exist three studies that involve head starts in multiplayer all-pay auction or lottery contest frameworks with heterogeneous players. 4 While Siegel (2014), [34], considers head starts in multi-prize all-pay auctions, Wasser (2013), [37], incorporates homogeneous head starts in a lottery contest framework. However, both studies are rather interested in equilibrium characterization and not in the optimal head start design. The only paper that explicitly addresses the design of optimal head starts and biases in a multi-player lottery contest framework is Dasgupta and Nti (1998), [8]. However, they assume that players and head starts are homogeneous and find that in their specific case no head start at all is optimal. Our study extends this framework by considering heterogeneous contestants with idiosyncratic head starts and biases and shows under which conditions these previously established results on optimal favoritism still hold under heterogeneity. The paper is organized as follows. In Section 2 we introduce the model setup and the two contest frameworks. In Section 3 we characterize the equilibrium in the lottery contest framework with bias and head starts. Based on this characterization we derive the optimal head starts, summarize the results on the optimal bias and analyze the simultaneous application of bias and head starts. We collect our result for the lottery contest and provide a complete revenue ranking of all instrument combinations in this framework. In Section 4 we analyze the all-pay auction framework by deriving equilibrium bid functions and the corresponding revenue in the restricted class of bias-head-start combinations which involve two active players. We then derive the optimal head start, summarize results on the optimal bias, and determine the optimal combination of bias and head starts in the restricted class. We also show that the resulting lower bound on revenue for the unrestricted class coincides with the upper bound on revenue, which allows us to establish a complete revenue ranking of all instrument combinations in the all-pay auction framework. In Section 5 we collect our results to compare equilibrium revenue between the two frameworks and provide a complete revenue ranking among all instrument combinations and both frameworks. In Section 6 we discuss the robustness of our results with respect to alternative instruments of favoritism and more general bid transformation. Section 7 concludes. 4 There is also a recent literature on the revenue-enhancing effects of favoring ex-ante symmetric players in incomplete information contests, see Drugov and Ryvkin (forthcoming), [9], Pérez-Castrillo and Wettstein (forthcoming), [31], and Matros and Possajennikov (forthcoming), [28]. 4

6 2 The Model There are n 2 players of set N = {1,..., n} that compete for an indivisible prize. Players are heterogeneous with respect to their valuations of the prize and can be ordered decreasingly with respect to their valuations: v 1 v 2... v n > 0. The probability Pr i (x i, x i ) of player i N to win the prize depends positively on her bid x i [0, ) and negatively on the bids x i = (x 1,..., x i 1, x i+1,..., x n ) [0, ) n 1 of her rivals, where all bids are non-refundable. The expected payoff for player i N is therefore: π i (x i, x i ) := Pr i (x i, x i )v i x i The probability function Pr i (x i, x i ) also depends on the specific design of the contest framework. We focus in our analysis on the two most frequently used contest frameworks; that is, a deterministic all-pay auction and a probabilistic lottery contest. More precisely, we consider asymmetric versions of those two contest frameworks where the bid x i of each player i N is converted by an idiosyncratic affine transformation consisting of a multiplicative bias parameter α i (0, ) and a non-negative head start δ i [0, ) that is added to player i s bid without any costs. Once, the decision on the framework is made and the respective instrument combination of bias α = (α 1,..., α n ) and head start δ = (δ 1,..., δ n ) is designed (given the specific framework), the probability for player i N to win the prize can be expressed as follows: 1. For the all-pay auction with bias and head start (framework BHA): Pr BHA i (x i, x i ) = 1, if α i x i + δ i > α j x j + δ j for all j i, 1, if α k+1 ix i + δ i = α j x j + δ j for k players j i and α i x i + δ i > α l x l + δ l for all other players l i, 0, if α i x i + δ i < α j x j + δ j for some j i. 2. For the lottery contest with bias and head start (framework BHL): nj=1 α i x i +δ i Pri BHL (x i, x i ) =, if n (α j x j +δ j ) j=1 (α j x j + δ j ) 0, 0, if n j=1 (α j x j + δ j ) = 0. These probability functions are also called contest success functions (CSFs). It should be noted, that they include two special cases that are of interest in their own right for our analysis: 5

7 Applying a symmetric bias to all players, e.g., α = (1,..., 1), leads to an unbiased all-pay auction/lottery contest with head starts (framework HA/HL); applying zero head starts to all players, δ = (0,..., 0), leads to a biased all-pay auction/lottery contest without head starts (framework BA/BL). We evaluate the revenue potential of the two instruments by deriving and comparing the maximal contest revenue (the sum of expected equilibrium bids by all players) that can be induced in each framework by specifying optimal head starts and/or biases. We denote this maximal revenue by X,c := i N E[x,c i ], where E[x,c i ] denotes the expected equilibrium bid of player i N under the optimal bias and/or head start in framework c {BHA, BHL, HA, HL, BA, BL}. Analyzing all 2 3 frameworks separately allows us to isolate the revenue-enhancing effects of the two instruments in separation and to compare them with the optimal affine transformation where both instruments are used simultaneously. As bias and head starts are framework-specific, our analysis proceeds by firstly characterizing equilibrium and revenue for a given bias-head-start combination in each of the two frameworks. We then characterize optimal head starts, optimal biases, and determine the optimal affine transformation for each framework. These instruments are then compared with respect to the induced revenue with-in each framework and, in a last step, between the two frameworks. 5 3 Lottery Contests Our analysis of the lottery contest framework starts with an explicit characterization of the equilibrium for a given bias-head-start combination (α, δ). We then determine optimal head starts without bias (framework HL) and summarize the existing results concerning optimal biases without head starts (framework BL). We use these results to show that the optimal combination of head starts and bias (framework BHL) coincides with the optimal biases with zero head starts. Hence, using head starts in addition to bias is not conducive to generate additional revenue in a lottery contest. In other words, a revenue-maximizing contest organizer would choose to specify optimal biases and refrain from using head starts at all. 5 Our analysis can be alternatively framed as the design problem of a contest organizer, whose objective function depends positively on contest revenue, and whose strategy space consists of the type of contest framework with type-dependent instrument (i.e., bias and/or head starts). 6

8 3.1 Equilibrium We derive the equilibrium for the most general framework BHL; that is, a lottery contest with non-negative head starts δ [0, ) n and multiplicative positive bias α (0, ) n. Let y i := α i x i +δ i denote the score of player i and Y := i N y i the aggregate score. We will consider the equivalent game where each player i chooses his score y i [δ i, ) rather than his effort x i and obtains payoff π i (y 1,..., y n ) = y i Y v i y i δ i α i (if Y > 0). As π i is strictly concave in y i, player i s best response is characterized by the first-order condition Y y i Y 2 v i 1 α i 0, with equality if y i > δ i. (1) An effort profile x 1,..., x n in the original contest is a pure-strategy Nash equilibrium if and only if the corresponding score profile y 1,..., y n satisfies (1) for all i. In the following, we generalize the equilibrium characterization of Wasser (2013), [37], who considered lottery contests without biases and with symmetric head starts which makes use of the share function approach by Cornes and Hartley (2005), [7]. For all K N, define the function K 1 + ( K 1) ( i K 1 )( ) α i v i j K δ j Y(K) := 2 if K, i K 1 α i v i i N δ i if K =. This function from the power set of N to R + is central to our equilibrium characterization: As shown in the following proposition, maximizing Y(K) with respect to K yields the equilibrium aggregate score, with the maximizer being the set of active players that submit non-zero effort in equilibrium. Proposition 3.1 There is a unique pure-strategy Nash equilibrium in the BHL framework. Let (2) K = arg max Y(K). K N In equilibrium, the aggregate score is Y(K ), each player i K exerts effort x i = 1 α i ( Y(K ) Y(K ) 2 7 α i v i ) δ i > 0, (3)

9 and each player j K exerts zero effort. Proof. Note that condition (1) implies y i = max { } Y Y2 α i v i, δ i. Following Cornes and Hartley (2005), [7], we define i s share function as φ i (Y) := y i Y = max { f i (Y), g i (Y) } where f i (Y) := 1 Y α i v i and g i (Y) := δ i. Let Φ(Y) := n Y i=1 φ i (Y) be the aggregate share function. A score profile y 1,..., y n corresponds to a pure-strategy Nash equilibrium if and only if there is an aggregate score Y such that y i = φ i (Y )Y for each i N and Φ(Y ) = 1. Note that for players i who are active in equilibrium (x i > 0 and y i > δ i ) we have φ i (Y ) = f i (Y ), whereas for inactive players (x i = 0 and y i = δ i ) we have φ i (Y ) = g i (Y ). Φ(Y) is continuous and strictly decreasing. Moreover, Φ ( ) ( ) j δ j i g i j δ j = 1 and Φ(Y) < 1 for Y large enough. Hence, there is a unique Y that solves Φ(Y ) = 1, implying existence of a unique pure-strategy Nash equilibrium. Now, consider an equilibrium where the players in K N are active, whereas the remaining players are inactive. Hence, Φ(Y ) = 1 is equivalent to Φ(Y ) = f i (Y ) + g j (Y ) = K 1 Y + 1 δ α i K j K i K i v i Y j = 1. j K Solving this equation for Y we obtain Y = Y(K ), where Y( ) was defined in (2). We will now show that the function Y(K) is maximized at the set of players K that are active in equilibrium. Consider a set of players M K. By the definition of Y( ), we have ( ) ( ) f i Y(M) + g j Y(M) = 1. i M j M However, since M does not correspond to the set of players that are active in equilibrium, we must have at least one i M where f i ( Y(M) ) < gi ( Y(M) ) or one j M where g j ( Y(M) ) < f j ( Y(M) ). Hence, Φ ( Y(M) ) > 1. Since Φ is strictly decreasing, we must have Y(M) < Y(K ), because K is the unique subset of players that satisfies Φ(Y(K )) = 1. Finally, the equilibrium efforts of active players i K given in (3) are obtained from the equilibrium scores y i = φ i ( Y(K ) ) Y(K ) = Y(K ) Y(K ) 2 α i v i. Proposition 3.1 is essential to derive the optimal bias-head-start combination in the lottery contest framework because equilibrium revenue can be characterized based on equilibrium 8

10 scores. However, before we proceed, we address two important special cases in which we consider each of the two instruments separately. Firstly, we analyze lottery contests with head starts where biases are absent. In this case Proposition 3.1 can be used directly to derive the optimal head start in explicit form. Secondly, we consider lottery contests with biases where head starts are absent and summarize previous results from the literature. Using these results we are able to proof that the optimal combination of simultaneous bias and head starts will never entail positive head starts; that is, the optimal bias-head-start combination coincides with the optimal bias in the lottery contest framework. 3.2 Optimal head starts We analyze lottery contests with head starts (framework HL) by setting α = (1,..., 1) and take advantage of the modified equilibrium characterization from Proposition 3.1. To clarify the dependence on head starts δ, we will slightly change the notation from the previous subsection: We will write Y(δ, K) for the function defined in (2), K (δ) for the set of players that are active in equilibrium, and x i (δ) for the equilibrium effort of player i. The revenue induced under head starts δ without bias simplifies to the equilibrium aggregate score minus the sum of the head starts. Making use of Proposition 3.1, we have X HL (δ) := x i (δ) = Y(δ, K (δ)) δ i. i N An important benchmark is the revenue in the standard lottery contests without head starts. We will represent this case by the zero head start vector δ 0 simplifies to i N := (0,..., 0). The induced revenue X HL (δ 0 ) = Y(δ 0, K (δ 0 )) = max Y(δ k 1 0, K) = max K N k>1 k. (4) i=1 1 v i The last equality follows from (2) and has also been shown by Fang (2002), [14]. As a first step towards finding optimal head starts, the following lemma provides an upper bound on revenue X HL (δ), which depends on the number of players that are active in equilibrium. Lemma 3.2 Let δ δ 0. If K (δ) = 1, then X HL (δ) 1 4 v 1. If K (δ) 2, then X HL (δ) < X HL (δ 0 ). Proof. First, suppose δ is such that only one player is active, i.e., K (δ) = {i} for some i. Then X HL (δ) = Y(δ,{i}) δ j = j N v i δ j δ j δ i j i j i δ j vi j i j i δ j v i 4 v

11 The first inequality is implied by δ i 0, the second inequality follows from the expression on the LHS being maximized at j i δ j = 1 2 vi, and the third inequality uses v 1 v i for all i. Now, suppose δ induces K (δ) 2. Observe that for any K N with K 2, we have 0 if i K, Y(δ, K) = 1 δ i ( K 1) ( l K 1 )( ) 1 1 if i K, v l j K δ K 1 j where Y(δ,K) ( ) δ i < 1 for i K with δ i > 0. Hence, δ j Y(δ, K) i N δ i 0 for all j N, with strict inequality for at least one j. This leads to the following chain of (in-)equalities, which proves the second result: X HL (δ) = Y (δ, K (δ)) δ i < Y (δ 0, K (δ)) Y (δ 0, K (δ 0 )) = X HL (δ 0 ). i N Lemma 3.2 differentiates between two cases. If at least two players are active under head starts δ, then revenue can be increased by reducing any individual head start δ i irrespective of the fact whether player i is active or inactive. For active players head starts are basically perfect substitutes for own effort exertion because (1) implies that the score of an active player will not be affected by marginal variation in the respective head start. For inactive players the proof of Lemma 3.2 reveals that reducing head starts is also revenue-enhancing. Hence, zero head starts δ 0 = (0,..., 0) will induce higher revenue than any other vector of head starts that involves at least two active players. If however exactly one player is active under head starts δ, then setting positive head starts for inactive players can lead to higher revenue. Intuitively, increasing the head starts of inactive players implies that the active player will best-respond and potentially increase effort as well. Lemma 3.2 shows that revenue in this case is bounded above by 1 4 v 1. The existence of this upper bound has an immediate implication. If revenue without head starts is higher than the upper bound, then Lemma 3.2 implies that optimal head starts are zero. In the opposite case the following result shows that there exist head starts that induce revenue at the upper bound; that is, player 1 submits effort 1 4 v 1 and all other players remain inactive. Proposition 3.3 If 1 4 v 1 X HL (δ 0 ), then all head starts δ where δ 1 = 0 and n j=2 δ j = 1 4 v 1 are optimal and induce revenue X HL (δ ) = x 1 (δ ) = 1 4 v 1. Otherwise, zero head starts δ 0 are the unique optimal head starts. 10

12 Proof. Suppose 1 4 v 1 X HL (δ 0 ). We will show that any head starts δ as defined in the proposition result in equilibrium efforts x 1 (δ ) = 1 4 v 1 and x j (δ ) = 0 for all j > 1. The aggregate score in such an equilibrium is Y = x 1 (δ ) + n j=2 δ j = 1 2 v 1. Recall that player i s best response in terms of individual score is given by (1), which implies { ) } { ( y i = max Y (1 Yvi 1, δ i = max 2 v 1 1 v ) } 1, δ i. 2v i Hence, the best-response effort of player 1 is indeed x 1 = y 1 = 1 4 v 1. Moreover, recall (4) and note v 1 2v j 3 2 that 1v k X HL (δ 0 ) = max k>1 ki= implies 3v vi v v 1. Therefore, for all j > 1, 1 v 2 implying y j = δ j. Consequently, the best-response effort of each player j > 1 is indeed x j = 0. Now, suppose 1 4 v 1 < X HL (δ 0 ). In this case, Lemma 3.2 implies that X HL (δ 0 ) > X HL (δ) for all δ δ 0, rendering δ 0 uniquely optimal. Proposition 3.3 implies that non-zero head starts are optimal if player 1 s valuation is sufficiently higher than the other players valuations. A simple sufficient condition for 1 4 v 1 X HL (δ 0 ) is v 1 4v 2 (whereas v 1 3v 2 is necessary, as shown in the proof of Proposition 3.3). If nonzero head starts are optimal, then they are set such that the strongest contestant, player 1, competes against the sum of head starts of all the other inactive players. Hence, from the perspective of player 1 it is as if he would face just one equally strong opponent with the same valuation v 1 like himself. If however the difference in valuations between player 1 and the other players is less pronounced, then head starts are not a suitable instrument for increasing revenue. Proposition 3.3 can also be combined with Lemma 3.2 which allows us to determine the maximal revenue under the optimal head start in the following corollary. Corollary 3.4 Optimal head starts in the HL framework yield equilibrium revenue 3.3 Optimal biases XHL = max v 1 4, max k>1 k 1 k i=1 1 v i. We now turn to framework BL, where the organizer can only set a bias α but no head starts. The revenue-maximizing bias for this case has been determined in Franke et al. (2013), [17], which is summarized in the following proposition. 11

13 Proposition 3.5 (Franke et al. (2013), [17]) An optimal bias in the BL framework yields revenue XBL = 1 k 4 v i (k 2) 2 { k i=1 i=1 1, where k = max k N v i The set of optimal biases consists of all α where, for some c > 0, α i = 2c v i + k 2 k j=1 1 v j 1 k 2 v k < for i k and α i < c v i for i > k. k 1 Under an optimal bias, each player i k is active and each player i > k is inactive. In an optimally biased lottery contest, the k contestants with the highest valuations are active in equilibrium. Compared with the unbiased lottery contest, an optimal bias α typically encourages additional entry (k is greater or equal to the number of active players in the unbiased lottery contest. For example, provided that n 3, at least three players will always be active in the optimally biased lottery contest while the lower bound is two in the unbiased contest. In this respect, the way in which optimal biases work is in stark contrast to our characterization of optimal head starts from the preceding subsection: If nonzero head starts are optimal, then they are designed to discourage entry, reducing the number of active players to one. i=1 v i }. 3.4 Optimal combinations of head starts and bias We now consider lottery contests where both instruments, head starts and bias, can be used simultaneously. Our main result for this case is that an organizer will never find it profitable to implement non-zero head starts in addition to an optimally chosen bias. In other words, any contest with head starts and bias can be replaced by a contest without head starts and adjusted bias that yields strictly higher revenue. Consider a contest with strictly positive head starts for some players and suppose we remove them. Each player i that is active under a positive head start will increase his effort due to the fact that head starts are perfect substitutes for own effort exertion. In addition, we prove that by sufficiently increasing α j, each inactive player j can be induced to actively invest effort that is equal to the removed head start. Proposition 3.6 For any combination of head starts and bias (δ, α) where δ δ 0, there exists a bias ˆα such that (δ 0, ˆα) yields strictly higher revenue than (δ, α). 12

14 Proof. Let y i and Y denote the individual score of player i and the aggregate score induced in the equilibrium under (δ, α) where δ δ 0. Moreover, let K (δ, α) denote the set of players that are active in this equilibrium (i.e., all i where y i > δ i ). Define a new bias ˆα such that ˆα i = α i for all i K (δ, α) and ˆα j = Y2 (Y δ j )v j > 0 for all j K (δ, α). We will now show that each individual equilibrium score ŷ i under (δ 0, ˆα) is identical to the equilibrium score y i under (δ, α). For the active players i K (δ, α), it is immediate from (1) that the best-response score is unaffected by setting the head start to zero (while keeping the bias constant), resulting in ŷ i = y i. For the players j K (δ, α) that are inactive in the original contest, condition (1) under (δ 0, ˆα) becomes i j ŷ i ( i N ŷ i ) 2 v j (Y δ j)v j Y 2 0, with equality if ŷ j > 0. Given ŷ i = y i for all i j and recalling that y j = δ j, the solution to the above is indeed ŷ j = y j. Let x i and ˆx i denote i s equilibrium effort under (δ, α) and (δ 0, ˆα), respectively. Since y i = ŷ i, ˆx i = x i + δ i ˆα i if i K (δ, α) x i + δ i ˆα i if i K (δ, α) Consequently, i N ˆx i > i N x i because δ i > 0 for at least one i. The crucial implication of Proposition 3.6 is, that an optimal bias α as provided by Proposition 3.5 combined with zero head starts is also optimal in framework BHL, resulting in maximal revenue XBHL = X BL. Summarizing our results, we obtain the following revenue ranking for the different instruments in the lottery contest framework. Proposition 3.7 For any given set of valuations v 1 v 2... v n 0 there is an unambiguous revenue-ranking of lottery contests with optimal head starts, multiplicative biases, or affine bid transformations: X BHL = X BL X HL. (5) Moreover, if v 1 > v 2 then X BHL = X BL > X HL. Proof. The equalities and the last inequality of (5) are immediate from Proposition 3.6. It remains to prove the strict inequality X BL > X HL = max { 1 4 v 1, X HL (δ 0 ) } if v 1 > v 2. If X HL = 1 4 v 1, then according to Proposition 3.3 at least one head start is strictly positive and the strict inequal- 13

15 ity follows from Proposition 3.6. If XHL = X HL(δ 0 ), then the strict inequality follows from Franke et al. (2013), [17] who have shown that XBL is strictly higher than the revenue in absence of any bias if v 1 > v 2. From the perspective of a revenue-maximizing contest organizer biasing the contest rule is the preferred instrument of favoritism. Under the optimal bias more contestants are induced to participate and the playing field is more balanced. Both effects imply more effort exertion and therefore higher revenue. In contrast, head starts are less suitable to induce additional revenue because contestants who are favored by positive head starts use them to reduce their effort level while maintaining their score. In equilibrium the score of other contestants is therefore not affected and revenue is generically lower under positive head starts. Moreover, for any head start there exists a corresponding bias that yields even higher revenue by inducing non-active contestants to participate. As this substitution works for any degree of heterogeneity in valuations, the respective revenue ranking is unambiguous and confirms the dominance of optimal biases over head starts in the lottery contest framework. 4 The All-Pay Auction The theoretical analysis of all-pay auctions with several heterogeneous players faces some typical intricacies that complicate equilibrium characterization: First, even in the standard framework without head starts or bias there might exist multiple equilibria that are not revenue-equivalent if players tie for the highest or second highest possible bid, see Baye et al. (1993) and (1996), [3] and [4]. Second, also in more general generic all-pay auctions without ties the equilibrium bid function can often only be derived in algorithmic form, see Siegel (2014), [34], which makes optimal contest design complicated. We therefore use an alternative approach to derive the optimal combination of bias and head starts based on a restricted class of bias-head-start combinations under which only two players are active. Under this restriction there exists a unique equilibrium which can be characterized in closed form such that the expected equilibrium revenue can be derived explicitly. Using this equilibrium characterization we first solve for the revenue-maximizing head start in an unbiased all-pay auction framework, extending Li and Yu (2012), [26]. A comparison with the revenue-maximizing bias in an all-pay auction framework without head starts, derived in Franke et al. (2014), [18], suggests that head starts are more conducive to induce additional revenue than biases which can be attributed to the strong competitive pressure in the all-pay 14

16 auction. Finally, we derive the optimal combination of bias and head starts and calculate the respective revenue. Given that the optimal bias-head-start combination belongs to the restricted class in which only two players are active, the derived revenue therefore constitutes a lower bound for maximal revenue in the unrestricted class. Moreover, we proof that it coincides with the upper bound in the unrestricted class where any number of players might be active. Hence, using this detour we are able to characterize the optimal bias-head-start combination in the all-pay auction and to determine the respective maximal revenue. This also allows us to establish an unambiguous revenue-ranking among all instruments and combinations in the all-pay auction framework which holds for any distribution of valuations. 4.1 Equilibrium We consider combinations of bias and head starts under which only two players are active. We denote these players by i and j and assume that v i v j. Given any degree of heterogeneity with respect to players valuations, these bias-head-start combinations can be easily constructed by granting players i and j head starts that are sufficiently high such that other players will never bid in the auction. 6 For this restricted class of bias and head starts the contest success function can therefore be reduced to the following expression: Pr BHA i (x i, x j ) = 1, if α i x i + δ i > α j x j + δ j, 1, if α 2 ix i + δ i = α j x j + δ j, 0, otherwise, and Pr BHA j (x j, x i ) = 1 Pri BHA (x i, x j ). For notational convenience we normalize bias and head starts for the two active players by defining α i j := α j α i and δ i j := δ j δ i α i. The contest success function then simplifies as follows: Pr BHA i (x i, x j ) = 1, if x i > α i j x j + δ i j, 1, if x 2 i = α i j x j + δ i j, 0, otherwise. 6 The following specification is an example. Let δ i = δ j = v 1, δ k = 0, and α k 1 for all k i, j. In this specification no player k i, j will bid a positive amount because they have to overcome the head start for player i and j to guarantee a positive probability to win the auction. However, bidding more than v 1 would result in a negative payoff such that all players k i, j refrain from bidding at all. 15

17 Hence, the design problem of optimal favoritism is reduced to find that combination (α i j, δ i j ), that maximizes expected revenue in an all-pay auction based on the modified CSF. The advantage of using this restricted class of bias and head starts is that standard arguments from the all-pay auction literature can be used to construct the (unique) equilibrium bidding function of the two active agents which are provided in the following lemma. Lemma 4.1 For the restricted class of bias and head starts where only two players i and j are active, there exists a unique equilibrium which is in mixed strategies and characterized by the following equilibrium bidding distribution functions: 1. If v i α i j v j + δ i j then F i (x) = F j (x) = 0, if 0 x < δ i j, δ i j α i j v j + x α i j v j, if δ i j x < α i j v j + δ i j, 1, otherwise. v 1 α j x i j v i + α i j v i, if 0 x < v j, 1, otherwise. (6) (7) 2. If v i α i j v j + δ i j then F i (x) = F j (x) = 1 v i α i j v j + δ i j α i j v j, if 0 x < δ i j, 1 v i α i j v j + x α i j v j, if δ i j x < v i, 1, otherwise. δ i j v i + α i jx v i if 0 x < v i δ i j α i j, 1, otherwise. (8) (9) Proof. Given the all-pay auction framework it is obvious that an equilibrium in pure strategies cannot exist. Hence, if an equilibrium exists it must be in mixed strategies. We now proof that the distribution functions (6) - (9) constitute a mixed strategy equilibrium. 7 For this we show that a player is indifferent among all bids from her active bidding support if her opponent bids according to the specified distribution function because all her bids would yield the same nonnegative payoff. At the same time bidding out of the active bidding support would result in a 7 For uniqueness, standard arguments from the all-pay auction literature can be applied, see for instance, Baye et al. 1996, [4], or Hillman and Riley 1989, [20]. 16

18 lower or even negative payoff. 1. Assume that v i α i j v j + δ i j and consider player i. The expected payoff for player i is π i (x i, x j ) = Pr(x i > α i j x j + δ ( xi δ i j)v i x i = F i j ) j vi α i j x i, provided that player j randomizes according to distribution function F j (x). Given that player j uses her equilibrium distribution function (7), Player i s payoff is constant and non-negative for any bid in the active bidding support x [δ i j, α i j v j + δ i j ]: π i (x i, x j ) = v i α i j v j δ i j 0. However, it is either negative or lower than the former for any positive bid outside the bidding support: For any positive x i < δ i j her payoff is π i ( x i, x j ) = x i < 0 because bidding less than the head start implies zero winning probability and for any ˆx i > α i j v j + δ i j her payoff is lower than under x : π i ( ˆx i, x j ) = v i ˆx i < v i α i j v j δ i j = π i (x i, x j ).8 Now consider player j. The expected payoff for player j is π j (x j, x i ) = Pr(α i jx j + δ i j > x i )v j x j = F i ( αi j x j + δ i j ) v j x j, provided that player i randomizes according to distribution function F i (x). Given that player i uses her equilibrium distribution function (6), player j s payoff is constant and equal to zero for any bid in the active bidding support x j [0, v j]: π j (x j, x i ) = 0. However, it is negative for any bid outside the bidding support: π j ( x j, xi ) = v j x j < 0 for any x j > v j. 2. Assume that v i α i j v j + δ i j and consider player i. As before the expected payoff for player i is π i (x i, x j ) = F ( xi δ i j ) j vi α i j x i. Given that player j uses her equilibrium distribution function (9), Player i s payoff is constant and equal to zero for any bid in the active bidding support x {0} [δ i j, v i ]: π i (xi, x j ) = 0. However, it is negative for any positive bid outside the bidding support: π i ( x i, x j ) = x i < 0 for any x i (0, δ i j ) because bidding less than the head start implies zero winning probability and π i ( ˆx i, x j ) = v i ˆx i < 0 for any ˆx i > v i. Now consider player j. The expected payoff for player j is π j (x j, xi ) = F ( ) i αi j x j + δ i j v j x j. Given that player i uses her equilibrium distribution function (8), player j s payoff is constant and non-negative for any bid in the active bidding support x j π j (x j, x i ) = v j v i δ i j α i j [ 0, v i δ i j ] α i j : 0. However, it is lower than the former for any higher bid ˆx j > v i δ i j α i j : π j ( ˆx j, x i ) = v j ˆx j < v j v i δ i j α i j = π j (x j, x i ). 8 The case v i = α i j v j + δ i j is degenerate in the sense that distribution functions (6) and (8), as well as (7) and (9) coincide. In this case both players obtain a payoff of zero in equilibrium. 17

19 Based on these equilibrium bidding functions it is straight forward to derive an explicit expression for equilibrium revenue. Lemma 4.2 In the restricted class of bias and head starts where only player i and j are active, the expected equilibrium revenue is Proof. X(α i j, δ i j ) = ( 1 + v j v i ) + δi j, if v i α i j v j + δ i j, α i j v j 2 v 2 i δ2 i j 2α i j v j + (v i δ i j ) 2 2α i j v i, if v i α i j v j + δ i j. Using the distribution functions (6) - (9) the expected equilibrium revenue can be calculated by summing the expected bids of the active players i and j: 1. If v i α i j v j + δ i j then X(α i j, δ i j ) = E[xi ] + E[x j ] = α i j v j +δ i j δ i j αi j v j +δ i j x α i j v j dx + v j α i j x 0 v i dx = α i jv j ( v j ) v i + δi j. δ i j x F i(x) dx + v j x F j(x) dx = x 0 x 2. If v i α i j v j +δ i j then X(α i j, δ i j ) = v i v 2 i δ2 i j 2α i j v j + (v i δ i j ) 2 2α i j v i. x F i(x) δ i j x dx+ vi δi j α i j 0 x F j(x) x dx = v i δ i j x α i j v j dx+ vi δi j α i j 0 α i j x v i dx = Based on Lemma 4.2 the maximal revenue in the all-pay auction with two active players can be obtained by maximizing the respective function with respect to head start δ i j and bias α i j subject to the relevant constraints and then deciding about the identity of the active players. Before we proceed as suggested, we consider first two important special cases; that is, the (unbiased) allpay auctions with head starts (framework HA), and the biased all-pay auction without head starts (framework BA). Addressing these cases separately allows us to evaluate the revenue-enhancing potential of each instrument in isolation which can be contrasted with the optimal combination of bias and head starts that is derived subsequently. 4.2 Optimal head starts As the unbiased all-pay auction with head starts (framework HA) is a special case of the biased all-pay auction with head starts (framework BHA), we can use the equilibrium characterization from Lemma 4.1 and 4.2 by setting, for instance, (α 1,..., α n ) = (1,..., 1), such that all bids are weighted by the same multiplicative bias and the all-pay auction becomes unbiased. 18

20 To find the optimal head start in the restricted class (where exactly two contestants are active) we first solve the maximization problem max δi j 0 X(1, δ i j ) to obtain the optimal head start δ i j and then max {i, j} N X(1, δ i j ) to determine the optimal subset {i, j } of active players. The following result is based on this two-step maximization procedure and shows that depending on the heterogeneity of valuations either the two strongest contestants or the strongest and weakest contestants will be active under the optimal head start. As already mentioned, the respective value for equilibrium revenue under optimal head starts in the restricted class constitutes a lower bound on maximal revenue in the unrestricted class where more than two contestants could be active. 9 We state these results in the subsequent proposition. Proposition 4.3 Optimal head starts in framework HA yield equilibrium revenue that satisfies Proof. { v 2 XHA 2 max + v 1 v 2 2v 1 2, v 2 n + v 1 v } n. 2v 1 2 Setting (α 1,..., α n ) = (1,..., 1) and using the expression in Lemma 4.2 leads to the following maximization problem: v j ( v max X(1, δ j ) v i j) = max i + δi j s.t. δ i j v i v j, δ i j 0 δ i j 0 v 2 i δ2 i j 2v j + (v i δ i j ) 2 2v i, s.t. δ i j v i v j. In the first case the objective function is clearly increasing in δ i j which implies that revenue is maximized by setting the highest feasible head start δ i j = v i v j. In the second case the objective function is decreasing in δ i j because both fractions are non-negative (note that v i δ i j because otherwise player i would not be active) and decreasing. The optimal head start is therefore as low as possible: δ i j = v i v j. Hence, the optimal head start is identical in both cases and balances exactly the difference in valuations, comp. Li and Yu (2012), [26]. The resulting revenue under δ i j = v i v j is then X(1, δ i j ) = v2 j 2v i + v i v j. 2 It remains to determine the identity of players i and j. Note that X(1,δ i j ) ( v i = 1 1 v j ) 2 2 v i > 0, which implies that the head start should be specified such that i = 1; that is, the player with the highest valuation is always active. Expected revenue is then X(1, δ 1 j ) = v2 j 2v 1 + v 1 v j. Note that 2 X(1,δ 1 j ) = v j v 1 1, which is increasing for relatively high values of v 2 j and decreasing for low values v j 9 An alternative approach would be to construct the mixed-strategy equilibria in explicit form for the unrestricted class where more than two player could be potentially active. However, this is challenging because there presumably exists a continuum of potentially not revenue-equivalent equilibria (similar to the symmetric all-pay auction in Baye et al. (1996), [4]). It should also be mentioned that our setup can neither be transformed into a symmetric all-pay auction, nor is it possible to apply Siegel (2014), [34], to (a transformed version of) our setup because his genericity assumption would preclude the use of the optimal head start δ. 19 (10)

21 of v j. Hence, the bias must be specified such that either j = 2 or j = n; that is, either the player with the second highest valuation or the one with the lowest valuation should be active. The first case revenue-dominates the second if X(1, δ 12 ) > X(1, δ 1n ) v2 2 2v 1 + v 1 v 2 > v2 n 2 2v 1 + v 1 v n 2 v 1 v 2 < v n. Hence, if v 1 v 2 < v n then the head start should be specified such that only player 1 and 2 are active and δ 12 = v 1 v 2 0 which yields expected revenue of v2 2 2v 1 +v 1 v 2 2. If v 1 v 2 > v n then the head start should be specified such that player 1 and n are active with δ 1n = v 1 v n 0, which results in expected revenue of v2 n 2v 1 + v 1 v n 2. Allowing any feasible head start (outside of the restricted class where potentially more than two { players are active in equilibrium) } must therefore result in maximal revenue of at least v 2 max 2 2v 1 + v 1 v 2, v2 n 2 2v 1 + v 1 v n. 2 Proposition 4.3 suggests that head starts can be used in two different ways to induce additional revenue. Consider the first case where player 1 and 2 have similar valuations such that v 1 v 2 < v n. In this case a standard all-pay auction involving player 1 and 2 already induces high competitive pressure, both players bid aggressively, and the resulting revenue is high even without a positive head start. Setting an additional small positive head start for player 2 (such that the original ranking of the players is not altered 10 ) will make player 1 slightly more aggressive without having any effect on player 2, see equations eqs. (6) and (7). Hence, in this case a small head start, that neutralizes the (small) difference in valuations between player 1 and 2, is optimal. Consider now the second case where the valuation of the weakest player is low such that v n < v 1 v 2. In this case the expected bid of the weak player n, competing in a standard all-pay auction against the strong player 1, is comparatively low. Hence, most of the revenue in this all-pay auction is due to the bidding of player 1. The fact that one of the active players is weak allows the contest organizer to set a large head start for this player without changing the relative ranking of the two active players. This makes player 1 s bidding more aggressive because she has to overcome the large head start for the weak player which induces additional revenue. Hence, also in this case the difference in valuations of the active players is neutralized under the optimal head start, but in contrast to the previous case the head start is now comparatively large. In this case most of the revenue is therefore due to the aggressive bidding of player 1 who basically competes against the large head start. It is also instructive to analyze how equilibrium payoff of the players is related to revenue extraction. Consider the restrictive class with two active players and an arbitrary head start δ i j < 10 The proof of Proposition 4.3 reveals that it cannot be optimal to choose a head start that reverses the original ranking of players in the sense that v j + δ i j > v i although v i > v j. 20