CContest Architecture with Public Disclosures ∗ Toomas Hinnosaar † May 30, 2019 ‡ Abstract
I study optimal disclosure policies in sequential contests. A contest designerchooses at which periods to publicly disclose the efforts of previous contestants.I provide results for a wide range of possible objectives for the contest designer.While different objectives involve different trade-offs, I show that under many cir-cumstances the optimal contest is one of the three basic contest structures widelystudied in the literature: simultaneous, first-mover, or sequential contest.
JEL : C72, C73, D72, D82
Keywords : contests, sequential games, contest design, rent-seeking, R&D, advertising
In this paper, I study sequential contests and contest-like economic interactions, wherethe payoffs increase with participants’ own efforts and decrease with the total effort. Forexample, lobbyists exert efforts to influence politicians towards desired outcomes. Thekey regulatory decision is transparency—how much and what kind of information shouldbe collected and disclosed to limit the rent-seeking efforts? A fully transparent lobbyingdisclosure rule would lead to sequential effort choices, whereas a non-transparent policyleads to independent choices of lobbying efforts. Similarly, firms entering oligopolisticmarkets invest to increase capacity and the market price decreases with the total capacity.Again, transparency plays a crucial role in these investment decisions and it is naturalto ask whether the investments should be publicly observable or not. Many economicinteractions satisfy the assumptions of the model discussed here, including oligopolies,public goods provision, rent-seeking, research and development, advertising, and sports.The objective of the contest designer depends on the specific economic problem. Twostandard goals discussed in the literature are maximizing the total effort (such as in ∗ I would like to thank Federico Boffa, Juan Carlos Carbajal, Dino Gerardi, Marit Hinnosaar, IgnacioMonzón, and Emil Temnyalov for their comments and suggestions. † Collegio Carlo Alberto, [email protected] . ‡ The latest version: toomas.hinnosaar.net/contest_architecture.pdf . Different countries have adopted different transparency rules regulating lobbying. For example, inthe United States lobbying efforts are all recorded and reported quarterly (Lobbying Disclosure Act,1995; Honest Leadership and Open Government Act, 2007), whereas in the European Union reporting isarranged on a more voluntary basis and on a yearly frequency (European Transparency Initiative, 2005). a r X i v : . [ ec on . T H ] M a y esearch and development contests) or minimizing the total effort (as in rent-seekingcontests). However, other objectives are more natural in other situations. For example,a school’s objective is not to maximize the average effort, but rather to make sure thatall students learn. In fact, the education reform in the US was named the No ChildLeft Behind Act of 2001. Similarly, in firms where efforts are complementary, it is oftenimportant to incentivize the lowest-performing employees, referred to in the businessterminology as to as the bottleneck or the weakest link. In welfare economics, there aretwo standard assumptions. The utilitarian social planner maximizes the total welfare ofthe participants, whereas the Rawlsian social planner maximizes the lowest utility.Different objectives lead to different trade-offs and it is natural that the optimal con-tests in various economic interactions may differ. However, the literature has so faralmost exclusively focused on three standard types of contests. First, simultaneous con-tests , where the players make their choices independently. This is the least informativeform of contest, where no efforts are disclosed. It is the most common type of contestconsidered by the literature, starting with Cournot (1838) in oligopoly theory and Tul-lock (1967, 1974) in contest theory. The second type of contest that has been extensivelystudied is the first-mover contest, where a single first-mover chooses the effort first andthe rest of the players move simultaneously in the second period. In oligopoly theory itwas first analyzed by von Stackelberg (1934) and in contest theory by Dixit (1987). Thethird type of contest that has been considered is the sequential contest , where all effortchoices are public. Then each player observes all the previous effort choices. Simultane-ous contests have been studied by Robson (1990) in the case of large oligopolies, Glazerand Hassin (2000), Hinnosaar (2018), and Kahana and Klunover (2018) in the case ofsequential Tullock contests, and Hinnosaar (2018) with more general payoff functions.The goal of this paper is to understand which types of contests are optimal underdifferent circumstances. On one hand, the paper aims to provide a menu of optimality re-sults that researchers and practitioners could build upon. Perhaps even more importantly,finding contest structures that are optimal under many different sets of assumptions mayexplain why these types of contests are often used.The main difficulty in studying sequential contests is tractability. Sequential gameswith non-quadratic payoffs are typically difficult to solve because best-response functionsare non-linear and therefore the standard backward-induction approach leads to increas-ingly complex expressions. This problem is even more pronounced in contest design, wherethe goal is to compare all possible contests. In this paper, I build on recent progress inaggregative games (Jensen, 2010; Martimort and Stole, 2012; Acemoglu and Jensen, 2013)and sequential contests (Hinnosaar, 2018, 2019) to overcome the tractability issues.The main finding of the paper is that under many circumstances the optimal contestis one of the three basic contest structures widely studied in the literature. In particular, Iconsider minimizing and maximizing eight different objectives: total effort, total welfare,lowest effort, lowest payoff, highest effort, highest payoff, effort inequality, and payoffinequality. I show that nine out of the sixteen maximization problems are solved eitherby the simultaneous contest or by the sequential contests. With the four out of theremaining five objectives I show that when the number of players is sufficiently large,they are maximized by one of the three basic contest structures mentioned above. I also For literature reviews on dynamic contests, see Konrad (2009) and Vojnović (2015). The paper contributes to both the contest design and the information design litera-tures. Earlier papers on contest design have focused on contests with private informationand have studied how to arrange contests into subcontests and which prizes to offer tomaximize either total effort or highest effort (Glazer and Hassin, 1988; Taylor, 1995; Cheand Gale, 2003; Moldovanu and Sela, 2001, 2006; Olszewski and Siegel, 2016; Bimpikiset al., 2019). In this paper, I study contest design with full information. The designermay have different objectives and can choose to disclose less or more information aboutthe players’ efforts. There is growing literature on information design (Kamenica andGentzkow, 2011; Bergemann and Morris, 2019), which has mostly focused on revealinginformation about the state of the world or private information. This paper is more inline with recent papers that have also studied disclosure of the actions of other players(Doval and Ely, 2019; Ely and Szydlowski, 2019; Gallice and Monzón, 2019).The paper is organized as follows. Section 2 describes the model. Section 3 providesall results, describing the contests that minimize and maximize eight different objectivefunctions. Section 4 summarizes and concludes. Proofs are in the appendix.
There is a finite number n players, N = { , . . . , n } . Players arrive sequentially andeach player i ∈ N chooses effort level x i ≥ x = ( x , . . . , x n ), the payoff of player i is u i ( x ) = x i h ( X ), where X = P ni =1 x i denotes thetotal effort and h ( X ) is the marginal benefit of effort, which is common to all players (Idiscuss its properties below).The contest designer chooses points of time at which to publicly disclose the cumulativeeffort of all players that have already arrived. The disclosure rule is chosen before effortchoices and is commonly known. Each such disclosure rule induces a partition of players I = ( I , . . . , I T ), where I t is the set of players arriving between disclosures number t − t . I refer to the set of players I t as players arriving in period t . Then player i ∈ I t observes the cumulative effort of all players arriving in previous periods, X t − = P t − s =1 P j ∈I s x j , and makes a choice simultaneously with all players arriving in period t .As all players are identical, the disclosure rule is equivalently defined by a vector n =( n , . . . , n T ), where n t = I t , so that P Tt =1 n t = n . The marginal benefit function h ( X ) is a strictly decreasing and continuously differ-entiable function in [0 , X ], where X > h ( X ) = 0.Moreover, I impose two technical assumptions, which are discussed in appendix A. Thefirst assumption guarantees that the best-response functions are well-behaved so that asubgame-perfect Nash equilibrium exists and is unique. The second assumption guar-antees that the efforts are higher-order strategic substitutes near equilibrium, i.e. highereffort of an earlier-mover discourages efforts by followers both directly and indirectly I provide conditions under which it is maximized by a contest where the first two players are movingsimultaneously and all the remaining players are arranged sequentially. Alternatively, the contest designer’s problem can be interpreted as dividing players between periods.
Example 1 (Sequential Tullock Contest) . Players choose efforts to compete for a prizewith value v > and have constant marginal cost c > . The probability of winning isproportional to efforts. The payoffs can be expressed as u i ( x ) = x i X v − cx i = x i h ( X ) , where h ( X ) = vX − c . Example 2 (Tragedy of the Commons) . The total amount of resources is and eachplayer chooses private consumption x i ≥ . Marginal benefit of consumption is linearlydecreasing in the total consumption, i.e. h ( X ) = 1 − X . Example 3 (Oligopoly with a Non-linear Demand Function) . Suppose that n oligopolistshave identical constant marginal costs c ≥ and produce identical good. Each oligopolistchooses a quantity (capacity) and the inverse demand function is P ( X ) = a − b X for some a > and b > . The profit of firm i is u i ( x ) = x i h ( X ) , where h ( X ) = P ( X ) − c . To compare different contests, I define informativeness as a partial order on all n -player contests. Intuitively, a contest n is equivalently defined by a partition I of players.Each additional disclosure divides the players in one period I t into two groups. Thereforethe new contest provides strictly more information to some players than the previouscontest, whereas no player receives less information. Formally, I say that the contest b n ismore informative than the contest n , if the corresponding partition b I is finer than I .In particular, the simultaneous contest n = ( n ) that does not provide any informationabout the efforts of other players is less informative than any other contest. In the otherextreme, the sequential contest n = (1 , , . . . ,
1) discloses efforts after each player andis therefore more informative than any other contest. Finally, the first-mover contest n = (1 , n −
1) discloses information after the first player, whereas all other players maketheir moves simultaneously. The first-mover contest is less informative than any other single-leader contest n = (1 , n , . . . , n T ). The most common objectives considered in the literature are minimizing or maximizingthe total equilibrium effort X ∗ . Hinnosaar (2018) showed that the total effort is strictlyincreasing in the informativeness of the contest, i.e. each additional disclosure increasesthe total equilibrium effort X ∗ . Therefore the simultaneous contest minimizes and the se-quential contest maximizes the total equilibrium effort. The intuition for this result comesfrom the discouragement effect. Efforts in contests are strategic substitutes. Therefore, This example can be interpreted as private provision of public goods, where each player dividesendowment ω i between private consumption x i and public good contribution g i = ω i − x i . Suppose thatthe total endowment is 1, so that the total quantity of public good is P ni =1 g i = 1 − X . The payoff isa multiplicative function of both private consumption and public good, for example u i ( · ) = x i (1 − X ).The marginal benefit function h ( X ) can be non-linear. x i of an earlier-mover i would decrease the total effort X and therefore increase i ’s payoff.This would be a profitable deviation and thus violate equilibrium conditions.Let us now look at total equilibrium welfare. For example, in the case of a normalizedTullock contest the total welfare is W ( x ∗ ) = P ni =1 u i ( x ∗ ) = 1 − X ∗ . This expressionis decreasing in X ∗ , so maximized by the simultaneous contest and minimized by thesequential contest. The intuition is simple—in a Tullock contest, the total prize is fixed.Additional players and additional disclosures lead to a higher total effort, which is costlyand therefore welfare-reducing. The following proposition shows that these conclusionsgeneralize to arbitrary payoffs and an arbitrary number of players. Proposition 1 (Total Effort and Total Welfare) . The total equilibrium effort X ∗ is min-imized by the simultaneous contest and maximized by the sequential contest. The totalequilibrium welfare W ( x ∗ ) = P ni =1 u i ( x ∗ ) is minimized by the sequential contest and max-imized by the simultaneous contest. We need to first determine who is the player choosing the lowest effort in equilibriumand who gets the lowest payoff. The answer comes from the earlier-mover advantage—equilibrium efforts and payoffs of earlier-movers who are observed by more players arehigher than the corresponding efforts and payoffs of later-movers. Therefore player n chooses the lowest effort and gets the lowest payoff with any n -player contest n . Thereason for this result is again the discouragement effect—earlier players choose higherefforts to discourage later players from exerting effort whenever their efforts are madepublic. By doing so, earlier players ensure higher and later players get lower payoffs.The following proposition shows that both the lowest effort and the lowest payoffare minimized and maximized by the same contests as the total welfare—sequential andsimultaneous contests respectively. The intuition is simple. As discussed above, eachadditional disclosure leads to higher efforts by earlier-movers and lower efforts by later-movers. Therefore the discouragement effect to the last player n is increased with eachdisclosure. Moreover, additional disclosures lead to higher total equilibrium effort X ∗ ,which reduces the payoff of the last player even further. Proposition 2 (Lowest Effort and Lowest Payoff) . Lowest equilibrium effort min i { x ∗ i } and lowest equilibrium payoff min i { u i ( x ∗ ) } are minimized by the sequential contest andmaximized by the simultaneous contest. The earlier-mover advantage result shows that the player who exerts the highest effortand gets the highest payoff is the first player. It is easy to see that the effort of player 1 See appendix A for a formal discussion.
5s minimized by the simultaneous contest. This is the contest which minimizes the totaleffort X ∗ and where all efforts are equal. In all other contests, the total effort is higherand the effort of player 1 is higher than the average, due to the discouragement effect,which means higher than in the simultaneous contest. Indeed, this is what figure 1aconfirms—for each n , the highest equilibrium effort is minimized by the simultaneouscontest.Maximization of the highest effort x ∗ is the first objective, where the optimal solution isnot always one of the extremes, i.e. simultaneous or sequential contest. For example, let usconsider three-player Tullock contests with a single leader. There are two such contests.The first-mover contest n = (1 ,
2) has a total equilibrium effort X ∗ = 0 .
75 and the highesteffort x ∗ = 0 . b n = (1 , , c X ∗ ≈ . b x ∗ ≈ . < x ∗ . Numerical calculations summarized by figure 1a showthat the contest that maximizes the highest equilibrium effort in a Tullock contest with2 < n ≤
12 players is always one with a single leader and followers that are arrangedpairwise, i.e. n = (1 , , . . . ,
2) when n is odd or n = (1 , , . . . , ,
1) when n is even. Arranging the followers pairwise leads to a strictly higher effort by the first player thanin any other contest, including the sequential contest.The observation that there should be a single first-mover is general and intuitive.Making first-movers’ effort observable to all players increases total effort only becauseof increased effort by the first player, so there is no trade-off. As discussed above, eachadditional disclosure has two opposite effects and the effect on highest effort dependson the weights to direct and indirect influences. For example, with exponential function h ( X ) = h − X − − i the first-mover contest n = (1 ,
2) gives highest effort x ∗ ≈ . b n = (1 , ,
1) gives strictly larger highest effort b x ∗ ≈ . > x ∗ . The literature on sequential oligopolies (starting from (Daughety,1990)) has found that if the demand function is linear, then Stackelberg leaders behaveas if there are no followers, i.e. x ∗ would be independent on the number and arrangementof followers. Hinnosaar (2019) provides the most general formulation of this Stackelbergindependence result, but also shows that this result holds only when h ( X ) function islinear. Therefore the formal statement, in this case, provides only a qualitative result,showing that n = 1. Indeed, as any single-leader is optimal in the case of linear h ( X ),the result cannot exclude any such contest from being optimal.Finally, in the case when n is large enough, we can say more regardless of the exactdetails of the payoff function. Namely, in this case, X ∗ is close to its upper bound andtherefore assuming that h ( X ) function is smooth, it can be closely approximated by a In numerical examples, I normalize the prize and marginal cost to 1, so that the payoff of player i is u i ( x ) = x i X − x i . For figures, I consider P n =2 n − = 4094 possible contests with 2 to 12 players. I usethe Matlab code available at toomas.hinnosaar.net/contests/ . The location of the second single-player period does not affect x ∗ . (a) Highest effort. Green dash-dotted line indi-cates the single-leader contest with pairwise fol-lowers. The slightly lower red dash-dotted lineindicates the first-mover contest. (b) Highest payoff. Green dash-dotted line indi-cates contests n = (1 , n −
1) and red long-dashedline indicates contests (2 , , . . . , Figure 1: Highest effort max i { x ∗ i } and highest payoff max i { u i ( x ∗ ) } in all normalizedTullock contests with 2 ≤ n ≤
12 players. The solid black line indicates sequentialcontests and dashed blue line simultaneous contests.linear function. In particular, this means that the conclusion from linear demand extendshere—any large single-leader contest leads to the highest effort that is arbitrarily closeto the maximized highest effort. Figure 1a also illustrates that while all single-leadercontests are optimal in the limit, for the sequential contest the convergence is very fast,whereas for the first-mover contest the convergence is much slower. Proposition 3 (Highest Effort) . The highest effort max i { x ∗ i } is minimized by simulta-neous contest. If contest n maximizes the highest effort then n = 1 . If h ( X ) is linear or n is large enough, then any single-leader contest ensures highest effort that is arbitrarilyclose to maximum. Maximizing highest equilibrium payoff u ( x ∗ ) balances a similar trade-off, but thedownward force through reduced marginal benefit h ( X ∗ ) is more pronounced. Not onlydoes it push effort down, but it also reduces h ( X ∗ ), which directly reduces all payoffs. Thismeans that we would naturally expect the optimal contest to have fewer disclosures thanthe contest that maximizes the highest equilibrium effort. Indeed, figure 1b shows thatamong all Tullock contests with up to 12 players, the first-mover contest n = (1 , n −
1) isalways optimal.The proposition below formalizes this qualitative property—a contest that maximizesthe highest equilibrium effort must have a single leader and has to be (weakly) less infor-mative than a contest that maximizes the highest payoff. For example, in the case of linear h ( X ) function, one maximizer of the highest effort is the first-mover contest n = (1 , n − The same arguments can be used to find the contest that minimizes or maximizes the i -th highesteffort or payoff when h ( X ) is linear or n is large enough. Appendix D provides these results.
7f linear h ( X ) function or a large number of players, we can therefore uniquely determinethe optimal contest. It is always n = (1 , n − u ( x ∗ ) requires balancing a different trade-off. As wesaw above, additional disclosures lead to larger highest effort due to a stronger discour-agement effect. This would imply that minimizing the highest payoff requires a relativelyuninformative contest. On the other hand, additional disclosures lead to a higher totaleffort, which reduces all payoffs. Let us compare for example three-player Tullock con-tests n = (2 ,
1) and b n = (1 , , n gives total effort X ∗ = 0 .
75, highest effort x ∗ ≈ . u ( x ∗ ) ≈ . b n gives total effort c X ∗ ≈ . > X ∗ , but highest effort b x ∗ ≈ . x ∗ andtherefore highest payoff u ( b x ∗ ) = 0 . > u ( x ∗ ). Figure 1b shows that in the case ofa Tullock contest with up to 12 players, contest n = (2 , , . . . ,
1) minimizes the highestpayoff.The formal result below provides only one general qualitative property for the contestthat minimizes the highest payoff: the number of players in the first period must beweakly higher than in any other period, i.e. n ≥ n t for all t . If this would not be thecase, the designer could rearrange the groups while keeping the total effort unchangedand decreasing the highest effort, which would reduce the highest payoff. Why can’t wesay more? Other possible manipulations of the contests, such as moving a player fromperiod 1 to period 2 or splitting up the period by an additional disclosure, lead to twoopposite effects: on one hand they increase the total effort in the contest, which reducesall efforts as well as the payoffs directly, but on the other hand they increase the influencethat the first player has through the discouragement effect. The relative magnitudes ofthese effects depend on the shape of the function h ( X ).In the case of linear h ( X ) or large contests, the optimal contest is again uniquelydetermined: it is always (2 , , . . . , This confirms the finding from figure 1b.
Proposition 4 (Highest Payoff) . If contest n = ( n , . . . , n T ) minimizes the highest equi-librium payoff max i { u i ( x ∗ ) } , then n ≥ n t for all t . If h ( X ) is linear or n is large enough,then the optimal contest is n = (2 , , , . . . , .If contest n maximizes the highest equilibrium payoff, then n = 1 and n is not moreinformative than any maximizer of the highest effort. If h ( X ) is linear or n is largeenough, then the optimal contest is n = (1 , n − . In this final part I study equilibrium effort inequality, defined as max i { x ∗ i } − min i { x ∗ i } ,and equilibrium payoff inequality, defined as max i { u i ( x ∗ ) } − min i { u i ( x ∗ ) } . By the earlier-mover advantage result discussed above, these expressions are equivalent to x ∗ − x ∗ n and u ( x ∗ ) − u n ( x ∗ ) respectively. It is easy to see that both expressions are minimized by the This comes from the independence of permutations result from Hinnosaar (2018), which shows thattotal equilibrium effort X ∗ is independent of permutations of vector n = ( n , . . . , n T ). Remark: when n is very large, h ( X ∗ ) → h ( X ) = 0 and therefore u i ( x ∗ ) → i , which meansthat technically all contests are approximately optimal. However, as figure 1b illustrates, in some conteststhis convergence is much faster than in others. By optimality I mean here the contest for which the valueof the objective is maximized for large but finite values of n . n = (1 , b n = (1 , , x ∗ = 0 . x ∗ = 0 . b n leads to slightly lowerhighest effort b x ∗ ≈ . b x ∗ ≈ . h ( X ) is linear or n is large, the most informative maximizer of the highest effort is the sequential contest.Therefore, we can immediately conclude that the sequential contest must also be theunique maximizer of the equilibrium effort inequality, which confirms the result fromfigure 2a. (a) Effort inequality (b) Payoff inequality. Green dot-dashed line in-dicates the first-mover contest. Figure 2: Effort inequality max i { x ∗ i } − min i { x ∗ i } and payoff inequality max i { u i ( x ∗ ) } − min i { u i ( x ∗ ) } in all normalized Tullock contests with 2 ≤ n ≤
12 players. The solid blackline indicates sequential contests and simultaneous contests lie on the horizontal axis.Finally, maximizing payoff inequality has an additional first-order effect: disclosuresincrease X ∗ and thus reduce all payoffs. This effect mechanically reduces payoff inequalityas u ( x ∗ ) − u ( x ∗ ) = ( x ∗ − x ∗ n ) h ( X ∗ ). In the three-player examples above, x ∗ − x ∗ n =0 . < b x ∗ − b x ∗ ≈ . h ( X ∗ ) ≈ . < c X ∗ ≈ . n = (1 ,
2) is the three-player contest that maximizes payoffinequality. Figure 2b shows that in the case of Tullock contests, this effect is strongenough so that for all n ≤
12, the contest that maximizes payoff inequality is always the9rst-mover contest n = (1 , n − h ( X ) function islinear or the contest is large, these conditions, unfortunately, are not very informative. Allsingle-leader contests are more informative than the highest payoff maximizer (1 , n − n = (1 , n − Proposition 5 (Effort Inequality and Payoff Inequality) . Both the equilibrium effortinequality max i { x ∗ i } − min i { x ∗ i } and the equilibrium payoff inequality max i { u i ( x ∗ ) } − min i { u i ( x ∗ ) } are minimized by the simultaneous contest.If contest n maximizes the equilibrium effort inequality, then n = 1 and n is notless informative than any maximizer of the highest effort. If h ( X ) is linear or n is largeenough, then n is the sequential contest.Any contest n that maximizes the equilibrium payoff inequality must satisfy n = 1 .Moreover, it cannot be less informative than any maximizer of the highest payoff and itcannot be more informative than any maximizer of the effort inequality. If h ( X ) is linearor n is large enough, then n is the first-mover contest. In this paper, I studied contest architecture with public disclosures. Additional disclosuresinduce the discouragement effect that leads to higher efforts by earlier-movers, lowerefforts by later-movers, and higher total effort. These effects have different implicationsto different objectives a contest designer may have. The optimal contests are summarizedby table 1. Perhaps the most surprising finding of the paper is that for most objectivesthe optimal contest is one of the three standard contests the previous literature has beenfocused on—the simultaneous contest, the sequential contest, or the first-mover contest. Again, subject to a remark: when n becomes large then all payoffs converge to zero and thereforeany contest approximately maximizes the payoff inequality. However as figure 2b clearly illustrates, u ( x ∗ ) − u n ( x ∗ ) in the first-mover contest converges to zero much faster than other contests. The optimalcontest here means that it maximizes the payoff inequality with finite but arbitrarily large n . X ∗ = P ni =1 x ∗ i Simultaneous SequentialTotal welfare P ni =1 u i ( x ∗ ) Sequential SimultaneousLowest effort min i { x ∗ i } Lowest payoff min i { u i ( x ∗ ) } Highest effort max i { x ∗ i } Simultaneous Single-leader † Highest payoff max i { u i ( x ∗ ) } (2 , , . . . , † First-mover † Effort inequality max i { x ∗ i } − min i { x ∗ i } Simultaneous Sequential † Payoff inequality max i { u i ( x ∗ ) } − min i { u i ( x ∗ ) } First-mover † Table 1: Summary of the optimal contests for the 8 × † are optimal at least for (1) Tullock contests with n ≤ h ( X ) function for any n , and (3) large contests. The simultaneouscontest is ( n ), the sequential contest is (1 , , . . . , , n − , n , . . . , n T ) for all n , . . . , n T (including the sequentialand the first-mover contests). 11 eferences Acemoglu, D. and M. K. Jensen (2013): “Aggregate comparative statics,”
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A Notation, Assumptions, and Useful Results
In this appendix, I describe additional notation and formalize technical assumptions thatare necessary for formal proofs. I then describe a few useful results from the literature.
A.1 Notation
Instead of working directly with the marginal benefit function h ( X ) it is more convenientto work with the following functions g, g , . . . , g T . Each of these function captures aparticular curvature property of h function and can be interpreted as a higher-orderstrategic substitutability term. Definition 1 (Functions g and g k ) . Let g ( X ) = − h ( X ) h ( X ) . With this, let us define g , . . . , g T recursively as g ( X ) = g ( X ) and g k +1 ( X ) = − g k ( X ) g ( X ) . For example, in the case of linear h ( X ) = a ( X − X ), g ( X ) = X − X = g k ( X ) for all k . In the case of normalized Tullock payoffs u i ( x ) = x i X − x i and so h ( X ) = X − g ( X ) = X (1 − X ). Therefore g k ( X ) is a polynomial of degree k + 1.For a contest n let us define its measures of information. Intuitively, it is a vector ofintegers, that captures the number of direct and indirect observations of other players’efforts. Let me describe the construction with an example of contest n = (1 , , S ( n ) = 1 + 2 + 3 = 6.The second measure is the number of direct observations of other players’ efforts, S ( n ) =1 · · · S ( n ) = 1 · · S ( n ) = (6 , ,
6) in theexample.
Definition 2 (Measures of Information S ( n )) . For a given contest n = ( n , . . . , n T ) , itsmeasures of information are defined as S ( n ) = ( S ( n ) , . . . , S T ( n )) , where S k ( n ) is thesum of products of all possible k -combinations of the set { n , . . . , n T } . For each period t , let n t denote the subcontest starting after period t , i.e. n t =( n t +1 , . . . , n T ). Then S ( n t ) are denoted analogously to the original contest. For example,if n = (1 , , n = (2 ,
3) and n = (3), therefore S ( n ) = (5 ,
6) and S ( n ) = (3).Finally, let me define functions f , . . . , f T as follows f t ( X ) = X − T − t X k =1 S k ( n t ) g k ( X ) . (1)These functions take the role of inverted best-response functions that I discuss below.Intuitively, the inverse functions f − t ( X t ) describe what is the total effort X when afterperiod t the cumulative effort is X t and all players after period t behave optimally. Note that when t = 0, then n t = n is the whole contest. .2 Technical Assumptions The first technical assumption is sufficient to guarantee that the best-response functions ofplayers are well-behaved. This is a sufficient assumption to guarantee both the existenceand uniqueness of the subgame-perfect Nash equilibrium for contest n . Assumption 1 (Inverted Best-Responses are Well-Behaved) . For all t = 0 , . . . , T − ,the function f t defined by equation (1) has the following properties:1. f t ( X ) = 0 has a root in [ X t +1 , X ] . Let X t be the highest such root.2. f t ( X ) < for all X ∈ [ X t +1 , X t ) .3. f t ( X ) > for all X ∈ [ X t , X ] .Moreover, X ∈ (0 , X ) . The second assumption states that the efforts are higher-order strategic substitutesnear equilibrium. This assumption is stronger than the standard assumption of strategicsubstitutes, which requires that higher effort of an opponent reduces the incentive fora player to exert effort. Higher-order influences matter in sequential games, where theearlier-mover may not only influence a follower directly but indirectly through changingthe behavior of some players between them.
Assumption 2 (Higher-Order Strategic Substitutes) . g k ( X ∗ ) > for all k = 2 , . . . , T . Hinnosaar (2018) shows that the two assumptions are satisfied for Tullock contestswith at least three players as well as in many other situations with well-behaving marginalbenefit functions.
A.3 Useful Results
The proofs rely on a few useful results from earlier works. The main tool for the analysis isthe characterization theorem, which characterizes the unique equilibrium for any contests.
Theorem 1 (Characterization Theorem (Hinnosaar, 2018)) . Under assumption 1 thetotal equilibrium effort X ∗ is the highest root of f ( X ) = 0 , where f ( X ) = X − T X k =1 S k ( n ) g k ( X ) . (2) The equilibrium effort of player i in period t is x ∗ i = g ( X ∗ ) + P T − tk =1 S k ( n t ) g k +1 ( X ∗ ) . Informativeness and equilibria:
As Hinnosaar (2018) shows, this result has strongimplications for equilibrium behavior. The total equilibrium effort X ∗ increases with theinformativeness of the contest. Therefore it is minimized by the simultaneous contest (theleast informative contest) and maximized by the sequential contest (the most informativecontest). Function f T ( X ) = X , therefore it has exactly one root, X T = 0. ndependence of permutations: permutations in n do not change the total equilib-rium effort X ∗ . For example, the contest n = (1 , ,
3) has exactly the same total effortas the contest b n = (1 , ,
2) because S ( b n ) = (6 , ,
6) = S ( n ). Of course, individual effortsmay be affected, as they also depend on subcontest. But note that permutations withinthe subcontest do not affect individual equilibrium effort either: in the examples here, n = (2 ,
3) and b n = (3 ,
2) is a permutation. Therefore b x ∗ = x ∗ . Earlier-mover advantage: take two players, i from period t and j from a later period s > t . Then x ∗ i > x ∗ j and u i ( x ∗ ) > u j ( x ∗ ). As explained in the text, this means thathighest effort is always max i { x ∗ i } = x ∗ , lowest effort min i { x ∗ i } = x ∗ n , highest payoffmax i { u i ( x ∗ ) } = u ( x ∗ ), and lowest playoff min i { u i ( x ∗ ) } = u n ( x ∗ ). Large contests:
Moreover, Hinnosaar (2019) shows that in the limit when the numberof players becomes large, the equilibrium behavior converges to specific functional form.Intuitively, as the number of players becomes large and the total equilibrium effort in-creases with the number of players, it is quite natural that the total equilibrium effortconverges to its full dissipation limit X . As the function h ( X ) is smooth, we can approx-imate it with a linear function near X arbitrarily closely when the number of players islarge. Therefore it is not surprising that the equilibrium behavior of individual playersconverges to equilibrium behavior of the same game but with linear h ( X ) function. Theorem 2 (Competitive Limits (Hinnosaar, 2019)) . Fix a sequence n = ( n , . . . , n T ) and let us increase n t at a particular period t , while keeping other elements of the sequence n unchanged. Then lim n t →∞ X ∗ = X and for each player i arriving in period s , lim n t →∞ x ∗ i = ∀ s ≥ t, X Q sk =1 (1+ n k ) ∀ s < t. (3)I use this result as follows. Note that n t is not determined, as with different n the con-test designer may want to use different disclosures. I use equation (3) as an approximationfor the equilibrium behavior, i.e. each player i ∈ I s chooses x ∗ i ≈ X Q sk =1 (1 + n k ) . (4)Note that this approximation also implies that X ∗ = n X i =1 x ∗ i ≈ X − X Q Tk =1 (1 + n k ) . (5)Since h ( X ) is a smooth function and X ∗ → X , we can approximate h ( X ∗ ) linearly near X , i.e. h ( X ∗ ) ≈ h ( X ) + h ( X )( X − X ∗ ) = αX Q Tk =1 (1 + n k ) , where α = − h ( X ) > . (6)16 Proofs
B.1 Proof of Proposition 1 (Total Effort and Total Welfare)
Results for X ∗ follow from theorem 1. The total welfare is W ( x ∗ ) = X ∗ h ( X ∗ ), so dW ( x ∗ ) dX ∗ = h ( X ∗ ) + X ∗ h ( X ∗ ) ≥ ⇐⇒ X ∗ ≥ − h ( X ∗ ) h ( X ∗ ) = g ( X ∗ ) . (7)Note that by theorem 1, each x ∗ i ≥ g ( X ∗ ), therefore this condition is always satisfied.Each disclosure increases X ∗ and therefore decreases total welfare. B.2 Proof of Proposition 2 (Lowest Effort and Lowest Payoff)
As argued in the text, min i { x ∗ i } = x ∗ n and by theorem 1, x ∗ n = g ( X ∗ ). Since g ( X ∗ ) isstrictly decreasing in X ∗ , the lowest effort x ∗ i is minimized when X ∗ is maximized, i.e. bysequential contest, and maximized when X ∗ is minimized, i.e. by simultaneous contest.Similarly, min i { u i ( x ∗ ) } = u n ( x ∗ ) = x ∗ n h ( X ∗ ) = g ( X ∗ ) h ( X ∗ ), where both g and h aredecreasing in X ∗ , which leads to the same conclusion. B.3 Proof of Proposition 3 (Highest Effort)
Let us first consider minimization of the highest equilibrium effort max i { x ∗ i } = x ∗ . Let X sim be the total equilibrium effort from simultaneous n -player contest ( n ). Then clearlythe highest equilibrium effort x simi = X sim n . Now take any non-simultaneous contest.As this contest is strictly more informative than the simultaneous contest, X ∗ > X sim .Moreover, by the earlier-mover advantage result, the highest effort is strictly larger thanthe average, which proves the claim, since x ∗ > X ∗ n > X sim n = x sim . (8)Suppose now that n maximizes the highest effort, which by theorem 1 ismax i { x ∗ i } = x ∗ = g ( X ∗ ) + T − X k =1 S k ( n ) g k +1 ( X ∗ ) . (9)I first claim that n < n t for all t . If this is not the case, we can take a permutation of n ,which leaves X ∗ unchanged, but increases the information measures of the subcontest n ,therefore increasing the highest effort. Next, I claim that n = 1. Suppose that this is notthe case, i.e. 2 ≤ n But the previous observation, n ≥ n ≥
2. Consider an alternativecontest b n = ( n − , n + 1 , n , . . . , n T ). Then b n = ( n + 1 , n , . . . , n T ). Since n ≥ n ,we have ( n − n + 1) = n n − n + n − < n n . Therefore S ( b n ) < S ( n ) and S ( b n ) < S ( n ). Therefore c X ∗ < X ∗ . This means that each g k ( c X ∗ ) > g k ( X ∗ ). Moreover, clearly S ( b n ) > S ( n ). Therefore b x ∗ > x ∗ , which meansthat x ∗ was not the maximal highest effort.17inally, when the h ( X ) is linear or when number of players is large, then by the argu-ments above, the highest effort is (approximately) x ∗ ≈ X n . Clearly, this is maximizedby setting n = 1, i.e. a disclosure right after the first player. All other disclosures haveno impact when h ( X ) is linear and a negligible impact when n is large enough. B.4 Proof of Proposition 4 (Highest Payoff)
Let n be a contest that minimizes the highest equilibrium payoff max i { u i ( x ∗ ) } = u ( x ∗ ) = x ∗ h ( X ∗ ). I claim that n ≥ n t for all t . If this is not the case, then there exists apermutation of n such that x ∗ is increased and X ∗ and thus h ( X ∗ ) is unchanged. Thiswould violate the optimality of n . When n is large, then by the highest equilibrium payoffis approximately u ( x ∗ ) = x ∗ h ( X ∗ ) ≈ X n αX Q Ts =1 (1 + n s ) . (10)Minimizing this objective is equivalent to maximizing (1 + n ) Q Ts =2 (1 + n s ). We can doit in two steps. First, fix n ≥ Q Ts =2 (1 + n s ) subject to constraint that P Ts =1 n s = n − n . Splitting each n s ≥ n s > n s > n s )(1 + n s ) =1 + n s + n s + n s n s > n s + n s . Therefore the maximized product Q Ts =2 (1 + n s ) = 2 n − n and so the maximization problem is2 n max ≤ n ≤ n (1 + n ) − n . (11)Treating n as a continuous variable and differentiating the objective gives − − n (1 + n ) ((1 + n ) log 2 − ≤ ⇐⇒ n ≤ − ≈ . . (12)The objective is decreasing in n for all n ≥
2, so there are only two candidates for themaximizer: either n = 1 or n = 2. Direct comparison gives(1 + 1) − = 2 < (1 + 2) − = 94 . (13)Therefore the optimal large contest is n = (2 , , . . . , n be a maximizer of the highest payoff u ( x ∗ ) = x ∗ h ( X ∗ ). The same argumentsas in the proof of proposition 3 show that n must be equal to 1 (first n ≤ n t for all t ,otherwise can take a permutation, and then if n >
1, splitting it makes the contest lesshomogeneous and thus decreases X ∗ , which increases total payoff while also increasingplayer 1’s share in it). Now, let b n be a contest that maximizes the highest effort. Let thecorresponding effort profile be b x ∗ . Then by definition x ∗ ≤ b x ∗ and u ( x ∗ ) = x ∗ h ( X ∗ ) ≥ b x ∗ h ( c X ∗ ) = u ( b x ∗ ), which cannot be satisfied unless X ∗ ≤ c X ∗ . Therefore n cannot bemore informative than b n .The final claim follows from the arguments above. With linear h ( X ) or large n , onecontest that maximizes the highest effort is b n = (1 , n − n = 1 and cannot be more informative than b n , it must coincide with b n .18 .5 Proof of Proposition 5 (Effort Inequality and Payoff In-equality) As argued in the text, the simultaneous contest is the only contest where the efforts andpayoffs of all players are equal, so it is the unique minimizer of both the equilibrium effortinequality max i { x ∗ i } − min i { x ∗ i } = x ∗ − x ∗ n as well as the equilibrium payoff inequalitymax i { u i ( x ∗ ) } − min i { u i ( x ∗ ) } = u ( x ∗ ) − u n ( x ∗ ).By the same arguments as with the highest effort and with the highest payoff, we getthat n = 1 both with effort inequality and payoff inequality maximizers. Note that since x ∗ is independent of permutations of the subcontest n and x ∗ n = g ( X ∗ ) is independentof permutations in the whole contest, both inequalities are independent of permutationsin n .Suppose that n is an effort inequality maximizer and let the corresponding equilibriumeffort profile be x ∗ . Take any highest effort maximizer b n and let the corresponding effortprofile be b x . Then by construction, x ∗ ≤ b x ∗ and x ∗ − x ∗ n ≥ b x ∗ − b x ∗ n , which impliesthat g ( X ∗ ) = x ∗ n ≤ b x ∗ n = g ( c X ∗ ). As g ( X ∗ ) is decreasing in X ∗ it implies X ∗ ≥ c X ∗ ,which means that n cannot be less informative than b n . Finally, if h ( X ) is linear or n islarge, then any single-leader contest maximizes the highest effort. One such contest is thesequential contest, which is more informative than any other contest. Therefore it mustbe the unique maximizer of the effort inequality in these cases.Now, let n be the maximizer of the equilibrium payoff inequality. As argued above, n = 1. Take any maximizer of the highest payoff b n . Then u ( x ∗ ) − u n ( x ∗ ) ≥ u ( b x ∗ ) − u n ( b x ∗ ) ≥ u ( x ∗ ) − u n ( b x ∗ ) and therefore g ( X ∗ ) h ( X ∗ ) = u n ( x ∗ ) ≤ u n ( b x ∗ ) = g ( c X ∗ ) h ( c X ∗ ),which implies that X ∗ ≥ c X ∗ . This means that n cannot be less informative than b n . Next,suppose that b x maximizes the effort inequality. Then ( x ∗ − x ∗ n ) h ( X ∗ ) ≥ ( b x ∗ − b x ∗ n ) h ( c X ∗ ) ≥ ( x ∗ − x ∗ n ) h ( c X ∗ ), which implies h ( X ∗ ) ≥ h ( c X ∗ ) or equivalently X ∗ ≤ c X ∗ . Therefore n cannot be more informative than b n .When h ( X ) is linear or n is large, the results above do not tell us much, since none ofthe single-leader contests is less informative than (1 , n −
1) (the maximizer of the highestpayoff) and no contest is more informative than the sequential contest (the maximizer ofthe effort inequality). Therefore we have to proceed with direct proof. The equilibriumpayoff inequality is (approximately) u ( x ∗ ) − u n ( x ∗ ) ≈ X n αX Q Ts =1 (1 + n s ) − X Q Ts =1 (1 + n s ) αX Q Ts =1 (1 + n s )= αX (1 + n ) Q Ts =2 (1 + n s ) " − Q Ts =2 (1 + n s ) . (14)I already proved that n = 1 for any payoffs, including linear. The remaining questionis how to arrange the remaining n − n = ( n , . . . , n T ). Let usdenote Y ( n ) = Q Ts =2 (1+ n s ) for brevity. Then Y ( n ) is decreasing with each disclosure.Combining this with the fact that the optimal contest is not simultaneous, we get Y ( n ) ≤ n − n ≤ . Now, note that the expression Y ( n )[1 − Y ( n )] is strictly increasing in Y ( n )for all Y ( n ) < . Therefore maximization of payoff inequality requires that there are nodisclosures after period 1, i.e. Y ( n ) = n − n = n . This implies that the optimal contestis the first-mover contest n = (1 , n − Additional Figures
For completeness, I include here the figures describing the total equilibrium effort (fig-ure 3a), total equilibrium welfare (figure 3b), lowest equilibrium effort (figure 4a), andlowest equilibrium payoff (figure 4b) in all Tullock contests with 2 to 12 players. Thefigures confirm fully general findings discussed in the text—in each of these eight possiblecases the optimal contest is either the sequential contest or the simultaneous contest. (a) Total effort (b) Total welfare
Figure 3: Total equilibrium effort X ∗ and total equilibrium welfare W ( x ∗ ) in all normal-ized Tullock contests with 2 ≤ n ≤
12 players. The solid black line indicates sequentialcontests and dashed blue line simultaneous contests. (a) Lowest effort (b) Lowest payoff
Figure 4: Lowest effort min i { x ∗ i } and lowest payoff min i { u i ( x ∗ ) } in all normalized Tullockcontests with 2 ≤ n ≤
12 players. The solid black line indicates sequential contests anddashed blue line simultaneous contests. 20
Minimizing or Maximizing the i -th Highest Effortand Payoff In the case of linear h ( X ) or large n , the problem is simple enough to consider otherobjectives. Consider for example i -th highest effort. By the earlier-mover advantageresult, it is the effort of player i ∈ I t . If h ( X ) is linear or n is large, it is (approximately) x ∗ i ≈ X Q tk =1 (1 + n k ) . (15)Remember the general trade-off: each disclosure reduces the marginal benefit of effort forall players and therefore pushes all efforts downwards, but it increases the efforts of earlier-movers whose effort is now made public. Therefore each disclosure before the arrival ofplayer i only has the former effect and therefore reduces x ∗ i . Therefore minimizing x ∗ i requires players before i to be arranged sequentially and maximizing requires them tobe simultaneous (i.e. no disclosures prior to i ). Disclosures after player i have the twoopposite effects, so their impact depends on the relative magnitudes. However, in thecase considered here (linear demand), they are exactly equal and therefore balance out.Therefore x ∗ i is independent on the choice of disclosures after the period player i arrives. This allows us to state the following simple proposition.
Proposition 6.
When h ( X ) is linear or n is large1. The contest that has i − first players arranged sequentially and the rest of theplayers arriving simultaneously, i.e. n = (1 , . . . , , n + 1 − i ) , minimizes the i -thlargest effort.2. Any contest that has i players in the first period, i.e. n = ( i, n , . . . , n T ) , maximizesthe i -th largest effort. When i = n and i = 1 this result confirms propositions 2 and 3: lowest effort ( i = n ) isminimized by sequential and maximized by sequential contest, and highest effort ( i = 1)is minimized by the simultaneous contest and maximized by any single-leader contest.We can analogously study the i -th highest payoff u i ( x ∗ ), which is (approximately) u i ( x ∗ ) = x ∗ i h ( X ∗ ) ≈ X Q tk =1 (1 + n k ) αX Q Tk =1 (1 + n k ) . (16)Now there are three effects. First, all disclosures before the arrival of player i decreaseboth x ∗ i and h ( X ∗ ) and thus decrease u i ( x ∗ ). Second, all disclosures after period t (when i arrives) leave x ∗ i unaffected, but decrease h ( X ∗ ), thus decreasing u i ( x ∗ ). And third, thelength of period t , i.e. delaying the first disclosure after the arrival of player i , has twoopposite effects—it reduces x ∗ i but increases h ( X ∗ ). Combining these effects gives us thefollowing proposition. Proposition 7.
When h ( X ) is linear or n is large This is the
Stackelberg Independence property of a standard sequential homogeneous goods oligopoly,studied in detail in Hinnosaar (2019). . The contest that minimizes the i -th highest payoff has the following structure: i − sequential players first, then two players simultaneously, and then the remaining n − i − players sequentially.2. The contest that maximizes the i -th highest payoff is n = ( i, n − i ) when i ≤ i = − + q + n and the simultaneous contest when i ≥ i . This confirms the findings from proposition 2 (the lowest payoff is minimized by se-quential and maximized by simultaneous contest) and proposition 4 (the highest payoffis minimized by contest (2 , , . . . ,
1) and maximized by the first-mover contest).
Proof.
Let us first consider minimization of the i -th highest payoff u i ( x ∗ ). The argumentsstate that the players before i should be sequential as well as players after period t . Theremaining question is n t , i.e. how many player to pool with i . This gives a maximizationproblem min ≤ n t ≤ n +1 − i i − (1 + n t ) 1(1 + n t )2 n − n t ⇐⇒ max ≤ n t ≤ n +1 − i (1 + n t ) − n t . (17)Treating n t as a continuous variable and differentiating the objective gives necessarycondition for optimality 2 − n t (1 + n t ) [2 − (1 + n t ) log 2] = 0 . (18)Solving this gives n t = − ≈ . ≤ n t ≤ n , so the only two candidates for the optimum are n = 1and n = 2. The latter is bigger than the former as (1 + 2) − = > (1 + 1) − = 2.Therefore optimal n t = 2. We get that the optimal contest is n = (1 , . . . , | {z } i − , , , . . . , | {z } n − i − ).Now let us consider maximization of the i -th highest payoff u i ( x ∗ ). As argued above,all players before i should be pooled with i , so that player i arrives in period 1. Allplayers after period t should be pooled together as well, which would lead to a two-periodcontest ( n , n − n ) with i ≤ n ≤ n (in case n = n it is in fact a simultaneous contest).Therefore the maximization problem is equivalent tomax i ≤ n ≤ n
11 + n n )(1 + n − n ) ⇐⇒ min i ≤ n ≤ n (1 + n ) (1 + n − n ) . (19)This problem is concave in the interior, therefore one of the two corners is optimal, either n = i , which gives (1 + i ) (1 + n − i ) as the value of the objective, or n = n which gives(1 + n ) . The former is the minimizer if and only if(1 + i ) (1 + n − i ) ≤ (1 + n ) . (20)Equalizing left-hand-side with the right-hand-side gives and solving for i gives three roots n and − ± q + n . Lowest root is always negative, and the second one i = − + q + n always strictly between 1 and n . Moreover, at i = 0 the condition is clearly violated.Therefore, we get that for low values i ≤ i the condition is violated and therefore theoptimal contest is n = ( i, n − i ), whereas for high values i ≥ i the condition is satisfiedand therefore the optimal contest is simultaneous n = ( n ). Since 1 < i = − + q + n ⇐⇒ < n and n > i = − + q + n ⇐⇒ n >1.