Optimal sequential contests

Hinnosaar, Toomas Article Optimal sequential contests Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Hinnosaar, Toomas (2024) : Optimal sequential contests, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 1, pp. 207-244, https://doi.org/10.3982/TE5536 This Version is available at: https://hdl.handle.net/10419/296458 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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I show that information about other players’ efforts increases the total effort. Thus, the total effort is maximized with full transparency and minimized with no transparency. I also show that in addition to the first-mover advantage, there is an earlier-mover advantage. Finally, I derive the limits for large contests and discuss the limit to perfectly competitive outcomes under different disclosure rules. Keywords. Contest design, oligopoly, public goods, rent-seeking, R&D. JEL classification. C72, C73, D72, D74, D82. 1. Introduction Many economic interactions have contest-like structures, with payoffs that increase in players’ own efforts and decrease in the total effort. Examples include oligopolies, public goods provision, tragedy of the commons, rent seeking, R&D, advertising, and sports. The literature typically assumes that effort choices are simultaneous. Simultaneous contests have convenient properties: the equilibrium is unique, is in pure strategies, and is relatively easy to characterize. In this paper, I study contests where the effort choices are not necessarily simultaneous. In many real-life situations, some players can observe their competitors’ efforts and respond appropriately to those choices. However, earlier movers can also anticipate these subsequent responses and, therefore, influence the behavior of later movers. Each Toomas Hinnosaar: [email protected] I am grateful to Stefano Barbieri, Federico Boffa, Jeff Ely, Alex Frankel, Andrea Gallice, Dan Garrett, Dino Gerardi, Marit Hinnosaar, Johannes Hörner, Martin Jensen, Kai Konrad, Dan Kovenock, Nenad Kos, Ignacio Monzón, Peter Norman, Marco Ottaviani, Mallesh Pai, Alessandro Pavan, Alex Possajennikov, Debraj Ray, Christian Seel, Marco Serena, Ron Siegel, Andy Skrzypacz, Martin Szydlowski, Jean Tirole, and Asher Wolinsky as well as seminar participants at Bocconi University, Collegio Carlo Alberto, Universidad Carlos III de Madrid, University of Bern, Rice University, University of North Carolina at Chapel Hill, University of Bristol, University of Bonn, Humboldt University of Berlin, Canadian Economic Theory Conference, Conference on Economic Design, Contests: Theory and Evidence Conference, SAET, PET, Stony Brook Game Theory Festival, SITE at Stanford, EARIE, Midwest Economic Theory Meeting, Lancaster Game Theory Conference, Markets with Informational Asymmetries workshop (Turin), IIOC, SEA, Lingnan Workshop (Guangzhou), QuaGaTCo virtual conference, and China Meeting of the Econometric Society for their comments and suggestions. I would also like to thank the Kellogg School of Management at Northwestern University and the Collegio Carlo Alberto where some of this work was carried out. ©2024 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5536 208 Toomas Hinnosaar Theoretical Economics 19 (2024) additional period in a sequential contest adds complexity to the analysis, which might explain why previous studies have focused mainly on simultaneous models. I characterize equilibria for a general class of sequential contests and analyze how the information about other players’ efforts influences the equilibrium behavior. Contests may be sequential by nature or by design. For example, in rent-seeking contests, firms lobby the government to achieve market power. One tool that regulators can use to minimize such rent-seeking is a disclosure policy. A nontransparent disclosure policy would lead to simultaneous effort choices, but a full transparency policy would lead to a fully sequential contest. There may be potentially intermediate solutions as well, where the information is revealed only occasionally. Over the last few decades, many countries have introduced new legislation regulating transparency in lobbying activity. This list includes the United States (Lobbying Disclosure Act, 1995; Honest Leadership and Open Government Act, 2007), the European Union (European Transparency Initiative, 2005), and Canada (Lobbying Act, 2008). However, there are significant crosscountry differences in regulations. For example, lobbying efforts in the US must be reported quarterly, whereas in the EU, reporting occurs annually and on a more voluntary basis. Another classic example of a contest is research and development (R&D), where the probability of a scientific breakthrough is proportional to agents’ research efforts. The question is how to best organize the disclosure rules to maximize aggregate research efforts. In some academic fields, it is common to present early findings in working papers and conferences. In other fields, these efforts are kept confidential until the work has been vetted and published in a journal. Similarly, when announcing an R&D contest, the organizer can choose a transparency level: whether to use a public leaderboard or perhaps keep the entries secret until the deadline. To address such questions, I study a model of sequential contests. First, I characterize all equilibria for any given sequential contest, i.e., for any fixed disclosure rule. The standard backward-induction approach requires finding best-response functions every period and substituting them recursively. This solution method is not generally tractable or even feasible. Instead, I use an alternative approach, in which I characterize best-response functions by inverse functions. This method pools all the optimality conditions into one necessary condition and solves the resulting equation just once. I prove that for any contest the equilibrium exists and is unique. Importantly, the characterization theorem shows how to compute the equilibrium. The main result of the paper shows that the information about other players’ efforts strictly increases the total effort. Consequently, the optimal contest is always one of the extremes. When efforts are desirable (as in R&D competitions), the optimal contest is one with full transparency. When the efforts are undesirable (as in rent-seeking), the optimal contest is one with hidden efforts. The intuition behind this result is simple. While players’ efforts could be strategic substitutes or complements, I show that efforts are strategic substitutes sufficiently close to the equilibrium. Therefore, earlier-moving players have an additional incentive to exert effort to discourage later players’ efforts. If the discouragement effect were strong enough to reduce the total effort, this would offer profitable deviations for some players. Therefore, the discouragement effect is less than Theoretical Economics 19 (2024) Optimal sequential contests 209 one-to-one. It increases earlier-movers efforts more than it reduces later-movers efforts, therefore increasing total effort. While there could be indirect effects that change the conclusions, I show that (again, near the equilibrium) efforts are higher-order strategic substitutes and, therefore, the result still holds. The information about other players’ efforts is important both qualitatively and quantitatively. For example, the sequential contest with 5 players ensures a higher total effort than the simultaneous contest with 24 players. The differences become even larger with larger contests. For example, a contest with 14 sequential players achieves a higher total effort than a contest with 16,000 simultaneous players. Therefore, the information about other players’ efforts is at least as important as other characteristics of the model, such as the number of players. I also generalize the first-mover advantage result by Dixit (1987), who showed that a player who pre-commits chooses a greater level of effort and obtains a higher payoff than his followers. This leader exploits two advantages: he moves earlier and has no direct competitors. With the characterization result, I can further explore this question and compare players’ payoffs and effort levels in sequential contests. I show that there is a strict earlier-mover advantage—earlier players choose greater efforts and obtain higher payoffs than later players. Finally, I provide insights for large contests. I derive an approximation result for contests with an infinitely large number of players. This result allows me to show that as the number of players becomes large, the total effort converges to the prize’s value (or perfectly competitive outcome more generally) regardless of the contest structure. However, the speed of convergence to this level is different under different disclosure policies. In simultaneous contests, the rate of convergence is linear, whereas in sequential contests it is exponential. These results paint a different picture of highly competitive strategic interactions. In simultaneous contests, a high degree of competitiveness requires a large number of players, all choosing a minuscule effort level. Contrastingly, in a sequential contest, the same total effort requires a much smaller number of players, each exerting different effort levels. The first player chooses a much higher effort than anyone else, the second one much higher than the first, and so on. By any definition, this is a highly concentrated market. However, the early movers cannot capitalize on their position, as later movers would react by increasing their efforts. Therefore, despite the different effort levels, their payoffs are still close to zero. These results thus provide an alternative foundation for the contestability theory (Baumol (1982)). Instead of introducing a separate class of inactive players—the competitive fringe—in this model, the competitive fringe arises endogenously through the order of moves. Literature: The simultaneous version of the model has been studied extensively, starting from Cournot (1838). The literature on Tullock contests was initiated by Tullock (1967,1974) and motivated by rent-seeking (Krueger (1974)).1The most general treatment of simultaneous contests is provided by the literature on aggregative games (Selten 1See Nitzan (1994), Konrad (2009), and Vojnovi´ c(2015) for literature reviews on contests. 210 Toomas Hinnosaar Theoretical Economics 19 (2024) (1970), Acemoglu and Jensen (2013), Jensen (2018)). My model is an aggregative game only in the simultaneous case. The only sequential contest that has been studied extensively is the first-mover contest. It was introduced by von Stackelberg (1934), who studied quantity leadership in an oligopoly. Dixit (1987) showed that there is a first-mover advantage in contests. Relatively little is known about (Tullock) contests with more than two periods. The only paper prior to this that studied sequential Tullock contests with more than two periods is Glazer and Hassin (2000), which characterized the equilibrium in the sequential threeplayer Tullock contest. Kahana and Klunover (2018) is an independent and concurrent work that uses a similar approach to characterize the equilibrium in an important special case of my model: an n-player fully sequential Tullock contest. In contrast to my paper, they do not study any of the questions that are the main focus of my paper, such as the optimal contests, earlier-mover advantage, and large contests. Moreover, as I argue in Section 8, the characterization alone is not sufficient to answer these questions. The only class of contests where equilibria are fully characterized for sequential contests are oligopolies with linear demand.2 More is known about large contests. Perfect competition (Marshall equilibrium) is a standard assumption in economics, and it is a baseline with which to understand its foundations. Novshek (1980) showed that Cournot equilibrium exists in large markets and converges to the Marshall equilibrium. Robson (1990) provided further foundations for Marshall equilibrium by proving an analogous result for large sequential oligopolistic markets. In this paper, I take an alternative approach. Under stronger assumptions about payoffs, I provide a full characterization of equilibria with any number of players and any disclosure structure, including simultaneous and sequential contests as opposite extremes. This allows me not only to show that the large contest limit is the Marshall equilibrium but also to study the rates of convergence under any contest structure. The paper also contributes to the contest design literature. Previous papers on contest design include Taylor (1995), Che and Gale (2003), Moldovanu and Sela (2001,2006), and Olszewski and Siegel (2016), which have focused on contests with private information. Halac, Kartik, and Liu (2017) studied contest design in the presence of informational externalities when players learn about the feasibility of the project. In this paper, I study contest design on a different dimension: how to optimally disclose other players’ efforts, when players move sequentially, to minimize or maximize total effort.3 Similar connections between disclosures and subsequent actions have been found in other settings. For example, Fershtman and Nitzan (1991), Varian (1994), and Wirl (1996) used a model of dynamic voluntary public goods provision to show that if contributions are adjusted after observing earlier contributions, this may increase the freeriding problem. Admati and Perry (1991)andBonatti and Hörner (2011) showed similar 2Daughety (1990) used such a model to show that an oligopoly where players are divided between two periods is more concentrated but also closer to competitive equilibrium than an oligopoly where all players move at once. Hinnosaar (2021) provides a literature review and shows that the linear oligopoly model has unique properties that fail when the demand is not linear. 3Recently, Ely, Georgiadis, Khorasani, and Rayo (2022) studied feedback design in a continuous-time model where the designer wants to prolong participation for as long as possible and contestants do not observe their successes. Theoretical Economics 19 (2024) Optimal sequential contests 211 effects in dynamic team production problems. While the driving forces in these papers are similar to the discouragement effect studied herein, none of these works addressed higher-order effects and their implications for resulting equilibria.4 The paper also helps to explain empirical findings. For example, there is widespread empirical evidence of earlier-mover advantage in consumer goods markets. According to a survey by Kalyanaram, Robinson, and Urban (1995), there is a negative relationship between a brand’s entry time and the brand’s market share in many mature markets, including pharmaceutical products, investment banks, semiconductors, and drilling rigs. For example, Bronnenberg, Dhar, and Dubé (2009) studied brands of typical consumer packaged goods and found a significant early entry advantage. The advantage is strong enough to drive the rank order of market shares in most cities. Lemus and Marshall (2021) used observational data and a lab experiment to study the impact of public leaderboards in prediction contests. They found that public leaderboards encouraged some players and discouraged others, but the overall effect was positive, improving the prediction contest’s quality. The rest of the paper unfolds as follows. Section 2introduces the model. Section 3 uses a three-player example to illustrate why the standard backward induction is not tractable and shows how the inverted best-response approach solves the tractability problem. Section 4provides the characterization result. Section 5discusses the second main result, connecting information and total effort, and discusses its implications. Section 6studies earlier-mover advantage and Section 7analyzes large contests. Section 8shows how the analysis applies to a broader class of models. Finally, Section 9 concludes. All proofs are in Appendix A. 2. Model There are nidentical players N={1, ,n}who arrive to the contest sequentially and make effort choices on arrival. At T−1 points in time, the sum of efforts by previous players is publicly disclosed. These disclosures partition players into Tgroups, denoted by I=(I1,,IT). In particular, all players in I1arrive before the first disclosure and, therefore, have no information about other players’ efforts. All players in Itarrive between disclosures t−1andtand, therefore, have exactly the same information: they observe the total effort of players arriving prior to disclosure t−1.5I refer to the time interval in which players in the group Itarrived as period t. As all players are identical, the disclosure rule of the contest is fully described by the vector n=(n1,,nT),where nt=|It|is the number of players arriving in period t.6 4More broadly, there is a connection with the sequential information design literature. For example, Doval and Ely (2020) and Makris and Renou (2023) provide characterization results in sequential models where information design may involve signals about players’ actions in addition to unknown types or states. Li and Norman (2021) study a sequential persuasion model and find that players generally want to move only once. 5As the payoffs depend on the total effort of other players and not their individual efforts, observing the sumofpreviousplayers’effortsisequivalenttoobservingtheirindividualefforts. 6Equivalently, the model can be stated as follows: nplayers are divided across Ttime periods, either exogenously or by the contest designer. 212 Toomas Hinnosaar Theoretical Economics 19 (2024) Figure 1. A contest with 7 players and 3 disclosures. Players 1 to 3 choose efforts x1,x2,andx3 independently; player 4 observes X1=x1+x2+x3,player5observesX2=X1+x4,andplayers 6and7observeX3=X2+x5. Each player ichooses an individual effort xi≥0 at the time of arrival. I denote the profile of effort choices by x=(x1,,xn), the total effort in the contest by X=n i=1xi, and the cumulative effort up (and including) to period tby Xt=t s=1i∈Isxi.Byconstruction, the cumulative effort before the contest is X0=0, and the cumulative effort after period Tis the total effort exerted during the contest, i.e., XT=X. Figure 1illus- trates the notation with an example of the four-period contest n=(3, 1, 1, 2): Players compete for a prize of size one, the probability of winning is proportional to the level of effort, and the marginal cost of effort is one. I therefore assume the normalized Tullock payoffs,with ui(x)=xi X−xi.(1) I study pure-strategy subgame-perfect equilibria, a natural equilibrium concept in this setting: there is no private information, and earlier arrivals can be interpreted as having greater commitment power. I show that there always exists a unique equilibrium. Throughout the paper, I maintain a few assumptions that simplify the analysis. First, there is no private information. Second, the arrival times and the disclosure rules are fixed and common knowledge. Third, each player makes an effort choice just once upon arrival. Fourth, disclosures make cumulative efforts public.7In Sections 8and 9, I discuss the extent to which the results rely on each of these assumptions and explain how the results extend to more general sequential games. 3. Example The standard Tullock contest has nidentical players who make their choices in isolation. Each player ichooses effort xito maximize payoff (1). The optimal efforts have to satisfy the first-order condition 1 X−xi X2−1=0, (2) 7Specifically, each player observes the sum of earlier-movers’ efforts with certainty and unconditionally. More complex disclosure rules would change the conclusions. For example, probabilistic disclosures may limit the earlier-movers commitment power (Bagwell (1995)) and conditional disclosures may substantially expand the set of possible outcomes (Bizzotto, Hinnosaar, and Vigier (2023)). Theoretical Economics 19 (2024) Optimal sequential contests 213 where X2is the total effort squared. Combining the optimality conditions leads to a total equilibrium effort X∗=(n−1)/n and individual efforts x∗ i=(n−1)/n2.Theequilibrium is unique, easy to compute, and easy to generalize in various directions, which may explain the widespread use of this model in various branches of economics. 3.1 The problem with standard backward induction Consider next a three-player version of the same contest, but the players arrive sequentially and their efforts are instantly publicly disclosed. That is, players 1, 2, and 3 make their choices x1,x2,andx3after observing the efforts of previous players. I will first try to find equilibria using the standard backward-induction approach. Player 3 observes the total effort of the previous two players, X2=x1+x2<1and maximizes the payoff. The optimality condition for player 3 is 1 X2+x3−x3 (X2+x3)2−1=0. (3) Solving it for x3gives the best-response function x∗ 3(X2)=√X2−X2.8Now, player 2 observes x1<1 and knows x∗ 3(X2)and, therefore, solves the maximization problem max x2≥0 x2 x1+x2+x∗ 3(x1+x2)−x2=max x2≥0 x2 √x1+x2−x2. The optimality condition for player 2 is 1 √x1+x2−x2 2(x1+x2)3 2−1=0. For each x1∈[0, 1), this equation defines a unique best-response, x∗ 2(x1)=1 12 −x1+827x3 1(27x1+1)+216x2 1+36x1+12 3+24x1+1 12827x3 1(27x1+1)+216x2 1+36x1+11 3 .(4) Finally, player 1’s problem is max x1≥0 x1 x1+x∗ 2(x1)+x∗ 3x1+x∗ 2(x1)−x1, where x∗ 2(x1)and x∗ 3(X2)are defined by equations (3)and(4). Although the problem is not complex, it is not tractable. Moreover, the direct approach is not generalizable for an arbitrary number of players. In fact, the best response function does not have an explicit representation for contests with a larger number of periods. 8In this example, I focus only on interior solutions. It is straightforward to verify that corner solutions cannot occur in equilibrium, as they require that at least one player chooses an effort level giving inducing a nonpositive payoff, and there is always a deviation with a strictly positive payoff. 214 Toomas Hinnosaar Theoretical Economics 19 (2024) 3.2 Inverted best-response approach In this paper, I use a different approach. Instead of characterizing individual (reduced) best-responses x∗ i(Xt−1), or the total efforts induced by Xt−1, i.e., X∗(Xt−1), I characterize the inverse of X∗(Xt−1). For any level of total effort X, the inverted best-response function ft−1(X)specifies the cumulative effort Xt−1up to period t−1(i.e.,beforethe move of players in period t), that is consistent with total effort being X, given that the players in periods t,,Tbehave optimally. To see how the characterization works, consider the three-player sequential contest again. In the last period, player 3 observes X2and chooses x3. Equivalently, we can think of his problem as choosing the total effort X≥X2by setting x3=X−X2, i.e., max X≥X2 X−X2 X−(X−X2). Differentiating the objective with respect to Xgives us the optimality condition 1 X−X−X2 X2−1=X2 X2−1=0, which implies X2=X2. That is, if the total effort in the contest is X,thenbefore player 3’s action, the cumulative effort had to be f2(X)=X2; otherwise, player 3 would not be behaving optimally. We can now think of player 2’s problem as choosing X≥X1=x1, which he can induce by making sure that the cumulative effort up to his move is X2=f2(X), setting x2=f2(X)−X1. Therefore, his maximization problem can be written as max X≥X1 f2(X)−X1 X−f2(X)−X1. Again, differentiating with respect to X, we get the optimality condition f 2(X) X−f2(X)−X1 X2−f 2(X)=0. (5) This is the key equation that shows the advantage of the inverted best-response approach. Equation (5)isnonlinearinXand, therefore, in x2, which causes the difficulty for the standard backward-induction approach. Solving this equation every period for the best-response function leads to complex expressions, and the complexity increases with each step of the recursion. However, (5) is linear in X1, making it easy to derive the inverted best-response function f1(X)=X1=f2(X)−f 2(X)X(1−X)=X2(2X−1). The condition X1=f1(X)aggregates the two necessary conditions of equilibrium into one, by capturing the best responses of players 2 and 3. It simply states that if the total effort at the end of the contest is X, then the cumulative effort X1had to be f1(X)after player 1. Otherwise, either player 2 or player 3 is not behaving optimally. Theoretical Economics 19 (2024) Optimal sequential contests 221 The equilibrium payoff of a player iis in ui(x∗)=x∗ i(1/X∗−1), and since X∗is the same for all the players, payoffs are proportional to efforts. Therefore, it suffices to show that the efforts of earlier players are strictly higher. Using Theorem 1and equation (9), I can express the difference between the equilibrium efforts of players iand jfrom consecutive periods tand t+1as x∗ i−x∗ j= T−t  k=1Sknt−Sknt+1gk+1X∗(11) where nt+1=(nt+2,,nT)is the subcontest starting after period t+1andnt= (nt+1,nt+1)is the subcontest starting after period t. Clearly, Sk(nt)>S k(nt+1)for all k; i.e., there is more information on all levels in a strictly longer contest. As gk+1(X∗)>0, for each kthe whole sum is strictly positive. The intuition of the result is straightforward: players in earlier periods are observed by strictly more followers than the players from the later periods. Therefore, in addition to the incentives that later players have, the earlier players have an additional incentive to exert more effort to discourage later players. 7. Large contests Numeric comparison of simultaneous and sequential contests highlights that the information about other players’ efforts is at least as important in determining the total effort as other parameters, such as the number of players. For example, the total effort in the simultaneous contest with 10 players is 0.9, whereas the total effort with four sequential players is 0.9082. A fifth sequential player increases the total effort to 0.9587. A simultaneous contest with the same total effort requires 24 players. Figure 2shows that the comparison becomes even more favorable for sequential contests with large n. The following proposition gives the reason for this connection. As the number of players becomes large, the total effort converges to 1 no matter the contest structure, but the convergence is different depending on the structure. For large simultaneous contests, the convergence is linear, with 1 −X∗≈1/n, while for large sequential contests, the convergence is exponential, with 1 −X∗≈1/2n.13 It is also worth noting that, although the individual payoffs converge to zero, the individual efforts may not. Proposition 2 (Large Contests). Fix T∈Nand a sequence of contests (nn)∞ n=3,suchthat contest nnis n-player contest with at most Tperiods. Let Xn=X∗(S(nn)) and for each player i, let xn ithe equilibrium effort in contest nn. For all t≤Tand all i∈In t, lim n→∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1−Xn−1 T  t=11+nn t ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦=0and lim n→∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ xn i−1 t  s=11+nn s ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦=0. (12) 13Proposition 2is stated for arbitrary fixed T. Therefore, it is straightforward to apply it to the limit of contests where Titself becomes infinitely large (e.g., large fully sequential contests) by taking another limit with respect to T. 222 Toomas Hinnosaar Theoretical Economics 19 (2024) Figure 2. Number of players in a sequential contest that leads to the same total effort as a simultaneous contest with nplayers. These results shed new light on the meaning of a highly competitive contest or market. In a large simultaneous contest, each contestant chooses a minuscule effort level. Such a market is clearly not concentrated. For example, with n=16,000 the standard measure of concentration, the Herfindahl–Hirschman Index is HHIsim ≈0. In contrast, a sequential contest requires only a limited number of players to achieve the same aggregate results, and players behave differently. In a large sequential contest, the individual equilibrium efforts are x∗≈(1/2, 1/4, ,1/2n). The earlier movers choose much larger efforts and achieve larger payoffs than the followers. For example, with n=14 sequential players, the corresponding concentration index HHIseq ≈1/3, which is typically interpreted as a highly concentrated market. However, in terms of outcomes, this market is highly competitive: as total effort is close to one, we have full dissipation of rents, and thus all players earn equilibrium profits that are close to zero. This effect is similar to contestability theory (Baumol, Panzar, and Willig (1988)), where a small number of firms cannot capitalize on their market power due to the presence of a competitive fringe—a large number of potential competitors, who could frictionlessly enter when a profit opportunity arises. In my model, the later movers are endogenously taking the role of the competitive fringe. In equilibrium, they decide to put in very little effort. However, if the earlier movers were to try and exploit their position by reducing their efforts, the later movers would be there to respond. 8. Generalization In this section, I discuss how to implement the methodology for a general class models. I also provide sufficient conditions under which the results above remain unchanged. Theoretical Economics 19 (2024) Optimal sequential contests 223 Specifically, I define a class of linearly multiplicative payoff functions and show that if it satisfies Property 1,Theorem1remains valid without any modifications. By adding another sufficient condition, Property 2, nearly all other results in the paper hold as well. The differences between Property 1and Property 2also suggest that Theorem 2 and most other results in the paper are not direct implications of Theorem 1. Suppose that each player chooses an action xifrom a set Xiand if the profile of actions is x=(x1,,xn), then player igets a payoff Ui(x)=ui(xi,X). (13) Take a player ifrom the last period T.Playeriobserves cumulative effort XT−1before period Tand knows that other players in period Tare choosing efforts simultaneously to him. Therefore, he solves the maximization problem max xi∈Xi uixi,xi+XT−1+ j∈IT\{i} xj. The standard best-response function would be x∗ i(XT−1).14 But suppose we can express the optimal effort xichoice as a function of total effort, φi(X). Then adding up individual efforts in period Tconsistent with total effort Xgives us a necessary condition for equilibrium, XT−1=X− i∈IT φi(X). I denote the function on the right-hand side by fT−1(X). Its inverse function (assuming it exists), f−1 T−1(XT−1)is the total effort induced by cumulative effort XT−1, if all players in period Tbehave optimally.15 Suppose by induction that the same argument holds starting from period t, i.e., if cumulative effort after tis Xtthen the total effort induced is f−1 t(Xt). Then player iin period tsolves the following problem: max xi∈Xi uixi,f−1 t(Xt). If again, we can express the optimal xionly as a function φi(X), then adding up the conditions would give us a necessary condition for equilibrium Xt−1=Xt− i∈It xi=ft(X)− i∈It φi(X), which I denote by ft−1(X). Finally, in the beginning of the game cumulative total action is X0=0, which gives us an equilibrium condition for the whole game. 14This function is also called the reduced best-response function as it only depends on the sum. 15When T=1, the game becomes a linearly aggregative game, as introduced in Selten (1970), with a known equilibrium condition X=n i=1φi(X),wheren i=1φi(X)is the aggregate backward correspondence. See Jensen (2018) for a literature review. If T>1, the game is not aggregative, so the analysis presented here is a dynamic generalization of linearly aggregative games. 224 Toomas Hinnosaar Theoretical Economics 19 (2024) There are some gaps in this analysis that need to be filled. I have already shown that with the Tullock contest payoffs, ui(xi,X)=xi/X −xiand Xi=R+, this approach characterizes the unique equilibrium. It is equally clear that the approach is not valid for all payoff functions, as interior optimums may not exist or be unique. Next, I introduce a more restricted class of payoff functions and sufficient conditions where all results hold and the analysis remains tractable. Linearly multiplicative payoffs: Assume that the payoff functions are identical and the utility is linearly multiplicative with respect to players’ own actions, ui(xi,X)=xih(X),xi∈Xi=R+. (14) For Tullock contest payoffs, h(X)=v/X −c,wherevrepresents the prize value and cdenotes the marginal cost of effort. This class of games also includes oligopolies with linear costs, where h(X)=P(X)−c,withxias the firm’s own quantity, Xas the total quantity, P(X)as the inverse demand function, and cas the marginal cost. Additionally, this class includes public goods games, in which xidenotes private consumption and h(X)represents the marginal benefit of private consumption, which decreases with public good contributions and, therefore, with total private consumption. It is natural to assume in these applications that h(X)is strictly decreasing up to some upper bound X, at which it takes value h(X)=0andabovewhichh(X)≤0. Therefore, effectively the action space is Xi=[0, X]. Without loss of generality, we can change the scale of actions so that X=1. The first-order optimality condition for players in period Tis then h(X)+xih(X)=0⇐⇒ xi=g1(X), where g1(X)=−h(X)/h(X). Therefore, we can write the inverted best-response function as fT−1(X)=X−nTg1(X). Similarly, if the inverted best-response functions at period tis ft(X), which is invertible in the relevant range, the payoff function of player iin period tis ui(xi,f−1 t(Xt)) = xih(f−1 t(Xt)) and, therefore, the first-order condition for players in period tis h(X)+xih(X)1 f t(X)=0⇐⇒ xi=g1(X)f t(X). (15) Therefore, ft−1(X)=ft(X)−ntf t(X)g1(X). This shows that we can use the characterization derived in the paper, with two modifications. First, instead of specific expression X(1−X),wehaveafunctiong1(X)=−h(X)/h(X). And second, we need to impose some conditions on the function h(X)so that the conditions for the existence and uniqueness are satisfied. In Appendix A, I define Property 1, which is a sufficient condition for all ftfunctions to be well behaved so that the characterization theorem (Theorem 1) holds without any modifications. Intuitively, Property 1puts two restrictions on ftfunctions. First, for Theoretical Economics 19 (2024) Optimal sequential contests 225 sufficiently high X, they are strictly increasing and, therefore, invertible in the relevant range. Second, at least one of the ftfunctions is taking a negative value for lower values of X, which eliminates such Xas a candidate for equilibrium. Proposition 3in Appendix Aproves that Tullock payoffs satisfy Property 1and below I discuss some other cases when it is satisfied. Therefore, under Property 1, the equilibrium is still unique and can be computed as the highest root of f0(X)in [0, 1]. Moreover, the limit for large contests (Proposition 2) holds as well, with a particular adjustment in formulas. Let α=−g 1(1)>0. Then the formulas in equation (12) would be adjusted as lim n→∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1−Xn−1 T  t=11+αnn t ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦=0and lim n→∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ xn i−α t  s=11+αnn s ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦=0. (16) For the information theorem (Theorem 2) its corollaries (Corollary 1), as well as the earlier-mover advantage result (Proposition 1) I need an additional assumption. First, let us adjust gkfunctions by defining these as g1(X)=−h(X)/h(X)and gk+1(X)= −g k(X)g1(X)for all k. The additional assumption, Property 2in Appendix A, states essentially that each gk(X∗)>0 near equilibrium. This assumption can be interpreted as actions being higher-order strategic substitutes. Proposition 4proves that Tullock payoffs satisfy Property 2and below I discuss some functional forms that satisfy this assumption. The only result that does not generalize is Lemma 1that showed that with Tullock payoffs, the weights gk(X∗)are decreasing in k. It is easy to see that this result depends on the function h(X). For example, consider the case when h(X)=α √1−Xfor all X∈ [0, 1]and 0 otherwise, where α>0 is a constant. Then g1(X)=α(1−X),g2(X)=α2(1− X),andsoon,gk(X)=αk(1−X).Wheneverα>1, the weights are increasing in this case. The remaining question is when are properties 1and 2satisfied? For example, one special class of functions where these assumption are satisfied, is the class of functions, where g1(X)=−h(X)/h(X)is completely monotone, i.e., (−1)kdkg1(X)/dXk≥0for all k∈N.16 This includes many functions, including linear h(X), power function h(X)=α √1−X, but also many other natural functions. For example, the following functions are all completely monotone: g(X)=α(1−Xm),g(X)=α(1−X)m,forallm∈N, g(X)=α((X+γ)s−(1+γ)s)for all s<0, γ>0, g(X)=α[e−rX −e−r]for all r>0, and g(X)=−αlog(X),allwithanyα>0. Also, all sums and products of completely monotone functions are completely monotone.17 16It suffices that g1(X)is only T-times monotone, which is less restrictive, but perhaps harder to verify. 17In the working paper version (https://arxiv.org/pdf/1802.04669.pdf), I give more examples: (1) An oligopoly with logarithmic demand, where the analysis can be directly extended, even with non-monotonic g1(X); (2) An example where properties 1and 2are violated, and equilibrium may not exist or be unique; 226 Toomas Hinnosaar Theoretical Economics 19 (2024) Note that linearly multiplicative payoffs combined with properties 1and 2are sufficient and convenient assumptions, but they are not necessary. These assumptions ensure that the first-order conditions are linear in the cumulative action of preceding players, and as a result, the inverted best-response functions can be easily characterized. This raises the question: is the functional form assumption relevant only for tractability or does it have economic implications? A simple continuity argument demonstrates that the results will not change with small perturbations near the original model.18 9. Discussion I showed that each contest has a unique equilibrium. It is in pure strategies and simple to compute. The main result of the paper shows that the total equilibrium effort is strictly increasing in information. This implies that the optimal contest for maximizing total effort is fully sequential, e.g., R&D contests benefit from full transparency. On the other hand, if the goal is to minimize the total effort, such as rent-seeking contests, the optimal contest is nontransparent, i.e., the simultaneous contest. Further, there is a strict earlier-mover advantage: players in earlier periods exert strictly greater efforts and obtain strictly higher payoffs. Total effort converges to full dissipation linearly with the number of players in large simultaneous contests but exponentially in large sequential contests. The results in this paper hold much more generally than the model discussed herein. In addition to the generalization discussed in Section 8, some assumptions about the timing of arrivals can be relaxed. I assumed that players exert efforts only at their arrival and that their efforts are publicly observable for players in the following periods. Given that players benefit from the discouragement effect, they would not hide or delay their actions. Thus, the outcomes would be unchanged if players could take hidden actions or take actions over multiple periods. This was shown by Yildirim (2005) in the two-player case. The analysis can also be extended to heterogeneous players. However, there is a new complication: players may find it optimal to stay inactive at different thresholds. This means that earlier movers may sometimes find it optimal to deter entry by followers and the order of players becomes an important determinant of outcomes. Xu, Zhang, and Zhang (2020) use the approach introduced here to study the three-player asymmetric sequential contests.19 Hinnosaar (2023) extends the methodology to another type of player heterogeneity, where the game is played on a network. Players only observe the choices of players they are linked to. This analysis shows that there is a connection between weighted measures of information and standard centrality measures from network theory. (3) An example where efforts may be direct strategic substitutes in the standard sense but strategic complements due to indirect effects. In this instance, the equilibrium with two periods behaves as one would expect with strategic substitutes, while introducing a third period alters the conclusions as a result of indirect effects. 18In the working paper version, I show that in Tullock contests with quadratic costs, the analysis still applies when the parameter multiplying the quadratic term is sufficiently close to zero. 19The two-player case has been studied by Morgan (2003) and Serena (2017), who also considered endogenous order of moves. Theoretical Economics 19 (2024) Optimal sequential contests 227 Appendix A: Proofs A.1 Proof of the characterization theorem (Theorem 1) Before proving Theorem 1, it is useful to define the following property. Property 1 (Inverted Best Responses are Well Behaved). Clearly, fT(X)=Xhas unique root XT=0. For all t=0, ,T−1, the function fthas the following properties: (a) ft(X)=0 has a root in [Xt+1,1 ].LetXtbe the highest such root. (b) ft(X)<0forallX∈[Xt+1,Xt). (c) f t(X)>0forallX∈[Xt,1 ]. Moreover, X0∈(0, 1). The proof of Theorem 1has two parts. The first part is Proposition 3in Appendix A.2 that shows that ftfunctions satisfy Property 1. The proof relies on keeping track of the roots of ftfunctions. The second part in Appendix A.3 establishes the theorem’s claims. Briefly, it shows that behavior where each player iin each period tbehaves according to equation (8) and expects that total effort induced by cumulative effort Xtto be f−1 t(X), is an equilibrium and in fact it is the only equilibrium. The proof is divided into five lemmas: 1. Lemma 5shows that in all histories where Xt−1<1, each player in period tchooses strictly positive effort, but these added efforts in period tare small enough so that the cumulative effort after period tremains strictly below one, Xt<1. On the other hand, in histories where Xt−1≥1, the players in period texert no effort. Therefore, on the equilibrium path Xt<1forallt. 2. Lemma 6shows that Xt=ft(X)is a necessary condition for equilibrium. In particular, f0(X)=0 is a necessary condition for equilibrium and, therefore, X∗must be a root of f0(X). 3. Lemma 7shows that under Property 1, the inverse function f−1 t−1(Xt−1)is welldefined and strictly increasing, f−1 t−1(0)=Xt−1and f−1 t−1(1)=1. 4. Lemma 8shows that the best-response function of player i∈Itafter cumulative effort Xt−1is x∗ i(Xt−1)=n−1 t[ft(f−1 t−1(Xt−1)) −Xt−1]for all Xt−1<1and x∗ i(Xt−1)=0forallXt−1≥0. On the equilibrium path, the individual efforts are x∗ i=n−1 t[ft(X∗)−ft−1(X∗)]. Note that this step in the proof implicitly also shows that strategies of all players in period tare identical. 5. Finally, Lemma 9verifies that the unique candidate for equilibrium, i.e., x∗specified in the theorem, is indeed an equilibrium, which completes the proof. The combination of these results proves Theorem 1. 228 Toomas Hinnosaar Theoretical Economics 19 (2024) A.2 Proof that Property 1is satisfied (Proposition 3) Proposition 3. Inverted best responses f0,,fTdefined by equation (7) are well behaved. Before giving the proof of Proposition 3, let me briefly describe its key idea. The function ft+1is a polynomial of degree r=T−t, so it can have at most rroots. By keeping track of all the roots, I show by induction that all rroots are real and in [0, 1), with the highest being Xt+1. Therefore, all r−1 roots of the derivative f tare also real and in [0, Xt+1). Evaluating ftat Xt+1and 1, we get ft(Xt+1)=ft+1(Xt+1)   =0 −nt+1f t+1(Xt+1)   >0 Xt+1(1−Xt+1)   >0 <0 ft(1)=ft+1(1)−nt+1f t+1(1)1(1−1)   =0=ft+1(1)=···=fT(1)=1>0. This implies that ftmust have a root Xt∈(Xt+1,1 ). Moreover, since the highest root of its derivative is again below Xt, it is strictly increasing in [Xt,1 ]. Finally, I show that the second highest root of ftis strictly below Xt+1,sothatft(X)<0forall[Xt+1,Xt). Proving this requires keeping track of all the roots. Proof of Proposition 3.First,notethatfT(X)=Xis a polynomial of degree 1, and each step of the recursion adds one degree, so ft(X)is a polynomial of degree T+1−t, which I denote by rfor brevity. The following two technical lemmas describe the values of the polynomials ftat 1 and the number of roots at 0. Lemma 2. ft(1)=1for all t=0, ,T. Proof.ft−1(1)=ft(1)−ntf t(1)1(1−1)=ft(1)=fT(1)=1. Lemma 3. ft(0)=0for all t=0, ,T. Depending on n, there could be either one or two roots at zero: (a) If ns=1for some s>t,thenft(X)hasexactlytworootsatzero. (b) Otherwise, i.e., if ns=1for all s>t,thenft(X)has exactly one root at zero. Proof.Asft(X)is a polynomial of degree r=T+1−t, it can be expressed as ft(X)= r  s=0 ct sXs⇒f t(X)= r  s=1 ct ssXs−1, where ct 0,,ct rare the coefficients. Therefore, ft−1(X)=ct 0+ct 1(1−nt)+ r  s=2ct s(1−snt)+ntct s−1(s−1)Xs+ntct T+1−t(T+1−t)XT+2−t. Theoretical Economics 19 (2024) Optimal sequential contests 229 As fT(X)=X,wehavethatcT 0=0andsoct 0=0forallt. Therefore, each fthas at least one root at 0. Next, ft−1(X)has two roots at zero if and only if ct−1 1=ct 1(1−nt)=0. This can happen only if either ct 1=0(i.e.,ft(X)has two roots at zero) or nt=1. As fT(X)=X, we have that cT 1=1 and, therefore, ft(X)does indeed have two roots at zero if and only if ns=1forsomes>t. Finally, ft−1(X)would have three roots at zero only if ct−1 2=ct−1 1=0=ct−1 0.This would require that ct−1 2=ct 2(1−2nt)+ntct 1=ct 2(1−2nt)=0. Since 2nt= 1, this can happen only when ct 2=0. But note that fT−1(X)=nTX2−(1−nT)X,sothatcT−1 2= nT=0. Therefore, ft(X)cannot have more than two roots at zero. Lemma 4. The leading coefficient of ftis (T−t)!T s=t+1ns>0. Proof. Using the same notation as in Lemma 3, the leading coefficient of ft−1(X)is ct−1 r+1=rntct r=r!T s=tns. Now I can proceed with the proof of Proposition 3itself. The proof uses that fact that the ftis a polynomial of degree r=T+1−tand keeps track of all of its roots. In particular, it can be expressed as ft(X)=ct r  s=1 (X−Xs,t), (17) where ct>0 is the leading coefficient and X1,t,,Xr,tare the rroots. By Lemma 3, either one or two of these roots are equal to zero. I show by induction that all other roots are distinct and in (0, 1). Let us consider the case of a single zero root first, i.e., assume that 0 =X1,t<X 2,t< ···<X r,t<1. We can express the derivative of ftas f t(X)=ct r  i=1 s=i (X−Xs,t). Therefore, at root Xj,t, the polynomial f t(X)takes value f t(Xj,t)=ct s=j (Xj,t−Xs,t). (18) In particular, at the highest root, f t(Xr,t)>0, and at the second highest f t(Xr−1,t)< 0; therefore, f tmust have a root Yr−1,t∈(Xr−1,t,Xr,t). By the same argument, there mustbearootYs,tof f tbetween each of the two adjacent distinct roots of ft.Asf tis a polynomial of degree r−1, this argument implies that all the roots of f tare distinct and such that X1,t=0<Y 1,t<X 2,t<Y 2,t<···<X r−1,t<Y r−1,t<X r,t<1. In particular, sgn f t(Xs,t)=sgnft(Ys,t)for all s∈{1, ,r−1}. Next, note that ft(1)= 1>0 and, as the highest root of f tis Yr−1,t<X r,t, this implies f t(Xr,t)>0, and so ft−1(Xr,t)=ft(Xr,t)−ntf t(Xr,t)Xr,t(1−Xr,t)<0. 230 Toomas Hinnosaar Theoretical Economics 19 (2024) Therefore, ft−1must have a root Xr+1,t−1∈(Xr,t,1 ). Now, for each s∈{2, r−1} ft−1(Ys,t)=ft(Ys,t)and ft−1(Xs,t)=−ntf t(Xs,t)Xs,t(1−Xs,t). Hence, sgn ft−1(Ys,t)=sgn ft(Ys,t)=sgn f t(Xs,t)=−ft−1(Xs,t). Thismeansthatft−1 must have a root Xs+1,t−1∈(Xs,t,Ys,t). This argument determines r−2 distinct roots in (X2,t,Yr−1,t). By Lemma 3,ft−1also has at least one root X1,t−1=0. We have therefore found 1+r−2−1=rdistinct real roots of ft−1that is a polynomial of degree r+1. Thus, the final root X2,tmust also be real. By Lemma 3,ifnt=1, then the ft−1must have two roots at zero; so, X2,t=0. Let us consider the remaining case where nt>1. By Lemma 3,X2,t= 0. To determine its location, consider the function fX t−1(X)=ft−1(X)/X.Notethat fX t(X)=ft(X) X=ct s>0 (X−Xs,t)⇒fX t(0)=ct s>0 (−Xs,t) and f t(0)=ct s>0 (−Xs,t). Therefore, fX t−1(0)=fX t(0)−ntf t(0)(1−0)=ct s>0 (−Xs,t)[1−nt]=f t(0)[1−nt]. We assumed that nt>1; so, sgnfX t−1(0)=−sgn f t(0). Evaluating the function sgn fX t−1at Y1,tgives sgnfX t−1(Y1,t)=sgn ft(Y1,t)=sgn f t(X1,t)=−sgn fX t−1(0). Hence, fX t−1must have a root X2,t−1∈(0, Y1,t).Asft−1(X)=Xf X t−1(X), it must be a root of ft−1as well. We have therefore located all r+1 roots of ft−1, which are all distinct in this case. Letusnowgetbacktothecasewherefthad two roots at zero. By the same argument as above, there must be a root of f tbetween each positive root of ft.Asthere are r−2 positive roots, this determines r−3 distinct positive roots of f t.Itisalso clear that f tmust have exactly one root at zero. Polynomial f thas r−1 roots, and we have determined that r−2 of them are real and distinct. Thus, the remaining root must be real. To determine its location, using the above approach, let fX t(X)= f t(X)/X.Thenasf t(Xr,t)>0, we have fX t(Xr,t)>0. Similarly, fX t(Xr−1,t)<0, and so on. In particular, fX t(X3,t)<0ifris even, and fX t(X3,t)>0ifris odd. Now, fX t(0)=2ct s>2 (−Xs,t), which is strictly positive if ris odd and strictly negative if ris even, so that sgn fX t(0)= −sgnfX t(X3,t).Hence,fX tmust have a root Y2,t∈(0, X3,t). Clearly, this Y2,tis Theoretical Economics 19 (2024) Optimal sequential contests 237 2. Lemma 12 establishes a connection between the total equilibrium effort X∗and Zk−1:k. It shows that if we take the sequential n-player contest n=(1, ,1 ),then fn−k(X)=gk(X)X/(1−X)for all k=1, ,n. Therefore, if we take the fully sequential contest with nplayers, we get f0(X)=gn(X)X/(1−X),andsothetotal equilibrium effort X∗of this contest is exactly equal to the second highest root of gn, i.e., Zn−1:n. This proves the “weak” part of the proposition, i.e., if nis fully sequential, then X∗=Zn−1:n, which is a root of gnand, therefore, gk(X∗)=0. 3. Lemma 13 shows directly20 that X∗is strictly increasing in each nt. Therefore, if the contest is not sequential (nt>1forsomet), then the total effort in this contest is strictly higher than in the fully sequential T-player contest. Thus, X∗>Z T−1:Tand gT(X∗)>0. 4. Finally, Lemma 11 also shows that the adjacent gk’s are interlaced; i.e., the second highest roots are increasing in k,sothatforallk<T,Zk−1:k<Z T−1:T≤X∗and, therefore, gk(X∗)>0forallk<T. Lemma 11. Each gkhas the following properties: (a) gk(1)=g k(1)=−1. (b) gkcan be expressed as gk(X)=− k  j=0 (X−Zj:k), (21) where 0=Z0:k<Z 1:k<···<Z k:k=1. (c) Zs:k+1∈(Zs−1:k,Zs:k)for all s=1, ,k. Proof.First,notethatg1(X)=g(X)=X(1−X)is a polynomial of degree 2. Each step of the recursion gives a polynomial of one degree higher; i.e., gk(X)is a polynomial of degree k+1, so g k(X)is a polynomial of degree kand, therefore, gk+1(X)= −g k(X)X(1−X)is a polynomial of degree k+2. 1. gk+1(1)=−g k(1)g(1)=0, because g(1)=1(1−1)=0. Therefore, g k(1)= −g k−1(1)g(1)−g(1)g k−1(1)=g k−1(1)=···=g 1(1)=g(1)=1−2·1=−1. 2. The claim clearly holds for g1(X)=X(1−X)with Z0:1 =0<Z 1:1 =1. Suppose it holds for k. Since all k+1 roots of gkare real and in [0, 1], by the Gauss–Lucas theorem all kroots of g kare in (0, 1).Thengk+1(X)=−g k(X)X(1−X)clearly has roots at 0 and 1 and kroots in (0, 1). To see that the roots are all distinct, note that g k(X)=− k  s=0 j=s (X−Zj:k). 20Note the first part of Corollary 1proves the same claim, but since Proposition 4establishes a sufficient condition for Theorem 2, and hence its Corollary 1, to avoid a circular argument I prove it here directly. 238 Toomas Hinnosaar Theoretical Economics 19 (2024) Therefore, g k(Zs:k)=−j=s(Zs:k−Zj:k), which is strictly negative for s=k, strictly positive for s=k−1, and so on. Therefore, for each s=1, ,k,functiong k;hence, gk+1also has a root Zs:k+1=(Zs−1:k,Zs:k). This determines the kinterior roots. 3. The previous argument also proves the last claim. Lemma 12. If n=(1, ,1 ),thenfn−k(X)=gk(X)X/(1−X)for all k=1, ,T. Proof. Suppose that n=(1, ,1 ).First,fn−1(X)=X−X(1−X)=X2=g1(X)X/ (1−X). Now, suppose that fn−k(X)=gk(X)X/(1−X). Then since dX 1−X dX X(1−X)=1 1−X−−X (1−X)2X(1−X)=X 1−X, we get that fn−(k+1)(X)=gk(X)X 1−X−gk(X) dX 1−X dX X(1−X)−g k(X)X 1−XX(1−X) =gk+1(X)X 1−X. Lemma 13. X∗is strictly increasing in each nt. Proof. I first show that X∗is independent of permutations of n.Fixacontestnand a period t>1. To shorten the notation, let φt(X)=f t(X)X(1−X): ft−1(X)=ft(X)−ntφt(X), f t−1(X)=f t(X)−ntφ t(X)=φt(X) g(X)−ntφ t(X), ft−2(X)=ft(X)−[nt−1+nt]φt(X)+nt−1ntφ t(X)X(1−X). Switching nt−1and ntin ndoes not affect ft−2and, therefore, it also does not affect f0. This means that any such switch leaves X∗unaffected, which means that X∗is independent of permutations of n. To prove that X∗is strictly increasing in each nt, it therefore suffices to prove that it is strictly increasing on n1.Take n=(n1+1, n2,,nT).Thenf1is unchanged and the corresponding  f0at the original equilibrium X∗is  f0X∗=f1X∗−(n1+1)f 1X∗X∗1−X∗=f0X∗−f 1X∗X∗1−X∗<0, because f0(X∗)=0andf1(X∗)>0byProperty1.ByProperty1, f0is strictly increasing between its highest root  X∗and 1, thus  X∗>X∗. Theoretical Economics 19 (2024) Optimal sequential contests 239 A.6 Proof of decreasing weights lemma (Lemma 1) This lemma allows to order some contests, which cannot be ranked according to their information measures. For example, two 10-player contests n=(5, 5)and  n=(8, 1, 1) have corresponding information measures S(n)=(10, 25)and S( n)=(10, 17, 8).Contest nhas more second-order information, but  nhas one more disclosure, and thus more third-order information. However, the sum of all information measures is 10 + 25 =10 +17 +8=35. Since the weights are higher in lower-order information, this implies that the total effort is higher in the first contest. Indeed, direct application Theorem 1confirms this, as X∗=(13 +√41)/20 ≈0.9702 > X∗=(31 +√241)/48 ≈0.9693. Proof of Lemma 1. By Lemma 12,gk(X)= f n−k(X)(1−X)/X,where f n−kis defined for a sequential  n≥k-player contest. Similarly, gk−1(X)= fn+1−k(X)(1−X)/X.Therefore, gk−1X∗−gkX∗= f n+1−kX∗− f n−kX∗1−X∗ X∗= f  n+1−kX∗1−X∗2. Now, take  n=T. Then by Lemma 13,X∗is weakly higher than the highest root of  f0. By Property 1, the highest root of  fT+1−kis even (weakly) lower and  fT+1−kis strictly increasing above its highest root, so that  f  n+1−k(X∗)>0. This proves that gk−1(X∗)> gk(X∗). A.7 Proofs of implications of the information theorem (Corollary 1) Proof of Corollary 1. Take two contests nand nand let Xand  Xbe the corresponding total equilibrium efforts. 1. Suppose that n< n.ThenS(n)<S( n)and, therefore, X< X. 2. If nis a permutation of  n,thenS(n)=S( n)and, therefore, X= X. 3. If Iis a coarser partition than  I,thenS(n)<S( n)and, therefore, X< X. 4. If tnt=t nt=nand there exist t,tsuch that ntnt< nt ntand ns= nsfor all s= t,t,thenbyconstructionS1(n)=S1( n)=nand Sk(n)<S k( n)for all k>1. Therefore, X< X. 5. Let n=(n).Thenforany n=n,S(n)<S( n), so indeed Xis the unique minimum of X∗over all contests. Similarly, if  n=(1, 1, ,1 ), any other contest has strictly lower measures of information and, therefore,  Xis the unique maximum of X∗ over all contests. To establish the final claim of the optimality of equal division of players, let nbe n-player contests where players are distributed among at most Tperiods. Suppose by contradiction that the corresponding total equilibrium effort X∗is a maximum over all such contests and ndoes not split players as equally as possible. In particular, let k=n/T . Equal split requires that each period has either nt∈{k,k+1} players. Since this is not the case, there exists a period twhere nt≤k−1anda 240 Toomas Hinnosaar Theoretical Economics 19 (2024) period swhere ns≥k+1(ort,ssuch that nt≤kand ns≥k+2, then the proof is analogous). We can now construct a new contest,  n, where we have moved one player from period sto period t.Thenasns−1≥k>n t,  nt ns=(nt+1)(ns−1)=ntns−nt+ns−1>n tns. Therefore, the contest  nis more homogeneous than nand so X< Xby the previous step. Thus, we found a contradiction with the assumption that Xis a maximal total effort among such contests. A.8 Proof of the earlier-mover advantage (Proposition 1) Proof of Proposition 1. The equilibrium payoff of player iis ui(x∗)=x∗ i(1/X∗−1), so the payoffs are ranked in the same order as the individual efforts (in fact they are proportional to individual efforts). Therefore, it suffices to prove that if i∈Itand j∈ It+1,thenx∗ i>x ∗ j. Using Theorem 1and equation (9), the difference in equilibrium efforts can be expressed as x∗ i−x∗ j= T−t  k=1Sknt−Sknt+1gk+1X∗. Now, note that S(nt)≥S(nt+1)as there is less information remaining in the game that starts one period later. Moreover, S1(nt)>S 1(nt+1)as ntincludes player j, whereas nt+1 does not. Finally, note that by Proposition 4,g2(X∗)>0 and, therefore, x∗ i−x∗ j>0. A.9 Proof of the large contests limit (Proposition 2) Proof of Proposition 2.ByTheorem1, each Xn<1. Meanwhile, by Theorem 2, Xn≥(n−1)/n, which is the total equilibrium effort of the simultaneous n-player contest (see Section 3). Therefore, limn→∞ Xn=1. The total equilibrium effort of a censored contest nnis the highest root of f0(X), which can be expressed by equation (10)as Xn= T  k=1 SknngkXn. (22) For each k,functiongk(X)is a twice continuously differentiable function (a polynomial), gk(1)=0, and g k(1)=−g k−1(1)g(1)−g k−1(1)g(1)=g k−1(1)=···=g 1(1)=−1, as g1(X)=X(1−X). Therefore, for all k>1, lim X→1 gk(X) X(1−X)=lim X→1−g k−1(X)X(1−X) X(1−X)=−g k−1(1)=1. Taking limits from both sides of equation (22) and using the result that limn→∞ Xn=1, 1=lim n→∞Xn=lim n→∞ T  k=1 SknngkXn Xn1−XnXn1−Xn=lim n→∞1−XnT  k=1 Sknn. Theoretical Economics 19 (2024) Optimal sequential contests 241 To shorten the notation, let Sn=T k=1Sk(nn). Rearranging the previous equation gives 0=lim n→∞1−1−XnSn=lim n→∞Xn−1−1 SnSn. (23) We can express Sn=T k=1Sk(nn)=T t=1(1+nk t)−1. As limn→∞ Sn=∞,equation(23) implies that lim n→∞Xn−1−1 Sn=lim n→∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Xn−⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1−1 T  t=11+nn t ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦=0. For individual effort of player i∈In t, we can use Theorem 1and equation (9)toget xn i=g1Xn+ T−t  k=1 Skntgk+1Xn. Taking the limit, again using the facts that Xn→1andgk+1(Xn)/[Xn(1−Xn)] →1, lim n→∞xn i=lim n→∞1−Xn%1+ T−t  k=1 Sknt&. Now, note that 1 +T−t k=1Sk(nt)=T s=t(1+nn s). 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