DeepAI

# Optimal sequential contests

I study sequential contests where the efforts of earlier players may be disclosed to later players by nature or by design. The model has a range of applications, including rent seeking, R&D, oligopoly, public goods provision, and tragedy of the commons. 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 study the advantages of moving earlier and the limits of large contests.

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## 1 Introduction

Many economic interactions have contest-like structures, where payoffs are increasing in players’ own efforts and decreasing in the total effort. Examples include oligopolies, public goods provision, tragedy of the commons, rent seeking, R&D, advertising, and sports. Most of the previous literature assumes that effort choices are simultaneous. Simultaneous contests often have convenient properties: the equilibrium is unique, in pure strategies, and is relatively easy to characterize.

In this paper, I study contests where the effort choices are not necessarily simultaneous. In most real-life situations some players can observe their competitors’ efforts. Later movers can respond appropriately to the choices earlier movers make. However, earlier movers can anticipate these responses and affect the behavior of later movers. Each additional period in a sequential contest adds a level of complexity, which might explain why most of the previous work studies only simultaneous and two-period models. I characterize equilibria for an arbitrary sequential contest and analyze how the information about other players’ efforts affects the equilibrium behavior.

Contests may be sequential by nature or by design. Contest designers often can choose how much information about players’ efforts to disclose. For example, in rent-seeking contests, firms lobby to achieve market power. One tool that regulators can use to minimize rent-seeking is to design a disclosure policy.111In the last decades, many countries have introduced new legislation regulating transparency in lobbying, including 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 cross-country differences in regulations—for example, in the US, lobbying efforts must be reported quarterly, whereas in the EU, reporting is arranged on a more voluntary basis and on yearly frequency.

A non-transparent policy would lead to simultaneous effort choices, full transparency to a fully sequential contest, and there may be potentially intermediate solutions such as revealing information only occasionally. In research and development, the probability of a scientific breakthrough may be proportional to research efforts, which are typically considered socially desirable. The question is again how to organize the disclosure rules. A transparent policy could be implemented as a public leaderboard or early working papers, whereas a non-transparent policy would encourage teams to work in isolation.

I provide two main results. First, in the characterization theorem (Theorem 1) I characterize all equilibria for any given disclosure rule. The standard backward-induction approach requires finding best-response functions every period and substituting them recursively. I show that this approach is not generally tractable, or even feasible. Instead, I introduce an alternative approach, in which I characterize best-response functions by their inverses. This allows me to pool all the optimality conditions into one necessary condition and solve the resulting equation just once. I prove that the equilibrium exists and is unique, and I show how to compute it.

In my second main result (Theorem 2) I show that the information about other players’ efforts strictly increases the total effort. This means 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, whereas when the efforts are undesirable (as in rent-seeking), the optimal contest is one with hidden efforts. The intuition behind this result is simple. Near the equilibrium, the efforts are strategic substitutes. Therefore, players whose actions are disclosed have an additional incentive to exert more effort to discourage later players. This discouragement effect is less than one-to-one near the equilibrium because if players can increase their effort in a way that diminishes total effort, this provides them with a profitable deviation.

The information about other players’ effort is important both qualitatively and quantitatively. Under typical assumptions (Tullock contest payoffs), the sequential contest with 5 players ensures higher total effort than the simultaneous contest with 24. The differences become even larger with larger contests. For example, a contest with 14 sequential players achieves 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 from Dixit (1987). Dixit showed that a player who can pre-commit chooses greater effort and obtains a higher payoff than the followers. The leader has two advantages: he moves earlier and has no direct competitors. With the new characterization, I can explore this idea further and compare players’ payoffs and efforts in an arbitrary sequential contest. I show that there is a strict earlier-mover advantage—earlier players choose greater efforts and obtain higher payoffs than later players.

I provide additional results for large contests. Although the characterization result holds for any number of players, calculation of equilibrium becomes cumbersome with large contests, especially when the number of periods becomes large. I show that there is a convenient approximation method in which the equilibrium efforts can be directly computed using a simple formula. This allows me to study the rates of convergence under different disclosure policies. I show that the convergence is much faster in the case of sequential versus simultaneous contests.

My results have applications in various branches of economic literature, including oligopoly theory, contestability, rent-seeking, research and development, public goods provision, and parimutuel betting. For example, they provide a natural foundation for the contestability theory: if the moves are sequential, then a market could be highly concentrated but still be very close to competitive equilibrium. The early players produce most of the output and get most of the profits, but later players behave almost as an endogenous competitive fringe by being ready to produce more as soon as earlier players try to exploit their market power. Similarly, my results provide an explanation to a paradox in the rent-seeking literature: to explain rent dissipation with strategic agents, we need an unrealistically large number of players. I show that in a sequential rent-seeking contest it is sufficient to have only a small number of players to achieve almost full rent dissipation.

#### Literature:

The simultaneous version of the model has been studied extensively in many branches of the literature, starting from Cournot (1838). I use the Tullock contest as my leading example. The literature on this type of contest was initiated by Tullock (1967, 1974) and motivated by rent-seeking (Krueger, 1974; Posner, 1975).222See Tullock (2001); Nitzan (1994); Konrad (2009) for literature reviews on contests. Bell et al. (1976) provided an axiomatization under which market shares are proportional to advertising spending so that this model could be directly applied to advertising. The most general treatment of simultaneous contests is provided by the literature on aggregative games (Jensen, 2010; Acemoglu and Jensen, 2013; Martimort and Stole, 2012).333See Jensen (2017) for a literature review on aggregative games. My model is an aggregative game only in the simultaneous case.

Two-period contests have also been studied extensively, starting with von Stackelberg (1934), who studied quantity leadership in an oligopoly. In Tullock contests, the outcomes in a two-player sequential contest coincide with that of a simultaneous contest (Linster, 1993). With more than two players, a first mover has a strict advantage (Dixit, 1987).444In asymmetric contests even the two-player sequential contest differs from the simultaneous contest (Morgan, 2003; Serena, 2017). Moreover, when the order of moves is endogenous, sequential contests arise in equilibrium in both two-player Tullock contests (Leininger, 1993; Morgan, 2003) and in oligopoly (Spence, 1977; Fudenberg and Tirole, 1985; Anderson and Engers, 1992; Amir and Grilo, 1999).

Relatively little is known about contests with more than two periods. The only paper I am aware of that has studied sequential Tullock contests with more than two periods is Glazer and Hassin (2000), who characterized the equilibrium in the sequential three-player Tullock contest. The only class of contests where equilibria are fully characterized for sequential contests are oligopolies with linear demand (Daughety, 1990; Ino and Matsumura, 2012; Julien et al., 2012). Linear demand implies quadratic payoffs; this is the only case where the first-order conditions are linear in all variables and therefore the equilibrium is simple to characterize.555However, linearity is a special assumption with perhaps undesirable consequences. For example, the leader’s quantity in Stackelberg model with linear demand is independent of the number of followers (Anderson and Engers, 1992). See appendix H for more detailed discussion of contests with linear payoffs.

More is known about large contests. Perfect competition (Marshall equilibrium) is a standard assumption in economics and it is important to understand its foundations. Novshek (1980) showed that although under some assumptions the Cournot equilibrium may not exist with a finite number of players, in large markets the Cournot equilibrium exists and converges to the Marshall equilibrium. Robson (1990) provided further foundations for Marshall equilibrium by proving an analogous result for large sequential oligopolies. In this paper, I take an alternative approach and under stronger assumptions about payoffs I provide a full characterization of equilibria with any number of players and any disclosure structure, which includes simultaneous and sequential contests as extremes. This allows me not only to show that the large contest limit is Marshall equilibrium but study the rates of convergence under any contest structures.

There are three somewhat distinct branches of literature on contests: Tullock contests, all-pay contests, and tournaments.666For detailed literature reviews, see Konrad (2009) and Vojnovic (2015). These three branches differ in terms of the contest success function, i.e., the criteria for allocating prizes. First, in Tullock contests the probabilities of receiving the prizes are proportional to efforts (Tullock, 1967, 1974, 2001). In the static framework, they typically give unique equilibria, which is in pure strategies. This model is often used to study rent-seeking, R&D races, advertising, and elections. My model extends this class of models to sequential settings.

The second class of contests includes all-pay auctions and wars of attrition, where the player with the highest effort always wins. These models are often used to study lobbying, military activities, and auctions. The equilibria in this class are typically in mixed strategies. Baye et al. (1996) characterize the equilibria in the static common-value (first-price) all-pay auction and Hendricks et al. (1988) in the second-price all-pay auction (also called the war of attrition). Siegel (2009) provides a general payoff characterization for the static all-pay contests. The third class of contests is rank-order tournaments in which prizes are allocated according to the highest output rather than the highest effort. Output is a noisy measure of effort. Tournaments were introduced by Lazear and Rosen (1981) and Rosen (1986) and are most often used to model principal-agent relationships and contract design in personnel economics and labor economics. My assumptions exclude all-pay contests and tournaments.

The paper also contributes to the contest design literature. Previous papers on contest design include Glazer and Hassin (1988), Taylor (1995), Che and Gale (2003), Moldovanu and Sela (2001, 2006), Olszewski and Siegel (2016), and Bimpikis et al. (2017), which have focused on contests with private information. In this paper I study contest design on a different dimension: how to disclose other players’ efforts optimally to minimize or maximize total effort.

The paper is organized as follows. Section 2 introduces the model and discusses the main assumptions. Section 3 discusses the difficulties with the standard solution method, describes the approach introduced in this paper, and provides the characterization result. Section 4 gives the second main result, the information theorem, which shows that the total effort is increasing in information in a general sense. It also discusses the corollaries of the result, including its implications for optimal contests. Section 5 studies earlier-mover advantage and section 6 discusses large contests. Section 7 analyses applications and policy implications of the results. Finally, section 8 concludes. All proofs are presented in the appendixes.

## 2 Model

#### Players and timing:

There are identical players who arrive to the contest sequentially and make effort choices on arrival. At points in time, the sum of efforts by previous players is publicly disclosed.777The points of disclosure could be exogenous or chosen by a contest designer. In the following sections I first characterize the equilibria of arbitrary contests with fixed disclosures and then study the relationship between the equilibrium efforts and disclosures. These disclosures partition players into groups, denoted by . In particular, all players in arrive before the first disclosure and therefore have no information about other players’ efforts. All players in arrive between disclosures and and therefore have exactly the same information: they observe the total effort of players arriving prior to disclosure . I refer to the time interval in which players in group arrived as period

. As all players are identical, the disclosure rule of the contest is fully described by the vector

, where is the number of players arriving in period .888Equivalently the model can be stated as follows: there are time periods and players are distributed among these periods and either exogenously or by the contest designer.

#### Efforts:

I assume that each player chooses an individual effort at the time of arrival. I denote the profile of effort choices by , the total effort in the contest by , and the cumulative effort after period by . As the payoffs only depend on the sum of other players’ efforts, it is sufficient to keep track of the cumulative effort, even if the players are able to observe individual efforts. By construction, the cumulative effort before the contest is , and the cumulative effort after period is the total effort exerted during the contest, i.e. . Figure 1 summarizes the notation with an example of a four-period contest.

#### Payoffs:

As the leading example of this paper I use normalized Tullock payoffs, with

 ui(x)=xiX−xi. (1)

This can be interpreted as a contest where players compete for prizes of total size one, the probability of winning is proportional to efforts, and the marginal cost of effort is one. Alternatively, this could be a model of an oligopoly with unit-elastic inverse demand function and marginal cost , or a public goods provision (or tragedy of the commons) with the marginal benefit of private consumption .

Note that all of the results apply to a more general class of contests than Tullock contests. As the proofs of the general results are analogous, it is useful to focus on the example with Tullock payoffs first. I discuss general payoff functions in section 7.

#### Equilibrium concept:

I study pure-strategy subgame-perfect equilibria (SPE), which is 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 pure-strategy SPE.

#### Restrictive assumptions:

Throughout this 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—on arrival. Fourth, disclosures make cumulative efforts public. In section 8, I discuss the extent to which the results rely on each of these assumptions.

## 3 Characterization

### 3.1 The problem with standard backward induction

Before describing the solution, let me use a simple example to illustrate the difficulty of using the standard backward-induction approach. I will then describe the alternative approach I introduce in this paper.

The standard Tullock contest with identical players is a simple game. If players make their choices in isolation, i.e., no information is revealed to them, then each player chooses effort simultaneously to maximize the payoff (1). The optimal efforts have to satisfy the first-order condition

 1X−xiX2−1=0. (2)

Combining the optimality conditions leads to a total equilibrium effort and individual efforts . The equilibrium is unique, in pure strategies, easy to compute, and easy to generalize in various directions, which may explain the widespread use of this model in various branches of economics.

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 , and after observing the efforts of previous players. I will first try to find equilibria using the standard backward-induction approach.

Player observes the total effort of the previous two players, 999Players get non-positive payoffs in histories with or , which cannot be an equilibrium. and maximizes the payoff (1), which in this case is

 maxx3≥0x3X2+x3−x3.

The optimality condition for player is

 1X2+x3−x3(X2+x3)2−1=0⇒x∗3(X2)=√X2−X2. (3)

Now, player observes and knows , and therefore maximizes

 maxx2≥0x2x1+x2+x∗3(x1+x2)−x2=maxx2≥0x2√x1+x2−x2.

The optimality condition is

 1√x1+x2−x22(x1+x2)32−1=0.

For each , this equation defines a unique best-response,

 x∗2(x1)=112−x1+(8√27x31(27x1+1)+216x21+36x1+1)23+24x1+112(8√27x31(27x1+1)+216x21+36x1+1)13. (4)

Finally, player ’s problem is

 maxx1≥0x1x1+x∗2(x1)+x∗3(x1+x∗2(x1))−x1,

where and are defined by equations 4 and 3. Although the problem is not complex, it is not tractable. Moreover, it is clear that the direct approach is not generalizable for an arbitrary number of players.101010Glazer and Hassin (2000) characterized the equilibrium in the contest with three sequential players. To my knowledge, no existing papers have characterized equilibria for sequential contests with more than three periods. In fact, the best response function typically does not have an explicit representation for contests with a larger number of periods.111111The representation from theorem 1 implies that best-response functions include roots of higher-order polynomials that are not solvable (see appendix C for details). Therefore, the best-response functions cannot be expressed explicitly in terms of standard mathematical operations.

### 3.2 Inverted best-response approach

In this paper I introduce a different approach. Instead of characterizing individual best-responses or the total efforts induced by , denoted by , I characterize the inverse of this function. For any level of total effort , the inverted best-response function is the cumulative effort prior to period , that is consistent with total effort being , given that the players in periods behave optimally.

To see how this characterization works, consider the three-player sequential contest again. In the last period, player observes and chooses . Equivalently we can think of his problem as choosing the total effort by setting . His maximization problem is

 maxX≥X2X−X2X−(X−X2)⇒1X−X−X2X2−1=X2X2−1=0,

which implies . That is, if the total effort in the contest is , then before the move of player , the cumulative effort had to be ; otherwise, player is not behaving optimally.

We can now think of player ’s problem as choosing , which he can induce by making sure that the cumulative effort after his move is , which means setting . Therefore, his maximization problem can be written as

 maxX≥X1f2(X)−X1X−(f2(X)−X1)⇒f′2(X)X−f2(X)−X1X2−f′2(X)=0. (5)

This is the key equation to examine in order to see the advantages of the inverted best-response approach. Equation 5 is non-linear in and therefore in , 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 accumulates with each step of the recursion. But equation (5) is linear in , and it is therefore easy to derive the inverted best-response function

 f1(X)=X1=f2(X)−f′2(X)X(1−X)=X2(2X−1).

Again, we know that if the total effort of the contest is , then after player the cumulative effort has to be ; otherwise, at least one of the later players is not behaving optimally.

Moreover, note that cannot be induced by any , as even if , the total effort chosen by players and would be . Inducing total effort below would require player 1 to exert negative effort, which is not possible. Therefore, is defined over the domain , and it is strictly increasing in this interval.

Finally, player maximizes a similar problem of choosing , which he can induce by setting , to maximize

 maxX≥12f1(X)X−f1(X)⇒f′1(X)X−f1(X)X2−f′1(X)=0,

which implies

 0=f0(X)=f1(X)−f′1(X)X(1−X)=X2(6X2−6X−1). (6)

Solving equation 6 gives three candidates for the total equilibrium effort . It is either , , or . As we already found that in equilibrium , only the highest root constitutes an equilibrium.121212I did not check that the individual necessary conditions are also sufficient in this example, but in the characterization result in the next section, I verify this for arbitrary contests.

The main advantage of the inverted best-response approach is that instead of finding the solution of a non-linear and increasingly complex equation, the approach allows for combining all of the first-order necessary conditions into one, which is then solved only once.

Note that in each period, there is a simple recursive dependence that determines how the inverted best-response function evolves. At the end of the contest, i.e., after period , the total effort is . In each of the previous periods it is equal to . Extending the analysis from three sequential players to four or more sequential players is straightforward. It simply requires applying the same rule more times and solving the somewhat more complex equation at the end.

### 3.3 Characterization theorem

Theorem 1 formalizes this approach, generalizes the result for more general payoff functions, and allows for multiple players who make simultaneous decisions. It shows that each contest has a unique equilibrium. The total equilibrium effort is the highest root of , and the individual equilibrium effort of player from period can be computed directly as .

The characterization builds on the recursively defined inverted best-response functions , where is the cumulative effort after period , consistent with the total effort in the contest being . The total effort after the last period is , and the cumulative efforts in all previous periods are recursively defined by

 ft−1(X)=ft(X)−ntf′t(X)g(X),∀X∈[X––t,1],∀t=1,…,T, (7)

where is the highest root of in and .

The recursive rule is similar to the example, but with two natural differences. First, it allows simultaneous decisions. When players make simultaneous decisions in period , then each of them has only a fractional impact on the best-responses of the following players, which means that the impact on the inverted best-response function is magnified by . Second, in the proof of the theorem, I allow more general payoff functions, which are captured by function .131313I discuss sufficient conditions on general payoffs in section 7.

The first part of the proof shows that the inverted best-response functions are well-behaved. As figure 2 illustrates, all of these functions take positive values at , their highest roots are always in and decreasing in (i.e., moving closer to as decreases). Moreover, each is strictly increasing between its highest roots and , which means that in this relevant range it is invertible and therefore defines the best-response function uniquely, and is strictly negative in the between the two consecutive highest roots, i.e. , which excludes all the other candidates for equilibria. These properties are formalized as condition 1.

###### Condition 1 (Inverted best responses are well-behaved).

For all , the function has the following properties:

1. has a root in .141414Function has only one root . Let be the highest such root.

2. for all .

3. for all .

Moreover, .

The following proposition 1 shows that condition 1 is satisfied in case of Tullock payoffs.

###### Proposition 1.

If , then condition 1 is satisfied.

The second part of the proof shows that the properties highlighted in condition 1 are sufficient for the existence and uniqueness of the equilibrium, and the equilibrium is characterized as described above. The idea of the proof is analogous to the example in section 3.2.

###### Theorem 1 (Characterization theorem).

Suppose condition 1 holds. Each contest has a unique equilibrium. The equilibrium strategy of player in period is151515Function denotes the inverse function of in the interval , where is the highest root of in . By condition 1, is strictly increasing in this interval.

 x∗i(Xt−1)={1nt[ft(f−1t−1(Xt−1))−Xt−1]∀Xt−1<1,0∀Xt−1≥1. (8)

In particular, the total equilibrium effort is , i.e., the highest root of , and the equilibrium effort of player is .

The characterization theorem provides a straightforward method for computing the equilibria in any contest . Appendix K provides several examples demonstrating how it can be applied in the case of Tullock payoffs and other payoff functions.

## 4 Information and effort

In this section, I present the second of the two main result of this paper. I show that the information increases total effort in contests. The result has strong implications for both the comparative statics and for optimal contests.

Before giving the formal result, I need to introduce the notation to keep track of the relevant information in contest . Let me use a contest to illustrate the construction. This contest has four players: a first-mover, two simultaneous followers, and a last-mover. At the most basic level, all players observe their own efforts (regardless of the disclosure rule). There are observations of this kind and they clearly affect the outcomes of the contest. I call this the first level of information and denote as . More importantly, some players directly observe the efforts of some other players. In the example, players 2 and 3 observe the effort of player 1 and player 4 observes the efforts of all three previous players. Therefore, there are five direct observations of other players’ efforts. I call this the second level of information and denote as . Finally, player 4 observes players 2 and 3 observing player 1. There are two indirect observations of this kind, which I call the third level of information and denote as . In contests with more periods, there would be more levels of information—observations of observations of observations and so on.

I call a vector the measure of information in a contest . In the example described above, . Formally, is the sum of all products of -combinations of set .161616For example, in a sequential -player contest , is simply the number of all -combinations, i.e., Note that we can also think of as an infinite sequence with for all .

Theorem 2 shows that the total effort is strictly increasing in vector . The key step in proving this is showing that the inverted best-response function can be expressed as

 f0(X)=X−T∑k=1Sk(n)gk(X), (9)

where are recursively defined as and . Each is independent of and clearly is independent of . As the total equilibrium effort, is the highest root of ; it therefore depends on only through information measures .

Each describes the substitutability between efforts. In particular, if , then the efforts are strategic substitutes near equilibrium in the standard sense, i.e., the effort of an earlier player discourages the effort of a later player who observes player . If it is also the case that , then the indirect impact through observations of observations also leads to discouragement, i.e., not only does player discourage a later player directly, but if this player moves after player , then the indirect effect of player ’s effort through player on player is discouraging. Similarly describes the discouragement effect through -th level of information.171717My assumptions also guarantee that , i.e., individual effort discourages one’s own effort, which is a standard concavity assumption that guarantees the interior optimum.

In particular, a sufficient condition that guarantees is strictly increasing in is that each , i.e., the efforts are higher-order strategic substitutes near equilibrium, which is formalized by condition 2.

###### Condition 2 (Higher-order strategic substitutes).

for all .

In the case of Tullock payoffs, , which means that with low total effort , the efforts are strategic complements; with high total effort , the efforts are strategic complements; and in the knife-edge case , they are independent. This is why information about the first mover’s effort in the two-player Tullock contest does not change the equilibrium efforts. Indeed, in both and the total effort is and the individual efforts are . I show that for any other contests the total equilibrium effort is strictly higher than and therefore, the efforts are strategic substitutes in the standard sense.

As the following proposition shows, this observation generalizes to -player contests. The efforts are higher-order strategic substitutes for any contest except in the fully sequential case, where the highest . However, if there are at least three players, this exception does not affect the conclusions because the fully sequential contest also has strictly higher lower-level information measures than any other contest. For example, when , , and , i.e., contest has a strictly higher than any other three-player contest.

###### Proposition 2.

Suppose . Then

1. if and , then ; otherwise,

2. if or , then .

Now, if condition 2 holds, then the increase181818 means that for all and for at least one . in would lower near the original equilibrium and therefore, due to condition 1, the total equilibrium effort must increase. These arguments jointly imply the second main result of the paper, theorem 2. It shows that the total equilibrium effort of any contest only depends on the information in the contest and is strictly increasing in each level of information.

###### Theorem 2 (Information theorem).

Suppose conditions 2 and 1 hold. Total effort is a strictly increasing function .

Therefore, information in contests defines a partial order over contests—if then . Theorem 2 has many direct implications. I highlight some of the most important ones in the following corollaries. Note that these corollaries hold whenever conditions 2 and 1 are satisfied, which includes all contests with Tullock payoffs and at least three players.

###### Corollary 1.

Under conditions 2 and 1

1. Comparative statics of : if 191919Including when , i.e., has more periods with strictly positive number of players., then and therefore .

2. Independence of permutations: if is a permutation of , then and therefore .

3. Informativeness increases total effort: if is a finer partition than , then and therefore .

4. Homogeneity increases total effort: if and there exist such that and for all , then and therefore .

5. Full dissipation in large contests: .

The first implication is natural—if the number of players increases in any particular period or a period with a positive number of players is added, then the total effort increases. Note that this does not mean that the total effort is a strictly increasing function of the number of players. For example, as we saw above, the three-player sequential Tullock contest gives total effort , whereas the four-player simultaneous Tullock contest gives total effort . The conclusion only holds when we add players while keeping the positions of other players unchanged, because this increases information.

The second implication is perhaps much more surprising—reallocating disclosures in a way that creates a permutation of (or equivalently, reordering periods together with the corresponding players) does not affect the total effort. For example, a contest with a first mover and followers leads to the same total effort as a contest with first movers and one last mover. The result comes from the property that all observations of the same level have the same impact on the total effort; i.e., it does not matter whether players observe one player or one player observes players.

The third implication is perhaps the most important in terms of its consequences. It shows that disclosures strictly increase total effort. In particular, adding disclosures makes the contest strictly more informative: all players observe everything that they observed before, but some observe the efforts of more players. Formally, the new contest is a finer partition of players than the old contest.

To see the intuition of this result, consider contests and . Now, player 2’s effort is made visible for player 3. Therefore, in addition to all marginal costs and benefits of effort, player 2 has an additional benefit of effort: as efforts are strategic substitutes, exerting more effort discourages player 3. This discouragement effect leads to added effort by player 2 and reduced effort by player 3, and the remaining question is how these effects compare. Player 2’s payoff is , where is a strictly decreasing function of total effort, which means that if he could increase his own effort without increasing the total effort, he would certainly do so and it would not be an equilibrium. Therefore, near equilibrium we would expect the discouragement effect to be less than one-to-one, i.e., player 2 exerts more effort and player 3 exerts less but the total increases. Of course, this argument works only for small changes, keeping the efforts of indirectly affected players 1 and 4 unchanged. However, since the efforts are strategic substitutes of higher order, the indirect effects have the same signs and therefore the result still holds.

The fourth implication gives even clearer implications for the optimal contest. Namely, more homogeneous contests give higher total effort. Intuitively, a contest is more homogeneous if its disclosures are spread out more evenly (or equivalently, players are divided more evenly across periods). For example, a contest is more homogeneous than . It has also more direct observations as . I define the more homogeneous contest as one which can be achieved by pairwise increases of products of group sizes while keeping everything else fixed. Therefore, by construction, it increases strictly and all other measures weakly.

The final implication is that in large contests rents are fully dissipated. This result follows from the fact that simultaneous contests are the least informative and give total effort , which converges to as . Therefore, the total effort from any contest converges to .202020This limit result is known in simultaneous contests since Novshek (1980) and (fully) sequential contests since Robson (1990).

These results give strong implications for the contest design, which I summarize in the following corollary.

###### Corollary 2.

Assume that conditions 2 and 1 hold and fix . A simultaneous contest minimizes the total effort, and a fully sequential contest maximizes the total effort. Moreover, if the contest designer can only make a fixed number of disclosures212121Or equivalently, there is a fixed number of periods., then contests that allocate players into groups that are as equal as possible maximize the total effort.

Therefore, if the goal is to minimize the total effort (such as in rent-seeking contests), then the optimal policy is to minimize the available information, which is achieved by a simultaneous contest. Transparency gives earlier players incentives to increase efforts to discourage later players, but this discouragement effect is less than one-to-one and therefore increases total effort.

On the other hand, if the goal is to maximize the total effort (such as in research and development), then the optimal contest is fully sequential as it maximizes the incentives to increase efforts through this discouragement effect. If the number of possible disclosures is limited (for example, collecting or announcing information is costly), then it is better to spread the disclosures as evenly as possible.222222It is easy to see that parts 2 and 4 of corollary 1 imply that the contest that maximizes total effort with disclosures ( periods) and players is such that there are periods with players and periods with players.

Vector comparison defines a partial order over contests. To complete the order, we would have to know how to weigh different measures of information. Equation 9 provides a clue: the correct weights are ; i.e., by magnitudes of discouragement effects near equilibrium. Unfortunately, these weights are endogenous and generally depend on a specific contest, but there are a few cases where we can say more.

I argue in section 6 that if the number of players is large, then is approximately , with in Tullock payoffs case. Therefore, in large sequential Tullock contests all measures of information are produced approximately equally, which gives a complete order on contests: if and only if .

However, with a smaller number of players, the following lemma shows that lower information measures have a higher weight.

###### Lemma 1.

For , for each , .

For example, let us compare two 10-player contests and . Then and . The two contests cannot be ranked according to the information measures, because the first contest has more second-order information, whereas the second contest has one more disclosure and thus more third-order information. However, the sum of all information measures is . Since the weights are higher in lower-order information, this implies that the total effort is higher in the first contest. Indeed, direct application of the characterization theorem confirms this, as .

In this section, I revisit Dixit’s first-mover advantage result. Dixit showed that in a contest with at least three players, if one player can pre-commit, this first mover chooses a strictly higher effort and achieves a strictly higher payoff than the followers. Using the tools developed in this paper, I can explore this idea further. Namely, in Dixit’s model, the first mover has two advantages compared to the followers. First, he moves earlier, and his action may impact the followers. Second, he does not have any direct competitors in the same period.

I can now distinguish these two aspects. For example, what would happen if players chose simultaneously first and the remaining player chose after observing their efforts? Or more generally, in an arbitrary sequence of players, which players choose the highest efforts and which ones get the highest payoffs? The answer to all such questions turns out to be unambiguous. As proposition 3 shows, it is always better to move earlier.

Suppose conditions 2 and 1 hold. The efforts and payoffs of earlier232323By earlier I mean players who belong to the strictly earlier group, i.e., player is earlier than if and only if . players are strictly higher than for later players.

Let us consider payoff comparisons first. Note that the equilibrium payoff of a player is in the form , and since 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. In the proof, I show that we can express the difference between the efforts of players and from consecutive periods and as

 x∗i−x∗j=T−t∑k=1[Sk(nt)−Sk(nt+1)]gk+1(X∗), (10)

where is the sub-contest starting after period and is the sub-contest starting after period . Clearly, for each ; i.e., there is more information on all levels in a strictly longer contest. By condition 2, for each as well, and therefore the 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 additional incentives to exert more effort to discourage later players.

## 6 Large contests

In this section, I show that as the number of players becomes large, the total effort converges to 242424Corollary 1 showed that the total effort converges to in any contest. much faster in sequential contests than in simultaneous contests. Figure 3 illustrates that although all contest types converge to , the rate of convergence depends significantly on the type of contest.

To compare the rates of convergence formally I need to discuss how to compute equilibria in large contests. Although the characterization theorem holds for an arbitrary contest, since is a polynomial of degree , its highest root may sometimes be difficult to compute. In appendix H I show that the total equilibrium effort and individual equilibrium efforts with a large number of players can be approximated by252525In case of general payoff functions, if , then these formulas are adjusted by a constant . See appendix H for details.

 (11)

That is, the total equilibrium effort converges to a strictly increasing function of . Therefore, in large contests, information of all degrees carries equal weight.

The reason for this result is simple. As the payoff function is smooth, it can be closely approximated by a linear function. More precisely, payoffs are represented by function . In large contests, the total effort , and therefore near we have that and . This gives us and so on; i.e., each near . Equation 9 can be approximated by

 f0(X∗)≈1−(1−X∗)S(n)=0⇒1−X∗≈1S(n).

From (11) we can make a few observations. As already argued, all measures of information carry approximately equal weight in determining the total equilibrium effort. Moreover, from (11) we see that the convergence is with rate . With simultaneous contests the sum of all levels of information is simply , so that the convergence is linear. On the other extreme, with sequential contests , we get that , so that the convergence is exponential. When there are at most periods, the rate of convergence is bounded by by the same argument.

These results highlight the importance of information in contests. The information provided is at least as important as the number of players in determining the total effort in contests. For example, the total effort in the simultaneous contest with ten players is , whereas the total effort with four sequential players is . Adding a fifth sequential player increases the total effort to , and to achieve this in a simultaneous contest with the same payoffs, we need 24 players. This comparison becomes even more favorable for the sequential contests with a large number of players, as figure 4 illustrates.

For large contests, we can use the approximation from equation 11 to get a relationship between the number of players in simultaneous contests and the number of players in sequential contests with approximately the same total revenue:

 11+nsim−1=1nsim≈1−X∗≈1(1+1)nseq⇒nsim≈2nseq.

## 7 Applications

In this section, I discuss some implications of the results in the various branches of economic literature. To apply the results to a wider range of applications, it is useful to note that all the results presented in the earlier sections apply more generally than Tullock payoffs. The approach and the results generalize to payoff functions in the form

 ui(x)=xih(X), (12)

where is a strictly decreasing function that satisfies sufficient conditions 2 and 1. In particular, and . Appendix I discusses the sufficient conditions in detail and describes a class of payoff functions, where the conditions are satisfied.262626A program that verifies condition 1 and, if it is satisfied, computes the equilibrium for any contest is available at http://toomas.hinnosaar.net/contests/. Appendix K provides some examples of specific functional forms.

### 7.1 Oligopolies

In case of an oligopoly with a homogeneous product, the profit function of firm is

 ui(x)=xiP(X)−cxi=xih(X),

where is firm’s own quantity, the total quantity, the inverse demand function, and the marginal cost of production, which is assumed to be constant and equal for all firms. The simultaneous contest is the standard Cournot oligopoly 272727In the case of Tullock payoffs, it is also known as the Cournot-Puu oligopoly after Puu (1991). and is the Stackelberg quantity-leader model. In general, the model is a hierarchical oligopoly model, where firms in earlier periods have more market power.

By theorem 1, the equilibrium exists and is unique, and the total equilibrium quantity is always below .282828The threshold is determined by as or equivalently , so it can be interpreted as the Marshall equilibrium quantity. It can be normalized to without loss of generality. Proposition 3 shows that earlier firms produce strictly more and earn strictly higher profits than the later firms. Corollary 2 shows that the market structure that maximizes total quantity is fully sequential and the market structure that minimizes is simultaneous (Cournot).

The social planner maximizes the total surplus

 S(X)=∫X0(P(t)−c)dt=∫X0h(t)dt. (13)

The surplus is maximized at and is strictly increasing in below that. By theorem 2 the total equilibrium quantity is strictly increasing in information in this range; therefore, the socially optimal oligopoly would be fully sequential.292929This implication has been studied in an oligopoly with linear demand and constant marginal costs by Daughety (1990).

For the firms, this outcome is not as appealing. The total profit of all firms is

 U(X)=n∑i=1ui(x)=n∑i=1xi(P(X)−c)=Xh(X). (14)

The joint profit maximization would lead to optimality condition

 dπ(X)dX=h(X)+Xh′(X)=h′(X)[X−g(X)]=0, (15)

where ; so the monopoly’s optimal total quantity satisfies . The joint profit is strictly decreasing in . Equation 9 and condition 2 imply that the total equilibrium quantity regardless of market structure, and therefore, a collusive agreement would choose a market structure that minimizes , which is the simultaneous contest (Cournot).

In practice, we may think of quantity competition as capacity competition.303030This idea was formalized by Kreps and Scheinkman (1983). The results in this paper imply that credible commitments to large capacities may lead to larger total capacity and thus to lower prices and higher welfare. Pre-commitments to large capacities allow discouraging capacity investments by the later movers. Since this discouragement effect is less than one-to-one, i.e., increased earlier-mover capacity is not fully canceled out through capacity reduction by later movers, the pre-commitments increase total capacity. Kalyanaram et al. (1995) survey the widespread empirical evidence that a negative relationship exists between brands’ entry to the market and market share. The negative relationship holds in many mature markets, including pharmaceutical products, investment banks, semiconductors, and drilling rigs. For example, Bronnenberg et al. (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.

### 7.2 Contestability

The theory of contestability (Baumol, 1982; Baumol et al., 1988) is widely used in practice; it postulates that with frictionless reversible entry, a market may be concentrated and contestable at the same time. In particular, if firms have no entry and exit barriers, no sunk costs, and access to the same technology, then the market is contestable. In a contestable market, firms operate at a zero-profit level, regardless of the number of incumbent firms in the market. The reason is that when they would try to use their dominance to extract rents, the competitive fringe would enter for a short period, undercut them, and capture the extra rents. Therefore, the theory postulates that the threat of entry works as a disciplining device, and that firms operate at zero profits.

This theory has been widely used, but also criticized (Brock, 1983; Shepherd, 1984; Dasgupta and Stiglitz, 1988; Gilbert, 1989), partly because it is sensitive to even small sunk costs and partly because it requires the existence of a competitive fringe that is not observed in equilibrium.

The results in this paper provide a natural foundation for contestability. As a benchmark, a competitive equilibrium would mean a large number of identical players who each choose low quantities. In contrast, in a fully sequential contest, we need a much lower number of firms to achieve the same total quantity and the individual behavior is very different. The first firm chooses a quantity which is about of the total quantity.313131In the Tullock payoffs case , so the first player’s quantity is about of the total. The second firm chooses and so on. Therefore, the market is highly concentrated323232Herfindahl-Hirschman Index is . With Tullock payoffs, . For any , , which is a highly concentrated market. In contrast, in a simultaneous contest, . and the first few firms have significant market power which they use to achieve higher profits than the followers. However, the market is quite competitive and this competitiveness comes from the later players. They produce very little in equilibrium, but as soon as earlier players deviate to lower quantities, the later players respond with higher quantities. Therefore, they act as a competitive fringe but in a standard oligopoly model. In this paper, I do not include fixed costs and entry decisions, but it would be straightforward to extend the analysis in this direction and endogenize the number of players.333333The observation that high concentration does not necessarily coincide with low competitiveness has been made by Demsetz (1968), Daughety (1990), Ino and Matsumura (2012) among others.

### 7.3 Rent dissipation in rent-seeking

Early papers by Tullock (1967), Krueger (1974), and Posner (1975) proposed that competitive rent-seeking leads to full dissipation of rents. However, strategic modeling of rent-seeking (Tullock, 1974, 2001) showed that strategic behavior leads to rent underdissipation.

As the results in this paper show, the significant underdissipation result may be an implication of mainly focusing on simultaneous rent-seeking contests. As discussed in section 6, it suffices to have 5–10 players in a sequential rent-seeking contest to achieve outcomes that are very close to full dissipation, whereas in a simultaneous contest the same outcomes are achieved with a very high number of players.343434According to Murphy et al. (1993), rent-seeking activities exhibit increasing returns, which means that the differences between simultaneous and sequential behavior may be magnified even further.

### 7.4 Research and development

Closely related models have been used to study patent races. The classic models, such as Loury (1979) and Dasgupta and Stiglitz (1980), assume that firms compete to innovate. Each firm chooses lump-sum investment , and its probability of making a discovery on or before time is with some hazard that is constant over time and increasing and concave function of investment. The probability that a firm makes a discovery at time is therefore , and the conditional probability that it is the firm who makes the discovery is . The first firm that innovates gets a patent with value , discounted at rate , and the payoff of the other firms is normalized to . Therefore, the expected payoff of firm is

 πi(x)=∫∞0vρ(xi)e−∑nj=1ρ(xj)te−rtdt−