DeepAI

# From Monopoly to Competition: Optimal Contests Prevail

We study competition among contests in a general model that allows for an arbitrary and heterogeneous space of contest design, where the goal of the contest designers is to maximize the contestants' sum of efforts. Our main result shows that optimal contests in the monopolistic setting (i.e., those that maximize the sum of efforts in a model with a single contest) form an equilibrium in the model with competition among contests. Under a very natural assumption these contests are in fact dominant, and the equilibria that they form are unique. Moreover, equilibria with the optimal contests are Pareto-optimal even in cases where other equilibria emerge. In many natural cases, they also maximize the social welfare.

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07/11/2019

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## 1 Introduction

Many important economic and social interactions may be viewed as contests. The designer aims to maximize her abstract utility (e.g. workers’ productivity, sales competitions, innovative ideas for new projects, useful information from contestants) by forming a contest, and contestants exert effort in hopes of winning a prize. The design of optimal contests is by now well understood in the monopolistic (single-contest) setting. In particular, in many cases, a winner-takes-all contest is optimal in terms of maximizing either the sum of contestants’ efforts or the single maximal effort (e.g. Barut and Kovenock, 1998; Kalra and Shi, 2001; Moldovanu and Sela, 2001; Terwiesch and Xu, 2008; Chawla et al., 2019).

While most of the existing literature on contest design focuses on a monopolistic contest with an exogenously given set of participants, in reality, many times, there are multiple contests on a market and these contests must compete to attract participants, which induces a participation vs. effort trade-off. Although the optimal contest in the single-contest setting induces maximal effort exertion after contestants choose to participate in their contests, contestants might at the same time be discouraged from choosing the more demanding contests. To attract participants, it seems helpful to design lucrative and easy contests that leave a large fraction of the total surplus to contestants. Thus, these two aspects appear to be contradicting.

Previous literature has already started to acknowledge this issue with few models that formally studied it (e.g., Azmat and Möller (2009); Stouras et al. (2020)). In particular, Azmat and Möller (2009) conclude that, despite the competitive environment, contest designers should still choose effort-maximizing contests since the effort aspect dominates the participation aspect. However, it is not clear how robust this conclusion really is, since these previous papers analyze models that are restricted in two main aspects. First, they assume that all contests have the same total prize to offer.111Previous analysis seems to significantly rely on this assumption. It also formally assumes exactly two competing contests, although this aspect may be more easily generalized.

Second, and perhaps even more important, they restrict the choice of a contest and assume that designers choose a multiple-prize contest where contestants’ winning probabilities for each prize are determined by a Tullock success function that is exgonenous and identical for all contest designers. A main appeal of a Tullock contest as a model to winner-determination in real-life contests is that it captures the fact that the efforts

of the contestants in a certain contest cannot be fully observed by the contest designer. The winning probability may be following a Tullock contest success function using some parameter to capture partial-observability by letting contestants with higher efforts win with higher probabilities (as increases, effort observability is better). With such a motivation, it seems that a contest designer always has the strategic option to artificially reduce her ability to observe effort, e.g., using a Tullock contest success function with some parameter . Combining various prize structures with a limited choice of the parameter is a natural way to expand the set of possible contests to consider. This is not captured in previous models. Alternatively, effort-observability can be endogenously determined (e.g., online contests could use better technology to increase effort observability). In this case, contest designers have the freedom to increase where in the limit as we have an all-pay auction where the contestant with the highest effort wins with certainty. Contest success functions may depend on the number of contestants in other complex ways and many additional examples of natural classes of contests exist (Corchón, 2007). Naturally, as the strategic flexibility of contest designers increases, existing outcomes may no longer be in equilibrium, and even if they do remain in equilibrium, additional more attractive equilibria might emerge.

This paper provides a more general framework and analysis of competition among multiple contests. To be consistent with previous literature our starting point is the model of Azmat and Möller (2009) which we generalize in order to capture the two main aspects discussed in the previous paragraph: a general contest design space and asymmetric contest designers. In a high-level, the model is composed of three phases. In the first phase, contest designers choose their contests (and commit to them) from a class of contests available to them which could be any arbitrary class of contests. In the second step, after seeing the contests chosen by designers, each contestant chooses (possibly in a random way) one contest to participate in. Finally, in each contest, contestants invest effort by playing a symmetric Nash equilibrium (which previous literature has shown to exist, see details in Section 2). Designers aim to maximize the sum of efforts exerted in their own contests and contestants aim to maximize the reward they receive minus their effort.

The bottom line of our analysis is that optimal (effort-maximizing) contests in the monopolistic setting are still in equilibrium even when significantly increasing the strategic flexibility of contest designers. In other words, effort indeed dominates participation in the aforementioned trade-off for the competing designers. In fact, these contests remain the unique equilibrium in many interesting and natural cases (although, as we show, not always). Moreover, even when additional contests emerge as equilibria, choosing effort-maximizing contests is a Pareto-optimal outcome for contest designers maintaining the attractiveness (for the contest designers) of these types of contests. These conclusions hold regardless of the number of designers, the total rewards they have, and the classes of contests they can choose from.

Technically, we identify two properties that characterize the class of effort-maximizing contests and show that if every contest designer chooses a contest that satisfies these two properties then we are at an equilibrium which is Pareto optimal for the designers. The first property, which we term Monotonically Decreasing Utility (MDU) simply says that a contestant’s symmetric-equilibrium utility in the single contest game decreases as the number of contestants increases. The second property, which we term Maximal Rent Dissipation (MRD), is defined with respect to the space of possible contests that contest designer has. A specific contest has maximal rent dissipation if, for any other contest that could be a possible choice for designer , and for any number of contestants, , the contestant’s symmetric-equilibrium utility in the single contest when there are contestants is not larger than the contestant’s equilibrium utility in the single contest when there are contestants. Thus, minimizes the contestants utilities and therefore maximizes the utility of the contest designer, among all contests available to designer . In this sense, is “optimal”.

Going back to Tullock contests and prize structures, the generality of our framework yields, as corollaries to our main result, that: (1) Choosing all-pay auctions is an equilibrium for designers who can only adjust their observability of effort (i.e. the Tullock parameter ) but must give the entire reward to the winner (i.e. winner-takes-all); in fact we show that choosing any is an equilibrium. (2) Choosing winner-takes-all contests is an equilibrium for designers who can only adjust prize structures while the parameter is exogenous. (3) Choosing the winner-takes-all all-pay auction is an equilibrium for the designers when they can adjust both prize structures and effort observability.

In this subsection we review two strands of literature on contest design. One strand of literature considers optimality of contests in a monopolistic (single contest) setting, in terms of revenue for the designer, agent participation, etc. A second, more recent strand, considers equilibrium outcomes when multiple contest designers compete over agents’ participation and effort. Our result is interesting in the way it ties together these domains: We show that effort-maximizing contests (those that were identified in the first literature strand) are in an equilibrium in our general model, that belongs to and follows the second strand. The takeaway message to a contest designer is that, in case she is interested to maximize the sum of contestants efforts, introducing competition does not change her basic goal of maximizing over effort extraction, given a fixed set of contestants.

#### 1.1.1 Optimal contests in the monopolistic setting

In the monopolistic setting, several works study optimal multiple-prize contest design with the objective of maximizing sum of efforts. Under different assumptions, most of these papers arrive at the same conclusion that the optimal contest is the winner-takes-all contest where the full prize is offered to the single contestant exerting the highest effort. For example, in all-pay auctions where contestants’ efforts are fully observable, Barut and Kovenock (1998); Moldovanu and Sela (2001) show that a winner-takes-all all-pay auction is optimal assuming contestants having either linear or concave cost functions (interestingly, for convex cost functions their results vary). An exception is Glazer and Hassin (1988) who show that the optimal contest should offer equal prizes to all players except for the player with the lowest effort if players value the prize money by a strictly concave utility function. We consider linear utilities and linear cost functions as in Barut and Kovenock (1998) so the winner-takes-all is optimal. When contestants’ efforts are not fully observable and their winning probabilities for different prizes are assumed to follow a Tullock success function, the optimal prize structure is once again a winner-takes-all (Clark and Riis, 1998). The winner-takes-all is also optimal in a stochastic-quality model (e.g., Kalra and Shi, 2001; Ales et al., 2017) where a contestant who exerts effort produces a submission with random quality where follows some noise distribution. In this case, prizes are allocated based on the submission qualities. If the designer must offer a single prize, then the all-pay auction is optimal among all possible contests, as it induces full rent dissipation – contestants’ utilities are reduced to zero and their sum of efforts is maximized (Baye et al., 1996).

A line of the literature including Schweinzer and Segev (2012); Baye et al. (1996); Alcalde and Dahm (2010); Ewerhart (2017) studies symmetric equilibria and rent dissipation in optimal contests. Our formal results, which analyze contests that admit symmetric equilibria and certain dissipation properties, have concrete applications thanks to existence and characterization results provided in these papers.

#### 1.1.2 Competition among contests

Previous works (e.g., Stouras et al., 2020; Azmat and Möller, 2009) on contest competition suggest that either participation or effort may dominate in the aforementioned trade-off under different assumptions. For example, Stouras et al. (2020) show that, for designers who wish to maximize the highest quality, participation outweighs effort and hence the designers will set multiple prizes when the quality of submission is sensitive to their effort. When the quality of submission is not sensitive to effort, they show that the effort aspect is dominating and hence the designers will offer a single prize which induces the maximum effort exertion in the single-contest setting. On the other hand, Azmat and Möller (2009) suggest that, for designers maximizing the sum of efforts, the effort aspect is always dominating, and hence a single prize should be offered. As mentioned before, we generalize their work in several aspects.

DiPalantino and Vojnovic (2009) consider an incomplete information competition among contests. They focus on participation issues rather than on the strategic choices of contest designers, by assuming that all contests are all-pay. They explicitly characterize the relationship between contestants’ participation behavior and contests’ rewards, and find that rewards yield logarithmically diminishing returns with respect to participation levels. Körpeoğlu et al. (2017) consider an incomplete information contest model where contestants can participate in multiple contests, and contest designers use winner-takes-all contests while strategically choosing rewards to maximize the maximal submission quality minus reward. They show that, in several cases, contest designers benefit from contestants’ participation in multiple contests.

## 2 Competition Among Contests: Model and Preliminaries

### 2.1 A single-contest game

A contest designer designs a contest among several contestants in order to maximize the sum of efforts exerted by the contestants in return for some reward to be divided among them according to some winning rule determined by the designer.

Formally, a contest is composed of a reward and a family of contest success functions for each number of contestants . Contestants exert efforts to compete for the reward. Each contestant receives a fraction of the reward, where is the

-th coordinate of the vector

. In a stochastic quality model (e.g., the additive-noise model222In an additive-noise contest model, a contestant who exerts effort produces a submission with random quality where follows some noise distribution. The contest designer observes but not and allocate rewards to contestants based on their ’s. ), is the expected fraction of reward received by contestant . We allow general functions and only require that . The utility of a contestant is the reward she gets minus the effort she exerts: . The utility of the contest designer is the sum of efforts . When the designer’s utility is 0.

###### Definition 2.1.

A contest is anonymous if its contest success functions satisfy, for any , for any and any permutation of ,

 fk(eπ(1),…,eπ(k))=(fkπ(1)(e1,…,ek),…,fkπ(k)(e1,…,ek)).\lx@notefootnoteThisisthesamedefinitionasin\@@cite[cite]\@@bibrefAuthorsPhrase1YearPhrase2alcalde2010rent\@@citephrase(\@@citephrase);itisequivalenttorequiringthat$fki(e1,…,ek)=fkπ(i)(~e1,…,~ek)$where$~eπ(j)=ej$forall$j$.
###### Definition 2.2.

A contest fully allocates the reward if its contest success functions satisfy, for any and any , .

###### Example 2.3.

A Tullock contest (or, more accurately, a single-prize Tullock contest) parameterized by has the following contest success function:

 fki(e1,…,ek)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩eτik∑j=1eτjif ej>0 for some j∈{1,…,k}1kotherwise

When , the contest becomes an “All Pay Auction (APA)” where the contestant with the highest effort wins with certainty (to maintain anonymity, if several contestants exert the highest effort, they all win with equal probability). A Tullock contest is anonymous and it fully allocates the reward.

###### Definition 2.4.

Denote by the set of all contests with reward that are anonymous, fully allocate the reward and have a symmetric Nash equilibrium among contestants for all .

For example, Alcalde and Dahm (2010) and Baye et al. (1996) show that Tullock contests with parameters and admit a symmetric Nash equilibrium; thus, contains all Tullock contests with reward . Other examples of contests that admit a symmetric Nash equilibria are given in e.g., the seminal works of Hirshleifer (1989); Nti (1997), a survey by Corchón (2007), as well as later works such as Amegashie (2012).

We assume throughout the paper that all contestants in the same contest will play a symmetric Nash equilibrium of that contest. Formally, for every contest we fix a (mixed strategy) symmetric Nash equilibrium, i.e.,

are i.i.d. random variables that follow a distribution

defined by a mixed strategy Nash equilibrium. Since is anonymous, in the symmetric Nash equilibrium all contestants get an equal expected fraction of the reward and hence their expected utilities are identical. We denote their identical expected utility by . Moreover, since fully allocates the reward, we must have and hence

 γC(k)=Rk−Eei∼F[ei]. (1)

We note that when , , because the single contestant will not exert any effort. We also have for any since a contestant can always choose to exert zero effort and guarantee non-negative utility. Moreover, since , we always have . We can use to express the utility of a contest designer in a contest with contestants by rearranging (1):

 Ee1,…,ek∼F[k∑i=1ei]=kEei∼F[ei]=R−kγC(k). (2)

Note that we assume that the utility of a contest designer is the expected sum of efforts, even if this is non-observable. This fits settings like workplace contests that aim to improve workers’ productivity. More generally, in the additive noise model, expected sum of qualities is equal to expected sum of efforts since the expected noise is usually assumed to be zero.

### 2.2 A contest competition game

In this paper we study a game where multiple contest designers compete by choosing their contest success functions. Contestants observe the different contests and choose in which one to participate.

###### Definition 2.5.

A complete-information contest competition game is denoted by , where is the number of contest designers, is the number of contestants, is the reward of contest , and . The game has two phases:

1. Designers choose contests. Each designer chooses a contest simultaneously. Contestants observe the chosen contests .

2. Contestants play a normal-form game of choosing in which contest to participate. A pure strategy of each contestant in this game is to choose one contest. Importantly, contestants may play a mixed strategy, meaning that each contestant participates in each contest with some probability , . We denote the vector of probabilities chosen by contestant by .

After Nature assigns contestants to contests, utilities are as follows. If there are contestants participating in contest , then each of these contestants gains utility and contest designer gains utility . If then the utility of the contest designer is .

The first important element of our model is the space of all possible contests a designer can (strategically) choose. For example, this could be the space of all Tullock contests, i.e., the parameter becomes a strategic choice. Some of the previous literature views as an exogenous parameter representing how accurately the designer is able to observe the ranking of efforts performed by contestants. Even so, it seems plausible that the designer chooses an “ignorance is bliss” approach where she lowers the value (thus, observes efforts’ ranking less accurately) in order to encourage participation. Alternatively, another example for could be the space of all prize structures where the ranking is determined according to a specific Tullock contest with a fixed exogenous , as in Azmat and Möller (2009).444Since we assume contests that fully allocate the reward, each can be derived from prize structures of at most prizes. Azmat and Möller (2009) allow for any number of prizes, although having more prizes than contestants is not beneficial for the contest designer.

We remark that this model implicitly assumes that when a contestant decides on the level of effort to exert in the contest she participates in, she knows the total number of contestants in the same contest. In practice, contestants can observe the number of participants when they are in physical contests (like sport contests) or when the contest designer chooses to reveal this information. Moreover, Myerson and Wärneryd (2006) show that contest designers have an incentive to do so because the expected aggregate effort in a contest with a commonly known number of participants is in general higher than that in a contest where the contestants do not see the number of participants.

In the second phase each contestant has a finite number of possible actions and the game is symmetric, hence there must exist at least one symmetric (mixed strategy) Nash equilibrium (Nash, 1951). We will assume in all our results that the contestants play this symmetric equilibrium, i.e., we will only consider equilibria in which the probability vector of every contestant is the same (). Example D.3 discusses the case where the contestants choose an asymmetric equilibrium. Formally, we denote by the probability vector chosen by the contestants at their symmetric equilibrium when the designers choose contests in the first phase.555If there are multiple symmetric equilibria, we allow to be any one of those. All our conclusions hold regardless of which symmetric equilibrium the contestants play. In addition, we show in Lemma 2.13 that the symmetric equilibrium is unique if a certain condition (that is satisfied, e.g., by all Tullock contests) holds.

Given that designers choose contests in the first phase of the game and contestants participate in contests with probabilities in equilibrium, the contestants’ utility is as follows. For a contestant who participates in , the number of contestants among the other contestants who also participate in

follows the binomial distribution

. Therefore, the expected utility of a contestant participating in , denoted by , equals

 β(Ci,pi)=Ek∼Bin(n−1,pi)[γCi(k+1)]=n−1∑k=0(n−1k)pki(1−pi)n−1−kγCi(k+1). (3)

Denote the set of indices of contests in which contestants participate with positive probability (i.e., the support of ) by

 Supp(C)={i:pi(C)>0}. (4)
###### Claim 2.6 (Equilibrium condition).

Suppose that designers choose contests in the first phase of the game and contestants participate in contests with probabilities in equilibrium. Then,

• If , then for any .

• Thus, if , then .

###### Proof.

is a symmetric equilibrium, i.e., given that all other participants play , a player’s best response is to play herself. Therefore, the expected utility of choosing to participate in contest is at least as high as choosing to participate in contest , as contest is assigned a positive probability . ∎

### 2.3 Equilibrium among contest designers

We use to denote the contests (strategies) chosen by all designers, where denotes the contests chosen by designers other than . Let be the expected utility of contest designer given that contestants use . Formally, by (2) the utility of the designer of contest equals when there are participants. Since each contestant participates in independently with probability , the total number of participants in follows the binomial distribution , and hence the designer’s expected utility equals

 (5)

Since a contest between the designer and the multiple contestants is a constant-sum game where the overall utility of all players (i.e., the welfare) equals the total reward whenever there is at least one contestant, designer ’s expected utility can be written as the expected welfare minus the sum of contestants’ expected utilities obtained from contest , . Formally,

###### Claim 2.7.
 ui(Ci,C−i)=Ri[1−(1−pi)n]−npiβ(Ci,pi). (6)
###### Proof.
 u(Ci,C−i) =Ek∼Bin(n,pi)[(Ri−kγCi(k))⋅1[k≥1]]=n∑k=1(nk)pki(1−pi)n−k(Ri−kγCi(k)) =n∑k=1(nk)pki(1−pi)n−kRi−n∑k=1(nk)pki(1−pi)n−kkγCi(k) =Ri[1−(1−pi)n]−n∑k=1(n)kpki(1−pi)n−kkγCi(k) =Ri[1−(1−pi)n]−npin∑k=1(n−1k−1)pk−1i(1−pi)n−kγCi(k) =Ri[1−(1−pi)n]−npiEk′∼Bin(n−1,pi)[γCi(k′+1)],

which equals by (3). ∎

In this paper we analyze the following solution concepts for the contest competition game:

###### Definition 2.8.

Given some ,

• A contest is dominant if , .

• A tuple of contests , where for all , is a contestant-symmetric subgame-perfect equilibrium if .

For simplicity and also for practical purposes, we do not consider the case where designers play mixed strategies (i.e., distributions over multiple contests).

### 2.4 Additional important properties of contests

Our results will use the following three properties of contests:

###### Definition 2.9.
• A contest has monotonically decreasing utility (MDU) if . In words, the symmetric Nash equilibrium expected utility of a contestant is decreasing as the number of contestants increases.

• A contest has maximal rent dissipation (MRD) in if for any and any , . Let denote the set of all contests with maximal rent dissipation in . In words, an MRD contest maximizes the designer’s utility regardless of the number of contestants which is equivalent to minimizing the symmetric Nash equilibrium expected utility of contestants.

• A contest has full rent dissipation if and for .

###### Claim 2.10.

If has monotonically decreasing utility, then for , .

The proof of the claim is in Appendix A.

###### Claim 2.11.

Let , and let be any other contest in . Then for any , .

###### Proof.

By the definition of maximal rent dissipation contest, for all , thus

 β(Ci,p)=Ek∼Bin(n−1,p)[γCi(k+1)]≥Ek∼Bin(n−1,p)[γTi(k+1)]=β(Ti,p).\qed

Note that a full rent dissipation contest has monotonically decreasing utility and has maximal rent dissipation in any set that contains it. It is known that has full rent dissipation (Baye et al., 1996) and in fact, as a corollary of Ewerhart (2017) we observe that every Tullock contest with parameter has full rent dissipation. Thus, the class of Tullock contests contains maximal rent dissipation contests (namely, those with ). Also, it is a class of contests that have monotonically decreasing utility:

###### Lemma 2.12 (Corollary of Baye et al., 1996; Schweinzer and Segev, 2012; Ewerhart, 2017).

Let be a Tullock contest with reward and with parameter . Then, if and if . For , and for . As corollaries,

• Every Tullock contest has monotonically decreasing utility.

• Every Tullock contest with parameter has full rent dissipation.

• If is the set of all Tullock contests with parameter in some range whose maximum is well defined and at most , then the Tullock contest with is the only contest in .

A proof of this lemma is given in Appendix A. Proposition 2 of Nti (1997) shows a large class of contests that generalize Tullock contests with and have the MDU property. MDU contests have another useful property:

###### Lemma 2.13.

If are MDU contests then the contestants’ symmetric equilibrium is unique.

###### Proof.

Let and be two symmetric equilibria for contestants. If they are different, then there exist such that and . Then we get the following contradiction

 β(Ci,p′i)> (p′i0,Claim~{}???) β(Cj,pj)> (pj0,Claim~{}???) β(Ci,p′i).

## 3 Main Results: Equilibria in Contest Competition Games

Our first main result shows that choosing a maximal rent dissipation contest with a monotonically decreasing utility is a subgame-perfect equilibrium of the CCG game, and, moreover, that such a maximal rent dissipation contest is a dominant contest when the set of all possible contests contains only contests with monotonically decreasing utilities:

###### Theorem 3.1.
1. Fix any where each contains a maximal rent dissipation contest that has monotonically decreasing utility, denoted by . Then, is a contestant-symmetric subgame-perfect equilibrium.

2. Moreover, if each only contains contests with monotonically decreasing utility, then is a dominant contest for each designer .

The full proof, as well as most other proofs in this section are deferred to Appendix B. In a very high-level, the argument for why MRD contests constitute an equilibrium for the contest competition game is the following. Consider any contest designer . Suppose each of the contestants participates in designer ’s contest with some probability (assuming a symmetric participation equilibrium). According to Claim 2.7, designer ’s expected utility equals

 ui(Ci,C−i)=Ri[1−(1−pi)n]−npiβ(Ci,pi),

where we recall that is each contestant’s expected utility conditioning on her already participating in . Now, suppose that contest designer switches to a contest that requires less effort from the contestants (namely, leaving more utility to the contestants) and hence increases the participation probability to . The welfare term is increased by . A contestant’s utility in contest will be increased (here we will use the condition that other contests , , are MDU contests, as explained in the formal proof) and suppose it is increased to , so the utility of each contestant is increased by

 p′iβ(C′i,p′i)−piβ(Ci,pi)=(pi+Δp)(β(Ci,pi)+Δβ)−piβ(Ci,pi) =Δpβ(Ci,pi)+piΔβ+ΔpΔβ >Δpβ(Ci,pi) ≥ΔpRi(1−pi)n−1,

where the last inequality is because a contestant obtains utility when no other contestants participate in , which happens with probability . Thus, the overall increase of contestants’ utility is greater than , outweighing the increase of the welfare term, so the designer’s utility is decreased.

Theorem 3.1 has the following implication regarding (or any other full rent dissipation contest):

###### Corollary 3.2.
1. Fix any where each contains a full rent dissipation contest (e.g., ), denoted by . Then, is a contestant-symmetric subgame-perfect equilibrium.

2. Let be the set of all Tullock contests with reward . Then, and any other Tullock contest with is a dominant contest for every designer in .

3. If is the set of all Tullock contests with parameter in some range whose maximum is well defined and at most . Then, the Tullock contest with is the only dominant contest for every designer in .

Theorem 3.1 shows a specific type of contestant-symmetric subgame-perfect equilibria. The following example shows that if the sets also contain contests that do not satisfy the condition of monotonically decreasing utility, other types of equilibria may exist, and the equilibria that the theorem shows is not dominant:

###### Example 3.3.

Let , , both and consist of two contests: the contest and a contest that gives the reward for free when the number of participants and runs otherwise. We thus have , which is not monotonically decreasing. We claim that is a contestant-symmetric subgame-perfect equilibrium and that is not a best-response to (and therefore not a dominant contest): When designers choose , by symmetry, contestants participate in either contest with equal probability . By direct computation (e.g., using (6)), the expected utility of each designers is . Now suppose designer switches to . The probabilities in the contestants’ symmetric mixed strategy Nash equilibrium must satisfy, according to Claim 2.6, (assuming ). By numerical methods, we find that . The expected utility of designers is then . Since , designer 1 will not switch to . By symmetry, designer 2 will not switch to . Hence, is an equilibrium, and is not a dominant contest.

In this example, for every , uses a Tullock contest with a parameter that depends on (namely, for and otherwise). An example where we use instead of for some could be constructed in a similar way.666In particular, , , both and consist of two contests: the contest and a contest with , that is, choosing Tullock contest with when and otherwise. Then is a contestant-symmetric subgame-perfect equilibrium, and is not a dominant contest for either designer. See Example D.1 for details. Moreover, Example D.2 shows that even if contains only contests with monotonically decreasing utility, may not be a dominant contest for designer 1.

When the sets contain only contests with monotonically decreasing utility, the contestant-symmetric subgame-perfect equilibria that Theorem 3.1 describes are the only possible equilibria:

###### Theorem 3.4.

Fix any where each only contains contests with monotonically decreasing utility. Assume for each . Pick , and let be the probability a contestant participates in contest in the equilibrium of contestants, and let be the set of indices of contests in which contestants participate with positive probability when the contests are . Then

1. for any contestant-symmetric subgame-perfect equilibrium , .

2. if , then is a contestant-symmetric subgame-perfect equilibrium if and only if .777If then contest could be anything: If , then the utility for agent is 0, which cannot be improved by choosing any other contest as is an equilibrium. Moreover, by Claim B.2, must be as well, so the choice of does not affect the choices of contests of other designers.

3. if , let , then is a contestant-symmetric subgame-perfect equilibrium if and only if .888As , with probability 1 there are contestants in contest , thus the contest success functions of contest for have no effect on the utility calculation for the contestants’ best response and could be anything.

In the symmetric-reward case we can show that which makes the statement shorter:

###### Corollary 3.5.

In the symmetric-reward case, i.e., , is a contestant-symmetric subgame-perfect equilibrium if and only if for all .

###### Proof.

We need to show that in this symmetric-reward case. Assume by contradiction that there exists such that . Since , there exists such that . Then by Claim 2.10, . However, this contradicts the equilibrium condition (Claim 2.6) which states that implies . Therefore, we conclude that for all , i.e., as required. ∎

Thus, the case of symmetric rewards is a “clear cut” while the general case is more involved. The following example demonstrates the need for this distinction using a setting with highly asymmetric rewards (see Appendix D.2 for a proof).

###### Example 3.6.

Consider contests and contestants. Contest 1 has reward , and each of others has reward . Each set contains all monotonically decreasing utility contests (hence contains ). Then for any contest , where for is a contestant-symmetric subgame-perfect equilibrium. In this equilibrium, , and for any .

Finally, the equilibria in Theorem 3.1 are Pareto optimal for the contest designers:

###### Definition 3.7.
• For two strategy profiles of the contest competition game , we say is a Pareto improvement of , if for all and for at least one .

• We say a strategy profile is Pareto optimal, if there is no Pareto improvement of it.

###### Theorem 3.8.

The equilibria in Theorem 3.1 are Pareto optimal.

## 4 Welfare Optimality

Throughout this section, let be a tuple of contests. Denote the sum of designers’ expected utilities, the sum of contestants’ expected utilities, and their sum by

 WD(C)=m∑i=1ui(C),WC(C)=nm∑i=1piβ(Ci,pi),WS(C)=WD(C)+WC(C)

where . By equilibrium condition (Claim 2.6) we have for all . Denote this constant by . This is a contestant’s expected utility in any contest in which she participates with positive probability. Note that by definition for any , so . As a result,

 WC(C)=nm∑i=1piβ(Ci,pi)=n∑i∈Supp(