Contest Copycats: Adversarial Duplication of Effort in Contests

ABSTRACT Participants in an innovation contest may steal their opponents’ ideas to enhance their chance of winning. To model this, I introduce the ability to copy another player’s effort in a Tullock contest between two players. I characterise the unique equilibrium in this game dependent on the cost of copying and one of the players’ productivity advantage. If the cost of copying is low, the less productive player is more likely to win the contest. The model’s comparative statics have important implications for governments who subsidise firms in contests and for contest designers.


Introduction
In innovation contests, participants may try to increase their chance of winning by spying on their opponents and copying their ideas. For example, the space race between the United States and the Soviet Union was accompanied by constant espionage, which went as far as 'kidnapping' a Soviet lunar spacecraft from an exhibition for a thorough inspection by the CIA (Wesley 1967). In the race to develop the first atomic bomb, the Soviet Union made significant efforts to gain access to the Manhattan Project (Haynes and Klehr 2000). Attempts to steal other contestants' ideas have also been documented in Formula 1 (Solitander and Solitander 2010). In 2004, back when it was a world leader in wireless technology, the Canadian company Nortel was hacked and had a large amount of its reports, design details, and top-secret source code stolen. While some have suspected an involvement of the Chinese government and the Chinese company Huawei, which subsequently gained a large share of the global wireless market, the hackers have never been identified (Pearson 2020). In general, evidence on economic espionage is scarce since firms who are spied on often are either ignorant of their situation or reluctant to report on it. But the US Intellectual Property Commission (2017) estimates the annual cost of IP theft to the US economy to be between $225 billion and $600 billion.
To enhance our understanding of how spying on other participants' ideas affects contests, I analyse a stylised model of a Tullock contest between two players. Both players can exert effortwhich I interpret as generating ideas -to increase their chance of winning the prize. I allow one player to be more productive in exerting contest effort. Both players can pay a fixed cost for the ability to copy their opponent's effort and add it to theirs -representing stealing the other's ideas and combining it with one's own. Let me briefly illustrate this concept: Without copying, if player 1 and player 2 exert efforts x 1 and x 2 , their respective probabilities of winning the contest are x 1 = x 1 þ x 2 ð Þ and x 2 = x 1 þ x 2 ð Þ. If player 2 copies the effort that player 1 exerts, her effective effort becomes x 1 þ x 2 , and the probability that player 2 wins the prize becomes x 1 þ x 2 ð Þ= 2x 1 þ x 2 ð Þ, while player 1ʹs win probability decreases to x 1 = 2x 1 þ x 2 ð Þ. I characterise the unique Nash equilibrium in this game and show how copying behaviour depends on the cost of copying and the stronger player's productivity advantage.
If the cost of copying is low, the weaker player is more likely to win the prize in equilibrium. Both the more productive player's utility as well as the aggregate effort players exert can decrease in the stronger player's productivity advantage. This is in contrast to the baseline without copying. It implies that a government who wants to increase a domestic firm's profit may not want to subsidise this firm's effort, even if the subsidy were costless. It also means that a contest designer who would like to maximise aggregate effort in a contest may in some circumstances want to exclude a more productive contestant in favour of a weaker one. Moreover, I show that the expected winner's effort -potentially including effort copied from an opponent -is generally increasing in the cost of copying. The designer of an innovation contest would like to make copying of effort prohibitively costly.
The article is structured as follows: First, I review the related literature. Second, I introduce and motivate a two-player contest model with copying of effort. Third, I characterise the unique Nash equilibrium, and subsequently discuss the model's comparative statics and its implications. Finally, I briefly sum up my contribution and present promising future avenues of research.

Related Literature
In my model, I interpret contest effort as generating ideas, and the sum of ideas in turn to determine the quality of a player's innovation. I assume a noisy contest, in which the player wins whose effort multiplied by an independent random variable is the highest. Following Hirshleifer and Riley (1992), this contest, including copying decisions, can be formulated as a standard lottery contest introduced by Tullock (1980). Whereas I focus on the adversarial duplication, or copying, of another player's effort, others have analysed harmful behaviour such as cheating, doping, and sabotage in contests. Enhancing one's own performance is, for example, treated by Eber and Jacques (1999), Berentsen (2002), Haugen (2004), Konrad (2005), Kräkel (2007), and Gilpatric (2011). The concept of sabotage -reducing the other's effective effort -was introduced into contests by Konrad (2000) and subsequently analysed by Kräkel (2005), Münster (2007), Gürtler (2008), and Gürtler and Münster (2010). For an overview of the literature on sabotage in contests, see Chowdhury and Oliver (2015). Spying on another player's ability in contests with uncertainty about the latter has been studied by Baik and Shogren (1995), though issues with inconsistent beliefs in their work have been pointed out by Bolle (1996). Chen (2019) analyses a setting in which contestants can spy on their opponent's valuation of the prize. In my model, players do not spy on contest productivities or prize valuations, but rather copy their opponent's exerted effort.
If players copy their opponents' effort, this introduces an additional effect of effort, which relates my work to that by Baye, Kovenock, and De Vries (2012) on spillovers in contests, although they focus on a rank-order contest, the effort spillover does not enter the contest function, and the spillover is exogenously given. This paper has direct applications to the field of innovation contests, which is surveyed in Adamczyk, Bullinger, and Möslein (2012), and is related to the study of innovation tournaments by Taylor (1995), Fullerton and Preston Mcafee (1999), and Che and Gale (2003), and patent races by Loury (1979), Dasgupta and Stiglitz (1980), and Lee and Wilde (1980). Spillovers in innovation races have been studied by Spence (1984) and Claude and Jacquemin (1988). There is a large literature on imitation in innovation races and its impact on economic growth and consumer welfare, with influential contributions by Scherer (1967), Reinganum (1982), Katz and Shapiro (1987), Grossman and Helpman (1991), Segerstrom (1991), Gallini (1992), Helpman (1993), and Aghion et al. (2001). The copying of effort that I consider is a more active mechanism than passive spillovers, and more immediate and more immediately targeted against another player than imitation. Moreover, in my model players can copy and exert effort themselves simultaneously. Cozzi (2001) and Cozzi and Spinesi (2006) conceptualise espionage in patent races as stealing an innovation from an innovator on their way to the patent office and focus on the implications for economic growth. Whereas I consider the duplication of a specific opponent's ideas, spies in their model can potentially steal all the ideas produced in an economy. Moreover, I focus on the effects of copying costs and productivity asymmetries on players' utilities and aggregate innovative effort.
A small game-theoretical literature on espionage models the ability to spy on other players' strategies (Matsui 1989;Solan and Yariv 2004;Alon et al. 2013;Barrachina, Tauman, and Urbano 2014), characteristics (Ho 2008;Wang 2020;Barrachina, Tauman, and Urbano 2021), or private signals (Kozlovskaya 2018;Pavan and Tirole 2021). In the literature on espionage in oligopolistic competition, there are some exceptions, in which players can duplicate their opponents' technologies. Gaisford (1996, 1999) consider firms and governments who can steal their opponent's production technology before entering a Cournot competition. Chen (2016) builds on this and endogenises additional innovation. Billand et al. (2010) model firms who can spy on their competitors to improve their product before competing in oligopolistic markets, Marjit and Yang (2015) model imitators who can steal the production technology of an innovator in a duopoly model with binary choices. Grabiszewski and Minor (2019) analyse a game in which a foreign firm can duplicate the effort a domestic firm exerts to innovate. Unfortunately, their model is not analytically tractable and leads to multiplicity of equilibria. In contrast, I investigate a contest in which both players can exert effort and copy their opponent's effort and derive closed-form solutions.

The Model
Two risk neutral players, i 2 1; 2 f g engage in a costly contest to win a prize of value V > 0. This prize may represent the value of being the first to patent a new invention, winning a research competition, or putting the first man on the moon. Both players simultaneously choose effort x i � 0, with effort cost defined as 1 À α i ð Þx i . Player 1 might have a productivity advantage: α 1 ¼ α 2 0; 1 ½ Þ and α 2 ¼ 0. Effort costs are common knowledge. I interpret effort as directly translating into ideas whose sum determines the quality of a player's innovation; the more effort a player exerts, the more ideas she generates, and the higher is the quality of her innovation.
In addition, players have the option to pay the fixed cost β to copy their opponent's effort and add it to theirs. This cost represents, for example, required expenditures -such as hiring hackers and spies -to be able to copy an opponent's ideas, or expected future costs, such as potential legal fees and penalties. 1 Copying decisions are made simultaneously and at the same time as efforts are chosen. 2 Denote a player's choice to copy by c i 2 0; 1 f g, where c i ¼ 1 means player i copies and c i ¼ 0 means she refrains from copying. Denote a player's effective effort as y i ¼ x i þ c i x j ; i�j. In addition to exerting effort and thus generating ideas, a player can also copy their opponent's ideas and add it to their own.
That the copying of an opponent's effort is successful with certainty is a simplifying assumption. While copying might be highly successful in contests with low protection barriers, such as innovation contests within a specific company, this might not be the case in other settings: attempts to hack into a competitor's IT system might be thwarted, and spies might be captured. Unfortunately, modelling copying success as uncertain in the presented framework would not allow me to solve for all equilibrium candidates. However, we can expect the model with certain copying success to be a good approximation for settings in which the probability of copying success is high. Moreover, in settings in which this probability is low, copying is not a very viable option unless its cost is sufficiently low too, making it a less relevant aspect of the contest anyway. I thus argue that certain copying success is not only a necessary assumption but also a reasonable one.
Let player i �j win the prize if θ i y i > θ j y j , where θ i and θ j are independent draws from an exponential distribution F θ ð Þ ¼ 1 À e À λθ with λ > 0. The more ideas a player has, the higher is the quality of her innovation, captured by her effective effort y i . However, there is randomness involved in the contest: The submissions in a research competition might be hard to evaluate. In the race to the moon, the more innovative contestant might face random setbacks such as accidents and illnesses. Following Hirshleifer and Riley (1992), it is straightforward to show that player i's probability to win this contest can be formulated as which corresponds to the standard Tullock contest success function; for a brief derivation, see Appendix A.
There are other ways to derive a Tullock contest from the microfoundations of an innovation contest. Namely, Baye and Hoppe (2003)show that under certain conditions, there is a strategic equivalence between innovation tournaments in the sense of Fullerton and Preston Mcafee (1999), patent races in the tradition of Loury (1979), and the Tullock rent-seeking game. However, in the case of Fullerton and Preston Mcafee (1999), this equivalence is not robust to the introduction of copying in the form that I use, since they consider a model in which a single drawn idea wins the game. 3 In the framework of Loury (1979), effort yields a hazard rate at which an innovation arrives. If we allow effort to be copied in order to simply increase one's own hazard rate in this framework, and if we assume a discount rate of zero, this game is strategically equivalent to the model I present here.
I make the usual assumption that p 1 ¼ p 2 ¼ 1=2 if y 1 ¼ y 2 ¼ 0. One way to motivate it is to interpret the status quo as a split prize. For example, if no competitor innovates, firms continue to share the market. If no country makes it to the moon, no country gains, but at the same time, no country loses. 4 Related is the fact that in the model, the level of effort does not play a role in the sense that the contest success function (1) is homogeneous of degree zero in effective efforts. In a game like the race to the moon, where a higher overall level of innovative effort can be expected to lead to a player winning the game earlier, this implies a discount rate of zero (which is the assumption required for the models' equivalence with the Loury (1979) framework).
If player i copies the effort x j of player j�i, player i's effective effort becomes y i ¼ x i þ x j . This assumes that efforts are additive. If we interpret effort as translating into innovative ideas, this means that these ideas are original. However, simultaneous research effort may sometimes lead to redundant innovations. 5 If a player copies an idea that she already has, this should arguably not increase her chance of winning as much as gaining access to an original idea. To test the importance of the assumption of additive efforts, I also analyse the model with redundant effort, where formally effective effort is y i ¼ max x i ; c i x j À � ; i�j, and show that my findings generalise; see Appendix C. Ideas and innovations (and the ways by which they are generated) are rarely exactly the same. For instance, the different Covid-19 vaccines that have recently been developed most likely have different strengths and weaknesses, and may even be more effective when used in combination. It is highly unlikely that two active contestants in an innovation contest can learn nothing from each other. The assumption of additive efforts is thus both intuitive, as well as uncritical to my findings.
Given the players' effort and copying decisions, a player's expected utility is where i�j. Denote by q i ¼ P c i ¼ 1 ð Þ 2 0; 1 ½ � the probability with which player i decides to copy. Additionally, denote by x ijc i ¼0 ¼ x n i the effort a player chooses if she does not copy and by x ijc i ¼1 ¼ x c i the potentially different effort she chooses if she does. If a player chooses not to copy, her first-order condition with respect to exerted effort x n i is where i �j. If player i chooses to copy, her first-order condition with respect to exerted effort x c i is If both players copy, effort is wasted and thus has no marginal benefit. If player j always copies, q j ¼ 1, player i optimally exerts zero effort when copying. Lastly, if player i mixes, 0 < q i < 1, it must hold in equilibrium that she is indifferent between not copying and copying: It cannot be an equilibrium to have both players play a pure strategy and always copy, i.e. q 1 ¼ q 2 ¼ 1. This would imply zero efforts and u 1 ¼ u 2 ¼ V=2 À β. It is clear that a player would have an incentive to deviate and not copy and receive V=2 for sure. This means that at least one player at least sometimes does not copy in any equilibrium. Moreover, it cannot be an equilibrium for a player to exert zero effort in expectation. To see this, assume x n i ¼ x c i ¼ 0 in equilibrium. If q i < 1, player j�i optimally responds by not copying, c j ¼ 0, and exerting infinitesimal effort x j ¼2 where 2! 0 cannot be pinned down. 6 If q i ¼ 1, player j's best response is then to not copy c j ¼ 0 and exert zero effort x j ¼ 0. But then, i would have an incentive to deviate to q i ¼ 0 and x i ¼2 with 2! 0. This means that in any equilibrium, both players must exert strictly positive effort in expectation. The second-order conditions of Equation (3) and, if q j < 1, Equation (4) holds for any non-zero effort of the opposing player j and on the whole domain x i � 0. Hence, players' best reply functions with respect to effort are single peaked given a strictly positive expected effort by their opponent. 7 In the following section, I derive the unique solution to Equations (3) to (5) for all combinations of productivity advantage α 2 0; 1 ½ Þ and copying cost β > 0, which pins down the unique Nash equilibrium.

Equilibrium
Both players can never copy, mix between copying and not copying, or always copy. As stated above, it cannot be an equilibrium for both players to always copy. Moreover, I show in Appendix B that it cannot be an equilibrium for player 1 to copy with a higher probability in equilibrium than player 2. This leaves us with five combinations as equilibrium candidates: both players never copy, player 1 never copies and player 2 mixes, player 1 never copies and player 2 always copies, player 1 mixes and player 2 always copies, and both players mix. Figure 1 anticipates the results of this section and illustrates in which parametric regions the five equilibrium candidates exist. In what follows, I characterise these equilibria and derive the boundaries that separate them. At the end of this section, I briefly sum up the results and formulate the first proposition.

Both Players Never Copy
Start by assuming that both players never copy, c 1 ¼ c 2 ¼ 0. This case corresponds to the standard model without copying. The first-order condition (3) reduces to We can solve this for equilibrium efforts Intuitively, as player 1ʹs productivity advantage α increases, her utility increases as well, while the utility of player 2 decreases.
This equilibrium exists if no player has an incentive to deviate by copying the effort of the other player. If player 2 deviates and copies, c 2 ¼ 1, her best-response effort is x c 2 ¼ 0. This gives her the utility V=2 À β. Thus, player 2 has an incentive to copy if player 1 never copies if If player 1 deviates and copies, her best-response effort is x c 1 ¼ αV= 2 À α ð Þ 2 , yielding the utility This means that player 1 has an incentive to copy given that player 2 never copies if (6) always holds if condition (7) holds. Thus, both players never copying is a Nash equilibrium if which follows from (6). The term 1=2 À β=V is the utility a player receives if she copies and does not exert effort, normalised by the value of the prize. 8 You can see in Figure 1 that this holds if the cost of copying, normalised by the value of the prize, β=V, is sufficiently large. If player 1ʹs productivity advantage α is high, player 2 is willing to pay a higher price for the ability to copy.

Player 1 Never Copies, Player 2 Mixes
Consider an equilibrium in which player 1 never copies, q 1 ¼ 0 and thus x 1 ¼ x n 1 , and player 2 plays a mixed strategy, q 2 2 0; 1 ð Þ. This means that player 2 copies the effort of player 1 with probability q 2 and refrains from doing so with probability 1 À q 2 ð Þ. For player 2, condition (3) reduces to and condition (4) becomes If x 1 > x n 2 , we must have that x c 2 ¼ 0, and x c 2 ¼ x n 2 À x 1 otherwise. Since the marginal cost of effort is constant, a player would like to have the same equal marginal benefit of effort, regardless whether she copies or not.
First, assume x 1 > x n 2 . Then, condition (9) cannot bind and in equilibrium we must have x c� 2 ¼ 0. We can solve for the probability that player 2 copies and get q � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V Equilibrium efforts are x � 1 ¼ 1 À ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V p � � 2 V and x n� 2 ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V This equilibrium can only exist if β=V < 1=2. Additionally, we must have that q � 2 > 0. This is the case if α > 1 À 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V p � � , which corresponds to (8). Further, x 1 > x n 2 is true if β=V > 1=4. Player 1 never has an incentive to deviate and copy if these conditions hold. 9 Now, assume x 1 � x n 2 . First-order conditions in this case yield equilibrium efforts ffi ffi ffi ffi ffi ffi βV p À β, and x c� 2 ¼ ffi ffi ffi ffi ffi ffi βV p À 2β. Win probabilities are independent of whether or not player 2 copies, and player 2 is (weakly) more likely to win. Player 2 copies with probability q � 2 ¼ 1= ffi ffi ffi ffi ffi ffi ffi ffi β=V p þ α À 2 � 0 in this equilibrium. For x � 1 � x n� 2 to hold, we must have that β=V � 1=4. For q � 2 < 1 to hold, we must have that Again, an increase in player 1ʹs productivity advantage α increases player 2ʹs inclination to copy. Player 1 does not have an incentive to deviate and copy if Intuitively, an increase in her productivity advantage makes player 1 less inclined to copy. The light blue region in Figure 1 depicts the combinations of the normalised cost of copying β=V and player 1ʹs productivity advantage α that make player 1 never copy and player 2 mix between copying and not copying in equilibrium.

Player 1 Never Copies, Player 2 Always Copies
Consider an equilibrium in which player 1 never copies, q 1 ¼ 0 and x 1 ¼ x n 1 , and player 2 always copies, q 2 ¼ 1 and x 2 ¼ x c 2 . Efforts are then x � 1 ¼ V= 3 À α ð Þ 2 and x � 2 ¼ 1 À α ð ÞV= 3 À α ð Þ 2 . Player 1 has no incentive to deviate and copy if her productivity advantage is large enough: Player 2 does not have an incentive to deviate and not copy if α � 3 À 1= ffi ffi ffi ffi ffi ffi ffi ffi β=V p , which corresponds to (10). The region on the β=V-α-plane where this Nash equilibrium in asymmetric pure strategies exists is shown in dark blue in Figure 1.

Player 1 Mixes, Player 2 Always Copies
Assume that in equilibrium, player 1 mixes between copying and not copying, q 1 2 0; 1 ð Þ, and player 2 always copies, q 2 ¼ 1 and x 2 ¼ x c 2 . If player 1 copies, she has no incentive to exert effort since she wins the price with probability 1=2 in any case, x c� 1 ¼ 0. We can solve for the probability that player 1 copies as q � 1 ¼ 1 À ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2= 1=2 À β=V ð Þ p À 2 À � = 1 À α ð Þ < 1 and for her equilibrium effort if she does not copy as x n� ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2= 1=2 À β=V ð Þ p , which corresponds to (12). Further, player 2 has no incentive to deviate and not copy if her opponent's productivity advantage is large enough: These conditions hold if the normalised cost of copying β=V is low and player 1ʹs productivity advantage α is high, but not too high, as illustrated by the violet region in Figure 1.

Both Players Mix
Lastly, consider an equilibrium in which both players play a mixed strategy, q i 2 0; 1 ð Þ "i 2 1; 2 f g. We can solve for equilibrium efforts as x n� The equilibrium probabilities with which players copy are given by q ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V ð Þ=2 p À � = β=V ð Þ, which corresponds to (13). Both players mixing between copying and not copying is an equilibrium if the normalised cost of copying β=V and player 1ʹs productivity advantage α are both sufficiently low. This region is depicted in orange in Figure 1.

Summary
As is evident in Figure 1, the five described equilibria are mutually exclusive and cover the entire area where the game is defined. This means that for player 1ʹs productivity advantage α 2 0; 1 ½ Þ and the normalised cost of copying β=V > 0, the Nash equilibrium of the contest with copying exists and is unique. Moreover, the transition between the different types of equilibria is 'smooth': for example, at the switch from player 2 mixing to copying, her probability of copying converges to zero as α approaches the threshold characterised by (8). This means that there are no discontinuous jumps in the decision and outcome variables. Some intuitive patterns emerge: In general, players copy more often if the cost of copying, normalised by the value of the prize, β=V is lower. Additionally, players copy more often if the relative productivity with which they exert effort declines.
Having solved for the equilibrium for all combinations of normalised copying cost β=V and productivity advantage α of player 1, we can formulate the first proposition. The expected probability of winning the contest for player 1 in equilibrium is p � 1 2 0; 1 ½ �. We have: Proposition 1. (Probability of winning) If β=V < 1=4 and α > ; p � 1 < 1=2. If the normalised cost of copying is sufficiently low, player 1 is less likely to win the contest in equilibrium if she has a productivity advantage. If β=V > 1=4 and α > 0, p � 1 > 1=2. Naturally, Proof 1. If β=V > 1=4, both players do not copy if condition (8) holds. The probability that player 1 wins in this baseline case is If condition (8) does not hold, player 1 does not copy, and player 2 mixes between copying and not copying in equilibrium. The probability that player 1 wins the contest is then ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V p > 1=2 if β=V > 1=4, we have that p � 1 > 1=2.
If β=V < 1=4, we know from Figure 1 that there are four cases to consider. If player 1 never copies and player 2 mixes, p � 1 ¼ ffi ffi ffi ffi ffi ffi ffi ffi β=V p < 1=2. If player 1 never copies and player 2 always copies, we have If player 1 mixes and player 2 always copies, p � 1 ¼ 1 À q � 1 À � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V ð Þ=2 p þ q � 1 =2 < 1=2. Finally, if both players mix between copying and not copying, the probability that player 1 wins the contest is given by p In sharp contrast to the baseline contest model without copying, a player with a productivity advantage is less likely to win the contest if the cost of copying is low. If player 2 copies, her chance of winning is at least 1=2 since her effort when she copies is positive, x c 2 � 0. Copying by player 2 reduces the incentive for player 1 to exert effort. This in turn can be exploited by player 2 when she does not copy, in which case she exerts more effort than player 1, x n 2 > x n 1 if β=V < 1=4 and α > 0, giving her a higher probability of winning than player 1. Despite this, player 1 does not have an incentive to copy more often than she does, since effort is not very costly to her relative to copying due to her productivity advantage. 10 If the cost of copying is high, i.e. β=V > 1=4, player 2 does not have an incentive to copy often enough to cause the same effect and player 1 is more likely to win. Note that both players have an equal chance of winning in equilibrium, p � i ¼ 1=2, if they are symmetric, i.e. α ¼ 0. The same holds if β=V ¼ 1=4. Further, from our results, we can easily deduct the probability of winning for the potentially weaker player 2 since p 2 ¼ 1 À p 1 .

Comparative Statics
Suppose that a government wants to enhance the winning chances or the profit of a domestic firm that competes in an innovation contest by subsidising the firm's innovative effort. Or imagine that a contest designer considers changing the cost of copying to maximise the innovative effort exerted by all participants. In these situations, it is important to know how changes in player 1ʹs productivity advantage α and the cost of copying β affect the equilibrium outcome. In this section, I go through the comparative statics of the model and highlight the most surprising effects that arise when players can copy effort in contests.

Decision Variables
Let us first consider the players' decision variables: the probability with which a player copies q i , the effort she exerts if she does not copy x n i , and the effort if she does x c i . In general, and unsurprisingly, we have that dq � 1 =dα � 0 and dq � 2 =dα � 0, and dq � 1 =d β � 0 and dq � 2 =dβ � 0. The probability with which a player copies decreases in their relative productivity advantage and the cost of copying.
The effort player 1 exerts when she does not copy weakly increases in both her productivity advantage and the cost of copying, dx n� 1 =dα � 0 and dx n� 1 =dβ � 0. The latter effect is driven by the fact that player 2 copies less often if β increases, making it more attractive for player 1 to exert high effort. Since player 1 never exerts effort if she does copy, we naturally have dx c� 1 =dα ¼ dx c� 1 =dβ ¼ 0. If player 2 does never copy in equilibrium, q � 2 ¼ 0, her effort is decreasing in player 1ʹs productivity advantage. If she copies at least sometimes however, her effort if she does not copy is weakly increasing, dx n� 2 =dα jq � 2 > 0 � 0. Copying by player 2, which is increasing in α, lowers the incentive for player 1 to exert high effort, which in turn increases the incentive for player 2 to exert high effort herself when she does not copy. If the normalised cost of copying is sufficiently high, β=V > 1=4, player 2ʹs effort when she does not copy is decreasing in said cost, dx n� 2 =dβ jβ=V > 1=4 < 0. In contrast, we have dx n� 2 =dβ jβ=V < 1=4 > 0. If player 1 never copies, player 2ʹs effort if she does copy is weakly decreasing in both α and β, dx c� 2 =dα jq � 1 ¼0 � 0 and dx c� 2 =dβ jq � 1 ¼0 � 0. If player 1 mixes between copying and not copying, the opposite is the case, dx c� 2 =dα jq � 1 > 0 > 0 and dx c� 2 =dβ jq � 1 > 0 > 0. In this latter case, a decrease in player 1ʹs probability to copy incentivises player 2 to exert higher effort.

Win Probabilities
In the baseline case without copying, player 1ʹs probability of winning the contest is intuitively increasing in her productivity advantage α. If at least one player copies, this only holds if player 1 never copies and player 2 always copies, dp � 1 =dα jq � i ¼0;q � 2 ¼1 > 0. In both cases, both players do not or cannot react to a change in α by adjusting their probability of copying. Otherwise, player 1ʹs win probability is weakly decreasing in α, dp � 1 =dα � 0 if q � 1 2 0; 1 ð Þ _ q � 2 2 0; 1 ð Þ. This surprising effect is closely related to Proposition 1 and follows the same intuition. If player 1 does not copy, her win probability is weakly increasing in the cost of copying, dp � 1 =dβ jq � 1 ¼0 � 0. If player 1 mixes between copying and not copying, an increase in the cost of copying leads to a decrease in her chance of winning the contest, dp � 1 =dβ jq � 1 > 0 < 0.

Utilities
The changes in the players' equilibrium behaviour in reaction to changes in player 1ʹs productivity advantage α and the cost of copying β lead to some counter-intuitive effects on the game's outcomes. Let us consider the players' expected utilities and first discuss the most surprising effect that arises.
Proposition 2. (Player 1ʹs utility can decrease in α) If player 2 mixes between copying and not copying while player 1 never copies, and if the cost of copying is sufficiently high, β=V > ffi ffi ffi 2 p À 1, an increase in player 1ʹs productivity advantage leads to a decrease in her expected utility. Otherwise, we have du � 1 =dα � 0.
Proof 2. In the baseline without copying, the expected utility of player 1 is u � 1 ¼ V= 2 À α ð Þ 2 . It is clear that then du � 1 =dα > 0. If player 2 mixes between copying and not copying and player 1 never copies, and if additionally β=V > 1=4, we have that We see that du � 1 =dα < 0 if 1 À ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V If β=V � 1=4 and player 2 mixes between copying and not copying, while player 1 never copies, player 1ʹs utility is u � 1 ¼ ffi ffi ffi ffi ffi ffi βV It is easy to see that here du � 1 =dα > 0. If payer 2 always copies and player 1 never copies in equilibrium we have that u � 1 ¼ 2V= 3 À α ð Þ 2 and it is clear that du � 1 =dα > 0. If player 1 mixes between copying and not copying, she exerts zero effort when she copies. This means that her utility when she copies is V=2 À β. Since she must be indifferent between copying and not copying, this means her expected utility is u � 1 ¼ V=2 À β. It is clear that then du � 1 =dα ¼ 0.
Intuition 2. An increase in her productivity advantage α benefits player 1 in the baseline without copying and in equilibria with copying if the cost of copying is sufficiently low. However, if the cost of copying is high, β=V > ffi ffi ffi 2 p À 1, and the productivity advantage high enough to incentivise player 2 to copy, i.e. condition (8) does not hold, a further increase in α harms player 1. This means for example that a government who subsidises a domestic firm's effort in a contest can end up harming the firm, even without taking into account the additional cost such a subsidy would incur. The effect can be explained by the fact that if the cost of copying is high but not prohibitive, an increase in α eventually leads to a sharp increase in player 2ʹs probability of copying q 2 , which more than offsets the benefit for player 1 of the decrease in her effort cost.
More intuitively, player 1ʹs expected equilibrium utility is weakly increasing in the cost of copying β if player 1 never copies, and decreasing in β if player 1 mixes between copying and not copying. We can write du � 1 =dβ jq 1 ¼0 � 0 and du � 1 =dβ jq 1 > 0 < 0. This result is similar for player 2, for whom we have du � 2 =dβ jq 2 ¼0 ¼ 0, since then no player ever copies, and du � 2 =dβ jq 2 > 0 < 0. The effect of an increase in α on player 2ʹs utility is negative if player 1 never copies, du � 2 =dα jq 1 ¼0 � 0, and positive if player 1 mixes, du � 2 =dα jq 1 > 0 > 0. The latter effect is also somewhat counter-intuitive. It is due to the fact that an increase in α leads player 1 to copy less often, which benefits player 2. If we combine these effects and look at aggregate expected utility in equilibrium, U � ¼ P i u � i , we find that an increase in α leads to an increase in U � and, if at least one player copies at least sometimes, an increase in β leads to a decrease in U � . For each effect, there is one exception: First, we have dU � =dα jq � 2 > 0^β=V > ffi ffi 2 p À 1 < 0, which is driven by the decrease in player 1ʹs utility discussed in Proposition 2. Second, it is the case that dU � =dβ jq � 2 > 0^α < γ > 0, where

Aggregate Effort
A contest designer might not take into account the participants' utilities. Rather, she might aim to maximise aggregate exerted contest effort X ¼ P i x i . This reflects the sum of generated original ideas. Why would a contest designer care about aggregate exerted effort? For instance, original research generated within a specific contest often has major positive externalities beyond this contest. Examples include the race to the moon, which lead to innovations for instance in freezedrying of foods, fireproof materials, and integrated circuits, and The Netflix Prize, a machine learning competition with a $1 million award announced by the content platform Netflix in 2006, which spurred innovation in recommender systems and machine learning in general. 11 As with players' utilities, when considering aggregate exerted effort, copying of effort in contests can change some of the dynamics compared to the baseline without copying.
Proposition 3. (Aggregate exerted effort can decrease in α) If player 2 mixes between copying and not copying while player 1 never copies, expected aggregate exerted effort in equilibrium X � decreases in α. This is in contrast to the baseline case without copying and the cases in which player 2 always copies. If both players mix, the sign of dX � =dα is ambiguous.
Proof 3. In the baseline without copying, the aggregate exerted effort in equilibrium is X � ¼ V= 2 À α ð Þ, and it is clear that then dX � =dα > 0. If player 1 never copies in equilibrium while player 2 mixes between copying and not copying, expected aggregate exerted effort is Since dx � 1 =dα ¼ dx n� 2 =dα ¼ dx c� 2 =dα ¼ 0, x n� 2 > x c� 2 , and dq � 2 =dα > 0, we have that then dX � =dα < 0. If player 2 always copies, while player 1 never copies, X � ¼ V 2 À α ð Þ= 3 À α ð Þ 2 . Then, dX � =dα > 0. If player 2 always copies and player 1 mixes in equilibrium, aggregate effort is X � ¼ 1 À q � 1 À � x n� 1 þ x � 2 . Since in this case dq � 1 =dα < 0, dx n� 1 =dα > 0, and dx � 2 =dα > 0, we have that dX � =dα > 0. If both players mix between copying and not copying, we can write Intuition 3. A contest designer interested in maximising aggregate exerted effort in a contest might not always benefit from a contest participant becoming more productive. This is due to the fact that an increase in productivity for player 1 incentivises player 2 to copy more frequently, which in turn lowers the expected effort for player 2 and lowers the incentive for player 1 to exert effort. For intermediary costs of copying, this means that such a contest designer would like to exclude a 'strong' player in favour of a 'weaker' player whose productivity is more similar to her opponent's. 12 More intuitively, expected aggregate exerted effort is generally weakly increasing in the cost of copying, dX � =dβ � 0. This implies that an international coalition of governments who would like to increase global research efforts has an incentive to make copying more costly, for example, by increasing the punishment of intellectual property law infringements.

Winner's Effort
While some contest designers might take into account positive externalities of innovation and thus try to maximise aggregate exerted effort, others might only care about the innovation that wins the contest. For example, a company running a research contest might only implement the winning entry. 13 The natural objective of such a contest designer is the expected winner's effort, Recall that y i denotes player i's effective effort and is defined as For the contest designer, it is irrelevant whether the innovation is completely original or at least partly copied. While intuitively we might expect copying of effort to be potentially beneficial for a contest designer who wants to maximise the winner's effort, we can formulate our last proposition, which disagrees with this intuition.
Proposition 4. (The expected winner's effort increases in β). The expected winner's effort weakly increases in the cost of copying, dy � w =dβ � 0.
Intuition 4. An increase in the cost of copying β makes players less likely to copy and thus less likely to have access to the other player's effort. However, this is offset by the fact that players usually also increase their exerted effort in response to an increase in β. This means that a contest designer who wants to maximise the expected winner's effort would like to increase the cost of copying, for example, by punishing players who are caught copying or by making it technically more difficult to copy. 14 The expected winner's effort in equilibrium y � w is also generally weakly increasing in player 1ʹs productivity advantage α, with the exception of the case that both players mix between copying and not copying and the normalised cost β=V being very small. Specifically, we can write dy � w =dα jβ=V > 1=4 > 0, dy � w =dα jq � 1 ¼0;q � 2 2 0;1 ð Þ^β=V�1=4 ¼ 0, and dy � w =dα jq � 2 ¼1 > 0. If both players mix, q � i 2 0; 1 ð Þ "i, dy � w =dα is strictly negative as β=V tends to zero and positive as β=V tends to 1=4. 15 This implies that a contest designer aiming to maximise the expected winner's effort in most cases benefits from one player gaining a productivity advantage.

Conclusion
In innovation contests, participants may try to enhance their chance of winning by spying on and copying their opponents' effort. I investigate this mechanism in a stylised model and characterise the Nash equilibrium in dependence on the cost of copying and the potential productivity advantage of one of the players. I show that, when copying of effort is possible, the weaker player is more likely to win the contest if copying costs are low, that the more productive player's utility and the aggregate effort may decrease in the productivity advantage of the more productive player, and that the expected winner's effort generally increases in the cost of copying. Hence, a government trying to increase the profit of a domestic firm competing in an innovation contest may not want to subsidise this firm's efforts, even if this subsidy were costless. The designer of a contest who tries to maximise aggregate innovative effort may want to handicap a player with a productivity advantage or substitute her for a weaker one. And a contest designer whose objective is the maximisation of the expected winner's effort generally has an incentive to make copying of effort more costly.
There are multiple avenues for future research, which relate to some of the potential limitations of my model I have discussed. I assume players are successful to copy their opponent's effort with certainty. The model becomes mostly intractable if copying success is uncertain, unfortunately. Specifying a model that is fully tractable also with uncertain copying success may allow us to test the robustness of the stylised model I have presented and may offer additional insights. 16 One may even go further and endogenise the probability of copying success by allowing players to take protective action, such as, for example, increasing IT security. Another improvement in realism might be to make the cost of copying dependent on the magnitude of the effort that is to be copied. Moreover, I solve the model in its main specification with additive efforts and briefly show that the resulting findings are robust to modelling efforts as redundant. It might still be interesting to analyse an intermediate case, in which some of the ideas the players generate are redundant by chance, while the rest are not. Finally, the game I describe is static. Large innovation contests, such as the space race or the nuclear arms race often have multiple stages and involve dynamic interaction of the players' strategies. It might be fruitful to see how the integration of such dynamic interactions affect the results I have presented.

Notes
1. See Crane (2005) for examples of such legal fees and penalties. 2. This corresponds to effort and copying decisions being made sequentially if the effort decision by the other player is unknown at the time of the copying decision. This is intuitive: the level of the effort is unknown to the opposing player before it is copied. Another form of sequentiality is copying decisions being made before efforts are chosen. Again, this is congruent with the simultaneous model if players are uninformed about the other player's choices. This reflects a reality in which espionage is often hard to detect. 3. In Fullerton and Preston Mcafee (1999), effort translates into random draws of ideas and the player with the best idea wins. In this framework, if we assume that a winner is picked at random if both players have access to the (same) best idea, the success function with copying becomes p alt i ¼ innovations or achievements are a combination of a number of ideas rather than a single one, especially the most significant ones. The first iPhone combined innovations in display technology, user interface, design, and many more. The moon landing required, among others, innovations in rocket and computer technology and the development of new fabrics and other materials. The model reflects this interpretation. 4. In the model, the issue of zero total effort only arises in equilibrium with symmetric players. In reality, productivity of players is often heterogeneous. Furthermore, in some contests, such as for example Kaggle machine-learning challenges, there is often a benchmark publicly available whose submission might represent exerting no effort, but cannot easily be ruled out as a winning approach by a contest designer. For an example of explicit modelling of draws, see Blavatskyy (2010). 5. Famous historical examples include the independent formulation of calculus by Newton and Leibniz, the development of the theory of natural selection by Darwin and Wallace, and the invention of the telephone by Gray and Bell. See Ogburn and Thomas (1922) for an early and fascinating account of almost 150 multiple inventions and discoveries. It is noteworthy that Gray and Bell ended up in a controversy over who had been first to invent the telephone, a controversy which did not remain free of accusations of copying the other's ideas. 6. Even if we could fix a very small 2 , player i would then have an incentive to deviate and exert strictly positive effort. 7. This also holds if the opponent were to play a mixed strategy with respect to effort, given her copying decision. It follows that players, given their copying decision, do not play a mixed strategy with respect to the effort they exert in equilibrium. 8. The term ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1=2 À β=V p is only well defined if β=V � 1=2. If β=V > 1=2, copying of effort is never optimal for any player. 9. Note that 1=4 is the effort players exert if they are symmetric and do not copy, normalised by the value of the prize. If players are symmetric, they start to copy if the normalised cost of copying β=V falls below this threshold. See Figure 1.
10. To some extent, this mechanism resembles the 'paradox of power' discussed by Hirshleifer (1991), which describes the phenomenon in a guns-and-butter model that poorer contenders appropriate a larger share of the pie relative to their endowment than richer contenders. Although in the present model there are no resource constraints, one player is more productive in generating innovative contest effort, which leads the 'weaker' player to be more motivated to 'appropriate' this effort by copying. 11. Related are arguments made by Terwiesch and Yi (2008) and Bessen and Maskin (2009) for why diversity of ideas might be good from the perspective of a contest designer. 12. Note that a contest designer would still prefer two strong players, however. 13. See Serena (2017) for a deeper exploration of this idea and more examples of such settings, which he calls 'quality contests'. 14. Note that an increase in the cost of copying β might not be costless to the contest designer, complicating her optimal policy. 15. The derivative of the expected winner's effort is given by the rather tedious term in this case. 16. For example, at the limit of always unsuccessful copying, we approach the baseline model without copying, and we can expect Proposition 1 to be overturned. It is intriguing to know at which point this overturning takes place.