Designing Dynamic Contests

Transcription

1 Designing Dynami Contests Kostas Bimpikis Shayan Ehsani Mohamed Mostagir This version: Otober, 2014 Abstrat Innovation ontests have emerged as a viable alternative to the standard researh and development proess. They are partiularly suited for settings that feature a high degree of unertainty regarding the atual feasibility of the end goal. Partiipants rae towards ompleting an innovation projet and learn about the underlying environment from their own efforts as well as from their ompetitors gradual progress. Learning about the status of ompetition an alleviate some of the unertainty inherent in the ontest, but it an also adversely affet effort provision from the laggards as they beome disouraged about their likelihood of winning. Thus, the ontest s information provision mehanism is ritial for its suess. This paper explores the problem of designing the award struture of a ontest as well as its information dislosure poliy in a dynami framework, and provides a number of guidelines with the objetive of maximizing the designer s expeted payoff. In partiular, we show that intermediate awards, apart from diretly affeting the partiipants inentives to exert ostly effort, may also be used as a way for the designer to appropriately disseminate information about the status of ompetition. Interestingly, our proposed design mathes many features observed in real-world innovation ontests. Keywords: Contests, learning, dynami ompetition, open innovation, information. Preliminary and Inomplete Comments Welome We are partiularly thankful to Alessandro Bonatti, Kimon Drakopoulos, Karim Lakhani, Mohammad Mousavi, and Alireza Tahbaz-Salehi. Any remaining errors are ours. Graduate Shool of Business, Stanford University. Department of Management Siene and Engineering, Stanford University. Ross Shool of Business, University of Mihigan.

2 1 Introdution Innovation ontests are fast beoming a tool that firms and institutions use to outsoure their innovation tasks to the publi. In these ontests, an open all is plaed for an innovation projet that partiipants ompete to finish, and the winners, if any, are awarded a prize. 1 Reent suessful ontests inlude The NetFlix Prize and the Heritage Prize 2, and a growing number of ventures like Innoentive, TopCoder, and Kaggle provide online platforms to organize and onnet innovation seekers with potential innovators. The objetive of the ontest designer is to maximize the probability of reahing the innovation goal while minimizing the time it takes to omplete the projet. Obviously, the suess of an innovation ontest ruially depends on the pool of partiipants and the amount of effort and experimentation they deide to provide, and a growing literature onsiders the question of how to best design these ontests. Three key modeling features are ruial for our analysis and distinguish our work from prior literature. First, in our model, an agent s progress towards the goal is not a deterministi funtion of effort. As is typially the ase in real-world settings, progress is positively orrelated with effort but the mapping involves a stohasti omponent. Seondly and quite importantly, it is possible that the innovation in question is not attainable, either beause the goal is atually infeasible or beause it requires too muh effort and resoures that it makes little eonomi sense to pursue. We model suh a senario by having an underlying state of the world whether the innovation is attainable or not over whih partiipants have some prior belief. Taken together, these two features imply that an an agent s lak of progress an be attributed to either an undesirable underlying state the innovation is not attainable or simply to the fat that the agent was unluky in how her effort was stohastially mapped to progress. Thirdly, we onsider a dynami framework to apture both how ompetition between the agents evolves over time as well as to inorporate the fat that they learn from eah other s partial progress. In partiular, we inlude well-defined intermediate milestones as part of the modeling setup whih onstitute partial progress towards the end goal. Apart from being an integral feature of most innovation ontests, expliitly aounting for intermediate milestones allows us to study the omplex role that information plays in this ontext. The previous disussion implies that information about the progress of one of the partiipants has the following interesting dual role: it makes partiipants more optimisti about the state of the world, as the goal is more likely to be attainable and thus agents have a higher inentive to exert ostly effort. Following the literature on strategi experimentation, we all this the enouragement effet. At the same time, suh information implies that one of the partiipants is leading the ontest, whih might negatively affet effort provision from the rest of the agents as the likelihood of them 1 We use the terms partiipant, ontestant, ompetitor, and agent interhangeably throughout. 2 The NetFlix Prize offered a million dollars to anyone who sueeded in improving the ompany s movie reommendation algorithm by a ertain margin, and was awarded in The Heritage Prize was a multi-year ontest whose goal was to provide an algorithm that better predits patient readmissions to hospitals. A suessful breakthrough was obtained in June of

3 beating the leader is now onsiderably lower. We refer to this as the ompetition effet. These two effets interat with eah other in subtle ways and understanding this interation is of paramount importane to the design of an innovation ontest. The primary ontribution of this paper is twofold. First, to the best of our knowledge, our framework is the first that expliitly models the three features of innovation ontests desribed above, i.e., unertainty regarding the feasibility of the end goal, stohasti mapping between effort and progress, and intermediate milestones. Doing so allows us to fous on the information dislosure poliy of the ontest designer and show how this poliy depends on whether the ompetition or the enouragement effet dominates. In partiular, we expliitly onsider the question of whether and when should the ontest designer dislose information regarding the ompetitors partial progress with the goal of maximizing her expeted payoff. Interestingly, we show that the optimal design features a non-trivial information dislosure poliy thus illustrating the ative role information about the status of ompetition an have in inentivizing agents to partiipate in the ontest. We then identify the role of intermediate awards as a way for the designer to implement the desired information dislosure poliy and thus inrease the agents effort provision and onsequently the hanes of innovation taking plae in the proess. Intermediate awards are very ommon in innovation ontests the aforementioned NetFlix and Heritage prizes are examples of ontests that have employed intermediate awards, but to the best of our knowledge, the exat role they play as information revelation devies has not yet been studied. A simple illustration of the main ideas in this paper is the following. Consider an innovation ontest that onsists of well-defined milestones towards the end goal. In this ase, reahing a milestone onstitutes partial progress and we assume that agents and the designer are able to verifiably ommuniate this. Assume for now that the innovation is attainable with ertainty and the agents are fully aware of that. The lak of progress towards the goal is then solely a result of the stohasti return on effort. If no information is dislosed about the agents progress, then they beome progressively more pessimisti about the prospet of them winning as they believe that they may be lagging behind in the rae towards the end goal. This may possibly lead them to abandon the ontest, thus dereasing the aggregate level of effort and onsequently inreasing the time to omplete the ontest. In ontrast, when there is unertainty on whether the end goal is attainable, agents that have made no or little progress towards the goal beome pessimisti about whether it is even possible to omplete the ontest. If this persists, an agent may hoose to drop out of the ompetition as she believes that it is not worth putting the effort for what is likely an unattainable goal, reduing the aggregate experimentation in the proess and dereasing the hanes of reahing a possibly feasible innovation. This disussion highlights the omplex role that information about the agents progress may play in this environment. In the first senario, when the ompetition effet is dominant sine there is no unertainty regarding the attainability of the end goal, dislosing that one of the partiipants 3

4 is ahead may deter future effort provision as it implies that the probability of winning is lower for the laggards. On the other hand, in the seond ase, when the enouragement effet dominates, news about an agent s progress are onsidered good news, sine it redues the unertainty involved in the ontest. The information dislosure mehanism is only one of the levers that the designer has at her disposal to affet the agents effort provision deisions. Another is obviously the ompensation sheme that, in the ontext of an innovation ontest, takes the form of an award struture. In a setting with potentially multiple milestones a design may involve ompensating agents for reahing a milestone or having them ompete for a grand prize given out for ompleting the entire ontest. Our analysis sheds light on the interplay between information dislosure and the ontest s award struture by omparing different mehanisms in terms of their expeted payoff for the designer, assuming a fixed budget for the awards given out to the ontestants. This way we essentially bring a ontest s information provision mehanism to the forefront and interestingly we show that the probability of obtaining the innovation as well as the time it atually takes to omplete the projet are largely affeted by when and what information the designer hooses to dislose. Related Literature Several papers study the design of innovation ontests mostly fousing on stati models that resemble all-pay autions. Typially, these models take the form of a one-shot game in whih agents hoose their effort levels inurring the assoiated ost of effort whih an be thought of as their bid. The agent that puts most effort typially wins the ontest and laims an award. Several questions have been explored in this setup. For example, Taylor 1995 finds that a poliy of free and open entry is not optimal beause high partiipation may give rise to low levels of effort at equilibrium. Thus, the sponsor of the innovation ontest restrits partiipation and extrats all expeted surplus by taxing ontestants through an entry fee. Moldovanu and Sela 2001onsider the ase when the agents ost of effort is their private information. They show that when the ost is linear or onave in effort, it is optimal to alloate the entire prize sum to the winner whereas when it is onvex several prizes may be optimal. Che and Gale 2003 find that for a set of prourement settings it is optimal to restrit the number of ompetitors to two and, in the ase that the two ompetitors are asymmetri, handiap the most effiient one. Moldovanu and Sela 2006 explore the performane of ontest arhitetures that may involve splitting the partiipants among several sub-ontests whose winners ompete against eah other. They identify onditions under whih it is optimal to have a single grand ontest or split the ompetitors into two divisions and have a final between the two winners. Siegel 2009 provides a general framework to study suh stati all-pay ontests that allows for several features suh as differing prodution tehnologies, attitudes toward risk, and onditional and unonditional investments. Terwiesh and Xu 2008 and Ales et al explore stati ontests in whih there is unertainty regarding the value of an agent s ontribution and explore the effet of the award struture and the number of ompetitors on the ontest s expeted performane. Finally, Boudreau et al examine related questions empirially using 4

5 data from software ontests. 3 Unlike the papers mentioned above, a entral feature in our model is the fat that there is unertainty with respet to the attainability of the end goal. In addition, agents dynamially adjust their effort provision levels over time responding to the information they reeive regarding the status of the ompetition and the state of the world, i.e., whether the ontest an be ompleted. Early papers that onsider the dynamis of ostly effort provision in the presene of unertainty are Choi 1991 and Malueg and Tsutsui They both assume that agents an perfetly observe the experimentation outomes of their ompetitors and the value of the innovation is fixed thus abstrating away from the main fous of our study whih is the role of the designer s information dislosure mehanism and the ontest s award struture on the suess of the innovation ontest. There is also a reent stream of papers Bolton and Harris 1999, Keller et al. 2005, Keller and Rady 2010, and Bonatti and Horner 2011 that study the dynamis of ollaboration within a team of agents that work towards ompleting a projet. Sine all agents reeive the same payoff upon the projet s ompletion, the aforementioned papers fous on free-riding and how this affets the team s aggregate experimentation level. Bimpikis and Drakopoulos 2014 also onsider ollaboration within a team and show that having agents work independently and then ombining their efforts stritly inreases the probability of having the projet ompleted ompared to the ase when the entire team works together from the onset. Although our model builds on the exponential bandits framework that was introdued in Keller et al. 2005, our setup and fous are onsiderably different than prior literature in strategi experimentation. In partiular, agents ompete with one another for a set of awards that are set ex-ante by the designer. Furthermore, we allow for imperfet monitoring of the agents progress experimentation outomes. This, in ombination with the fat that agents dynamially learn about the attainability of the end goal as well as the status of ompetition, signifiantly ompliates the analysis as agents not only form beliefs about whether they an omplete the ontest but also they may need to form beliefs about their progress relative to their ompetitors. The latter is not an issue in the strategi experimentation literature sine experimentation outomes are typially assumed to be perfetly observable. Our paper is also related to the literature on dynami ompetition when there is unertainty in the mapping between effort and progress. For example, Harris and Vikers 1987 show that in a one-dimensional model of a rae between two ompetitors, the leader provides more effort than the follower and her effort inreases as the gap between the ompetitors dereases. Building on this work and very losely in spirit with our motivation, Mosarini and Smith 2007 study the design of optimal inentives in a two-player dynami ontest. Theirs is a model of perfet monitoring where the fous is on the design of a soring funtion in whih the leader is appropriately taxed whereas the laggard is subsidized. The goal of suh a funtion is to inrease overall effort provision by arefully trading off the ex-ante inentives of beoming a leader and getting taxed with the ex- 3 Several other papers, e.g., DiPalantino and Vojnovi 2009 and Chawla et al. 2012, study rowdsouring ontests as all-pay autions and exploit this onnetion to provide design guidelines that maximize their performane. Slivkins and Vaughan 2013 provide a reent survey of the theoretial hallenges related to rowdsouring. 5

6 post inentives of the laggard giving up. Unlike both of these papers we allow the ontest designer to hoose what information and when to dislose it, thus putting more emphasis on how the designer an inentivize agents to take a ertain set of ations by ontrolling the information they have aess to. A general premise of our work is that a prinipal may inentivize an agent or agents to take a given set of ations by ontrolling what information they have aess to. Relatedly, Kamenia and Gentzkow 2011 onsider the problem of a sender persuading a reeiver to take an ation by ommitting ex-ante to whih of a potentially large pool of noisy signals to reveal albeit in a stati setting. Furthermore, in a soial learning ontext, Che and Hörner 2013 and Kremer et al study a framework in whih a prinipal provides reommendations to a set of short-lived agents that deide sequentially on whih of two ations to take. The prinipal is interested in maximizing the disounted sum of the agents expeted payoffs whereas the agents take the ation that maximizes their own payoff based on the information they an infer from the prinipal s reommendation. Closest to our work are Lang et al as well as the independent and onurrent ontribution of Hala et al Lang et al study a two-player ontinuous time ontest in whih there is no unertainty about the underlying environment but agents exert ostly effort to omplete as many milestones as they an before a predetermined deadline. They haraterize equilibrium behavior and establish a lose relation with the outomes of stati all-pay autions. Hala et al study an experimentation ontest that ends after the ourrene of a single suess, thus not inorporating the possibility of partial progress and effetively not allowing for the enouragement effet. They ompare winner-takes-all ontests that feature full information dislosure with hidden-equal-sharing ontests in whih agents are guaranteed a fration of the total monetary prize if they omplete the ontest within a deadline. Although our work shares some features with theirs, partiularly, the unertainty regarding the attainability of the end goal, a major part of our study fouses on exploring the interplay between the ontest s award struture and the information dislosure poliy that it implies in relation to the enouragement and ompetition effets. This beomes extremely relevant in the presene of partial progress towards the end goal and disounting, whih are two features that are unique to our model and that, we believe, apture realisti aspets of ontests where innovation seekers are interested in ompleting their projets as quikly as possible. As Hala et al onsider a ontest with no intermediate milestones and assume that the designer and partiipants do not disount future outomes, the time it takes to omplete the ontest is immaterial for their analysis. Outline of the Paper The rest of the paper is organized as follows. Setion 2 presents our basi model and assumptions. Setion 3 provides the analysis for the senarios disussed above and highlights the role of intermediate awards. Finally, Setion 4 summarizes our findings and disusses diretions for future researh. 6

7 2 Model Our benhmark model is an innovation ontest with two sequential stages, Stage A and Stage B, and two ompetitors, Agent 1 and Agent 2. 4 Innovation happens if both stages are suessfully ompleted by a ompetitor, at whih time the ontest also ends. Stage j = A, B, is assoiated with a binary state θ j that desribes whether that stage an be ompleted θ j = 1 or not θ j = 0. If θ j = 0, then the breakthrough required to omplete stage j is not feasible and therefore innovation overall is not possible. If θ j = 1, then a breakthrough is feasible and has an arrival rate that is desribed by a Poisson proess with parameter j. 5 We assume that agents have a ommon prior on θ j and we denote that prior by p j = Pθ j = 1. We also assume that the events that stages A and B an be ompleted are independent, i.e., Pθ B = 1 θ A = 1 = Pθ B = 1. Agents hoose their effort levels ontinuously over time. Agent i {1, 2} hooses effort level x i,t [0, 1] at time t and inurs an instantaneous ost of effort equal to x i,t for some > 0. An agent in stage j who puts effort x t at time t obtains a breakthrough with instantaneous probability θ j j x t. Agent i is endowed with an information set Γ i,t = p j i,t, q i,t, Rt that summarizes her information about the tournament at time t, where p j i,t is her belief about the feasibility of stage j, q i,t is her belief about whether the ompeting agent is in Stage B for example, q i,t = 1 implies that agent i believes with ertainty that her ompetitor is in Stage B, and Rt is the award struture at time t. A strategy x i t = {x i,t } t 0 for agent i maps the agent s information set at time t to an effort provision level in [0, 1], i.e., x i,t : Γ i,t [0, 1]. Finally, let Π i x i t, x i t denote the payoff to agent i when she uses strategy x i t = {x i,t } t 0 and her ompetitor uses strategy x i t = {x i,t } t 0. The strategy profile x t = x 1 t, x 2 t is an equilibrium if for i {1, 2} the following holds x i t = arg max x it Π ix i t, x it. One an see from this formulation that agents effort deisions are a diret funtion of their information sets whih in turn summarize the agents beliefs about the feasibility of the end goal and the status of the ompetition. Our goal in this paper is to explore how the designer s information dislosure poliy along with the ontest s award struture affet the agents information sets and onsequently influene their effort provision deisions. Understanding these effets allows us to presribe guidelines for maximizing the designer s expeted payoff. Note that sine in our setting the tournament is over one one of the agents ompletes Stage B, the information provision mehanism is entered around partial progress, i.e., the ompletion or not of Stage A. 6 Formally, the designer s information provision strategy takes the following form: m t : H 1 t H 2 t {0, 1}, 4 Setion 4 disusses how our insights apply to multi-stage tournaments and a setting with multiple ompetitors. 5 It is possible that agents have aess to different tehnologies and therefore have different progress rates. This introdues a new set of interesting questions espeially when the agents skills, i.e., progress rates, are their private information. We further disuss this point in Setion 4. 6 It should be lear by now that progress in this setting takes the form of disrete breakthroughs, and it is not a smooth funtion of the agents efforts. 7

8 where Ht i {0, 1} for i = 1, 2 represents the designer s information about whether agent i has ompleted Stage A by time t 0 implies that she has not, 1 that she has ompleted the stage, and implies that the designer does not have any information. In turn, the designer an take one of three ations at every time t: announe nothing, announe that a ompetitor has ompleted Stage A 1, or announe that none of the ompetitors has ompleted Stage A 0. We assume that 0 or 1 announements are truthful, e.g., beause partial progress is verifiable, but the designer an hoose not to dislose the information she possesses. We do not allow for mixed strategies, i.e., probability distributions over the set of available ations at a given time instant t. In addition to simplifying the analysis, the fous on pure strategies stems from the desire to keep the ontest rules unambiguous and easy to interpret, with the impliit assumption that simple rules are more onduive to partiipation. Intertwined with the information provision mehanism is an award struture that speifies the size and timing of the awards handed out to ontestants. Apart from diretly affeting the agents effort provision inentives larger awards lead to higher effort levels from the agents, they an also be used as a lever to implement the designer s information provision poliy. In partiular, there is no a priori reason for a ontestant to reveal her partial progress to the designer unless this revelation leads to an inrease in her expeted utility. As we show in Setion 3, the ontest designer may inentivize agents to provide information about their progress by appropriately giving out intermediate awards. In partiular, the problem we study is designing a mehanism by whih awards are given out to ontestants as a funtion of their progress towards the end goal and, quite importantly, time. Formally, the deision problem that the designer faes is determining the award poliy Rt that maximizes the expeted payoff from running the ontest. Denoting this payoff by Π d and letting U be the utility that the designer gets from obtaining the innovation, we have Π d = E[e rτb U], 1 where τ B is the random time at whih Stage B and the tournament is ompleted. Throughout the paper we assume that U is normalized to 1. In order to failitate the omparison between different ontest designs and illustrate the role of the designer s information dislosure poliy on the ontest s expeted outome, we assume that the designer s budget for running a ontest is fixed and equal to B. In other words, the expeted disounted sum of the awards given out by the designer has to satisfy the following onstraint E [ ] e rτa R A + e rτb R B B, 2 where τ A is the random time at whih the award for ompleting Stage A is given out, and R A and R B are the sizes of the awards for ompleting stages A and B, respetively. In other words, the objetive for the designer is to maximize her expeted disounted utility subjet to a budget onstraint by hoosing how to alloate this budget to the ompetitors in the form of awards and impliitly designing an information provision mehanism in the proess. 8

9 Finally, we note that we use this two-stage tournament as a way of providing a reasonable approximation of the dynamis of multi-stage ontests. Naturally, the more progress being made, the less unertain agents are about the feasibility of the end goal. Thus, at a high level a multi-stage ontest an be thought of as having two distint phases. First, during the early stages, unertainty regarding the attainability of the end goal is the main driving fore behind the ompetitors ations. Competition is of seondary importane as there is still plenty of time for the laggards to ath up. We apture this situation as a two-stage ontest in whih the feasibility of the disovery required to omplete the first stage is unertain, i.e., p A < 1. On the other hand, the seond stage whih models the remainder of the ontest takes on average a muh longer time to omplete, i.e., the arrival rate assoiated with Stage B is muh lower than that of Stage A, i.e., B < A. As the ontest draws to an end, the dynamis beome quite different. Agents are more optimisti about the feasibility of the end goal, but the hanes for the laggards to ath up with the leader are slimmer. Thus, the agents behavior is mainly presribed by the ompetition effet. We apture this senario by examining two suessive stages that feature little or no unertainty, i.e., p A, p B 1. Studying these two extreme ases, namely, the behavior of ontestants early and late in a tournament, allows us to abstrat away from a multi-period setting and simplifies the analysis. This way we are able to provide sharp insights about the interplay of the ompetition and enouragement effets and disuss the role of information provision and information provision devies suh as intermediate awards in the suess of a ontest. 3 Analysis Our analysis fouses on the two ases disussed above that apture the dynamis near the beginning and near the end of a ontest. The main goal is to study how the designer s information dislosure mehanism and the ontest s award struture affet the agents effort provision and onsequently the expeted payoff for the designer. Having an understanding of this relationship allows us to provide onrete presriptions for the design of dynami ontests. We start with the ase when the ontest is still at its early stages and there is signifiant unertainty regarding the feasibility of the end goal. 3.1 Early Stages: Unertainty is Dominant As disussed earlier, when there is unertainty about whether the end goal is attainable or not, agents may beome more optimisti about their hanes of finishing the ontest by observing the partial progress of others. This setion explores how this enouragement effet an be utilized by the designer to improve the outome of the ontest. To simplify the analysis and exposition, we assume that p A < 1 and that P θ B = 1 θ A = 1 = 1, so that there is no unertainty in the seond stage provided that the first stage an be ompleted. Conditional on θ A = 1, the nominal arrival rate of a breakthrough in the first stage is for both agents. In keeping with our fous on modeling the 9

10 situation near the beginning of a ontest, we let the arrival rate in the seond stage, whih we denote by µ, to be suh that µ <. This assumption together with the assumption that p A < 1 provide a good approximation of the dynamis at the early stages of a ontest, when there is both a signifiant amount of unertainty as well as plenty of time before a ompetitor reahes the end goal. Before we proeed with the analysis, we desribe the belief update proess under the two extremes of information dislosure. These are full dislosure, i.e., when information regarding an agent s progress is immediately shared among all ompetitors, and no dislosure, i.e., when an agent observes only the outomes of her own experimentation as well as whether the ontest has ome to an end. - Full dislosure: If agents an observe eah others outomes, then the ommon posterior belief at time t regarding the feasibility of stage A in the absene of any progress is given by Bayes rule as: p 1,t = p 2,t = p A e 2t p A e 2t + 1 p A, 3 when both ompetitors put full effort up to time t, i.e., x 1,τ = x 2,τ = 1 for τ t. Obviously, in the event that one of the ompetitors finishes stage A then p 1,t and p 2,t jump to one as the unertainty regarding stage A is resolved. Finally, if agents do not put full effort up to t but rather use effort provision strategies {x 1,t } t 0 and {x 2,t } t 0, then we replae term e 2t in Expression 3 by e t 0 x1,τ +x2,τ dτ. - No dislosure: In the absene of any partial progress and assuming that her ompetitor has not laimed the final award, agent i s belief at time t is given by: p A e t e µt µe t µ p i,t =, 4 p A e t e µt µe t µ + 1 p A when both ompetitors put full effort up to time t. Note that the term e µt µe t aptures the information agent i infers from the fat that her ompetitor has not ompleted Stage B up to time t. In partiular, Pontest has not ended until t θ A = 1 = e t + t = e µt µe t, µ 0 µ e τ e µt τ dτ 5 where the first term in Equation 5 is equal to the probability that the ompetitor has not ompleted Stage A onditional on θ A = 1 and the seond term is equal to the probability that the ompetitor has ompleted stage A but not B. It is instrutive to ompare the expeted outomes of a ontest under these two extreme information dislosure poliies. Under full dislosure, an agent ompleting Stage A is onsidered good 10

11 news, sine it provides a positive signal about the feasibility of the end goal, and therefore instantly affets the ompetitors effort provision. On the flip side, absene of progress makes agents beome pessimisti at a faster rate than when information is not publi. Indeed, omparing Expressions 3 and 4 one an easily dedue that agents beliefs move downwards at a faster rate under full dislosure. This omparison learly illustrates one of the designer s main tradeoffs: on one hand, sharing progress between ompetitors allows for timely dissemination of good news. On the other hand, the absene of partial progress early on in the proess an make agents pessimisti about the feasibility of the underlying projet and, as we will see below, adversely affet their effort provision. Our goal for the remainder of this setion is to provide guidelines for the design of an innovation ontest. Speifially, we onsider the use of awards as a means of inentivizing agents to remain ative in the ontest as well as dislose their experimentation outomes. The designer has to balane several onfliting trade-offs. First, the size of the awards diretly affets the inentives of the agents to exert effort in the ontest. Seond, the timing of the awards impliitly indues an information dislosure poliy. For example, the ases disussed above full and no dislosure an be implemented using a simple award struture. In the ase of full dislosure, the designer an offer a large enough intermediate award R A for the agent who first ompletes Stage A, and no dislosure an be implemented by simply offering a single final award to the first agent who ompletes the entire ontest. Finally, the designer has to take the agents inentives to reveal information about their progress into aount. Speifially, in the presene of an enouragement effet, agents may be relutant to share their partial progress in the hope that their ompetitors beome more pessimisti and drop out of the rae. Thus, it is not lear whether and when agents will dislose information even when the designer allows them to do so. As should be lear by the disussion above, the spae of potential designs for an innovation ontest is quite large. Throughout the paper, we restrit attention to the sub-spae of designs that feature intermediate awards of onstant size that may or may not be offered at a given time period, i.e., we assume that R A t {0, R A }. We also assume that R B t = R B, i.e., the award for ompleting the ontest is always available to the agents. The seond assumption is without loss of generality when the designer disounts future at a high enough rate. However, the designer may potentially benefit if she allows the size of the intermediate awards to vary over time. We hose not onsider suh a senario mainly beause of two reasons. First, optimizing the timing of awards even when their values are fixed is already quite hallenging analytially. More importantly, we believe that in real-world tournaments simple award shemes are preferable and more likely to be implemented than awards whose sizes are a omplex funtion of time. Indeed, most if not all real-world tournaments feature awards of onstant size that are given out for reahing pre-speified milestones. Furthermore, this assumption allows us to learly illustrate the interation between the size of the awards and the time at whih they are given out and how this interation affets the agents beliefs about the underlying environment and onsequently their effort provision. 11

12 As a natural starting point, Subsetions and explore equilibrium behavior in ontests that indue full and no information dislosure respetively. Our analysis learly highlights the tradeoffs involved and motivates Subsetion in whih we show that more sophistiated information dislosure poliies that may be implemented by appropriately timing intermediate awards lead to a higher expeted payoff for the designer. These results provide a omplete piture regarding the question of optimal ontest design at least in the spae of determisti dislosure poliies as a funtion of the parameters in the model, i.e., the designer s budget, the disount rate, and the ompetitors skills as aptured by rates and µ Full Information Dislosure: Real-Time Leaderboard In this subsetion, we study equilibrium behavior in a ontest where when an agent ompletes stage i {A, B}, she has the inentive to immediately report it and laim the assoiated award. Before desribing in detail the award struture that indues suh behavior, we turn our attention to the subgame that results when one of the agents, the leader, ompletes Stage A. The leader s optimal effort provision takes a very simple form for t τ A, where τ A is the random time at whih Stage A is ompleted. In partiular, if we index the leader by i, we have x i,t = { 1 if RB µ 0 otherwise. Thus, the designer should set R B to be at least as high as µ to ensure that the ontest is finished reall that Stage B an be finished with probability 1, so as soon as an agent breaks through to that stage, she will ontinue putting effort if the value of the award is high enough. Similarly, the laggard ontinues putting effort in the ontest if her expeted payoff is higher than the instantaneous ost of effort, i.e., x j,t = 1 for τ A t τ B if µr B 2µ + r R B 1 + 2µ + r. µ where j is the index of the laggard, and τ B is the time at whih Stage B is finished and the ontest ends. In other words, upon ompletion of stage A, both the leader and the laggard remain in the ontest and put full effort until one of them ompletes stage B if the final award R B is at least R B RB min 1 + 2µ + r. 6 µ If, on the other hand, µ R B < RB min, the laggard drops out of the ontest whereas the leader ontinues putting full effort until the end. If we let V i k, l denote the expeted payoff of agent i 12

13 when she is in stage k and her ompetitor is in stage l, we have 2µ V i +µ+r 2µ+r R B A, B = and V i B, A = +µ+r µr B µ+r 2µ 2µ+r + µ +µ+r 2µ++r +µ+r2µ+r if R B R min B 0 otherwise R B 2µ++r +µ+r2µ+r if R B R min B otherwise. 7 8 We are now ready to desribe the award struture that indues full information dislosure. Note that when the leader deides whether to laim an intermediate award and reveal her partial progress she is faing the following trade-off. On the one hand, she would be ompensated by R A for her revelation. On the other hand though, her progress enourages her ompetitor to stay in the ontest and thus dereases her own likelihood of winning the final award of size R B. The inrease in the leader s expeted payoff when her ompetitor quits providing effort is given by R B = µr B 2µ µ + r + µ + r 2µ + r R 2µ + + r B + µ + r2µ + r. Thus, if the intermediate award R A is at least as high as R B the leader will have an inentive to laim it and reveal her partial progress. This is formally stated in the next proposition the proof is omitted as it follows from the arguments in the text. Proposition 1. The following award shedule implements full information dislosure when R B R min B : R A t = { RA R B for t t 0 otherwise, 9 where t is a utoff time desribed below. If µ R B < RB min, then full information dislosure an be implemented for any R A t. Following the disussion above agent i s optimization problem an be written as follows when R A R B, i.e., information regarding partial progress is dislosed as soon as it ours. max {x i,t} t 0 0 [ xi,t p i,t R A + V i B, A + x j,t p i,t V i A, B ] e t 0 pi,sx1,s+x2,s+rds dt. We are interested in haraterizing the unique symmetri equilibrium in Markovian strategies in this setting. Proposition 2 below states that agents follow a utoff experimentation poliy in stage A and the aggregate amount of experimentation inreases with the size of the intermediate award. Proposition 2. When the award for ompleting stage A is suh that R A R B, there exists a unique symmetri equilibrium in whih agents experiment as follows: 13

14 i Agents follow a utoff experimentation poliy in stage A, i.e., 1 for t t = 1 x 2 ln 1 p p i,t = 0 otherwise. The utoff belief p is given as follows R A + µrb µ+r p = R A + +µ+r p A 1 p A if µ R B < R min B 2µ 2µ+r + µ +µ+r R B 2µ++r +µ+r2µ+r ii If stage A is ompleted, experimentation ontinues as follows 10 if R B RB min. 11 a If R B RB min : Both agents experiment with rate one until the end of the ontest. b If µ R B < RB min : The laggard drops out of the ontest whereas the leader experiments with rate one until the end. Proposition 2 illustrates the tradeoff between setting R B at a higher value and thus providing an inentive for the laggard to stay ative in the ontest and inreasing the size of R A in whih ase agents experiment more in Stage A. In partiular, the utoff belief is dereasing in R A, implying that the aggregate amount of experimentation in Stage A is inreasing with the size of the intermediate award. Note also that having both agents ompete in the ontest until its end adversely affets the aggregate amount of experimentation in Stage A, sine the leader s expeted payoff for moving on to Stage B is smaller. Taken together, these fats imply that aggregate experimentation in Stage A and thus the probability that the ontest will be ompleted eventually is maximized by setting R B to the smallest possible value that ensures that an agent that ompletes Stage A has the inentive to omplete Stage B as well, i.e., µ. On the other hand, setting R B to µ implies that only one of the agents will put effort in stage B as opposed to when R B RB min, in whih ase both the leader and the laggard ontinue to ompete for the final award until the ontest is over. Thus, when setting the sizes of the two awards the designer trades off a higher probability of obtaining the innovation by setting R B = /µ with getting it sooner by setting R B RB min. To resolve the issue of whether it is optimal to have both agents ompete until the end of the ontest or set the intermediate award at a higher value requires that we obtain a haraterization of the expeted ost of running a ontest sine the designer is budget onstrained reall that R A and R B should be set suh that the budget onstraint in Equation 2 is not violated in expetation. Lemma 1 below provides an expression for the expeted ost of a ontest with awards R A and R B. Lemma 1. The ost of running a ontest with awards R A and R B with R A R B and θ A = 1 is given as follows 14

15 a If R B RB min, it is equal to 2 1 e 2+rt R A + R B 2 + r b If µ R B < RB min, it is equal to + µ + r 2 µ 1 e 2+rt R A + R B 2 + r µ + r 2µ 2µ + r + µ. + µ + r The experimentation time utoff t is given by expressions 10 and 11. We onlude this subsetion by providing onditions under whih the optimal full information dislosure ontest features a high intermediate award and thus the design indues more experimentation in stage A or a high final award and thus the ontest is ompleted sooner in expetation onditional on it being ompleted. In partiular, Proposition 3 desribes the optimal full information dislosure ontest as a funtion of the disount rate r and the ompetitors skill. It states that when the designer is suffiiently patient, i.e., she has a suffiiently low disount rate, it is optimal to set the final award R B to its minimum value /µ, and maximize the agents aggregate experimentation in stage A reall that a higher intermediate award indues more experimentation in stage A. If, on the other hand, the designer disounts future at a suffiiently high rate, then it is optimal to alloate a larger fration of the designer s budget to the final award, and thus inentivize both agents to ompete until the ontest is over. Although this leads to lower aggregate experimentation in Stage A and thus implies that the probability that the ontest will be ompleted is lower, it also implies that onditional on the ontest being ompleted the time it takes is lower sine both agents ompete until the end. Let D 1 denote the design for whih the final award is equal to R B = µ whereas let D2 denote the design for whih the final award is equal to R B = RB min and the rest of the designer s budget is alloated to the award for partial progress R A here the supersript refers to the number of agents that ontinue putting effort if stage A is ompleted. Then, Proposition 3. When the intermediate award R A R B, the two ontest designs that maximize the designer s expeted payoff are D 1 and D 2. In partiular, i Contest design D 1 outperforms design D 2 when the designer is suffiiently patient, i.e., r 0. ii When r > 0 there exists r suh that for > r, ontest design D 2 outperforms design D 1. Proposition 3 ompletes the disussion on optimal ontest design under the onstraint that the award struture is suh that agents are inentivized to dislose their progress as soon as it ours, i.e., R A R B. Induing immediate dissemination of good news may be advantageous to the designer when the disount rate is high as we elaborate towards the end of the setion, but it also implies that in the absene of progress agents beome pessimisti at a fast rate. Subsetion onsiders the alternative of having just one final award and not dislosing any information about partial progress. 15.

16 3.1.2 No Information Dislosure: Grand Prize At the other extreme, we have ontests that feature a single final award and no information dislosure in the interim. The single final award R B is set so that E[e rτb R B ] = B, i.e., the designer s budget is entirely onsumed onditional on the ontest being ompleted. Agent i s optimization problem an be then expressed as follows: max {x i,t} t 0 t=0 xi,t p i,t qi,t V i B, B + 1 q i,t V i B, A, {x i,t } x i,t e rt e t τ=0 pi,τ qi,τ µ+xi,τ dτ dt, where reall that V i B, A denotes the expeted payoff for agent i when she has ompleted stage A whereas her ompetitor has not. Agent i annot observe agent j s progress as long as agent j has not ompleted stage B. Therefore, she forms beliefs about whether or not her ompetitor has ompleted stage A up to time t. In partiular, q i,t denotes agent i s belief that her ompetitor has advaned to stage B by time t. As we show in Proposition 4 equilibrium behavior takes the form of a threshold poliy as in the ase of full information dislosure. However, in this ase the time threshold t after whih an agent stops experimenting in the absene of partial progress depends on both her own effort provision as well as her onjeture about her ompetitor s strategy that olletively determine the speed at whih the agent updates her beliefs about the attainability of the end goal as well as whether her ompetitor has already ompleted stage A. Note that as before as soon as the agent ompletes stage A then it is optimal for her to put effort until the ontest is over. Proposition 4. When R B RB min and agents progress is not observable, there exists a unique symmetri equilibrium in whih agents experiment as follows: i Agents follow a utoff experimentation poliy in stage A, i.e., 12 x i,t = { 1 for t t 0 otherwise, 13 where the utoff time t is given as the unique solution to the following equation p A e t p A e t e µt µe t µ e µr t B e + e µt t + 1 p A µ + r µ µr B = 2µ + r. 14 ii If the agent ompletes stage A, then she experiments with rate one until the end of the ontest. Proposition 4 is stated under the assumption that the single final award R B is suh that R B RB min, i.e., ompleting Stage A is seen as good news for both ompetitors, the leader and the laggard. In other words, suh an award ensures that the enouragement effet dominates. Note that this assumption is in line with our fous on the dynamis at the beginning of the ontest. 16

17 The juxtaposition of Propositions 2 and 4 learly illustrates the main tradeoff that the designer faes. In partiular, equilibrium experimentation takes the form of a threshold poliy under both full and no information dislosure. The threshold under full dislosure satisfies the following equation p A e 2t R A + p A e 2t + 1 p A + µ + r 2µ 2µ + r + µ R B + µ + r 2µ + + r + µ + r2µ + r =, 15 when both agents ompete until the end of the ontest given that stage A is ompleted, i.e., when R B RB min. The threshold under no dislosure satisfies p A e t p A e t e µt µe t µ e µr t B e + e µt t + 1 p A µ + r µ µr B = 2µ + r. 16 Comparing Equations 15 and 16 implies that agents stop experimenting earlier when information is fully dislosed and > µ. Thus, the probability that the ontest is ompleted is higher when the ontest has only one final award. On the other hand though, there is a positive probability that in the ase when information about partial progress is not dislosed one of the agents drops out even though the other has ompleted stage A. The latter never ours under full information dislosure when R B RB min. Thus, onditional on one agent ompleting stage A early enough, the ontest is ompleted earlier in expetation when experimentation outomes are publily observable. This tradeoff motivates the searh for alternative information dislosure poliies that ould ombine the benefits of these two extremes. The final part of this setion shows that appropriately restriting the times at whih an agent an dislose information about her progress leads to stritly better outomes for the designer. Finally, we show that intermediate awards are ruial in implementing this more sophistiated information dislosure poliy and we make a onnetion between our results and design features of real-world innovation ontests Design with Silent Periods As disussed in the beginning of the setion, although full dislosure allows for the fast dissemination of good news, it may also adversely affet effort provision as agents beome pessimisti regarding the feasibility of stage A in the absene of partial progress early in the proess. The fat that no news is negative news motivates onsidering alternative designs that may feature silent periods, i.e., time intervals in whih the designer does not dislose any information regarding the ompetitors progress obviously, a speial ase is the design that features no information dislosure. In partiular, for the remainder of this subsetion we explore the performane of ontest designs as they are defined by the timing of the designer s announement about the status of the ompetition between the agents as well as by the award struture. Speifially, we parameterize designs by T [0, and R = R A, R B, i.e., the set of times the designer disloses information about the ompetitors progress, i.e., whether one or both have ompleted stage A and the size of the awards for ompleting stage A and B respetively. For example, T = [0, orresponds to full dislosure, T = to no 17

18 dislosure, whereas T = [t 1, t 2 ] to a design in whih information is dislosed only between t 1 and t 2. We say that the design features a silent period when T [0,, i.e., the set of time periods that the designer does not dislose any information has non-zero measure. As a first indiation that silent periods may inrease the designer s expeted payoff, Proposition 5 below provides a suffiient ondition on the designer s budget for suh designs to outperform full information dislosure. Proposition 5. Assume that the designer has budget B > 2 2µ 2+r +µ+r 2µ+r + µ +µ+r RB min. Then, a design with T = [t A,, i.e., a design that features a silent period of length t A > 0, always outperforms full information dislosure. Proposition 5 shows that designs with T = [t A, and t A > 0 outperform full dislosure. Essentially, agents may require enouragement that takes the form of a positive announement regarding their ompetitors progress but only after some time has elapsed and their own experimentation has not generated any results. The lower bound on B, i.e., B > 2 2µ 2+r +µ+r 2µ+r + µ +µ+r RB min ensures that the designer has enough budget to set the award for finishing stage B at R B RB min whih as we mentioned in the previous subsetion it is neessary for the enouragement effet to be relevant in this setting if R B < RB min then being informed that a ompetitor has ompleted stage A makes the laggard quit the ontest. Although this result establishes that full dislosure is never optimal, it does not prelude the optimality of a no dislosure information poliy, i.e., t A =. On the one hand, when t A is finite positive news may be shared between the agents albeit with some delay but on the other in the absene of a positive announement agents beome pessimisti about the feasibility of the goal and drop out. No dislosure does not allow for sharing of information but agents beliefs are updated downwards at a slower rate. Interestingly, we show under some additional mild assumptions that a design that features a silent period of finite length outperforms the no information dislosure poliy. Theorem 1. Let B > r + µ + r 2µ 2µ + r + µ RB min + µ + r and, r r for onstants, r. Then, the design that features T = [t A, and R B = RB min outperforms both the design that implements full information dislosure as well as the one that implements no information dislosure. Announement time t A is given as the solution to Equation 17 below: p A e ta p A e ta e µt A µe t A µ e e ta R A + V B, A + e µt t RA + 1 p A µ 2 + µr B = 2µ + r. 17 Theorem 1 states that the design in whih information regarding the status of ompetition is first dislosed at time t A > 0 outperforms both full information dislosure as well as no dislosure. 18