The last shall be first: Innovation as a head-to-head race

Arnold, Patrick; Möller, Marc; Roux, Catherine Working Paper The last shall be first: Innovation as a head-to-head race Discussion Papers, No. 25-05 Provided in Cooperation with: Department of Economics, University of Bern Suggested Citation: Arnold, Patrick; Möller, Marc; Roux, Catherine (2025) : The last shall be first: Innovation as a head-to-head race, Discussion Papers, No. 25-05, University of Bern, Department of Economics, Bern This Version is available at: https://hdl.handle.net/10419/333489 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Faculty of Business, Economics and Social Sciences Department of Economics The Last Shall Be First: Innovation as a Head-to-Head Race Patrick Arnold, Marc Möller, Catherine Roux 25-05 September, 2025 Schanzeneckstrasse 1 CH-3012 Bern, Switzerland http://www.vwi.unibe.ch DISCUSSION PAPERS The Last Shall Be First: Innovation as a Head-to-Head Race* Patrick Arnold†Marc M¨ oller‡Catherine Roux§ Abstract Uncertainty about the value of a contested innovation induces leaders and laggards to update their expectations in opposite directions. We characterize situations in which firms that have obtained an initial advantage are not the most likely to achieve final success. In spite of amplifying a leader’s advantage, greater contest intensity facilitates this effect, challenging the view that laggards require support to remain competitive. Keywords: Innovation contests; learning; competitive balance; leapfrogging. JEL classification: C72, D82. 1. Introduction While Neil Armstrong’s iconic words—“That’s one small step for man, one giant leap for mankind”—are etched into history, it is easily overlooked that only a few years earlier, the USSR had established a significant lead in the Space Race. After the successful launch of Vostok, which made Yuri Gagarin the first human to orbit our planet, President Kennedy acknowledged that “[. . . ]the Soviet Union gained an important advantage by securing these large boosters” (Logsdon 2010). However, instead of discouraging U.S. efforts, the Vostok *This work was supported by the Swiss National Science Foundation [grant number 100018 207874]. †University of Basel. E-mail: [email protected] ‡University of Bern. E-mail: [email protected], corresponding author §University of Basel. E-mail: [email protected] 1 marked a turning point in Kennedy’s support of the space program. After years of downplaying its importance, Kennedy made NASA the largest civilian agency in history which helped the U.S. secure ultimate victory in the race to the Moon.1 Examples where an innovator’s success sparks the R&D efforts of its rivals are ubiquitous, ranging from the design of twin aircraft-engines by Boeing and Douglas (Sutton 1998) and commercial mainframe computers by Remington Rand and IBM (Khanna 1995), to the more recent development of touchscreen phones by Apple and Samsung or conversational artificial intelligence by OpenAI and Google. Lerner (1997) provides further historical evidence suggesting that, in technology-intensive industries, catch-up behavior is the norm and firms deliberately increase their R&D expenditures when trailing behind. This observation contrasts with the common concern that a laggard’s incentive to innovate might be muted by a rival’s lead, even if that lead is small (ϵ-preemption) and many steps remain to be completed (discouragement effect). In this paper, we argue that it is an innovator’s ability to observe a rival’s outputs (e.g, technological advancements, patents) but not inputs (e.g., R&D spending, efforts), that induces the intense competition for innovation we often observe. Our argument is based on the idea that, in a race for an innovation of uncertain value, leading serves as a signal for a large investment from a leader motivated by high expectations, whereas lagging indicates a small investment by a laggard, likely to be the consequence of a more pessimistic view. Leaders and laggards thus update their beliefs in opposite directions, and a laggard can have a stronger incentive to invest, despite a leader’s initial technological advantage. We show that, as a result, innovation can become a head-to-head race, in which laggards are equally or even more likely than initial leaders to achieve final success. Our main contribution is to characterize the environments most conducive to making innovation a head-to-head race. Such environments offer sufficient rewards for intermediate technological advances, for example, in the form of easily filed patents, and they eliminate noise from the determination of the final winner. Perhaps surprisingly, and contrasting with existing results (Wang 2010), noise is not suitable for a head-to-head race, because although noise makes a leader’s technological advantage less decisive, it also obstructs the informational effect that induces a laggard to catch-up. A policy implication is that innovation contests should reward contestants for intermediate advances and offer clear-cut specification of the desired product.2 Our findings challenge a long-held view that laggards require external support to maintain competitive balance in innovative industries (Fudenberg et al. 1983; Harris and Vickers 1. At its peak, NASA employed up to 400,000 people and its spending increased to 4.4% of federal budget (Kantor and Whalley 2025). 2. An example meeting these requirement is the XPRIZE Carbon Removal Competition which offered $1million to the 15 most advanced contestants after the first year and a thirty page competition guideline specifying how to measure each contestant’s performance with great precision. 2 1987; Lippman and McCardle 1987). This view is in line with the more general idea of “leveling the playing field” in competitive environments and the associated call for affirmative action (M¨ oller 2012; Chowdhury, Esteve-Gonz´ alez, and Mukherjee 2023). We provide a countervailing argument by showing that innovation races may be more likely to be won by laggards than by leaders, even without external support, suggesting the existence of a selfcorrecting mechanism. This mechanism complements leapfrogging (Fudenberg et al. 1983; Aoki 1991), the opportunity for a laggard to overtake a leader through a sufficiently large technological jump, in securing competitive balance. Rather than being strategic, our mechanism is informational. We thereby contribute to a more recent literature highlighting the relevance of learning effects in dynamic innovation contests (Choi 1991; Doraszelski 2003; Halac, Kartik, and Liu 2017; Awaya and Krishna 2021).3A distinguishing feature of our setting, crucial to the insight that learning improves competitive balance, is that contest outcomes induce leaders and laggards to update their beliefs in opposite directions. Although this feature is present in M¨ oller and Beccuti (2025), a head-to-head race cannot arise. In their best-of-N-battle contest, a laggard might invest more than a leader but is never more likely to win. 2. The model Two homogeneous, risk-neutral players engage in a two-stage innovation contest that awards a prize Vt≥0 to the winner of stage t∈ {1,2}. We let ν=V1 V2denote the value-ratio of winning the initial stage (being the first in space) relative to winning the final stage (being the first on the Moon). The probability that player i∈ {A,B}wins stage tis given by the generalized Tullock (1980) contest success function: pi,t(ei,t,ej,t)=er i,t er i,t+er j,t ,(1) where ei,tdenotes player i’s R&D expenditure, or effort, in stage tand r∈[0,1] is a parameter determining the sensitivity of the contest outcome with respect to players’ efforts as opposed to noise. We denote the first-stage winner and the first-stage loser as the leader (L) and the follower (F), respectively. Assumption 1. The leader wins stage 2 with probability pL,2(1 αeL,2,eF,2), where 1 α>1denotes a technological advantage the leader obtains by winning stage 1. 3. Learning effects also play a role in a related literature on entryand exit-dynamics in competitive industries (e.g., Awaya and Krishna 2021; Chen, Ishida, and Mukherjee 2023; Cetemen and Margaria 2024). 3 A key feature of our model is that players are privately informed about the uncertain value of innovation. Following M¨ oller and Beccuti (2025), we model this uncertainty by postulating the existence of a “good” state S=gand a “bad” state S=b, equally likely and such that V1,V2>0 if and only if S=g, and we make the following assumption: Assumption 2. Each player i privately observes a signal si∈ {B,G}. Players’ signals are independent with P(si=G|S=g)=1and P(si=B|S=b)=σ∈(0,1). Note that signals are informative and that a bad signal is even conclusive, since si=B can only be received if S=b. Together with the fact that innovation has no value in the bad state, this assumption greatly simplifies the analysis because it implies that upon the observation of si=B, player iwill choose ei,1=ei,2=0. In addition to their private signals, players observe contest outcomes, i.e., who won and who lost each stage, but cannot observe their rival’s efforts. This assumption is important as it implies that players will learn about their rival’s signals only through their observation of contest outcomes. All uncertainty is resolved at the end of stage 2 when values V1and V2 materialize.4 Player i’s payoffis given by the difference between his prize earnings and his aggregate R&D expenditure, ei,1+ei,2. Our setting constitutes a dynamic Bayesian game, with players’ types given by their signals. We thus use Perfect Bayesian equilibrium as our solution concept. As players choose zero effort conditional on a bad signal, a symmetric equilibrium is completely characterized by a triplet (e∗ 1,e∗ L,e∗ F) denoting players’ first-stage effort, leader’s effort, and follower’s effort, all upon receipt of a good signal. 3. Learning How do players learn about their rival’s signal? Consider a player who received a good signal. Let βdenote the player’s prior belief that his rival’s signal is also good: β≡P(sj=G|si=G)= P(si=sj=G) P(si=G)=1+(1 −σ)2 2−σ∈(0,1).(2) After observing the stage 1 outcome, the player updates this belief as follows. After losing, the player concludes that his rival exerted positive effort in stage 1 which only happens after the rival received a good signal. Hence, the follower updates his belief upwards to 4. In an extension, we allow V1and V2to be correlated and assume players learn the value of V1before stage 2. Results remain qualitatively unchanged unless prizes are perfectly correlated. 4 β∗ F=1> β. In contrast, after winning stage 1, the leader updates his belief downwards to β∗ L= 1 2β 1 2β+1−β< β, (3) because, in equilibrium, he would have defeated a rival with a good signal only with probability one half. Note that innovation is expected to have positive value only when both players’ signals are good and that this value is VG t≡E[V|si=sj=G]=Vt 1+(1 −σ)2.(4) It thus follows from β∗ F> β > β∗ Lthat learning induces players’ expectations about the value of innovation to move in opposite directions. In particular, learning induces the follower’s expectation about the value of innovation to be higher than the leader’s: β∗ FVG t> β∗ LVG t. 4. Head-to-head races With the aim to identify learning about the uncertain value of innovation as a self-correcting mechanism, this section derives the conditions under which innovation becomes a head-tohead race. We formalize the notion of a head-to-head race as follows: Definition 1. Innovation constitutes a head-to-head race if, conditional on the value of innovation being positive, the follower’s probability of winning the race is at least as high as the leader’s: p∗ F≡(αe∗ F)r (e∗ L)r+(αe∗ F)r≥(e∗ L)r (e∗ L)r+(αe∗ F)r≡p∗ L.(5) The fact that we condition on the innovation having positive value captures the idea that a “winner” of an innovation race would not be observed as such if he had obtained an innovation with no value. To derive the conditions under which (5) holds, we now characterize the equilibrium (e∗ 1,e∗ L,e∗ F). In stage 2 it has to hold that e∗ F∈arg max eF≥0 (αeF)r (e∗ L)r+(αeF)rβ∗ FVG 2−eFand e∗ L∈arg max eL≥0 er L er L+(αe∗ F)rβ∗ LVG 2−eL.(6) The unique effort levels satisfying these conditions are e∗ L=VG 2β∗ Lr(αβ∗ L)r (αr+(β∗ L)r)2and e∗ F=VG 2r(αβ∗ L)r (αr+(β∗ L)r)2,(7) 5 and they imply that p∗ F p∗ L =αr (β∗ L)rand p∗ F≥p∗ L⇔β∗ L≤α. (8) This means that for a head-to-head race to exist, learning has to be sufficiently strong relative to the leader’s technological advantage. However, condition (8) is only necessary but not sufficient for a head-to-head race to exist. Remember from Section 3 that learning requires that players condition their stage 1 efforts on their private information. More specifically, in stage 1, a player should exert positive effort if and only if the player has observed a good signal. To determine the condition under which e∗ 1>0, let U∗ Land U∗ Fdenote the players’ continuation values of entering stage 2 as the leader or the follower, conditional on having received a good signal. We have U∗ L−U∗ F=(β∗ L)2r+1−α2r−(1 −β∗ L)(αβ∗ L)r(1 −r) [αr+(β∗ L)r]2VG 2,(9) and the equilibrium effort in stage 1 has to satisfy e∗ 1∈arg max e1≥0βer 1 er 1+(e∗ 1)rU∗ L−U∗ F+VG 1−e1=⇒e∗ 1=r 4βU∗ L−U∗ F+VG 1.(10) Note that, offthe equilibrium path, the leader’s continuation value ULdepends on stage 1 efforts through the updated belief βL=e1β e1β+(1−β)(e1+e∗ 1), but (10) can be obtained easily by use of the envelop theorem.5Also note that if p∗ F≥p∗ L, then (8) implies U∗ L−U∗ F≤0. Hence, for innovation to be a head-to-head race, the prize for winning stage 1 has to be sufficiently large. Only when VG 1>U∗ F−U∗ L, players exert positive effort in stage 1 conditional on having received a good signal, making the contest’s stage 1 outcome informative. We are now ready to formulate our result: Proposition 1. There exist a non-empty interval (σ, σ)⊂(0,1) of information structures for which learning induces innovation to be a head-to-head race if and only if 1 α≤√2and 1+1 ν =V1+V2 V1 <4 (1 −α)(2 −r)+1,(11) that is, when both the technological advantage of winning the initial stage and the fraction of innovation value reserved to the winner of the final stage are not too large. 5. Equilibrium existence can be shown along the lines of M¨ oller and Beccuti (2025). 6 Proposition 1 characterizes the conditions under which innovation becomes a head-tohead race. Learning effects have to be sufficiently strong to overcome the leader’s technological advantage. This requires the players’ information to be neither too precise nor too imprecise, because for σ→0 or σ→1, the rival’s signal adds no additional information to a player’s own signal. However, in an intermediate range of σ, losing stage 1 induces the follower to exert more effort than the leader in stage 2, eventually overcoming the leader’s technological advantage and equalizing both players’ chances to claim final victory. Proposition 1 shows that this requires players to be able to secure a sufficiently large part of the value of innovation upfront. It suggests that environments, where inexpensive and rapidly filed patents for intermediate technological advances are available, are conducive to head-tohead competition. How does noise influence whether innovation is head-to-head? Our model sheds light on this question, because an increase in the parameter rof the contest success function makes contest outcomes less noisy and more sensitive to players’ efforts. Intuitively, one might expect that increasing rtilts the race towards the leader by making his technological advantage more relevant. However, this intuition neglects the fact that an increase in ralso raises players’ incentives to provide effort in the initial stage of the contest, which, as argued above, constitutes a prerequisite for learning. The following result shows that this indirect informational effect is strong enough to overcome the direct technological effect, making a reduction in noise conducive for head-to-head competition. Corollary 1. Reducing the noisiness with which the innovation contest determines its winners is suitable for inducing a head-to-head race, i.e., an increase in r relaxes (11). In innovation contests, intermediate and final winners are determined in accordance with evaluation criteria specified by the contest’s organizer. Corollary 1 suggests that such criteria should leave little indeterminacy to promote the head-to-head nature of innovation. 5. Concluding remark Admittedly, the present model is stylized, but it enhances our understanding of a basic learning mechanism through which innovation can become a head-to-head race. Our main result highlights that, to promote a balanced race, technological advances should be sufficiently credited and attributed with minimal error. While the basic implications for patent policy are straightforward, future research should extend the model to explore how learning varies with the number of competitors and the length of the time horizon. 7