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How competition propels scientific risk-taking

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Scarce scientific prizes push even risk-averse researchers toward riskier projects, and raising the stakes pulls more scientists into the race, intensifying the effect.

desk verdict A clean extension of the silent-duel game, but the central risk-taking result relies on perfect evaluation and the paper's noise-robustness claim is likely wrong. read the letter →

arxiv 2509.06718 v1 pith:XSKJDYJM submitted 2025-09-08 physics.soc-ph

Kevin Gross , Carl T. Bergstrom This is my paper · ORCID
classification physics.soc-ph
keywords scientificrisk-takingcompetitionsilentduelmixed-strategyNashequilibriumcontestspublicationincentivesacademiccareersgametheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that competition for scarce scientific rewards—elite publications, prizes, and faculty jobs—is a causal driver of scientific risk-taking, counterbalancing the usual incentives toward cautious, incremental work. Using a game-theoretic model in which researchers choose how risky a project to attempt, it shows that when there are fewer prizes than contestants, even risk-averse scientists must select riskier projects to have a chance of winning. A notable finding is that even a small amount of competition induces substantial risk-taking: when almost everyone wins a prize, most competitors still choose projects with success probability below the prize ratio. If correct, competition does not merely screen who succeeds; it shapes what kind of science gets done.

What carries the argument

The machinery is a multi-prize extension of the silent-duel game, a classic game of timing in which contestants choose how bold an action to take and the boldest successful player wins. Each researcher selects a project certainty $p\in[0,1]$, with success value $v(p)$ decreasing in $p$, and prizes go to the most valuable successful projects. The mandatory-contest payoff is $\pi(p)=p\,\zeta(p)$, the probability of succeeding times the probability that fewer than $K$ rivals succeed with riskier projects; the voluntary-contest payoff adds the elite-journal premium, $\pi(p)=p[1+\beta\zeta(p)]$. Standard existence results for discontinuous games guarantee an atomless mixed-strategy Nash equilibrium, and the equilibrium cumulative distribution follows numerically from the constant-payoff condition, with the pure equilibrium $p=K/N$ emerging in the population limit.

What would settle it

Run the contest in the laboratory with $N$ players and $K$ prizes, letting a judge rank successful projects with controllable noise: the model predicts that as evaluation noise increases, the equilibrium distribution of chosen success probabilities shifts toward the safest project $p=1$; if risk-taking stays high under maximally noisy ranking, the perfect-ranking assumption fails.

Watch

Extended reading notes

Core claim

The central discovery is that mandatory scientific contests have no pure-strategy equilibrium: if all rivals choose the same project certainty $p$, any focal researcher can nearly guarantee a prize by picking an infinitesimally riskier project, since successful risky work is judged more valuable. Equilibrium must therefore be a mixed strategy spread over a range of risk levels, and as the ratio of prizes to participants falls, that equilibrium shifts toward riskier projects. In the large-community limit where $N$ and $K$ grow with $K/N\to\phi$, the mixed equilibrium collapses to the pure strategy $p=\phi$, but finite communities retain a spread of risk-taking behavior. For voluntary contests, raising the premium of the elite venue draws more researchers in, crowds the contest, and further pushes entrants toward high-risk, high-return projects; when the premium is modest, the population splits into a group that plays it safe and a group that competes boldly.

Load-bearing premise

The model assumes judges can perfectly rank every successful project by scientific value, so that a slightly riskier successful project always outranks a slightly safer one; the authors themselves note that noisy evaluation weakens this incentive and removes it entirely in the infinite-noise limit.

Editorial extensions

If this is right

  • As competition intensifies—fewer prizes relative to contestants—equilibrium research portfolios skew toward high-risk, high-reward projects even when every scientist is individually risk-averse.
  • Even slight prize scarcity produces appreciable risk-taking, so competition need not be severe to alter scientific practice.
  • Raising the premium of elite publication venues attracts more participants, crowds the contest, and indirectly forces all entrants to take larger risks.
  • As the number of competitors and prizes grows together, researchers become more homogeneous in the risk they adopt, converging to the pure strategy $p=K/N$.
  • Researchers at different career stages face different prize ratios and therefore different optimal risk levels, creating internal tension in training and collaboration over how risky a shared project should be.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The perfect-ranking assumption is the hinge: under real-world peer review with noisy evaluation, the risk-taking effect should weaken, so the model predicts that more precise evaluation amplifies risk-taking while noisier evaluation mutes it.
  • A testable empirical extension is to compare bibliometric risk proxies across fields or periods where the ratio of prizes (faculty jobs, grants, elite slots) to applicants differs, expecting higher risk-taking where the ratio is lower.
  • The model implies that policies reducing competition—such as expanding funding or hiring—could shift equilibrium research toward safer projects, a trade-off against the common assumption that competition boosts productivity.
  • In voluntary contests with modest premiums, the two-part mixture equilibrium suggests aggregate risk-taking can be bimodal, with a bold elite-facing segment and a conservative segment coexisting in the same field.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper models scientific competition as a multi-prize contest in which researchers simultaneously choose the riskiness of their projects, indexed by the success probability p, with successful riskier projects generating higher scientific value. The authors analyze two versions: mandatory participation (all researchers compete for K of N prizes) and voluntary participation (researchers can opt for a safe discipline-specific journal or enter an elite-journal contest with additive premium β). Using the silent-duel framework and Dasgupta-Maskin existence results, they characterize the symmetric mixed-strategy Nash equilibrium for finite N and K, show that in the mandatory contest the equilibrium places mass on riskier projects than the naive K/N benchmark, and derive a large-N pure equilibrium for the voluntary case. They also discuss heterogeneous ability, implications for training and collaboration, and a Harsanyi purification argument.

Significance. The paper gives a clean game-theoretic mechanism by which competition induces scientific risk-taking, counterbalancing well-documented conservative incentives. Its strengths include a careful equilibrium derivation, exact numerical characterization, an analytic large-N result for voluntary participation, and a useful connection to the classic silent-duel literature. The central conclusion—that even mild competition pushes researchers toward riskier projects than the prize ratio suggests—is a genuine derivation, not an artifact of parameter fitting. The main weakness is an unsupported claim about robustness to evaluation noise, which is load-bearing for the paper's real-world message.

major comments (3)
  1. [Discussion, evaluation-noise paragraph] The claim that 'Intermediate levels of noise in evaluation lead to equilibria that intergrade smoothly between this case and the MSNE under perfect evaluation' is unsupported and, as far as the presented analysis goes, likely false. With i.i.d. assessment noise of scale σ, the probability that a project with success probability p−ε outranks a competitor's successful project at p is approximately 1/2 when ε is small relative to σ, so the infinitesimal-undercutting advantage that drives the mixed-strategy equilibrium disappears. For σ above a finite threshold, the all-safe profile p=1 can be a strict Nash equilibrium, so the equilibrium correspondence in σ can be discontinuous. Because the headline result that even small competition induces substantial risk-taking depends precisely on the undercutting mechanism, this is a load-bearing gap: the authors should either provide an analysis of noisy evaluation or substantially qualify the robustness claim.
  2. [Section III.A and footnote 5] The derivation of the equilibrium cumulative distribution function assumes that the support of the MSNE is a connected interval (a,1). Footnote 5 sketches why b=1 and says a 'nearly identical argument' rules out disjoint intervals, but the latter step is not shown. Since all subsequent numerical results and figures depend on this characterization, the proof should be completed or a citation provided for a complete proof in the silent-duel literature.
  3. [Section III.A, paragraph after Fig. 2] The statement that in the large-N limit with K/N→φ a pure-strategy Nash equilibrium is p=φ, and that the finite-N MSNE converges smoothly to it, is asserted without proof. The 'easy to see' justification is not a proof, and the finite-N payoff computations in the mixed equilibrium do not transparently converge to the payoff at p=φ. Please either supply a rigorous argument for the convergence or label it explicitly as a numerical observation or conjecture.
minor comments (4)
  1. [Appendix A.2, first paragraph] The word 'playoff' appears where 'payoff' is intended; please correct the typo.
  2. [Appendix A.2] The purification illustration is for a binary action space with a particular type distribution; the paper should state explicitly that this is a special case and does not by itself establish purification for the continuous-action game.
  3. [Introduction (first paragraph)] The phrase 'even risk-averse scientists will have to attempt riskier projects' goes beyond the model, which analyzes risk-neutral expected-payoff maximizers. The Discussion mentions risk aversion but does not formally incorporate it; please clarify that the model's prediction is about risk-neutral players and that the risk-aversion statement is an interpretation.
  4. [Figure 3] The median is a discontinuous functional of the equilibrium distribution; consider also reporting the mean or the full distribution for small N, which would make the 'substantial risk-taking' claim easier to assess.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the risk-taking results are theorems of an explicitly specified game, not fitted inputs or self-citation renames.

full rationale

The paper's derivation chain is self-contained. Section II specifies the game: each researcher chooses a success probability p, successful projects have value v(p) with v'<0, and the top K successful projects win prizes. Sections III.A and III.B then characterize the symmetric mixed-strategy Nash equilibrium using only these primitives. The headline conclusions—that mandatory competition induces riskier play and that voluntary competition crowds entrants toward risk—are equilibrium properties derived from the payoff functions (Eqs. 1 and 2), not quantities fitted to data or supplied by normalization. The only imported mathematical results are external and non-self-citational: Dasgupta & Maskin's existence/atomlessness theorems, Harsanyi's purification theorem, and Karlin/Dresher's silent-duel solutions for the illustrative heterogeneous-ability case, whose use the paper explicitly flags. The self-citations to Gross & Bergstrom (2019, 2021, 2024) and Bergstrom et al. (2016) appear only as background motivation for why scientists are conservative or how contests work; none of these citations supplies a load-bearing premise in the equilibrium proof. The 'undercutting' argument is stated directly in the text: if all rivals play p, a focal player can guarantee a prize by playing p-epsilon with negligible added failure risk. This is a derivation from the model's ordering assumption, not an import. The Discussion's claim that intermediate evaluation noise 'intergrade[s] smoothly' between the noisy and perfect-evaluation equilibria is unsupported and may be false, but it is an extrapolation beyond the model rather than a circular step. No prediction is statistically forced by a fitted parameter, and no known result is merely renamed as a new one; the multi-prize and voluntary-participation equilibria are genuine extensions derived from the stated game.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model's central claims rest on standard game theory (existence theorems) plus a small set of clearly stated domain assumptions about the risk-reward trade-off and prize allocation. The parameters N, K, and β are inputs varied for comparative statics, not fitted to data. No hidden entities or unstated tuning is required.

free parameters (3)
  • Number of competitors N = Varied: 2, 4, 10, 20, 200
    Model input controlling competition intensity; not fitted to data. Conclusions are qualitative across these values.
  • Number of prizes K = Varied: K=N-1, N/2, 1, 3, 5, 10, 100
    Model input controlling reward scarcity; varied in figures. Not data-fitted.
  • Elite-journal premium β = Varied: 1, 2, 4
    Model input in voluntary-participation game; chosen to illustrate comparative statics. Not fitted to data.
assumptions (5)
  • domain assumption The risk-reward frontier v(p) is strictly decreasing (v'(p) < 0): riskier projects, if successful, produce more scientific value.
    Invoked at the start of Section II (Model); this trade-off is the core of the model's notion of risk.
  • domain assumption Each project either succeeds (producing value v(p)) or fails (producing zero value); prizes go to the researchers with the most valuable successful outcomes.
    Section II; defines the contest structure.
  • domain assumption All prizes have equal value, normalized to 1; in the voluntary game, winning the glossy journal adds premium β over the base reward of 1.
    Section II and Section III.B; simplifies analysis.
  • domain assumption Judges perfectly rank successful projects by their value, so that any infinitesimal increase in risk (decrease in p) yields a strictly more valuable successful project.
    Assumed implicitly in the payoff functions; the authors note in the Discussion that relaxing this (adding evaluation noise) erodes the risk-taking result.
  • standard math Existence of a symmetric mixed-strategy Nash equilibrium follows from Dasgupta & Maskin (1986), Theorem 5.
    Appendix A.1 relies on DM86; the paper fills in the verification of the theorem's conditions.

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Pith. "Pith review of How competition propels scientific risk-taking." pith.science (2026). https://pith.science/paper/XSKJDYJM

@misc{pith2026250906718,
  author       = {Pith},
  title        = {Pith review of: How competition propels scientific risk-taking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSKJDYJM}},
  note         = {Machine review of arXiv:2509.06718}
}
read the original abstract

In science as elsewhere, attention is a limited resource and scientists compete with one another to produce the most exciting, novel and impactful results. We develop a game-theoretic model to explore how such competition influences the degree of risk that scientists are willing to embrace in their research endeavors. We find that competition for scarce resources -- for example, publications in elite journals, prestigious prizes, and faculty jobs -- motivates scientific risk-taking and may be important in counterbalancing other incentives that favor cautious, incremental science. Even small amounts of competition induce substantial risk-taking. Moreover, we find that in an ``opt-in'' contest, increasing the stakes induces increased participation -- which crowds the contest and further impels entrants to pursue higher-risk, higher-return investigations. The model also illuminates a source of tension in academic training and collaboration. Researchers at different career stages differ in their need to amass accomplishments that distinguish them from their peers, and therefore may not agree on what degree of risk to accept.

Figures

Figures reproduced from arXiv: 2509.06718 by the authors.

Figure 1
Figure 1. FIG. 1. Cdfs of mixed-strategy Nash equilibria when [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cdfs of mixed-strategy Nash equilibria when [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Median certainty, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Cdfs of mixed-strategy Nash equilibria when [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Nash equilibria when a large number of researchers [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. MSNEs for a postdoc (green) engaged in a com [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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Reference graph

Works this paper leans on

56 extracted references · 54 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. D. Adams, G. C. Black, J. R. Clemmons, and P. E. Stephan. Scientific teams and institutional collabora- tions: Evidence from us universities, 1981–1999.Re- search Policy, 34(3):259–285, 2005

    work page 1981

  2. [2]

    Alpern and J

    S. Alpern and J. Howard. A short solution to the many- player silent duel with arbitrary consolation prize.Eu- ropean Journal of Operational Research, 273(2):646–649, 2019

    work page 2019

  3. [3]

    K. Arrow. Economic welfare and the allocation of re- sources for invention. InThe rate and direction of inven- tive activity: Economic and social factors, pages 609–626. Princeton University Press, 1962

    work page 1962

  4. [4]

    Azoulay and D

    P. Azoulay and D. Li.Scientific grant funding, volume 26889. National Bureau of Economic Research Cam- bridge, MA, 2020

    work page 2020

  5. [5]

    G. S. Becker and K. M. Murphy. The division of labor, coordination costs, and knowledge.The Quarterly Jour- nal of Economics, 107(4):1137–1160, 1992

    work page 1992

  6. [6]

    C. T. Bergstrom, J. G. Foster, and Y. Song. Why scien- tists chase big problems: individual strategy and social optimality.arXiv:1605.05822, 2016

    work page Pith review arXiv 2016

  7. [7]

    D. A. Blackwell and M. A. Girshick.Theory of games and statistical decisions. Wiley, New York, 1954

    work page 1954

  8. [8]

    Bozeman and E

    B. Bozeman and E. Corley. Scientists’ collaboration strategies: implications for scientific and technical hu- man capital.Research Policy, 33(4):599–616, 2004

    work page 2004

Show all 56 references
  1. [9]

    Cyranoski, N

    D. Cyranoski, N. Gilbert, H. Ledford, A. Nayar, M. Yahia, et al. Education: the PhD factory.Nature, 472(7343):276–279, 2011

    2011

  2. [10]

    Dasgupta and P

    P. Dasgupta and P. A. David. Information disclosure and the economics of science and technology. InArrow and the Ascent of Modern Economic Theory, pages 519–542. Springer, 1987. 13 Which it would if, for example, successful projects are published and unsuccessful projects are not

    1987

  3. [11]

    Dasgupta and E

    P. Dasgupta and E. Maskin. The existence of equilibrium in discontinuous economic games, I: Theory.The Review of Economic Studies, 53(1):1–26, 1986

    1986

  4. [12]

    Dasgupta and E

    P. Dasgupta and E. Maskin. The simple economics of research portfolios.The Economic Journal, 97(387):581– 595, 1987

    1987

  5. [13]

    Dresher.Games of strategy: theory and applications

    M. Dresher.Games of strategy: theory and applications. Prentice-Hall, Englewood Cliffs, NJ, 1961

    1961

  6. [14]

    S. K. Ethiraj and P. Garg. The division of gains from complementarities in human-capital-intensive activ- ity.Organization Science, 23(3):725–742, 2012

    2012

  7. [15]

    Fitzenberger and U

    B. Fitzenberger and U. Schulze. Up or out: Research incentives and career prospects of postdocs in germany. German Economic Review, 15(2):287–328, 2014

    2014

  8. [16]

    J. G. Foster, A. Rzhetsky, and J. A. Evans. Tradition and innovation in scientists’ research strategies.American Sociological Review, 80(5):875–908, 2015

    2015

  9. [17]

    M. F. Fox and P. E. Stephan. Careers of young scientists: Preferences, prospects and realities by gender and field. Social Studies of Science, 31(1):109–122, 2001

    2001

  10. [18]

    Franzoni and P

    C. Franzoni and P. Stephan. Uncertainty and risk-taking in science: Meaning, measurement and management in peer review of research proposals.Research Policy, page 104706, 2023

    2023

  11. [19]

    Franzoni, P

    C. Franzoni, P. Stephan, and R. Veugelers. Funding risky research.Entrepreneurship and Innovation Policy and the Economy, 1(1):103–133, 2022

    2022

  12. [20]

    Fudenberg and J

    D. Fudenberg and J. Tirole.Game theory. MIT press, Cambridge, MA, 1991

    1991

  13. [21]

    Govindan, P

    S. Govindan, P. J. Reny, A. J. Robson, et al. A short proof of Harsanyi’s purification theorem.Games and Economic Behavior, 45(2):369–374, 2003

    2003

  14. [22]

    Gross and C

    K. Gross and C. T. Bergstrom. Contest models highlight inherent inefficiencies of scientific funding competitions. PLoS Biology, 17(1):e3000065, 2019. ISSN 1544-9173

    2019

  15. [23]

    Gross and C

    K. Gross and C. T. Bergstrom. Why ex post peer re- view encourages high-risk research while ex ante review 9 discourages it.Proceedings of the National Academy of Sciences, 118(51):e2111615118, 2021

    2021

  16. [24]

    Gross and C

    K. Gross and C. T. Bergstrom. Rationalizing risk aver- sion in science: Why incentives to work hard clash with incentives to take risks.Plos Biology, 22(8):e3002750, 2024

    2024

  17. [25]

    J. C. Harsanyi. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory, 2(1):1–23, 1973

    1973

  18. [26]

    J. J. Heckman and S. Moktan. Publishing and promotion in economics: The tyranny of the top five.Journal of Economic Literature, 58(2):419–470, 2020

    2020

  19. [27]

    R. Heesen. Why the reward structure of science makes reproducibility problems inevitable.The Journal of Phi- losophy, 115(12):661–674, 2018

    2018

  20. [28]

    silent duel

    and is, to our knowledge, the first to consider silent- duel-type games with several prizes or with voluntary participation. Alpern & Howard [2] have also noted the connection between silent-duel-type games and competi- tion among scientific researchers. III. ANALYSIS A. Oblig...

  21. [29]

    Henig and B

    M. Henig and B. O’Neill. Games of boldness, where the player performing the hardest task wins.Operations Re- search, 40(1):76–86, 1992

    1992

  22. [30]

    Hilhorst and C

    H.-J. Hilhorst and C. Appert-Rolland. Mixed-strategy Nash equilibrium for a discontinuous symmetricN-player game.Journal of Physics A: Mathematical and Theoret- ical, 51(9):095001, 2018

    2018

  23. [31]

    D. L. Hull.Science as a process: an evolutionary account of the social and conceptual development of science. Uni- versity of Chicago Press, 2010

    2010

  24. [32]

    Karlin.Mathematical methods and theory in games, programming, and economics, volume 2

    S. Karlin.Mathematical methods and theory in games, programming, and economics, volume 2. Pergamon Press, London, 1959

    1959

  25. [33]

    P. Kitcher. The division of cognitive labor.The Journal of Philosophy, 87(1):5–22, 1990

    1990

  26. [34]

    Kleinberg and S

    J. Kleinberg and S. Oren. Mechanisms for (mis) allo- cating scientific credit. InProceedings of the forty-third annual ACM symposium on theory of computing, pages 529–538, 2011

    2011

  27. [35]

    R. K. Merton. Priorities in Scientific Discovery: A Chapter in the Sociology of Science.American Socio- logical Review, 22(6):635, 1957. ISSN 00031224. doi: 10.2307/2089193

    1957 doi

  28. [36]

    F. M. Neffke. The value of complementary co-workers. Science Advances, 5(12):eaax3370, 2019

    2019

  29. [37]

    M. J. Osborne and A. Rubinstein.A course in game theory. MIT Press, Cambridge, MA, 1994

    1994

  30. [38]

    Owen.Game theory

    G. Owen.Game theory. Emerald Group Publishing, Bin- gley, UK, 4th edition, 2013

    2013

  31. [39]

    R. E. Ployhart and O. R. Cragun. Human capital re- source complementarities.The Oxford Handbook of Tal- ent Management, 115:134–149, 2017

    2017

  32. [40]

    Presman and I

    E. Presman and I. Sonin. The choice of a decision mo- ment in a game. InStochastic control problems in eco- nomics, pages 145–166. Nauka, Moscow, 1977

    1977

  33. [41]

    Presman and I

    E. Presman and I. Sonin. The existence and uniqueness of Nash equilibrium point in anm-player game ‘shoot later, shoot first!’.International Journal of Game Theory, 34: 185–205, 2006

    2006

  34. [42]

    T. Radzik. Results and problems in games of timing. In Statistics, probability, and game theory: Papers in honor of David Blackwell, volume 30, pages 269–292. Institute of Mathematical Statistics, Hayward, CA, 1996

    1996

  35. [43]

    Rzhetsky, J

    A. Rzhetsky, J. G. Foster, I. T. Foster, and J. A. Evans. Choosing experiments to accelerate collective discovery. Proceedings of the National Academy of Sciences, 112 (47):14569–14574, 2015

    2015

  36. [44]

    Sakaguchi

    M. Sakaguchi. Marksmanship contests: Non-zero-sum game of timing.Mathematica Japonica, 22:457–469, 1977/78

    1977

  37. [45]

    Semrau and C

    T. Semrau and C. Hopp. Complementary or compen- satory? A contingency perspective on how entrepreneurs’ human and social capital interact in shaping start-up progress.Small Business Economics, 46:407–423, 2016

    2016

  38. [46]

    Stephan.How economics shapes science

    P. Stephan.How economics shapes science. Harvard Uni- versity Press, 2015

    2015

  39. [47]

    Strevens

    M. Strevens. The role of the priority rule in science.The Journal of Philosophy, 100(2):55–79, 2003

    2003

  40. [48]

    Thelwall and N

    M. Thelwall and N. Maflahi. Research coauthorship 1900–2020: Continuous, universal, and ongoing expan- sion.Quantitative Science Studies, 3(2):331–344, 2022

    1900

  41. [49]

    I. M. Verma. Impact, not impact factor.Proceedings of the National Academy of Sciences, 112(26):7875–7876,

  42. [50]

    K. H. Wapman, S. Zhang, A. Clauset, and D. B. Lar- remore. Quantifying hierarchy and dynamics in us faculty hiring and retention.Nature, 610(7930):120–127, 2022

    2022

  43. [51]

    C. J. Waaijer, C. Teelken, P. F. Wouters, and I. C. van der Weijden. Competition in science: Links between publication pressure, grant pressure and the academic job market.Higher Education Policy, 31:225–243, 2018

    2018

  44. [52]

    Zhang, K

    S. Zhang, K. H. Wapman, D. B. Larremore, and A. Clauset. Labor advantages drive the greater produc- tivity of faculty at elite universities.Science Advances, 8 (46):eabq7056, 2022

    2022

  45. [53]

    Wuchty, B

    S. Wuchty, B. F. Jones, and B. Uzzi. The increasing dominance of teams in production of knowledge.Science, 316(5827):1036–1039, 2007

    2007

  46. [55]

    K. J. Zollman. The credit economy and the economic rationality of science.The Journal of Philosophy, 115(1): 5–33, 2018. 10 APPENDIX A.1. Proof of the existence of a MSNE That a MSNE exists in our game follows from Theorem 5 of Dasgupta & Maskin [11] (henceforth DM86). Here, w...

    2018

  47. [56]

    Suppose now that if playeriplaysR, then their probability of success isp+ϵt i, wheret i∼Unif(0,1),t 1 andt 2 are independent, andϵis small, such that p+ϵ <1

    then the game has a symmetric MSNE in whichSis played with probabilityq= 1−(2p−1)/(p 2−2p+ 1) = (p 2−4p+ 2)/(1−p) 2 We now use Harsanyi’s argument to show that the MSNE can be interpreted as the limit of a Bayesian Nash equilibrium (BNE) in a nearby perturbed game of incomplet...

  48. [2015]

    doi:10.1073/pnas.1509912112

    doi

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