## wp18231 ## Source details **Canonical URL:** [wp18231](https://www.imf.org/-/media/files/publications/wp/2018/wp18231.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp18231.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp18231.pdf.json) --- ### I. Introduction — main claim and intuition - Meritocracy measured by precision 1/σ (σ>0 scale of noise); perfect meritocracy as σ→0. - Main claim: more meritocracy is not always better — too much meritocracy can reduce aggregate output and be Pareto inefficient under increasing marginal costs. - Key intuition: - With pure noise (large σ) no effort; some meritocracy raises output via competition. - Beyond a critical precision (σ̄), intensified competition leads some contestants to “drop out” (zero effort) and aggregate output can fall. - Cost curvature determines the sign: - If C''(·)>0 (increasing marginal costs) → optimal σ = σ̄; perfect meritocracy (σ→0) is not output-maximizing nor Pareto efficient. - If C''(·)=0 (constant marginal costs) → any σ∈(0,σ̄] maximize output. - If C''(·)<0 (decreasing marginal costs) → perfect meritocracy uniquely maximizes output and is Pareto efficient. ### II. Baseline model — setup and equilibrium characterization - Primitives and definitions: - Unit mass of homogeneous, risk-neutral agents i∈I=[0,1]. - Measured performance Y_i = X_i · E_i with X_i≥0, E_i i.i.d.; log-variables x=ln X, y=ln Y, ε=ln E, χ(x)=C(e^x). - Noise ε_i drawn from a location-scale family with scale σ>0 (precision = 1/σ). - Mass μ∈(0,1) of contestants with highest measured performances receive prize v>0; equilibrium standard α clears μ winners. - Individual expected payoff (in log-output x_i and standard α): - π(x_i; α) = v · ̄Φ((α−x_i)/σ) − χ(x_i). - First-order condition for interior optimum: - v/σ · φ((α−x_i)/σ) = χ'(x_i). (Equation (3) form) - Best-response structure (Lemma 2): - For any α, best responses are either drop-out (x = −∞) or a single interior maximizer x_s(α); mixed responses mix on x_s. - Equilibrium conditions (FOC, SOC, PC, market-clearing MCC): - FOC: v/σ · φ((α−x_s)/σ) = χ'(x_s) - SOC: (1/σ) φ'((α−x_s)/σ)/φ((α−x_s)/σ) + χ''(x_s)/χ'(x_s) < 0 - Market-clearing: ˆθ · ̄Φ((α−x_s)/σ) = μ - Existence and uniqueness (Proposition 1): - Unique symmetric equilibrium exists. - Threshold σ̄>0 separates pure-strategy equilibrium (σ>σ̄: all produce x_s>−∞) from mixed equilibrium (σ<σ̄: agents mix between x_s and −∞). ### III. Limits of meritocracy — key elasticity and Theorem 1 - Elasticity of aggregate output O* with respect to σ (Lemma 4): - dO*/O* · dσ/σ = - −1/(ε_C0 + 1) for σ > σ̄ - ε_C − 1 / (−soc) for σ < σ̄ - Notation: ε_C0 = output elasticity of marginal costs; ε_C denotes output elasticity of costs; soc as in (4). - Theorem 1 (homogeneous contestants, fixed prize structure (μ,v)): - Aggregate output is maximized at σ = σ̄. - If C''(·)>0 then σ̄ is unique maximizer; perfect meritocracy (σ→0) is not output maximizing nor Pareto efficient. - If C''(·)=0 then all σ∈(0,σ̄] are output maximizing. - Economic interpretation: - For σ>σ̄: reducing σ (more meritocracy) raises output via competition effect only (no attrition). - At σ=σ̄: contestants’ expected payoffs fall to zero; organizer should maximize the pie. - For σ<σ̄: attrition effect can dominate under increasing marginal costs → output falls. ### IV. Closed-form threshold σ̄ and comparative statics - Exact expression preserved: - σ̄ = φ(̄Φ^{-1}(μ)) · ̄Φ(̄Φ^{-1}(μ)) · χ'(χ^{-1}(μv)) / χ(χ^{-1}(μv)) - Comparative statics (Proposition 2 consequences): - σ̄ is strictly decreasing in μ and weakly decreasing in v. - More, smaller prizes (higher μ holding budget fixed) require higher meritocracy (lower σ) to maximize output. - Perfect meritocracy optimal iff μ→1 (participation-prize contests). - For μ→0 (winner-take-all), σ̄ may diverge depending on tail behavior (hazard rate φ/̄Φ); logistic vs. normal tail examples discussed. - Organizer’s budget β = μv affects output only via χ^{-1}(β): O*_{σ̄} = χ^{-1}(μv). ### V. Multiple prize levels - Proposition 3 (homogeneous contestants): - Offering multiple prize levels cannot outperform equal-value prizes for maximizing aggregate output. - Intuition: equal prizes avoid heterogeneous effort under convex costs (Jensen’s inequality); the organizer appropriates surplus and chooses output-maximizing behavior. ### VI. Heterogeneous contestants — forces, decomposition, and results - Extension primitives: - Agents indexed by ability i with distribution Γ(·); cost C(X,i) with C_X > 0, C_XX ≥ 0, and C_Xi < 0. - Equilibrium properties (Proposition 4): - Existence and uniqueness of equilibrium. - Participation single-crossing in ability: at most one boundary type i* with i>i* participating and i 0 when a boundary type exists; indirect ˜A_ind ambiguous via standard change). - ˜H: heterogeneity effect > 0 (meritocracy reduces marginal benefit relative to homogeneous case by causing complacency of strong and discouragement of weak that dominate spurring of the middle). - Proposition 6: signs of components established; ˜H→0 as heterogeneity → homogeneity. - Examples (Example 2 summary): - Low heterogeneity: optimal σ at critical point where attrition begins (σ̄). - Medium heterogeneity: competition effect dominates → perfect meritocracy optimal. - High heterogeneity: heterogeneity effect dominates → imperfect meritocracy optimal. - Robustness (Proposition 5): - Homogeneous baseline is not a knife-edge; for sufficiently homogeneous contests, perfect meritocracy is strictly suboptimal. - Very heterogeneous contests (Theorem 2 / propositions): - For sufficiently heterogeneous contests, perfect meritocracy can be strictly suboptimal even with linear costs due to complacency of high-ability types. ### VII. Discrete-number-of-contestants results - 2-player Lazear-Rosen (LR) contests: - Errors: difference of log errors location-scale with scale ρ analogous to σ. - Lemma 5 summary: - Symmetric equilibrium exists for all ρ>0. - Pure-strategy equilibrium (PSE) exists iff ρ≥ρ̄; PSE unique and symmetric. - If ρ<ρ̄ only mixed-strategy equilibria exist; symmetric MSE payoffs are zero. - Proposition (LR): Aggregate output among PSEs is uniquely maximized at ρ = ρ̄; if C''>0 then ρ̄ is unique maximizer; perfect meritocracy (ρ→0) is not output maximizing nor Pareto efficient. - n-player Tullock contests: - EVTIM(0,σ) errors yield isomorphism to Tullock contest with discriminatoriness r = 1/σ: - Winning probability = X_i^r / Σ_j X_j^r (Lemma 6). - Theorem (Tullock): Aggregate output maximized in PSE with discriminatoriness r̄ = χ'(χ^{-1}(v/n)) / χ(χ^{-1}(v/n)) · (1 − 1/n) (exact expression preserved). - r̄ strictly decreasing in n; perfect discrimination (r→∞) not output maximizing under convex costs. ### VIII. Allowing concave costs and relaxed regularity (Appendix II) - Model relaxation: - Drop requirement C''(·)≥0; allow C''(·) possibly <0 while keeping analyticity and C(0)=0, C'(X)>0 for X>0. - Require output elasticity of marginal costs ×_C0 ≡ d ln C'(X) / d ln X > −1. - Key implications (Proposition 8 summary and Lemma 43): - For sufficiently small σ, sign of dO/(dσ/σ) coincides with sign of χ'/χ − 1 which, via identities, corresponds to sign of C'': - If C''(·)>0 → dO/(dσ/σ) < 0 for small σ → perfect meritocracy suboptimal. - If C''(·)=0 → neutrality: any σ≤σ̄ optimal. - If C''(·)<0 → dO/(dσ/σ) > 0 for small σ → perfect meritocracy uniquely output-maximizing and Pareto efficient. - Under strictly concave costs, the organizer optimally eliminates noise (σ→0); abstract optimal contest becomes winner-take-all (μ→0, v→∞, σ→0). ### IX. Policy-relevant takeaways and examples - Central takeaway: the socially optimal degree of meritocracy depends on cost curvature and heterogeneity. - Practical implications: - Firms: with homogeneous workforces and increasing marginal costs, committing to noisier evaluations can increase aggregate output by avoiding excessive competition and attrition. - Education/admissions: mechanisms like weighted lotteries can raise effort by reducing discouragement and complacency relative to perfect meritocracy. - Empirical notes: historical changes increasing test noisiness (e.g., SAT modifications) have ambiguous welfare effects; optimality depends on context-specific cost structures and heterogeneity. _Italic: Source — IMF Working Paper wp18231 (https://www.imf.org/-/media/files/publications/wp/2018/wp18231.pdf)._ ### References...................................... 34 ### wp18231 - References...................................... 34 ### Appendix ...................................... 37 - A Online Appendix:............................... 37 ### 1. Proofs.................................. 37 - A.BaselineModel.......................... 37 - B. Heterogeneous Contestants . . ................. 49 - C.DiscreteNumberofContestants ................ 66 ### 2. Allowing for Concave Costs and  00  0 ®−1............ 78 - A.ModelandResults........................ 78 - B.Proofs .............................. 80 ### Figures - 1. Marginal BenefitversusMarginalCost................... 12 - 2. Best Response as Function of....................... 14 - 3. Equilibrium as Function of......................... 16 - 4. LocusofOptimalMeritocracyandNumberofPrizes........... 18 - 5. Example2 .................................. 27 *Source: wp18231 - References...................................... 34 — https://www.imf.org/-/media/files/publications/wp/2018/wp18231.pdf* ### 6. InteriorBestResponses ........................... 72 ### 6. InteriorBestResponses ### I. Introduction - Meritocracy defined as accuracy of performance ranking; extremes: a-meritocratic (random winners) vs. perfectly meritocratic (best performers win with certainty); intermediate = partially meritocratic. - Main claim: more meritocracy is not always better—too much meritocracy can reduce aggregate output and be Pareto inefficient. - Key setup assumptions for initial analysis: - Contestants homogeneous (level playing field). - Output costs with increasing marginal costs (and cases of constant or decreasing marginal costs considered). - Performance measurement can be noisy; noise precision interpreted as meritocracy. - High-level intuition: - With pure noise, nobody exerts effort; some meritocracy increases output. - Beyond a critical precision, competition intensifies and contestants begin “dropping out” (zero effort). - For strictly increasing marginal costs, output loss from drop-outs exceeds gains from remaining contestants working harder → optimal meritocracy equals the critical level; perfect meritocracy reduces aggregate output and is Pareto inefficient. - For strictly decreasing marginal costs, perfect meritocracy uniquely maximizes output and is Pareto efficient. - For constant marginal costs, any meritocracy greater than the critical level maximizes output (threshold result). - Joint optimization with fixed prize budget: - With homogeneous contestants, equal-value prizes are weakly optimal. - The optimal level of meritocracy is positively related to the number of prizes: more (smaller) prizes → contest should be more meritocratic. - Perfect meritocracy is optimal iff the contest has a “participation-prize” structure (almost everybody wins). - Winner-take-all optimal iff contest is almost entirely a-meritocratic. - Heterogeneous contestants: - Effects of a rise in meritocracy split additively into: attrition effect (negative), competition effect (positive), heterogeneity effect (negative). - Heterogeneity tends to reduce benefits of meritocracy; complacency of strong and discouragement of weak dominate spurring-on of the middle. - Perfect meritocracy can be optimal for intermediate heterogeneity, but strictly suboptimal for very high heterogeneity (even with linear costs). - Practical implications and examples: - Firms may choose whether to use detailed micro-performance data depending on workforce heterogeneity. - Dutch weighted lottery for university admissions may increase studiousness by reducing discouragement and complacency relative to perfect meritocracy. - Empirical and theoretical complements: Holmstrom and Milgrom (1991), Goodhart’s Law, Lazear (1989). ### III. Baseline Model — Set-Up and Equilibrium - Model primitives: - Unit mass of homogeneous, risk-neutral agents i∈I=[0,1]. - Measured performance Y_i = X_i · E_i with X_i≥0 output, E_i i.i.d. noise. - Cost C(X_i) analytic, C(0)=0, C'(·)>0 for X>0, C''(·)≥0. - Define log variables x = ln X, y = ln Y, ε = ln E, χ(x)=C(e^x). - Noise ε_i drawn from a location-scale family with scale σ>0 (precision = 1/σ measures meritocracy); perfect meritocracy as σ→0. - Mass μ∈(0,1) of contestants with highest measured performances receive prize v>0. - Standard (performance threshold) α determines winning; equilibrium clears μ winners. - Individual expected payoff (log-output x_i, standard α): - π(x_i; α) = v · ̄Φ((α−x_i)/σ) − χ(x_i). (Equation (2) form) - First-order condition for interior optimum: - v/σ · φ((α−x_i)/σ) = χ'(x_i). (Equation (3)) - Structure of best responses (Lemma 2): - For any α, best responses are either drop-out (x = −∞) or a single interior maximizer x_s(α). Mixed responses characterized by probability of playing x_s. - Symmetric equilibria suffice for aggregate-output analysis (Lemma 3). - Equilibrium characterization (FOC, SOC, PC, MCC): - FOC: v/σ · φ((α−x_s)/σ) = χ'(x_s) - SOC: (1/σ) φ'((α−x_s)/σ)/φ((α−x_s)/σ) + χ''(x_s)/χ'(x_s) < 0 - Participation constraint depends on whether mix is pure or mixed. - Market-clearing: ˆθ · ̄Φ((α−x_s)/σ) = μ - Existence and uniqueness (Proposition 1): - Unique symmetric equilibrium exists. - For large σ (low meritocracy), equilibrium in pure strategies: all agents produce x_s(α;σ)>−∞. - For small σ (high meritocracy), agents mix between x_s and dropping out. - Threshold σ̄ > 0 separates pure-strategy equilibrium from mixed. ### III.C Limits of Meritocracy — Main result for homogeneous contestants - Elasticity of output w.r.t. σ (Lemma 4): - dO*/O* · dσ/σ = - −1/(ε_C0 + 1) for σ > σ̄ - ε_C − 1 / (−soc) for σ < σ̄ - Notation: ε_C0 = output elasticity of marginal costs; ε_C denotes output elasticity of costs; soc as in (4). - Theorem 1 (homogeneous contestants, fixed prize structure (μ,v)): - Aggregate output maximized at σ = σ̄. - If C''(·)>0 then σ̄ is unique maximizer; perfect meritocracy (σ→0) is not output maximizing nor Pareto efficient. - If C''(·)=0 then all σ∈(0,σ̄] are output maximizing. - Economic interpretation: - For σ>σ̄: reducing σ (more meritocracy) raises output via competition effect only (no attrition). - At σ=σ̄: contestants’ expected payoffs fall to zero; organizer should maximize pie size. - For σ<σ̄: attrition effect dominates competition effect when marginal costs increasing (loss from drop-outs > gain from remaining working harder by Jensen’s inequality) → output falls. - For constant marginal costs, gains and losses offset. - For decreasing marginal costs, perfect meritocracy maximizes output. - Comparative statics for σ̄ (equation (10) and Proposition 2): - σ̄ = φ(̄Φ^{-1}(μ)) · ̄Φ(̄Φ^{-1}(μ)) · χ'(χ^{-1}(μv)) / χ(χ^{-1}(μv)) (exact expression preserved) - σ̄ is strictly decreasing in μ and weakly decreasing in v; more, smaller prizes require higher meritocracy. - Perfect meritocracy optimal iff μ→1 (participation-prize contests). - For μ→0 (winner-take-all), σ̄ may diverge depending on tail behavior (hazard rate φ/̄Φ); logistic vs. normal examples discussed. - Output depends on budget β = μv only via χ^{-1}(β): O*_σ̄ = χ^{-1}(μv). ### III.D Multiple Prize Levels - Proposition 3: With homogeneous contestants, offering multiple prize levels (m, v) cannot outperform equal-value prizes (v1 = v2 = … = v_ℓ) for maximizing aggregate output. - Intuition: organizer appropriates surplus and chooses output-maximizing behavior; multiple prize levels induce heterogeneous effort and are suboptimal under convex costs (Jensen’s inequality). ### IV. Heterogeneous Contestants — forces and results - Extension: agents indexed by ability i with distribution Γ(·); cost C(X,i) with C_X > 0, C_XX ≥ 0, and C_Xi < 0 (higher ability reduces marginal costs). - Equilibrium properties (Proposition 4): - Existence and uniqueness of equilibrium. - Participation satisfies single-crossing in ability: there is at most one boundary type i* such that agents with i>i* participate and those with i 0 when boundary type exists; indirect ˜A_ind ambiguous via standards change). - ˜H: heterogeneity effect > 0 (meritocracy reduces marginal benefit relative to homogeneous case); discouragement of weak + complacency of strong dominate spurring of middle. - Proposition 6: signs of components established; ˜H→0 as heterogeneity → homogeneity. - Examples (Example 2): - Low heterogeneity: optimal σ at critical point where attrition begins. - Medium heterogeneity: competition effect dominates → perfect meritocracy optimal. - High heterogeneity: heterogeneity effect dominates → imperfect meritocracy optimal. ### V. Discrete Number of Contestants - Continuum results carry over qualitatively to finite-player contests. - 2-player Lazear-Rosen (LR) contests: - Errors: difference of log errors must be location-scale, log-concave; scale parameter ρ plays role analogous to σ. - Lemma 5: symmetric equilibrium exists for all ρ>0; pure-strategy equilibrium (PSE) exists iff ρ≥ρ̄; PSE unique and symmetric; for ρ<ρ̄ only mixed-strategy equilibria with zero expected payoffs in symmetric equilibria. - Theorem (LR): aggregate output maximized in PSE at ρ = ρ̄; if C''>0 then ρ̄ unique maximizer; perfect meritocracy (ρ→0) is not output maximizing nor Pareto efficient. - n-player Tullock contests: - Special case: if errors are EVTIM(0,σ) then n-player LR contest is isomorphic to a Tullock contest with discriminatoriness r = 1/σ and winning probability X_i^r / Σ_j X_j^r (Lemma 6). - Theorem (Tullock): aggregate output maximized in PSE with discriminatoriness r̄ = χ'(χ^{-1}(v/n)) / χ(χ^{-1}(v/n)) · (1 − 1/n) (preserved exact expression) - r̄ strictly decreasing in n (Remark 2); perfect discrimination (r→∞) is not output maximizing under convex costs. ### VI. Conclusion — policy-relevant takeaways - Central finding: too much meritocracy (perfect accuracy of performance ranking) can reduce aggregate output and be Pareto inefficient, especially with homogeneous contestants and increasing marginal costs. - For heterogeneous contestants, the optimal degree of meritocracy depends on the balance of competition, attrition, and heterogeneity effects. - Practical implications: - Firms with homogeneous workforces may benefit from committing to noisier performance evaluation (reducing meritocracy) to avoid excessive competition and attrition. - Education systems and admission mechanisms (e.g., weighted lotteries) might use imperfect meritocracy to prevent discouragement and complacency, potentially increasing overall effort. - Empirical notes: - Changes in standardized tests (e.g., SAT) that increase noisiness may have ambiguous welfare effects; historical changes cited (test length/time, essay, penalty removal, multiple-choice options) that increased noisiness but the optimality is open. *Source: IMF Working Paper wp18231 — Chapter 6 (InteriorBestResponses) — extracted content.* ### References ### References ### Bibliographic entries - Bagnoli, M., and T. Bergstrom, 2004, “Log-concave probability and its applications,”Economic Theory26: 445-469. - Baik, Kyung Hwan, 1998, “Difference-form Contest Success Functions and Effort Levels in Contests,”European Journal of Political Economy14: 685-701. - Baye, Michael R., Dan Kovenock, and Casper G. de Vries, 1994, “The Solution to the Tullock Rent-seeking Game When2: Mixed-strategy Equilibria and Mean Dissipation Rates,”Public Choice81: 363-380. - Che, Yeon-Koo, and Ian Gale, 2000, “Difference-form Contests and the Robustness of All-pay Auctions,”Games and Economic Behavior30: 22-43. - Clark, D. J., and C. Riis, 1998, “Competition Over More Than One Prize,” American Economic Review88(1), 276—289. - Corchón, Luis C., 2007, “The Theory of Contests: A Survey,”Review of Economic Design11: 69-100. - Cornes, R. and R. Hartley, 2005, “Asymmetric Contests With General Technologies,”Economic Theory26: 923—946. - Cowgill, Bo. 2015, “Competition and Productivity in Employee Promotion Contests,”Working Paper. - Dasgupta, P. and E. Maskin, 1986, “The existence of equilibrium in discontinuous economic games, I: Theory,”The Review of Economic Studies53: 1-26. - Frank, Robert H., 2016,Success and Luck: Good Fortune and the Myth of Meritocracy, Princeton: Princeton University Press. - Friedersdorf, 2012, “The Cult of Smartness: How Meritocracy Is Failing America,” Atlantic Monthly, June 14, 2012, available at: www.theatlantic.com/politics/archive/2012/06/the-cult-of-smartness-how-meritocracy-is-failing-america/258492/ - Fu, Qiang, and Jingfeng Lu, 2012, “MicroFoundations of Multi-Prize Lottery Contests: A Perspective of Noisy Performance Ranking,”Social Choice and Welfare38: 497-517. - Fu, Qiang, Qian Jiao, and Jingfeng Lu, 2015, “Contests with endogenous entry,” International Journal of Game Theory44: 387—424. - Guesnerie, R. and J.J. Laffont, 1984, “A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm,”Journal of public Economics25: 329-369. - Hayes, C., 2012,Twilight of the Elites: America After Meritocracy,NewYork: Crown Publishing. - Holmstrom, Bengt, and P. Milgrom, 1991, “Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design,”Journal of Law, Economics, & Organization7: 24-52. - Hirshleifer, J., 1989, “Conflict and Rent-Seeking Success Functions: Ratio vs. Difference Models of Relative Success,”Public Choice63: 101-112. - Jacobsen, Erik, 2017, “A (Mostly) Brief History Of The SAT And ACT Tests,” available at:www.erikthered.com/tutor/sat-act-history.html. - Jensen, M. K., 2016, “Existence, uniqueness, and comparative statics in contests.” InEquilibrium Theory for Cournot Oligopolies and Related Games(pp. 233-244), Cham: Springer. - JROTC, 2008, “What You Need to Know About the Revised Scholastic Aptitude Test,” available at: www.jrotccollegeprep.com/parents.php?content=incl_content_gen_adv_14. - KNAW, 2013, “Is Loting (On)rechtvaardig?” available at: www.knaw.nl/nl/actueel/agenda/is-loting-on-rechtvaardig. - Konrad, Kai, 2009,Strategy and Dynamics in Contests, Oxford: Oxford University Press. - Lazear, Edward, 1989, “Pay Equality and Industrial Politics,”Journal of Political Economy97: 561-80. - Lazear, Edward and Sherwin Rosen, 1981, “Rank-Order Tournaments as Optimal Labor Contracts.”JournalofPoliticalEconomy89: 841-864. - List, Benjamin, 2017. “Crowd-based peer review can be good and fast.”Nature News, 546 (7656): 9. - Lohr, Steve, 2012. “The age of big data.”New York Times, February 11, 2012, available at:www.nytimes.com/2012/02/12/sunday-review/big-datas-impact-in-the-world.html - Lohr, Steve, 2014, “Unblinking Eyes Track Employees: Workplace Surveillance Sees Good and Bad,”New York Times, June 21, 2014, available at: www.nytimes.com/2014/06/22/technology/workplace-surveillance-sees-good-and-bad.html?_r=0 - May, Theresa, 2017, Speech in Nottinghamshire on April 6 to Launch the Conservative Campaign Ahead of Local Elections in May 2017. available at: www.politicshome.com/news/uk/political-parties/conservative-party/theresa-may/news/84903/full-theresa-mays-speech - Moldovanu, B. and A. Sela, 2001, “The Optimal Allocation of Prizes in Contests,”American Economic Review91: 542-558. - Morgan, John, Dana Sisak, and Felix Várdy, 2018, “The Ponds Dilemma,”The Economic Journal128: 1634-1682. - Myerson, R. B., 1981, “Optimal Auction Design,”Mathematics of Operations Research,6(1), 58-73. - O’Keeffe, W. Kip Viscusi, and Richard J. Zeckhauser, 1984, “Economic Contests: Comparative Reward Schemes,”Journal of Labor Economics2: 27-56. - PISA, 2016, “PISA 2015: Results in Focus,” available at: http://www.oecd.org/pisa/pisa-2015-results-in-focus.pdf. - Olszewski, W., and Siegel, R., 2016, “Large contests,”Econometrica84(2): 835-854. - Olszewski, W., and Siegel, R., 2018, “Performance-Maximizing Contests,” Working Paper. - Scaggs, Alexandra, 2018, “How to Win aDebate in the Cult of Meritocracy,” Financial Times, March 18, 2018, available at: ftalphaville.ft.com/2018/03/28/1522201766000/How-to-win-a-debate-in-the-cult-of-meritocracy/ - Solomon.“Ecclesiastes” 9:11. - Strathern, Marilyn, 1997, “‘Improving ratings’: audit in the British University system,”European Review5: 305-321. - Tullock, Gordon, 1980,Towards a Theory of the Rent-seeking Society,Edward Elgar Publishing. - Wade, Laura, 2010, “Interview: Laura Wade, playwright,”The Scotsman, 8 April. ### APPENDIX I — Online Appendix: Proofs ### Baseline Model — Setup and Lemma 1 (properties of (·)) - Part 1: (−∞)=(0) = 0. The claim that 0(−∞)=0 follows from the fact that, for ∈[−∞∞), ()= ∫ −∞ 0()d is finite. - Part 2: 0()=d()/d = d(e)/de · de/d = d()/d |=e · e. For =ln−∞, we have 0 and 0()0. Hence, 0()0. Next, 00()=0()+200(). Because 0()0 and 00()≥0 by assumption, 00()0. - Part 3: d[00()/0()]/d = [ d[00()/0()]/d + 00()/0()] e. Therefore d[00()/0()]/d≥0 (so 0() is weakly log-convex) iff  d[00()/0()]/d ≥ − 00()/0(). This condition is equivalent to 00()=0 or d[00()/0()]/d = [00()/0()]/ ≥ −1. The inequality holds by assumption. Log-convexity of 0(·) implies log-convexity of (·) (Lemma 3 in Bagnoli and Bergstrom (2004)). - Part 4: Observations yield identities: 00()/0()=1+00()/0()=1+0, and 0()/()=0()/()=. Because (·) is log-convex, 1+0 = 00()/0() ≥ 0()/()=. Because (0)=0, ()=∫0  0()d () ≤ 0(), depending on whether 00(·) (=) 0. Hence  = 0()/() = 0()/() (=)  0()/0()=1. Together, these imply the result. - Part 5: Follows from straight-forward calculation. ### Equilibrium — Existence and Uniqueness (key lemmas and statements) - Lemma 7: For all , the pure best-response correspondence ˆ() is non-empty, compact-valued, and upper hemicontinuous (UHC). Moreover: - (i) there exists 0 such that for all 0, ˆ()=−∞, - (ii) lim→∞ −ˆ()=∞, - (iii) lim→−∞ −ˆ()=−∞, - (iv) lim→−∞ ˆ()=−∞. - Lemma 8: If ˆ(·) is single-valued at 0, then it is locally differentiable and ˆ0(0)1. In particular, - ˆ0(0)∈ (−∞0) if ˆ(0)0, - ˆ0(0)∈ (01) if ˆ(0)≥0. - If ˆ(0)=−∞ and single-valued, derivative is zero in a neighborhood. If ˆ(0)−∞, apply the implicit function theorem to the FOC: dˆ/d = (/2) 0 / [(/2) 0 + 00], denominator > 0 by SOC. Hence signs/bounds follow. - Lemma 9: The ratio / · (−)/0() is either single-peaked or strictly decreasing in . Furthermore, lim→∞ / · (−)/0()=0. - Proof uses log-concavity: / · (−) is log-concave in  by assumption; 1/0() is weakly log-concave by Lemma 1; product is log-concave → unimodality. For large  the ratio → 0 because numerator is density with finite variance and 0() remains bounded away from zero. - Definition: Denote the largest crossing point between / · (−) and 0() by . - Lemma 2 (proved): For fixed , ˆ() ⊂ {−∞, ()}. - Lemma 10: Either ˆ()=() for all , or there exists a unique ̄∈R such that - ˆ()=() for  ̄, - ˆ()={−∞, ()} for = ̄, - ˆ()=−∞ for  ̄. - Proof uses Lemma 2 and envelope theorem showing d/d [(),] = − (/) ( (−())/ ) 0, so at most one crossing ̄ can equate (−∞,) and ((),). - Definitions for Ω() and inverse Ω−1(): - Ω() = set of masses of winners ω that can arise when agents symmetrically best respond to standard : Ω() ≡ {ω: ∃ˆΓ() st. [,ˆ()] = ω }. - Ω−1(ω) ≡ {∈R: ∃ˆΓ() st. [,ˆ()] = ω }. - Lemma 11: Properties of Ω() 1. If ̄ does not exist, then Ω() is single-valued, differentiable, and strictly decreasing for all ∈R. 2. If ̄ exists then: - i) Ω() is single-valued, differentiable, and strictly decreasing for all  ̄; - ii) Ω(̄) = [0, lim↑ ̄ Ω()]; - iii) Ω()=0 for  ̄. 3. lim→−∞ Ω()=1 and lim→∞ Ω()=0. - Proof sketches: - If ̄ does not exist then ˆ()=() for all , so Ω()= ̄[(−())/], differentiable and strictly decreasing since dΩ/d = −(1− d/d)(1/)  0 by Lemma 8. - If ̄ exists apply Lemma 10: for  ̄ same argument; for  ̄, ˆ()=−∞ so Ω()=0; for = ̄ the set of masses is a closed interval [0, ̄((̄−(̄))/)] = [0, lim↑̄ Ω()]. *Source: wp18231 - References (PDF chapter/section).* ### Part 3.From the second part of Lemma 7 we know thatlim ### wp18231 - Part 3.From the second part of Lemma 7 we know thatlim ### Existence and uniqueness of symmetric equilibrium - From Lemma 11 and the properties of Ω(·): - lim_{α→∞} α−x̂(α)=∞ and lim_{α→−∞} α−x̂(α)=−∞. - Hence lim_{α→∞} Ω(α)=0 and lim_{α→−∞} Ω(α)=1. - Lemma 12: There exists a unique symmetric equilibrium. The equilibrium standard, α^*, is unique. - Proof outline: - Lemma 11 ⇒ Ω^{-1}(μ) exists and is single-valued for μ∈(0,1). Thus α^*=Ω^{-1}(μ). - If α^*< ᾱ or ᾱ does not exist, then x̂(α^*)=x_s(α^*) and Ω(α^*)=μ, so (α^*, x_s(α^*)) is the unique symmetric equilibrium. - If α^*=ᾱ, mixing occurs at boundary outputs: contestants mix between x_s(ᾱ) and −∞ with probabilities θ̂^*∈(0,1] and 1−θ̂^*, where market clearing requires θ̂^̄Φ[(ᾱ−x_s(ᾱ))/σ]=μ. Uniqueness of θ̂^* follows from monotonicity and the intermediate value theorem. ### Shape of equilibrium strategies and limits as σ→0 - Lemma 13: For any sequence α_σ, lim_{σ→0} (α_σ − x_s(α_σ;σ)) = 0. - Reason: 1/σ φ(·/σ) → Dirac at 0, and x_s is intersection between χ'_0(·) and v/σ φ((α−x)/σ). - Lemma 14: As σ→0, the equilibrium standard α^*_σ remains bounded (both above and below). - Claim 1 (bounded above): If α^*_σ →∞ then participation constraint violated since payoff from x_s would approach −χ(·)<0. - Claim 2 (bounded below): If α^*_σ →−∞ then payoffs converge to v and Ω(α^*_σ)→1>μ, contradiction. - Bolzano-Weierstrass ⇒ subsequential convergence; denote limit α^*_{σ→0}. - Lemma 15: lim_{σ→0} ̄Φ[(α^*_σ − x_s(α^*_σ))/σ]=1 (probability a non-drop-out wins →1). - Proof by contradiction using profitable deviation x_σ = α^*_σ + ϵ√σ. - Lemma 16: For σ sufficiently large, x̂(α^*) = x_s(α^*). (No dropout in equilibrium.) - Using Lemma 15 and the FOC: χ'(x_s(α^*)) = (v/σ) φ((α^*−x_s)/σ) →0 as σ→∞ ⇒ x_s(α^*)→−∞; payoff→μv>0 ⇒ dropping out not best response. - Lemma 17: For σ sufficiently small, x̂(α^*) = {−∞, x_s(α^*)}; both are played with strictly positive probabilities. Equilibrium payoffs are zero. - When σ small, ̄Φ[(α^*−x_s)/σ]→1 so everyone playing x_s with prob 1 is inconsistent with μ<1; thus mixing and zero-profit. ### Comparative statics in σ and mixing probability - Lemma 18: - (i) If x̂(α^*)=x_s(α^*), then (d x_s(α^*) / dσ)/σ = −1 / (χ''/χ'). - Sign: <0. - (ii) If x̂(α^*)={−∞, x_s(α^*)} then - (d x_s(α^*) / dσ)/σ = −1/(−σ_oχ) <0, - (d θ̂^* / θ̂^*)/dσ = χ'/(χ − σ_oχ) >0, - lim_{σ→0} θ̂^* = μ. - Lemma 19 (single-crossing in σ): There exists a unique σ^>0 such that - x̂(α^*)={−∞, x_s(α^*)} iff σ ≤ σ^, - x̂(α^*)=x_s(α^*) iff σ>σ^. - Intuition: profits strictly increase in σ when all participate, preventing return to zero profits; combined with Lemmas 16 and 17. ### Closed-form threshold σ and organizer's objectives - Lemma 20: The threshold σ satisfies - σ = v φ[ ̄Φ^{-1}(μ) ] / χ'[ χ^{-1}(μ v) ]. - Derivation: At σ where all participate with zero profit and FOC holds; use ̄Φ^{-1}(μ) and χ^{-1}(μ v). - Limits of Meritocracy (proof of Lemma 4 summary): - If σ>σ^: x̂(α^*)=x_s(α^*), all participate ⇒ O^* = e^{x_s}; elasticity (d O^*/O^*)/(dσ/σ) = −1/(χ''/χ') = −1×(C'_0 + 1). - If σ<σ^: x̂(α^*) mixes, O^* = θ̂^* e^{x_s}; elasticity equals χ'/(χ − 1) − σ_oχ factor as given (preserved algebraic expression in source). ### Multiple-prize and heterogeneous contestants (key results) - Multiple prize levels (Proposition 3): The optimal single-prize-level contest is first-best for the organizer at σ=σ^: - 1) Organizer appropriates all rents (zero profits for contestants). - 2) Production is efficient because all contestants produce same output x_s(α^*)>−∞ and C is strictly convex. - Heterogeneous contestants: - Lemma 21: Properties of C(·, i) preserved per-type: C(−∞, i)=0, derivatives positive; C_x, C_{xx}>0; C_i, C_{x i}<0; weak log-convexity statements; multiplicative separability special case. - Lemma 22–26: Existence and uniqueness of equilibrium with heterogeneous contestants: - For all (α, i), x̂(α,i) non-empty, compact-valued, UHC; limits α−x̂→±∞ as α→±∞. - Ω(α) single-valued, differentiable, strictly decreasing, with lim_{α→∞}Ω(α)=0 and lim_{α→−∞}Ω(α)=1. - Therefore equilibrium exists and is unique (α^* solves Ω(α)=μ). - Lemma 27 (single-crossing in ability): If type i_0 participates, then all i>i_0 participate; if i_0 drops out, all i1−μ exert arbitrarily more effort than the marginal participant; aggregate output at σ→0 can exceed the perfectly meritocratic benchmark, so perfect meritocracy is suboptimal. - Conclusion (Proposition 2): For sufficiently heterogeneous contests, perfect meritocracy is suboptimal. ### Attrition, heterogeneity, and limits - Lemmas 30–33: Detailed formulas (semi-elasticities) for d x_s / dσ, d i / dσ, and components of the elasticity decomposition: - Direct attrition effect Ã_­dir and indirect attrition Ã_ind expressions (see eqns (38)–(39)). - Lemma 31: As ability distribution becomes degenerate (types concentrate), direct and indirect attrition effects converge to their homogeneous counterparts; explicit limiting values involve  and σ_oχ. - Lemma 32: Aggregate attrition effect Ã>0 if boundary type exists; its limit matches homogeneous attrition effect under degeneracy. - Lemma 33: Heterogeneity effect Ĥ strictly positive; Ĥ→0 under degenerate ability distribution. ### Discrete-contest results (finite players) - 2-Player LR contests: - Lemma 34: If ρ<ρ̄ then in all symmetric mixed-strategy equilibria (MSEs), supports are unbounded below. - Lemma 35: In any symmetric MSE with unbounded lower support, players’ equilibrium payoffs are zero. - Lemma 5 (summary): - (i) For all ρ>0, a symmetric equilibrium exists. - (ii) A unique PSE exists iff ρ≥ρ̄; it is symmetric with x^_ρ = χ'^{-1}[ v/ρ · h(0) ]; x^_ρ increases in meritocracy. - (iii) If ρ<ρ̄ only MSEs exist and payoffs are zero in symmetric MSEs. - (iv) Aggregate output comparisons to 2 x̄ given convexity/linearity conditions on C. - Proposition 1.LR: Among PSEs, output is uniquely maximized at ρ=ρ̄ with aggregate output O^*=2 x̄; for C''(·)>0 this is unique maximizer. - n-Player Tullock contests: - Lemma 6: EVTIM noise yields closed-form ranking probabilities; probability player i ranked first equals X_i^r / Σ_j X_j^r. - Lemma 36: PSE of an n-player Tullock contest is unique. - Proposition 1.T: PSE characterized; zero-profit and FOC conditions yield explicit threshold r̄: - r̄ = [χ'(χ^{-1}(v/ν)) / χ(χ^{-1}(v/ν))] · (1−1/ν)^{−1} (as algebraically given in source). ### Allowing concave costs and relaxed regularity (Appendix II) - Model relaxation: - Drop assumption C''(·)≥0 and allow C''(·) possibly <0; require analyticity, C(0)=0, C'(X)>0 for X>0. - Require output elasticity of marginal costs satisfies: - ×_C0 ≡ d ln C'(X) / d ln X > −1. (Preserves that raising standard reduces winners.) - Notation and equilibrium concept extended to best-response distributions H_i(·; α) with supports in x̂(α). Symmetric profiles H_sym(·; α) average over types. - Proposition 7: Equilibrium exists for all σ; for σ sufficiently small equilibrium is unique and x̂(α^*)={−∞, x_s(α^*)}. - Proposition 8: Whether organizer should aim for perfect meritocracy (σ→0) depends on convexity vs concavity of C(·): - Concave costs (C''<0) lead to different optimality conclusions (specifically: perfect meritocracy can be uniquely output maximizing and Pareto efficient under strict concavity). - (Proposition 8 is stated but its detailed statement and proof follow beyond the supplied excerpt.) _Italic: Source — IMF Working Paper wp18231 (Appendix I & II material provided in supplied content)._ ### 1. For ### 1. For ### Main results on optimality and efficiency - If''(·)0, perfect meritocracy is neither output maximizing nor Pareto efficient. - If''(·)=0, then all∈(0] are output maximizing. - If''(·)0, only perfect meritocracy is output maximizing. It is also Pareto efficient. - Relaxing the assumption   0 / 0 ≥−1 leaves conclusions essentially unchanged: - For strictly convex costs, perfect meritocracy is suboptimal. - For linear costs, any≤ is optimal. - For strictly concave costs, perfect meritocracy is uniquely output maximizing; decreasing marginal costs (economies of scale) make output loss from drop-outs smaller than output gain from remaining contestants working harder, implying the contest organizer should fully eliminate noise in performance ranking. Under economies of scale the optimal contest is winner-take-all: in the abstracted optimal contest, a minimum number of contestants produce maximum output under perfect meritocracy — i.e.,→0,→∞,and→0. ### Properties of the cost-transform function (·) (Lemma 37) - 1) (−∞)= 0 (−∞)=0. - 2)  0 (·)0,  00 (·)0, for −∞. - 3) Convexity (concavity) of (·) is equivalent to: - ∀0:  00 () () 0 ⇐⇒ ∀−∞:  0 ()() () 1 ⇐⇒ ∀−∞:  00 () 0 () () 1. - Key identities used: -  0 ()()= 0 (). (49) - For functions through the origin: ∀0:  00 () () 0 ⇐⇒ ∀0:  0 ()() () 1. (50) -  00 () 0 ()=1+ 00 () 0 (). (51) ### Best-response correspondence ˆ() (Lemma 38) - 1. If ˆ() is single-valued at = 0 , then ˆ() is single-valued and differentiable in a neighborhood of  0 . - 2. If ˆ() is multi-valued at = 0 , then ˆ() is single-valued and differentiable in a left and right neighborhood of  0 . Furthermore, - lim ↑ 0 ˆ()=maxˆ( 0 ) - lim ↓ 0 ˆ()=minˆ( 0 ). - Key proof components: - Use of implicit function theorem (IFT) when ˆ( 0 )−∞ and single-valued. - Berge’s maximum theorem and continuity of [ˆ()] when ˆ( 0 )=−∞. - For multi-valued case, show payoff from playing minˆ( 0 ) rises strictly faster than from any other element of ˆ( 0 ) using derivative comparisons and  00 (·)0. - Envelope theorem and re-optimization arguments to restrict possible best responses in neighborhoods. ### Existence and uniqueness of inverse mapping Ω - Lemma 39: For all ∈(01), Ω −1 (·) is non-empty and single-valued. - lim →∞ −ˆ()=∞ and lim →−∞ −ˆ()=−∞. - lim →∞ Ω()=0 and lim →−∞ Ω()=1, so inf ∈R Ω()=0 and sup ∈R Ω()=1. - Ω(·) is strictly decreasing: when ˆ() single-valued, - dΩ() d =−·[1−dˆ()/d] 0, using Lemma 8 where dˆ()/d1. - Lemma 40: Equilibrium exists. The equilibrium standard is unique. ### Equilibrium multiplicity and uniqueness for small  (Lemmas 41–42) - Lemma 41 (fixed ): For  sufficiently small, the marginal cost  0 () intersects marginal revenue    ¡ −  ¢ exactly twice: - first from above at   , - then from below at   . - As →0:   ↑ and   ↓. - Proof uses: as →0,    converges to Dirac mass at zero, lim →0   (0)=∞, and tail behavior to rule out additional intersections. - Technical condition (52): derivative sign condition equivalent to violation of soc in (4). - Corollary 2: For fixed  and  sufficiently small, ˆ()⊂{−∞  ()}. - Lemma 42: For  sufficiently small, equilibrium is unique and ˆ( ∗ )={−∞  ( ∗ )}. Both elements of ˆ( ∗ ) are played with strictly positive probability. - There exists a unique mixture (1−ˆ ∗ ˆ ∗ ), ˆ ∗ ∈(01) that clears the market; contestants mix between   ( ∗ ) and −∞ with probabilities ˆ ∗ and 1−ˆ ∗ . ### Comparative statics: effect of noise parameter  on output (Lemma 43 and Proposition 8) - Lemma 43 (for 0 sufficiently small): - d d =  0 −1 1   0 + 00  0 . - Proof of Proposition 8: - From Lemma 43 and SOC, for small : - d(d)  0 ⇐⇒  0   1. - From Lemma 37,  0   =  1 ⇐⇒  00  =  0. Hence: - d(d)  0 ⇐⇒  00  0. - Because ˆ( ∗  ;)={−∞  ( ∗  ;)} for small , we know  ∗ =0. - Parts 1 and 3 follow from these observations (signs of  00 ). Part 2 follows by observing that for  00 =0 the baseline model conditions are satisfied and the relevant result from Theorem 1 carries over unchanged. *Appendix II: proofs and lemmas supporting equilibrium characterization and comparative statics (from wp18231 - 1. For).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18231.pdf_