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Single- and Double-Elimination Tournaments under Psychological Momentum

  • Bo Chen ORCID logo , Xiandeng Jiang EMAIL logo and Zijia Wang
Published/Copyright: October 25, 2021
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 22 Issue 2

Abstract

This paper studies the effects of “psychological momentum” on strategic behavior in single- and double-elimination tournaments. We show that in presence of both positive and negative momentum a single-elimination tournament elicits a higher total effort than that of a double-elimination tournament if and only if the positive momentum is insignificant and the negative momentum is significant. Regardless of momentum effects, a single-elimination tournament elicits a higher average effort per match than a double-elimination tournament does.

Keywords: elimination tournament; psychological momentum; tournament design
JEL Classifications: C7; D4; D7; D9
Corresponding author: Xiandeng Jiang, School of Public Finance and Taxation, Southwestern University of Finance and Economics, Chengdu, China, E-mail:

Funding source: National Natural Science Foundation of China; Department of Education of Guangdong Province; Ministry of Education of the People’s Republic of China; Renmin University of China

Award Identifier / Grant number: 71973040

Award Identifier / Grant number: 20YJC790051

Award Identifier / Grant number: 2021WQNCX085

Award Identifier / Grant number: 72104204

Acknowledgment

We are grateful to three anonymous referees for valuable comments which improved the manuscript substantially.

  1. Research funding: Chen acknowledges the financial support from the National Natural Science Foundation of China (No. 71973040) and the Department of Education of Guangdong Province (No. 2021WQNCX085). Jiang acknowledges the financial support from the National Natural Science Foundation of China (No. 72104204) and the Ministry of Education of China (No. 20YJC790051). Wang acknowledges the support of the Digital Economy Platform, Major Innovation & Planning Interdisciplinary Platform for the “Double-First Class” Initiative, and the Research Funds of Renmin University of China (the Fundamental Research Funds for Central Universities).

Appendix

Proof of Proposition 3.2

It is straightforward that TESE − TEsDE < 0 and TESE − TEvDE < 0 when α ≥ 2. Hence, we only analyze the case in which α < 2 below.

Consider TESE − TEsDE. First, (α 4 − 2α 3)λ 2 + (5α 2 − 3α)λ + 2 is a quadratic function with respect to λ, whose value is strictly positive for λ ∈ [0, 1] and goes to be −∞ as λ approaches +∞. Thus, there is a unique value λ ̂ α s > 1 such that TESE − TEsDE > 0 if and only if λ > λ ̂ α s .

Consider TESE − TEvDE. We need to analyze G(λ). First, the third order derivative is G‴(λ) = 60(α 10 − 2α 9)λ 2 + 24(11α 8 − 8α 7)λ + 3(22α 6 − 14α 5), with G‴(0) > 0 and lim λ→+∞  G‴(λ) < 0. Because G‴ is a quadratic function, there is a cutoff λ 3 > 1 such that G‴(λ) < 0 iff λ > λ 3. Hence, the second order derivative G″ firstly increases and then decreases in λ for λ > 0.

Second, G″(0) > 0 and lim λ→+∞  G″(λ) < 0. Due to the property of G‴, there is a cutoff λ 2 > 0 such that G″(λ) < 0 iff λ > λ 2. Hence, G′ firstly increases and then decreases in λ for λ > 0.

Third, G′(0) > 0 and lim λ→+∞  G′(λ) > 0. Due to the property of G″, there is a cutoff λ 1 > 0 such that G′(λ) < 0 iff λ > λ 1. Hence, the first order derivative G firstly increases and then decreases in λ for λ > 0.

Last, G(1) > 0 for all λ ∈ (0, 1] and lim λ→+∞  G(λ) < 0. Due to the property of G′, there is a cutoff λ ̂ > 1 such that G(λ) < 0 iff  λ > λ ̂ . That is, TESE − TEvDE > 0 iff  λ > λ ̂ . □

Proof of Proposition 3.5

The result is derived by making direct comparisons as below:

AE SE − AE sDE = TE SE / 3 − AE sDE / 6 = 3 α 3 β 2 + 4 α 2 β 2 + 3 α 2 β + 7 α β + 2 α + 2 48 ( α β + 1 ) 2 > 0 ; AE sDE − AE vDE = AE sDE / 6 − TE vDE / [ 6 + e h / ( e WW + e h ) ] = 5 α 6 β 6 + 4 α 5 β 5 − 35 α 4 β 5 + 132 α 4 β 4 − 115 α 3 β 4 + 254 α 3 β 3 − 147 α 2 β 3 + 235 α 2 β 2 − 113 α β 2 + 114 α β − 34 β + 24 ⋅ α 48 ( α β + 1 ) 5 ( 13 α β + 12 ) > 5 α 5 β 4 ( β 2 − 7 β + 10 ) 48 ( α β + 1 ) 5 ( 13 α β + 12 ) > 0 .

□

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Received: 2019-12-21
Accepted: 2021-07-02
Published Online: 2021-10-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Skepticism and Credulity: A Model and Applications to Political Spin, Belief Formation, and Decision Weights
  4. Expert Panels with Selective Investigation
  5. Self-Control Preferences and Status-Quo Bias
  6. Passive Cross-Holding in a Stackelberg Oligopoly
  7. A Rehabilitation of the Law of Diminishing Marginal Utility: An Ordinal Marginal Utility Approach
  8. Workplace Heterogeneity and the Returns to Versatility
  9. Single- and Double-Elimination Tournaments under Psychological Momentum
  10. Cheap Talk with Multiple Experts and Uncertain Biases
  11. Legal Environment and Contractual Choice
  12. Politically Connected Firms and the Environment
  13. Injurers versus Victims: (A)Symmetric Reactions to Symmetric Risks
  14. Notes
  15. On an “Important Principle” of Arrow and Debreu
  16. A Note on the Existence of the Competitive Equilibrium in Grossman and Shapiro (1984)
  17. Endogenous Expectations Management with Network Effects: A Note
https://doi.org/10.1515/bejte-2019-0187
Keywords for this article
elimination tournament; psychological momentum; tournament design

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Skepticism and Credulity: A Model and Applications to Political Spin, Belief Formation, and Decision Weights
  4. Expert Panels with Selective Investigation
  5. Self-Control Preferences and Status-Quo Bias
  6. Passive Cross-Holding in a Stackelberg Oligopoly
  7. A Rehabilitation of the Law of Diminishing Marginal Utility: An Ordinal Marginal Utility Approach
  8. Workplace Heterogeneity and the Returns to Versatility
  9. Single- and Double-Elimination Tournaments under Psychological Momentum
  10. Cheap Talk with Multiple Experts and Uncertain Biases
  11. Legal Environment and Contractual Choice
  12. Politically Connected Firms and the Environment
  13. Injurers versus Victims: (A)Symmetric Reactions to Symmetric Risks
  14. Notes
  15. On an “Important Principle” of Arrow and Debreu
  16. A Note on the Existence of the Competitive Equilibrium in Grossman and Shapiro (1984)
  17. Endogenous Expectations Management with Network Effects: A Note
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