# Point Score Systems and Cooperative Incentives: The 3-1-0 Curse

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. A Simple Game Model

- N: Play normally
- A: Make an agreement sharing two victories—a home win for each

- $\mathcal{W}$: ⇒ $\omega $ points
- $\mathcal{D}$: ⇒ $\delta $ points
- $\mathcal{L}$: ⇒ 0 points

- A,A
- If both teams choose the A-action, they share the total amount of points after one win each. Hence, $\omega $ to both players.
- N,N
- Without any agreement (choosing N), both teams plays as usual, and by being perfect clones, will have a probability of $\frac{1}{3}$ for each outcome $\mathcal{W},\mathcal{D}$ and $\mathcal{L}$. Hence, $2\xb7\frac{1}{3}\left(\omega +\delta \right)$ to both players (Recall 2 games are played).
- A,N
- (or N,A). Now, our model is a bit too simplified. We overlook the obvious inherent sequentiality here; one team may observe that the other deviates, and hence punish back. Furthermore, there may very well be some effects of incomplete information as well, making it even more complex. Anyway, what we can establish is that the remaining pay-offs must be located within the following interval:$$\left[min\left\{\omega ,\frac{2}{3}\left(\omega +\delta \right)\right\},max\left\{\omega ,\frac{2}{3}\left(\omega +\delta \right)\right\}\right]$$

**not**collapse to a non-gaming situation.

## 3. Theoretical Discussion and Conclusions

## 4. Empirical Analysis

## 5. Discussion, Conclusions and Suggestions for Further Research

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Alternative Derivation of the ω = 2δ Result

**not**OK, as $3\ne 2\xb71$. However, the former point score system in football – (2-1-0) is OK as $2=2\xb71$.

## Appendix B. Empirical Analysis, Definitions and Results

#### Appendix B.1. Definitions

- ${\delta}_{ij}$: Number of drawn matches played by team i in league j
- ${N}_{j}$: Number of teams playing in league j
- ${\rho}_{L}^{j}$: Calculated measure for uncertainty of outcome in league j
- $A{P}_{ij}$: Actual point score for team i in league j
- $LC{P}_{ij}$: Least competitive point score for team i in league j
- $MC{P}_{J}$: Maximally competitive point score for team i in league j

#### Appendix B.2. Calculations

#### Appendix B.3. Data

#### Appendix B.4. Empirical Analysis

## References

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**Table 1.**Estimated uncertainty of outcome $\left({\rho}_{L}^{j}\right)$, and draw ratio $\left({\delta}_{j}\right)$ for some European leagues—2017.

2017 | ${\mathit{\rho}}_{\mathit{L}}^{\mathit{j}}$ | ${\mathit{\delta}}_{\mathit{j}}$ |
---|---|---|

ALBANIA | 28.0% | 32.8% |

AUSTRIA | 34.2% | 18.9% |

BELARUS | 15.1% | 25.8% |

BOSNIA & HERC. | 19.5% | 28.8% |

CROATIA | 10.1% | 23.9% |

CZECH REP. | 25.2% | 28.8% |

ENGLAND | 21.0% | 22.1% |

FINLAND | 30.3% | 27.8% |

FRANCE | 27.9% | 25.0% |

GERMANY | 30.9% | 24.2% |

GREECE | 21.4% | 27.5% |

HOLLAND | 24.8% | 23.9% |

HUNGARY | 40.3% | 27.3% |

ICELAND | 35.1% | 27.3% |

IRELAND | 25.9% | 23.2% |

ITALY | 16.4% | 22.2% |

LUXEMBOURG | 26.8% | 19.8% |

MACEDONIA | 25.6% | 31.7% |

NORWAY | 38.8% | 24.6% |

PORTUGAL | 23.6% | 26.1% |

SPAIN | 16.4% | 23.4% |

SWEDEN | 28.0% | 28.8% |

SLOVENIA | 29.6% | 30.6% |

SWITZERLAND | 28.9% | 21.1% |

TURKEY | 26.6% | 21.9% |

**Table 2.**Regression with draw ratio ($\delta $) as dependent variable and uncertainty of outcome $\left({\rho}_{L}\right)$ as independent variable.

Coefficient | Estimate | t-Value |
---|---|---|

$Constant$ | 24.35107 *** | 8.653 |

${\rho}_{L}$ | 0.04416 | 0.423 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Haugen, K.K.; Heen, K.P. Point Score Systems and Cooperative Incentives: The 3-1-0 Curse. *Sports* **2018**, *6*, 110.
https://doi.org/10.3390/sports6040110

**AMA Style**

Haugen KK, Heen KP. Point Score Systems and Cooperative Incentives: The 3-1-0 Curse. *Sports*. 2018; 6(4):110.
https://doi.org/10.3390/sports6040110

**Chicago/Turabian Style**

Haugen, Kjetil K., and Knut P. Heen. 2018. "Point Score Systems and Cooperative Incentives: The 3-1-0 Curse" *Sports* 6, no. 4: 110.
https://doi.org/10.3390/sports6040110