# Instability of Mixed Nash Equilibria in Generalised Hawk-Dove Game: A Project Conflict Management Scenario

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## Abstract

**:**

## 1. Introduction

- The divergence of interests may induce conflict, which in return can lead to greater risk and lengthened project duration.
- The participants’ behaviour is often tacit, requiring clarifications which may be obtained through rigorous modelling.

## 2. Evolutionary Game Theory: Generalised Hawk-Dove Model

- The owner has a higher power and assumes the governance of the project.
- The owner pays an agreed amount to the contractor.

- A hawk owner does not incur an additional cost but a hawk contractor would. The cost of negotiation varies with the attitude of the two parties.
- Owner’s payoff is represented as a (negative) “loss” (constituting the payment amount), while the contractor payoff, conversely, is positive and includes the received payment amount.

## 3. Analytical Solutions

## 4. Results

#### 4.1. Mixed Strategy Equilibrium under Stationary Conditions

#### 4.1.1. $\beta =1$: Classical Hawk-Dove Model

- The increase of $\alpha $ encourages the owner to act as a hawk while suppressing the contractor to act as a hawk.
- The higher the value of $\alpha $, the more likely is the transition from $({A}_{2},{B}_{1})$ with small $\alpha $ to $({A}_{1},{B}_{2})$ with large $\alpha $.

#### 4.1.2. Other Values of $\beta <1$

- p* reaches 1 when $\beta $ approaches 1 with $\alpha $ at minima.
- With large $\alpha $, p* is close to 0 regardless of which value $\beta $ takes.
- q* approaches 1 when both $\beta $ and $\alpha $ move towards 1, indicating that q* is positively related to both $\beta $ and $\alpha $.
- Sharing ratios do impact on both p* and q* but the manipulation of $\alpha $ and $\beta $ cannot create scenario $({A}_{1},{B}_{1})$ with high p* and q*: the former requires small $\alpha $ and the latter requires greater $\alpha $, where $\alpha $ clearly cannot meet both requirements.

#### 4.2. Mixed Strategy Equilibrium under Noisy and Latent Conditions

#### 4.2.1. Stability of Nash Equilibria

#### 4.2.2. Noise and Delay

- Pure Nash equilibria retain their robustness, and the combination of $\alpha $ and $\beta $ do not affect the resultant expected probability.
- The addition of noise and delay postpones but does not challenge the convergence to pure Nash equilibria.
- Noise creates noticeable oscillations but does not have a lasting impact once it is removed (in this case, after 200 days).
- The mixed Nash equilibrium, (p*, q*), is not attained.
- The combination of $\alpha $ and $\beta $, as shown earlier, determines whether the expected probability converges to 0 or 1.

## 5. Application in Context of Project Management

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Expected probability of contractor p* and owner q*. Ten equally spaced $\alpha $ between 0.48 and 0.9.

**Figure 2.**Expected payment of contractor ${U}_{B}$ and owner ${U}_{A}$. (

**a**) payment with ${p}^{*}$ and ${q}^{*}$; (

**b**) payment with $\alpha $.

**Figure 4.**Expected payment ${U}_{A}$ and ${U}_{B}$ as a function of expected probability p* and q*. (

**a**) 3d view; (

**b**) 2d view.

**Figure 5.**Expected payment ${U}_{A}$ and ${U}_{B}$ as a function of sharing ratios $\alpha $ and $\beta $.

**Figure 6.**Original simulated expected probability. (

**a**) $\beta $ = 1 and varying $\alpha $; (

**b**) varying $\beta $ and $\alpha $.

**Figure 8.**Phase portrait: mixed combination. (

**a**) $\alpha $ = 0.48 $\beta $ = 1; (

**b**) $\alpha $ = 0.5 $\beta $ = 0.5; (

**c**) $\alpha $ = 0.7 $\beta $ = 0.5; (

**d**) $\alpha $ = 0.9 $\beta $ = 0.1.

**Figure 9.**Expected probability with varying $\alpha $ and $\beta $. (

**a**) $\beta =1$; (

**b**) varying $\alpha $ and $\beta $.

**Figure 10.**Phase portrait: delay and noise under different initial conditions. (

**top left**) $({A}_{1},{B}_{2})$; (

**top right**) $({A}_{1},{B}_{1})$; (

**bottom left**) $({A}_{2},{B}_{2})$; (

**bottom right**) $({A}_{2},{B}_{1})$.

Player 2 | Hawk | Dove | |
---|---|---|---|

Player 1 | |||

Hawk | $\left({\displaystyle \frac{v-c}{2}},{\displaystyle \frac{v-c}{2}}\right)$ | $(v,0)$ | |

Dove | $(0,v)$ | $\left({\displaystyle \frac{v}{2}},{\displaystyle \frac{v}{2}}\right)$ |

Project Owner | Hawk ${\mathit{A}}_{1}$ | Dove ${\mathit{A}}_{2}$ | |
---|---|---|---|

Project Contractor | |||

Hawk ${B}_{1}$ | $\left({\displaystyle \frac{v-c}{2}},-{\displaystyle \frac{v-c}{2}}\right)$ | $(v,-v)$ | |

Dove ${B}_{2}$ | $(v,-v)$ | $\left({\displaystyle \frac{v}{2}},-{\displaystyle \frac{v}{2}}\right)$ |

Project Owner | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | |
---|---|---|---|

Project Contractor | |||

${B}_{1}$ | $({V}_{1}-(1-\alpha ){L}_{1},-{V}_{1}-\alpha {L}_{1})$ | $({D}_{1}-(1-\beta ){L}_{2},-{D}_{1}-\beta {L}_{2})$ | |

${B}_{2}$ | $({V}_{2},-{V}_{2})$ | $({D}_{2},-{D}_{2})$ |

Parameter | ${V}_{1}$ | ${L}_{1}$ | ${D}_{1}$ |

Value | 2600 | 1050 | 2950 |

Parameter | ${L}_{2}$ | ${V}_{2}$ | ${D}_{2}$ |

Value | 150 | 2500 | 2550 |

Project Owner | Hawk $\left({\mathit{A}}_{1}\right)$ | Dove $\left({\mathit{A}}_{2}\right)$ | |
---|---|---|---|

Project Contractor | |||

$Hawk\left({B}_{1}\right)$ | $(2600-1050(1-\alpha ),-2600-1050\alpha )$ | $(2950-150(1-\beta ),-2950-150\beta )$ | |

$Dove\left({B}_{2}\right)$ | $(2500,-2500)$ | $(2550,-2550)$ |

Best Scenario | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{U}}_{\mathit{A}}$ | ${\mathit{U}}_{\mathit{B}}$ |
---|---|---|---|---|

Owner | 0.65–0.9 | 0–0.8 | −2600 | 2505 |

Contractor | 0.35 | 0 | −2950 | 2535 |

Tested Combination | p* | q* |
---|---|---|

$\alpha $ = 0.48 $\beta $ = 1 | 0.926 | 0.473 |

$\alpha $ = 0.60 $\beta $ = 1 | 0.278 | 0.556 |

$\alpha $ = 0.80 $\beta $ = 1 | 0.128 | 0.784 |

$\alpha $ = 0.90 $\beta $ = 1 | 0.101 | 0.987 |

$\alpha $ = 0.34 $\beta $ = 0 | 0.877 | 0.297 |

$\alpha $ = 0.38 $\beta $ = 0.3 | 0.926 | 0.349 |

$\alpha $ = 0.42 $\beta $ = 0.6 | 0.980 | 0.400 |

Tested Combination | p* | q* |
---|---|---|

$\alpha $ = 0.48 $\beta $ = 1 | 0.926 | 0.473 |

$\alpha $ = 0.5 $\beta $ = 0.5 | 0.3333 | 0.4333 |

$\alpha $ = 0.7 $\beta $ = 0.5 | 0.1388 | 0.6019 |

$\alpha $ = 0.9 $\beta $ = 0.1 | 0.0794 | 0.9815 |

© 2017 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**

Le Chang, S.; Prokopenko, M. Instability of Mixed Nash Equilibria in Generalised Hawk-Dove Game: A Project Conflict Management Scenario. *Games* **2017**, *8*, 42.
https://doi.org/10.3390/g8040042

**AMA Style**

Le Chang S, Prokopenko M. Instability of Mixed Nash Equilibria in Generalised Hawk-Dove Game: A Project Conflict Management Scenario. *Games*. 2017; 8(4):42.
https://doi.org/10.3390/g8040042

**Chicago/Turabian Style**

Le Chang, Sheryl, and Mikhail Prokopenko. 2017. "Instability of Mixed Nash Equilibria in Generalised Hawk-Dove Game: A Project Conflict Management Scenario" *Games* 8, no. 4: 42.
https://doi.org/10.3390/g8040042