# Equilibrium Selection in Hawk–Dove Games

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Tracing Procedure Method

**Lemma 1.**

**Proposition 1.**

- i.
- Low demand $(0<\theta \le {\widehat{\theta}}_{h})$: Both players submit a bid equal to the minimum bid allowed by the auctioneer.
- ii.
- Intermediate demand $({\widehat{\theta}}_{h}<\theta \le {\widehat{\theta}}_{l})$: The player with the higher production capacity submits the maximum bid, and the player with the lower production capacity submits the minimum bid allowed by the auctioneer.
- iii.
- High demand $({\widehat{\theta}}_{l}<\theta \le {k}_{h}+{k}_{l})$: Both players submit a bid equal to the maximum bid.

**Proposition 2.**

**Proposition 3.**

- i.
- The parameter t for which the players deviate from the equilibrium when $t=0$ increases.
- ii.
- The parameter t for which the players coordinate in one of the Nash equilibria of the original game increases.

## 4. Robustness to Strategic Uncertainty Method

## 5. Quantal Response Method

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma A1.**

**Proposition A1.**

**Proposition A2.**

**Figure A3.**Payoff matrix in a uniform-price auction. ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =12.5$, ${b}_{min}=1$, ${b}_{max}=10$, $N=11$.

**Figure A4.**Payoff matrix in a uniform-price auction. ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{max}=10$, $N=11$.

**Proposition A3.**

## Appendix B

**Table A1.**t parameter using different methods. ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$, $N=110$, $\u03f5={\displaystyle \frac{{b}_{max}-{b}_{min}}{N}}=0.0826$.

${\mathit{b}}_{\mathit{h}}^{*}$ | ${\mathit{b}}_{\mathit{l}}^{*}$ | m | M | ${\mathit{t}}_{1}$ (Nash Equilibrium) | ${\mathit{t}}_{2}$ (Fminsearch) | ${\mathit{t}}_{3}$ (Equations (A7) and (A11)) |
---|---|---|---|---|---|---|

3.72 | 1 | 34 | 35 | 0.2330 | 0.2356 | 0.2409 |

4.63 | 1 | 45 | 46 | 0.261 | 0.2647 | 0.2679 |

5.54 | 1 | 56 | 57 | 0.2870 | 0.2901 | 0.2931 |

6.44 | 1 | 67 | 68 | 0.31 | 0.3138 | 0.3165 |

7.35 | 1 | 78 | 79 | 0.3330 | 0.3359 | 0.3385 |

8.26 | 1 | 89 | 90 | 0.3540 | 0.3566 | 0.3591 |

9.17 | 1 | 100 | 101 | 0.3740 | 0.3762 | 0.3784 |

10 | 1 | 109 | 110 | 0.39 | 0.3929 | 0.3934 |

- For h, set $\lambda $ to ${\lambda}_{i}$.
- Given ${\lambda}_{i}$, define a system of equations given by $\pi (x;{\lambda}_{i})-x=0$.
- Define optimization starting parameters$${x}_{0}=\left\{\begin{array}{ll}(1,\dots ,1)/s& \mathrm{if}\phantom{\rule{4.pt}{0ex}}i=0\\ {\gamma}^{*}\left({\lambda}_{i-1}\right)& \mathrm{otherwise}\end{array}\right.$$

## Appendix C

**Figure A6.**Tie-breaking rule 1: Suppliers dispatched in proportion to their production capacities. Parameters: ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$.

**Figure A7.**Tie-breaking rule 2: The larger supplier is dispatched first. Parameters: ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$. Tie-breaking rule 3: The smaller supplier is dispatched first. Parameters: ${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$.

**Figure A8.**Tie-breaking rule 4: The efficient supplier is dispatched first (the efficient supplier is also the larger supplier). Parameters: ${k}_{h}=8.7$, ${k}_{l}=6.5$, ${c}_{h}=0$, ${c}_{l}=0.5$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$. Tie-breaking rule 5: The efficient supplier is dispatched first (the efficient supplier is also the smaller supplier). Parameters: ${k}_{h}=8.7$, ${k}_{l}=6.5$, ${c}_{h}=0.5$, ${c}_{l}=0$, $\theta =10$, ${b}_{min}=1$, ${b}_{max}=10$.

## Notes

1 | We frame the paper as a uniform-price auction because we are interested in studying the strategic behaviour of firms in an industrial organization context. However, there are many other real examples such as trade wars, military battles, collective bargaining, and legal disputes—of which the hawk–dove game is the prototypical example. |

2 | As we have pointed out in note 1, we frame the analysis as an industrial organization problem. However, by using an experiment, Ref. [4] study a similar hawk–dove game framing the asymmetries as the relative strength of the players. In that sense, their outline is more clean, since they need a neutral framing to avoid any possible setting-induced bias in the players’ behaviour during the experiment. |

3 | The inelastic demand assumption is not a critical one, but it simplifies the analysis. When the demand is elastic, the set of strategies changes, since the maximum bid is determined by the bid that maximizes the residual demand profit function. When the demand is elastic, the rationing rule plays a crucial role in determining the profitability of the residual demand profit function, and therefore, defining the maximum bid set in the auction. Moreover, the rationing rule could also foster or hinder the coordination in one of the equilibria. A complete discussion of the role of the rationing rule determining the equilibrium in models of price competition with elastic demand can be found in [5,6]. |

4 | We provide a complete description of the uniform-price auction in the model section and in Appendix C. In Appendix C, we explain that with the tie-breaking rule implemented in the uniform-price auction, the uniform-price auction has the structure of a “hawk–dove” game as defined in [7]. In this appendix, we also discuss the relationship between the “hawk–dove” and the “Battle-of-Sexes” games and work out the equilibrium in a uniform-price auction when five different tie-breaking rules are implemented. |

5 | We believe that our assumptions are less stringent, since we are not imposing any restriction on the set of strategies that the players can select. This assumption is validated by experimental evidence [4], where the authors find that the players select strategies outside the set of Nash equilibrium strategies. Moreover, by not restricting the set of strategies, we follow the same approach as in the robustness to strategic uncertainty method [2] and quantal response method [3], making the predictions of the three theoretical methods comparable. |

6 | As we have presented in note 2, in [4], the inequalities represent the relative strength of the players. |

7 | The minimum bid in the auction (${b}_{min}$) and the maximum bid (${b}_{max}$) are determined by the auctioneer. The minimum bid guarantees a minimum profit for the players. The maximum bid represents the reservation price for the consumers of the good. |

8 | It is important to emphasize that ${q}_{h}(b;\theta ,k)$ is only valid under the assumption that $\theta \in [{k}_{l},{k}_{h}+{k}_{l}]$. When $\theta <{k}_{l}$, ${q}_{h}(b;\theta ,k)$ is slightly different, since in this case both players have enough production capacity to satisfy the entire demand and the equilibrium is unique. For a complete analysis of the uniform-price auction when the demand is low, see [17]. |

9 | The profit function in Equation (2) follows the classical profit function definition in a uniform-price auction [17]. In Section 3, Section 4 and Section 5, we define other profit functions following the definitions of [4] that we are studying. To avoid confusion, we formally define these functions when we apply them. |

10 | As with ${q}_{h}(b;\theta ,k)$, ${\pi}_{h}(b;\theta ,k)$ is slightly different when the assumptions of the model are relaxed. |

11 | There appears to be no clear consensus in the literature as to the the definition and relationship between “hawk–dove” and “Battle-of-Sexes” games. According to [15], the Battle of the Sexes game defined in [18] and the hawk–dove game defined in [19] are equivalent. However, the payoff matrix in [7] and the one in [20] show that those games are different. In this paper, we assume that a game has the structure of a hawk–dove game when it follows the structure presented in [7]. In Appendix C, we apply five different tie-breaking rules to the uniform-price auction, and we show that for the applied tie-breaking rules, the uniform-price auction has the structure of a “hawk–dove” game as defined in [7]. For a complete discussion, we refer the reader to Appendix C. |

12 | |

13 | As we have discussed in the model section, the number of bids can be arbitrary large by raising N. When the number of bids tends to infinity, the weight of the payoffs in the diagonal tend to zero and can be “neglected” when equilibrium selection techniques are applied. |

14 | We restrict our attention to the low-demand case, since when the demand is intermediate or high, an increase in the lower bid does not affect the equilibrium selected by the tracing procedure. |

15 | |

16 | In this section, we have changed slightly the original notation in [3], since in our paper we use $\overline{\pi}$ to denote players’ profits, while in their paper they denote those profits as $\overline{u}$. We made that change to help the reader compare the three methods that we are studying by having a coherent notation through the paper. It is also important to note that in the quantal response method, the subindexes i and j refer to the strategies, and not to the players. |

17 | It is important to note that our results, as are those of [3], are based on numerical simulations. In our case, the numerical simulations presented in Figure 6 are based on the two matrices introduced in Appendix B, Figure A4, which are the matrices that we have used in all the numerical examples in the paper. |

18 | |

19 | We assume that N is large. This guarantees that the first and second derivatives of $\pi $ adequately approximate the change in payoffs as players adjust their strategy slightly and the probability of a tie goes to 0, so that $\pi ({b}_{i},{p}_{j})$ does not have a term for that possibility. |

20 | |

21 | Choosing ${\lambda}_{0}$ to be roughly zero helps find good starting values for subsequent values of $\lambda $ as we have that when $\lambda =0$, each strategy is played with equal probability. |

22 | We use Matlab’s non-linear equation solver fsolve, with function and x tolerances both set to 10 × 10 ^{−10}. |

23 | An alternative termination criteria is to look at the distance between ${\gamma}^{*}\left({\lambda}_{i}\right)$ and ${\gamma}^{*}\left({\lambda}_{i-1}\right)$. In our application, we define our sequence ${\left\{{\lambda}_{i}\right\}}_{i=0}^{n}$ such that the ${\lambda}_{i}$ are increasing exponentially rather than linearly and choose therefore to terminate at a large predefined $\lambda $. |

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**Figure 5.**Players’ best response functions (${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{max}=10$, $N=110$).

**Figure 6.**Quantal response method (${k}_{h}=8.7$, ${k}_{l}=6.5$, $\theta =10$, ${b}_{max}=10$, $N=11$).

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Blázquez de Paz, M.; Koptyug, N.
Equilibrium Selection in Hawk–Dove Games. *Games* **2024**, *15*, 2.
https://doi.org/10.3390/g15010002

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Blázquez de Paz M, Koptyug N.
Equilibrium Selection in Hawk–Dove Games. *Games*. 2024; 15(1):2.
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**Chicago/Turabian Style**

Blázquez de Paz, Mario, and Nikita Koptyug.
2024. "Equilibrium Selection in Hawk–Dove Games" *Games* 15, no. 1: 2.
https://doi.org/10.3390/g15010002