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Network Creation Games with Traceroute-Based Strategies^{ †}

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Standard Model for NCGs

#### 1.2. Other Models for NCGs

#### 1.3. Our New Local-View Models for NCGs

- (${\mathcal{M}}_{1}$)
- Distance vector: in addition to his/her incident edges, each player u knows only the distances in $G\left(\sigma \right)$ between u and all the agents (this is the minimal knowledge needed by a player in order to compute her current cost).
- (${\mathcal{M}}_{2}$)
- Shortest-Path Tree (SPT) view: each player u knows the edges of some SPT of $G\left(\sigma \right)$ rooted at u.
- (${\mathcal{M}}_{3}$)
- Layered view: each player u knows the set of edges belonging to at least one SPT of $G\left(\sigma \right)$ rooted at u.

## 2. Convergence

#### 2.1. Model ${\phantom{\rule{3.33333pt}{0ex}}\mathcal{M}}_{1}$

**Lemma**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 2.2. Models ${\phantom{\rule{3.33333pt}{0ex}}\mathcal{M}}_{2}$ and ${\mathcal{M}}_{3}$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Complexity of Computing a Best Response

#### 3.1. Model ${\phantom{\rule{3.33333pt}{0ex}}\mathcal{M}}_{1}$

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

#### 3.2. Model ${\phantom{\rule{3.33333pt}{0ex}}\mathcal{M}}_{2}$

**Theorem**

**6.**

**Proof.**

#### 3.3. Model ${\phantom{\rule{3.33333pt}{0ex}}\mathcal{M}}_{3}$

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 4. Price of Anarchy

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Cycle of best responses for MaxNCG in ${\mathcal{M}}_{2}$ for $\alpha \ge 6$. The cycle is obtained by first traversing the configurations (

**a**), (

**b**), (

**c**), and (

**d**), in order, and then traversing the configurations obtained from (

**b**), (

**c**), and (

**d**) by exchanging the roles of vertices a and b.

**Figure 2.**Cycle of best responses for SumNCG in ${\mathcal{M}}_{2}$ for $\alpha \ge 6$. The cycle traverses configurations (

**a**), (

**b**), (

**c**), (

**d**), and (

**a**), in this order.

**Figure 3.**Cycle of best responses for SumNCG in ${\mathcal{M}}_{3}$ for $\alpha =15$. The cycle is obtained by first traversing the configurations (

**a**), (

**b**), (

**c**), and (

**d**), in order, and then traversing the configurations obtained from (

**b**), (

**c**), and (

**d**) by exchanging the roles of vertices a and c.

**Figure 4.**Cycle of best responses for MaxNCG in ${\mathcal{M}}_{3}$ for $\alpha =2-\u03f5$. The cycle traverses configurations (

**a**), (

**b**), (

**c**), (

**d**), and (

**a**), in this order.

**Table 1.**Summary of our results (and open problems). In the first column, convergence (and divergence) are reported w.r.t. either improving or best-response dynamics (improving response dynamics (IRD) and best response dynamics (BRD), resp.). In the second column, we report the time complexity of selecting a best-response strategy.

Convergence | Best-Response Complexity | PoA | |
---|---|---|---|

${\mathcal{M}}_{1}$ | Sum: Yes (∀ improving response dynamics (IRD)) Max: Yes (∀ IRD) | Sum: Open Max: Polynomial | Sum: $\Theta (min\{1+\alpha ,n\left\}\right)$ Max: $\Theta \left(n\right)$ if $\alpha =\Omega \left(1\right)$ $\Theta (1+\alpha n)$ if $\alpha =O\left(1\right)$ |

${\mathcal{M}}_{2}$ | Sum: No (∃ best response dynamics (BRD) cycle) Max: No (∃ BRD cycle) | Sum: Polynomial Max: Polynomial | Sum: $\Theta (min\{1+\alpha ,n\left\}\right)$ Max: $\Theta \left(n\right)$ if $\alpha =\Omega \left(1\right)$ $\Theta (1+\alpha n)$ if $\alpha =O\left(1\right)$ |

${\mathcal{M}}_{3}$ | Sum: No (∃ BRD cycle) Max: No (∃ BRD cycle) | Sum: NP-hard Max: NP-hard | Sum: $\Theta (min\{1+\alpha ,n\left\}\right)$ Max: $\Theta \left(n\right)$ if $\alpha =\Omega \left(1\right)$ $\Theta (1+\alpha n)$ if $\frac{1}{n-1}\le \alpha =O\left(1\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Bilò, D.; Gualà, L.; Leucci, S.; Proietti, G.
Network Creation Games with Traceroute-Based Strategies. *Algorithms* **2021**, *14*, 35.
https://doi.org/10.3390/a14020035

**AMA Style**

Bilò D, Gualà L, Leucci S, Proietti G.
Network Creation Games with Traceroute-Based Strategies. *Algorithms*. 2021; 14(2):35.
https://doi.org/10.3390/a14020035

**Chicago/Turabian Style**

Bilò, Davide, Luciano Gualà, Stefano Leucci, and Guido Proietti.
2021. "Network Creation Games with Traceroute-Based Strategies" *Algorithms* 14, no. 2: 35.
https://doi.org/10.3390/a14020035