# Game Theoretic Foundations of the Gately Power Measure for Directed Networks

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## Abstract

**:**

## 1. Introduction

- Relationship to the literature

- Structure of the paper

## 2. Game Theoretic Representations of Hierarchical Networks

- Notation: Representing hierarchical networks

- (i)
- The map ${s}_{D}:N\to \mathbb{N}$ counts the number of successors of a node defined by ${s}_{D}\left(i\right)=\#D\left(i\right)$ for $i\in N$;
- (ii)
- The map ${p}_{D}:N\to \mathbb{N}$ counts the number of predecessors of a node defined by ${p}_{D}\left(i\right)=\#{D}^{-1}\left(i\right)$ for $i\in N$;

- ${s}_{D}^{a}\left(i\right)=\#\left[\phantom{\rule{0.166667em}{0ex}}D\left(i\right)\cap {N}_{D}^{a}\phantom{\rule{0.166667em}{0ex}}\right]$ and ${s}_{D}^{b}\left(i\right)=\#\left[\phantom{\rule{0.166667em}{0ex}}D\left(i\right)\cap {N}_{D}^{b}\phantom{\rule{0.166667em}{0ex}}\right]$, resulting in the conclusion that ${s}_{D}\left(i\right)={s}_{D}^{a}\left(i\right)+{s}_{D}^{b}\left(i\right)$.
- From the definitions, above we conclude immediately that$$\sum _{i\in N}{s}_{D}^{a}\left(i\right)=\#{N}_{D}^{a}$$$$\sum _{i\in N}{s}_{D}^{b}\left(i\right)=\sum _{j\in {N}_{D}^{b}}{p}_{D}\left(j\right).$$

- Classes of hierarchical networks

**Definition 1.**

- (a)
- The network D is
**weakly regular**if for all nodes $i,j\in {N}_{D}^{b}:{p}_{D}\left(i\right)={p}_{D}\left(j\right)$.The collection of weakly regular hierarchical networks is denoted by ${\mathbb{D}}_{w}^{N}\subset {\mathbb{D}}^{N}$. - (b)
- The network D is
**regular**if for all nodes $i,j\in {N}_{D}:{p}_{D}\left(i\right)={p}_{D}\left(j\right)$.The collection of regular hierarchical networks is denoted by ${\mathbb{D}}_{r}^{N}\subset {\mathbb{D}}_{w}^{N}$. - (c)
- The network D is
**simple**if for every node $i\in {N}_{D}:{p}_{D}\left(i\right)=1$.The collection of simple hierarchical networks is denoted by ${\mathbb{D}}_{s}^{N}\subset {\mathbb{D}}_{r}^{N}$.

**Definition 2.**

**simple subnetwork**of D if it satisfies the following two properties:

- (i)
- For every node $i\in N:T\left(i\right)\subseteq D\left(i\right)$, and
- (ii)
- For every node $j\in {N}_{D}:{p}_{T}\left(j\right)=1$.

- Measuring power in hierarchical networks

**Definition 3.**

**power gauge**for D is a vector $\delta =({\delta}_{1},\dots ,{\delta}_{n})\in {\mathbb{R}}_{+}^{N}$ such that ${\sum}_{i\in N}{\delta}_{i}={n}_{D}$.

**power measure**on ${\mathbb{D}}^{N}$ is a function $m:{\mathbb{D}}^{N}\to {\mathbb{R}}_{+}^{N}$ such that ${\sum}_{i\in N}\phantom{\rule{0.166667em}{0ex}}{m}_{i}\left(D\right)={n}_{D}$ for every hierarchical network $D\in {\mathbb{D}}^{N}$.

#### 2.1. Game Theoretic Representations of Hierarchical Networks

**Definition 4.**

**cooperative game with transferable utilities**—or a “TU-game”—on the node set N is a map $v:{2}^{N}\to \mathbb{R}$ such that $v(\u2300)=0$.

**worth**$v\left(H\right)\in \mathbb{R}$. A group of nodes $H\subseteq N$ is also called a

**coalition**of nodes.

**Definition 5.**

- (a)
- The
**successor representation**of D is the TU-game ${\rho}_{D}:{2}^{N}\to \mathbb{N}$ for every coalition $H\subseteq N$ given by ${\rho}_{D}\left(H\right)=\#D\left(H\right)$, the number of successors of the coalition H in the network D. - (b)
- The
**partial successor representations**of D are given as the two TU-games ${\rho}_{D}^{a}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{\rho}_{D}^{b}:{2}^{N}\to \mathbb{N}$, which for every node coalition $H\subseteq N$ are given by ${\rho}_{D}^{a}\left(H\right)=\#\left[\phantom{\rule{0.166667em}{0ex}}D\left(H\right)\cap {N}_{D}^{a}\phantom{\rule{0.166667em}{0ex}}\right]$ and ${\rho}_{D}^{b}\left(H\right)=\#\left[\phantom{\rule{0.166667em}{0ex}}D\left(H\right)\cap {N}_{D}^{b}\phantom{\rule{0.166667em}{0ex}}\right]$. - (c)
- The
**strong successor representation**of D is the TU-game ${\sigma}_{D}:{2}^{N}\to \mathbb{N}$ for every coalition $H\subseteq N$ given by ${\sigma}_{D}\left(H\right)=\#{D}^{*}\left(H\right)$, the number of strong successors of the coalition H in the network D.

**Example 1.**

Coalition H | ${\rho}_{D}\left(H\right)$ | ${\sigma}_{D}\left(H\right)$ | Coalition H | ${\rho}_{{D}^{\prime}}\left(H\right)$ | ${\sigma}_{{D}^{\prime}}\left(H\right)$ |

$\left\{1\right\}$ | 0 | 0 | $\left\{4\right\}$ | 1 | 0 |

$\left\{2\right\}$ | 2 | 2 | $\left\{5\right\}$ | 2 | 1 |

$\left\{3\right\}$ | 0 | 0 | $\left\{6\right\}$ | 0 | 0 |

$\{1,2\}$ | 2 | 2 | $\{4,5\}$ | 2 | 2 |

$\{1,3\}$ | 0 | 0 | $\{4,6\}$ | 1 | 0 |

$\{2,3\}$ | 2 | 2 | $\{5,6\}$ | 2 | 1 |

$N=\{1,2,3\}$ | 2 | 2 | $N=\{4,5,6\}$ | 2 | 2 |

- Properties of successor representations

**Proposition 1.**

- (i)
- For every node, $i\in N:{\rho}_{D}\left(\phantom{\rule{0.166667em}{0ex}}\left\{i\right\}\phantom{\rule{0.166667em}{0ex}}\right)={s}_{D}\left(i\right)$, and the worth of the whole node set is determined as ${\rho}_{D}\left(N\right)={n}_{D}=\#{N}_{D}$.
- (ii)
- For every node, $i\in N:{\sigma}_{D}\left(\phantom{\rule{0.166667em}{0ex}}\left\{i\right\}\phantom{\rule{0.166667em}{0ex}}\right)={s}_{D}^{a}\left(i\right)$, and the worth of the whole node set is determined as ${\sigma}_{D}\left(N\right)={n}_{D}=\#{N}_{D}$.
- (iii)
- The successor game is the sum of the two partial successor games in the sense that ${\rho}_{D}\left(H\right)={\rho}_{D}^{a}\left(H\right)+{\rho}_{D}^{b}\left(H\right)$ for every node coalition $H\in {2}^{N}$.
- (iv)
- For every node coalition, $H\subseteq N:{\rho}_{D}^{a}\left(H\right)={\sum}_{i\in H}{s}_{D}^{a}\left(i\right)$, implying that the partial successor representation ${\rho}_{D}^{a}$ is an additive game.
- (v)
- For every node coalition, $H\subseteq N:{\rho}_{D}^{b}\left(H\right)\u2a7d{\sum}_{i\in H}{s}_{D}^{b}\left(i\right)$.
- (iv)
- ${\sigma}_{D}={\rho}_{D}^{a}+{\widehat{\sigma}}_{D}$, where for every coalition $H\subseteq N:{\widehat{\sigma}}_{D}\left(H\right)={\sigma}_{D}\left(H\right)-{\rho}_{D}^{a}\left(H\right)\u2a7d{\rho}_{D}^{b}\left(H\right)$.

**Theorem 1.**

- (i)
- The strong successor representation ${\sigma}_{D}$ is the dual of the successor representation ${\rho}_{D}$ in the sense that$${\sigma}_{D}\left(H\right)={\rho}_{D}\left(N\right)-{\rho}_{D}(N\setminus H)\phantom{\rule{2.em}{0ex}}forallH\subseteq N.$$
- (ii)
- The strong successor representation ${\sigma}_{D}$ is decomposable into unanimity games with$${\sigma}_{D}=\sum _{j\in {N}_{D}}{u}_{{D}^{-1}\left(j\right)}.$$
- (iii)
- The strong successor representation ${\sigma}_{D}$ is a convex TU-game [7] in the sense that ${\sigma}_{D}\left(H\right)+{\sigma}_{D}\left(K\right)\u2a7d{\sigma}_{D}(H\cup K)+{\sigma}_{D}(H\cap K)$ for all $H,K\subseteq N$.
- (iv)
- The successor representation ${\rho}_{D}$ is a concave TU-game in the sense that ${\rho}_{D}\left(H\right)+{\rho}_{D}\left(K\right)\u2a7e{\rho}_{D}(H\cup K)+{\rho}_{D}(H\cap K)$ for all $H,K\subseteq N$.

**Proof.**

#### 2.2. Some Standard Solutions of the Successor Representations

**Definition 6.**

**core power gauge**for a given hierarchical network $D\in {\mathbb{D}}^{N}$ is a power gauge $\delta \in {\mathbb{R}}_{+}^{N}$ that satisfies the normalisation property ${\sum}_{i\in N}{\delta}_{i}={n}_{D}$ and for every node coalition $H\subseteq N:{\sum}_{j\in H}{\delta}_{j}\u2a7e{\sigma}_{D}\left(H\right)=\#{D}^{*}\left(H\right)$.

**Example 2.**

**Proposition 2.**

- (i)
- If D is a simple hierarchical network, then there a unique core power gauge ${\delta}^{D}\in {\mathbb{R}}_{+}^{N}$ such that $\mathcal{C}\left(D\right)=\left\{{\delta}^{D}\right\}$ exists, where ${\delta}_{i}^{D}={s}_{D}\left(i\right)$ for every node $i\in N$.
- (ii)
- More generally, $\mathcal{C}\left(D\right)$ is equal to the Weber set of ${\sigma}_{D}$, which is the convex hull of the unique core power gauges of all simple subnetworks of D given by $\mathcal{C}\left(D\right)=\mathrm{Conv}\phantom{\rule{0.166667em}{0ex}}\left\{{\delta}^{T}\mid T\in \mathcal{S}\left(D\right)\phantom{\rule{0.166667em}{0ex}}\right\}\ne \u2300$.

**Proof.**

- The $\beta $-measure

**Proposition 3.**

- (i)
- The β-measure is a core power gauge: $\beta \left(D\right)\in \mathcal{C}\left(D\right)$.
- (ii)
- $\beta \left(D\right)=\phi \left({\rho}_{D}\right)=\phi \left({\sigma}_{D}\right)$, where φ is the Shapley value7 on the collection of all cooperative games on N.

## 3. The Gately Power Measure

**Definition 7.**

**Gately power measure**on the class of hierarchical networks ${\mathbb{D}}^{N}$ on N is the power measure $\xi :{\mathbb{D}}^{N}\to {\mathbb{R}}^{N}$ with

**Theorem 2.**

**Proof.**

#### 3.1. Properties of the Gately Power Measure

- The Gately measure is not necessarily a core power gauge

**Example 3.**

**Theorem 3.**

- (i)
- If $\#\phantom{\rule{0.166667em}{0ex}}\{i\in N\mid D(i)\ne \u2300\}\u2a7d3$, then $\xi \left(D\right)\in \mathcal{C}\left(D\right)$.
- (ii)
- If D is weakly regular, i.e., for all $i,j\in {N}_{D}^{b}:{p}_{D}\left(i\right)={p}_{D}\left(j\right)$, then $\xi \left(D\right)\in \mathcal{C}\left(D\right)$.

**Proof.**

- An axiomatic characterisation of the Gately power measure

**Theorem 4.**

- (i)
**Normalisation:**m is ${n}_{D}$-normalised in the sense that ${\sum}_{N}{m}_{i}\left(D\right)={n}_{D}$ for all $D\in {\mathbb{D}}^{N}$;- (ii)
**Normality:**For every hierarchical network $D\in {\mathbb{D}}^{N}$, it holds that$$m\left(D\right)={s}_{D}^{a}+m\left({P}_{D}\right)$$- (iii)
**Restricted proportionality:**For every principal network $D\in {\mathbb{D}}^{N}$ with $D={P}_{D}$, it holds that$$m\left(D\right)={\lambda}_{D}\phantom{\rule{0.166667em}{0ex}}{s}_{D}\phantom{\rule{2.em}{0ex}}forsome{\lambda}_{D}0.$$

**Proof.**

- As indicated above, with regard to the axiomatisation devised in Theorem 4, the $\beta $-measure satisfies the normalisation property (i) as well as the normality property (ii), but not the restricted proportionality property (iii).The restricted egalitarian power measure $\epsilon $ is an example of a power measure on ${\mathbb{D}}^{N}$ that satisfies (i) as well as (ii), but not the restricted proportionality property (iii). We remark that the restricted egalitarian power measure is in the same family of power measures as the Gately measure.
- Consider the proportional power measure $\pi $ on ${\mathbb{D}}^{N}$ with for every $D\in {\mathbb{D}}^{N}:$$$\pi \left(D\right)=\frac{{n}_{D}}{{\sum}_{i\in N}{s}_{D}\left(i\right)}\phantom{\rule{0.166667em}{0ex}}{s}_{D}.$$Then, this proportional power measure satisfies the normalisation property (i) as well as the restricted proportionality property (iii) but not the normality property (ii) stated in Theorem 4.
- Finally, consider the direct power measure $\alpha $ on ${\mathbb{D}}^{N}$ with for every $D\in {\mathbb{D}}^{N}:$$$\alpha \left(D\right)={s}_{D}\in {\mathbb{R}}_{+}^{N}.$$This direct power measure $\alpha $ satisfies the restricted proportionality property (iii) as well as the normality property (ii) stated in Theorem 4 but not the stated normalisation property (i).

#### 3.2. A Comparison between the $\beta $-Measure and the Gately Measure

**Theorem 5.**

**Proof.**

**Example 4.**

**Example 5.**

## 4. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | We refer to Tarkowski et al. [4] for an overview of the literature on cooperative game theoretic constructions of centrality measures in directed networks. |

2 | This authority can also be psychological and can be an influence on reputational features in the relationship. An example might be the relationship between two chess players. One of these players can have a psychological advantage over the other based on the outcomes of past games between the two players and/or the Elo rating differential between the two players. |

3 | |

4 | We remark that other distribution principles can also be applied such as distributions founded on egalitarian fairness considerations. This falls outside the scope of the present paper. |

5 | We emphasise that in our setting, hierarchical networks are not necessarily tiered or top-down. Hence, we allow these networks to contain cycles and even binary relationships. This allows for the incorporation of sports competitions and other social activities to be represented by these hierarchical networks. |

6 | We recall that the unanimity game of node coalition $H\ne \u2300$ is defined by ${u}_{H}:{2}^{N}\to \{0,1\}$ such that ${u}_{H}\left(T\right)=1$ if and only if $H\subseteq T\subseteq N$. This implies that ${u}_{H}\left(T\right)=0$ for all other node coalitions $T\subseteq N$. |

7 | It is well-established that every TU-game $v:{2}^{N}\to {\mathbb{R}}^{N}$ can be written as $v={\sum}_{H\subseteq N}{\Delta}_{v}\left(H\right)\phantom{\rule{0.166667em}{0ex}}{u}_{H}$, where ${\Delta}_{v}\left(H\right)$ is the Harsanyi dividend of coalition H in the game v [39]. Now, the Shapley value is for every $i\in N$ defined by ${\phi}_{i}\left(v\right)={\sum}_{H\subseteq N:i\in H}{\textstyle \frac{{\Delta}_{v}\left(H\right)}{\#\phantom{\rule{0.166667em}{0ex}}H}}$. Hence, the Shapley value fairly distributes the generated Harsanyi dividends over the players that generate these dividends. The Shapley value was seminally introduced by Shapley [8]. |

8 | We use the definition that for $x,y\in {\mathbb{R}}^{N}:x>y$ if and only if ${x}_{i}\u2a7e{y}_{i}$ for all $i\in N$ and ${x}_{j}>{y}_{j}$ for some $j\in N$. |

9 | We point out that there are two simple subnetworks of D that result in exactly the same power gauge, namely, the subnetwork in which node 1 dominates node 4, node 2 dominates node 5, and vice versa. The resulting power gauge is $(3,1,0,0,0)$. |

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Gilles, R.P.; Mallozzi, L.
Game Theoretic Foundations of the Gately Power Measure for Directed Networks. *Games* **2023**, *14*, 64.
https://doi.org/10.3390/g14050064

**AMA Style**

Gilles RP, Mallozzi L.
Game Theoretic Foundations of the Gately Power Measure for Directed Networks. *Games*. 2023; 14(5):64.
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**Chicago/Turabian Style**

Gilles, Robert P., and Lina Mallozzi.
2023. "Game Theoretic Foundations of the Gately Power Measure for Directed Networks" *Games* 14, no. 5: 64.
https://doi.org/10.3390/g14050064