Notation, Definitions and Preliminaries

Most of the notation and the model definition are introduced in Section 8. Next, we introduce some additional notation and discuss some important preliminaries.

We recall that $`\N=\{1, \ldots, n\}`$, with $`n \geq 2`$, is the set of agents, $`U`$ is the strategy universe, and $`Z \subseteq U`$, with $`|Z| \geq 2`$, is the strategy space of the agents. $`S = \{ s_1, \ldots, s_n \} \subseteq U`$ is the set of the agents’ preferred strategies, and $`N_S = \{ i \in N: s_i \in Z\}`$ is the set of agents with preferred strategy in $`Z`$. So, $`N_S = N`$ denotes an unrestricted game, while $`N_S = \emptyset`$ denotes a restricted one. We recall that $`w_{ij} \in [0, 1]`$ is the amount of influence agent $`j`$ imposes on agent $`i`$. Unless stated otherwise, $`w_{ij}`$ may be different than $`w_{ji}`$. We always assume that $`w_{ii} = 0`$ and that $`\sum_{j=1}^n w_{ij} = 1`$. We recall that $`\alpha \in [0, 1]`$ is the confidence-level of the agents. We say that the agents are stubborn, if $`\alpha \in (1/2, 1]`$, and compliant, otherwise.

We refer to any $`\vec{z} \in Z^n`$ as a state of the game. For any state $`\zz`$ and any strategy $`z`$, we let $`[\zz_{-i}, z]`$ be the new state obtained from $`\zz`$ by replacing its $`i`$-component $`\vec{z}(i)`$ with $`z`$ and keeping the remaining components unchanged. If $`\vec{z} = (z, \ldots, z)`$, we say that $`\vec{z}`$ is a consensus on $`z`$.

We say that $`\d: \U \times \U \mapsto \reals_{\geq 0}`$ is a $`\rho`$-approximate metric, for some $`\rho \geq 1`$, if it satisfies (i) $`d(x, x) = 0`$, for all $`x \in U`$; (ii) symmetry, i.e., $`d(x, y) = d(y, x)`$, for all $`x, y \in U`$; and (iii) (approximate) triangle inequality, i.e., $`d(x, y) \leq \rho(d(x, z) + d(z, y))`$, for all $`x, y, z \in U`$. We say that $`\d`$ is an exact metric (or simply metric) if $`\rho = 1`$. We say that $`\d`$ is uniform if it is an exact metric such that $`d(x, y) = 1`$ for all $`x \neq y`$. We assume that $`d`$ is a $`\rho`$-approximate metric, for some $`\rho \geq 1`$, unless stated otherwise.

We consider aggregation functions that satisfy (i) unanimity, i.e., if $`\vec{x}_{-i}`$ is a consensus on $`x`$, then $`\aggr_i(\vec{x}_{-i}) = x`$; and (ii) consistency, i.e., for all $`\vec{x}_{-i}`$, $`\vec{y}_{-i}`$ with $`\sum_{j \neq i} w_{ij} d(\vec{x}(j), \vec{y}(j)) = 0`$, $`\aggr_i(\vec{x}_{-i}) = \aggr_i(\vec{y}_{-i})`$. We say that such aggregation functions are feasible. In this work we focus on feasible aggregation functions. When $`\d`$ is an exact metric on $`\Z`$, notable examples of feasible aggregation functions are the Fréchet mean and the Fréchet median.

Given any state $`\zz`$, the Fréchet mean of agent $`i`$ in $`\zz`$, denoted by $`\Fmean_i(\zz_{-i})`$, is any strategy in $`\Z`$ that minimizes the weighted sum of its squared distances to the strategies in $`\zz_{-i}`$. Formally,

\begin{equation}
\label{def:mean}
    \Fmean_i(\zz_{-i}) \in  {\arg\min}_{\y\in \Z} \sum_{j \neq i}\w_{ij}\d^2\big(\y, \zz(j)\big)\,.
\end{equation}

The Fréchet median of agent $`i`$ in $`\zz`$, denoted by $`\Fmedian_i(\zz_{-i})`$, is any strategy that minimizes the weighted sum of its distances to the strategies in $`\zz_{-i}`$ :

\begin{equation}
\label{def:median}
    \Fmedian_i(\zz_{-i}) \in  {\arg\min}_{\y\in \Z} \sum_{j\neq i}\w_{ij}\d\big(\y, \zz(j)\big)\,.
\end{equation}

We can show that the Fréchet mean and the Fréchet median are indeed feasible aggregation functions.

The following proposition shows that both aggregation functions are feasible.

\begin{align*}
    \sum_{j\in \N\setminus\{i\}}\w_{ij}\d^2\big(\y, \xx(j)\big) 
    & = 
    \sum_{
    \begin{subarray}{c} 
        j\in \N\setminus\{i\} \\  \w_{ij} > 0
    \end{subarray}
    }
    \w_{ij}\d^2\big(\y, \xx(j)\big) %\\
    %& = 
    =
    \sum_{
    \begin{subarray}{c} 
        j\in \N\setminus\{i\} \\  \w_{ij} > 0
    \end{subarray}
    }
    \w_{ij}\d^2\big(\y, \yy(j)\big) %\\
    %& = 
    =
    \sum_{j\in \N\setminus\{i\}}\w_{ij}\d^2\big(\y, \yy(j)\big),
\end{align*}

A pure Nash equilibrium (or equilibrium, for brevity) is a state $`\vec{z}`$ such that for every agent $`i`$ and every strategy $`z \in Z`$, $`c_i(\zz) \leq c_i([\zz_{-i}, z])`$. $`\E \subseteq \Z^n`$ denotes the set of all pure Nash equilibria of a given preference game. A strategy $`z^\ast \in \Z`$ is a best response of agent $`i`$ to a state $`\zz`$, if $`z^\ast \in \arg\min_{z\in \Z} c_i([\zz_{-i}, z])`$. We say that a strategy $`\x\in \Z`$ is a (strictly) dominant strategy of agent $`i`$, if $`c_i\big([\xx_{-i},\x]\big) \leq c_i\big([\xx_{-i},\y]\big)`$ ($`<`$, if strictly), for all states $`\xx_{-i}`$ and strategies $`\y \in \Z`$.

We measure the efficiency of each state $`\zz`$ according to a social objective. We consider two social objectives, the social cost $`\SUM(\zz) = \sum_{i\in \N}c_i(\zz)`$ and the maximum cost $`\MAX(\zz) = \max_{i\in \N} c_i(\zz)`$. A state $`\oo`$ is optimal wrt. $`\SUM`$, if $`\SUM(\oo) \leq \SUM(\zz)`$, for all states $`\zz`$. We denote by $`\OSUM \subseteq \Z^n`$ the set of optimal states wrt. $`\SUM`$, i.e., $`\OSUM = \arg\min_{\zz \in \Z^n}\SUM(\zz)`$. Similarly, a state $`\oo`$ is optimal wrt. $`\MAX`$, if $`\MAX(\oo) \leq \MAX(\zz)`$, for all states $`\zz`$. We let $`\OMAX = \arg\min_{\zz \in \Z^n}\MAX(\zz)`$ be the set of all optimal states wrt. $`\MAX`$.

The price of anarchy of a game wrt. $`\SUM`$ is $`\POASUM = \max_{\ee \in \E}\frac{\SUM(\ee)}{\SUM(\oo)}`$, if $`\SUM(\oo) > 0`$ for some state $`\oo\in \OSUM`$. If $`\SUM(\oo) = 0`$, then $`\POASUM = +\infty`$, if $`\E \neq \OSUM`$, and $`\POASUM = 1`$, if $`\E = \OSUM`$. The definition of $`\POAMAX`$ is similar.

For every pair of states $`\xx`$, $`\yy`$, $`\D(\xx,\yy) = \{j\in \N : \xx(j) \neq \yy(j)\}`$ is the set of agents with different strategies in $`\xx`$ and $`\yy`$. If $`\D(\xx,\yy) = \emptyset`$, we say that $`\xx`$ and $`\yy`$ are globally equivalent. For every agent $`i`$ and all pairs of states $`\xx`$, $`\yy`$, we define the relative distance of $`\xx`$ and $`\yy`$ for $`i`$ as $`\ds_i(\xx,\yy) = \sum_{j\neq i}\w_{ij}\d\big(\xx(j),\yy(j)\big)`$. If $`\ds_i(\xx,\yy) = 0`$, $`\xx`$ and $`\yy`$ are equivalent for $`i`$. We observe that $`\D(\xx,\yy) = \emptyset`$ implies $`\ds_i(\xx,\yy) = 0`$ (while converse may not be true). Moreover, $`\ds_i\big([\xx_{-i}, x],[\xx_{-i},y]\big) = 0`$, for all strategies $`\x`$ and $`\y`$.

The social impact of agent $`i`$ is $`\SI_i = \sum_{j\in \N}\w_{ji}`$ and quantifies the intensity by which $`i`$ influences the environment. The global social impact is $`\SI = \max_{j \in N} \SI_j`$. We observe that $`\SI \in [1, n-1]`$ and $`\sum_{i \in N} \SI_i = n`$. We refer to the case where influence weights are symmetric, i.e., $`w_{ij} = w_{ji}`$ for all $`i`$ and $`j`$, and $`\SI = 1`$ as the fully symmetric case.

The stretch $`\tau_i`$ of agent $`i`$ quantifies how sensitive the aggregation function is wrt. changes in the state of the game. Formally, we let

\hat{\tau}_i =\!\! \inf_{\begin{subarray}{c}
                    \vec{e} \in \E\\
                    \vec{y} \in Z^n\\
                    \ds_i(\vec{e}, \vec{y}) > 0
                  \end{subarray}}
\!\!\!\!\left\{ r \geq 0\,\Big|\,\frac{d(\aggr_{i}(\vec{e}_{-i}),\aggr_{i}(\vec{y}_{-i}))}{\ds_i(\vec{e}, \vec{y})} \leq r \right\}.

We define $`\tau_i = \max\{\hat{\tau_i}, 1\}`$, so as to account for the case where $`\ds_i(\vec{e}, \vec{y}) = 0`$ and $`d(\aggr_{i}(\vec{e}_{-i}),\aggr_{i}(\vec{y}_{-i})) = 0`$ (recall the definition of feasible aggregation functions). Therefore, existence of $`r`$ is always guaranteed and $`\tau_i`$ is well defined. Since $`\tau_i`$ is used in the price of anarchy bounds, we can restrict its definition to optimal states $`\vec{o}`$, instead of arbitrary states $`\vec{y}`$. We use this more restricted definition in the proofs of propositions [prop:strech:o-consensus] and [prop:strech:e-consensus], in Section 6.

The (global) stretch is $`\tau = \max_{j \in N} \tau_j`$ . At the conceptual level, the stretch quantifies how much the price of anarchy increases because agents only have access to an aggregate of the strategies in $`\vec{z}`$, instead of $`\vec{z}`$ itself.

The boundary of agent $`i`$, denoted by $`\beta_i`$ quantifies how much closer a strategy $`x`$ can be to $`s_i`$ compared against an equilibrium strategy $`\vec{e}(i)`$ of $`i`$. Formally,

\begin{equation}
\label{def:boundary} 
\beta_i = \min_{\vec{e} \in E, x \neq \vec{e}(i)} \frac{d(x, s_i)}{d(x, \vec{e}(i))}.
\end{equation}

For nontrivial games, there always exists a strategy $`x`$ with $`d(x, \vec{e}(i)) > 0`$. Thus, $`\beta_i`$ is well defined. The (global) boundary is $`\beta = \min_{j \in N} \beta_j`$ .