profile ${\bf \tilde{g}}$ is called *socially optimal* if it maximizes the social welfare (achieves the social optimum $\tilde{U}$), i.e. $$\tilde{U}{\vcentcolon=}U({\bf \tilde{g}})\geq U({\bf g}),\forall{\bf g}\in{\bf G}.\label{eq8}$$ When there are multiple equilibria, we use two metrics to assess the equilibrium efficiency. First, we adopt the *Price of Anarchy* (PoA) to quantify the impact of the agents’ selfish behavior on the social welfare. The PoA is defined as the ratio between the social optimum and the lowest social welfare achieved at equilibrium, i.e. $$\mbox{PoA}=\frac{\tilde{U}}{\min_{{\bf g^{*}}\in{\bf G^{*}}}U({\bf g^{*}})}.\label{eq9}$$

In addition, we analyze the impact of the agents selfish behavior on the information gathering process by defining a novel metric that we term the *Maximum Information Loss* (MIL). The MIL is defined as the maximum difference between the amount of information gathered by any agent at two different equilibria as shown in (\[regn1\]). Unlike the PoA, the MIL quantifies the maximum information loss without considering the link cost. In addition, while the PoA considers the welfare of *all agents*, the MIL quantifies the highest information loss incurred by an *agent* in the worst case.

$$\mbox{MIL}=\max_{i}\left(\sup_{{\bf g}_{u}^{*}\in G^{*}}H(X_{i}\cup X_{\mathcal{R}_{i}({\bf g}_{u}^{*})})-\inf_{{\bf g}_{v}^{*}\in G^{*}}H(X_{i}\cup X_{\mathcal{R}_{i}({\bf g}_{v}^{*})})\right).\label{regn1}$$

Nash Equilibrium Analysis for Homogeneous Link Costs
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In this section, we 