Inequality and Network Formation Games

This paper introduces the Nash Inequality Ratio (NIR) as a natural characterization of the extent to which inequality is permitted between individual agents in Nash equilibrium outcomes of a given strategic setting. For any particular strategy, the inequality ratio is defined as the ratio between the highest and lowest costs incurred to individual agents in the outcome dictated by that strategy. The NIR of a game is defined as the maximal inequality ratio over all Nash equilibrium strategy profiles. It indicates quantitatively how intrinsically fair (or unfair) a game can be for the agents involved. Moreover, the NIR allows us to quantify the relationship between efficiency and (in)equality, by establishing whether there exist efficient Nash equilibrium outcomes that maximize and/or minimize the inequality ratio. We analyze the NIR for two distinct network formation games: the Undirected Connections (UC) game of Fabrikant et al. (PODC ’03) and the Undirected Bounded Budget Connections (UBBC) game of Ehsani et al. (SPAA ’11). In both settings, agents unilaterally build edges to other agents and incur a usage cost equal to the sum of shortest-path distances to every other agent. Additional costs associated with the creation of edges are treated differently in both games – in the UC model, agents incur a construction cost of α > 0 per edge they create, and in the UBBC model, agents are endowed with an edge budget that determines the maximum number of edges they can build. In the UC model, we establish the NIR parameterized on α, showing that (i) when α < 1, the NIR is at most 1+α; (ii) when 1 ≤ α < 2, the NIR is at most 2; and (iii) when 2 ≤ α, the NIR is at most 2 + α. In the UBBC model, it is shown that the NIR is upper-bounded by 2. The UBBC upper-bound is shown to hold even in the restricted uniform setting in which every agent is endowed with the same budget, establishing that the NIR of 2 is intrinsic to the game itself and not a consequence of nonuniform budgets. The relationship between efficiency and (in)equality is analyzed for both games. In the UC model it is shown that (i) when α < 1, there exist efficient Nash equilibrium that maximize inequality and others that achieve perfect equality; (ii) when 1 ≤ α < 2, there exist Nash equilibrium that achieve perfect equality but no Nash equilibrium is efficient; and (iii) when 2 ≤ α, there exist efficient Nash equilibrium with perfect equality but no efficient Nash equilibrium that maximizes inequality. For the UBBC model it is shown that when edge budgets are sufficiently large, there are efficient Nash equilibrium strategies that achieve perfect equality; and