Timing Matters: Online Dynamics in Broadcast Games

TL;DR

This paper investigates how the order and control of agent moves in network design games affect the efficiency of resulting Nash equilibria, revealing significant differences between decentralized and controlled dynamics.

Contribution

It demonstrates that natural, uncontrolled dynamics can lead to highly inefficient equilibria, while controlled sequences of moves significantly improve efficiency in network design games.

Findings

Uncontrolled agent dynamics can produce polynomially large inefficiencies.

Controlled move sequences lead to exponentially better, logarithmic ratio of equilibrium to optimal costs.

Efficiency depends critically on whether agents join or leave arbitrarily or under designer control.

Abstract

A central question in algorithmic game theory is to measure the inefficiency (ratio of costs) of Nash equilibria (NE) with respect to socially optimal solutions. The two established metrics used for this purpose are price of anarchy (POA) and price of stability (POS), which respectively provide upper and lower bounds on this ratio. A deficiency of these metrics, however, is that they are purely existential and shed no light on which of the equilibrium states are reachable in an actual game, i.e., via natural game dynamics. This is particularly striking if these metrics differ significantly, such as in network design games where the exponential gap between the best and worst NE states originally prompted the notion of POS in game theory (Anshelevich et al., FOCS 2002). In this paper, we make progress toward bridging this gap by studying network design games under natural game dynamics. First we show that in a completely decentralized setting, where agents arrive, depart, and make improving moves in an arbitrary order, the inefficiency of NE attained can be polynomially large. This implies that the game designer must have some control over the interleaving of these events in order to force the game to attain efficient NE. We complement our negative result by showing that if the game designer is allowed to execute a sequence of improving moves to create an equilibrium state after every batch of agent arrivals or departures, then the resulting equilibrium states attained by the game are exponentially more efficient, i.e., the ratio of costs compared to the optimum is only logarithmic. Overall, our two results establish that in network games, the efficiency of equilibrium states is dictated by whether agents are allowed to join or leave the game in arbitrary states, an observation that might be useful in analyzing the dynamics of other classes of games with divergent POS and POA bounds.