Playing the Wrong Game: Smoothness Bounds for Congestion Games with Heterogeneous Biases

In many situations a player may act so as to maximize a perceived utility that does not coincide with the objective of the system analyst or designer. Such discrepancies may arise due to rational factors (such as tolls or restricted budgets), or due to cognitive biases such as those that are common in behavioral economics: risk attitudes, altruism, present-bias and so on. When analyzing a game, one may ask how inefficiency, as measured by the Price of Anarchy (PoA), is affected by the perceived utilities. This question is especially challenging in the presence of heterogeneous biases, where each player essentially plays a different game. The smoothness method [Roughgarden '04,'09] naturally extends to games with homogeneous perceived utilities or costs, regardless of the game or the behavioral bias. We show that such biased-smoothness is broadly applicable in the context of nonatomic congestion games: In games with arbitrary heterogeneous biases, we bound the agents' equilibrium costs purely based on their own biased-smoothness parameters, as well as on structural parameters of the underlying network. For symmetric heterogeneous games, we provide a {\em biased PoA} bound that depends on the average biased-smoothness of all participating types. We complement our positive results with examples showing tightness or almost-tightness of the upper bounds. We also identify various classes of cost functions and biases that are biased-smooth, thereby substantially improving some recent results from the literature.