Nicolás E. Stier-Moses

The Impact of Oligopolistic Competition in Networks

In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the systems point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria.

On the inefficiency of equilibria in congestion games

We present a short geometric proof for the price of anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks. This novel proof also facilitates two new types of results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive improved bounds on the inefficiency of Nash equilibria for situations in which the equilibrium travel times are within reasonable limits of the free-flow travel times. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of congestion games, which provides the framework in which we describe this work.

Investment in Two Sided Markets and the Net Neutrality Debate

This paper develops a game-theoretic model based on a two-sided market framework to compare Internet service providers’ (ISPs) investment incentives, content providers’ (CPs) participation, and social welfare between neutral and non-neutral network regimes. We find that ISPs’ investments are driven by the trade-off between softening consumer price competition and increasing revenues from CPs. Specifically, investments are higher in the non-neutral regime because it is easier to extract revenue through appropriate CP pricing. On the other hand, participation of CPs may be reduced in a non-neutral network due to higher prices. The net impact of non-neutrality on social welfare is determined by which of these two effects is dominant. Overall, we find that the non-neutral network is always welfare superior in a “walled-gardens” model, while the neutral network is superior in a “priority lanes” model when CP-quality heterogeneity is large. These results provide useful insights that inform the net-neutrality debate.

A Geometric Approach to the Price of Anarchy in Nonatomic Congestion Games

We present a short, geometric proof for the price-of-anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.

Network games with atomic players

We study network and congestion games with atomic players that can split their flow. This type of games readily applies to competition among freight companies, telecommunication network service providers, intelligent transportation systems and manufacturing with flexible machines. We analyze the worst-case inefficiency of Nash equilibria in those games and conclude that although self-interested agents will not in general achieve a fully efficient solution, the loss is not too large. We show how to compute several bounds for the worst-case inefficiency, which depend on the characteristics of cost functions and the market structure in the game. In addition, we show examples in which market aggregation can adversely impact the aggregated competitors, even though their market power increases. When the market structure is simple enough, this counter-intuitive phenomenon does not arise.

Fast, Fair, and Efficient Flows in Networks

We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, an optimal flow and an equilibrium flow share a desirable property in this situation: All flow-carrying paths have the same length, i.e., these solutions are “fair,” which is in general not true for optimal flows in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances with multiple sources and a single sink for which the price of anarchy is unbounded, even in networks with linear latencies. Furthermore, we show that an s-t-flow that is optimal with respect to the average latency objective is near-optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency.

Wardrop Equilibria with Risk-Averse Users

Network games can be used to model competitive situations in which players select routes to maximize their utility. Common applications include traffic, telecommunication and distribution networks. Although traditional network models have assumed that utilities only depend on congestion, in most applications they also have an uncertain component. In this work, we extend Wardrops network game (1952) by explicitly incorporating uncertainty in utility functions. Players are utility maximizers and select their route by solving a robust optimization problem, which takes the uncertainty into account. We define a robust Wardrop equilibrium as a solution under which all players are assigned to an optimal solution to their robust problems. Such a solution always exists and can be computed through efficient column generation methods. We show through a computational study that a robust Wardrop equilibrium tends to be more fair than the classic Wardrop equilibrium which ignores the uncertainty. Hence, a robust Wardrop equilibrium is more stable than the nominal counterpart as it reduces the regret that players experience after the uncertainty is revealed. Finally, we show that a pricing mechanism allows the network planner to coordinate players into a socially optimal solution, and show how the necessary tolls can be computed.

Robust wardrop equilibrium

Agents competing in a network game typically prefer the least expensive route to their destinations. However, identifying such a route can be difficult when faced with uncertain cost estimates. We introduce a novel solution concept called robust Wardrop equilibria (RWE) that takes into account these uncertainties. Our approach, which generalizes the traditional Wardrop equilibrium, considers that each agent uses distribution-free robust optimization to take the uncertainty into account. By presenting a nonlinear complementary problem that captures this user behavior, we show that RWE always exist and provide an efficient algorithm based on column generation to compute them. In addition, we present computational results that indicate that RWE are more stable than their nominal counterparts because they reduce the regret experienced by agents.

Stackelberg Routing in Atomic Network Games

We consider network games with atomic players, which indicates that some players control a positive amount of flow. Instead of studying Nash equilibria as previous work has done, we consider that players with considerable market power will make decisions before the others because they can predict the decisions of players without market power. This description fits the framework of Stackelberg games, where those with market power are leaders and the rest are price-taking followers. As Stackelberg equilibria are difficult to characterize, we prove bounds on the inefficiency of the solutions that arise when the leader uses a heuristic that approximate its optimal strategy.

The cost of moral hazard and limited liability in the principal-agent problem

In the classical principal-agent problem, a principal hires an agent to perform a task. The principal cares about the tasks output but has no control over it. The agent can perform the task at different effort intensities, and that choice affects the tasks output. To provide an incentive to the agent to work hard and since his effort intensity cannot be observed, the principal ties the agents compensation to the tasks output. If both the principal and the agent are risk-neutral and no further constraints are imposed, it is well-known that the outcome of the game maximizes social welfare. In this paper we quantify the potential social-welfare loss due to the existence of limited liability, which takes the form of a minimum wage constraint. To do so we rely on the worst-case welfare loss--commonly referred to as the Price of Anarchy--which quantifies the (in)efficiency of a system when its players act selfishly (i.e., they play a Nash equilibrium) versus choosing a socially-optimal solution. Our main result establishes that under the monotone likelihood-ratio property and limited liability constraints, the worst-case welfare loss in the principal-agent model is exactly equal to the number of efforts available.