Computing large market equilibria using abstractions

Computing market equilibria is an important practical problem for market design (e.g. fair division, item allocation). However, computing equilibria requires large amounts of information (e.g. all valuations for all buyers for all items) and compute power. We consider ameliorating these issues by applying a method used for solving complex games: constructing a coarsened abstraction of a given market, solving for the equilibrium in the abstraction, and lifting the prices and allocations back to the original market. We show how to bound important quantities such as regret, envy, Nash social welfare, Pareto optimality, and maximin share when the abstracted prices and allocations are used in place of the real equilibrium. We then study two abstraction methods of interest for practitioners: 1) filling in unknown valuations using techniques from matrix completion, 2) reducing the problem size by aggregating groups of buyers/items into smaller numbers of representative buyers/items and solving for equilibrium in this coarsened market. We find that in real data allocations/prices that are relatively close to equilibria can be computed from even very coarse abstractions

Pacing Equilibrium in First-Price Auction Markets

In the isolated auction of a single item, second price often dominates first price in properties of theoretical interest. But, single items are rarely sold in true isolation, so considering the broader context is critical when adopting a pricing strategy. In this paper, we study a model centrally relevant to Internet advertising and show that when items (ad impressions) are individually auctioned within the context of a larger system that is managing budgets, theory offers surprising endorsement for using a first price auction to sell each individual item. In particular, first price auctions offer theoretical guarantees of equilibrium uniqueness, monotonicity, and other desirable properties, as well as efficient computability as the solution to the well-studied Eisenberg-Gale convex program. We also use simulations to demonstrate that a bidder's incentive to deviate vanishes in thick markets

The Cost of Moral Hazard and Limited Liability in the Principal-Agent Problem

Abstract. In the classical principal-agent problem, a principal hires an agent to perform a task. The principal cares about the task's output but has no control over it. The agent can perform the task at different effort intensities, and that choice affects the task's output. To provide an incentive to the agent to work hard and since his effort intensity cannot be observed, the principal ties the agent's compensation to the task's 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

Selfish Routing in Capacitated Networks

According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer extensions of recent positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable and even discontinuous. The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hither