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

Uniformly Bounded Regret in Dynamic Fair Allocation

We study a dynamic allocation problem in which $T$ sequentially arriving divisible resources need to be allocated to $n$ fixed agents with additive utilities. Agents' utilities are drawn stochastically from a known distribution, and decisions are made immediately and irrevocably. Most works on dynamic resource allocation aim to maximize the utilitarian welfare of the agents, which may result in unfair concentration of resources at select agents while leaving others' demands under-fulfilled. In this paper, we consider the egalitarian welfare objective instead, which aims at balancing the efficiency and fairness of the allocation. To this end, we first study a fluid-based policy derived from a deterministic approximation to the underlying problem and show that it attains a regret of order $\Theta(\sqrt{T})$ against the hindsight optimum, i.e., the optimal egalitarian allocation when all utilities are known in advance. We then propose a new policy, called Backward Infrequent Re-solving with Thresholding ($\mathsf{BIRT}$), which consists of re-solving the fluid problem at most $O(\log\log T)$ times. We prove the $\mathsf{BIRT}$ policy attains $O(1)$ regret against the hindsight optimum, independently of the time horizon length $T$ and initial welfare. We also present numerical experiments to illustrate the significant performance improvement against several benchmark policies

Robust Auction Design with Support Information

A seller wants to sell an item to $n$ buyers. The buyer valuations are drawn i.i.d. from a distribution, but the seller does not know this distribution; the seller only knows the support $[a,b]$. To be robust against the lack of knowledge of the environment and buyers' behavior, the seller optimizes over DSIC mechanisms, and measures the worst-case performance relative to an oracle with complete knowledge of buyers' valuations. Our analysis encompasses both the regret and the ratio objectives. For these objectives, we derive an optimal mechanism in closed form as a function of the support and the number of buyers $n$. Our analysis reveals three regimes of support information and a new class of robust mechanisms. i.) With "low" support information, the optimal mechanism is a second-price auction (SPA) with a random reserve, a focal class in the earlier literature. ii.) With "high" support information, we show that second-price auctions are strictly suboptimal, and an optimal mechanism belongs to a novel class of mechanisms we introduce, which we call $\textbf{pooling auctions}$ (POOL); whenever the highest value is above a threshold, the mechanism still allocates to the highest bidder, but otherwise the mechanism allocates to a uniformly random buyer, i.e., pools low types. iii.) With "moderate" support information, a randomization between SPA and POOL is optimal. We also characterize optimal mechanisms within nested central subclasses of mechanisms: standard mechanisms (only allocate to the highest bidder), SPA with random reserve, and SPA with no reserve. We show strict separations across classes, implying that deviating from standard mechanisms is necessary for robustness. Lastly, we show that the same results hold under other distribution classes that capture "positive dependence" (mixture of i.i.d., exchangeable and affiliated), as well as i.i.d. regular distributions.Comment: An abstract of this work appeared in Proceedings of the 24th ACM Conference on Economics and Computation (EC'23