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This paper studies delegation in a model of discrete choice. In the delegation problem, an uninformed principal must consult an informed agent to make a decision. Both the agent and principal have preferences over the decided-upon action which vary based on the state of the world, and which may not be aligned. The principal may commit to a mechanism, which maps reports of the agent to actions. When this mechanism is deterministic, it can take the form of a menu of actions, from which the agent simply chooses upon observing the state. In this case, the principal is said to have delegated the choice of action to the agent. We consider a setting where the decision being delegated is a choice of a utility-maximizing action from a set of several options. We assume the shared portion of the agent's and principal's utilities is drawn from a distribution known to the principal, and that utility misalignment takes the form of a known bias for or against each action. We provide tight approximation analyses for simple threshold policies under three increasingly general sets of assumptions. With independently-distributed utilities, we prove a 3-approximation. When the agent has an outside option the principal cannot rule out, the constant-approximation fails, but we prove a log ρ/ log log ρ-approximation, where ρ is the ratio of the maximum value to the optimal utility. We also give a weaker but tight bound that holds for correlated values, and complement our upper bounds with hardness results. One special case of our model is utility-based assortment optimization, for which our results are new. 

We study a dynamic model of procurement auctions in which the agents (sellers) will abandon the auction if their utility does not satisfy their private target, in any given round. We call this “abandonment” and analyze its consequences on the overall cost to the mechanism designer (buyer), as it reduces competition in future rounds of the auction and drives up the price. We show that in order to maintain competition and minimize the overall cost, the mechanism designer has to adopt an inefficient (per-round) allocation, namely to assign the demand to multiple agents in a single round. We focus on threshold mechanisms as a simple way to achieve ex-post incentive compatibility, akin to reserves in revenue-maximizing forward auctions. We then consider the optimization problem of finding the optimal thresholds. We show that even though our objective function does not have the optimal substructure property in general, if the underlying distributions satisfy some regularity properties, the global optimal solution lies within a region where the optimal thresholds are monotone and can be calculated with a greedy approach, or even more simply in a parallel fashion. 

We propose a methodology to carry out vertex-frequency analyses of graph signals, with the goal of unveiling the signal’s frequency occupancy over a localized region in the network. To this end, we first introduce localized graph signals in the vertex domain, by defining windows that are localized around each node by construction. Recent directed graph Fourier transform (DGFT) advances facilitate the frequency analysis of said localized signals, to reveal the signal’s energy distribution in a way akin to a spectrogram in the vertex-frequency plane. We then learn a set of windows by applying gradient descent method to an optimization problem governed by penalty parameters in the spectral domain. We also argue about the tradeoff between the resolution in the vertex and frequency domains based on the said parameters. We evaluate the performance of the proposed windowed GFT approach through numerical experiments on synthetic and real-world graphs.