Bayesian Algorithmic Mechanism Design Jason Hartline Northwestern University Brendan Lucier University of Toronto

A central authority wishes to provide service to a group of rational agents. Each agent has a private value for service. Agents declare values, and the mechanism chooses a subset of agents to satisfy. – subject to problem-specific outcome costs and/or feasibility constraints. Goal: maximize the efficiency (social welfare) of the outcome: Agents are rational and can misrepresent their values. Mechanism can charge payments; we assume agents want to maximize utility = value - payment. Social Welfare Problems Social Welfare = total value of satisfied agents - cost of outcome $10 $1 $5 $20 $2 $15 cost: $10 Social Welfare: $12

Examples Combinatorial auctions Connectivity problems (e.g. Steiner tree) Scheduling with deadlines Facility location etc. This talk: a general way to solve strategic issues for a given algorithmic solution, when agent values are drawn from publicly known distributions. This talk: a general way to solve strategic issues for a given algorithmic solution, when agent values are drawn from publicly known distributions. $20 $10 $15 $10 $3 $2 $5

The Vickrey-Clarke-Groves construction converts any optimal algorithm into a mechanism where each agent maximizes his utility by reporting his value truthfully. Maximizes social welfare and solves the strategic issues. The VCG mechanism is infeasible for computationally hard problems! The construction does not apply to approximation or ad hoc algorithms. Question: can we turn any approximation or ad hoc algorithm into a mechanism for strategic agents? The VCG Mechanism But optimal

Algorithmic Incentive Compatibility The VCG mechanism implies the following: A primary goal of algorithmic mechanism design for the past decade has been to extend this result to approximation algorithms. We would like: Theorem (VCG): Given optimal algorithm A for a social welfare problem, one can construct an optimal truthful mechanism M. The runtime of M is polynomial in and the runtime of A. Theorem? Given any algorithm A for a social welfare problem, one can construct a truthful mechanism M such that SW(M) ≥ SW(A). The runtime of M is polynomial in and the runtime of A. ?

The problem: we are insisting on dominant strategies. Agents tell the truth regardless of their beliefs. This is a strong requirement that can come at a loss. A more standard approach is to model the information that agents have about each other, then require that truth-telling be optimal given this knowledge. Standard model: agent values are private but drawn independently from publicly known distributions. The appropriate notion of truthfulness in this model is Bayesian incentive compatibility (BIC): – A mechanism is BIC if every agent maximizes his expected utility by declaring his value truthfully. – Expectation is over the distribution of other agents’ values. A Bayesian Solution Concept ∼

The Bayesian optimization problem: – Input: ∈ ℝ, where ∼ i.d. – Output: () ∈ ℝ – Goal: maximize E [ () · − (()) ] Motivating question: can an arbitrary algorithm for the optimization problem be made BIC without loss of performance? The Bayesian Setting

Main Result Reduces the problem of designing a BIC mechanism to the problem of designing an approximation or ad hoc algorithm. Any approximation factor that can be obtained with a (non-BIC) algorithm can also be obtained with a BIC mechanism. Theorem: Given algorithm A for a single-parameter social welfare problem, one can construct a BIC mechanism M such that E[SW(M)] ≥ E[SW(A)]. The runtime of M is polynomial in and the runtime of A. Theorem: Given algorithm A for a single-parameter social welfare problem, one can construct a BIC mechanism M such that E[SW(M)] ≥ E[SW(A)]. The runtime of M is polynomial in and the runtime of A.

We think of algorithm A as a mapping from to (). Write () = E[ () | ], the expected allocation to agent i if he declares value. Theorem [Myerson, ‘81]: There is a BIC mechanism implementing algorithm A if and only if () is a monotone non-decreasing function for each agent. Our goal: given a (possibly non-monotone) algorithm A, we must construct a “monotonized” version of A. Bayesian Incentive Compatibility Expected allocation to agent i () () Not BIC BIC

Monotonizing Allocation Curves Pick an interval on which the curve () is non-monotone. If ∈, pick some ′ ∈. Pretend agent declared ′. How we choose ′ depends only on. This flattens the allocation curve! We would like to do this independently for each agent, but… Problem: this changes the distribution of values, which affects the allocation curves of the other agents. The main idea: ′

Monotonizing Allocation Curves Pick an interval on which the curve () is non-monotone. If ∈, pick some ′ ∈. Pretend agent declared ′. How should we pick ′ ? Choose ′ according to distribution restricted to ! Then ′ is distributed according to. Other agents’ allocation curves remain unchanged. How should we choose which interval(s) to iron? The main idea: E [ () | ∈ ] ′ ~|′ ~| ′ ~|′ ~| ′

Monotonizing Allocation Curves is monotone precisely when is convex. Take the convex hull of. Iron the intervals corresponding to the added line segments in the convex hull. Why? Replacing with E[ () | ∈ ] on interval is equivalent to replacing that portion of curve with a line segment. (Actually true only when = U[0,1]; more generally we require a change of variables). () ()

The Full Construction Algorithm A’ Input: ∈ ℝ For each agent : Construct cumulative allocation curve and convex hull ′. Let 1, …, be the intervals where ≠ ′. If ∈, draw ′ ~|. Otherwise set ′ =. Return A( ′ ) Algorithm A’ Input: ∈ ℝ For each agent : Construct cumulative allocation curve and convex hull ′. Let 1, …, be the intervals where ≠ ′. If ∈, draw ′ ~|. Otherwise set ′ =. Return A( ′ ) Claim 1: A’ is BIC. Claim 2: E[SW(A’)] ≥ E[SW(A)]. E[ ()∗] () () () ()

Practical Issues Our construction requires that we know the allocation curves for algorithm A under distribution 1,…,. If A is provided as a black box, we can estimate the allocation rules of A by sampling, then run our ironing procedure using the estimated curves. Theorem: Given any > 0 and black-box access to algorithm A, we can construct BIC mechanism M such that E[SW(M)] ≥ E[SW(A)] −. Mechanism M uses poly(, 1/) calls to A.

Conclusions We consider single-parameter social welfare problems when agent values are drawn independently from commonly-known distributions. In this setting, any algorithm can be made Bayesian incentive compatible without loss of performance. This applies even to ad hoc algorithms that are tailored to a particular input distribution. The key to this transformation is an ironing procedure that monotonizes allocation rules.

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