Bayesian Algorithmic Mechanism Design

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

2. Social Welfare Problems • 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. $2 $1 $5 cost: $10 $15 $20 $10 Social Welfare: $12 Social Welfare = total value of satisfied agents - cost of outcome

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

4. The VCG Mechanism • 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? optimal But

5. 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 𝑛 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 𝑛 and the runtime of A.

6. A Bayesian Solution Concept • 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. 𝑣𝑖 ∼ 𝐹𝑖

7. The Bayesian Setting • The Bayesian optimization problem: • Input: 𝒗 ∈ ℝ𝑛, where 𝑣𝑖 ∼ 𝐹𝑖i.d. • Output: 𝑥(𝒗) ∈ ℝ𝑛 • Goal: maximize E[𝑥(𝒗) · 𝒗 − 𝑐(𝑥(𝒗))] • Motivating question: can an arbitrary algorithm for the optimization problem be made BIC without loss of performance?

8. 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 𝑛 and the runtime of A.

9. Bayesian Incentive Compatibility • We think of algorithm A as a mapping from 𝒗 to 𝑥(𝒗). • Write 𝑥𝑖(𝑣𝑖) =E𝑣[ 𝑥𝑖(𝒗) | 𝑣𝑖], the expected allocation to agent i if he declares value 𝑣𝑖. • Theorem[Myerson, ‘81]: There is a BIC mechanism implementing algorithm A if and only if 𝑥𝑖(𝑣𝑖) 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. Not BIC 𝑥𝑖(𝑣𝑖) Expected allocation to agent i BIC 𝑣𝑖

10. Monotonizing Allocation Curves • Pick an interval 𝐼 on which the curve 𝑥𝑖(𝑣𝑖) is non-monotone. • If 𝑣𝑖∈ 𝐼, pick some𝑣′𝑖 ∈ 𝐼.Pretend agent 𝑖 declared 𝑣′𝑖. • How we choose 𝑣′𝑖 depends only on 𝐼. 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. 𝑣′𝑖 𝑣𝑖 The main idea:

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

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

13. The Full Construction Algorithm A’ Input: 𝒗 ∈ ℝ𝑛 For each agent 𝑖: Construct cumulative allocation curve 𝐺𝑖 and convex hull 𝐺′𝑖. Let 𝐼1 , … , 𝐼𝘬 be the intervals where 𝐺𝑖≠ 𝐺′𝑖. If 𝑣𝑖∈ 𝐼𝘫, draw 𝑣′𝑖 ~𝐹𝑖|𝐼𝘫 . Otherwise set 𝑣′𝑖 = 𝑣𝑖. Return A(𝒗′) Claim 1: A’ is BIC. Claim 2: E[SW(A’)] ≥ E[SW(A)]. 𝑥𝑖(𝑣𝑖) 𝐺𝑖(𝑣𝑖) E[𝑥𝑖(𝑣𝑖)∗𝑣𝑖]

14. Practical Issues • Our construction requires that we know the allocation curves for algorithm A under distribution 𝐹1 ,…, 𝐹𝑛. • 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 𝜀 > 0 and black-box access to algorithm A, we can construct BIC mechanism M such that E[SW(M)] ≥ E[SW(A)] − 𝜀. Mechanism M uses poly(𝑛, 1/𝜀) calls to A.

15. 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.

16. Thank You