[Paper Review] Mixture Selection, Mechanism Design, and Signaling

Yu Cheng, Ho Yee Cheung+4 more|arXiv (Cornell University)|Aug 14, 2015

Auction Theory and Applications17 references9 citations

TL;DR

This paper introduces a novel algorithmic framework for mixture selection—optimizing a function $g(Ax)$ over a probability simplex—by leveraging Lipschitz continuity and a new notion of noise stability. It presents an additive polynomial-time approximation scheme (PTAS) when both properties are bounded, resolving key open problems in mechanism design and optimal signaling.

ABSTRACT

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function $g$ from the $n$-dimensional hypercube to the bounded interval $[-1,1]$, and an $n imes m$ matrix $A$ with bounded entries, maximize $g(Ax)$ over $x$ in the $m$-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling). We present an approximation algorithm for this problem when $g$ simultaneously satisfies two smoothness properties: Lipschitz continuity with respect to the $L^\infty$ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which low-probability errors in the inputs of $g$ can impact its output. When $g$ is both $O(1)$-Lipschitz continuous and $O(1)$-stable, we obtain an (additive) PTAS for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive FPTAS. We apply our algorithm to different game-theoretic applications from mechanism design and optimal signaling. We make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen and Sheffet; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and Câmara.

Motivation & Objective

• To formalize and study the mixture selection problem as a central algorithmic primitive in mechanism design and Bayesian game theory.
• To identify two smoothness properties—Lipschitz continuity in $L^∞$ and noise stability—that govern the tractability of mixture selection.
• To develop a general approximation algorithm for mixture selection that degrades gracefully with the smoothness parameters of $g$.
• To resolve open problems in revenue-maximizing signaling, lottery design, and optimal signaling in voting and normal-form games.
• To establish tight complexity bounds by proving that neither smoothness condition alone suffices for an additive PTAS, and both together do not yield an FPTAS.

Proposed method

• Define noise stability as a measure controlling the impact of low-probability, correlated input errors on $g$'s output, distinct from prior stability notions.
• Use empirical sampling from a distribution $x$ to construct a sparse approximation $\widetilde{x}$, bounding the expected error in $g(Ax)$ via two components: large errors (controlled by noise stability) and small errors (controlled by Lipschitz continuity).
• Apply Fourier analysis on Boolean extensions of $g$ to bound the error in terms of low-degree Fourier coefficients, assuming algebraic stability.
• Instantiate the general framework into a PTAS by setting sample size $s = \Omega(\frac{1}{\delta^2} \log \frac{k}{\epsilon})$, ensuring concentration of $\widetilde{x}$ around $x$.
• Prove that for $c$-Lipschitz and $\beta$-noise stable $g$, the expected objective $\mathbb{E}[g(A\widetilde{x})] \geq g(Ax) - \epsilon - c\delta$, enabling additive approximation.
• Leverage the connection between noise stability and non-negative, low-degree Fourier spectra to extend results to algebraically stable functions.

Experimental results

Research questions

• RQ1 Can mixture selection be approximated efficiently when the objective function $g$ is both Lipschitz continuous and noise stable?
• RQ2 Is noise stability necessary for an additive PTAS in mixture selection, or can it be replaced by other smoothness conditions?
• RQ3 Can the mixture selection framework resolve open problems in mechanism design and optimal signaling, such as revenue-maximizing signaling in second-price auctions?
• RQ4 Does the combination of Lipschitz continuity and noise stability suffice for an additive fully polynomial-time approximation scheme (FPTAS)?
• RQ5 Which canonical game-theoretic objectives (e.g., lottery design, voting, revenue maximization) satisfy the required smoothness and algebraic stability properties?

Key findings

• An additive PTAS exists for mixture selection when $g$ is both $O(1)$-Lipschitz continuous and $O(1)$-noise stable, with approximation error $\epsilon + c\delta$.
• Neither Lipschitz continuity nor noise stability alone suffices for an additive PTAS, indicating both are necessary for the result.
• The combination of both properties does not yield an additive FPTAS, showing inherent limitations in achieving fully polynomial approximation.
• The framework resolves the small-menu lottery design problem posed by Dughmi, Han, and Nisan (2014) as a special case.
• An additive PTAS is obtained for revenue-maximizing signaling in Bayesian second-price auctions, resolving an open problem by Emek et al. (2012) and Miltersen and Sheffet (2012).
• A quasipolynomial-time approximation scheme is designed for optimal signaling in normal-form games, and an approximation algorithm is developed for the voting model of Alonso and Câmara (2014).