Consider a standard decision making setting that includes a set of possible actions (decisions or policies), and a set of individuals who assign utilities to the actions.

In contrast, the mathematics needed to analyze such schemes is what forms the focus in Stochastic Approximation (SA) theory [2, 4]. More generally, SA refers to an iterative scheme that helps find zeroes or optimal points of a function, for which only noisy evaluationsarepossible.

Theyaredesigned tohaveagame-theoretic equilibrium where everyone reveals their private information truthfully.

A.1 ProofofTheorem3.1 The proof can be found in (Hardt et al., 2016). We provide the proof in Appendix for reference. Firstly, it is easy to see that h1(w) and h2(w) are both η-approximate β-gradient Lipschitz, which satisfiesinequatilyinEq. (A.1). A.5 ProofofTheorem5.1 The proof follows the standard techniques for uniform stability. We need to replace the nonexpansive property used in standard analysis by the approximately non-expansive property.