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Learning under Imitative Strategic Behavior with Unforeseeable Outcomes

Tian Xie, Zhiqun Zuo, Mohammad Mahdi Khalili, Xueru Zhang

TL;DR

This paper proposes a Stackelberg game to model the interplay between individuals and the decision-maker, and theoretically illustrates how a decision-maker with adjusted preferences may simultaneously disincentivize manipulation, incentivize improvement, and promote fairness.

Abstract

Machine learning systems have been widely used to make decisions about individuals who may behave strategically to receive favorable outcomes, e.g., they may genuinely improve the true labels or manipulate observable features directly to game the system without changing labels. Although both behaviors have been studied (often as two separate problems) in the literature, most works assume individuals can (i) perfectly foresee the outcomes of their behaviors when they best respond; (ii) change their features arbitrarily as long as it is affordable, and the costs they need to pay are deterministic functions of feature changes. In this paper, we consider a different setting and focus on imitative strategic behaviors with unforeseeable outcomes, i.e., individuals manipulate/improve by imitating the features of those with positive labels, but the induced feature changes are unforeseeable. We first propose a Stackelberg game to model the interplay between individuals and the decision-maker, under which we examine how the decision-maker's ability to anticipate individual behavior affects its objective function and the individual's best response. We show that the objective difference between the two can be decomposed into three interpretable terms, with each representing the decision-maker's preference for a certain behavior. By exploring the roles of each term, we theoretically illustrate how a decision-maker with adjusted preferences may simultaneously disincentivize manipulation, incentivize improvement, and promote fairness. Such theoretical results provide a guideline for decision-makers to inform better and socially responsible decisions in practice.

Learning under Imitative Strategic Behavior with Unforeseeable Outcomes

TL;DR

This paper proposes a Stackelberg game to model the interplay between individuals and the decision-maker, and theoretically illustrates how a decision-maker with adjusted preferences may simultaneously disincentivize manipulation, incentivize improvement, and promote fairness.

Abstract

Machine learning systems have been widely used to make decisions about individuals who may behave strategically to receive favorable outcomes, e.g., they may genuinely improve the true labels or manipulate observable features directly to game the system without changing labels. Although both behaviors have been studied (often as two separate problems) in the literature, most works assume individuals can (i) perfectly foresee the outcomes of their behaviors when they best respond; (ii) change their features arbitrarily as long as it is affordable, and the costs they need to pay are deterministic functions of feature changes. In this paper, we consider a different setting and focus on imitative strategic behaviors with unforeseeable outcomes, i.e., individuals manipulate/improve by imitating the features of those with positive labels, but the induced feature changes are unforeseeable. We first propose a Stackelberg game to model the interplay between individuals and the decision-maker, under which we examine how the decision-maker's ability to anticipate individual behavior affects its objective function and the individual's best response. We show that the objective difference between the two can be decomposed into three interpretable terms, with each representing the decision-maker's preference for a certain behavior. By exploring the roles of each term, we theoretically illustrate how a decision-maker with adjusted preferences may simultaneously disincentivize manipulation, incentivize improvement, and promote fairness. Such theoretical results provide a guideline for decision-makers to inform better and socially responsible decisions in practice.
Paper Structure (39 sections, 8 theorems, 12 equations, 19 figures, 7 tables)

This paper contains 39 sections, 8 theorems, 12 equations, 19 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Individual's best response.
  4. Decision-maker's optimal policy
  5. Decomposition of the Objective Difference
  6. Strategic decision-maker with adjusted preferences
  7. Impacts of Adjusting Preferences
  8. Preferences shift the optimal threshold
  9. Preferences as (dis)incentives for manipulation
  10. Preferences shape algorithmic fairness
  11. Experiments
  12. FICO data Hardt2016b.
  13. Robustness of results when $q, \epsilon$ are noisy.
  14. Gaussian Data.
  15. Conclusions & Limitations
  16. ...and 24 more sections

Key Result

Theorem 2.3

Under Assumption assumption: cost, $P_M(\theta)$ is continuous and satisfies the following: (i) If $q+\epsilon \ge 1$, then $P_M(\theta)$ strictly increases. (ii) If $q+\epsilon < 1$, then $P_M(\theta)$ first increases and then decreases, thereby existing a unique maximizer $\theta_{max}$. Moreover,

Figures (19)

  • Figure 1: Illustration of the strategic interaction
  • Figure 2: Illustration of scenario 1 (left) and scenario 2 (right) in Thm. \ref{['theorem: incentivize']}: adjusting preferences decreases manipulation probability $P_M(\theta)$.
  • Figure 3: $P_M(\theta)$ of Caucasian and African American (left plot) and of Asian and Hispanic (right plot).
  • Figure 4: Impact of adjusted preferences: Caucasian and African American (left plot), Asian and Hispanic (right plot)
  • Figure 5: Impact of adjusted preferences (FICO data) when there is a Gaussian noise on $q$. The noises have $0$ mean, and $0.05, 0.1$ standard deviation from the left two plots to the right two plots.
  • ...and 14 more figures

Theorems & Definitions (8)

  • Theorem 2.3: Manipulation Probability
  • Theorem 4.1: Comparison of strategic and non-strategic policy
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 4.5: Preferences serve as (dis)incentives
  • Theorem 4.6: Promote fairness while disincentivizing manipulation
  • Corollary 4.7