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Algorithmic Decision-Making under Agents with Persistent Improvement

Tian Xie, Xuwei Tan, Xueru Zhang

TL;DR

This paper develops a dynamic model to characterize persistent improvements and constructs a Stackelberg game to model the interplay between agents and the decision-maker, and analytically characterize the equilibrium strategies and identifies conditions under which agents have incentives to invest efforts to improve their qualifications.

Abstract

This paper studies algorithmic decision-making under human's strategic behavior, where a decision maker uses an algorithm to make decisions about human agents, and the latter with information about the algorithm may exert effort strategically and improve to receive favorable decisions. Unlike prior works that assume agents benefit from their efforts immediately, we consider realistic scenarios where the impacts of these efforts are persistent and agents benefit from efforts by making improvements gradually. We first develop a dynamic model to characterize persistent improvements and based on this construct a Stackelberg game to model the interplay between agents and the decision-maker. We analytically characterize the equilibrium strategies and identify conditions under which agents have incentives to improve. With the dynamics, we then study how the decision-maker can design an optimal policy to incentivize the largest improvements inside the agent population. We also extend the model to settings where 1) agents may be dishonest and game the algorithm into making favorable but erroneous decisions; 2) honest efforts are forgettable and not sufficient to guarantee persistent improvements. With the extended models, we further examine conditions under which agents prefer honest efforts over dishonest behavior and the impacts of forgettable efforts.

Algorithmic Decision-Making under Agents with Persistent Improvement

TL;DR

This paper develops a dynamic model to characterize persistent improvements and constructs a Stackelberg game to model the interplay between agents and the decision-maker, and analytically characterize the equilibrium strategies and identifies conditions under which agents have incentives to invest efforts to improve their qualifications.

Abstract

This paper studies algorithmic decision-making under human's strategic behavior, where a decision maker uses an algorithm to make decisions about human agents, and the latter with information about the algorithm may exert effort strategically and improve to receive favorable decisions. Unlike prior works that assume agents benefit from their efforts immediately, we consider realistic scenarios where the impacts of these efforts are persistent and agents benefit from efforts by making improvements gradually. We first develop a dynamic model to characterize persistent improvements and based on this construct a Stackelberg game to model the interplay between agents and the decision-maker. We analytically characterize the equilibrium strategies and identify conditions under which agents have incentives to improve. With the dynamics, we then study how the decision-maker can design an optimal policy to incentivize the largest improvements inside the agent population. We also extend the model to settings where 1) agents may be dishonest and game the algorithm into making favorable but erroneous decisions; 2) honest efforts are forgettable and not sufficient to guarantee persistent improvements. With the extended models, we further examine conditions under which agents prefer honest efforts over dishonest behavior and the impacts of forgettable efforts.
Paper Structure (50 sections, 10 theorems, 29 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 50 sections, 10 theorems, 29 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Strategic Manipulation
  4. Improvement
  5. Other Agent Dynamics
  6. Problem Formulation
  7. Decision-maker's policy.
  8. Agent qualification dynamics.
  9. Agent's utility & action.
  10. Stackelberg game.
  11. Manipulation & forgetting.
  12. Objective.
  13. Improvement & Optimal Effort
  14. Decision-maker's policy to incentivize improvement
  15. Impact of Manipulative Behavior
  16. ...and 35 more sections

Key Result

Lemma 3.2

Consider an agent with initial similarity $x_0:=q_0^Td > 0$. If he/she makes an effort $k$ and improves qualification profile $q_t$ based on dynamics in eq:dynamics, then $q_t$ converges to the desired profile $d$. The evolution of the similarity $x_t:=q_t^Td$ is given by:

Figures (12)

  • Figure 1: Dynamics of agent's qualification $q_t$.
  • Figure 2: Impact of effort $k$ on agent utility $U$ under different $C:=C(\theta,r,x_0)$: $\exists m>0$ such that agents have incentives to invest and improve their qualifications if $C< m$.
  • Figure 3: Optimal thresholds $\theta^*$ under different density functions $f$ and discounting factors $r$.
  • Figure 4: Upper bound $\widehat{k}$ of the optimal effort as a function of $x_0$.
  • Figure 5: From the left to the right are: optimal thresholds to incentivize improvement for males/females; manipulation probability under the thresholds for males/females for Exam data.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Lemma 3.2: Convergence of qualification
  • Theorem 4.1: Improvement & optimal effort
  • Theorem 5.1: Existence of optimal threshold
  • Corollary 5.2
  • Theorem 6.1
  • Theorem 7.1: Convergence of qualification under forgetting
  • Theorem 7.2
  • Theorem A.1
  • Lemma F.1
  • Lemma F.2