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Fair Incentives for Repeated Engagement

Daniel Freund, Chamsi Hssaine

TL;DR

The paper addresses designing fair, unconditional monetary incentives for repeated engagement when agent retention is stochastic and heterogeneous by type. It shows that under stringent fairness constraints, a static fluid-optimal policy is asymptotically optimal in a large-market limit, while dynamic policies may outperform static ones only in discriminatory ways; a key two-reward structure enables efficient computation of the fluid optimum and yields convergence rates of $O(1/\theta)$ under mild conditions. The authors develop a deterministic relaxation (fluid model) and prove that the fluid heuristic achieves near-optimal long-run profit, with numerical experiments confirming the theory and illustrating when dispersion (lotteries) is beneficial. The work contributes to fair retention design by characterizing when fairness constraints restrict dynamic gains and by providing a fast, tractable policy construction applicable to large populations, with practical implications for loyalty programs and service platforms.

Abstract

We study a decision-maker's problem of finding optimal monetary incentive schemes for retention when faced with agents whose participation decisions (stochastically) depend on the incentive they receive. Our focus is on policies constrained to fulfill two fairness properties that preclude outcomes wherein different groups of agents experience different treatment on average. We formulate the problem as a high-dimensional stochastic optimization problem, and study it through the use of a closely related deterministic variant. We show that the optimal static solution to this deterministic variant is asymptotically optimal for the dynamic problem under fairness constraints. Though solving for the optimal static solution gives rise to a non-convex optimization problem, we uncover a structural property that allows us to design a tractable, fast-converging heuristic policy. Traditional schemes for retention ignore fairness constraints; indeed, the goal in these is to use differentiation to incentivize repeated engagement with the system. Our work (i) shows that even in the absence of explicit discrimination, dynamic policies may unintentionally discriminate between agents of different types by varying the type composition of the system, and (ii) presents an asymptotically optimal policy to avoid such discriminatory outcomes.

Fair Incentives for Repeated Engagement

TL;DR

The paper addresses designing fair, unconditional monetary incentives for repeated engagement when agent retention is stochastic and heterogeneous by type. It shows that under stringent fairness constraints, a static fluid-optimal policy is asymptotically optimal in a large-market limit, while dynamic policies may outperform static ones only in discriminatory ways; a key two-reward structure enables efficient computation of the fluid optimum and yields convergence rates of under mild conditions. The authors develop a deterministic relaxation (fluid model) and prove that the fluid heuristic achieves near-optimal long-run profit, with numerical experiments confirming the theory and illustrating when dispersion (lotteries) is beneficial. The work contributes to fair retention design by characterizing when fairness constraints restrict dynamic gains and by providing a fast, tractable policy construction applicable to large populations, with practical implications for loyalty programs and service platforms.

Abstract

We study a decision-maker's problem of finding optimal monetary incentive schemes for retention when faced with agents whose participation decisions (stochastically) depend on the incentive they receive. Our focus is on policies constrained to fulfill two fairness properties that preclude outcomes wherein different groups of agents experience different treatment on average. We formulate the problem as a high-dimensional stochastic optimization problem, and study it through the use of a closely related deterministic variant. We show that the optimal static solution to this deterministic variant is asymptotically optimal for the dynamic problem under fairness constraints. Though solving for the optimal static solution gives rise to a non-convex optimization problem, we uncover a structural property that allows us to design a tractable, fast-converging heuristic policy. Traditional schemes for retention ignore fairness constraints; indeed, the goal in these is to use differentiation to incentivize repeated engagement with the system. Our work (i) shows that even in the absence of explicit discrimination, dynamic policies may unintentionally discriminate between agents of different types by varying the type composition of the system, and (ii) presents an asymptotically optimal policy to avoid such discriminatory outcomes.

Paper Structure

This paper contains 63 sections, 33 theorems, 102 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Summary of contributions
  3. Structure of the paper.
  4. Related work
  5. Workforce capacity planning.
  6. Customer retention.
  7. Empirical work on effectiveness of monetary incentives.
  8. Algorithmic fairness.
  9. Preliminaries
  10. Technical notation.
  11. Basic setup
  12. Agents.
  13. Periods.
  14. Objective.
  15. Large-market regime
  16. ...and 48 more sections

Key Result

Proposition 2.2

Suppose $\widetilde{N}_i(0) = \mathbb{E}\left[\frac{N_i^\theta(0)}{\theta}\right]$ for all $i \in [K]$. Then, there exists a constant $C_0 > 0$ (independent of $T$, $\theta$) such that $\lim_{T\to\infty}\frac{\Delta(T)}{T} \leq C_0/\sqrt{\theta}$.

Figures (7)

  • Figure 1: Additive loss of the fluid (dark blue curve), deterministic (dark orange curve), and lottery (green curve) heuristics.
  • Figure 2: Departure probability functions for convex and concave mixtures
  • Figure 3: Impact of population composition on dispersion of optimal reward distribution
  • Figure 4: Departure probability function $\ell(r)$ for various values of $\epsilon$, with $K = 1$ and $v = 25$.
  • Figure 5: Dependence of profit on $\epsilon$ for three types, with $\lambda_1 + \lambda_2 + \lambda_3 = 10$, $v_1 = 25, v_2 = 30, v_3 = 40$, and $R(N) = 40\min\{N,D\}$, with $D \in \{25,100,400\}$.
  • ...and 2 more figures

Theorems & Definitions (75)

  • Remark 2.1
  • Proposition 2.2
  • Definition 3.1: Group-fair policy
  • Theorem 3.2
  • Proposition 3.3
  • Definition 3.4: Cyclic policy
  • Proposition 3.5
  • Proposition 4.1
  • Theorem 4.2
  • Proposition 4.3
  • ...and 65 more