How to guide a present-biased agent through prescribed tasks?

TL;DR

This work studies how to guide a present-biased agent, modeled via the Kleinberg-Oren time-inconsistent planning framework, through prescribed tasks by graph modifications. It formulates two problems, $T$-path-Deletion and $T$-path-Addition, and analyzes them through parameterized complexity, establishing both hardness (NP-hardness and $\mathsf{W}[1]$-hardness) and tractable regimes. For deletion, the authors prove $\mathsf{W}[1]$-hardness parameterized by $k$ and $\mathsf{NP}$-hardness in general, yet obtain $\mathsf{FPT}$ algorithms when the overall path length is bounded by $m$ and offer a polynomial kernel parameterized by the graph's $\operatorname{fes}(G)$ via weight compression. For addition, they show $\mathsf{W}[1]$-hardness on a path with detours but identify fixed-parameter tractable cases when the added arc set has a separable structure, with time $2^{\tau} n^{O(1)}$ where $\tau$ is the largest intersection component size; a dynamic-programming approach generalizes these results. Together, the results map when principal interventions to direct time-inconsistent planning are computationally feasible, linking behavioral economics models to parameterized graph algorithms.

Abstract

The present bias is a well-documented behavioral trait that significantly influences human decision-making, with present-biased agents often prioritizing immediate rewards over long-term benefits, leading to suboptimal outcomes in various real-world scenarios. Kleinberg and Oren (2014) proposed a popular graph-theoretical model of inconsistent planning to capture the behavior of present-biased agents. In this model, a multi-step project is represented by a weighted directed acyclic task graph, where the agent traverses the graph based on present-biased preferences. We use the model of Kleinberg and Oren to address the principal-agent problem, where a principal, fully aware of the agent's present bias, aims to modify an existing project by adding or deleting tasks. The challenge is to create a modified project that satisfies two somewhat contradictory conditions. On one hand, the present-biased agent should select specific tasks deemed important by the principal. On the other hand, if the anticipated costs in the modified project become too high for the agent, there is a risk of the agent abandoning the entire project, which is not in the principal's interest. To tackle this issue, we leverage the tools of parameterized complexity to investigate whether the principal's strategy can be efficiently identified. We provide algorithms and complexity bounds for this problem.

Figures (6)

• Figure 1: For $\beta =1/3$, the agent will follow the path $sadet$ instead of selecting the shortest path $sabct$.
• Figure 2: Let $P_1=sabct$ and $P_2=sadt$. For $\beta =1/3$, the agent will follow the path $P_2$. Indeed, in node $s$, the estimated cost is $6+1/3(2+2+2)=8$, which is exactly the value $1/3 \cdot r$ of discounted reward, so the agent proceeds to $a$. When standing in $a$, the estimated cost of the remaining part of $P_1$ is now $3\frac{1}{3}$ and of $P_2$ is $3$. Both costs are less than the discounted reward, so the agent follows $P_2$.
• Figure 3: The construction of the graph $G'$ for \ref{['th_w1_k']}.
• Figure 4: The construction of the graph $G'$ for Theorem 2.
• Figure 5: Construction of the graph for reduction in Theorem \ref{['thm-hard-addition']}.

Theorems & Definitions (23)

• Theorem 1
• proof
• Corollary 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof