Motivating Time-Inconsistent Agents: A Computational Approach

In this paper we investigate the computational complexity of motivating time-inconsistent agents to complete long term projects. We resort to an elegant graph-theoretic model, introduced by Kleinberg and Oren, which consists of a task graph $G$ with $n$ vertices, including a source $s$ and target $t$, and an agent that incrementally constructs a path from $s$ to $t$ in order to collect rewards. The twist is that the agent is present-biased and discounts future costs and rewards by a factor $β\in [0,1]$. Our design objective is to ensure that the agent reaches $t$ i.e.\ completes the project, for as little reward as possible. Such graphs are called motivating. We consider two strategies. First, we place a single reward $r$ at $t$ and try to guide the agent by removing edges from $G$. We prove that deciding the existence of such motivating subgraphs is NP-complete if $r$ is fixed. More importantly, we generalize our reduction to a hardness of approximation result for computing the minimum $r$ that admits a motivating subgraph. In particular, we show that no polynomial-time approximation to within a ratio of $\sqrt{n}/4$ or less is possible, unless ${\rm P}={\rm NP}$. Furthermore, we develop a $(1+\sqrt{n})$-approximation algorithm and thus settle the approximability of computing motivating subgraphs. Secondly, we study motivating reward configurations, where non-negative rewards $r(v)$ may be placed on arbitrary vertices $v$ of $G$. The agent only receives the rewards of visited vertices. Again we give an NP-completeness result for deciding the existence of a motivating reward configuration within a fixed budget $b$. This result even holds if $b=0$, which in turn implies that no efficient approximation of a minimum $b$ within a ration grater or equal to $1$ is possible, unless ${\rm P}={\rm NP}$.