The tendency to overestimate immediate utility is a common cognitive bias. As a result people behave inconsistently over time and fail to reach long-term goals. Behavioral economics tries to help affected individuals by implementing external incentives. However, designing robust incentives is often difficult due to imperfect knowledge of the parameter @math quantifying a person's present bias. Using the graphical model of Kleinberg and Oren, we approach this problem from an algorithmic perspective. Based on the assumption that the only information about @math is its membership in some set @math , we distinguish between two models of uncertainty: one in which @math is fixed and one in which it varies over time. As our main result we show that the conceptual loss of efficiency incurred by incentives in the form of penalty fees is at most @math in the former and @math in the latter model. We also give asymptotically matching lower bounds and approximation algorithms.
The graphical model is of particular interest to us as it provides a natural framework for a design problem frequently encountered in behavioral economics. Given a certain social or economic setting, the problem is to improve a time-inconsistent person's performance via various sorts of incentives , such as monetary rewards, deadlines or penalty fees, see e.g. @cite_7 . Using the graphical model, Kleinberg and Oren demonstrate how a strategic choice reduction can incentivize people to reach predefined goals @cite_4 . To implement their incentives, they simply remove the corresponding edges from the task graph. However, there is a computational drawback to this approach. As we have shown in previous work, an optimal set of edges to remove from a task graph with @math nodes is NP-hard to approximate within a factor less than @math @cite_8 . A more general form of incentives avoiding these harsh complexity theoretic limitations are penalty fees. In the graphical model penalty fees are at least as powerful as choice reduction and admit a polynomial time @math -approximation @cite_3 .
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People tend to behave inconsistently over time due to an inherent present bias. As this may impair performance, social and economic settings need to be adapted accordingly. Common tools to reduce the impact of time-inconsistent behavior are penalties and prohibition. Such tools are called commitment devices. In recent work Kleinberg and Oren [EC, 2014] connect the design of a prohibition-based commitment device to a combinatorial problem in which edges are removed from a task graph G with n nodes. However, this problem is NP-hard to approximate within a ratio less than n^(1 2) 3 [Albers and Kraft, WINE, 2016]. To address this issue, we propose a penalty-based commitment device that does not delete edges, but raises their cost. The benefits of our approach are twofold. On the conceptual side, we show that penalties are up to 1 beta times more efficient than prohibition, where 0 < beta <= 1 parameterizes the present bias. On the computational side, we improve approximability by presenting a 2-approximation algorithm for allocating penalties. To complement this result, we prove that optimal penalties are NP-hard to approximate within a ratio of 1.08192. In many settings, people exhibit behavior that is inconsistent across time ' we allocate a block of time to get work done and then procrastinate, or put effort into a project and then later fail to complete it. An active line of research in behavioral economics and related fields has developed and analyzed models for this type of time-inconsistent behavior. Here we propose a graph-theoretic model of tasks and goals, in which dependencies among actions are represented by a directed graph, and a time-inconsistent agent constructs a path through this graph. We first show how instances of this path-finding problem on different input graphs can reconstruct a wide range of qualitative phenomena observed in the literature on time-inconsistency, including procrastination, abandonment of long-range tasks, and the benefits of reduced sets of choices. We then explore a set of analyses that quantify over the set of all graphs; among other results, we find that in any graph, there can be only polynomially many distinct forms of time-inconsistent behavior; and any graph in which a time-inconsistent agent incurs significantly more cost than an optimal agent must contain a large 'procrastination' structure as a minor. Finally, we use this graph-theoretic model to explore ways in which tasks can be designed to help motivate agents to reach designated goals. We study the complexity of motivating time-inconsistent agents to complete long term projects in a graph-based planning model as proposed by Kleinberg and Oreni¾?[5]. Given a task graph G with n nodes, our objective is to guide an agent towards a target node t under certain budget constraints. The crux is that the agent may change its strategy over time due to its present-bias. We consider two strategies to guide the agent. First, a single reward is placed at t and arbitrary edges can be removed from G. Secondly, rewards can be placed at arbitrary nodes of G but no edges must be deleted. In both cases we show that it is NP-complete to decide if a given budget is sufficient to guide the agent. For the first setting, we give complementing upper and lower bounds on the approximability of the minimum required budget. In particular, we devise a @math 1+n-approximation algorithm and prove NP-hardness for ratios greater than @math n 3. Finally, we argue that the second setting does not permit any efficient approximation unless @math P=NP.