A Computational issues in time-inconsistent planning

Time-inconsistency refers to a paradox in decision making where agents exhibit inconsistent behaviors over time. Examples are procrastination where agents tends to postpone easy tasks, and abandonments where agents start a plan and quit in the middle. These behaviors are undesirable in that agents make clearly suboptimal decisions over optimal ones. To capture such behaviors and more importantly, to quantify inefficiency caused by such behaviors, [Kleinberg and Oren 2014] propose a graph model which is essentially the same as the standard planning model except for the cost structure. Using this model, they initiate the study of several interesting computation problems: 1) cost ratio: the worst ratio between the actual cost of the agent and the optimal cost, over all the graph instances; 2) motivating subgraph: how to motivate the agent to reach the goal by deleting nodes and edges; 3) Intermediate rewards: how to motivate agents to reach the goal by placing intermediate rewards. Kleinberg and Oren give partial answers to these questions, but the main problems are still open. In fact, they raise these problems as open problems in their original paper. In this paper, we give answers to all three open problems in [Kleinberg and Oren 2014]. First, we show a tight upper bound of cost ratio for graphs without Akerlof ’s structure, thus confirm the conjecture by Kleinberg and Oren that Akerlof ’s structure is indeed the worst case for cost ratio. Second, we prove that finding a motivating subgraph is NP-hard, showing that it is generally inefficient to motivate agents by deleting nodes and edges in the graph. Last but not least, we show that computing a strategy to place minimum amount of total reward is also NP-hard.