Abstract
We present a novel graph-theoretic principal-agent model in which the agent is present biased (a bias that was well studied in behavioral economics). Our model captures situations in which a principal guides an agent in a complex multi-step project. We model the different steps and branches of the project as a directed acyclic graph with a source and a target, in which each edge has the cost for completing a corresponding task. If the agent reaches the target it receives some fixed reward R. We assume that the present-biased agent traverses the graph according to the framework of Kleinberg and Oren (EC’14) and as such will continue traversing the graph as long as his perceived cost is less than R. We further assume that each edge is assigned a value and if the agent reaches the target the principal’s payoff is the sum of values of the edges on the path that the agent traversed. Our goal in this work is to understand whether the principal can efficiently compute a subgraph that maximizes his payoff among all subgraphs in which the agent reaches the target. For this central question we provide both impossibility results and algorithms.
The work was done while D. Soker was a student at Ben-Gurion University.
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Notes
- 1.
In quasi-hyperbolic discounting a cost that will be incurred t steps from now is discounted by \(\beta \cdot \delta ^t\). Where \(\beta \in [0,1]\) is a present bias parameter and \(\delta \in [0,1]\) is an exponential discounting parameter. Our model is equivalent to the \((\beta ,\delta )\) model for \(\beta =1/b\) and \(\delta =1\). Similar to [9] and some of its follow-ups we focus on the case that \(\delta =1\) to highlight the effects of present bias.
- 2.
XP is the class of parameterized problems that can be solved in time \(n^{f(k)}\), where k is the parameter, n is the input’s size and f is a computable function.
- 3.
In order to simplify the analysis we require the set of potential shortcuts to include a path for each node \(u_i\in S(P)\). As a result, in some cases the path \(\hat{P}(u_i)\) is not a proper shortcut and all its edges are in \(P \cup (\mathcal {\hat{P}} - \hat{P}(u_i))\).
- 4.
We can easily extend this set to a set of potential shortcuts.
- 5.
We assume that in case that \(b\cdot w^{-} + d_H(u_{i+1},t) = b\cdot w^{+} + (d_H(u_i,t)-w^{+})\) the agent will break ties in favor of taking the shortcut.
- 6.
In this construction we ignore the costs of the edges.
- 7.
For example, we can go over the graph in reverse topological order and update for each node the length of the maximum path connecting it to t.
- 8.
By construction \(\forall i,j\in X\) we have that \(x_i=x_j\) and \(x_i=y_j\).
- 9.
By replacing the list with a binary matrix of size \((|V|\times |V|)^k\) in which a cell \((h_1,l_1,\ldots ,h_k,l_k)\) represents whether there are k converging paths such that each path i has \(h_i\) edges of weight \(w^+\) and \(l_i\) edges of weight \(w^-\), we can reduce the running time to \(O(k\cdot |V|^{4k})\).
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Oren, S., Soker, D. (2019). Principal-Agent Problems with Present-Biased Agents. In: Fotakis, D., Markakis, E. (eds) Algorithmic Game Theory. SAGT 2019. Lecture Notes in Computer Science(), vol 11801. Springer, Cham. https://doi.org/10.1007/978-3-030-30473-7_16
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