Abstract
We introduce and study incentive equilibria for multi-player mean-payoff games. Incentive equilibria generalise well-studied solution concepts such as Nash equilibria and leader equilibria. Recall that a strategy profile is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally. In the setting of incentive and leader equilibria, there is a distinguished player—called the leader—who can assign strategies to all other players, referred to as her followers. A strategy profile is a leader strategy profile if no player, except for the leader, can improve his payoff by changing his strategy unilaterally, and a leader equilibrium is a leader strategy profile with a maximal return for the leader. In the proposed case of incentive equilibria, the leader can additionally influence the behaviour of her followers by transferring parts of her payoff to her followers. The ability to incentivise her followers provides the leader with more freedom in selecting strategy profiles, and we show that this can indeed improve the leader’s payoff in such games. The key fundamental result of the paper is the existence of incentive equilibria in mean-payoff games. We further show that the decision problem related to constructing incentive equilibria is NP-complete. On a positive note, we show that, when the number of players is fixed, the complexity of the problem falls in the same class as two-player mean-payoff games. We present an implementation of the proposed algorithms, and discuss experimental results that demonstrate the feasibility of the analysis.
This work was supported by the EPSRC through grant EP/M027287/1 (Energy Efficient Control), by DARPA under agreement number FA8750-15-2-0096 and by the US National Science Foundation (NSF) under grant numbers CPS-1446900.
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Notes
- 1.
When translating a strategy in the presence of incentives \((\sigma ,\iota )\), the translation can be done by simulating the runs. The translation of the strategy profile \(\sigma \) is straight forward; it determines the choices at the respective last states from the gadgets. The translation of the incentives refer to the choices within the gadgets. They can be obtained by letting the leader make the decision to transfer h to follower p if, and only if, the sum of the incentives this follower p has collected in the game on the original MMPG is at least h higher than the sum of the utilities the leader has so far transferred to p in the gadgets. If passing through a gadget is counted as one step, all \(\liminf \) values agree on the original and its simulation. The back translation is even more direct: it suffices to wait till the end of each gadget, and then assign incentives accordingly.
- 2.
Including the first iteration of \(\pi _{i+1}\) is a technical necessity, as a complete iteration of \(\pi _{i+i}\) provides better guarantees, but without the inclusion of this guarantee, the \(\pi _j\)’s might grow too fast, preventing the existence of a limes.
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Gupta, A., Schewe, S., Trivedi, A., Deepak, M.S.K., Padarthi, B.K. (2016). Incentive Stackelberg Mean-Payoff Games. In: De Nicola, R., Kühn, E. (eds) Software Engineering and Formal Methods. SEFM 2016. Lecture Notes in Computer Science(), vol 9763. Springer, Cham. https://doi.org/10.1007/978-3-319-41591-8_21
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