A Stay-in-a-Set Game without a Stationary Equilibrium Mikhail Raskin - PowerPoint PPT Presentation

A Stay-in-a-Set Game without a Stationary Equilibrium Mikhail Raskin , raskin@mccme.ru 1 Kristoffer Arnsfelt Hansen September 3, 2019 1 The author has received funding from ERC under the EU H2020 programme grant No 787367 (PaVeS) and from the ANR project GraphEn / ANR-15-CE40-0009. K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 1 / 11

Setting Turn-based (perfect information) Finite Stochastic Independent safety goals K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 2 / 11

Secchi, Sudderth 2002 Exact Nash equilibrium exists with bounded memory Remember who has lost K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 3 / 11

Existence proof: overview Induction 1 player: Markov decision process Arbitrary positional strategy for each player in case of loss Uniform bound on time until someone loses or loss becomes impossible Payoffs at that point: by inductive assumption (or obvious) K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 4 / 11

Question For 2 players, is there always zero memory Nash equilibrium (instead of 2 bits of memory)? (i.e. equilibrium in stationary strategies) K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 5 / 11

Similar questions Discounted payoffs: stationary equilibrium Deterministic game: stationary equilibrium Recursive/imperfect information games: no stationary ε -Nash equilibrium K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 6 / 11

Example without stationary equilibrium 3 / 4 c 1 / 4 c 1 L 2 2 q q 1 / 4 1 / 4 1 / 2 1 / 4 W 1 / 4 1 / 2 L K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 7 / 11

Example without stationary equilibrium: incentives If cycle is left, both P1 and P2 prefer that P1 leaves For P1 best case is staying For P2 staying means loss Note that ε -equilibrium exists K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 8 / 11

Open questions (for two players) Stationary ε -equilibrium? Stationary equilibrium for reachability conditions? Stationary equilibrium for deterministic recursive games with reachability conditions? K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 9 / 11

Thanks for your attention! Questions? K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 10 / 11

Summary Discounted payoffs: stationary equilibrium Deterministic game: stationary equilibrium Recursive/imperfect information games: no stationary ε -Nash equilibrium Two-player turn-based stay-in-a-set game: no stationary equilibrium Stationary ε -equilibrium? Stationary equilibrium for reachability conditions? Stationary equilibrium for deterministic recursive games with reachability conditions? K. A. Hansen & M. Raskin Stay-in-Set Game without Stationary NE September 3, 2019 11 / 11