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Cops and Robbers: The Cost of Drunkenness Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thanasis Kehagias, Thessaloniki Faculty of Engineering, The Cost of Aristotle

  1. Cops and Robbers: The Cost of Drunkenness The Cost of Drunkenness Thanasis Kehagias, Faculty of In “classic” CR game both cop and robber are adversarial Engineering, Aristotle University of The cop moves to capture the robber as fast as possible Thessaloniki The robber moves to avoid capture for as long as possible The Cost of What if the robber is drunk ?, i.e., he performs a random Drunkenness walk . Outline Visible The adversarial cop should be able to capture the drunk Adversarial Robber robber no later than the adversarial one. Visible Drunk We will try to quantify this intuition Robber Cost of Two papers: Drunkenness Some remarks on cops and drunk robbers (Kehagias + Algorithm + Video Pralat, TCS 2012) Cops and invisible robbers: The cost of drunkenness Invisible Adversarial (Kehagias + Mitsche + Pralat, TCS 2013) Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 2 / 33 Invisible Cost

  2. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  3. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  4. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  5. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  6. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  7. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  8. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  9. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  10. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  11. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  12. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  13. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  14. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  15. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Rescue in Contaminated Environment Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  16. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Rescue in Contaminated Environment Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  17. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  18. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  19. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  20. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  21. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  22. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  23. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  24. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  25. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  26. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  27. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  28. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  29. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Then ct ( G ) = T ( σ ∗ C , σ ∗ R ) is the capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  30. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Then ct ( G ) = T ( σ ∗ C , σ ∗ R ) is the capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  31. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  32. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  33. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  34. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  35. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Theorem Cost of Drunkenness There exists a memoryless strategy σ ∗ C s.t. Algorithm + Video E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  36. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Theorem Cost of Drunkenness There exists a memoryless strategy σ ∗ C s.t. Algorithm + Video E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Invisible Adversarial Robber Then dct ( G ) = E ( T | σ ∗ C ) is the drunk capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  37. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  38. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  39. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  40. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Theorem Adversarial Robber For every real constant c ∈ [ 1, ∞ ) there exists a sequence of Visible Drunk n ) ∞ graphs ( G c n = 1 s.t. lim n → ∞ F ( G n ) = c. Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  41. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Theorem Adversarial Robber For every real constant c ∈ [ 1, ∞ ) there exists a sequence of Visible Drunk n ) ∞ graphs ( G c n = 1 s.t. lim n → ∞ F ( G n ) = c. Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  42. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  43. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  44. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  45. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible We have computed ct ( G ) , dct ( G ) , H ( G ) for specific Adversarial Robber graph families Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  46. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible We have computed ct ( G ) , dct ( G ) , H ( G ) for specific Adversarial Robber graph families Visible Drunk F ( G ) Robber Graph Cost of 2 + o ( 1 ) Path Drunkenness 2 + o ( 1 ) Cycle Algorithm + Video 1 + o ( 1 ) Tree Invisible 8 / 3 + o ( 1 ) Grid Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  47. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  48. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Here is another video Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  49. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Here is another video ; something is wrong with it ... Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  50. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  51. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  52. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  53. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  54. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Then ct i ( G ) = E ( T | σ ∗ C , σ ∗ Robber R ) is the invisible capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  55. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Then ct i ( G ) = E ( T | σ ∗ C , σ ∗ Robber R ) is the invisible capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  56. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  57. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  58. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  59. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  60. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Theorem Robber There exists a memoryless strategy σ ∗ Cost of C s.t. Drunkenness Algorithm + E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  61. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Theorem Robber There exists a memoryless strategy σ ∗ Cost of C s.t. Drunkenness Algorithm + E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Video Invisible Adversarial Then dct i ( G ) = E ( T | σ ∗ Robber C ) is the invisible drunk capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  62. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  63. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  64. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  65. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  66. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Question: Is there a G s.t. F i ( G ) ∈ ( 1, 2 ) ? ( COD gap ) Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  67. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Question: Is there a G s.t. F i ( G ) ∈ ( 1, 2 ) ? ( COD gap ) Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  68. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  69. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  70. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  71. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  72. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  73. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial We have computed ct i ( G ) , dct i ( G ) , F i ( G ) for specific Robber Visible Drunk graph families Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  74. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial We have computed ct i ( G ) , dct i ( G ) , F i ( G ) for specific Robber Visible Drunk graph families Robber F i ( G ) Cost of Graph Drunkenness 2 + o ( 1 ) Path Algorithm + Video 2 + o ( 1 ) Cycle Invisible Tree Θ ( log n ) Adversarial √ n ) Robber ≤ O ( n 3 / 2 5 Grid Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  75. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33 Invisible Cost

  76. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33 Invisible Cost

  77. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  78. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  79. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Drunk COV: H d ( G ) = dct i ( G ) Visible dct ( G ) Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  80. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Drunk COV: H d ( G ) = dct i ( G ) Visible dct ( G ) Adversarial Robber The Role of Visibility in Pursuit / Evasion Games , Visible Drunk Kehagias + Mitsche + Pralat, arXiv:1402.6136 Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  81. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  82. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  83. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  84. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  85. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  86. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  87. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline For every state there is a (trivial) next state transition Visible Adversarial probability. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  88. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline For every state there is a (trivial) next state transition Visible Adversarial probability. Robber The lazy robber is also a FSA (how many states?) Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

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