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Infinite Games Martin Zimmermann Saarland University October 21st, 2014 Research Training Group SCARE, Oldenburg, Germany Martin Zimmermann Saarland University Infinite Games 1/30 Lets Play S T You move at circles and want to reach T

  1. Winning A game G = ( A , Win ) consists of an arena A and a set Win ⊆ V ω of winning plays for Player 0. Set of winning plays for Player 1: V ω \ Win . Strategy σ for Player i is winning strategy from v , if every play that starts in v and is consistent with σ is winning for him. Winning region W i ( G ) : set of vertices from which Player i has a winning strategy. Always: W 0 ( G ) ∩ W 1 ( G ) = ∅ . G determined, if W 0 ( G ) ∪ W 1 ( G ) = V . Solving a game: determine the winning regions and winning strategies. Martin Zimmermann Saarland University Infinite Games 8/30

  2. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Martin Zimmermann Saarland University Infinite Games 9/30

  3. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Martin Zimmermann Saarland University Infinite Games 9/30

  4. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Martin Zimmermann Saarland University Infinite Games 9/30

  5. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } Martin Zimmermann Saarland University Infinite Games 9/30

  6. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } There are many other winning conditions. Martin Zimmermann Saarland University Infinite Games 9/30

  7. What Are We Interested in? Given a type of winning condition (e.g., reachability, parity, Muller),.. .. are games with this condition always determined? .. what kind of strategy do the players need (e.g., positional, finite-state)? .. if finite-state strategies are necessary, how large do they have to be? How hard is it to solve the game? Martin Zimmermann Saarland University Infinite Games 10/30

  8. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 11/30

  9. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  10. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  11. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  12. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  13. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  14. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Martin Zimmermann Saarland University Infinite Games 13/30

  15. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Martin Zimmermann Saarland University Infinite Games 13/30

  16. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  17. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 R Martin Zimmermann Saarland University Infinite Games 13/30

  18. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 A 2 R Martin Zimmermann Saarland University Infinite Games 13/30

  19. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 A 2 · · · A n = A n + 1 R Martin Zimmermann Saarland University Infinite Games 13/30

  20. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  21. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  22. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Remark: Attractors can be computed in linear time in | E | . Martin Zimmermann Saarland University Infinite Games 13/30

  23. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 14/30

  24. Parity Games Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } 4 3 2 1 0 1 2 3 0 Martin Zimmermann Saarland University Infinite Games 15/30

  25. Parity Games Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } 4 4 3 3 2 2 1 1 0 0 1 1 2 2 3 3 0 0 Martin Zimmermann Saarland University Infinite Games 15/30

  26. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Martin Zimmermann Saarland University Infinite Games 16/30

  27. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Martin Zimmermann Saarland University Infinite Games 16/30

  28. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. Martin Zimmermann Saarland University Infinite Games 16/30

  29. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. n = 1 trivial: c or c Martin Zimmermann Saarland University Infinite Games 16/30

  30. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. n = 1 trivial: c or c Player i wins iff Par ( c ) = i Martin Zimmermann Saarland University Infinite Games 16/30

  31. Proof Sketch Now n > 1 and min Ω( V ) = 0. Attr 0 (Ω − 1 ( 0 )) Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  32. Proof Sketch Martin Zimmermann Saarland University Infinite Games 17/30

  33. Proof Sketch Induction hypothesis applicable.. Martin Zimmermann Saarland University Infinite Games 17/30

  34. Proof Sketch .. yields winning regions W ′ i and positional strategies σ ′ , τ ′ . W ′ 0 W ′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  35. Proof Sketch W ′ 1 empty: W ′ 0 Martin Zimmermann Saarland University Infinite Games 17/30

  36. Proof Sketch W ′ 1 empty: Player 0 wins from everywhere. Winning strategy: combine σ ′ and attractor strategy, play arbitrarily at Ω − 1 ( 0 ) . W ′ 0 Attr 0 (Ω − 1 ( 0 )) Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  37. Proof Sketch W ′ 1 non-empty: W ′ 0 Attr 0 (Ω − 1 ( 0 )) W ′ 1 Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  38. Proof Sketch W ′ 1 non-empty: Attr 1 ( W ′ 1 ) W ′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  39. Proof Sketch W ′ 1 non-empty: Martin Zimmermann Saarland University Infinite Games 17/30

  40. Proof Sketch W ′ 1 non-empty: Induction hypothesis applicable.. Martin Zimmermann Saarland University Infinite Games 17/30

  41. Proof Sketch W ′ 1 non-empty:.. yields winning regions W ′′ i and positional strategies σ ′′ , τ ′′ . W ′′ 0 W ′′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  42. Proof Sketch W ′ 1 non-empty:.. yields winning regions W ′′ i and positional strategies σ ′′ , τ ′′ . W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  43. Proof Sketch W ′ 1 non-empty: Player 0 wins from W ′′ 0 with σ ′′ . W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  44. Proof Sketch W ′ 1 non-empty: Player 1 wins from W ′′ 1 ∪ Attr 1 ( W ′ 1 ) . Winning strategy: combine τ ′ , τ ′′ , and attractor strategy. W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  45. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Martin Zimmermann Saarland University Infinite Games 18/30

  46. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Intriguing complexity-theoretic status: in NP ∩ Co-NP (even in UP ∩ Co-UP and thus unlikely to be complete for NP or Co-NP ). Martin Zimmermann Saarland University Infinite Games 18/30

  47. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Intriguing complexity-theoretic status: in NP ∩ Co-NP (even in UP ∩ Co-UP and thus unlikely to be complete for NP or Co-NP ). Open problem: is solving parity games in polynomial time? Martin Zimmermann Saarland University Infinite Games 18/30

  48. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 19/30

  49. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } Martin Zimmermann Saarland University Infinite Games 20/30

  50. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } A 1 B 2 C 3 D 4 F ∈ F iff | F ∩ { A , B , C , D }| = max ( F ∩ { 1 , 2 , 3 , 4 } ) Martin Zimmermann Saarland University Infinite Games 20/30

  51. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } A 1 B 2 in general: DJW n here: DJW 4 C 3 D 4 F ∈ F iff | F ∩ { A , B , C , D }| = max ( F ∩ { 1 , 2 , 3 , 4 } ) Martin Zimmermann Saarland University Infinite Games 20/30

  52. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } Martin Zimmermann Saarland University Infinite Games 21/30

  53. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # Martin Zimmermann Saarland University Infinite Games 21/30

  54. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  55. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D Martin Zimmermann Saarland University Infinite Games 21/30

  56. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 Martin Zimmermann Saarland University Infinite Games 21/30

  57. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 D D B A C # Martin Zimmermann Saarland University Infinite Games 21/30

  58. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  59. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  60. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  61. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  62. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  63. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 Martin Zimmermann Saarland University Infinite Games 21/30

  64. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  65. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  66. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  67. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  68. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  69. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  70. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 2 Martin Zimmermann Saarland University Infinite Games 21/30

  71. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 2 From some point onwards only vertices that are visited infinitely often are in front of # , and infinitely often exactly the set of vertices that are visited infinitely often is in front of # . Martin Zimmermann Saarland University Infinite Games 21/30

  72. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Martin Zimmermann Saarland University Infinite Games 22/30

  73. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Theorem Muller games are determined with finite-state strategies of size n · n ! . Martin Zimmermann Saarland University Infinite Games 22/30

  74. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Theorem Muller games are determined with finite-state strategies of size n · n ! . Matching lower bounds via DJW n games. Complexity depends on encoding of F : P , if F is given as list of sets. NP ∩ Co-NP , if F is encoded by a tree. Pspace -complete, if F is encoded by circuit or boolean formula (with variables V ). Martin Zimmermann Saarland University Infinite Games 22/30

  75. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 23/30

  76. Concurrent Games Both players choose their moves simultaneously Matching pennies: (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) Martin Zimmermann Saarland University Infinite Games 24/30

  77. Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) Martin Zimmermann Saarland University Infinite Games 24/30

  78. Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) The “Snowball Game”: (hide, wait) (run, wait) (run, throw) (*,*) (*,*) (hide, throw) Martin Zimmermann Saarland University Infinite Games 24/30

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