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Specification of Dynamic Strategy Switching Soumya Paul Joint work - - PowerPoint PPT Presentation

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Specification of Dynamic Strategy Switching Soumya Paul Joint work with R. Ramanujam and S. Simon The Institute of Mathematical Sciences Taramani, Chennai - 600 113 January 8, 2009 Soumya Paul Joint work with R. Ramanujam and S. Simon (The

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SLIDE 1

Specification of Dynamic Strategy Switching

Soumya Paul

Joint work with R. Ramanujam and S. Simon

The Institute of Mathematical Sciences Taramani, Chennai - 600 113

January 8, 2009

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 1 / 64

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SLIDE 2

Motivation

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 2 / 64

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SLIDE 3

Motivation

  • Should I bowl a short pitch delivery?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 2 / 64

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SLIDE 4

Motivation

  • Should I bowl a short pitch delivery?
  • Should I bowl a slower ball?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 2 / 64

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SLIDE 5

Motivation

  • Should I bowl a short pitch delivery?
  • Should I bowl a slower ball?
  • Should I bowl to the off or on his legs?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 2 / 64

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SLIDE 6

Motivation

  • Should I bowl a short pitch delivery?
  • Should I bowl a slower ball?
  • Should I bowl to the off or on his legs?
  • If I bowl a wrong ’un, will it be too predictable?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 2 / 64

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SLIDE 7

Motivation[2]

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 3 / 64

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SLIDE 8

Motivation[2]

  • Should I attack or defend?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 3 / 64

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SLIDE 9

Motivation[2]

  • Should I attack or defend?
  • If he bowls to my legs, should I pelt him for a boundary?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 3 / 64

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SLIDE 10

Motivation[2]

  • Should I attack or defend?
  • If he bowls to my legs, should I pelt him for a boundary?
  • Or should I just score a single so that he doesn’t change his line?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 3 / 64

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SLIDE 11

Motivation[2]

  • Should I attack or defend?
  • If he bowls to my legs, should I pelt him for a boundary?
  • Or should I just score a single so that he doesn’t change his line?
  • If I score too many runs, will he be taken off the attack?

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 3 / 64

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SLIDE 12

Motivation[3]

  • (goodlength-legs followed by short-off . . .) or (short-legs followed by

goodlength-off . . .) or . . .. In addition bowl a slower delivery now and again.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 4 / 64

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SLIDE 13

Motivation[3]

  • (goodlength-legs followed by short-off . . .) or (short-legs followed by

goodlength-off . . .) or . . .. In addition bowl a slower delivery now and again.

  • (goodlength-defend, short-single, . . .) or (goodlength-single,

short-defend, . . .) or . . .. Hit a boundary now and again.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 4 / 64

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SLIDE 14

Motivation[4]

  • Strategies are structured.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 5 / 64

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SLIDE 15

Motivation[4]

  • Strategies are structured.
  • Complex strategies are built from simpler strategies.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 5 / 64

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Motivation[4]

  • Strategies are structured.
  • Complex strategies are built from simpler strategies.
  • A player doesn’t decide on an exact strategy beforehand but may

change her strategy dynamically as the game progresses depending on the outcome so far.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 5 / 64

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SLIDE 17

Formalising

  • N = {1, 2, . . . , n} is the set of players.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 18

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 19

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

  • Arena G = (W , →, w0).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 20

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

  • Arena G = (W , →, w0).
  • At any position w each player i chooses an action ai ∈ Ai.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 21

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

  • Arena G = (W , →, w0).
  • At any position w each player i chooses an action ai ∈ Ai.This

defines an edge in the arena and the play moves along this edge to a new position.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 22

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

  • Arena G = (W , →, w0).
  • At any position w each player i chooses an action ai ∈ Ai.This

defines an edge in the arena and the play moves along this edge to a new position.

  • Thus a play is just a sequence ρ ∈ Aω.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 23

Formalising

  • N = {1, 2, . . . , n} is the set of players.
  • For each i ∈ N, Ai is a finite set of actions. A = A1 × . . . × An are

the action tuples.

  • Arena G = (W , →, w0).
  • At any position w each player i chooses an action ai ∈ Ai.This

defines an edge in the arena and the play moves along this edge to a new position.

  • Thus a play is just a sequence ρ ∈ Aω.
  • The tree unfolding of G is called TG.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 6 / 64

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SLIDE 24

Example

N = {1, 2}, A1 = {a, b}, A2 = {c, d}, G : w0

(b,c),(b,d)

  • (a,c),(a,d)
  • w1

(b,c),(b,d)

  • (a,c),(a,d)
  • Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani,

Specification of Dynamic Strategy Switching January 8, 2009 7 / 64

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SLIDE 25

Example[2]

Let t1 = (a, c), t2 = (a, d), t3 = (b, c), t4 = (b, d). The tree unfolding TG

  • f G is:

ǫ

  • t1
  • t2
  • t3
  • t4
  • t1

t2 t3 t4 t1 t2 t3 t4 t1 t2 t3 t4 t1 t2 t3 t4 . . . . . . . . . . . .

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 8 / 64

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SLIDE 26

Strategies

  • A strategy for player i is a partial function:

σ : TG ⇀ Ai from the nodes of the tree unfolding TG of the arena G (histories) to her action set.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

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SLIDE 27

Strategies

  • A strategy for player i is a partial function:

σ : TG ⇀ Ai from the nodes of the tree unfolding TG of the arena G (histories) to her action set.

  • If σ is not defined for some history, she may play any action there.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

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SLIDE 28

Strategies

  • A strategy for player i is a partial function:

σ : TG ⇀ Ai from the nodes of the tree unfolding TG of the arena G (histories) to her action set.

  • If σ is not defined for some history, she may play any action there.
  • Σi denotes the set of all strategies of player i.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

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SLIDE 29

Strategies

  • A strategy for player i is a partial function:

σ : TG ⇀ Ai from the nodes of the tree unfolding TG of the arena G (histories) to her action set.

  • If σ is not defined for some history, she may play any action there.
  • Σi denotes the set of all strategies of player i.
  • A strategy σ may be viewed as a subtree T σ

G of TG.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 9 / 64

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SLIDE 30

Example

Let σ be the strategy of player 1 which is undefined at the empty history but prescribes her to play the action a for all subsequent histories. Then T σ

G looks like:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 10 / 64

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SLIDE 31

Example

Let σ be the strategy of player 1 which is undefined at the empty history but prescribes her to play the action a for all subsequent histories. Then T σ

G looks like:

ǫ

  • (a, c)
  • (a, d)
  • (b, c)
  • (b, d)
  • (a, c)

(a, d) (a, c) (a, d) (a, c) (a, d) (a, c) (a, d) . . . . . . . . . . . .

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 10 / 64

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SLIDE 32

Strategy Specifications

  • Players change/compose/form strategies based on certain observable

properties of the game.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

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SLIDE 33

Strategy Specifications

  • Players change/compose/form strategies based on certain observable

properties of the game.

  • P is a countable set of propositions that talk about the observables in

the game.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

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SLIDE 34

Strategy Specifications

  • Players change/compose/form strategies based on certain observable

properties of the game.

  • P is a countable set of propositions that talk about the observables in

the game.

  • V : W → 2P is a valuation function.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

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SLIDE 35

Strategy Specifications

  • Players change/compose/form strategies based on certain observable

properties of the game.

  • P is a countable set of propositions that talk about the observables in

the game.

  • V : W → 2P is a valuation function.
  • V may be lifted to TG, V : TG → 2P in the usual way.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 11 / 64

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SLIDE 36

Strategy Specifications[2]

An observable property of the game may be of the form:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 37

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 38

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 39

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 40

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 41

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 42

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 43

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

  • TG, t |

= p iff p ∈ V (t).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 44

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

  • TG, t |

= p iff p ∈ V (t).

  • TG, t |

= ψ1 ∨ ψ2 iff TG | = ψ1 or TG | = ψ2.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 45

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

  • TG, t |

= p iff p ∈ V (t).

  • TG, t |

= ψ1 ∨ ψ2 iff TG | = ψ1 or TG | = ψ2.

  • TG, t |

= ¬ψ iff TG, t | = ψ.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 46

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

  • TG, t |

= p iff p ∈ V (t).

  • TG, t |

= ψ1 ∨ ψ2 iff TG | = ψ1 or TG | = ψ2.

  • TG, t |

= ¬ψ iff TG, t | = ψ.

  • TG, t |

= ⊖ψ iff k > 0 and TG, tk−1 | = ψ.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 47

Strategy Specifications[2]

An observable property of the game may be of the form: Ψ ::= p ∈ P | ψ1 ∨ ψ2 | ¬ψ | ⊖ ψ | ψ1Sψ2 The truth of a formula at a node t = ¯ a1 . . . ¯ ak of the game tree is defined as:

  • TG, t |

= p iff p ∈ V (t).

  • TG, t |

= ψ1 ∨ ψ2 iff TG | = ψ1 or TG | = ψ2.

  • TG, t |

= ¬ψ iff TG, t | = ψ.

  • TG, t |

= ⊖ψ iff k > 0 and TG, tk−1 | = ψ.

  • TG, t |

= ψ1Sψ2 iff ∃l : 1 ≤ l < k such that TG, tl | = ψ2 and ∀m : l < m ≤ k, TG, tm | = ψ1.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 12 / 64

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SLIDE 48

Strategy Specifications[3]

A strategy of player i can be of the form:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 49

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 50

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 51

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 52

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 53

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2 | (π1 + π2)

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 54

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2 | (π1 + π2) | ψ?π where ψ ∈ Ψ.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 55

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2 | (π1 + π2) | ψ?π where ψ ∈ Ψ.Intuitively:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 56

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2 | (π1 + π2) | ψ?π where ψ ∈ Ψ.Intuitively:

  • π1 ∪ π2 means that the player plays according to the strategy π1 or

the strategy π2.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 57

Strategy Specifications[3]

A strategy of player i can be of the form: Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1π2 | (π1 + π2) | ψ?π where ψ ∈ Ψ.Intuitively:

  • π1 ∪ π2 means that the player plays according to the strategy π1 or

the strategy π2.

  • π1 ∩ π2 means that if at a history t ∈ TG, π1 is defined then the

player plays according to π1; else if π2 is defined at t then the player plays according to π2.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 13 / 64

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SLIDE 58

Strategy Specifications[4]

  • π1π2 means that the player plays according to the strategy π1 and

then after some history, switches to playing according to π2. The position at which she makes the switch is not fixed in advance.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

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SLIDE 59

Strategy Specifications[4]

  • π1π2 means that the player plays according to the strategy π1 and

then after some history, switches to playing according to π2. The position at which she makes the switch is not fixed in advance.

  • (π1 + π2) says that at every point, the player can choose to follow

either π1 or π2.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

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SLIDE 60

Strategy Specifications[4]

  • π1π2 means that the player plays according to the strategy π1 and

then after some history, switches to playing according to π2. The position at which she makes the switch is not fixed in advance.

  • (π1 + π2) says that at every point, the player can choose to follow

either π1 or π2.

  • ψ?π says at every history, the player tests if the property ψ holds of

that history. If it does then she plays according to π.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

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SLIDE 61

Strategy Specifications[4]

  • π1π2 means that the player plays according to the strategy π1 and

then after some history, switches to playing according to π2. The position at which she makes the switch is not fixed in advance.

  • (π1 + π2) says that at every point, the player can choose to follow

either π1 or π2.

  • ψ?π says at every history, the player tests if the property ψ holds of

that history. If it does then she plays according to π. For a specification π for player i, let [ [π] ]G ⊆ Σi be the strategies (partial functions), it denotes.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 14 / 64

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SLIDE 62

Examples

  • Σbowler = {σshort, σgood, σoutside−off , σlegs}

P = {p(short,single), p(short,boundary ), . . . , p(legs,sixer), . . .} ¬✸

  • (p(good,sixer) ∧ p(legs,sixer))?(σshort + σgood + σoutside−off + σlegs) ∪

✸

  • (p(good,sixer) ∧ p(legs,sixer))?(σshort + σoutside−off )

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 15 / 64

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SLIDE 63

Examples

  • Σbowler = {σshort, σgood, σoutside−off , σlegs}

P = {p(short,single), p(short,boundary ), . . . , p(legs,sixer), . . .} ¬✸

  • (p(good,sixer) ∧ p(legs,sixer))?(σshort + σgood + σoutside−off + σlegs) ∪

✸

  • (p(good,sixer) ∧ p(legs,sixer))?(σshort + σoutside−off )
  • Σbowler = {σ5, σ2, . . .}

σ5σ2

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 15 / 64

slide-64
SLIDE 64

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-65
SLIDE 65

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

  • A finite set M, the memory of the strategy.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-66
SLIDE 66

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

  • A finite set M, the memory of the strategy.
  • mI ∈ M, the initial memory.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-67
SLIDE 67

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

  • A finite set M, the memory of the strategy.
  • mI ∈ M, the initial memory.
  • A function δ : A × M → M, the memory update.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-68
SLIDE 68

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

  • A finite set M, the memory of the strategy.
  • mI ∈ M, the initial memory.
  • A function δ : A × M → M, the memory update.
  • A function g : A × M → A, the action update.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-69
SLIDE 69

Bounded Memory Strategies

A strategy σ is said to be bounded memory if there exists:

  • A finite set M, the memory of the strategy.
  • mI ∈ M, the initial memory.
  • A function δ : A × M → M, the memory update.
  • A function g : A × M → A, the action update.

such that when ¯ a1 . . . ¯ ak−1 is a play and the sequence m0, m1, . . . , mk is determined by m0 = mI and mi+1 = δ(¯ ai−1, mi) then σ(¯ a1 . . . ¯ ak−1) = g(¯ ak−1, mk).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 16 / 64

slide-70
SLIDE 70

Finite State Transducer

A finite state transducer FST over input alphabet A and output alphabet Ai is a tuple A = (Q, →, I, f ) such that

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

slide-71
SLIDE 71

Finite State Transducer

A finite state transducer FST over input alphabet A and output alphabet Ai is a tuple A = (Q, →, I, f ) such that

  • Q is a finite set (of states).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

slide-72
SLIDE 72

Finite State Transducer

A finite state transducer FST over input alphabet A and output alphabet Ai is a tuple A = (Q, →, I, f ) such that

  • Q is a finite set (of states).
  • →: Q × A → 2Q is the transition function.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

slide-73
SLIDE 73

Finite State Transducer

A finite state transducer FST over input alphabet A and output alphabet Ai is a tuple A = (Q, →, I, f ) such that

  • Q is a finite set (of states).
  • →: Q × A → 2Q is the transition function.
  • I ⊆ Q is the set of initial states.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

slide-74
SLIDE 74

Finite State Transducer

A finite state transducer FST over input alphabet A and output alphabet Ai is a tuple A = (Q, →, I, f ) such that

  • Q is a finite set (of states).
  • →: Q × A → 2Q is the transition function.
  • I ⊆ Q is the set of initial states.
  • f : Q → Ai is the output function.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 17 / 64

slide-75
SLIDE 75

FST for Bounded Memory Strategy

Given a bounded memory strategy σ for player i we can construct an FST Aσ = (Q, →, I, f ) over A and Ai such that the output of the transducer correctly reflects whatever the strategy σ prescribes.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

slide-76
SLIDE 76

FST for Bounded Memory Strategy

Given a bounded memory strategy σ for player i we can construct an FST Aσ = (Q, →, I, f ) over A and Ai such that the output of the transducer correctly reflects whatever the strategy σ prescribes.

  • Q = M × A ∪ {ǫ} × Ai × W .

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

slide-77
SLIDE 77

FST for Bounded Memory Strategy

Given a bounded memory strategy σ for player i we can construct an FST Aσ = (Q, →, I, f ) over A and Ai such that the output of the transducer correctly reflects whatever the strategy σ prescribes.

  • Q = M × A ∪ {ǫ} × Ai × W .
  • →: Q × A → 2Q such that if for any (m, ¯

a, a, w) ∈ Q, δ(¯ a, m) = m′ and g(¯ a, m′) = a′ then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) such that ¯ a′(i) = a and w ¯

a′

→ w′. If g is not defined at (¯ a, m′) then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) for all a′ ∈ Ai and w ¯

a′

→ w′.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

slide-78
SLIDE 78

FST for Bounded Memory Strategy

Given a bounded memory strategy σ for player i we can construct an FST Aσ = (Q, →, I, f ) over A and Ai such that the output of the transducer correctly reflects whatever the strategy σ prescribes.

  • Q = M × A ∪ {ǫ} × Ai × W .
  • →: Q × A → 2Q such that if for any (m, ¯

a, a, w) ∈ Q, δ(¯ a, m) = m′ and g(¯ a, m′) = a′ then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) such that ¯ a′(i) = a and w ¯

a′

→ w′. If g is not defined at (¯ a, m′) then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) for all a′ ∈ Ai and w ¯

a′

→ w′.

  • I = {(mI , ǫ, a, w0) | a = g(ǫ, mI ) if defined, else a ∈ Ai}.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

slide-79
SLIDE 79

FST for Bounded Memory Strategy

Given a bounded memory strategy σ for player i we can construct an FST Aσ = (Q, →, I, f ) over A and Ai such that the output of the transducer correctly reflects whatever the strategy σ prescribes.

  • Q = M × A ∪ {ǫ} × Ai × W .
  • →: Q × A → 2Q such that if for any (m, ¯

a, a, w) ∈ Q, δ(¯ a, m) = m′ and g(¯ a, m′) = a′ then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) such that ¯ a′(i) = a and w ¯

a′

→ w′. If g is not defined at (¯ a, m′) then (m, ¯ a, a, w) ¯

a′

→ (m′, ¯ a′, a′, w′) for all a′ ∈ Ai and w ¯

a′

→ w′.

  • I = {(mI , ǫ, a, w0) | a = g(ǫ, mI ) if defined, else a ∈ Ai}.
  • f : Q → Ai such that f ((m, ¯

a, a, w)) = a.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 18 / 64

slide-80
SLIDE 80

Language of an FST

  • A run χ of an fst A = (Q, →, I, f , λ) on a (total) strategy µ is a

labelling of the nodes of strategy tree T µ

G with the states of Q such

that the transitions of A are respected. That is, if there is an edge from node from ¯ a1 . . . ¯ ak to ¯ a1 . . . ¯ ak+1 in T µ

G then

χ(¯ a1 . . . ¯ ak) ∈→ (χ(¯ a1 . . . ¯ ak), ¯ ak+1).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

slide-81
SLIDE 81

Language of an FST

  • A run χ of an fst A = (Q, →, I, f , λ) on a (total) strategy µ is a

labelling of the nodes of strategy tree T µ

G with the states of Q such

that the transitions of A are respected. That is, if there is an edge from node from ¯ a1 . . . ¯ ak to ¯ a1 . . . ¯ ak+1 in T µ

G then

χ(¯ a1 . . . ¯ ak) ∈→ (χ(¯ a1 . . . ¯ ak), ¯ ak+1).

  • A strategy µ is said to be accepted by A if there exists a run χ of A
  • n µ such that ∀t = ¯

a1 . . . ¯ ak ∈ T µ

G , ¯

a(i) = f (χ(t)).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

slide-82
SLIDE 82

Language of an FST

  • A run χ of an fst A = (Q, →, I, f , λ) on a (total) strategy µ is a

labelling of the nodes of strategy tree T µ

G with the states of Q such

that the transitions of A are respected. That is, if there is an edge from node from ¯ a1 . . . ¯ ak to ¯ a1 . . . ¯ ak+1 in T µ

G then

χ(¯ a1 . . . ¯ ak) ∈→ (χ(¯ a1 . . . ¯ ak), ¯ ak+1).

  • A strategy µ is said to be accepted by A if there exists a run χ of A
  • n µ such that ∀t = ¯

a1 . . . ¯ ak ∈ T µ

G , ¯

a(i) = f (χ(t)).

  • The language of A, L(A) is defined to be the set of all strategies that

are accepted by it.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 19 / 64

slide-83
SLIDE 83

Transducer Lemma

Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Πi, there is a transducer Aπ such that for all µ ∈ Ωi, µ ∈ [ [π] ]G iff µ ∈ L(Aπ).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

slide-84
SLIDE 84

Transducer Lemma

Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Πi, there is a transducer Aπ such that for all µ ∈ Ωi, µ ∈ [ [π] ]G iff µ ∈ L(Aπ). Proof Sketch: By induction on π

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

slide-85
SLIDE 85

Transducer Lemma

Lemma Given game arena G, a player i ∈ N and a strategy specification π ∈ Πi, there is a transducer Aπ such that for all µ ∈ Ωi, µ ∈ [ [π] ]G iff µ ∈ L(Aπ). Proof Sketch: By induction on π

  • For σ ∈ Σi we have an FST Aσ from the construction above.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 20 / 64

slide-86
SLIDE 86

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 21 / 64

slide-87
SLIDE 87

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 22 / 64

slide-88
SLIDE 88

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 23 / 64

slide-89
SLIDE 89

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 24 / 64

slide-90
SLIDE 90

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 25 / 64

slide-91
SLIDE 91

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 26 / 64

slide-92
SLIDE 92

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 27 / 64

slide-93
SLIDE 93

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 28 / 64

slide-94
SLIDE 94

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 29 / 64

slide-95
SLIDE 95

Proof Sketch

  • π1 ∪ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 30 / 64

slide-96
SLIDE 96

Proof Sketch[2]

  • π1 ∩ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 31 / 64

slide-97
SLIDE 97

Proof Sketch[2]

  • π1 ∩ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 32 / 64

slide-98
SLIDE 98

Proof Sketch[2]

  • π1 ∩ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 33 / 64

slide-99
SLIDE 99

Proof Sketch[2]

  • π1 ∩ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 34 / 64

slide-100
SLIDE 100

Proof Sketch[2]

  • π1 ∩ π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 35 / 64

slide-101
SLIDE 101

Proof Sketch[3]

  • π1π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 36 / 64

slide-102
SLIDE 102

Proof Sketch[3]

  • π1π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 37 / 64

slide-103
SLIDE 103

Proof Sketch[3]

  • π1π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 38 / 64

slide-104
SLIDE 104

Proof Sketch[3]

  • π1π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 39 / 64

slide-105
SLIDE 105

Proof Sketch[3]

  • π1π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 40 / 64

slide-106
SLIDE 106

Proof Sketch[4]

  • π1 + π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 41 / 64

slide-107
SLIDE 107

Proof Sketch[4]

  • π1 + π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 42 / 64

slide-108
SLIDE 108

Proof Sketch[4]

  • π1 + π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 43 / 64

slide-109
SLIDE 109

Proof Sketch[4]

  • π1 + π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 44 / 64

slide-110
SLIDE 110

Proof Sketch[4]

  • π1 + π2:

Aπ1 · · · Aπ2 · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 45 / 64

slide-111
SLIDE 111

Proof Sketch[5]

  • ψ?π′:

Aπ′ · · · MCS(CL(ψ)) · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 46 / 64

slide-112
SLIDE 112

Proof Sketch[5]

  • ψ?π′:

Aπ′ · · · MCS(CL(ψ)) · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 47 / 64

slide-113
SLIDE 113

Proof Sketch[5]

  • ψ?π′:

Aπ′ · · · MCS(CL(ψ)) · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 48 / 64

slide-114
SLIDE 114

Proof Sketch[5]

  • ψ?π′:

Aπ′ · · · MCS(CL(ψ)) · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 49 / 64

slide-115
SLIDE 115

Proof Sketch[5]

  • ψ?π′:

Aπ′ · · · MCS(CL(ψ)) · · ·

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 50 / 64

slide-116
SLIDE 116

Stability

  • A strategy is switch-free if it does not have any of the , + or the

ψ?π′ constructs.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 51 / 64

slide-117
SLIDE 117

Stability

  • A strategy is switch-free if it does not have any of the , + or the

ψ?π′ constructs.

  • Given a strategy π ∈ Πi of player i, let the set of substrategies of π

be Sπ.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 51 / 64

slide-118
SLIDE 118

Stability

  • A strategy is switch-free if it does not have any of the , + or the

ψ?π′ constructs.

  • Given a strategy π ∈ Πi of player i, let the set of substrategies of π

be Sπ.

  • Let SF(Sπ) be the set of switch-free strategies of Sπ.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 51 / 64

slide-119
SLIDE 119

Stability[2]

Let π ∈ Πi be a strategy of player i and Aπ = (Q, →, I, f , λ) be the FST for π. Let G↾Aπ = (W ′, →′, w′

0) where

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 52 / 64

slide-120
SLIDE 120

Stability[2]

Let π ∈ Πi be a strategy of player i and Aπ = (Q, →, I, f , λ) be the FST for π. Let G↾Aπ = (W ′, →′, w′

0) where

  • W ′ = W × Q

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 52 / 64

slide-121
SLIDE 121

Stability[2]

Let π ∈ Πi be a strategy of player i and Aπ = (Q, →, I, f , λ) be the FST for π. Let G↾Aπ = (W ′, →′, w′

0) where

  • W ′ = W × Q
  • →′⊆ W ′ × W ′ such that (w1, q1)

¯ a

→′ (w2, q2) iff w1

¯ a

→ w2, q

¯ a

→ q2 and f (q1) = ¯ a(i)

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 52 / 64

slide-122
SLIDE 122

Stability[2]

Let π ∈ Πi be a strategy of player i and Aπ = (Q, →, I, f , λ) be the FST for π. Let G↾Aπ = (W ′, →′, w′

0) where

  • W ′ = W × Q
  • →′⊆ W ′ × W ′ such that (w1, q1)

¯ a

→′ (w2, q2) iff w1

¯ a

→ w2, q

¯ a

→ q2 and f (q1) = ¯ a(i)

  • w′

0 = {w0} × I

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 52 / 64

slide-123
SLIDE 123

Stability[3]

G

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 53 / 64

slide-124
SLIDE 124

Stability[3]

G↾Aπ

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 54 / 64

slide-125
SLIDE 125

Results

Theorem Given a game arena G = (W , →, w0) with a valuation V : W → 2P, a subarena R of G and strategy specifications π1, . . . , πn for players 1 to n, the question, Does the game eventually settle down to R is decidable.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 55 / 64

slide-126
SLIDE 126

Proof

  • Construct

Gπ = (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπn) = (Wπ, →π, wπ)

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 56 / 64

slide-127
SLIDE 127

Proof

  • Construct

Gπ = (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπn) = (Wπ, →π, wπ)

  • Let F ⊆ Gπ = (W ′, →′) such that

W ′ = {(w, q1, . . . , qn) | w ∈ R, q1 ∈ Qπ1, . . . , qn ∈ Qπn} Let →′=→π ∩(W ′ × W ′).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 56 / 64

slide-128
SLIDE 128

Proof

  • Construct

Gπ = (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπn) = (Wπ, →π, wπ)

  • Let F ⊆ Gπ = (W ′, →′) such that

W ′ = {(w, q1, . . . , qn) | w ∈ R, q1 ∈ Qπ1, . . . , qn ∈ Qπn} Let →′=→π ∩(W ′ × W ′).

  • Check if F is a maximal connected component in Gπ. If so proceed,

else output a ‘NO’.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 56 / 64

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SLIDE 129

Proof

  • Construct

Gπ = (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπn) = (Wπ, →π, wπ)

  • Let F ⊆ Gπ = (W ′, →′) such that

W ′ = {(w, q1, . . . , qn) | w ∈ R, q1 ∈ Qπ1, . . . , qn ∈ Qπn} Let →′=→π ∩(W ′ × W ′).

  • Check if F is a maximal connected component in Gπ. If so proceed,

else output a ‘NO’.

  • Check if all paths starting at w′ ∈ wπ reach F. If so, output a ‘YES’,
  • therwise, output a ‘No’.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 56 / 64

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SLIDE 130

Proof[2]

Gπ

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 57 / 64

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SLIDE 131

Proof[2]

F

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 58 / 64

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SLIDE 132

Proof[2]

F c

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 59 / 64

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SLIDE 133

Results[2]

Theorem Given a game arena G = (W , →, w0) with a valuation V : W → 2P, a subarena R of G and strategy specifications π1, . . . , πn for players 1 to n, the question, If the game converges to R, does the strategy of player i become eventually stable with respect to switching is decidable.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 60 / 64

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SLIDE 134

Proof

  • Check if the game eventually settles down to R.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 61 / 64

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SLIDE 135

Proof

  • Check if the game eventually settles down to R.
  • For all initial nodes w′ ∈ wπ repeat:

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 61 / 64

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SLIDE 136

Proof

  • Check if the game eventually settles down to R.
  • For all initial nodes w′ ∈ wπ repeat:

Let w⋆

π = (w⋆, q⋆ 1, . . . , q⋆ n) be the state of F which is first reachable

from w′. Let GR = (R, →↾ R × R, w⋆). For each π′ ∈ SF(Sπi) construct G ′

π = (· · · (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπ′)↾· · · )↾Aπn)

and check if the entire graph is a connected component. If it is not the case, output a ‘NO’ and halt. Else repeat.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 61 / 64

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SLIDE 137

Proof

  • Check if the game eventually settles down to R.
  • For all initial nodes w′ ∈ wπ repeat:

Let w⋆

π = (w⋆, q⋆ 1, . . . , q⋆ n) be the state of F which is first reachable

from w′. Let GR = (R, →↾ R × R, w⋆). For each π′ ∈ SF(Sπi) construct G ′

π = (· · · (· · · ((G↾Aπ1)↾Aπ2 · · · )↾Aπ′)↾· · · )↾Aπn)

and check if the entire graph is a connected component. If it is not the case, output a ‘NO’ and halt. Else repeat.

  • Output a ‘YES’.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 61 / 64

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SLIDE 138

Complexity

  • Size of Aπ is O(2|π|).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 62 / 64

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SLIDE 139

Complexity

  • Size of Aπ is O(2|π|).
  • The size of the restricted graph |G↾Aπ| is O(|G| · 2|π|).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 62 / 64

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SLIDE 140

Complexity

  • Size of Aπ is O(2|π|).
  • The size of the restricted graph |G↾Aπ| is O(|G| · 2|π|).
  • Checking for the maximal connected components of a graph can be

done is time polynomial in the size of the graph.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 62 / 64

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SLIDE 141

Complexity

  • Size of Aπ is O(2|π|).
  • The size of the restricted graph |G↾Aπ| is O(|G| · 2|π|).
  • Checking for the maximal connected components of a graph can be

done is time polynomial in the size of the graph.

  • Thus the running time of the decision procedures is O(|G| · 2|π|).

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 62 / 64

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SLIDE 142

Extensions

  • Restriction of choice.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 63 / 64

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SLIDE 143

Extensions

  • Restriction of choice.
  • Imposing neighbourhood structures on the players.

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 63 / 64

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SLIDE 144

Thanks

Soumya PaulJoint work with R. Ramanujam and S. Simon (The Institute of Mathematical Sciences Taramani, Specification of Dynamic Strategy Switching January 8, 2009 64 / 64