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Specification of Dynamic Strategy Switching
Soumya PaulJoint work with R. Ramanujam and S. Simon
The Institute of Mathematical SciencesTaramani, 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
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
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
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
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
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
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
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
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
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
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
Motivation[3]
• (goodlength-legs followed by short-off . . .) or (short-legs followed bygoodlength-off . . .) or . . .. In addition bowl a slower delivery now andagain.
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
Motivation[3]
• (goodlength-legs followed by short-off . . .) or (short-legs followed bygoodlength-off . . .) or . . .. In addition bowl a slower delivery now andagain.
• (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
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
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
Motivation[4]
• Strategies are structured.
• Complex strategies are built from simpler strategies.
• A player doesn’t decide on an exact strategy beforehand but maychange her strategy dynamically as the game progresses depending onthe 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
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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe 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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe 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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe 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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe action tuples.
• Arena G = (W ,→,w0).
• At any position w each player i chooses an action ai ∈ Ai .Thisdefines an edge in the arena and the play moves along this edge to anew 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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe action tuples.
• Arena G = (W ,→,w0).
• At any position w each player i chooses an action ai ∈ Ai .Thisdefines an edge in the arena and the play moves along this edge to anew 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
Formalising
• N = 1, 2, . . . , n is the set of players.
• For each i ∈ N, Ai is a finite set of actions. A = A1 × . . .× An arethe action tuples.
• Arena G = (W ,→,w0).
• At any position w each player i chooses an action ai ∈ Ai .Thisdefines an edge in the arena and the play moves along this edge to anew 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
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)
gg (a,c),(a,d)vv
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
Example[2]
Let t1 = (a, c), t2 = (a, d), t3 = (b, c), t4 = (b, d). The tree unfolding TG
of G is:
ǫ
qqddddddddddddddddddddddd
vvnnnnnn
((PPPPPP
--ZZZZZZZZZZZZZZZZZZZZZZZ
t1wwnnnnnn
~~~~ @@
t2wwnnnnnn
~~~~ @@
t3~~~~ @
@
''PPPPPP t4~~~~ @
@
''PPPPPP
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
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) toher 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
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) toher 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
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) toher 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
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) toher 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
Example
Let σ be the strategy of player 1 which is undefined at the empty historybut prescribes her to play the action a for all subsequent histories. ThenT 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
Example
Let σ be the strategy of player 1 which is undefined at the empty historybut prescribes her to play the action a for all subsequent histories. ThenT σ
G looks like:
ǫ
rrfffffffffffffffffff
xxxx
##FFF
F
,,XXXXXXXXXXXXXXXXXXX
(a, c)
yytttt
(a, d)
yytttt
(b, c)
%%JJJ
J(b, d)
%%JJJ
J
(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
Strategy Specifications
• Players change/compose/form strategies based on certain observableproperties 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
Strategy Specifications
• Players change/compose/form strategies based on certain observableproperties of the game.
• P is a countable set of propositions that talk about the observables inthe 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
Strategy Specifications
• Players change/compose/form strategies based on certain observableproperties of the game.
• P is a countable set of propositions that talk about the observables inthe 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
Strategy Specifications
• Players change/compose/form strategies based on certain observableproperties of the game.
• P is a countable set of propositions that talk about the observables inthe 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
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
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
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
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
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
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
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 definedas:
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
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 definedas:
• 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
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 definedas:
• 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
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 definedas:
• TG , t |= p iff p ∈ V (t).
• TG , t |= ψ1 ∨ ψ2 iff TG |= ψ1 or TG |= ψ2.
• TG , t |= ¬ψ iff TG , t 6|= 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
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 definedas:
• TG , t |= p iff p ∈ V (t).
• TG , t |= ψ1 ∨ ψ2 iff TG |= ψ1 or TG |= ψ2.
• TG , t |= ¬ψ iff TG , t 6|= 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
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 definedas:
• TG , t |= p iff p ∈ V (t).
• TG , t |= ψ1 ∨ ψ2 iff TG |= ψ1 or TG |= ψ2.
• TG , t |= ¬ψ iff TG , t 6|= 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
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
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
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
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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2
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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2 | (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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2 | (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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2 | (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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2 | (π1 + π2) | ψ?π
where ψ ∈ Ψ.Intuitively:
• π1 ∪ π2 means that the player plays according to the strategy π1 orthe 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
Strategy Specifications[3]
A strategy of player i can be of the form:
Πi ::= σ ∈ Σi | π1 ∪ π2 | π1 ∩ π2 | π1aπ2 | (π1 + π2) | ψ?π
where ψ ∈ Ψ.Intuitively:
• π1 ∪ π2 means that the player plays according to the strategy π1 orthe strategy π2.
• π1 ∩ π2 means that if at a history t ∈ TG , π1 is defined then theplayer plays according to π1; else if π2 is defined at t then the playerplays 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
Strategy Specifications[4]
• π1aπ2 means that the player plays according to the strategy π1 and
then after some history, switches to playing according to π2. Theposition 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
Strategy Specifications[4]
• π1aπ2 means that the player plays according to the strategy π1 and
then after some history, switches to playing according to π2. Theposition at which she makes the switch is not fixed in advance.
• (π1 + π2) says that at every point, the player can choose to followeither 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
Strategy Specifications[4]
• π1aπ2 means that the player plays according to the strategy π1 and
then after some history, switches to playing according to π2. Theposition at which she makes the switch is not fixed in advance.
• (π1 + π2) says that at every point, the player can choose to followeither π1 or π2.
• ψ?π says at every history, the player tests if the property ψ holds ofthat 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
Strategy Specifications[4]
• π1aπ2 means that the player plays according to the strategy π1 and
then after some history, switches to playing according to π2. Theposition at which she makes the switch is not fixed in advance.
• (π1 + π2) says that at every point, the player can choose to followeither π1 or π2.
• ψ?π says at every history, the player tests if the property ψ holds ofthat history. If it does then she plays according to π.
For a specification π for player i , let [[π]]G ⊆ Σi be the strategies (partialfunctions), 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
Examples
• Σbowler = σshort , σgood , σoutside−off , σlegsP = p(short,single), p(short,boundary ), . . . , p(legs,sixer), . . .
¬3- (p(good,sixer) ∧ p(legs,sixer))?(σshort + σgood + σoutside−off + σlegs) ∪3- (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
Examples
• Σbowler = σshort , σgood , σoutside−off , σlegsP = p(short,single), p(short,boundary ), . . . , p(legs,sixer), . . .
¬3- (p(good,sixer) ∧ p(legs,sixer))?(σshort + σgood + σoutside−off + σlegs) ∪3- (p(good,sixer ) ∧ p(legs,sixer))?(σshort + σoutside−off )
• Σbowler = σ5, σ2, . . .
σ5aσ2
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
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
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
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
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
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
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 isdetermined by m0 = mI and mi+1 = δ(ai−1,mi ) thenσ(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
Finite State Transducer
A finite state transducer FST over input alphabet A and output alphabetAi 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
Finite State Transducer
A finite state transducer FST over input alphabet A and output alphabetAi 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
Finite State Transducer
A finite state transducer FST over input alphabet A and output alphabetAi 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
Finite State Transducer
A finite state transducer FST over input alphabet A and output alphabetAi 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
Finite State Transducer
A finite state transducer FST over input alphabet A and output alphabetAi 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
FST for Bounded Memory Strategy
Given a bounded memory strategy σ for player i we can construct an FSTAσ = (Q,→, I , f ) over A and Ai such that the output of the transducercorrectly 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
FST for Bounded Memory Strategy
Given a bounded memory strategy σ for player i we can construct an FSTAσ = (Q,→, I , f ) over A and Ai such that the output of the transducercorrectly 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
FST for Bounded Memory Strategy
Given a bounded memory strategy σ for player i we can construct an FSTAσ = (Q,→, I , f ) over A and Ai such that the output of the transducercorrectly 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 wa′→ 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
FST for Bounded Memory Strategy
Given a bounded memory strategy σ for player i we can construct an FSTAσ = (Q,→, I , f ) over A and Ai such that the output of the transducercorrectly 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 wa′→ 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
FST for Bounded Memory Strategy
Given a bounded memory strategy σ for player i we can construct an FSTAσ = (Q,→, I , f ) over A and Ai such that the output of the transducercorrectly 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 wa′→ 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
Language of an FST
• A run χ of an fst A = (Q,→, I , f , λ) on a (total) strategy µ is alabelling of the nodes of strategy tree T µ
G with the states of Q suchthat the transitions of A are respected. That is, if there is an edgefrom node from a1 . . . ak to a1 . . . ak+1 in T µ
G thenχ(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
Language of an FST
• A run χ of an fst A = (Q,→, I , f , λ) on a (total) strategy µ is alabelling of the nodes of strategy tree T µ
G with the states of Q suchthat the transitions of A are respected. That is, if there is an edgefrom node from a1 . . . ak to a1 . . . ak+1 in T µ
G thenχ(a1 . . . ak) ∈→ (χ(a1 . . . ak), ak+1).
• A strategy µ is said to be accepted by A if there exists a run χ of Aon 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
Language of an FST
• A run χ of an fst A = (Q,→, I , f , λ) on a (total) strategy µ is alabelling of the nodes of strategy tree T µ
G with the states of Q suchthat the transitions of A are respected. That is, if there is an edgefrom node from a1 . . . ak to a1 . . . ak+1 in T µ
G thenχ(a1 . . . ak) ∈→ (χ(a1 . . . ak), ak+1).
• A strategy µ is said to be accepted by A if there exists a run χ of Aon 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 thatare 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Proof Sketch[3]
• π1aπ2:
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
Proof Sketch[3]
• π1aπ2:
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
Proof Sketch[3]
• π1aπ2:
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
Proof Sketch[3]
• π1aπ2:
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
Proof Sketch[3]
• π1aπ2:
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
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
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
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
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
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
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
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
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
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
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
Stability
• A strategy is switch-free if it does not have any of the a,+ or theψ?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
Stability
• A strategy is switch-free if it does not have any of the a,+ or theψ?π′ constructs.
• Given a strategy π ∈ Πi of player i , let the set of substrategies of πbe 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
Stability
• A strategy is switch-free if it does not have any of the a,+ or theψ?π′ constructs.
• Given a strategy π ∈ Πi of player i , let the set of substrategies of πbe 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
Stability[2]
Let π ∈ Πi be a strategy of player i and Aπ = (Q,→, I , f , λ) be the FSTfor π. Let GAπ = (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
Stability[2]
Let π ∈ Πi be a strategy of player i and Aπ = (Q,→, I , f , λ) be the FSTfor π. Let GAπ = (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
Stability[2]
Let π ∈ Πi be a strategy of player i and Aπ = (Q,→, I , f , λ) be the FSTfor π. Let GAπ = (W ′,→′,w ′
0) where
• W ′ = W × Q
• →′⊆ W ′ × W ′ such that (w1, q1)a
→′ (w2, q2) iff w1a→ 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
Stability[2]
Let π ∈ Πi be a strategy of player i and Aπ = (Q,→, I , f , λ) be the FSTfor π. Let GAπ = (W ′,→′,w ′
0) where
• W ′ = W × Q
• →′⊆ W ′ × W ′ such that (w1, q1)a
→′ (w2, q2) iff w1a→ 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
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
Stability[3]
GAπ
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
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
Proof
• Construct
Gπ = (· · · ((GAπ1)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
Proof
• Construct
Gπ = (· · · ((GAπ1)Aπ2 · · · )Aπn) = (Wπ,→π,wπ)
• Let F ⊆ Gπ = (W ′,→′) such thatW ′ = (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
Proof
• Construct
Gπ = (· · · ((GAπ1)Aπ2 · · · )Aπn) = (Wπ,→π,wπ)
• Let F ⊆ Gπ = (W ′,→′) such thatW ′ = (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
Proof
• Construct
Gπ = (· · · ((GAπ1)Aπ2 · · · )Aπn) = (Wπ,→π,wπ)
• Let F ⊆ Gπ = (W ′,→′) such thatW ′ = (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’,otherwise, 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
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
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
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
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
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
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
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 = (· · · (· · · ((GAπ1)Aπ2 · · · )Aπ′)· · · )Aπn)
and check if the entire graph is a connected component. If it is notthe 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
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 = (· · · (· · · ((GAπ1)Aπ2 · · · )Aπ′)· · · )Aπn)
and check if the entire graph is a connected component. If it is notthe 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
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
Complexity
• Size of Aπ is O(2|π|).
• The size of the restricted graph |GAπ| 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
Complexity
• Size of Aπ is O(2|π|).
• The size of the restricted graph |GAπ| is O(|G | · 2|π|).
• Checking for the maximal connected components of a graph can bedone 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
Complexity
• Size of Aπ is O(2|π|).
• The size of the restricted graph |GAπ| is O(|G | · 2|π|).
• Checking for the maximal connected components of a graph can bedone 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
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
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
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