Graph Games with Reachabillity Objectives:
Mixing Chess, Soccer and Poker
Krishnendu Chatterjee
5th Workshop on Reachability Problems,
Genova, Sept 30, 2011
Krishnendu Chatterjee
1
Games on Graphs
 Games on graphs.
 History
 Zermelo’s theorem about Chess in 1913
 From every configuration
 Either player 1 can enforce a win.
 Or player 2 can enforce a win.
 Or both players can enforce a draw.
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Chess: Games on Graph
 Chess is a game on graph.
 Configuration graph.
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Graphs vs. Games
Two interacting players in games: Player 1 (Box) vs Player 2 (Diamond).
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Game Graph
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Game Graphs
 A game graph G= ((S,E), (S1, S2))
 Player 1 states (or vertices) S1 and similarly player 2
states S2, and (S1, S2) partitions S.
 E is the set of edges.
 E(s) out-going edges from s, and assume E(s) nonempty for all s.
 Game played by moving tokens: when player 1
state, then player 1 chooses the out-going edge,
and if player 2 state, player 2 chooses the
outgoing edge.
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Game Example
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Game Example
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Game Example
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Strategies
 Strategies are recipe how to move tokens or how
to extend plays. Formally, given a history of play
(or finite sequence of states), it chooses a
probability distribution over out-going edges.
 ¾: S* S1  D(S).
 ¼: S* S2 ! D(S).
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Strategies

Strategies are recipe how to move tokens or how to extend plays. Formally,
given a history of play (or finite sequence of states), it chooses a probability
distribution over out-going edges.

¾: S* S1 ! D(S).

History dependent and randomized.

History independent: depends only current state (memoryless or positional).


Deterministic: no randomization (pure strategies).


¾: S* S1 ! S
Deterministic and memoryless: no memory and no randomization (pure and
memoryless and is the simplest class).


¾: S1 ! D(S)
¾: S1 ! S
Same notations for player 2 strategies ¼.
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Objectives
 Objectives are subsets of infinite paths, i.e., Ã µ
S!.
 Reachability: there is a set of good vertices
(example check-mate) and goal is to reach them.
Formally, for a set T if vertices or states, the
objective is the set of paths that visit the target T
at least once.
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Applications: Verification and Control of Systems
 Verification and control of systems
M
satisfies property (Ã)
 Environment
 Controller
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E
C
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Applications: Verification and Control of Systems
 Verification and control of systems
C
||
M
||
E
satisfies property (Ã)
 Question: does there exists a controller that
against all environment ensures the property.
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Game Models Applications
-synthesis [Church, Ramadge/Wonham, Pnueli/Rosner]
-model checking of open systems
-receptiveness [Dill, Abadi/Lamport]
-semantics of interaction [Abramsky]
-non-emptiness of tree automata [Rabin, Gurevich/ Harrington]
-behavioral type systems and interface automata [deAlfaro/ Henzinger]
-model-based testing [Gurevich/Veanes et al.]
-etc.
16
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Reachability Games
 Pre(X): given a set X of states, Pre(X) is the set
of states such that player 1 can ensure next state
in X.
X
T
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Reachability Games
 Pre(X): given a set X of states, Pre(X) is the set
of states such that player 1 can ensure next state
in X.
X
T
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Reachability Games
 Pre(X): given a set X of states, Pre(X) is the set
of states such that player 1 can ensure next state
in X.
X
T
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Reachability Games
 Pre(X): given a set X of states, Pre(X) is the set
of states such that player 1 can ensure next state
in X.
X
 Fix-point
T
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Reachability Games
 Winning set for a partition: Determinacy
 Player 1 wins: then no matter what player 2 does,
certainly reach the target.
 Player 2 wins: then no matter what player 1 does, the
target is never reached.
 Memoryless winning strategies.
 Can be computed in linear time [Beeri 81,
Immerman 81].
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Chess Theorem
 Zermelo’s Theorem
Win1
Win2
Both draw
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Game Graphs Till Now
 Game graphs we have seen till now
 Many rounds (possibly infinite).
 Turn-based.
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Simultaneous Games
 Theory of rational behavior as game theory
 von Neumann- Morgenstern games
 Matrix zero-sum games
R
R (0,0)
P (1,-1)
S (-1,1)
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P
(-1,1)
(0,0)
(1,-1)
S
(1,-1)
(-1,1)
(0,0)
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Simultaneous Games
 Theory of rational behavior as game theory
 von Neumann- Morgenstern games
 Matrix zero-sum games
R
R (0,0)
P (1,-1)
S (-1,1)
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P
(-1,1)
(0,0)
(1,-1)
S
(1,-1)
(-1,1)
(0,0)
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Simultaneous Games
 Example: Prisoners dilemma.
 Another example.
R
R (1,-1)
L (-1,1)
C (-1,1)
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L
(-1,1)
(1,-1)
(-1,1)
C
(-1,1)
(-1,1)
(1,-1)
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Simultaneous Games
 Example: Prisoners dilemma.
 Another example.
R
R (1,-1)
L (-1,1)
C (-1,1)
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L
(-1,1)
(1,-1)
(-1,1)
C
(-1,1)
(-1,1)
(1,-1)
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Simultaneous Games
 Another example: Penalty shoot-out (Soccer)
R
R (1,-1)
L (-1,1)
C (-1,1)
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L
(-1,1)
(1,-1)
(-1,1)
C
(-1,1)
(-1,1)
(1,-1)
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Chess Vs. Soccer (Penalty)
 Chess:
 Turn-based
 Possibly infinite rounds
 Theory of simultaneous games (Soccer)
 Concurrent
 One-shot (one-round)
 Mix chess and soccer
 Concurrent games on graphs
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Mixing Chess and Soccer:
Concurrent Graph Games
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Concurrent Game Graphs
A concurrent game graph is a tuple G =(S,M,¡1,¡2,±)
• S is a finite set of states.
• M is a finite set of moves or actions.
• ¡i: S ! 2M n ; is an action assignment function that assigns the non-empty
set ¡i(s) of actions to player i at s, where i 2 {1,2}.
• ±: S £ M £ M ! S, is a transition function that given a state and actions of
both players gives the next state.
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An Example: Snow-ball Game
run, wait
hide, throw
run, throw
s
hide, wait
[Everett 57]
R
Hide
Run
Throw
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Wait
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New Solution Concepts
 Sure winning for turn-based.
 New solution concepts
 Almost-sure winning.
 Limit-sure winning.
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Almost-sure Winning Example
head, head
tail, tail
s
R
head, tail
tail, head
Almost-sure winning strategy: say head and tail with probability ½.
Randomization is necessary.
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Concurrent reachability games: limit-sure
run, wait
hide, throw
run, throw
s
hide, wait
Move
run
hide
Probability
q
1-q
(q>0)
[Everett 57]
R
Hide
Run
Throw
Wait
Win at s with probability
1-q, for all q > 0.
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Concurrent reachability games: limit-sure
w=0
run, throw
1
run, wait
hide, throw
s
hide, wait
Move
run
hide
Probability
q
1-q
(q>0)
1
[Everett 57]
R
Hide
Run
Throw
Wait
Win at s with probability
1-q, for all q > 0.
Player 1 cannot achieve w(s) = 1, only w(s) = 1-q for all q > 0.
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Results for Concurrent Reachability Games
 Sure winning:
 Deterministic memoryless sufficient.
 Linear time.
 Almost-sure winning:
 Randomization is necessary.
 Randomized memoryless is sufficient.
 Quadratic time algorithm.
 Limit-sure winning:
 Randomization is necessary.
 Randomized memoryless is sufficient.
 Quadratic time algorithm.
 Results from [dAHK98, CdAH06, CdAH09]
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Games Till Now
 Turn-based graph games
 Concurrent graph games
 Applications: again verification and synthesis with
synchronous interaction.
 Both these games are perfect-information games.
Players know the precise state of the game.
 The game of Poker: players play but do not know the
perfect state of the game.
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Summary: Theory of Graph Games
Winning Mode/
Game Graphs
Sure
Almost-sure
Limit-sure
Turn-based Games
(CHESS)
Linear time
(PTIME-complete)
Linear-time
(PTIME-complete)
Linear-time
(PTIME-complete)
Concurrent Games
(CHESS+ SOCCER)
Linear time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Partial-information
Games
(CHESS + SOCCER+
POKER)
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Mixing Chess and Poker:
Partial-information Graph Games
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Why Partial-information
 Perfect-information: controller knows everything
about the system.
 This is often unrealistic in the design of reactive
systems because
• systems have internal state not visible to controller (private
variables)
• noisy sensors entail uncertainties on the state of the game
 Partial-observation
Hidden variables = imperfect information.
Sensor uncertainty = imperfect information.
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Partial-information Games
 A PIG G =(L, A, , O) is as follows
 L is a finite set of locations (or states).
 A is a finite set of input letters (or actions).
  µ L £ A £ L non-deterministic transition relation that
for a state and an action gives the possible next
states.
 O is the set of observations and is a partition of the
state space. The observation represents what is
observable.
 Perfect-information: O={{l} | l 2 L}.
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PIG: Example
b
a
a,b
b
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a
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New Solution Concepts
 Sure winning: winning with certainty (in perfect
information setting determinacy).
 Almost-sure winning: win with probability 1.
 Limit-sure winning: win with probability arbitrary
close to 1.
 We will illustrate the solution concepts with card
games.
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Card Game 1
 Step 1: Player 2 selects a card from the deck of
52 cards and moves it from the deck (player 1
does not know the card).
 Step 2:
 Step 2 a: Player 2 shuffles the deck.
 Step 2 b: Player 1 selects a card and view it.
 Step 2 c: Player 1 makes a guess of the secret card or
goes back to Step 2 a.
 Player 1 wins if the guess is correct.
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Card Game 1
 Player 1 can win with probability 1: goes back to
Step 2 a until all 51 cards are seen.
 Player 1 cannot win with certainty: there are
cases (though with probability 0) such that all
cards are not seen. Then player 1 either never
makes a guess or makes a wrong guess with
positive probability.
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Card Game 2
 Step 1: Player 2 selects a new card from an exactly
same deck and puts is in the deck of 52 cards
(player 1 does not know the new card). So the deck
has 53 cards with one duplicate.
 Step 2:
 Step 2 a: Player 2 shuffles the deck.
 Step 2 b: Player 1 selects a card and view it.
 Step 2 c: Player 1 makes a guess of the secret duplicate
card or goes back to Step 2 a.
 Player 1 wins if the guess is correct.
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Card Game 2
 Player 1 can win with probability arbitrary close
to 1: goes back to Step 2 a for a long time and
then choose the card with highest frequency.
 Player 1 cannot win probability 1, there is a tiny
chance that not the duplicate card has the
highest frequency, but can win with probability
arbitrary close to 1, (i.e., for all ² >0, player 1 can
win with probability 1- ², in other words the limit is
1).
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Sure winning for Reachability
 Result from [Reif 79]
 Memory is required.
 Exponential memory required.
 Subset construction: what subsets of states player 1
can be. Reduction to exponential size turn-based
games.
 EXPTIME-complete.
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Almost-sure winning for Reachability
 Result from [CDHR 06, CHDR 07]
 Standard subset construction fails: as it captures
only sure winning, and not same as almost-sure
winning.
 More involved subset construction is required.
 EXPTIME-complete.
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Summary: Theory of Graph Games
Winning Mode/
Game Graphs
Sure
Almost-sure
Limit-sure
Turn-based Games
(CHESS)
Linear time
(PTIME-complete)
Linear-time
(PTIME-complete)
Linear-time
(PTIME-complete)
Concurrent Games
(CHESS+ SOCCER)
Linear time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Partial-information
Games
(CHESS + SOCCER+
POKER)
EXPTIME-complete
EXPTIME-complete
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54
Limit-sure winning for Reachability
 Limit-sure winning for reachability is undecidable
[GO 10, CH 10].
 Reduction from
problem (PCP).
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the
Post-correspondence
55
Mixing Chess, Soccer and Poker
 Partial-information concurrent games
 Concurrency can be obtained for free (polynomial
reduction) for partial-information games.
 So all the results for partial-information turn-based
games also hold for partial-information concurrent
games.
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56
Summary: Theory of Graph Games
Winning Mode/
Game Graphs
Sure
Almost-sure
Limit-sure
Turn-based Games
(CHESS)
Linear time
(PTIME-complete)
Linear-time
(PTIME-complete)
Linear-time
(PTIME-complete)
Concurrent Games
(CHESS+ SOCCER)
Linear time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Quadratic time
(PTIME-complete)
Partial-information
Games
(CHESS + SOCCER+
POKER)
EXPTIME-complete
EXPTIME-complete
Undecidable.
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57
Strategy Complexity
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Classes of strategies
rand. action-visible
rand. action-invisible
Classification according to
the power of strategies
pure
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Classes of strategies
rand. action-visible
rand. action-invisible
Classification according to
the power of strategies
pure
Poly-time reduction from decision problem of rand. act.-vis.
to rand. act.-inv.
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Known results
Reachability - Memory requirement (for player 1)
Almost-sure
rand. act.-vis.
rand. act.-inv.
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
exponential (belief)
memoryless
exponential (belief)
[CDHR’06]
[BGG’09]
exponential (belief)
[BGG’09]
exponential (belief)
[CDHR’06(remark), GS’09]
[GS’09]
?
?
?
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
?
?
?
pure
Positive
pure
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Beliefs
Three prevalent beliefs:
• Belief is sufficient.
• Randomized action invisible or visible almost same.
• The general case memory is similar (or in some cases
exponential blow up) as compared to the one-sided case.
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Pure Strategies
Belief
• Belief is sufficient.
Proofs
• Doubts.
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63
Pure Strategies
Belief
• Belief is sufficient.
Proofs
• Doubts
Lesson:
Doubt your belief and believe in your doubts!!! See the
unexpected.
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64
New results
Reachability - Memory requirement (for player 1)
Almost-sure
rand. act.-vis.
rand. act.-inv.
pure
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
exponential (belief)
memoryless
exponential (belief)
[CDHR’06]
[BGG’09]
exponential (belief)
[BGG’09]
exponential (belief)
[CDHR’06(remark), GS’09]
[GS’09]
?
?
?
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
?
?
?
Positive
pure
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65
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
?
?
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
exponential (more
than belief)
?
?
Almost-sure
Positive
pure
Krishnendu Chatterjee
[CDHR’06]
[BGG’09]
[BGG’09]
exponential (belief)
[GS’09]
66
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
?
?
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
exponential (more
than belief)
?
?
Almost-sure
Positive
pure
Krishnendu Chatterjee
[CDHR’06]
[BGG’09]
[BGG’09]
exponential (belief)
[GS’09]
67
Pure Strategies: Player 1 Perfect, Player 2 Partial (positive)
Pl1 Perfect, Pl 2 Partial :
Stochastic, Randomized.
Memoryless
Pl1 Partial, Pl 2 Perfect:
Stochastic, Pure.
Exponential
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Pl1 Perfect, Pl 2 Partial :
Non-stochastic, Pure.
Memoryless
Pl1 Perfect, Pl 2 Perfect:
Stochastic, Pure.
Memoryless
68
Pure Strategies: Player 1 Perfect, Player 2 Partial (positive)
Pl1 Perfect, Pl 2 Partial :
Stochastic, Randomized.
Memoryless
Pl1 Perfect, Pl 2 Partial :
Non-stochastic, Pure.
Memoryless
Restrict to pure
Add probability
Pl1 Perfect, Pl 2 Partial:
Stochastic, Pure.
Pl 1 more informed,
Pl 2 less informed
Pl1 Partial, Pl 2 Perfect:
Stochastic, Pure.
Exponential
Krishnendu Chatterjee
Pl 2 less informed
Pl1 Perfect, Pl 2 Perfect:
Stochastic, Pure.
Memoryless
69
Pure Strategies: Player 1 Perfect, Player 2 Partial (positive)
Pl1 Perfect, Pl 2 Partial :
Stochastic, Randomized.
Memoryless
Restrict to pure
Pl1 Perfect, Pl 2 Partial :
Non-stochastic, Pure.
Memoryless
Add probability
Pl1 Perfect, Pl 2 Partial:
Stochastic, Pure.
Non-elementary complete
Pl 1 more informed,
Pl 2 less informed
Pl1 Partial, Pl 2 Perfect:
Stochastic, Pure.
Exponential
Krishnendu Chatterjee
Pl 2 less informed
Pl1 Perfect, Pl 2 Perfect:
Stochastic, Pure.
Memoryless
70
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
non-elementary
complete
?
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
exponential (more
than belief)
non-elementary
complete
?
Almost-sure
Positive
pure
Krishnendu Chatterjee
[CDHR’06]
[BGG’09]
[BGG’09]
exponential (belief)
[GS’09]
71
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
Almost-sure
Positive
[CDHR’06]
player 1 partial
player 2 perfect
rand. act.-vis.
memoryless
rand. act.-inv.
pure
Krishnendu Chatterjee
[BGG’09]
[BGG’09]
exponential (belief)
[GS’09]
non-elementary
complete
?
player 1 perfect
2-sided
Player
states,
player12wins
partialfrom more
both partial
but needs more memory !
memoryless
memoryless
memoryless
memoryless
memoryless
exponential (more
than belief)
non-elementary
complete
?
72
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
non-elementary
complete
finite (at least nonelementary)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
exponential (more
than belief)
non-elementary
complete
finite (at least nonelementary)
Almost-sure
Positive
pure
Krishnendu Chatterjee
[CDHR’06]
[BGG’09]
[BGG’09]
exponential (belief)
[GS’09]
73
Player 1 perfect, player 2 partial
More information:
• Win from more places.
• Winning strategy is very hard to implement.
Information is useful, but ignorance is bliss  !!!
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74
Reductions for equivalence
Equivalence of the decision problems for almost-sure reach
with pure strategies and rand. act.-inv. strategies
• Reduction of rand. act.-inv. to pure
choice of a subset of actions (support of prob. dist.)
• Reduction of pure to rand. act.-inv. (holds for almost-sure
only)
It follows that the memory requirements for
pure hold for rand. act.-inv. as well !
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75
New results
Reachability - Memory requirement (for player 1)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
exponential (belief)
memoryless
exponential (belief)
rand. act.-inv.
exponential (more
than belief)
pure
exponential (more
than belief)
non-elementary
complete
finite (at least nonelementary)
player 1 partial
player 2 perfect
player 1 perfect
player 2 partial
2-sided
both partial
rand. act.-vis.
memoryless
memoryless
memoryless
rand. act.-inv.
memoryless
memoryless
memoryless
exponential (more
than belief)
non-elementary
complete
finite (at least nonelementary)
Almost-sure
Positive
pure
Krishnendu Chatterjee
[CDHR’06]
[BGG’09]
[BGG’09]
finite (at least nonelementary)
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Beliefs
Three prevalent beliefs:
• Belief is sufficient.
• Randomized action invisible or visible almost same.
• The general case memory is similar (or in some cases
exponential blow up) as compared to the one-sided case.
Belief Fails!
[CD11] Chatterjee, Doyen. Partial-Observation Stochastic Games: How to Win when
Belief Fails. CoRR abs/1107.2141, July 2011.
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The Message
Play Chess; Play Soccer;
But stay away from Poker !!!
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Conclusion
 Theory of graph games
 Turn-based, concurrent, and partial-information games.
 Different solution concepts and different complexity.
 Several algorithmic questions open.
 Partial information games
 Problem with clear practical motivation.
 Challenging to establish the right frontier of complexity.
 Important generalization of perfect-information games.
 Unfortunately, undecidable and also high complexity.
 Current research: identifying decidable and more efficient sub-classes.
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Collaborators
 Luca de Alfaro
 Laurent Doyen
 Thomas A. Henzinger
 Jean-Francois Raskin
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The end
Thank you !
Questions ?
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