Recovering Incomplete Information
Dietmar Berwanger
Joint work with R. Ramanujam
Highlights, Bruxelles
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
1 / 11
Information Matters
perfect — imperfect
whether players know the play history
played actions ● observations (received by others) ● visited states
complete — incomplete
whether players know the game structure
winning condition ● observation function ● available actions (of others)
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
2 / 11
Incomplete Information
Game tree
1
Winning condition?
2
2
First player knows which
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
3 / 11
Incomplete Information
players: n processes — of type: alive or dead
one process is dead (x, initially unknown)
actions:
if alive: crash, idle, output_y
if dead: crash
1
2
3
8
4
7
6
5
observations: actions of neighbours
winning: all alive processes output_x, simultaneously
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
3 / 11
Key Issue: Recovering the Initial State
Recovering strategy: joint strategy s such that
any two plays that start from different states and follow s
are finally distinguishable, for each player
Strategy to make initial state common knowledge.
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
4 / 11
Incomplete vs Imperfect
Incomplete info: imperfect info only about first move
– finite amount of uncertainty
Recovering condition as reachability, with imperfect info:
G3
G1
33
11
Message: Synthesis
of
is solvable (finite-state).
≠33
≠11recovering strategies
+
D. Berwanger (LSV)
-
+
-
Recovering Incomplete Information
Highlights 2016
5 / 11
Monitoring: Perfect, Public, Private
Monitoring structure:
� perfect: observation identifies move
� public: same observation for all players
� private: not public
Theorem. The synthesis problem for recovering strategies
is finite-state solvable for games:
( ) with incomplete, but public monitoring, and moreover,
( ) with private monitoring, if states are public.
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
6 / 11
Case : Public Monitoring
The coordinator game:
� states: knowledge about initial state – equivalence structure
� moves: joint actions + (public) observation
� winning condition: following play in original game
free to include conditions about knowledge structure
A zero-sum game with perfect information.
Solution: joint strategy for original game
recovering incomplete information + winning.
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
7 / 11
Knowledge About Initial State (Public)
1
5
3
2
4
Update in a Move
1
5
3
2
4
b
a
1
5
1
3
2
5
3
4
2
4
Update in a Move
1
5
3
2
a
1
4
b
b
5
3
3
2
5
1
4
2
4
Joint Action
a
1
5
3
2
4
Joint Action
a
1
5
3
2
4
Private Knowledge
1
5
3
2
4
Update in a Move
1
5
3
2
4
b
a
1
5
1
3
2
5
3
4
2
4
Signals, Public and Private
Signal f ∶ histories → value
� Private signal of player i: depends on observed history
� Public signal: private to all players
Examples: observations, state of strategy automaton, chosen action
Action of i at a given history: private signal
However, the strategy of i, as a function, is public
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
8 / 11
Separating Public and Private Signals
� Decompose observations into a public view Y, private views (Z i )i
Strategy machine, in general:
public update: (Q × Y) → Q
private update: (Q i × B i ) → (Q i × Ai )
� Make private update conditional to public state:
Q → [(Q i × Z i ) → (Q i × Ai )]
Whenever this can be done in finite states, we’ve made it!
Under perfect monitoring, easy:
Q knowledge structures (finite), Q i partition
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
9 / 11
Private Monitoring
Private Monitoring
Private-Monitoring Strategy - Player 1
Private-Monitoring Strategy - Player 2
Putting them Toghether (What a Mess)
Public-Private Monitoring Strategy
Public component
Public component
Perfec
!
n
o
i
t
a
m
r
o
t inf
Case : Private Monitoring, but States are Public
Factoring out public signals does not help immediately:
private updates won’t be finite-state
� Insight:
“Nothing is more confusing than persistent responses from Nature”
If coalition wins against every persistent response of Nature,
then it has a winning strategy.
A game with incomplete information. Sovlable !
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
10 / 11
Bottomline
� Incomplete information: special kind of imperfect information
� Recovering: winning condition about plays in relation
� Synthesis possible with small doses of imperfect information
� Decomposition into public, private signals; factorisation
D. Berwanger (LSV)
Recovering Incomplete Information
Highlights 2016
11 / 11