Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
A Dynamic-Epistemic View on Two
Interpretations of Imperfect Information
Extensive-Form Games
(Ongoing work)
Yanjing Wang
Department of Philosophy, Peking University
SAET2014, Waseda University
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Yanjing Wang Department of Philosophy, Peking University:
Conclusions and future work
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Extensive Games with perfect Information
1
a
b
2
o
c
d
o0
o0
c
2
d
o
I
Game rules: an exact and exhaustive description of the
physical rules of a game.
I
Game runs: (branching-time) temporal structure of the
plays of the game.
I
How to play v.s. the actual plays
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Extensive games with imperfect information
1
a
b
2
o
I
I
I
I
I
c
d
o0
o0
c
2
d
o
‘Player 2 is not informed where he is’
‘Player 2 cannot distinguish two histories’,
‘Player 2 does not observe the actions by 1’
Game rules + external info to be provided to the players.
Game runs + players’ uncertainty when they play
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Extensive games with imperfect information
However, these two views may not coincide any more:
1
a
2
c
(0, 2)
b
{
#
d
(0, 1)
c
(1, 2)
2
d
(1, 3)
I
Rule-view: Information sets are primitive, players can
distinguish states in different info sets without making
inferences (they may distinguish more when playing).
I
Run-view: perfect/imperfect recall and uniform strategies.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Extensive games with imperfect information
From the run-view, the models may be inadequate:
I
I
I
I
Uncertainty of one player at decision nodes for others?
Higher-order belief and knowledge? (in solution concepts?)
Information sets are overloaded (observability, belief of the
games rules, memory, rationality...)
Finite representation of infinite games?
1
a
b
@2
ac
2
d
o0
How to draw the right run model?
Yanjing Wang Department of Philosophy, Peking University:
o0
c
2^
db
2
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Our proposal
Separate rules and runs, and study how to go from rules to
runs with explicit assumptions of agents’ information and ability
Occasionally, rules ≈ runs, and this may explain why many
people implicitly embrace the ‘mixed’ interpretation, but we
want to know when this can happen.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Logical tools to handle rules and runs
I
Epistemic Temporal Logic (ETL): knowledge in distributed
systems based on temporal logic.
[Fagin et al., 1995, Parikh and Ramanujam, 1985]
I
Dynamic Epistemic Logic (DEL): knowledge in multi-agent
interactions based on epistemic logic. [Plaza, 1989,
Gerbrandy and Groeneveld, 1997, Baltag et al., 1998]
ETL
DEL
language
time+K
K+action
model
temporal+epistemic
epistemic
semantics
Kripke
Kripke+dynamic
Initial epistemic model × DEL updates ≈ special ETL models
[van Benthem et al., 2009]. DEL is a special kind of ETL in
terms of logic [Wang and Cao, 2013, Wang and Aucher, 2013].
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
The big picture
The runs of a game are determined by the (physical) game
structure plus the assumptions of players.
I
Game structure
I
I
I
Game rules
External information pieces
Player assumptions
I
I
I
Initial uncertainty
Observational power of actions
Knowledge / belief update mechanism
Runs are induced by iterated updating observation of actions
from initial uncertainties, according to the game rules under the
external information.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Observational
Initial
uncertainty: power:
1,2
1,2
w:s
a
2
1,2
o1
o2 o3
o4
c
1,2
b
1,2
d
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Initial
uncertainty:
1,2
1,2
w:s
a
o1
o2 o3
o4
c
a
1,2
2
1,2
Run model:
w:s
Observational
power:
b
1,2
d
b
wa : t
c
2
d
c
d
wac wad wbc wbd
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
wb : t 0
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Initial
uncertainty:
Observational
power:
1,2
1,2
w:s
a
o1
o2 o3
o4
c
b
1,2
1,2
1,2
d
Run model:
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Initial
uncertainty:
1,2
1,2
w:s
a
o1
o2 o3
o4
c
a
1,2
b
1,2
Run model:
w:s
Observational
power:
1,2
d
wa : t
c
d
wb : t 0
c
d
wac wad wbc wbd
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
b
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Initial
uncertainty:
Observational
power:
1,2
1,2
w:s
a
1,2
2
1,2
o1
o2 o3
o4
c
b
1,2
d
Run model:
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s:1
a
b
t :2
t0 : 2
c d
c d
Initial
uncertainty:
1,2
1,2
w:s
a
o1
o2 o3
o4
c
a
1,2
2
1,2
Run model:
w:s
Observational
power:
b
1,2
d
wa : t
c
d
wb : t 0
c
d
wac wad wbc wbd
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
b
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s0 : 1
s:1
a
b
t :2
t0 : 2
c
c
o1
d
o2 o3
Initial
uncertainty:
1,2
1,2
w:s
1
Observational
power:
1,2
u:s
a
2
1,2
v7 : s 0
c
d
o4
1,2
1,2
b
1,2
d
Run model:
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
s0 : 1
s:1
a
b
t :2
t0 : 2
c
c
o1
d
o2 o3
Initial
uncertainty:
1,2
1,2
w:s
1
Observational
power:
1,2
u:s
a
2
1,2
v7 : s 0
c
d
o4
1,2
1,2
b
1,2
d
Run model:
w : s 1 u : s 2 v : s0
a
a
b
vb : t 0
wa : t
1
ua : t
The essence of the update mechanism:
I
layout the ‘first floor’ using initial uncertainty
I
previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Run model:
Game structure:
Initial
s:1
uncertainty:
a
7t :2
c
d
b
1,2
t0 : 2 c
c
w:s
d
o o0
Observational
power:
1,2
1,2
a
2
1,2
c
b
1,2
d
Previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Examples instead of formal definitions
Game structure:
Initial
s:1
uncertainty:
a
7t :2
c
d
b
1,2
t0 : 2 c
c
w:s
d
o o0
Run model:
w:s
Observational
power:
1,2
1,2
a
2
1,2
c
d
wa : t
b
1,2
a
c
2
wb : t 0
c
d
wac : t wad wbc wbd : t 0
c d
Previous floor × available actions ≈ new floor:
ua ∼i vb ⇐⇒ u ∼i v, a ∼i b and infoi (u) = infoi (v)
Yanjing Wang Department of Philosophy, Peking University:
d
b
c d
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
A logical language
Given a non-empty set of agents I, and a non-empty finite set Σ
of basic actions, the epistemic action language EAL is defined
as follows:
φ ::= TURNi | ¬φ | φ ∧ φ | Ki φ | [a]φ
where i ∈ I, pi ∈ P, and a ∈ Σ. The semantics is defined on
generated epistemic temporal structures:
M, w TURNi
M, w ¬φ
M, w φ ∧ ψ
M, w Ki ψ
M, w [a]φ
⇔
⇔
⇔
⇔
⇔
w is mapped to a decision node of i
M, w 2 φ
M, w φ and M, w ψ
∀v : w ∼i v =⇒ M, v ψ
a
∀v : w → v =⇒ M, v ψ
More primitive propositions about the outcome can be added.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Language and semantics
Let K̂i be the dual of Ki , and let Kwi φ be the short hand of
Ki φ ∨ Ki ¬φ. We can specify the properties of the generated
model using this language, e.g.,
_
^
TURNi →
¬TURNj , Kwi TURNi , TURNi →
Kwi hai>
j,i
Perfect Recall : haiK̂i φ →
a∈Σ
_
K̂i hbiφ
b:a∼i b
No miracles : K̂i hbi(φ ∧ ψIS ) → [a](ψIS → K̂i φ)) (if a ∼i b)
We may axiomatize the validities of EAL on generated
epistemic temporal structures [Wang and Aucher, 2013].
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Freedom in design
Our proposal is:
I
explicit in the assumptions of the games and players.
I
constructive in generating the run models.
I
good for higher-order epistemic reasoning.
To-do:
I
Axiomatize this logic to reveal the underlying assumptions
via ETL methods.
I
Characterize properties of games and agents that induces
a structure preserving mapping from game structures to its
run model, and back (rules ≈ runs).
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Generalizations
We can also relax some assumptions or generalize the
framework:
I
belief instead of knowledge
I
higher-order uncertainty about observational power
I
asynchronous runs via silent moves
I
imperfect recall via forgetting in updates
I
(discrete) probabilistic belief and Bayesian-like updates
Hard updates involving expectations of rationality
Hard learning via partial games inductively
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Take home message
Separate and then connect two interpretations of extensive
games with imperfect information via runs=rules × updates.
Thank you very much for your attention!
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Game structure (basic rules)
Game structure=Game rules+external information.
Given a set Σ of actions, a set I of players, a game structure
w.r.t. Σ and I is a tuple: G = hS, O, ι, Ri where:
I
S is a non-empty set of configurations (game states),
I
O is a possibly empty set of outcomes,
I
ι : S → I is the player assignment,
I
R is a partial function S × Σ → (S ∪ O) such that for any
s ∈ S there exists at least an a ∈ Σ such that R(s, a) is
defined.
a
To simplify notations, we write s → t for sRa t and write s ∈ i if
s ∈ P(i).
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Configuration dependent external information
An (external) information model w.r.t. G is a tuple V = hP, Vi
where:
I
P is a (finite) set of pieces of information.
I
V : SG × I → 2P is a function assigning each game sate
and each agent i a piece of external information that is
available to i.
In particular we are interested in the normal information models
satisfying the following conditions for all s, t ∈ G :
I
V(s, i) = ∅, if s < i and s < OG .
I
ei (s) , ei (t) implies V(s, i) , V(t, i).
I
s ∈ OG and t < OG implies V(s, i) , V (t, i).
Clearly V induces a partition over {s | s ∈ i} for each i.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Initial uncertainty: epistemic frame
Player assumptions=Initial uncertainty + Observational power
of actions + Knowledge / belief update mechanism
An epistemic frame w.r.t. G is a tuple: F = (W , f , ∼) where:
I W is a nonempty set
I f : W → SG ∪ OG assigns each w ∈ W a configuration in
the game.
I ∼: I → 2W ×W assigns an (equivalence relation) over W to
each i ∈ I such that w ∼i v implies ei (f (w)) = ei (f (v)).
1,2
1,2
w1 : s o
1
/ w2 : s o
1,2
2
/ w3 : s 0
Given G and P, an epistemic model is an epistemic frame with
an information structure (W , f , ∼, V).
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Observational power: action model
An action model U w.r.t. Σ is a tuple (U, g, ) where:
I
U is a set of epistemic actions including the empty action .
I
g : U → Σ assigns each epistemic action an action in Σ.
I
: I → 2U×U is a binary relation over U for each i ∈ I.
1,2
1,2
u1 : a o
1
1,2
/ u2 : a o 2 / u3 : b
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Update Product ⊗
Fixing G and thus Σ, an information model V, an epistemic
model M = (W , f , ∼, V) and an action model U = (U, g, ),
the product model is an epistemic model
(M ⊗ U) = (W 0 , f 0 , ∼0 , V 0 ) where:
W 0 = {(w, a) | w ∈ W , a ∈ e(f (w))}
f 0 (w, a) = R(f (w), a))
∼0i = {((w, a), (v, b)) | w ∼i v, a i b
and V (R(f (w), a), i) = V(R(f (w), b), i)
0
V ((w, a)) = V(R(f (w), a))
Iterated updating with the action model according to the game
rules induces the desirable ETL-like temporal structure.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Baltag, A., Moss, L., and Solecki, S. (1998).
The logic of public announcements, common knowledge,
and private suspicions.
In Proceedings of TARK ’98, pages 43–56. Morgan
Kaufmann Publishers Inc.
Fagin, R., Halpern, J., Moses, Y., and Vardi, M. (1995).
Reasoning about knowledge.
MIT Press, Cambridge, MA, USA.
Gerbrandy, J. and Groeneveld, W. (1997).
Reasoning about information change.
Journal of Logic, Language and Information, 6(2):147–169.
Parikh, R. and Ramanujam, R. (1985).
Distributed processes and the logic of knowledge.
In Proceedings of Conference on Logic of Programs, pages
256–268, London, UK. Springer-Verlag.
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
Plaza, J. A. (1989).
Logics of public communications.
In Emrich, M. L., Pfeifer, M. S., Hadzikadic, M., and Ras,
Z. W., editors, Proceedings of the 4th International
Symposium on Methodologies for Intelligent Systems,
pages 201–216.
Stalnaker, R. (1978).
Assertion.
In Cole, P., editor, Syntax and Semantics, volume 9. New
York Academic Press.
van Benthem, J., Gerbrandy, J., Hoshi, T., and Pacuit, E.
(2009).
Merging frameworks for interaction.
Journal of Philosophical Logic, 38(5):491–526.
Wang, Y. and Aucher, G. (2013).
Yanjing Wang Department of Philosophy, Peking University:
Two interpretations
A dynamic-epistemic proposal to connect the two
Conclusions and future work
An alternative axiomatization of del and its applications.
In IJCAI.
Wang, Y. and Cao, Q. (2013).
On axiomatizations of public announcement logic.
Synthese.
Online first: http://dx.doi.org/10.1007/s11229-012-0233-5.
Yanjing Wang Department of Philosophy, Peking University: