Introduction The Logic Aumann’s Theorem Conclusion
An Epistemic Logic of Extensive Games E. Lorini
F. Moisan
Universite´ de Toulouse, CNRS, Institut de Recherche en Informatique de Toulouse (IRIT)
LAMAS 2011 Osuna, Spain
1 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
What type of games?
= extensive / sequential / dynamic games players interact strictly sequentially no simultaneous moves
Games with perfect and complete information ⇒ only uncertainty about the opponents’ future moves!
2 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
What type of games?
= extensive / sequential / dynamic games players interact strictly sequentially no simultaneous moves
Games with perfect and complete information ⇒ only uncertainty about the opponents’ future moves!
2 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
Representation of games in extensive form (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1
Properties of the game Vertices: {w1 , w2 , w3 } Agents (one / vertex): {Alice, Bob} Actions: {α1 , β1 , α2 , β2 , α3 , β3 } Payoffs: e.g. (30, 20) 3 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
Representation of games in extensive form (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1
Defining strategy profiles: ⇒ specifies an action at every vertex of the game. 8 strategy profiles: e.g. (α1 , α2 , β3 ), (β1 , β2 , α3 ) 4 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
Background In Economics: Epistemic game theory / interactive epistemology [Aumann, Battigally, Brandenburger, Gintis, Bonanno, . . . ]
In Logic: Logics for reasoning about strategies: e.g. [Bonanno, 2001, 2002][van Benthem et al, 2006, 2010] [Ramanujan and Simon, 2008][Walther et al, 2007] No epistemic state!
Logics for reasoning about solution concepts and epistemic states: e.g. [Baltag et al, 2009][Lorini and Schwarzentruber, 2010] No temporal reasoning!
5 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
Motivation
Create a logic sufficiently general to: reason about the temporal factor reason about the players’ epistemic states ⇒ express economic concepts in the object language (e.g. backward induction, rationality)
Provide a formal analysis of Aumann’s theorem Aumann’s statement: ”for any non degenerate game (i.e. with 6= payoffs at all leaves) of perfect information, common knowledge of rationality implies the backward induction solution” ⇒ Identify specific assumptions to prove the theorem
6 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Extensive games Motivation
Motivation
Create a logic sufficiently general to: reason about the temporal factor reason about the players’ epistemic states ⇒ express economic concepts in the object language (e.g. backward induction, rationality)
Provide a formal analysis of Aumann’s theorem Aumann’s statement: ”for any non degenerate game (i.e. with 6= payoffs at all leaves) of perfect information, common knowledge of rationality implies the backward induction solution” ⇒ Identify specific assumptions to prove the theorem
6 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Outline
1
The Logic
2
Aumann’s Theorem
3
Conclusion
7 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
An Epistemic Logic of Extensive Games (ELEG)
The syntactic primitives of ELEG: A finite set of agents Agt A finite set of atomic propositions Atm A nonempty finite set of atomic action names Act = {α1 , α2 , . . . , α|Act| } A non-empty finite set of n integers I = {0, . . . , n}
8 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
The language of ELEG The set of atomic formulas: χ ::= p | α | turni | end | ki p ∈ Atm
α ∈ Act = “the action α is performed”
turni = “it is agent i’s turn to play (i ∈ Agt)”
end = “the current vertex of the game is an end vertex” ki = “the current strategy profile will ensure a payoff k ∈ I to agent i ∈ Agt” 9 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
The language of ELEG The set of all formulas: ϕ ::= χ | ¬ϕ | ϕ ∨ ϕ | Xϕ | AXϕ | ϕ | [Ki ]ϕ Xϕ = “ϕ will be true next according to the current strategy profile” AXϕ = “ϕ is true at every possible next vertex along the current strategy profile” ϕ = “ϕ holds for all strategy profiles of the current extensive game” [Ki ]ϕ = “agent i knows that ϕ is true” 10 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some useful definitions X0 ϕ
def
=
ϕ
Xn+1 ϕ
def
XXn ϕ
=
Xn ϕ = “ϕ will be true n steps from now according to the current strategy profile” AX0 ϕ AX
n+1
ϕ
AX≤n ϕ
def
=
ϕ
def
AX(AXn ϕ) V m 0≤m≤n AX ϕ
=
def
=
AX≤n ϕ = “ϕ is true at every vertex that can be reached within n step(s) from now, along the current strategy profile” 11 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some useful definitions
hα0 ; . . . ; αn iϕ
def
=
V
0≤l≤n
Xl αl ∧ Xn ϕ
hiϕ = “the sequence of actions ∈ Seq will occur and ϕ will be true afterwards” V def [EKC ]ϕ = i∈C [Ki ]ϕ
[EKC ]ϕ = “everyone in C knows that ϕ”
12 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some useful definitions
[EK0C ]ϕ
def
=
ϕ
[EKkC ]ϕ
def
=
[EKC ]([EKk−1 C ]ϕ)
[CK0C ]ϕ
def
[CKnC ]ϕ
def
ϕ V
= =
k 1≤k≤n [EKC ]ϕ
[CKnC ]ϕ = “It is common knowledge up to n iterations among agents in C that ϕ”
13 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Strategic structure) T = hV , Q, S, next, EndV i:
V is a non-empty set of vertices; Q is a total function Q : V −→ Agt; (30, 20)
V = {w1 , w2 , w3 } Q(w1 ) = Alice Q(w2 ) = Bob Q(w3 ) = Bob
α2
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 14 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Strategic structure) T = hV , Q, S, next, EndV i:
S is a nonempty set of strategy profiles on V , and every strategy profile s ∈ S is a total function s : V −→ Act; (30, 20)
s1 ∈ S s1 (w1 ) = α1 s1 (w2 ) = α2 s1 (w3 ) = β3
α2
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 15 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Strategic structure) T = hV , Q, S, next, EndV i: next is a partial function next : V × S −→ V such that: C1 if s(w) = s0 (w) then next(w, s) = next(w, s0 ); (30, 20) α2
next(w1 , s1 ) = w2
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 16 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Strategic structure) T = hV , Q, S, next, EndV i: EndV ⊆ V is the set of end vertices such that:
C2 w ∈ EndV if and only if, next(w, s) is undefined for every s. (30, 20) α2
EndV = {w2 , w3 }
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 17 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Successor) R is a relation on vertices such that: for every w, v ∈ V , wRv if and only if there is s ∈ S such that next(w, s) = v . (30, 20) α2
w1 Rw2 w1 Rw3
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 18 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Extensive game model) M = hT , π, {Ei | i ∈ Agt}, {Pi | i ∈ Agt}i: T is a strategic structure;
π : Atm −→ 2V ×S is a valuation function. (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1 19 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Extensive game model) M = hT , π, {Ei | i ∈ Agt}, {Pi | i ∈ Agt}i: Every Ei is an equivalence relation on S such that: C3 if sEi s0 and Q(w) = i, then s(w) = s0 (w);
(30, 20) α2
α3
β2
(30, 20)
(50, 5)
(5, 10) (15, 40)
EBob
β3
α2
w3
w2 α1
β1 w1
(5, 10) (15, 40) α3
β2
β3 w3
w2
s1
s2
(50, 5)
α1
β1 w1
20 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Extensive game model) M = hT , π, {Ei | i ∈ Agt}, {Pi | i ∈ Agt}i: Every Pi is a total function Pi : V × S −→ I such that:
C4 if next(w, s) = w 0 , then Pi (w, s) = k if and only if Pi (w 0 , s) = k; C5 if w ∈ EndV and s(w) = s0 (w) then Pi (w, s) = Pi (w, s0 ). (30, 20)
PAlice (w1 , s1 ) = 30 PAlice (w2 , s1 ) = 30 PAlice (w3 , s1 ) = 50
α2
⇔
(50, 5)
(5, 10) (15, 40) α3
β2
β3 w3
w2 α1
β1 w1 21 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Truth conditions) M, w, s |= p iff (w, s) ∈ π(p);
M, w, s |= ¬ϕ iff M, w, s 6|= ϕ;
M, w, s |= ϕ ∨ ψ iff M, w, s |= ϕ or M, w, s |= ψ; M, w, s |= α iff s(w) = α;
M, w, s |= turni iff Q(w) = i;
M, w, s |= end iff w ∈ EndV ; M, w, s |= ki iff Pi (w, s) = k; ...
22 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Truth conditions) M, w, s |= Xϕ iff if next(w, s) is defined then M, next(w, s), s |= ϕ; (30, 20) α2
M, w1 , s1 |= Xϕ
⇔
(50, 5)
(5, 10) (15, 40) α3
β2 w2
β3
ϕ
w3
α1
β1 w1
23 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Truth conditions) M, w, s |= AXϕ iff M, w 0 , s |= ϕ for all w 0 ∈ V such that wRw 0 ; (30, 20) α2
M, w1 , s1 |= AXϕ
⇔
(50, 5)
(5, 10) (15, 40) α3
β2 w2
β3
ϕ
ϕ
α1
w3
β1 w1
24 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Semantics Definition (Truth conditions) M, w, s |= ϕ iff M, w, s0 |= ϕ for all s0 ∈ S;
M, w, s |= [Ki ]ϕ iff M, w, s0 |= ϕ for all s0 such that sEi s0 . M, w1 , s1 |= [KBob ]ϕ ⇔ (30, 20) α2
α3
β2
(30, 20)
(50, 5)
(5, 10) (15, 40)
EBob
β3
α2
w3
w2 α1
β1 w1
ϕ
(5, 10) (15, 40) α3
β2
β3 w3
w2
s1
s2
(50, 5)
α1
β1 w1
ϕ
25 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some validities
All principles of classical propositional logic
(CPL)
All S5 principles for
(S5 )
All S5 principles for every [Ki ] All K principles for X All K principles for AX
(S5[Ki ] ) (KX ) (KAX )
26 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some validities About the structure of the game: Unchanging game structure: AXϕ ↔ AXϕ (α ∧ Xϕ) → (α → Xϕ) Preference consistency: ¬end → (ki ↔ Xki ) Complete information: (end ∧ α ∧ ki ) → (α → ki ) 27 / 44
Introduction The Logic Aumann’s Theorem Conclusion
The language Some useful definitions Semantics Validities
Some validities About the epistemic operator: Perfect information: ϕ → [Ki ]ϕ No learning / no forgetting: [Ki ]AXϕ ↔ AX[Ki ]ϕ Awareness: turni → (α → [Ki ]α)
28 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Outline
1
The Logic
2
Aumann’s Theorem
3
Conclusion
29 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Backward Induction
Backward induction = computational method to reach subgame perfect Nash equilibria Subgame perfect Nash equilibria = restriction on Nash equilibria ⇒ Rules out incredile threats ⇒ More realistic!
30 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Backward Induction
Backward induction = computational method to reach subgame perfect Nash equilibria Subgame perfect Nash equilibria = restriction on Nash equilibria ⇒ Rules out incredile threats ⇒ More realistic!
30 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Backward Induction An example: (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1
2 pure Nash equilibria: (α1 , α2 , α3 ), (β1 , β2 , α3 ) Unique subgame perfect Nash equilibrium: (α1 , α2 , α3 ) 31 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Backward Induction Definition in ELEG (assuming a game of uniform depth): M, w, s |= BIn iff at w, s is a backward induction solution that can be computed in n steps For the case n = 0: _ _ def BI0 = end ∧ (turni ∧ ki ∧ ( hi )) i∈Agt,k∈I
h∈I:h≤k
For every n > 0: def
BIn = ¬end ∧
_
i∈Agt,k∈I
(turni ∧ ki ∧ AX(BIn−1 ∧
_
hi ))
h∈I:h≤k
32 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality
Aumann’s definition: Rational player = payoff maximizer “No matter where a player finds himself - at which vertex he will not knowingly continue with a strategy that yields him less than he could have gotten with a different strategy” [Aumann, 1995] ⇒ Concept of substantive rationality
33 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality An example: (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1
Is it rational for Alice to play α1 ? β1 ? Is it rational for Bob to play (α2 , α3 )? (α2 , β3 )? 34 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality An example: (30, 20)
(50, 5)
(5, 10) (15, 40)
α2
α3
β2
β3 w3
w2 α1
β1 w1
Is it rational for Alice to play α1 ? β1 ? Is it rational for Bob to play (α2 , α3 )? (α2 , β3 )? 34 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality Definition of material rationality in ELEG: If current vertex = end vertex: def
Ratend = (end ∧ turni ) → i
_
k∈I
(ki ∧ (
_
If current vertex 6= end vertex: def
= (¬end ∧ turni ) → Rat¬end i
_
k∈I
hi ))
h∈I:h≤k
hKi i(ki ∧ AX(
_
hKi ihi ))
h∈I:h≤k
Generally: def
Rati = Ratend ∧ Rat¬end i i 35 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality
Definition of substantive rationality in ELEG: def
SRatni = AX≤n Rati
Agents are aware of their rationality: `ELEG Rati ↔ [Ki ]Rati `ELEG SRatni ↔ [Ki ]SRatni
36 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Epistemic Rationality
Definition of substantive rationality in ELEG: def
SRatni = AX≤n Rati
Agents are aware of their rationality: `ELEG Rati ↔ [Ki ]Rati `ELEG SRatni ↔ [Ki ]SRatni
36 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Aumann’s Theorem
Additional definitions: The game (of depth at most n) is in the general position: GenPosn
def
=
V
0≤h≤n
V
k∈I,i∈Agt,∈Seq h
AX≤n ((ki ∧ hiend)
→ (hiend ↔ ki )) The game is finite and has a uniform depth of degree n from the current vertex: def
Depthn = (X)n end
37 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Aumann’s theorem
Theorem For every n, m ∈ N such that n ≤ m, we have: `ELEG ([CKm Agt ]
^
i∈Agt
SRatni ∧ Depthn ∧ GenPosn ) → BIn
38 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Proving Aumann’s Theorem The syntactic proof in ELEG: ⇒ Hilbert style proof
Some results: KD4 principles for [Ki ] operator are sufficient ⇒ [Ki ] can be interpreted as a belief operator (without negative introspection)
agents are required to have perfect recall throughout the game Axiom [Ki ]AXϕ → AX[Ki ]ϕ necessary to the proof
Agents may learn through the gameplay
Axiom AX[Ki ]ϕ → [Ki ]AXϕ irrelevant to the proof ⇒ Not allowed with the current interpretation of [Ki ]! 39 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
A more convenient characterization of knowledge
Reinterpretation of the epistemic modal operator By means of an equivalence relation Eiw on strategy profiles S for every agent i ∈ Agt and vertex w ∈ W Perfect recall constraint: if sEiv s0 and wRv then sEiw s0 ⇒ Agents can learn! ⇒ Agents need not be aware of their future strategy!
40 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Backward Induction Epistemic Rationality The Theorem Analysis
Criticism about Aumann’s Theorem
Perfect recall constraint still very strong and not very realistic! ⇒ Substantive rationality requires belief revision [Stalnaker, 1998] Example from Philosophical literature [Bennett]: If Shakespeare had not written Hamlet, then one may believe that: it would never have been written (cf. Aumann) someone else would have written it (cf. Stalnaker)
41 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Summary Future Work
Outline
1
The Logic
2
Aumann’s Theorem
3
Conclusion
42 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Summary Future Work
Summary
We provide a logical framework sufficiently general to: define solution concepts (e.g. backward induction) define epistemic concepts (e.g. (bounded) rationality)
We demonstrate that a formal syntactic proof of some economic theorem: provides some in-depth analysis allows to identify needed/unnecessary assumptions
43 / 44
Introduction The Logic Aumann’s Theorem Conclusion
Summary Future Work
Future Work
To provide a complete axiomatization of the logic To define and analyse other economic concepts such as fairness and reciprocity To consider imperfect/incomplete information games To generalize the logic for reasoning about the past (e.g. forward induction reasoning, emotional reasoning)
44 / 44
