General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Models of Beliefs for Boundedly Rational Agents Mikaël Cozic IHPST (Paris I-ENS Ulm-CNRS), GREGHEC (HEC-CNRS) & DEC (ENS Ulm)
LogKCA - ILCLI 29-30/11/2007
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
General Introduction
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
doxastic models ◮
two main families of doxastic models:
(i) epistemic logic for full beliefs: • Pierre believes that φ • Pierre believes that ¬φ • Pierre neither believes that φ nor believes that ¬φ (ii) probability for partial beliefs: • Pierre believes that φ to degree r
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
doxastic models ◮
two main families of doxastic models:
(i) epistemic logic for full beliefs: • Pierre believes that φ • Pierre believes that ¬φ • Pierre neither believes that φ nor believes that ¬φ (ii) probability for partial beliefs: • Pierre believes that φ to degree r ◮
both families of doxastic models are used • in isolation as abstract representations of beliefs • as component in models of rational action since what it is rational to do depends on one’s beliefs (including, in a strategic context, what one believes that others believe) Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
cognitive idealizations ◮
it is well known that epistemic logic suffers from two main cognitive idealizations:
(i) logical omniscience (LO): closure under logical consequence, substituability of logically equivalent formulas, etc. (ii) full awareness(FA): full understanding of the state space An agent modelled by epistemic logic is both logically omniscient and fully aware.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
dealing with idealizations ◮
lots of attention has been devoted to the weakening of these two cognitive idealizations :
• (LO): impossible states (Hintikka 1975, Wansing 1991) c awareness structures (Fagin & Halpern 1988), neighborhood structures (Chellas 1980), see Stalnaker (1991 & 1999), FHMV (1995) c • (FA) : awareness structures (Fagin & Halpern 1988, Halpern 2001), generalized standard structures (Modica & Rustichini (MR) 1994, 1999), interactive unawareness (Heifetz, Meier & Schipper (HMS) 2006, HMS 2007a, HMS 2007b) see also Dekel, Lipman & Rustichini (1998)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
a plea for impossible states ◮
it is rarely recognized that probabilistic models of beliefs suffer from (LO) and (FA) as well
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
a plea for impossible states ◮
it is rarely recognized that probabilistic models of beliefs suffer from (LO) and (FA) as well
◮
to weaken these cognitive idealizations is intrinsically valuable and crucial for the development of bounded rationality in decision theory and game theory
◮
these models of rational action inherit the cognitive idealizations of their underlying doxastic models, especially the probabilistic one
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
a plea for impossible states ◮
it is rarely recognized that probabilistic models of beliefs suffer from (LO) and (FA) as well
◮
to weaken these cognitive idealizations is intrinsically valuable and crucial for the development of bounded rationality in decision theory and game theory
◮
these models of rational action inherit the cognitive idealizations of their underlying doxastic models, especially the probabilistic one
◮
Question: how to weaken (LO) and (FA) in models of partial beliefs ?
◮
broad aim: defend the impossible states (or worlds) approach as a unifying way to weaken cognitive idealizations = a plea for impossible states Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
a unifying solution
(i) what is a “unifying” solution ? (1) a solution to both logical omniscience and unawareness (2) a solution for models of full and partial beliefs
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
a unifying solution
(i) what is a “unifying” solution ? (1) a solution to both logical omniscience and unawareness (2) a solution for models of full and partial beliefs (ii) what are impossible states? states of the state space where a priori “anything goes”
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Outline of the talk
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Tutorial 1 : Logical Omniscience
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
Kripke structures ◮
Definition : The set of formulas of an epistemic propositional language LB(At) based on a set At of propositional variables Form(LB(At)), is defined by φ ::= p|¬φ|φ ∨ ψ|Bφ where p ∈ At.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
Kripke structures ◮
Definition : The set of formulas of an epistemic propositional language LB(At) based on a set At of propositional variables Form(LB(At)), is defined by φ ::= p|¬φ|φ ∨ ψ|Bφ where p ∈ At.
◮
Definition : let LB(At) an epistemic propositional language ; a Kripke structure for LB(At) is a 3-tuple M = (S, π, R) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation (iii) R ⊆ S × S is an accessibility relation
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
◮
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
Definition : π ¯ , called the satisfaction relation, extends π to every formula of the langage according to the following conditions : ◮
π ¯ (s, p) = π(s, p) if p ∈ At
◮
π ¯ (s, φ ∨ ψ) = 1 iff π ¯ (s, φ) = 1 or π ¯ (s, ψ) = 1
◮
π ¯ (s, ¬φ) = 1 iff π ¯ (s, φ) = 0
◮
π ¯ (s, Bφ) = 1 iff ∀s′ s.t. sRs′ , π ¯ (s, φ) = 1 (= possible-state analysis of belief = to believe something is to exclude that it could be false)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
epistemic logic, example p, q
p
s1
s2
s3
s4
q
Pierre believes that p, hence that p ∨ q. Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
logical omniscience ◮
Definitions : (i) the proposition expressed by φ, or informational content of φ is [[φ]]M = {s : π ¯ (φ, s) = 1} (ii) φ M-implies ψ if [[φ]]M ⊆ [[ψ]]M (iii) φ and ψ are M-equivalent if [[φ]]M = [[ψ]]M
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
logical omniscience ◮
Definitions : (i) the proposition expressed by φ, or informational content of φ is [[φ]]M = {s : π ¯ (φ, s) = 1} (ii) φ M-implies ψ if [[φ]]M ⊆ [[ψ]]M (iii) φ and ψ are M-equivalent if [[φ]]M = [[ψ]]M
◮
Proposition: for all Kripke structure M, ◮
◮
Deductive monotony : if φ M-implies ψ, then Bφ M-implies Bψ Intensionality : if φ and ψ are M-equivalent, then Bφ and Bψ are M-equivalent
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
Kripke structures, axiomatization
System K
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (K) ((Bφ ∧ B(φ → ψ)) → Bψ) (NEC) From φ infer Bφ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
neighborhood structures ◮
Definition : a neighborhood structure is a 3-tuple M = (S, π, V ) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation, (iii) V : S → ℘(℘(S)), called the agent’s neighborhood system, associates to every state a set of propositions (a neighborhood set)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
neighborhood structures ◮
Definition : a neighborhood structure is a 3-tuple M = (S, π, V ) where (i) S is a state space, (ii) π : At × S → {0, 1} is a valuation, (iii) V : S → ℘(℘(S)), called the agent’s neighborhood system, associates to every state a set of propositions (a neighborhood set)
◮
New doxastic satisfaction condition : π ¯ (Bφ, s) = 1 iff [[φ]]M ∈ V (s)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
neighborhood structures, example p, q
p
s1
s2
s3
s4
V(s1) = {{s1 , s2 }}
q
Pierre believes that p but not that p ∨ q since [[p ∨ q]] ∈ / V (s1 ). Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
neighborhood structures, axiomatization
System E (Chellas, 1980)
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (RE) From φ ↔ ψ infer Bφ ↔ Bψ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
c awareness structures
◮
c Definition : an awareness structure is a 4-tuple (S, π, R, A) where
(i) (ii) (iii) (iv)
S is a state space, π : At × S → {0, 1} is a valuation, R ⊆ S × S is an accessibility relation, A : S → Form(LB(At)) is a function which maps every state c in a set of formulas ("awareness set").
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
c awareness structures
◮
c Definition : an awareness structure is a 4-tuple (S, π, R, A) where
(i) (ii) (iii) (iv) ◮
S is a state space, π : At × S → {0, 1} is a valuation, R ⊆ S × S is an accessibility relation, A : S → Form(LB(At)) is a function which maps every state c in a set of formulas ("awareness set").
New doxastic satisfaction condition : π ¯ (Bφ, s) = 1 iff ∀s′ s.t. sRs′ , s′ ∈ [[φ]]M and φ ∈ A(s)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
c awareness structures, example p, q
p
s1
s2
s3
s4
A(s1) = {p}
q
Pierre believes that p but not that p ∨ q since p ∨ q ∈ / A(s1 ). Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
c awareness structures, axiomatization
Minimal Epistemic Logic (FHMV 1995)
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
non-standard structures ◮
Definition : a non-standard structure is a 4-tuple M = (S, S ′ , π ¯ , R) where (i) (ii) (iii) (iv)
S is a space of standard states, S ′ is a space of non-standard states, R ⊆ S ∪ S ′ × S ∪ S ′ is an accessibility relation, π : Form(LB(At)) × S → {0, 1} is a satisfaction relation standard on S
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
non-standard structures ◮
Definition : a non-standard structure is a 4-tuple M = (S, S ′ , π ¯ , R) where (i) (ii) (iii) (iv)
S is a space of standard states, S ′ is a space of non-standard states, R ⊆ S ∪ S ′ × S ∪ S ′ is an accessibility relation, π : Form(LB(At)) × S → {0, 1} is a satisfaction relation standard on S
◮
Definition : the subjective informational content of φ is the set of states where φ is true : [[φ]]∗M = {s ∈ S ∗ : π(φ, s) = 1}
◮
Doxastic satisfaction condition : for s ∈ S, π(s, Bφ) = 1 iff for all s′ s.t. sRs′ , s′ ∈ [[φ]]∗M Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
non-standard structures, example p, q
p
s1
s2
s3
s4
s5
q
Pierre believes that p but not that p ∨ q since π(s5 , p ∨ q) = 0 and s1 Rs5 . Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
non-standard structures, axiomatization
Minimal Epistemic Logic (Wansing 1991)
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
respective powers
AWARENESS STRUCTURES
NON-STANDARD STRUCTURES
NEIGHBORHOOD STRUCTURES
KRIPKE STRUCTURES
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
implicit probabilistic structures
◮
Definition : let L(At) a propositional language ; an implicit probabilistic structure for L(At) is a 3-tuple M = (S, π, P) where (i) S is a state space, (ii) π is a valuation, (iii) P is a probability distribution on S.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
implicit probabilistic structures
◮
Definition : let L(At) a propositional language ; an implicit probabilistic structure for L(At) is a 3-tuple M = (S, π, P) where (i) S is a state space, (ii) π is a valuation, (iii) P is a probability distribution on S.
◮
Let PB(ϕ) the degree to which the agent modelled believes ϕ ie PB(ϕ) = r iff P([[ϕ]]M ) = r
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
probabilistic logical omniscience ◮
Are there forms of LO in this probabilistic framework? Yes :
◮
Proposition : for all probabilistic structure M, (i) deductive monotony : if φ M-implies ψ, then PB(φ) ≤ PB(ψ). (ii) intensionality : if φ and ψ are M-equivalent, then PB(φ) = PB(ψ).
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
probabilistic logical omniscience ◮
Are there forms of LO in this probabilistic framework? Yes :
◮
Proposition : for all probabilistic structure M, (i) deductive monotony : if φ M-implies ψ, then PB(φ) ≤ PB(ψ). (ii) intensionality : if φ and ψ are M-equivalent, then PB(φ) = PB(ψ).
◮
Certainty (r = 1 ) (i) if φ M-implies ψ, if it is certain that φ, it is certain that ψ (ii) if φ and ψ are M-equivalent, φ is certain iff ψ is certain.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
non-standard implicit probabilistic structures (NSIPS)
◮
Definition : let L(At) a propositional language ; a non-standard implicit probabilistic structure for L(At) is a 4-tuple M = (S, S ′ , π, P) where (i) S is a standard state space, (ii) S ′ is a non-standard state space, (iii) π : Form(L(At)) × S ∪ S ′ → {0, 1} is a satisfaction relation which is standard on S, (iv) P is a probability distribution on S ∗ = S ∪ S ′
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
NSIPS, example p, q 1/8 π(p ∨ q, s5) = 0
p 1/8
s1
s2
s3
s4
q 1/8
1/8
1/2 s5
Pierre believes that p to degree 3/4, but that p ∨ q to degree 3/8. Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
deductive information and learning in NSIPS ◮
What does it mean to learn that φ implies ψ ? According to the possible-state analysis of belief : exclude states where φ is true but ψ false ; hence, to learn the event I(φ, ψ) = −[[φ]]∗M ∪ [[ψ]]∗M
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
deductive information and learning in NSIPS ◮
What does it mean to learn that φ implies ψ ? According to the possible-state analysis of belief : exclude states where φ is true but ψ false ; hence, to learn the event I(φ, ψ) = −[[φ]]∗M ∪ [[ψ]]∗M
◮
Compatibility between conditionalization on I and deductive monotony Proposition : if I(φ, ψ) is learned according to Bayes rule, then deductive monotony is regained, i.e. PBI (φ) ≤ PBI (ψ).
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
explicit probabilistic structures (EPS) ◮
Definition : the set of formulas of an explicit probabilistic language LL(At) based on a set At of propositional variables, Form(LL(At)) is defined by : φ ::= p|¬φ|φ ∨ ψ|La φ where p ∈ At and a ∈ [0, 1] ⊆ Q. La φ : P. believes that φ at least to degree a
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
explicit probabilistic structures (EPS) ◮
Definition : the set of formulas of an explicit probabilistic language LL(At) based on a set At of propositional variables, Form(LL(At)) is defined by : φ ::= p|¬φ|φ ∨ ψ|La φ where p ∈ At and a ∈ [0, 1] ⊆ Q. La φ : P. believes that φ at least to degree a
◮
Definition : an explicit probabilistic structure for LLa (At) is a 3-tuple M = (S, π, P) where P : S → ∆(S) assigns to every state a probability distribution on the state space.
◮
Satisfaction condition for La : π ¯ (s, La φ) = 1 iff P(s)([[φ]]) ≥ a Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, cont. ◮
Some Remarks on EPS :
(i) this language is the one proposed by Aumann (1999) and Heifetz and Mongin (2001). Fagin, Halpern and Meggido 1990 and Halpern 2003 use a different language (linear inequalities)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, cont. ◮
Some Remarks on EPS :
(i) this language is the one proposed by Aumann (1999) and Heifetz and Mongin (2001). Fagin, Halpern and Meggido 1990 and Halpern 2003 use a different language (linear inequalities) (ii) EPS correspond to the type spaces used in games of incomplete information, in the same way that Kripke structures (with R as an equivalence relation) corresponds to the information partitions
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, cont. ◮
Some Remarks on EPS :
(i) this language is the one proposed by Aumann (1999) and Heifetz and Mongin (2001). Fagin, Halpern and Meggido 1990 and Halpern 2003 use a different language (linear inequalities) (ii) EPS correspond to the type spaces used in games of incomplete information, in the same way that Kripke structures (with R as an equivalence relation) corresponds to the information partitions (iii) from the explicit probabilistic language, one can define • Ma φ = L1−a ¬φ (P. believes at most to degree a that φ) • Ea φ = Ma φ ∧ La φ (P. believes exactly to degree a that φ) Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, cont. (iv) the landscape of doxastic models is roughly this one : full beliefs
partial beliefs
non-iterated attitudes
belief set B⊆S
implicit probabilities structures P ∈ ∆(S)
iterated beliefs
Kripke structures B : S → ℘(S)
explicit probabilistic structures P : S → ∆(S)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, cont.
◮
EPS is not compact : • Σ = {L1/2−1/n φ : n ≥ 2, n ∈ N} • ψ = ¬L1/2 φ Clearly, • for each finite Σ′ ⊂ Σ, Σ′ ∪ ψ is coherent • but Σ ∪ ψ is not coherent
◮
consequence : no strong completeness (Meier (2001) provides a strong completeness proof with an infinitary language)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, axiomatization System HM (Heifetz and Mongin 2001) (PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A1) L0 φ (A2) La ⊤ (A5)La φ → ¬Lb ¬φ (a + b > 1) (Def M) Ma φ ↔ L1−a ¬φ (A8)¬La φ → Ma φ (RE) From φ ↔ ψ infer La φ ↔ La ψ (B) VFrom ((φ1 , ..., Vφnm ) ↔ (ψ1 , ..., ψn )) infer (( m L φ ) ∧ ( i=1 ai i j=2 Mbj ψj ) → L(a1+...+am)−(b1+...+bn) ψ1 ) Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
Epistemic Logic Implicit probabilistic structures Explicit probabilistic structures Dutch Books for (logical) dummies
EPS, axiomatization, cont.
◮
some theorems : (RN) From φ → ψ infer La φ → La ψ (A2+) ¬La ⊥ (a > 0) (A3) La (φ ∧ ψ) ∧ Lb (φ ∧ ¬ψ) → La+b φ (a + b ≤ 1) (finite superadditivity) (A4)¬La (φ ∧ ψ) ∧ ¬Lb (φ ∧ ¬ψ) → ¬La+b φ (a + b ≤ 1) (finite subadditivity) (A7) La φ → Lb φ (b < a)
Mikaël Cozic
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the rule (B) ◮
the sum of the probabilities of the members of two partitions of an event are equal: let • E1 ∪ ... ∪ Em = A with Ei pairwise disjoint • F1 ∪ ... ∪ Fn = A with Fj pairwise disjoint Clearly, Pn Pm i=1 P(Ei ) = j=1 P(Fj ) = P(A)
Remarks: Pm Pn (i) i=1 IEi = j=1 IFj = IA ◮
(ii) in the case m = n, it is clear that if P(Ei ) ≥ P(Fi ) for 1 ≤ i < n, then the two last events have to “compensate” the disequilibrium ie P(En ) ≤ P(Fn ) Mikaël Cozic
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the rule (B), cont. ◮
one does not need to have partitions : the more general condition is that Pn Pm i=1 IEi = j=1 IFj this means that any element of S belongs to as many Ei ’s as Fj ’s
◮
one can think about a “compensation” closer to EPS: if P(Ei ) ≥ αi for i = 1, ..., n and P(Fj ) ≤ βj for j = 2, ..., m then P(F1 ) ≥ (α1 + ... + αn ) − (β2 + ... + βm ) Mikaël Cozic
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the rule (B), cont. ◮
example: E1 = A ∪ B, F1 = A and F2 = B if P(E1 = A ∪ B) ≥ 1/2 and P(F2 = B) ≤ 1/6, then P(F1 = A) ≥ 1/3
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the rule (B), cont. ◮
example: E1 = A ∪ B, F1 = A and F2 = B if P(E1 = A ∪ B) ≥ 1/2 and P(F2 = B) ≤ 1/6, then P(F1 = A) ≥ 1/3
◮
let’s come back to the inference rule: (B) VFrom ((φ1 , ..., Vφnm ) ↔ (ψ1 , ..., ψn )) infer (( m L φ ) ∧ ( ai i i=1 j=2 Mbj ψj ) → L(a1+...+am)−(b1+...+bn)ψ1 )
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the rule (B), cont. ◮
the premiss of (B) : ((φ1 , ..., φm ) ↔ (ψ1 , ..., ψn )) is a syntactical rendering of the equality of sums of characteristic functions
◮
((φ1 , ..., φm ) ↔ (ψ1 , ..., ψn )) is an abbreviation for Vmax(m,n) (k ) φ ↔ ψ (k ) k =1 where W φ(k ) = 1≤l1 ≤...≤lk m, by convention φ(k ) = ⊥) φ(k ) says that at least k of the formulas φi are true
◮
the conclusion of (B) is a direct translation of the “compensation” principle in terms of inequalities Mikaël Cozic
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the rule (B), cont.
◮
(B) is a very powerful probabilistic inference rule. It allows e.g. to derive (A3) La (φ ∧ ψ) ∧ Lb (φ ∧ ¬ψ) → La+b φ (a + b ≤ 1) (finite superadditivity)
◮
Let φ1 = φ ∧ ψ, φ2 = φ ∧ ¬ψ, ψ1 = φ Suffices to notice that (φ1 , φ2 ) ↔ ψ1 ) ; hence La φ1 ∧ Lb φ2 → La+b ψ1 , which is nothing but (A3)
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non-standard explicit probabilistic structures (NSEPS) ◮
Définition : a non-standard explicit probabilistic structure for LLa (At) is a 4-tuple M = (S, S ′ π, P) where π is standard on S and P : S ∗ → ∆(S ∗ ) assigns to every state a probability distribution on the state space.
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non-standard explicit probabilistic structures (NSEPS) ◮
Définition : a non-standard explicit probabilistic structure for LLa (At) is a 4-tuple M = (S, S ′ π, P) where π is standard on S and P : S ∗ → ∆(S ∗ ) assigns to every state a probability distribution on the state space.
◮
Remark : ⊤ and ⊥ are added with the meanings : ◮
◮
⊤ is what the agent recognizes as necessarily true, then for every s ∈ S ∗ , π(s, ⊤) = 1 ⊥ is what the agent recognizes as necessarily false, then for no s ∈ S ∗ , π(s, ⊥) = 1
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NSEPS, axiomatization ◮
What becomes the axiom system with NSS ?
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NSEPS, axiomatization ◮
What becomes the axiom system with NSS ? Minimal Probabilistic Logic
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A1) L0 φ (A2) La ⊤ (A2+) ¬La ⊥ (a > 0) (A7) La φ → Lb φ (b < a)
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NSEPS, axiomatization ◮
What becomes the axiom system with NSS ? Minimal Probabilistic Logic
(PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A1) L0 φ (A2) La ⊤ (A2+) ¬La ⊥ (a > 0) (A7) La φ → Lb φ (b < a) Completeness Theorem : |=NSEPS φ iff ⊢MPL φ Mikaël Cozic
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axiomatization, cont. ◮
Proof of completeness : ◮ ◮
◮ ◮
method of canonical models with filtration for each formula φ, one defines a subset of formulas ; in this subset of formulas, atoms are maximal coherent set of formulas the hard stuff is to define a canonical probability distribution idea : let Γ be an atom ; in the (standard state) sΓ , P(sΓ ) is an equiprobability on non-standard states s.t. for every formula χ s.t. some La χ are in Γ, a proportion b∗ of non-standard states make χ true, where b∗ = maxb Lb χ ∈ Γ.
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Dutch Book and bayesianism ◮
Why model partial beliefs through probabilities ?
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Dutch Book and bayesianism ◮
Why model partial beliefs through probabilities ?
◮
Bayesian answer: probabilistic beliefs are the necessary consequences of practical rationality
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Dutch Book and bayesianism ◮
Why model partial beliefs through probabilities ?
◮
Bayesian answer: probabilistic beliefs are the necessary consequences of practical rationality
◮
The bayesian answer is supported by the Dutch Book argument (de Finetti, Ramsey): if Bob’s partial beliefs are not probabilistic and are mirrored in betting quotients, then Alice can devise from these very betting quotients a set of bets such that whatever happens, Bob will lose money. (From Gillies, 2000.)
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Dutch Book-coherence ◮
Basic Idea: Bob chooses betting quotient qφ toward formula φ; Alice chooses stake Sφ (positive or negative). Bob pays qφ .Sφ to Alice to enter the following bet: ◮ ◮
if φ is true, then Bob wins Sφ if φ is not true, then Bob wins nothing
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Dutch Book-coherence ◮
Basic Idea: Bob chooses betting quotient qφ toward formula φ; Alice chooses stake Sφ (positive or negative). Bob pays qφ .Sφ to Alice to enter the following bet: ◮ ◮
◮
if φ is true, then Bob wins Sφ if φ is not true, then Bob wins nothing
Definition: Bob is DB-coherent toward a set of formulas Γ iff he chooses betting quotients such that Alice cannot assign stakes to formulas such that for every possible state s ∈ S, if s is realized Bob loses money - a Dutch Book against Bob
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Example
◮
Claim If Bob chooses a betting quotient qφ > 1 for any formula φ, then Alice can devise a Dutch Book against Bob
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Example
◮
Claim If Bob chooses a betting quotient qφ > 1 for any formula φ, then Alice can devise a Dutch Book against Bob How ? It suffices for Alice to assign Sφ > 0.
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Example
◮
Claim If Bob chooses a betting quotient qφ > 1 for any formula φ, then Alice can devise a Dutch Book against Bob How ? It suffices for Alice to assign Sφ > 0.
◮
By the same token, one can justify that partial beliefs should be positive, additive, etc.
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imperfect logicians lose money ◮
Claim 1 Let θ be a formula that is logically valid but that Bob doesn’t recognizes as such. Suppose that Bob chooses the betting quotient qθ < 1. Then Bob is DB-incoherent.
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imperfect logicians lose money ◮
Claim 1 Let θ be a formula that is logically valid but that Bob doesn’t recognizes as such. Suppose that Bob chooses the betting quotient qθ < 1. Then Bob is DB-incoherent.
◮
Claim 2 Let φ, ψ two formulas such that φ implies ψ. Suppose that Bob chooses betting quotients qφ > qψ - Bob violates (probabilistic) deductive monotony. Then Bob is DB-incoherent. Example: - Bob chooses qp = 3/4 and qp∨q = 3/8 - Alice chooses Sp = 1 and Sp∨q = −1 Mikaël Cozic
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logical assumptions of Dutch Books ◮
If Bob is not omniscient, Bob is DB-incoherent
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logical assumptions of Dutch Books ◮
If Bob is not omniscient, Bob is DB-incoherent
◮
A crucial requirement in the Dutch Book argument is that Alice doesn’t have more information than Bob
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logical assumptions of Dutch Books ◮
If Bob is not omniscient, Bob is DB-incoherent
◮
A crucial requirement in the Dutch Book argument is that Alice doesn’t have more information than Bob
◮
But Alice is logically omniscient is the sense that she doesn’t have to take into account what happens in non-standard states
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logical assumptions of Dutch Books ◮
If Bob is not omniscient, Bob is DB-incoherent
◮
A crucial requirement in the Dutch Book argument is that Alice doesn’t have more information than Bob
◮
But Alice is logically omniscient is the sense that she doesn’t have to take into account what happens in non-standard states
◮
What happens if one extends to logical domain the requirement that Alice doesn’t have more information than Bob?
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non-standard Dutch Books ◮
Definition: Bob is NSDB-coherent with respect to state space S ∗ including non-standard states toward a set of formulas Γ iff he assigns to every formula in Γ a betting quotient such that Alice cannot assign stakes to formulas such that for every possible state s ∈ S ∗ , if s is realized Bob loses money.
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non-standard Dutch Books ◮
Definition: Bob is NSDB-coherent with respect to state space S ∗ including non-standard states toward a set of formulas Γ iff he assigns to every formula in Γ a betting quotient such that Alice cannot assign stakes to formulas such that for every possible state s ∈ S ∗ , if s is realized Bob loses money.
◮
Fact: NSDB-coherence doesn’t justify full probabilistic beliefs but justifies minimal probabilistic beliefs: ◮
justified: for every formula φ, 0 ≤ qφ ≤ 1 ; q⊤ = 1
◮
not justified: if φ implies ψ, qφ ≤ qψ ; if φ and ψ are logically incompatible, qφ∨ψ = qφ + qψ
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conclusion
◮
The shift to non-standard probabilistic structures is only a first step from the decision-theoretic point of view: one has still to "plug" the new doxastic model on an axiological model and a criterion of choice
◮
The next step is to provide a representation theorem à la Savage
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conclusion, cont. ◮
Subjective Expected Utility: if agent’s preferences conform to a set of conditions Π, then there exists a probability distribution P on S and a utility function u s.t. preferences can be represented by expected utility defined on P and u
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conclusion, cont. ◮
Subjective Expected Utility: if agent’s preferences conform to a set of conditions Π, then there exists a probability distribution P on S and a utility function u s.t. preferences can be represented by expected utility defined on P and u
◮
Subjective Expected Utility without LO: if agent’s preferences conform to a set of conditions Π′ , then there exists non-standard states S ′ , a probability distribution P on S ∪ S ′ and a utility function u s.t. preferences can be represented by expected utility defined on P and u
◮
Question: what is Π′ ? Mikaël Cozic
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Tutorial 2 : (Un)awareness
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warm-up
full beliefs
partial beliefs
non-iterated beliefs
belief set B⊆S
implicit probabilistic structures P ∈ ∆(S)
iterated beliefs
Kripke structures B : S → ℘(S)
explicit probabilistic structures P : S → ∆(S)
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warm-up ◮
Definition : the set of formulas of an explicit probabilistic language LL(At) based on a set At of propositional variables, Form(LL(At)) is defined by : φ ::= p|¬φ|φ ∨ ψ|La φ where p ∈ At and a ∈ [0, 1] ⊆ Q. La φ : P. believes that φ at least to degree a
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warm-up ◮
Definition : the set of formulas of an explicit probabilistic language LL(At) based on a set At of propositional variables, Form(LL(At)) is defined by : φ ::= p|¬φ|φ ∨ ψ|La φ where p ∈ At and a ∈ [0, 1] ⊆ Q. La φ : P. believes that φ at least to degree a
◮
Definition : an explicit probabilistic structure for LLa (At) is a 3-tuple M = (S, π, P) where P : S → ∆(S) assigns to every state a probability distribution on the state space.
◮
Satisfaction condition for La : π ¯ (s, La φ) = 1 iff P(s)([[φ]]) ≥ a Mikaël Cozic
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warm-up System HM (Heifetz and Mongin 2001) (PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A1) L0 φ (A2) La ⊤ (A5)La φ → ¬Lb ¬φ (a + b > 1) (Def M) Ma φ ↔ L1−a ¬φ (A8)¬La φ → Ma φ (RE) From φ ↔ ψ infer La φ ↔ La ψ (B) VFrom ((φ1 , ..., Vφnm ) ↔ (ψ1 , ..., ψn )) infer (( m L φ ) ∧ ( i=1 ai i j=2 Mbj ψj ) → L(a1+...+am)−(b1+...+bn) ψ1 ) Mikaël Cozic
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warm-up
◮
two main cognitive idealizations:
(1) Logical Omniscience (2) Full Awareness ◮
one common approach: impossible states
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what is (un)awareness ? ◮
Modica & Rustichini 1999: • “ignorance about the state space” • “some of the facts that determine which state of nature occurs are not present in the subject’s mind” • “the agent does not know, does not know that she does not know, does not know that she does not know that she does not know, and so on...”
◮
Heifetz, Meier & Schipper 2007b: • “Unawareness refers to lack of conception rather than to lack of information."
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example ◮
Pierre plans to rent a house for the holiday; three main factors from the modeler point of view:
• p: the house is no more than 1 km far from the sea • q: the house is no more than 1 km far from a bar • r : the house is no more than 1 km far from an airport
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example ◮
Pierre plans to rent a house for the holiday; three main factors from the modeler point of view:
• p: the house is no more than 1 km far from the sea • q: the house is no more than 1 km far from a bar • r : the house is no more than 1 km far from an airport ◮
“simple”, factual, ignorance of r : Pierre doesn’t know whether there is an airport no more than 1 km far from the house - there are both r -states and ¬r -states which are epistemically accessible
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example ◮
Pierre plans to rent a house for the holiday; three main factors from the modeler point of view:
• p: the house is no more than 1 km far from the sea • q: the house is no more than 1 km far from a bar • r : the house is no more than 1 km far from an airport ◮
“simple”, factual, ignorance of r : Pierre doesn’t know whether there is an airport no more than 1 km far from the house - there are both r -states and ¬r -states which are epistemically accessible
◮
unawareness: Pierre doesn’t ask to himself: “is there an airport no more than 1 km far from the house?” [See states in small worlds in Savage (1954/72)] Mikaël Cozic
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example, cont. ◮
the possibility that r is not simply excluded: it is out of Pierre’s state space
◮
modeler’s point of view: pqr p¬qr ¬pqr ¬p¬qr
◮
pq¬r ¬pq¬r
p¬q¬r ¬p¬q¬r
Pierre’s point of view: pq ¬pq
p¬q ¬p¬q
[See Savage: “...a smaller world is derived from a larger by neglecting some distinctions between states”]
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properties of (un)awareness
◮
some intuitive properties of (un)awareness: Aφ ↔ A¬φ A(φ ∧ ψ) ↔ Aφ ∧ Aψ Aφ ↔ AAφ Uφ → UUφ Uφ → ¬Bφ ∧ ¬B¬Bφ Uφ → (¬B)φn ∀n ∈ N ¬BUφ
(Symmetry) (Self-Reflection) (U-introspection) (Plausibility) (Strong Plausibility) (BU-introspection)
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the modeling of (un)awareness ◮
it is impossible to devise a non-trivial (un)awareness operator that satisfies most of the intuitively appealing properties above mentioned
◮
for instance, in “Standard State-Space Models Preclude Unawareness” (1998) , Dekel, Lipman & Rustichini show that it is impossible to have (i) a non-trivial awareness operator which satisfies Plausibility, U-introspection and BU-introspection (ii) a belief operator which satisfies either Necessitation or Monotonicity
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main models of unawareness ◮
two main ways to circumvent the issue:
(i) endogenous characterization: awareness defined in terms of beliefs : Modica & Rustichini (1999), Heifetz, Meier & Schipper (2006), (2007a) M, s Aφ ⇔ M, s Bφ ∨ B¬Bφ
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main models of unawareness ◮
two main ways to circumvent the issue:
(i) endogenous characterization: awareness defined in terms of beliefs : Modica & Rustichini (1999), Heifetz, Meier & Schipper (2006), (2007a) M, s Aφ ⇔ M, s Bφ ∨ B¬Bφ (ii) exogenous characterization: Fagin & Halpern (1988), c Halpern (2001) awareness structures M, s Aφ ⇔ φ ∈ A(s) where A : S → ℘(L(At))
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GSM structures, example ◮
state space S based on At = {p, q, r } : pqr ¬pqr
◮
p¬qr ¬p¬qr
pq¬r ¬pq¬r
p¬q¬r ¬p¬q¬r
in actual state s = pqr , Pierre believes that p, does not know whether q and is unaware of r ; his non-standard state space S{p,q} and accessibility correspondance in s are pq ¬pq
Mikaël Cozic
p¬q ¬p¬q
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
GSM structures, example pqr ¬pqr
p¬qr ¬p¬qr pq ¬pq
◮
pq¬r ¬pq¬r
p¬q¬r ¬p¬q¬r
p¬q ¬p¬q
suppose that pqr in projected in pq: ρ(pqr ) = pq • if pqr and pq¬r are projected in pq, pqr and pq¬r agree on p and q • if pqr and pq¬r are projected in pq, R(pqr ) = R(pq¬r ) • R(pqr ) ⊆ S{p,q}
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
GSM structures A GSM structure is a t-uple M = (S, S ′ , π, R, ρ) (i) S is a state space S (ii) S ′ = X ⊆At SX′ (where SX′ are disjoint) is a (non-standard) state space (iii) π : At × S → {0, 1} is a valuation for S (iv) R : S → ℘(S ′ ) is an accessibility correspondence
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
GSM structures A GSM structure is a t-uple M = (S, S ′ , π, R, ρ) (i) S is a state space S (ii) S ′ = X ⊆At SX′ (where SX′ are disjoint) is a (non-standard) state space (iii) π : At × S → {0, 1} is a valuation for S (iv) R : S → ℘(S ′ ) is an accessibility correspondence (v) ρ : S → S ′ is a projection s.t. (1) if ρ(s) = ρ(t) ∈ SX′ , then (a) for each atomic formula p ∈ X , π(s, p) = π(t, p) and (b) R(s) = R(t) and (2) if ρ(s) ∈ SX′ , then R(s) ⊆ SX′
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
GSM structures A GSM structure is a t-uple M = (S, S ′ , π, R, ρ) (i) S is a state space S (ii) S ′ = X ⊆At SX′ (where SX′ are disjoint) is a (non-standard) state space (iii) π : At × S → {0, 1} is a valuation for S (iv) R : S → ℘(S ′ ) is an accessibility correspondence (v) ρ : S → S ′ is a projection s.t. (1) if ρ(s) = ρ(t) ∈ SX′ , then (a) for each atomic formula p ∈ X , π(s, p) = π(t, p) and (b) R(s) = R(t) and (2) if ρ(s) ∈ SX′ , then R(s) ⊆ SX′ ◮
one can extend R and π to the whole state space with π ∗ : if s′ ∈ SX′ , then π ∗ (s′ , p) = 1 iff (a) p ∈ X and (b) for all s ∈ ρ−1 (s′ ), π(s, p) = 1. Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
satisfaction relation ◮
one may then define as follows the satisfaction relation for each s∗ ∈ S ∗ = S ∪ S ′ (Halpern’s 2001 version):
(i) M, s∗ p iff π ∗ (s∗ , p) = 1 (ii) M, s∗ φ ∧ ψ iff M, s∗ φ and M, s∗ ψ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
satisfaction relation ◮
one may then define as follows the satisfaction relation for each s∗ ∈ S ∗ = S ∪ S ′ (Halpern’s 2001 version):
(i) M, s∗ p iff π ∗ (s∗ , p) = 1 (ii) M, s∗ φ ∧ ψ iff M, s∗ φ and M, s∗ ψ (iii) M, s∗ ¬φ iff M, s∗ 2 φ and either s∗ ∈ S, or s∗ ∈ SX′ and φ ∈ L(X )
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
satisfaction relation ◮
one may then define as follows the satisfaction relation for each s∗ ∈ S ∗ = S ∪ S ′ (Halpern’s 2001 version):
(i) M, s∗ p iff π ∗ (s∗ , p) = 1 (ii) M, s∗ φ ∧ ψ iff M, s∗ φ and M, s∗ ψ (iii) M, s∗ ¬φ iff M, s∗ 2 φ and either s∗ ∈ S, or s∗ ∈ SX′ and φ ∈ L(X ) (iv) M, s∗ Bφ iff for each t ∗ ∈ R ∗ (s∗ ), M, t ∗ φ ◮
crucial point: (iii) introduces partiality: if p ∈ / X and s∗ ∈ SX′ then neither M, s∗ p nor M, s∗ ¬p (for short, M, s∗ ⇑ p). More generally, M, s∗ ⇓ φ for s∗ ∈ SX′ iff φ ∈ L(X ) Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
what (un)awareness is not ◮
MR 1999 and HMS 2006 define unawareness in terms of beliefs; this is OK given their assumption that accessibility correspondences are partitional
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
what (un)awareness is not ◮
MR 1999 and HMS 2006 define unawareness in terms of beliefs; this is OK given their assumption that accessibility correspondences are partitional
◮
but this endogenous characterization is not robust under change of the accessibility relation’s properties. In the general case, - it is plausible that if Pierre is unaware of φ, he doesn’t believe that φ nor believe that he doesn’t believe that φ (Uφ → (¬Bφ ∧ ¬B¬Bφ)) - it is not plausible that if Pierre doesn’t believe that φ nor believe that he doesn’t believe that φ, he is necessarily unaware of φ ((¬Bφ ∧ ¬B¬Bφ) → Uφ) Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
example
◮ ◮
′ the actual state s is projected in s1 ∈ S{p,q} ′ R(s1 ) = {s2 , s3 } (hence s2 , s3 ∈ S{p,q} as well) ; R(s2 ) = {s2 } ; R(s3 ) = {s3 }
◮
M, s2 ¬p, hence M, s2 ¬Bp ∧ B¬p
◮
M, s3 p, hence M, s3 Bp
◮
M, s ¬Bp ∧ ¬B¬Bp hence M, s U MR p But, for instance, M, s B(B¬p ∨ Bp)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
unawareness as partiality ◮
hence: keep the underlying GSM structure but change the definition of (un)awareness
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
unawareness as partiality ◮
hence: keep the underlying GSM structure but change the definition of (un)awareness
◮
the possible states that Pierre conceives do not “answer” to ?p, ?q and ?r : they answer only to ?p and ?q
◮
unawareness may be seen as partiality: the possible states that Pierre conceives make true neither r nor ¬r
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
unawareness as partiality ◮
hence: keep the underlying GSM structure but change the definition of (un)awareness
◮
the possible states that Pierre conceives do not “answer” to ?p, ?q and ?r : they answer only to ?p and ?q
◮
unawareness may be seen as partiality: the possible states that Pierre conceives make true neither r nor ¬r
◮
semantic characterization of unawareness in terms of partiality: M, s Aφ iff M, ρ(s) ⇓ φ
◮
Let’s call a P-GSM structure a GSM structure where the truth conditions of the unawareness operator are given in terms of partiality Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
c partiality and awareness ◮
◮
c Halpern 2001 relates GSM structures and awareness structures; one obtains a still closer connection with P-GSM c an awareness structure M = (S, R, A, π) is propositionally determined (pd) if (1) for each state s, A(s) is generated by some atomic formulas X ⊆ At i.e. A(s) = L(X ) and (2) if t ∈ R(s), then A(s) = A(t)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
c partiality and awareness ◮
◮
◮
c Halpern 2001 relates GSM structures and awareness structures; one obtains a still closer connection with P-GSM c an awareness structure M = (S, R, A, π) is propositionally determined (pd) if (1) for each state s, A(s) is generated by some atomic formulas X ⊆ At i.e. A(s) = L(X ) and (2) if t ∈ R(s), then A(s) = A(t)
Proposition (see Halpern 2001 Thm 4.1)
c 1. For every pd awareness structure M there exists a ′ P-GSM structure M based on the same state space S and the same valuation π s.t. for all formulas φ ∈ LBA (At) and each possible state s M, s a c φ iff M′ , s P−GSM φ Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
c partiality and awareness , cont.
c 2. For every P-GSM structure M there exists a awareness ′ structure M based on the same state space S and the same valuation π s.t. for all formulas φ ∈ LBA (At) and each possible state s M, s P−GSM φ iff M′ , s a c φ ◮
Corollary: the axiom system KX in Halpern 2001 is sound and complete for P-GSM structures.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
axiom system KX (Halpern 2001) (PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A0) Bφ ⇒ Aφ (K) Bφ ∧ B(φ → ψ) → Bψ (Gen) From φ infer Aφ → Bφ (A1) Aφ ↔ A¬φ (A2) A(φ ∧ ψ) ↔ (Aφ ∧ Aψ) (A3) Aφ ↔ AAφ (A4) ABφ ↔ Aφ (A5) Aφ → BAφ (Irr) If no atomic formulas in φ appear in ψ, from Uφ → ψ infer ψ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness
◮
as stressed by Cozic (2007) and Pucella & Halpern (2007), one of the main advantages of the impossible states framework is that it can be straightforwardly extended to the probabilistic case
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness
◮
as stressed by Cozic (2007) and Pucella & Halpern (2007), one of the main advantages of the impossible states framework is that it can be straightforwardly extended to the probabilistic case
◮
actually, P-GSM structures can be given probabilistic analogues for explicit probabilistic structures
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
explicit probabilistic structures (EPS), reminder ◮
Definition : an explicit probabilistic structure for LL (At) is a 3-tuple M = (S, π, P) where P : S → ∆(S) assigns to every state a probability distribution on the state space.
◮
Satisfaction condition for La : M, s La φ ⇔ P(s)([[φ]]) ≥ a
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
explicit probabilistic structures (EPS), reminder ◮
Definition : an explicit probabilistic structure for LL (At) is a 3-tuple M = (S, π, P) where P : S → ∆(S) assigns to every state a probability distribution on the state space.
◮
Satisfaction condition for La : M, s La φ ⇔ P(s)([[φ]]) ≥ a
◮
higher-order (partial) beliefs are induced in the same way that higher-order (full) beliefs are induced by the accessibility relation
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness, first attempt ◮
Definition : an P-GSM explicit probabilistic structure for LLA (At) is a t-tuple M = (S, S ′ , π, P, ρ) where
(i) S is a state space S ′ (where S ′ are disjoint) is a state space (ii) S ′ = Φ⊆At SΦ Φ (iii) π : At × S → {0, 1} is a valuation for S (iv) P : S → ∆(S ′ ) ′ , then (v) ρ : S → S ′ is a projection s.t. (1) if ρ(s) = ρ(t) ∈ SΦ (a) for each atomic formula p ∈ Φ, π(s, p) = π(t, p) and (b) ′ , then Supp(P(s)) ⊆ S ′ P(s) = P(t) and (2) if ρ(s) ∈ SΦ Φ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness, first attempt ◮
some good news: for all GSM-EPS M and all standard state s, unawareness precludes positive probability: M, s Uφ → ¬La φ for a > 0 M, s Uφ → ¬La ¬φ for a > 0 M, s ¬La Uφ for a > 0
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness, first attempt ◮
some good news: for all GSM-EPS M and all standard state s, unawareness precludes positive probability: M, s Uφ → ¬La φ for a > 0 M, s Uφ → ¬La ¬φ for a > 0 M, s ¬La Uφ for a > 0
◮
but some (very) bad news: for all GSM-EPS M and all standard state s, M, s Uφ → L0 φ M, s Uφ → L0 ¬φ M, s Uφ → L1 L0 φ (!!)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness, second attempt ◮
satisfaction condition for La φ : M, s La φ ⇔ P(s)([[φ]]) ≥ a and M, ρ(s) ⇓ φ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
probabilistic unawareness, second attempt ◮
satisfaction condition for La φ : M, s La φ ⇔ P(s)([[φ]]) ≥ a and M, ρ(s) ⇓ φ
◮
in this case, the following holds: Aφ ↔ A¬φ (Symmetry) Aφ ↔ AAφ (Self-Reflection) Uφ → UUφ (U-introspection) Uφ → ¬La φ ∧ ¬La ¬La φ (Plausibility) Uφ → (¬La )n φ ∀n ∈ N (Strong Plausibility) ¬La Uφ (a >0) (La U-introspection) L0 φ ↔ Aφ
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Models of Beliefs for Boundedly Rational Agents
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HMU axiom system ◮
an axiom system for explicit probabilistic structures with unawareness: System HMU , Part I (PROP) Instances of propositional tautologies (MP) From φ and φ → ψ infer ψ (A0) Aφ ↔ L0 φ (A1) Aφ ↔ A¬φ (A2) A(φ ∧ ψ) ↔ Aφ ∧ Aψ (A3) Aφ ↔ AAφ (A4) Aφ ↔ ALa φ (A5L ) Aφ → L1 Aφ Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
HMU axiom system System HMU , Part II (A6) La ⊤ (A7) La φ → ¬Lb ¬φ a + b > 1 (A8U ) (¬La φ ∧ Aφ) → Ma φ (REU ) From φ ↔ ψ infer A(φ ∧ ψ) → (La φ ↔ La ψ) (BV m ) ↔ (ψ1 , ..., ψn )) infer U ) From ((φ1 , ..., φV n (( m L φ ) ∧ ( → i=1 ai i j=2 Mbj ψj ) L(a1+...+am)−(b1+...+bn)ψ1 ))
Mikaël Cozic
(Aψ1
→
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
La U introspection ◮
the axiom of La U introspection ¬La Uφ (a > 0) may be derived from HMU
◮
intuitively, (1) La ¬Aφ → L0 ¬Aφ (2) La ¬Aφ → A¬Aφ (from (1) and (A0)) (3) La ¬Aφ → AAφ (from (2) and (A1)) (4) La ¬Aφ → Aφ (from (3) and (A3)) (5) La ¬Aφ → L1 Aφ (from (4) and (A5L ) (6) La ¬Aφ → ¬La ¬Aφ (from (5) and (A7)) (7) ⊥ (from (6) by propositional reasoning)
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
counter-example to (B) ◮
the famous inference rule (B) needs to be restricted with the help of the awareness operator.
◮
Let φ1 = (p ∨ ¬p) ψ1 = (q ∨ ¬q) ψ2 = (p ∨ ¬p) So, clearly the premiss of (B) - ((φ1 ) ↔ (ψ1 , ψ2 )) is satisfied
◮
Suppose that Pierre is aware of p but not of q ; in this case, it will be true that L1 φ1 and M1 ψ2 . Hence by the compensation principle, L0 ψ1 . But this will not be the case if Pierre is unaware of q.
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
completeness
◮
some steps remain to be checked in the proof (!), but one may confidently claim the following Completeness Theorem : |=GSM−EPS φ iff ⊢HMU φ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
further issues
1 becoming aware 2 multi-agent unawareness 3 applications to decision theory and game theory
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
becoming aware ◮
Pierre may be initially unaware of φ and become aware of φ:
(1) when someone gives Pierre an information that involves φ (2) when someone asks Pierre what he thinks about φ
Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
becoming aware ◮
Pierre may be initially unaware of φ and become aware of φ:
(1) when someone gives Pierre an information that involves φ (2) when someone asks Pierre what he thinks about φ ◮
when one thinks about (1) for full beliefs, things may look simple:
• initially, Pierre is only aware of p, neither q nor r : ρ(s) ∈ S{p} and R(s) ⊆ S{p} . He believes that p p ¬p
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Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
• Pierre is informed that q (i) first, the structure is modified such that ρ′ (s) ∈ S{p,q} R ′ (s) ⊆ S{p,q} pq ¬pq
p¬q ¬p¬q
(ii) then, the ¬q-states are eliminated pq ¬pq
Mikaël Cozic
p¬q ¬p¬q
Models of Beliefs for Boundedly Rational Agents
General Introduction Tutorial 1 : Logical Omniscience Tutorial 2: (Un)awareness
unawareness models of unawareness unawareness and partiality probabilistic extension
becoming aware, cont.
◮
but even for scenario of type (1), the probabilistic case is much more tricky
◮
one could reason like this:
• initially, Pierre is only aware of p, neither q nor r : ρ(s) ∈ S{p} and Supp(P(s)) ⊆ S{p} p ¬p
Mikaël Cozic
(α) (1 − α)
Models of Beliefs for Boundedly Rational Agents
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unawareness models of unawareness unawareness and partiality probabilistic extension
• Pierre is informed that q (i) first, the structure is modified: Supp(P ′(s)) ⊆ S{p,q} and for each φ of L({p}), P(s)([[φ]] = P ′ (s)([[φ]]) • state p has initially weight α • state p is splitted in pq and p¬q, each with weight α/2 pq ¬pq
(α/2) ((1 − α)/2)
p¬q ¬p¬q
(α/2) ((1 − α)/2)
(ii) then, Pierre conditionalizes on the information that q pq ¬pq ◮
(α) (1 − α)
p¬q ¬p¬q
but the new probability of p could be affected by the fact that the agent learns that q (intuitively, if p and q are not independent - think about p = “the house is quiet” and q = “the house is no more than 1 km far from an airport” Mikaël Cozic
Models of Beliefs for Boundedly Rational Agents
