Intuitive Beliefs Jawwad Noor 1 1 Department of Economics Boston - PowerPoint PPT Presentation

Intuitive Beliefs Belief Formation Conclusion Intuitive Beliefs Jawwad Noor 1 1 Department of Economics Boston University December 24, 2019 Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Introduction How do limited agents make decisions in a complex world? In the field, besides using tools for analysis, investors also talk about having an “instinct” or a “feel for the market” (Salas et al 2010, Hensman and Sadler-Smith 2011, Huang and Pearce 2015, Huang 2018). Keynes’ animal spirits is a “spontaneous urge to action” (Keynes, 1936) This paper: formal theory of intuition Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Introduction Identify intuition with reliance on associative memory The activation of a thought or mental image activates/inhibits another Word associations, triggered memory, image of stereotype Associative/Intuitive/System 1 process: Involuntary and costless mental processing Associations learnable, strength determined by frequency, salience, similarities, etc Can be strengthened (reinforcement) Can be weakened (counter-conditioning or decay) Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Introduction Identify intuition with reliance on associative memory Model associative memory as an associative network Inspiration from energy-based neural networks Hopfield net, Boltzmann machine Used for classification, object/speech recognition, etc. Key questions studied in the paper: Testable implications for likelihood judgements Formation of networks on the basis of data Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Outline Intuitive Beliefs 1 Model Characterization Results Bayesian Intuitive Beliefs Belief Formation 2 Model Illustration Conclusion 3 Related Literature Concluding Comments Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Outline Intuitive Beliefs 1 Model Characterization Results Bayesian Intuitive Beliefs Belief Formation 2 Model Illustration Conclusion 3 Related Literature Concluding Comments Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives Elements of uncertainty Γ = { 1 , .. i , j , ..., N } Index set, cardinality N < ∞ E.g. assets with uncertain return: Γ = { 1 , 2 } Elementary state space Ω i = { x i , y i , z i ... } Abstract set for each i ∈ Γ E.g. Asset i returns are high, medium or low: Ω i = { h i , m i , l i } Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives (Full) state space � Ω = Ω i i ∈ Γ Generic element x = ( x 1 , .., x N ) E.g. vectors of asset returns ( h 1 , l 2 ) ∈ Ω = { h 1 , m 1 , l 1 } × { h 2 , m 2 , l 2 } Complexity of the state space Tension in rational decision models Intuition is a non-conscious, cognitively costless process Associations form on arbitrarily complicated state space Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives Elementary event space : algebra of events Σ i = { x i , y i , .. } ⊂ 2 Ω i E.g. asset i gives a good return: x i lies in x i = { h i , m i } (Full) event space: � Σ = Σ i i ∈ Γ generic element x = ( x 1 , .., x N ) Good (resp. bad) return for asset 1 (resp. 2), x = ( { h 1 , m 1 } , { m 2 , l 2 } ) Vectors of elementary events vs subsets of the state space Each node of associative network is an elementary state Can be extended Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives Information: true state lies in z = ( z 1 , ..., z N ) ∈ Σ A belief is a normalized set-function p ( · ) over (Ω , Σ) conditional on z ∈ Σ: Assigns p ( x | z ) ∈ [0 , 1] to each event x ∈ Σ and satisfies: (i) p (Ω | z ) = 1 (ii) p ( x | z ) = 0 if x i = φ for some i (iii) p ( x ∩ z | z ) = p ( x | z ) Non-additive probability/capacity if satisfies monotonicity : x ⊂ y = ⇒ p ( x | z ) ≤ p ( y | z ) , Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives Primitive: a family p of beliefs p ( � | z ) over (Ω , Σ) for each z ∈ Σ + where Σ + = { z ∈ Σ : p ( z | Ω) > 0 } . Avoid conditioning on non-credible information Behavioral meaning of p Scoring rules in experiments Representation of bets on x given z Connect to choice through signed Choquet integration (Waegenaere and Wakker, 2001) Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Primitives Notation For x = ( x 1 , .., x N ) ∈ Σ and I ⊂ Γ, define the projection x I : = x I Ω − I . In particular x i = (Ω j , .., x i , .., Ω j ) x i x k = (Ω j , .., x i , Ω i +1 ..., Ω k − 1 , x k ., Ω j ) x ⊂ z is notation for x i ⊂ z i for all i x ∩ z is notation for x i ∩ z i for all i Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Network Inspired by energy-based neural networks (reviewed later) Nodes: elementary events z i ∈ Σ for each i ∈ Γ Nodes contribute to “associative energy” in the network � � Λ( x | z ) := − [ a ( x i x j ) + b ( x i | z )] . i < j i ∈ Γ Associative bias b ( x i | z ) is node x i association with information Symmetric associative weight a ( x i x j ) between pairs of nodes Energy: (negative of) sum of associative weights and biases Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Network Definition An associative network is a tuple ( a , b ) that consists of (i) an association function a that maps each x i , x j ∈ ∪ k ∈ Γ Σ k to a symmetric associative weight a ( x i x j ) ∈ R ∪ {−∞} , (ii) a bias function b that maps each x i ∈ ∪ k ∈ Γ Σ k and z ∈ Σ + to some bias b ( x i | z ) ∈ R ∪ {−∞} . Symmetry: a ( x i x j ) = a ( x j x i ) a ( x i x j ) = −∞ : occurrence of one associated with non-occurrence of other Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Network Definition An associative network ( a , b ) is regular if for all x i ∈ ∪ k ∈ Γ Σ k and z ∈ Σ + , the it satisfies b ( x i | z i ) = a ( x i z i ) and b ( x i | z ) > −∞ = ⇒ a ( x i z j ) > −∞ for all j ∈ Γ and b ( x i | Ω) > −∞ . Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: General Updating Definition Beliefs p are Intuitive Beliefs with General Updating (IBGU) if there exists a regular associative network ( a , b ) and a function Z : Σ + → R ++ such that for any ( x , z ) ∈ Σ × Σ + s.t. x ⊂ z ,   1 � �  . p ( x | z ) = Z ( z ) × exp a ( x i x j ) + b ( x i | z ) i < j i ∈ Γ The associative network ( a , b ) is said to represent p . 1 That is, p ( x | z ) = Z ( z ) × exp [ − Λ( x | z )] Note Z ( z ) = exp [ − Λ( z | z )] is determined by ( a , b ) Apply p ( x ∩ z | z ) = p ( x | z ) to non-nested x , z Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Intuitive Updating Definition Beliefs p are Intuitive Beliefs with Intuitive Updating (IBIU) if there exists a regular associative network ( a , b ) satisfying � b ( x i | z ) = a ( x i z j ) j ∈ Γ for all x i ∈ ∪ k ∈ Γ Σ k and z ∈ Σ + , and moreover, for any ( x , z ) ∈ Σ × Σ + s.t. x ⊂ z ,   1 � � �  . p ( x | z ) = Z ( z ) × exp a ( x i x j ) + a ( x i z j ) i < j i ∈ Γ j ∈ Γ The association function a is said to represent p . Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Intuitive Updating IBGU silent about how information z changes energy IBIU produces all beliefs from a Regularity of ( a , b ) becomes: for all x i ∈ Σ and z ∈ Σ + , � � a ( x i z j ) > −∞ = ⇒ a ( x i Ω j ) > −∞ , j ∈ Γ j ∈ Γ IBGU model: only matters whether a ( x i z i ) > −∞ IBIU model: exact value of a ( x i z i ) matters Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Model Belief Formation Characterization Results Conclusion Bayesian Intuitive Beliefs Model: Boltzmann Machine Each pixel i ∈ Γ on a screen is a node A node i can be off/on, value x i ∈ Ω i = { 0 , 1 } A state x = ( x 0 , ..., x N ) is a configuration of activations Define energy at state x by � � Λ( x ) = − [ a ( x i x j ) + b ( x i )] i < j i ∈ Γ where x i = 0 = ⇒ a ( x i x j ) = 0 and b ( x i ) = 0 Update rule for activation: prob x i = 1 is a logistic function of energy ψ ( x i = 1 | x ) = f (Λ( x )) Update nodes in random order − → dynamic evolution of state Jawwad Noor Intuitive Beliefs