Unrestricted Information Acquisition Tommaso Denti MIT June 7, 2016
Introduction ◮ A theory of information acquisition in games ◮ Endogenize assumptions on players’ information ◮ Common extra layer of strategic interaction ◮ Flexible learning about state and what others know ◮ � = Bergemann & Valimaki 2002, Hellwig & Veldkamp 2009, Myatt & Wallace 2012, Yang 2015,... ◮ Expose primitive incentives to acquire information ◮ Broad assumptions on cost of information ◮ Costly to learn state and what others know ◮ Example: Shannon mutual information ◮ Applications ◮ Investment games: risk-dominance selection ◮ Games on networks: Bonacich centrality ◮ Large games: endogenous informational smallness 1 / 19
Introduction ◮ A theory of information acquisition in games ◮ Endogenize assumptions on players’ information ◮ Common extra layer of strategic interaction ◮ Flexible learning about state and what others know ◮ � = Bergemann & Valimaki 2002, Hellwig & Veldkamp 2009, Myatt & Wallace 2012, Yang 2015,... ◮ Expose primitive incentives to acquire information ◮ Broad assumptions on cost of information ◮ Costly to learn state and what others know ◮ Example: Shannon mutual information ◮ Applications ◮ Investment games: risk-dominance selection ◮ Games on networks: Bonacich centrality ◮ Large games: endogenous informational smallness 1 / 19
Today Investment games: bunk runs, currency crises,... Exogenous information ◮ Common knowledge: multiplicity ◮ Diamond & Dybvig 1983, Obstfeld 1996,... ◮ Global games: risk-dominance selection ◮ Carlsson & van Damme 1993, Morris & Shin 1998,... ◮ Any perturbation: any selection ◮ Weinstein & Yildiz 2007 Endogenous information w/ mutual information ◮ Flexible info acquisition about state: multiplicity ◮ Yang 2015 ◮ Unrestricted info acquisition: risk-dominance selection ◮ Extend to potential games Endogenous information w/out mutual information: next talk 2 / 19
Outline ◮ Investment game with incomplete information ◮ Basic game: actions, states, utilities ◮ Exogenous Information structure ◮ Recap: common knowledge, global games,... ◮ Investment game with information acquisition ◮ Basic game: actions, states, utilities ◮ Information acquisition technology ◮ Flexible info acquisition about state: multiplicity ◮ Unrestricted info acquisition: risk-dominance selection 3 / 19
Basic Game ◮ N = { 1 , . . . , n } : finite set of players ◮ A i = { invest , not invest } : set of i ’s actions ◮ Θ ⊆ R : closed set of states ◮ P Θ ∈ ∆(Θ) : state distribution ◮ ρ (¯ a − i , θ ) ∈ R : i ’s non-decreasing return ◮ ¯ a − i : share of opponents who invest ◮ ρ integrable in θ w.r.t. P Θ ◮ P Θ ( { θ : ρ ( 1 , θ ) < 0 } ) > 0: dominance region ◮ P Θ ( { θ : ρ ( 0 , θ ) > 0 } ) > 0: dominance region ◮ u i ( a , θ ) = 1 { invest } ( a i ) ρ (¯ a − i , θ ) : i ’s utility invest not invest Today: θ, θ θ − 1 , 0 invest 0 , θ − 1 0 , 0 not invest 4 / 19
Exogenous Information Structure ◮ (Ω , F , P ) : underlying probability space ◮ θ : Ω → Θ : random variable with θ ∼ P Θ ◮ X i : Polish space of i ’s messages ◮ x i : Ω → X i : i ’s signal , random variable 5 / 19
Recap Common knowledge: multiplicity ◮ x i = θ for all i ◮ θ ∈ [ 0 , 1 ] : equilibrium indeterminacy Global games: risk-dominance selection ◮ x i = θ + λ ǫ i for all i ◮ λ > 0: scale factor ◮ ǫ i : idiosyncratic noise ◮ λ → 0: perturbation of complete information ◮ 1 / 2: risk-dominance threshold Any perturbation: any selection ◮ Weinstein & Yildiz 2007 6 / 19
Information Acquisition Technology ◮ (Ω , F , P ) : underlying probability space ◮ θ : Ω → Θ : random variable with θ ∼ P Θ ◮ X i : Polish space of i ’s messages ◮ X i : nonempty set of i ’s signals x i : Ω → X i ◮ C i : ∆( X × Θ) → [ 0 , ∞ ] : i ’s cost of information 7 / 19
Game with Information Acquisition Basic game + info technology = strategic form game: ◮ Set of players: N ◮ i ’s strategy: signal x i ∈ X i , contingency plan s i ∈ S i ◮ S i : set of all measurable s i : X i → A i ◮ i ’s payoff: E [ u i ( s ( x ) , θ )] − λ C i ( P ( x , θ ) ) ◮ λ > 0: scale factor ◮ P ( x , θ ) ∈ ∆( X × Θ) : joint distribution of x and θ Solution concept: pure-strategy Nash equilibrium To ease notation: C i ( x , θ ) = C i ( P ( x , θ ) ) λ → 0: multiplicity/selection of equilibria? 8 / 19
Flexible Info Acquisition about State Yang 2015: for all players i Assumption 0. | X i | ≥ | A i | . Moreover, if random variable x ′ i : Ω → X i is measurable w.r.t. some x i ∈ X i , then x ′ i ∈ X i . Assumption 1. Take any x ∈ X . Then ( x i ⊥ x − i ) | θ . Assumption 2. Take any P X i × Θ ∈ ∆( X i × Θ) . If θ ∼ marg Θ ( P X i × Θ ) , then ( x i , θ ) ∼ P X i × Θ for some x i ∈ X i . Assumption 3. For all x ∈ X , C i ( x , θ ) = I ( x i ; x − i , θ ) . Mutual information: for X i finite, p p.m.f. of x i , � log p ( x i | x − i , θ ) � I ( x i ; x − i , θ ) = E . p ( x i ) 9 / 19
Flexible Info Acquisition about State Yang 2015: for all players i Assumption 0. | X i | ≥ | A i | . Moreover, if random variable x ′ i : Ω → X i is measurable w.r.t. some x i ∈ X i , then x ′ i ∈ X i . Assumption 1. Take any x ∈ X . Then ( x i ⊥ x − i ) | θ . Assumption 2. Take any P X i × Θ ∈ ∆( X i × Θ) . If θ ∼ marg Θ ( P X i × Θ ) , then ( x i , θ ) ∼ P X i × Θ for some x i ∈ X i . Assumption 3. For all x ∈ X , C i ( x , θ ) = I ( x i ; x − i , θ ) . Mutual information: for X i finite, p p.m.f. of x i , � log p ( x i | ✟✟ x − i , θ ) � I ( x i ; ✟✟ x − i , θ ) = E . p ( x i ) 9 / 19
Revelation Principle Direct technology: X i = A i for all i Basic game + direct technology = strategic form game: ◮ Set of players: N ◮ i ’s strategy: direct signal x i ∈ X i ◮ i ’s payoff: E [ u i ( x , θ )] − λ C i ( x , θ ) Solution concept: pure-strategy Nash equilibrium Revelation principle (Yang 2015): “A0-A3 ⇒ w.l.o.g. technology and signals are direct.” 10 / 19
Multiplicity Theorem (Yang 2015) Assume A0-A3. Let P Θ be abs. continuous w.r.t. Lebesgue measure. Then ∀ ˆ θ ∈ [ 0 , 1 ] , ∃ equilibria ( x λ : λ > 0 ) : ∀ i ∈ N � if θ ≥ ˆ 1 θ, a . s . P ( x i ,λ = invest | θ ) − → if θ < ˆ 0 θ. Remark. Also non-monotone equilibria 11 / 19
Monotone Equilibria λ > 0: 12 / 19
Monotone Equilibria λ > 0: 12 / 19
Monotone Equilibria λ > 0: 12 / 19
Monotone Equilibria λ > 0: 12 / 19
Monotone Equilibria λ → 0: 12 / 19
Monotone Equilibria λ > 0: 12 / 19
Monotone Equilibria λ > 0: ◮ i ’s primitive incentive: learn { ρ (¯ x − i , θ ) ≥ 0 } x − i , θ ) ≥ 0 } � = { θ ≥ ˆ ◮ { ρ (¯ θ } since Var (¯ x − i | θ ) � = 0 12 / 19
Unrestricted Info Acquisition For all players i : Assumption 0. | X i | ≥ | A i | . Moreover, if random variable x ′ i : Ω → X i is measurable w.r.t. some x i ∈ X i , then x ′ i ∈ X i . Assumption 3. For all x ∈ X , C i ( x , θ ) = I ( x i ; x − i , θ ) . Assumption 4. Take any x − i ∈ X − i and P X × Θ ∈ ∆( X × Θ) . If ( x − i , θ ) ∼ marg X − i × Θ ( P X × Θ ) , then ( x , θ ) ∼ P X × Θ for some x i ∈ X i . Finite-game Construction General Construction 13 / 19
Unrestricted Info Acquisition For all players i : Assumption 0. | X i | ≥ | A i | . Moreover, if random variable x ′ i : Ω → X i is measurable w.r.t. some x i ∈ X i , then x ′ i ∈ X i . Assumption 3. For all x ∈ X , C i ( x , θ ) = I ( x i ; x − i , θ ) . Assumption 4. Take any x − i ∈ X − i and P X × Θ ∈ ∆( X × Θ) . If ( x − i , θ ) ∼ marg X − i × Θ ( P X × Θ ) , then ( x , θ ) ∼ P X × Θ for some x i ∈ X i . Finite-game Construction General Construction 13 / 19
Unrestricted Info Acquisition For all players i : Assumption 0. | X i | ≥ | A i | . Moreover, if random variable x ′ i : Ω → X i is measurable w.r.t. some x i ∈ X i , then x ′ i ∈ X i . Assumption 3. For all x ∈ X , C i ( x , θ ) = I ( x i ; x − i , θ ) . Assumption 4. Take any x − i ∈ X − i and P X × Θ ∈ ∆( X × Θ) . If ( x − i , θ ) ∼ marg X − i × Θ ( P X × Θ ) , then ( x , θ ) ∼ P X × Θ for some x i ∈ X i . Finite-game Construction General Construction Revelation principle: “A0, A3, A4 ⇒ w.l.o.g. technology and signals are direct.” 13 / 19
All Equilibria λ > 0: 14 / 19
All Equilibria λ > 0: θ = 1 P ( x i = invest ) ˆ 2 − λ log P ( x i = not invest ) 14 / 19
All Equilibria λ > 0: θ = 1 P ( x i = invest ) ˆ 2 − λ log P ( x i = not invest ) 14 / 19
All Equilibria λ > 0: θ = 1 P ( x i = invest ) ˆ 2 − λ log P ( x i = not invest ) 14 / 19
All Equilibria λ > 0: θ = 1 P ( x i = invest ) ˆ 2 − λ log P ( x i = not invest ) 14 / 19
All Equilibria λ → 0: 14 / 19
Risk-dominance Selection Theorem Assume A0, A3, and A4. Then ∀ equilibria ( x λ : λ > 0 ) : ∀ i ∈ N � if θ > 1 1 2 , a . s . P ( x i ,λ = invest | θ ) − → if θ < 1 0 2 . Theorem extends to potential games. 15 / 19
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