Unrestricted Information Acquisition

Transcription

1 Unrestricted Information Acquisition Tommaso Denti MIT June 7, 2016

2 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

3 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

4 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

5 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

6 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(ā i, θ) R: i s non-decreasing return ā i : share of opponents who invest ρ integrable in θ w.r.t. PΘ PΘ ({θ : ρ(1, θ) < 0}) > 0: dominance region PΘ ({θ : ρ(0, θ) > 0}) > 0: dominance region u i (a, θ) = 1 {invest} (a i )ρ(ā i, θ): i s utility Today: invest not invest invest θ, θ θ 1, 0 not invest 0, θ 1 0, 0 4 / 19

7 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

8 Recap Common knowledge: multiplicity x i = θ for all i θ [0, 1]: equilibrium indeterminacy Global games: risk-dominance selection x i = θ + λɛ i for all i λ > 0: scale factor ɛi : idiosyncratic noise λ 0: perturbation of complete information 1/2: risk-dominance threshold Any perturbation: any selection Weinstein & Yildiz / 19

9 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

10 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 Si : set of all measurable s i : X i A i i s payoff: E[u i (s(x), θ)] λc 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

11 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 Xi Θ (X i Θ). If θ marg Θ (P Xi Θ), then (x i, θ) P Xi 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, [ I (x i ; x i, θ) = E log p(x ] i x i, θ). p(x i ) 9 / 19

12 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 Xi Θ (X i Θ). If θ marg Θ (P Xi Θ), then (x i, θ) P Xi 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, [ I (x i ; x i, θ) = E log p(x ] i x i, θ). p(x i ) 9 / 19

13 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)] λc 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

14 Multiplicity Theorem (Yang 2015) Assume A0-A3. Let P Θ be abs. continuous w.r.t. Lebesgue measure. Then ˆθ [0, 1], equilibria (x λ : λ > 0): i N { P(x i,λ = invest θ) a.s. 1 if θ ˆθ, 0 if θ < ˆθ. Remark. Also non-monotone equilibria 11 / 19

15 Monotone Equilibria λ > 0: 12 / 19

16 Monotone Equilibria λ > 0: 12 / 19

17 Monotone Equilibria λ > 0: 12 / 19

18 Monotone Equilibria λ > 0: 12 / 19

19 Monotone Equilibria λ 0: 12 / 19

20 Monotone Equilibria λ > 0: 12 / 19

21 Monotone Equilibria λ > 0: i s primitive incentive: learn {ρ( x i, θ) 0} {ρ( x i, θ) 0} {θ ˆθ} since Var( x i θ) 0 12 / 19

22 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

23 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

24 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

25 All Equilibria λ > 0: 14 / 19

26 All Equilibria λ > 0: ˆθ = 1 2 λ log P(x i = invest) P(x i = not invest) 14 / 19

27 All Equilibria λ > 0: ˆθ = 1 2 λ log P(x i = invest) P(x i = not invest) 14 / 19

28 All Equilibria λ > 0: ˆθ = 1 2 λ log P(x i = invest) P(x i = not invest) 14 / 19

29 All Equilibria λ > 0: ˆθ = 1 2 λ log P(x i = invest) P(x i = not invest) 14 / 19

30 All Equilibria λ 0: 14 / 19

31 Risk-dominance Selection Theorem Assume A0, A3, and A4. Then equilibria (x λ : λ > 0): i N P(x i,λ = invest θ) a.s. Theorem extends to potential games. { 1 if θ > 1 2, 0 if θ < / 19

32 Proof (x λ ): family of equilibria v : A Θ R: potential s.t. for all i, a i, and a i u i (a i, ) u i (a i, ) = v(a i, ) v(a i, ) Investment games: for all a and θ v(a, θ) = a 1 m=0 ( ) m ρ n 1, θ with a number of players who invest a risk-dominant at θ: v(a, θ) > v(a, θ) for all a a Info acquisition w/ mutual information, λ > 0: Static 1-player: Csiszar 1974, Matejka & McKay 2016 Dynamic 1-player: previous talk Static n-player: next slide 16 / 19

33 Key Lemma P (x λ,θ): joint distribution of x λ and θ 1. Quality. Almost surely, dp (x λ,θ) e (a, θ) = d(p θ i N P x i,λ ) v(a,θ) λ A e v(a,θ) λ ( i N P x i,λ )(da ) 2. Quantity. P x 1,λ,..., P x n,λ eq. of potential game V : ˆ V (P A1,..., P An ) = Θ (ˆ log A ) e v(a,θ) λ ( i N P Ai )(da) P Θ (dθ).. P A1,..., P An : equilibrium of V Existence There is equilibrium x λ s.t. x i,λ P Ai for all i N. 17 / 19

34 Key lemma: for all θ and a a dp (x λ,θ) e (a, θ) = d(p θ i N P x i,λ ) v(a,θ) λ A e v(a,θ) λ ( i N P x i,λ )(da ) = e v(a,θ) λ e v(a,θ) λ ( i N P x i,λ )({a }) 1 e v(a,θ) v(a,θ) λ ( i N P x i,λ )({a }) 0. Recall: v(a, θ) > v(a, θ) for all a a Dominance regions: lim inf λ 0 P x i,λ ({ai }) > 0 for all i P(x λ = a θ = θ) 1. Extension Infinite N 18 / 19

35 Conclusion Today: multiplicity/selection of eq. in coordination games Exogenous: info players have about others info Endogenous: info players want about others info Unrestricted info acquisition Broad assumptions on cost of information: A5, A6 Games on networks, Bonacich centrality Large games, endogenous informational smallness Rich yet tractable language for info acquisition in games 19 / 19

36 Appendix

37 Finite-game Construction Θ and X finite (Ω, F, P): nonatomic probability space X i : all measurable functions x i : Ω X i Back x i x i1 x i2 x i1 x i2 x i1 x i2 x i1 x i2 x j x j1 x j2 x j1 x j2 θ Ω θ 1 θ 2 0 1

38 Finite-game Construction Θ and X finite (Ω, F, P): nonatomic probability space X i : all measurable functions x i : Ω X i Back x i x i1 x i2 x i1 x i2 x i1 x i2 x i1 x i2 x j x j1 x j2 x j1 x j2 θ Ω θ 1 θ 2 0 1

39 Finite-game Construction Θ and X finite (Ω, F, P): nonatomic probability space X i : all measurable functions x i : Ω X i Back x i x i1 x i2 x i1 x i2 x i1 x i2 x i1 x i2 x j x j1 x j2 x j1 x j2 θ Ω θ 1 θ 2 0 1

40 General Construction T : uncountable set. (Ω, F, P): θ : Ω Θ: θ P Θ. z t : Ω [0, 1], t T : θ and z t, t T, independent z t, t T, uniformly distributed i N, x i X i iff countable Q T : x i measurable w.r.t. θ and z t, t Q. Back

41 Existence A i, Θ: Polish spaces v : A Θ R: measurable function s.t. ˆ ˆ < inf v(a, θ)p Θ(dθ) < sup v(a, θ)p Θ (dθ) < a A a A Θ V : (A 1 ) (A n ) R: ˆ (ˆ ) V (P A1,..., P An ) = log e v(a,θ) ( i N P Ai )(da) P Θ (dθ) A Θ Θ Fact Function V has a maximum if: A i is compact for all i N. v is upper semi-continuous in a. Back

42 Unique Selection for Potential Games N = {1,..., n}: finite set of players A i : Polish space of i s actions Θ: Polish space of states P Θ (Θ): state distribution v : A Θ R: potential Integrability: Θ sup a A v(a, θ)p Θ (dθ) <

43 Theorem Assume A0, A3, and A4. Take any a A s.t. i N Θ ai Θ: P Θ (Θ ai ) > 0 inf θ Θai inf a i a i inf a i A i v(a, θ) v(a i, a i, θ) > 0 Then equilibria (x λ : λ > 0): almost surely v(a, θ) > sup v(a, θ) lim P(x λ = a θ = θ) = 1. a a λ 0 Back

44 Infinite N: Multiplicity Theorem Assume A0, A3, and A4. Let N = {1, 2,...}. Let P Θ be abs. continuous w.r.t. Lebesgue measure. Then ˆθ [0, r], equilibria (x λ : λ > 0): i N (x i x i ) θ P(x i,λ = invest θ) a.s. { 1 if θ ˆθ 0 if θ < ˆθ

45 Proof Assume (x i x i ) θ for all i N. Law of large numbers: Var( x θ) = 0. Hence, Var(ρ( x, θ) θ) = 0. Independent information acquisition is optimal. Multiplicity as in Yang Back

46 Assumption 5 Take any i N, x i X i, and x i, x i X i. Assume that (x i (x i, θ)) x i. Then C i (x i, x i, θ) C i (x i, x i, θ). Equality holds only if (x i (x i, θ)) x i. Back

47 Assumption 6 Take any i N, x i X i, and x i, x i X i. Assume there is measurable f : X i Θ Z: (x i, f (x i, θ)) (x i, f (x i, θ)) (x i (x i, θ)) f (x i, θ) Then C i (x i, x i, θ) C i (x i, x i, θ). Equality holds only if (x i (x i, θ)) f (x i, θ). Proof Set z = f (x i, θ): Back I (x i; x i, θ) = I (x i; z) = I (x i ; z) I (x i ; x i, θ).