Coordinating over Signals

Transcription

1 Coordinating over Signals Jeff S. Shamma Behrouz Touri & Kwang-Ki Kim School of Electrical and Computer Engineering Georgia Institute of Technology ARO MURI Program Review March 18, 2014 Jeff S. Shamma Coordinating over Signals 1/23

2 Coordination and information Rebelling against a regime is a coordination problem: each person is more willing to show up at a demonstration if many others do, perhaps because success is more likely and getting arrested is less likely. Chwe, Rational Ritual, Jeff S. Shamma Coordinating over Signals 2/23

3 General setting State of World Local Signals Agent Actions Coordination Effects Consequences Local signals: Private vs Public vs Overlapping How does information structure affect ability to coordinate? Jeff S. Shamma Coordinating over Signals 3/23

4 General setting State of World Local Signals Agent Actions Coordination Effects Consequences Local signals: Private vs Public vs Overlapping How does information structure affect ability to coordinate? For nearly thirty years, the price of a loaf of bread in Egypt was held constant; Anwar el-sadat s attempt in 1977 to raise the price was met with major riots. Since then, one government tactic has been to make the loaves smaller gradually...(jehl 1996). Chwe, Rational Ritual, Jeff S. Shamma Coordinating over Signals 3/23

5 Model: Global games Agents 1,..., n Actions: Risky a i = 1 (IN) Safe a i = 0 (OUT) Strength of regime: θ Payoffs: u(a i, a i, θ) = { π(#in, θ), a i = 1; 0, a i = 0. Payoff of risky action, π(k, θ): Increasing in k Decreasing in θ e.g, π(k, θ) = #IN θ Jeff S. Shamma Coordinating over Signals 4/23

6 Model: Global games Agents 1,..., n Actions: Risky a i = 1 (IN) Safe a i = 0 (OUT) Strength of regime: θ Payoffs: u(a i, a i, θ) = { π(#in, θ), a i = 1; 0, a i = 0. Payoff of risky action, π(k, θ): Increasing in k Decreasing in θ e.g, π(k, θ) = #IN θ Jeff S. Shamma Coordinating over Signals 4/23

7 Model: Global games Agents 1,..., n Actions: Risky a i = 1 (IN) Safe a i = 0 (OUT) Strength of regime: θ Payoffs: u(a i, a i, θ) = { π(#in, θ), a i = 1; 0, a i = 0. Payoff of risky action, π(k, θ): Increasing in k Decreasing in θ e.g, π(k, θ) = #IN θ Jeff S. Shamma Coordinating over Signals 4/23

8 Model: Global games Agents 1,..., n Actions: Risky a i = 1 (IN) Safe a i = 0 (OUT) Strength of regime: θ Payoffs: u(a i, a i, θ) = { π(#in, θ), a i = 1; 0, a i = 0. Payoff of risky action, π(k, θ): Increasing in k Decreasing in θ e.g, π(k, θ) = #IN θ Jeff S. Shamma Coordinating over Signals 4/23

9 Model: Global games Agents 1,..., n Actions: Risky a i = 1 (IN) Safe a i = 0 (OUT) Strength of regime: θ Payoffs: u(a i, a i, θ) = { π(#in, θ), a i = 1; 0, a i = 0. Payoff of risky action, π(k, θ): Increasing in k Decreasing in θ e.g, π(k, θ) = #IN θ Jeff S. Shamma Coordinating over Signals 4/23

10 Model: Information Noisy signals conditioned on θ: s = 1, 2,..., S X s = θ + ξ s Agent i observes a subset: I i {X 1,..., X S } Examples: Private signals: I i = X i Public signal: I i = X Shared/Overlapping/Networked... Jeff S. Shamma Coordinating over Signals 5/23

11 Model: Information Noisy signals conditioned on θ: s = 1, 2,..., S X s = θ + ξ s Agent i observes a subset: I i {X 1,..., X S } Examples: Private signals: I i = X i Public signal: I i = X Shared/Overlapping/Networked... Jeff S. Shamma Coordinating over Signals 5/23

12 Model: Equilibria Strategy: A i : I i {0, 1} Special case. Threshold strategy: { 1, E [ θ X ] τ; A(X ) = 0, otherwise Bayesian-Nash Equilibrium (BNE): A i ( ) is optimal with respect to A i ( ) Jeff S. Shamma Coordinating over Signals 6/23

13 Issues Uniqueness of equilibrium? State of World Local Signals Agent Actions Coordination Effects Consequences Jeff S. Shamma Coordinating over Signals 7/23

14 Issues Uniqueness of equilibrium? π(k, θ) = k 1 2 θ X s = θ + ξ s, ξ s N(0, σ 2 ) Private info: τ = 1/2 Public info: τ [0, 1] Shared 1 vs 2,3: τ [1/3, 2/3] for σ 1 State of World Local Signals Agent Actions Coordination Effects Consequences Jeff S. Shamma Coordinating over Signals 7/23

15 Issues Uniqueness of equilibrium? π(k, θ) = k 1 2 θ X s = θ + ξ s, ξ s N(0, σ 2 ) Private info: τ = 1/2 Public info: τ [0, 1] Shared 1 vs 2,3: τ [1/3, 2/3] for σ 1 Temporal coordination? Gathering of information over stages What to do? When to do it? State of World Local Signals Agent Actions Coordination Effects Consequences Jeff S. Shamma Coordinating over Signals 7/23

16 Global games: Group vs group Payoff function: Best response: Two groups: BR(X i ) = π(k, θ) = #IN θ { 1, 1 + E [ j i A j 0, otherwise Xi ] E [ θ Xi ] ; G 1 = {1,..., m} observes X = θ + ξ G 2 = {m + 1,...m + n} observes Y = θ + η ξ, η N(0, σ 2 ) Jeff S. Shamma Coordinating over Signals 8/23

17 Similar information in equilibrium Proposition If X i = X i, then in any equilibrium, A i ( ) = A i ( ) Group vs group implication: Equilibrium looks like (A 1, A 2 ) = (A 1,..., A }{{} 1, A 2,..., A 2 ) }{{} m times n times Issue: Multiple equilibria still possible...refinement? Jeff S. Shamma Coordinating over Signals 9/23

18 Best group response dynamics Let (A 1 [0], A 2 [0]) be threshold policies with thresholds (τ 1 [0], τ 2 [0]). For k = 1, 2,... A 1 [k] = BR(A 2 [k 1]) A 2 [k] = BR(A 1 [k 1]) Group best response assumes others in group will use same policy. Jeff S. Shamma Coordinating over Signals 10/23

19 Theorem: Equilibrium refinement in groups Threshold policies: (A 1[k], A 2[k]) are threshold policies with thresholds (τ 1[k], τ 2[k]) with «τ2[k 1] τ 1[k] nφ + m = τ 1[k] 2σ «τ1[k 1] τ 2[k] mφ + n = τ 2[k] 2σ Jeff S. Shamma Coordinating over Signals 11/23

20 Theorem: Equilibrium refinement in groups Threshold policies: (A 1[k], A 2[k]) are threshold policies with thresholds (τ 1[k], τ 2[k]) with «τ2[k 1] τ 1[k] nφ + m = τ 1[k] 2σ «τ1[k 1] τ 2[k] mφ + n = τ 2[k] 2σ Unique fixed point, (τ1, τ2 ): «τ (m + n)φ 2 τ1 = n (τ2 τ1 ) 2σ τ 1 «τ = nφ 2 τ1 + m 2σ Jeff S. Shamma Coordinating over Signals 11/23

21 Theorem: Equilibrium refinement in groups Threshold policies: (A 1[k], A 2[k]) are threshold policies with thresholds (τ 1[k], τ 2[k]) with «τ2[k 1] τ 1[k] nφ + m = τ 1[k] 2σ «τ1[k 1] τ 2[k] mφ + n = τ 2[k] 2σ Unique fixed point, (τ1, τ2 ): «τ (m + n)φ 2 τ1 = n (τ2 τ1 ) 2σ τ 1 «τ = nφ 2 τ1 + m 2σ Convergence rate: τ l [k] τ l 1 (1 + 2 πσ/m)(1 + 2 πσ/n) τ l[k 2] τ l Jeff S. Shamma Coordinating over Signals 11/23

22 Global games & noisy signal sharing Main focus: Threshold policies How general? E [ θ X ] τ Jeff S. Shamma Coordinating over Signals 12/23

23 Global games & noisy signal sharing Main focus: Threshold policies E [ θ X ] τ How general? Setup: Two agents: X 1 = θ + ξ 1 X 2 = θ + ξ 2 ξ 1, ξ 2 N(0, σ 2 ) Jeff S. Shamma Coordinating over Signals 12/23

24 Global games & noisy signal sharing Main focus: Threshold policies E [ θ X ] τ How general? Setup: Two agents: Noisy shared data: X 1 = θ + ξ 1 X 2 = θ + ξ 2 ξ 1, ξ 2 N(0, σ 2 ) Y 1 = X 2 + η 1 Y 2 = X 1 + η 2 η 1, η 2 N(0, ρ 2 ) Jeff S. Shamma Coordinating over Signals 12/23

25 Global games & noisy signal sharing Regime strength posterior: θ(x, y) = E [ θ Xi = x, Y i = y ] Jeff S. Shamma Coordinating over Signals 13/23

26 Global games & noisy signal sharing Regime strength posterior: θ(x, y) = E [ θ Xi = x, Y i = y ] Theorem There do not exist equilibrium threshold policies on θ. Jeff S. Shamma Coordinating over Signals 13/23

27 Temporal coordination Gathering of information over stages What to do? When to do it? State of World Local Signals Agent Actions Coordination Effects Consequences Jeff S. Shamma Coordinating over Signals 14/23

28 Recall: One-sided hypothesis testing Binary state of world: {0, 1} Daily decision: Declare 0 (with error cost L 0) Declare 1 (with error cost L 1) Continue (with cost c) Outcome: Posteriors uniformly bounded away from certainty Jeff S. Shamma Coordinating over Signals 15/23

29 Two-sided hypothesis testing As before... Costly observations, C o Costly erroneous declarations C e(θ, a) New terms: Costly disagreement: C d (a i, a i ) Costly separation: C s Jeff S. Shamma Coordinating over Signals 16/23

30 Two-sided hypothesis testing As before... Costly observations, C o Costly erroneous declarations C e(θ, a) New terms: Costly disagreement: C d (a i, a i ) Costly separation: C s Continuing to take data after partner s declaration. Jeff S. Shamma Coordinating over Signals 16/23

31 Two-sided hypothesis testing As before... Costly observations, C o Costly erroneous declarations C e(θ, a) New terms: Costly disagreement: C d (a i, a i ) Costly separation: C s Continuing to take data after partner s declaration. Compare: Common learning (Cripps, et al., 2008) Jeff S. Shamma Coordinating over Signals 16/23

32 Issues Strategy: A : X (1), X (2),..., X (t) {0, 1, w} What are optimal/equilibrium strategies? Threshold? Schedule? Jeff S. Shamma Coordinating over Signals 17/23

33 Issues Strategy: A : X (1), X (2),..., X (t) {0, 1, w} What are optimal/equilibrium strategies? Threshold? Schedule? What is best response? Jeff S. Shamma Coordinating over Signals 17/23

34 Issues Strategy: A : X (1), X (2),..., X (t) {0, 1, w} What are optimal/equilibrium strategies? Threshold? Schedule? What is best response? Setup: Leader (single hypothesis tester) & Follower Case I: Conditionally independent signals Case II: Conditionally dependent signals Jeff S. Shamma Coordinating over Signals 17/23

35 Computational approach Formulate DP on information state I t+1 = {I t, x t+1 } x t+1 α t f 0 ( ) + (1 α t )f 1 ( ) α t = Pr [ θ = 0 I t ] Given I t, can formulate beliefs on data & action sequence of partner Execute value iteration over I t (with exponentially growing state) Jeff S. Shamma Coordinating over Signals 18/23

36 Case I: Conditionally independent signals Leader s cost functions C e(0, 1) = C e(1, 0) = 1 and C e(0, 0) = C e(1, 1) = 0; C o = 0.005;» 1 Q x = + ɛ 1 ɛ ɛ 1 + ɛ with ɛ = Follower s cost functions C e(0, 1) = C e(1, 0) = 1 and C e(0, 0) = C e(0, 0) = 0; C o = 0.005; C d (0, 1) = C d (1, 0) = 0.05 and C d (0, 0) = C d (1, 1) = 0; C s = 0.05; Q z Q x. Horizon = 10 Jeff S. Shamma Coordinating over Signals 19/23

37 Jeff S. Shamma Coordinating over Signals 20/23 Optimal follower policy

38 Jeff S. Shamma Coordinating over Signals 20/23 Optimal follower policy

39 Jeff S. Shamma Coordinating over Signals 20/23 Optimal follower policy

40 Jeff S. Shamma Coordinating over Signals 20/23 Optimal follower policy

41 Case II: Conditionally dependent signals Noisy measurements: (x, z) (0, 0) (0, 1) (1, 0) (1, 1) θ = 0 θ = 1 q q , q = q 4 Cost functions Leader s cost functions C e(0, 1) = C e(1, 0) = 1 and C e(0, 0) = C e(1, 1) = 0; C o = 0.005; Follower s cost functions C(0, 1) = C(1, 0) = 1 and C(0, 0) = C(1, 1) = 0; C o = 0.005; C d (0, 1) = C d (1, 0) = 0.05 and C d (0, 0) = C d (1, 1) = 0; C s = 0.05; Horizon = 10 4 q 4 Jeff S. Shamma Coordinating over Signals 21/23

42 Jeff S. Shamma Coordinating over Signals 22/23 Optimal follower policy

43 Jeff S. Shamma Coordinating over Signals 22/23 Optimal follower policy

44 Recap & ongoing work Networked global games Structure of equilibria Suboptimal policies Temporal coordination Analytical characterizations Time as coordinating device Social influence, evolutionary dynamcs... State of World Local Signals Agent Actions Coordination Effects Consequences Jeff S. Shamma Coordinating over Signals 23/23