Coordination with Local Information Munther Dahleh Alireza Tahbaz-Salehi, John Tsitsiklis Spyros Zoumpouli

What Do I do? Fundamental limits of networked decision systems  Information theoretic Learning in large networks  Deterministic and stochastic networks Fragility and cascaded failures  Transportation and flow networks  Power Grid 2

How Information Is Shared Affects Outcomes 3 Motivation: self-fulfilling crises debt crises (PIGS) bank runs (Argentina ) social upheavals (Arab revolutions) … Information sharing (locality) enables coordination. How do equilibria depend on details of information sharing?

Model – Agents and Payoffs 4 Agents 1, …, n Actions: risky (α i = 1) and safe (α i = 0) Payoffs k: # agents who play risky π: continuous in θ θ: fundamentals Assumptions on π Strategic complementarities State monotonicity π is strictly decreasing in θ Strict dominance regions For sufficiently low (high) fundamentals, playing risky (safe) is strictly dominant

Model – Information 5 θ is realized, agents hold improper prior over R Conditional on θ, noisy signals (x 1, …, x m ) are generated x r = θ + ξ r, (ξ 1,..., ξ m ) - independent of θ - drawn from continuous density with full support over R m Agent i only observes a subset of the signals: observation set : the information structure of the game Example x r in I i for all i: public signal x r in I i for only one i: private signal all other cases: local signal Strategy:

Application to Networks 6 The collection of noisy signals are the idiosyncratic signals of each of the agents (m = n) Graph G = (V, E) represents social network Link i – j: agent i observes x j, agent j observes x i Agent i’s observation set: her idiosyncratic signal and the idiosyncratic signals of her neighbors What network topologies induce a unique equilibrium/multiple equilibria?

Problem 7 How does the number of equilibria (one vs. many) depend on the details of the information structure? Solution concept: Bayesian Nash equilibria (or Iterative Elimination of Strictly Dominated Strategies) Why is question of uniqueness vs multiplicity of BNE important? Uniqueness: predictive power over outcomes Multiplicity: a variety of outcomes is possible. Each outcome is a different self-fulfilling belief “It is a love-hate relationship: economists are at once fascinated and uncomfortable with multiple equilibria” ( Angeletos and Werning (2006))

A Three-Agent Example 8 common knowledge restores multiplicity strategic uncertainty refines set of equilibria (yet not to the extent of uniqueness) Public information: Profile “risky iff x < τ” is BNE for any Consider payoffs, noise i.i.d. Private information: Unique BNE: “risky iff x i < 1/2” Local information: As σ → 0 profile “risky iff x i < τ” is BNE for any

Results 9 Containment: Information structures where a collection of agents have identical observation sets: multiplicity Information locality is important in enabling coordination Common: Signals that are common knowledge between agents may lead to uniqueness Characterization of the set of BNE as a function of information structure

Results – Identical Observation Sets: Multiplicity 10 Proposition 1. Information structure with n ≥ 2 and collection C, such that all agents in C have same observation set I for all i not in C, Then multiple BNE. Interpretation No strategic uncertainty among agents in C enables them to coordinate Specifies how information homogeneity gives rise to multiplicity Extension: Information Containment

Results – Common Knowledge: May Have Uniqueness 11 Proposition 2. Information structure Then unique BNE as long as Consider payoffs, noise i.i.d. Proposition 3. Information structure Then unique BNE. Proper containment of observation sets does not necessarily lead to multiplicity! (Although information structure I 1 ={x 1 }, I 2 = {x 1,x 2 } does.)

Results – Local Information and the Set of Equilibria 12 Proposition 4. As σ → 0, strategy s i survives IESDS if and only if Consider payoffs, noise i.i.d. The larger is, the smaller becomes the set of strategies that survives IESDS. Examples: m = n (only private signals): τ΄ = τ΄΄ uniqueness m = 1 (one public signal): τ΄ = 0, τ΄΄ = 1 fixed m smallest gap: c r = n/m for all r largest gap: c 1 =… = c m-1 = 1, c n = n-(m-1) Each agent observes exactly one signal. c r : # agents who observe x r. c 1 +…+ c m = n where

Application to Networks 13 Characterization of strategies that survive IESDS in the case of unions of disconnected cliques Average of observations is sufficient statistic for each agent (assuming normality) Characterize thresholds for playing risky/safe  Multiplicity If all cliques grow linearly in n, then multiple strategies survive IESDS. Proposition 4. If network G n is a union of equally sized disconnected cliques, and all cliques grow at rate sublinear with n, then asymptotically there is a unique strategy that survives IESDS.

Final Remarks Interested in effects of networks or Information structure on decisions Promising results in the context of self-fulfilling formulations Move to more dynamic formulations (say multi-stage games). 14

Common Knowledge vs. No Common Knowledge 15 Complete information θ < 0: (R, R) unique NE θ > 1: (S, S) unique NE 0 < θ < 1: (R, R), (S, S), (θR + (1-θ)S, θR + (1-θ)S) are all NE 1-θ,1-θ-θ, 0 0, -θ0, 0 Player 1 Player 2 R S R S Incomplete information θ: improper prior over real line, x i = θ + ξ i, ξ 1,ξ 2 i.i.d. normal, independent of θ IESDS: Each player plays R if x i ½. Unique BNE Common knowledge of fundamentals induces multiple equilibria Failure of common knowledge through perturbation induces a unique equilibrium Today: Perturbation may or may not induce uniqueness, depending on how information is shared