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
Motivation: self-fulfilling crises
• debt crises (PIGS)
• bank runs (Argentina 1999-2002)
• social upheavals (Arab revolutions)
• …
Information sharing (locality) enables coordination. How do equilibria depend on
details of information sharing?
3
Model – Agents and Payoffs
Agents 1, …, n
Actions: risky (αi = 1) and safe (αi = 0)
Payoffs
 k ,  if ai  1
uai , ai ,   
if ai  0
0
k: # agents who play risky
θ: fundamentals
π: continuous in θ
Assumptions on π
• Strategic complementarities
 k ,    k  1,     0
• State monotonicity
π is strictly decreasing in θ
• Strict dominance regions
For sufficiently low (high) fundamentals, playing risky (safe) is strictly dominant
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Model – Information
θ is realized, agents hold improper prior over R
Conditional on θ, noisy signals (x1, …, xm) are generated
xr = θ + ξr , (ξ1, ..., ξm) - independent of θ
- drawn from continuous density with full support over Rm
Agent i only observes a subset of the signals: observation set
Strategy: si : R
Ii
I i  x1 ,..., xm 
 0,1
Ii in1 : the information structure of the game
Example
• xr in Ii for all i: public signal
• xr in Ii for only one i: private signal
• all other cases: local signal
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Application to Networks
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 xj, agent j observes xi
Agent i’s observation set:
her idiosyncratic signal and the idiosyncratic signals of her neighbors
I i  {xi , ( x j ) j:i , j E }
What network topologies induce a unique equilibrium/multiple equilibria?
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Problem
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))
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A Three-Agent Example
Consider payoffs
 k ,   
k 1
 ,
2
noise
r ~ N 0,  2  i.i.d.
Public information: I i  {x} i
Profile “risky iff x < τ” is BNE for any
  [0,1]
Private information: I i  {xi } i
Unique BNE: “risky iff xi < 1/2”
Local information:
•
•
I1  {x1}, I 2  I 3  {x2 }
As σ → 0 profile “risky iff xi < τ” is BNE for any   1 / 3,2 / 3
common knowledge restores multiplicity
strategic uncertainty refines set of equilibria (yet not to the extent of uniqueness)
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Results
Containment: Information structures where a collection of agents have identical
observation sets: multiplicity
Common: Signals that are common knowledge between agents may lead to
uniqueness
Characterization of the set of BNE as a function of information structure
Information locality is important in enabling coordination
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Results – Identical Observation Sets: Multiplicity
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, I i  I  
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
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Results – Common Knowledge: May Have Uniqueness
Consider payoffs  k ,   
k 1
 ,
n 1
noise
r ~ N 0,  2  i.i.d.
Proposition 2. Information structure I1  {x2 , x3}, I 2  {x1 , x3}, I 3  {x1 , x2 }
Then unique BNE as long as   1
3
Proposition 3. Information structure I1  {x1}, I 2  {x2 }, I 3  {x1 , x2 }
Then unique BNE.
Proper containment of observation sets does not necessarily lead to multiplicity!
(Although information structure I1={x1}, I2 = {x1,x2} does.)
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Results – Local Information and the Set of Equilibria


2
k 1
i.i.d.
  , noise  r ~ N 0, 
n 1
Each agent observes exactly one signal. cr: # agents who observe xr. c1+…+ cm = n
Consider payoffs  k ,   
Proposition 4. As σ → 0, strategy si survives IESDS if and only if
if x   
if x   
1
si  x   
0
The larger
c c
i j
i
j
where   
1
 ci c j ,    1   
2n n  1 i  j
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: cr = n/m for all r
• largest gap: c1 =… = cm-1 = 1, cn = n-(m-1)
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Application to Networks
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
Proposition 4. If network Gn is a union of equally sized disconnected cliques, and all
cliques grow at rate sublinear with n, then asymptotically ( n  ) there is a unique
strategy that survives IESDS.
If all cliques grow linearly in n, then multiple strategies survive IESDS.
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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).
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Common Knowledge vs. No Common Knowledge
Player 2
R
Player 1
R
S
S
1-θ,1-θ
-θ, 0
0, -θ
0, 0
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
Incomplete information
• θ: improper prior over real line, xi = θ + ξi, ξ1,ξ2 i.i.d. normal, independent of θ
• IESDS: Each player plays R if xi < ½, S if xi > 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
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