Robust Predictions in Games with Incomplete Information joint with - - PowerPoint PPT Presentation

Robust Predictions in Games with Incomplete Information

joint with Stephen Morris (Princeton University) November 2010

Payoff Environment

• in games with incomplete information, the agents are

uncertain about the payoff functions

• the payoff functions depend on some fundamental variable,

the payoff relevant state, over which there is uncertainty

• the payoff functions and the common prior over the payoff

relevant state define the payoff environment of the game

• the behavior of each agent depends on his information, the

posterior, about the fundamental variable...

• ...but also on his information about the other agents’ action

Games with Incomplete Information: Information Environment

• the strategy of agent 1 depends on his expectation about

payoff function of agent 2, as the nature of the latter will be an important determinant of agent 2’s behavior; his “first order expectation”

• but the strategy of agent 1 also depends on what he expects

to be agent 2’s first-order expectation about his own payoff function; his “second order expectation”, and so on...

• the resulting hierarchies of expectations, or in Harsanyi’s

re-formulation, the types of the agents, define the information environment of the game

• the optimal strategy of each agent (and in turn the

equilibrium) of the game is sensitive to the specification of the payoff environment and the information environment

Many Possible Informational Environments

• for a given payoff environment (payoff functions, common

prior of payoff relevant states ) there are many information environments which are consistent with the given payoff environment

• consistent in that, after integrating over the types, the

marginal over the payoff relevant states coincides with the common prior over the payoff relevant states

• the possible information environments vary widely:

from “complete uncertainty”, where every agent knows nothing beyond the common prior over the payoff relevant states to “complete information”, where every agents knows the realization of the payoff relevant state

• each specific information environment may generate specific

predictions regarding equilibrium behavior

Robust Predictions in Games with Incomplete Information

• yet, given that they share the same payoff environment, does

the predicted behavior share common features across information environments

• can analyst make predictions which are robust to the exact

specification of the information environment?

• we take as given a commonly known common prior over the

payoff relevant states...

• ... and that the agents share a common prior over some larger

type space (representing higher-order beliefs), but unknown to the analyst

• objective: predict the outcome of the game for all possible

common prior type spaces which project into the same common prior over payoff relevant states

• set prediction rather than point prediction about the

equilibrium outcomes

Revealed Preference and Robust Predictions

• the observable outcomes of the game are the actions and the

payoff relevant states

• the chosen action reveals the preference of the agent given his

interim information, but typically does not reveal his interim information

• thus we rarely observe or can infer the information

environment of the agents, but do infer (ex post) the payoff environment

Prediction and Identification

• for a given payoff environment, specified in terms of

preferences and common prior over fundamental variable and all possible higher order beliefs with respect to the given payoff environment, we pursue two related questions:

1 Predictions:

What restrictions are imposed by the structural model on the

• bservable endogenous variables?

2 Identification:

What restrictions can be imposed/inferred on the parameters

• f the structural model by the observations of the endogenous

variables?

Preview of Results: Epistemic Insight

• how to describe the set of Bayes Nash equilibrium outcomes

across all possible information environments?

• an indirect approach via the notion of Bayes correlated

equilibrium

• the object of the Bayes correlated equilibrium is simply a joint

distribution over actions and outcomes, independent of a type space and/or an information structure

• we establish an epistemic relationship between the set of

Bayes Nash equilibria and Bayes correlated equilibria

• we show that the Bayes Nash equilibria for all common prior

type spaces to be identical to the set of Bayes correlated equilibria

Preview of Results: Robust Prediction, Robust Identification, Robust Policy

• the set of Bayes correlated equilibria is a set of joint

distribution over actions and fundamentals, we ask what distributional (statistical) properties are shared by these joint distributions?

• characterize the outcome of the game in terms of the set of

moments of the individual and aggregate outcome of the correlated equilibria

• analyze how the outcome of the game is affected by given

private information of the agents

• compare the individual and social welfare across different

equilibria and/or belief systems

• analyze how identification is affected by concern for robustness

Payoff Environment

• continuum of players
• action ai ∈ Ai
• action profile a = (..., ai, ...) ∈ A
• payoff relevant state θ ∈ Θ
• payoff functions ui : A × Θ → R
• common prior over the payoff relevant states:

ψ ∈ ∆ (Θ)

• “payoff environment”:

(u, ψ)

• r “belief free game”: there is no information about players’

beliefs or higher order beliefs beyond the common prior ψ

Information Environment

• the private information of the agents is represented by an

information environment (information structure) T

• information environment T is a conditional probability system:

T =

• (Ti)I

i=1 , π

• each ti ∈ Ti represents private information (type) of agent i
• π is a conditional probability π [t] (θ) over type profiles

t = (t1, ..., tI ) : π : Θ → ∆ (T)

• a standard Bayesian game is described by (u, ψ, T )

Type and Posterior Beliefs

• ti ∈ Ti represents private information (type) of agent i
• ti ∈ Ti encodes information about payoff state -

“first order beliefs“ πi [θ] (ti) =

• t−i ψ (θ) π [t] (θ)
• θ

t−i ψ

• θ

π [t]

• θ
• ti ∈ Ti encodes information about types of other agents -

"higher order beliefs": πi [t−i] (ti) =

• θ ψ (θ) π [ti, t−i] (θ)
• θ
• t

−i ψ (θ) π

• ti, t

−i

• (θ)

Multitude of Information Environments

• every type ti of agent i could contain many pieces of

information ti = (s, si, sij, sijk,....) every agent i may observe a public (common) signal s centered around the state of the world θ: s ∼ N

• θ, σ2

s

• every agent i may observe a private signal si centered around

the state of the world θ: si ∼ N

• θ, σ2

i

• every agent i may observe a private signal si,j about the signal
• f agent j :

si,j ∼ N

• sj, σ2

i,j

• every agent i may observe a private signal si,j,k about ....:

si,j,k ∼ N

• sj,k, σ2

i,j,k

Bayes Nash Equilibrium

• a standard Bayesian game is described by (u, ψ, T )
• a behavior strategy of player i is defined by:

σi : Ti → ∆ (Ai)

Definition (Bayes Nash Equilibrium (BNE))

A strategy profile σ is a Bayes Nash equilibrium of (u, ψ, T ) if

• t−i,θ

ui ((σi (ti) , σ−i (ti)) , θ) ψ (θ) π [ti, t−i] (θ) ≥

• t−i,θ

ui ((ai, σ−i (t−i)) , θ) ψ (θ) π [ti, t−i] (θ) . for each i, ti and ai.

Bayes Nash Equilibrium Distribution

• given a Bayesian game (u, ψ, T ), a BNE σ generates a joint

probability distribution µσ over outcomes and states A × Θ , µσ (a, θ) = ψ (θ)

• t

π [t] (θ) I

• i=1

σi (ai|ti)

• equilibrium distribution µσ (a, θ) is specified without reference

to information structure T which gives rise to µσ (a, θ)

Definition (Bayes Nash Equilibrium Distribution)

A probability distribution µ ∈ ∆ (A × Θ) is a Bayes Nash equilibrium distribution (over action and states) of (u, ψ, T ) if there exists a BNE σ of (u, ψ, T ) such that µ = µσ.

Implications of BNE

• recall the original equilibrium conditions on (u, ψ, T ):
• t−i,θ

ui ((σi (ti) , σ−i (ti)) , θ) ψ (θ) π [ti, t−i] (θ) ≥

• t−i,θ

ui ((ai, σ−i (t−i)) , θ) ψ (θ) π [ti, t−i] (θ) .

• with the equilibrium distribution

µσ (a, θ) = ψ (θ)

• t

π [t] (θ) I

• i=1

σi (ai|ti)

• an implication of BNE of (u, ψ, T ) : for all

ai ∈ supp µσ (a, θ) :

• a−i,θ

ui ((ai, a−i) , θ) µσ (a, θ) ≥

• a−i,θ

ui

• a

i, a−i

• , θ
• µσ (a, θ) ;

Bayes Correlated Equilibrium

• joint distribution over actions and states:

µ (a, θ) ∈ ∆ (A × Θ)

Definition (Bayes Correlated Equilibrium (BCE))

A probability distribution µ ∈ ∆ (A × Θ) is a Bayes correlated equilibrium of (u, ψ) if for all i, ai and a

i;

• a−i,θ
• ui ((ai, a−i) , θ) − ui
• a

i, a−i

• , θ
• µ ((ai, a−i) , θ) ≥ 0;

and

• a∈A

µ (a, θ) = ψ (θ) , for all θ.

• Bayes correlated equilibrium is defined in terms of the payoff

environment and without reference to type spaces

• the equilibrium object is defined on “small” payoff space and

characterized as a solution of a linear programming problem

Basic Epistemic Result

• now given (u, ψ), what is the set of equilibrium distributions µ

across all possible information structures T

Theorem (Equivalence )

A probability distribution µ ∈ ∆ (A × Θ) is a Bayes correlated equilibrium of (u, ψ) if and only if it is a Bayes Nash Equilibrium distribution of (u, ψ, T ) for some information system T .

• BCE ⇒ BNE uses the richness of the possible information

structure to complete the equivalence result

• Aumann (1987) established the above characterization result

for complete information games

• in companion paper, “Correlated Equilibrium in Games with

Incomplete Information” we relate it to earlier definitions and establish comparative results with respect to information environments

Now: Quadratic Payoffs ...

• utility of each agent i is given by quadratic payoff function:
• determined by individual action ai ∈ R, state of the world

θ ∈ R, and average action A ∈ R: A = 1 aidi and thus: ui (ai, A, θ) = (ai, A, θ)   γaa γaA γaθ γaA γAA γAθ γaθ γAθ γθθ   (ai, A, θ)T

• game is completely described by interaction matrix Γ =
• γij

...and Normally Distributed Fundamental Uncertainty

• the state of the world θ is normally distributed

θ ∼ N

• µθ, σ2

θ

• with mean µθ ∈ R and variance σ2

θ ∈ R+

• the distribution of the state of the world is commonly known

common prior

Interaction Matrix

• given the interaction matrix Γ, complete information game is

a potential game (Monderer and Shapley (1996)): Γ =   γaa γaA γaθ γaA γAA γAθ γaθ γAθ γθθ  

• diagonal entries: γaa = γa, γAA, γθθ describe “own effects”
• off-diagonal entries: γaθ, γAθ, γaA “interaction effects”
• fundamentals matter, “return shocks”:

γaθ = 0;

• strategic complements and strategic substitutes:

γaA > 0

• vs. γaA < 0

Concave Game

• concavity at the individual level (well-defined best response):

γa < 0

• concavity at the aggregate level (existence of an interior

equilibrium) γa + γaA < 0

• concave payoffs imply that the complete information game

has unique Nash and unique correlated equilibrium (Neyman (1997))

Example 1: Beauty Contest

• continuum of agents: i ∈ [0, 1]
• action (= message): a ∈ R
• state of the world: θ ∈ R
• payoff function

ui = − (1 − r) (ai − θ)2 − r (ai − A)2 with r ∈ (0, 1)

• see Morris and Shin (2002), Angeletos and Pavan (2007)

Example 2: Competitive Market

• action ( = quantity): ai ∈ R
• cost of production c (ai) = 1

2γa (ai)2

• state of the world ( = demand intercept): θ ∈ R
• inverse demand ( = price):

p (A) = γaθθ − γaAA where A is average supply: A = 1 aidi

• see Guesnerie (1992) and Vives (2008)

Robust Prediction

• payoff environment with common prior:

u (ai, A, θ) , θ ∼ N

• µθ, σ2

θ

• information environment:

s ∼ N

• θ, σ2

s

• , si ∼ N
• θ, σ2

i

• , si,j ∼ N
• sj, σ2

i,j

• , si,j,k ∼ .....
• (Bayesian Nash) equilibrium play typically depends on payoff

and information environment

• outside observer, analyst may have limited information about

information environment,

• what can we say about equilibrium play irrespective of the

details of the information environment or ...

• ... what are the common features of equilibrium play across

all information environments?

Bayes Nash Equilibrium (BNE)

• begin with a specific information structure, say each agent i is
• bserving a private signal:

xi = θ + εi and a public signal y = θ + ε

• εi and ε are assumed to independently distributed normal

random variables, with zero mean and variances given by σ2

x

and σ2

y.

• this information structure appears in Morris and Shin (2002),

Angeletos and Pavan (2007), among many others

• the best response of each agent is:

a = − 1 γa (γaθE [θ |x, y ] + γAaE [A |x, y ])

Bayes Nash Equilibrium in Linear Strategies

• exploiting the linear first order conditions and the normality of

the information environment - in particular, the linearity of the conditional expectation E [θ |x, y ]

• suppose the equilibrium strategy is given by a linear function:

a (x, y) = α0 + αxx + αyy,

• now match the coefficients in the first order conditions to

construct the equilibrium strategy.

Unique Bayes Nash Equilibrium

• denote the sum of the precisions: σ−2 = σ−2

θ

+ σ−2

x

+ σ−2

y

Theorem

The unique Bayesian Nash equilibrium (given the bivariate information structure) is a linear equilibrium, α∗

0 + α∗ xx + α∗ yy, with

α∗

x = −

γaθσ−2

x

γAaσ−2

x

+ γaσ−2 , and α∗

y = −

γa γa + γaA γaθσ−2

y

γAaσ−2

x

+ γaσ−2 .

• sharp equilibrium prediction for given information structure...
• ...but what equilibrium properties are common to all bivariate

normal information structures

• ...to all multivariate normal information structure?
• ...to all information structures?

Bayes Correlated Equilibria

• the object of analysis: joint distribution over actions and

states: µ (a, A, θ)

• characterize the set of (normally distributed) BCE:

  ai A θ   ∼ N     µa µA µ   ,   σ2

a

ρaAσaσA ρaθσaσθ ρaAσaσA σ2

A

ρAθσAσθ ρaθσaσθ ρAθσAσθ σ2

θ

   

• σ2

A is the aggregate volatility (common variation)

• σ2

a − σ2 A is the cross-section dispersion (idiosyncratic variation)

• statistical representation of equilibrium in terms of first and

second order moments

Symmetric Bayes Correlated Equilibria

• with focus on symmetric equilibria:

µA = µa, σ2

A = ρaσ2 a,

ρaAσaσA = ρaσ2

a

where ρa is the correlation coefficient across individual actions

• the first and second moments of the correlated equilibria are:

  ai A θ   ∼ N     µa µa µ   ,   σ2

a

ρaσ2

a

ρaθσaσθ ρaσ2

a

ρaσ2

a

ρaθσaσθ ρaθσaσθ ρaθσaσθ σ2

θ

   

• correlated equilibria are characterized by:

{µa, σa, ρa, ρaθ}

Equilibrium Analysis

• in the complete information game, the best response is:

a = −θγaθ γa − AγAa γa

• best response is weighted linear combination of fundamental θ

and average action A relative to the cost of action: γaθ/γa, γAa/γa

• in the incomplete information game, θ and A are uncertain:

E [θ] , E [A]

• given the correlated equilibrium distribution µ (a, θ) we can

use the conditional expectations: Eµ [θ |a] , Eµ [A |a]

Equilibrium Conditions

• in the incomplete information game, the best response is:

a = −Eµ [θ |a] γaθ γa − Eµ [A |a] γAa γa

• best response property has to hold for all a ∈ supp µ (a, θ)
• a fortiori, the best response property has to hold in

expectations over all a : Eµ [a] = Eµ

• −
• Eµ [θ |a] γaθ

γa + Eµ [A |a] γAa γa

• by the law of iterated expectation, or law of total expectation:

Eµ [Eµ [θ |a]] = µθ, Eµ [Eµ [A |a]] = Eµ [A] = Eµ [a] ,

Equilibrium Moments: Mean

• the best response property implies that for all µ (a, θ) :

Eµ [a] = Eµ

• −
• Eµ [θ |a] γaθ

γa + Eµ [A |a] γAa γa

• r by the law of iterated expectation:

µa = −µθ γaθ γa − µa γAa γa

Theorem (First Moment)

In all Bayes correlated equilibria, the mean action is given by: E [a] = −µθ γaθ γa + γaA .

• result about “mean action” is independent of symmetry or

normal distribution

Equilibrium Moments: Variance

• in any correlated equilibrium µ (a, θ), best response demands

a = −

• E [θ |a] γaθ

γa + E [A |a] γAa γa

• ,

∀a ∈ supp µ (a, θ)

• or varying in a

1 = − ∂E [θ |a] ∂a γaθ γa + ∂E [A |a] ∂a γAa γa

• ,
• the change in the conditional expectation

∂E [θ |a] ∂a , ∂E [A |a] ∂a is a statement about the correlation between a, A, θ

Equilibrium Moment Restrictions

• the best response condition and the condition that Σa,A,θ

forms a multivariate distribution, meaning that the variance-covariance matrix has to be positive definite

• we need to determine:

{σa, ρa, ρaθ}

Theorem (Second Moment)

The triple (σa, ρa, ρaθ) forms a Bayes correlated equilibrium iff: ρa − ρ2

aθ ≥ 0,

and σa = − σθγaθρaθ ρaγAa + γa .

Moment Restrictions: Correlation Coefficients

• the equilibrium set is characterized by inequality ρa − ρ2

aθ ≥ 0

• ρa: correlation of actions across agents; ρaθ : correlation of

actions and fundamental

.

Set of correlated equilibria a a

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Bayes Correlated and Bayes Nash Equilibrium

• recall epistemic result on relationship between BCE and BNE
• now we can ask which type space / information structure

turns BNE into BCE ?

• consider the following bivariate information structure:

1 every agent i observes a public signal y about θ :

y ∼ N

• θ, σ2

y

• 2 every agent i observes a private signal xi about θ :

xi ∼ N

• θ, σ2

x

Equivalence between BCE and BNE

• bivariate information structure which generates volatility

(common signal) and dispersion (idiosyncratic signal)

Theorem

There is BCE with (ρa, ρaθ) if and only if there is a BNE with

• σ2

x, σ2 y

• .
• a public and a private signal are sufficient to generate the

entire set of correlated equilibria...

• ... but a given BCE does not uniquely identify the information

environment of a BNE.

Information Bounds

• the analyst may not know how much information the agents

have, yet may have a lower bound on how much information the agents have

• how does the set of BCE change with the lower bound
• assume that all agents observe a public signal y :

y = θ + ε and a private signal xi xi = θ + εi

• the given information of the agents is described by:

ε εi

• ∼ N
• ,

σ2

y

σ2

x

Information bounds and correlated equilibrium

• the equilibrium conditions are augmented by “for all a, x, y”:

a = − 1 γa (γaθE [θ |a, x, y ] + γAaE [A |a, x, y ]) , ∀a, x, y.

• we determine σa , ρax, ρay in terms of ρa, ρaθ, e.g.:

ρay = σθ σyρaθ γa + ρaγAa γa + γAa − ρ2

aθ

• set of correlated equilibria is given by the inequalities:

ρa − ρ2

aθ − ρ2 ay ≥ 0,

1 − ρa − ρax ≥ 0,

Given Public Information

• describe the equilibrium set C (σ) ∈ [0, 1]2 in terms of the

noise of the signal pair σ = (σx, σy)

• the interior of each level curve describes the correlated

equilibria with a given amount of public correlation

• movements along level curve are variations in σ−2

x

given σ−2

y

Correlated Equilibria of beauty contest with minimal precsion of

y 2 and r=.25

a a 2

.01 .2 .5 1 .1 .001 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Given Private Information

• each level curve describes the correlated equilibria with a

given amount of private correlation

• equivalent it can be understood in terms of BNE with a given

information structure σ = (σx, σy)

• movements along level curve are variations in σ−2

y

given σ−2

x

Correlated Equilibria of beauty contest with minimal precsion of

x 2 and r=.25

.01 .1 .2 .5 1

a 2 a

45o

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Given Private and Public Information

• the interior intersection of the level curves generates the

corresponding equilibrium set

• more information reduces the set of possible outcomes,

because it adds incentive constraints but does not remove any correlation possibilities

2 2

a 2

x 2

.5

x 2

.1

y 2

.1

y 2

.5

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Given Information and the Equilibrium Set

• describe the equilibrium set C (σ) ∈ [0, 1]2 in terms of the

noise of the signal pair σ = (σs, σi)

Theorem (Information Bounds)

1 For all σ < σ, C (σs) ⊂ C (σ s) ; 2 For all σ < σ:

min

ρa∈C (σ) ρa >

min

ρa∈C (σ) ρa; 3 For all σ < σ:

min

ρaθ∈C (σ) ρaθ >

min

ρa∈C (σ) ρaθ.

• more private information shrinks the equilibrium set and

makes predictions sharper

Identification

1 Predictions:

What restrictions are imposed by the structural model (u, ψ)

• n the observable endogenous statistics (µa, σa, ρa, ρaθ)?

2 Identification:

What restrictions can be imposed/inferred on the parameters Γ of the structural model by the observations of the endogenous variables (µa, σa, ρa, ρaθ)?

Identification with Complete Information

• classic problem of supply and demand identification:
• demand is given by:

Pd = d0 + d1Q + d2θd

• supply is given by:

Ps = s0 + s1Q + s2θs

• the exogenous random variables are θd and θs are demand

and supply shocks (”demand, supply shifters”)

• complete information: each firm observes shocks (θd, θs)

and makes supply decisions accordingly

• the econometrician uses instrumental variable (zd, zs):

zd = θd + νd, zs = θs + νs to estimate and identify (d1, s1); see Wright (1928), Koopmans (1949), Fisher (1966), ...

Incomplete Information

• incomplete information: each firm observes a vector of

signals: yi = (ys, ysi, yd, ydi) regarding the true cost and demand shocks: ys = θs + εs, ysi = θs + εsi, yd = θd + εd, ydi = θd + εdi before its supply decision

• maintain normality and independence of (θd, θs, εd, εid, εs, εis)
• the variance of the random variables:

I =

• σ2

d, σ2 s , λ2 d, λ2 id, λ2 s , λ2 is

• represents the information structure I in the economy

Competitive Equilibrium with Incomplete Information

• in Bayes Nash equilibrium of competitive economy, each firm

supplies qi (yi)

• n the basis of its private (but noisy) information
• the Bayes Nash equilibrium price is given by the equilibrium

condition: Pd = d0 + d1

• qi (yi) di + d2θd

with respect to the realized demand shock θd

Identification with Incomplete Information

• can we identify and estimate the slope of supply and demand

function in the presence of incomplete information

• for every information structure I each firm observes a noisy

signal of the true cost and demand shock and makes a supply decision on the basis of the noisy information

Theorem (Point Identification)

For every information structure I, the demand and supply functions are point identified if the firms have noisy information about their cost: min

• λ2

s , λ2 is

• < ∞.
• asymmetry arises as realized price varies with realized demand

Robust Identification

• earlier we asked what predictions can be made for all (or a

subset of) information structures,

• now we ask can identification be accomplished for all (or a

subset of) information structures, i.e. can we achieve “robust identification”

Theorem (Set Identification)

1 For every information bound, the demand and supply

functions are set identified.

2 If the information bounds increase, then the identified set

decreases.

• concern for robustness weakens the ability to identify the

structural parameter to allow for partial identification only

Beyond Demand and Supply

• consider the quadratic game environment
• suppose only the actions are observable:

(µa, σa, ρa) but the realization of the state is unobservable, and hence we do not have access to covariate information between a and θ :

• the identification then uses the mean:

µα = −µθ γaθ γa + γaA and variance σa = − σθγaθρaθ ρaγAa + γa .

Sign Identification

• relative to supply and demand identification, less observable

data and restrict attention to identify sign of interaction

• recall:γaθ informational externality, γaA strategic externality

Theorem (Sign Identification)

The Bayes Nash Equilibrium identifies the sign of γaθ and γaA.

• identification in Bayes Nash equilibrium uses

variance-covariance given information structure

• σ2

x, σ2 y

• Theorem (Partial Sign Identification)

The Bayes Correlated Equilibrium identifies the sign of γaθ but it does not identify the sign of γaA.

• failure to identify the strategic nature of the game, strategic

complements or strategic substitutes

Robust (Information) Policy

• information regarding fundamentals is widely dispersed in

society

• identify policies to improve the decentralized use of dispersed

information

• induce agents to internalize the informational externality
• for given structure of private information find the optimal

policy of public information revelation (Morris & Shin (2002))

Robust Information Policy: The Beauty Contest

• individual payoff of agent i is given by:

ui (a, θ) = − (1 − r) (a − θ)2 − r (a − A)2

• social payoff is given by:

w (a, θ) = − (a − θ)2

• the individual weight r ∈ (0, 1) on coordination is larger than

the social weight s = 0:

The Value of Public Information

• Morris & Shin (2002) showed that if r > 1/2, then the value
• f additional public information is negative
• public information allows agents to coordinate on a common

action at the expense of matching the state of the world θ

• denote the precision of public information by α, the precision
• f private information by β :

α = 1 σ2

s

, β = 1 σ2

i

• the social value of information in the beauty contest is:

W (α, β) = − α + β (1 − r)2 [α + β (1 − r)]2

• given r > 1/2 and β, it is U shaped in the precision α of

public information (informational transparency may not be socially optimal)

Robust Information Policy

• now robust welfare (for Given information) is given by

W (α, β) where: W (α, β) = min

α≥α,β≥β

• − α + β (1 − r)2
• α + β (1 − r)

2

• where α and β are the Given precision of public and private

information respectively

• with U shaped value function the lowest valuation α is either:
• in the interior: α < α or
• coincides with the lower bound: α = α

Theorem (Robust Information Policy)

The robust welfare is (weakly) increasing in the precision of the public signal everywhere.

Discussion

• Bayes correlated equilibrium encodes concern for robustness to

information environment

• identification: complete vs. incomplete information
• demand and supply identification: the market participants have

complete information, but the analyst has noisy information (uses instrument to recover the information)

• auction identification: each bidder has private information

about his valuation, but the bidders have the same information about each other as the analyst

• presumably neither informational assumption is valid,

suggesting a role for robust identification

• robust welfare improving policy
• how responsive is robust policy to informational conditions

An Aside: Mechanisms versus Games

• in earlier work we investigated the robustness of social choice

problems to beliefs and higher order beliefs

• in mechanism design, the game is designed to have favorable

properties Q1: Can we identify a mechanism such that for all beliefs and higher order beliefs an equilibrium which implements the social choice function exists? A1: Yes, if and only if the social choice function satisfies ex post incentive compatibility (Robust Mechanism Design, ECTA 2005) Q2: Can we identify a mechanism such that for all beliefs and higher order beliefs all equilibria implement the social choice function? A2: Yes, if and only if the social choice function satisfies strict ex post incentive compatibility and the preferences are not too interdependent, a contraction-like property, (Robust Implementation, RES 2009)