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 observable endogenous variables? 2 Identification: What restrictions can be imposed/inferred on the parameters of 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 a i A i action profile a = (..., a i,...) A payoff relevant state θ Θ payoff functions u i : A Θ R common prior over the payoff relevant states: ψ (Θ) payoff environment : (u, ψ) or 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 = ((T i ) I i=1, π each t i T i represents private information (type) of agent i π is a conditional probability π [t] (θ) over type profiles t = (t 1,..., t I ) : π : Θ (T ) a standard Bayesian game is described by (u, ψ, T )
Type and Posterior Beliefs t i T i represents private information (type) of agent i t i T i encodes information about payoff state - first order beliefs t π i [θ] (t i ) = i ψ (θ) π [t] (θ) θ t i ψ ( θ ) π [t] ( θ ) t i T i encodes information about types of other agents - "higher order beliefs": θ π i [t i ] (t i ) = ψ (θ) π [t i, t i ] (θ) θ t i ψ (θ) π [ ] t i, t i (θ)
Multitude of Information Environments every type t i of agent i could contain many pieces of information t i = (s, s i, s ij, s ijk,...) every agent i may observe a public (common) signal s centered around the state of the world θ: s N ( θ, σ 2 ) s every agent i may observe a private signal s i centered around the state of the world θ: s i N ( θ, σ 2 ) i every agent i may observe a private signal s i,j about the signal of agent j : s i,j N ( s j, σ 2 ) i,j every agent i may observe a private signal s i,j,k about...: s i,j,k N ( s j,k, σ 2 ) 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: σ i : T i (A i ) Definition (Bayes Nash Equilibrium (BNE)) A strategy profile σ is a Bayes Nash equilibrium of (u, ψ, T ) if u i ((σ i (t i ), σ i (t i )), θ) ψ (θ) π [t i, t i ] (θ) t i,θ t i,θ u i ((a i, σ i (t i )), θ) ψ (θ) π [t i, t i ] (θ). for each i, t i and a i.
Bayes Nash Equilibrium Distribution given a Bayesian game (u, ψ, T ), a BNE σ generates a joint probability distribution µ σ over outcomes and states A Θ, µ σ (a, θ) = ψ (θ) t ( I ) π [t] (θ) σ i (a i t i ) equilibrium distribution µ σ (a, θ) is specified without reference to information structure T which gives rise to µ σ (a, θ) i=1 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 = µ σ.
Implications of BNE recall the original equilibrium conditions on (u, ψ, T ): u i ((σ i (t i ), σ i (t i )), θ) ψ (θ) π [t i, t i ] (θ) t i,θ t i,θ u i ((a i, σ i (t i )), θ) ψ (θ) π [t i, t i ] (θ). with the equilibrium distribution µ σ (a, θ) = ψ (θ) ( I ) π [t] (θ) σ i (a i t i ) t an implication of BNE of (u, ψ, T ) : for all a i supp µ σ (a, θ) : (( ) ) u i a i, a i, θ µσ (a, θ) ; a i,θ u i ((a i, a i ), θ) µ σ (a, θ) a i,θ i=1
joint distribution over actions and states: µ (a, θ) (A Θ) Definition (Bayes Correlated Equilibrium (BCE)) Bayes Correlated Equilibrium A probability distribution µ (A Θ) is a Bayes correlated equilibrium of (u, ψ) if for all i, a i and a i ; [ (( ) )] ui ((a i, a i ), θ) u i a i, a i, θ µ ((ai, a i ), θ) 0; a i,θ and µ (a, θ) = ψ (θ), for all θ. a A 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
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 a i R, state of the world θ R, and average action A R: and thus: A = u i (a i, A, θ) = (a i, A, θ) 1 0 a i di γ aa γ aa γ aθ γ aa γ AA γ Aθ γ aθ γ Aθ γ θθ (a i, A, θ) T game is completely described by interaction matrix Γ = { γ ij }
...and Normally Distributed Fundamental Uncertainty the state of the world θ is normally distributed θ N ( µ θ, σ 2 ) θ with mean µ θ R and variance σ 2 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 = γ aa γ aa γ aθ γ aa γ AA γ Aθ γ aθ γ Aθ γ θθ diagonal entries: γ aa = γ a, γ AA, γ θθ describe own effects off-diagonal entries: γ aθ, γ Aθ, γ aa interaction effects fundamentals matter, return shocks : γ aθ 0; strategic complements and strategic substitutes: γ aa > 0 vs. γ aa < 0
Concave Game concavity at the individual level (well-defined best response): γ a < 0 concavity at the aggregate level (existence of an interior equilibrium) γ a + γ aa < 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 u i = (1 r) (a i θ) 2 r (a i A) 2 with r (0, 1) see Morris and Shin (2002), Angeletos and Pavan (2007)
Example 2: Competitive Market action ( = quantity): a i R cost of production c (a i ) = 1 2 γ a (a i ) 2 state of the world ( = demand intercept): θ R inverse demand ( = price): where A is average supply: p (A) = γ aθ θ γ aa A A = 1 0 a i di see Guesnerie (1992) and Vives (2008)
Robust Prediction payoff environment with common prior: u (a i, A, θ), θ N ( µ θ, σ 2 ) θ information environment: s N ( θ, σ 2 s ), si N ( θ, σ 2 i ), si,j N ( s j, σ 2 ) 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 observing a private signal: and a public signal x i = θ + ε i y = θ + ε ε i and ε are assumed to independently distributed normal random variables, with zero mean and variances given by σ 2 x and σ 2 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 γ a (γ aθ E [θ x, y ] + γ Aa E [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) = α 0 + α x x + α y y, now match the coeffi cients in the first order conditions to construct the equilibrium strategy.
denote the sum of the precisions: σ 2 = σ 2 θ Unique Bayes Nash Equilibrium + σ 2 x + σ 2 y Theorem The unique Bayesian Nash equilibrium (given the bivariate information structure) is a linear equilibrium, α 0 + α x x + α y y, with α γ x = aθ σ 2 x γ Aa σ 2 x + γ a σ, 2 and α y = γ a γ aθ σ 2 y γ a + γ aa γ Aa σ 2 x + γ a σ. 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: a i A θ N µ a µ A µ, σ 2 a ρ aa σ a σ A ρ aθ σ a σ θ ρ aa σ a σ A σ 2 A ρ Aθ σ A σ θ ρ aθ σ a σ θ ρ Aθ σ A σ θ σ 2 θ σ 2 A is the aggregate volatility (common variation) σ 2 a σ 2 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: µ A = µ a, σ 2 A = ρ aσ 2 a, ρ aa σ a σ A = ρ a σ 2 a where ρ a is the correlation coeffi cient across individual actions the first and second moments of the correlated equilibria are: a i µ a σ 2 a ρ a σ 2 a ρ aθ σ a σ θ A N µ a, ρ a σ 2 a ρ a σ 2 a ρ aθ σ a σ θ θ µ ρ aθ σ a σ θ ρ aθ σ a σ θ σ 2 θ correlated equilibria are characterized by: {µ a, σ a, ρ a, ρ aθ }
Equilibrium Analysis in the complete information game, the best response is: a = θ γ aθ γ a A γ Aa γ a best response is weighted linear combination of fundamental θ and average action A relative to the cost of action: γ aθ /γ a, γ Aa /γ a 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: E µ [θ a], E µ [A a]
Equilibrium Conditions in the incomplete information game, the best response is: a = E µ [θ a] γ aθ γ a E µ [A a] γ Aa γ a 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 : ( E µ [a] = E µ [ E µ [θ a] γ aθ + E µ [A a] γ )] Aa γ a γ a by the law of iterated expectation, or law of total expectation: E µ [E µ [θ a]] = µ θ, E µ [E µ [A a]] = E µ [A] = E µ [a],
Equilibrium Moments: Mean the best response property implies that for all µ (a, θ) : ( E µ [a] = E µ [ E µ [θ a] γ aθ + E µ [A a] γ )] Aa γ a γ a or by the law of iterated expectation: µ a = µ θ γ aθ γ a µ a γ Aa γ a Theorem (First Moment) In all Bayes correlated equilibria, the mean action is given by: γ aθ E [a] = µ θ. γ a + γ aa 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] γ aθ + E [A a] γ ) Aa, a supp µ (a, θ) γ a γ a or varying in a ( E [θ a] 1 = a γ aθ γ a + E [A a] a ) γ Aa, γ a 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 Σ a,a,θ forms a multivariate distribution, meaning that the variance-covariance matrix has to be positive definite we need to determine: {σ a, ρ a, ρ aθ } Theorem (Second Moment) The triple (σ a, ρ a, ρ aθ ) forms a Bayes correlated equilibrium iff: ρ a ρ 2 aθ 0, and σ a = σ θγ aθ ρ aθ ρ a γ Aa + γ a.
Moment Restrictions: Correlation Coeffi cients the equilibrium set is characterized by inequality ρ a ρ 2 aθ 0 ρ a : correlation of actions across agents; ρ aθ : correlation of actions and fundamental a 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Set of correlated equilibria. a
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, σ 2 ) y 2 every agent i observes a private signal x i about θ : x i N ( θ, σ 2 ) x
Equivalence between BCE and BNE bivariate information structure which generates volatility (common signal) and dispersion (idiosyncratic signal) Theorem There ( is BCE with (ρ a, ρ aθ ) if and only if there is a BNE with σ 2 x, σ 2 ) y. a public and a private signal are suffi cient 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 : and a private signal x i y = θ + ε x i = θ + ε i the given information of the agents is described by: ( ) (( ) ( )) ε 0 σ 2 N, y 0 ε i 0 0 σ 2 x
Information bounds and correlated equilibrium the equilibrium conditions are augmented by for all a, x, y : a = 1 γ a (γ aθ E [θ a, x, y ] + γ Aa E [A a, x, y ]), a, x, y. we determine σ a, ρ ax, ρ ay in terms of ρ a, ρ aθ, e.g.: ρ ay = σ ( ) θ γa + ρ a γ Aa ρ 2 aθ σ y ρ aθ γ a + γ Aa set of correlated equilibria is given by the inequalities: ρ a ρ 2 aθ ρ2 ay 0, 1 ρ a ρ ax 0,
Given Public Information describe the equilibrium set C (σ) [0, 1] 2 in terms of the noise of the signal pair σ = (σ x, σ y ) 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 2 a 1.0 0.8 0.6 1 0.4.5.2 0.2.1.001.01 0.2 0.4 0.6 0.8 1.0 Correlated Equilibria of beauty contest with minimal precsion of y 2 and r=.25 a
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 = (σ x, σ y ) movements along level curve are variations in σ 2 y given σ 2 x 2 a 1.0 0.8 45 o 0.6 1 0.4.5 0.2.2.1.01 0.2 0.4 0.6 0.8 1.0 a Correlated Equilibria of beauty contest with minimal precsion of x 2 and r=.25
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 a 1.0 0.8 0.6 0.4 x 2.5 y 2.5 0.2 x 2.1 y 2.1
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 = (σ s, σ i ) Theorem (Information Bounds) 1 For all σ < σ, C (σ s ) C (σ s) ; 2 For all σ < σ : 3 For all σ < σ : min ρ a > min ρ a; ρ a C (σ) ρ a C (σ ) min ρ aθ > min ρ aθ. ρ aθ C (σ) ρ a C (σ ) more private information shrinks the equilibrium set and makes predictions sharper
Identification 1 Predictions: What restrictions are imposed by the structural model (u, ψ) on the observable endogenous statistics (µ a, σ a, ρ a, ρ aθ )? 2 Identification: What restrictions can be imposed/inferred on the parameters Γ of the structural model by the observations of the endogenous variables (µ a, σ a, ρ a, ρ aθ )?
Identification with Complete Information classic problem of supply and demand identification: demand is given by: supply is given by: P d = d 0 + d 1 Q + d 2 θ d P s = s 0 + s 1 Q + s 2 θ s the exogenous random variables are θ d and θ s are demand and supply shocks ( demand, supply shifters ) complete information: each firm observes shocks (θ d, θ s ) and makes supply decisions accordingly the econometrician uses instrumental variable (z d, z s ): z d = θ d + ν d, z s = θ s + ν s to estimate and identify (d 1, s 1 ); see Wright (1928), Koopmans (1949), Fisher (1966),...
Incomplete Information incomplete information: each firm observes a vector of signals: y i = (y s, y si, y d, y di ) regarding the true cost and demand shocks: y s = θ s + ε s, y si = θ s + ε si, y d = θ d + ε d, y di = θ d + ε di before its supply decision maintain normality and independence of (θ d, θ s, ε d, ε id, ε s, ε is ) the variance of the random variables: I = ( σ 2 d, σ2 s, λ 2 d, λ2 id, λ2 s, λ 2 ) is represents the information structure I in the economy
Competitive Equilibrium with Incomplete Information in Bayes Nash equilibrium of competitive economy, each firm supplies q i (y i ) on the basis of its private (but noisy) information the Bayes Nash equilibrium price is given by the equilibrium condition: P d = d 0 + d 1 q i (y i ) di + d 2 θ d with respect to the realized demand shock θ d
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 { λ 2 s, λ 2 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: (µ a, σ a, ρ a ) 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: µ α = µ θ γ aθ γ a + γ aa and variance σ a = σ θγ aθ ρ aθ ρ a γ Aa + γ a.
Sign Identification relative to supply and demand identification, less observable data and restrict attention to identify sign of interaction recall:γ aθ informational externality, γ aa strategic externality Theorem (Sign Identification) The Bayes Nash Equilibrium identifies the sign of γ aθ and γ aa. identification in Bayes Nash equilibrium uses variance-covariance given information structure ( σ 2 x, σ 2 ) y Theorem (Partial Sign Identification) The Bayes Correlated Equilibrium identifies the sign of γ aθ but it does not identify the sign of γ aa. 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: u i (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 of 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 of private information by β : α = 1 σ 2, β = 1 s σ 2 i the social value of information in the beauty contest is: α + β (1 r)2 W (α, β) = [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 + β (1 r) 2 [ α α,β β 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)