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 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 i ) I T = 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 − i ψ ( θ ) π [ t ] ( θ ) � θ � � � θ � � � θ � � π i [ θ ] ( t i ) = t − i ψ π [ t ] • t i ∈ T i encodes information about types of other agents - "higher order beliefs": � θ ψ ( θ ) π [ t i , t − i ] ( θ ) � � � � π i [ t − i ] ( t i ) = t i , t � − i ψ ( θ ) π ( θ ) t � θ − i

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 : � � θ, σ 2 s ∼ N s • every agent i may observe a private signal s i centered around the state of the world θ : � � θ, σ 2 s i ∼ N i • every agent i may observe a private signal s i , j about the signal of agent j : � � s j , σ 2 s i , j ∼ N i , j • every agent i may observe a private signal s i , j , k about ....: � � s j , k , σ 2 s i , j , k ∼ N 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 � ≥ u i (( a i , σ − i ( t − i )) , θ ) ψ ( θ ) π [ t i , t − i ] ( θ ) . 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 , � I � � � µ σ ( a , θ ) = ψ ( θ ) π [ t ] ( θ ) σ i ( a i | t i ) t i = 1 • equilibrium distribution µ σ ( a , θ ) is specified without reference to information structure T which gives rise to µ σ ( 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 = 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 � ≥ u i (( a i , σ − i ( t − i )) , θ ) ψ ( θ ) π [ t i , t − i ] ( θ ) . t − i ,θ • with the equilibrium distribution � I � � � µ σ ( a , θ ) = ψ ( θ ) π [ t ] ( θ ) σ i ( a i | t i ) t i = 1 • an implication of BNE of ( u , ψ, T ) : for all a i ∈ supp µ σ ( a , θ ) : � � �� � � a � u i (( a i , a − i ) , θ ) µ σ ( a , θ ) ≥ u i i , a − i , θ µ σ ( a , θ ) ; a − i ,θ a − i ,θ