Bayes correlated equilibrium and inflationary bias
correlated equilibrium (CE). The function that translates signals into actions could be represented as 𝜎 :Σ → 𝐴 CB × 𝐴 EA , modifying the probabilities associated with Γ .
Bayes correlated equilibrium and its extensions in economic modelling
Above we outlined the concept of CE as it relates to the coordinated decision-making process between EAs and the CB. CE effectively captures how public signals can guide strategic interactions, yet it assumes a static information environment that all players observe. However, economic decisions are often made under conditions of incomplete and asymmetric information. To account for this, we shift our analysis to the Bayes correlated equilibrium (BCE). A BCE is a solution concept for static games of incomplete information. 15 It is both a generalization of the CE perfect information solution concept to Bayesian games, and also a broader concept than the standard Bayesian Nash equilibrium theorem. BCE allows players to coordinate their actions by correlating their strategies in a way that maintains individual rationality for each possible type they may possess. In other words, each player, considering their unique information and possible types, finds it optimal to follow the strategy prescribed by the BCE rather than deviating unilaterally. This was initially proposed by Bergemann & Morris 2013.
Foundational Notation 16
Let 𝐼 be a set of players, 1, 2,… , 𝐼 , and we write 𝑖 for a typical player, and Θ a set of possible states of the world. A basic game is defined as tuple 𝐺 = ⟨ ( 𝐴 i , 𝑢 i ) i ∈ I , Θ, 𝜓⟩ , where 𝐴 i (for which we write as 𝐴 = 𝐴 1 × … × 𝐴 1 ) is the set of possible actions (with 𝐴 = ∏ i ∈ I 𝐴 i ) 17 and 𝑢 i : 𝐴 × Θ → ℝ is the utility function for each player, mapping the combination of action and states of the world to real numbers. The probability distribution over the states of the world is given by 𝜓 ∈ ∆ ++ ( Θ ), where ∆ ++ represents the set of all probability distributions and 𝜓 is a specific probability distribution. An information structure 𝑆 consists of a finite set of signals (or types) 𝑇 i each player can receive (with 𝑇 = ∏ i ∈ I 𝑇 i ), where we write 𝑇 = 𝑇 1 × … × 𝑇 1 and 𝜋 ∶ Θ → Δ ( 𝑇 ) is a signal distribution function, informing the probability 𝜋 ( 𝑡 | 𝜃 ) of observing the joint signal 𝑡 ∈ 𝑇 when the state of the world 𝜃 ∈ Θ . 18 Thus, By joining those two definitions, the basic game 𝐺 and the information structure 𝑆 i.e. Γ = ( 𝐺 , 𝑆 ) define a standard incomplete information game.
A decision rule in the incomplete information game ( 𝐺 , 𝑆 ) is a mapping 𝜎 ∶
15 CE applies when all players have the ‘common knowledge’ about the state of the world; each player receives a signal from a correlation device and these signals help coordinate their actions. BCE comes into play when knowledge regarding the state of the world is imperfect amongst players – players have private information that may not be known to others, and they receive signals that help them update their beliefs about the state of the world and other players’ information/actions. These signals are processed in a Bayesian manner, meaning players use them alongside their private information to update their strategies and maximise their expected utility based on these updated beliefs. 16 The notation that follows borrows from Bergemann & Morris 2016. 17 The set 𝐴 is the Cartesian product of all sets 𝐴 i for each 𝑖 that is an element of 𝐼 . 18 That is, the signal distribution function dictates the likelihood of observing a particular signal given the true state of the world.
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