Bayes correlated equilibrium and inflationary bias
𝜎 ∶ 𝑇 × Θ → ∆ ( 𝐴 ) ( 𝟔 ).
‘ One way to mechanically understand the notion of the decision rule is to 𝜎 as the strategy of an omniscient mediator who first observes the realization of 𝜃 ∈ Θ chosen according to 𝜓 and the realization of 𝑡 ∈ 𝑇 chosen according to 𝜋 ( ∙ | 𝜃 ); and then picks an action profile 𝑎 ∈ 𝐴 , and privately announces to each player 𝑖 the draw of 𝑎 i . For players to have an incentive to follow the mediator’s recommendation in this scenario, it would have to be the case that the recommended action 𝑎 i was always preferred to any other action 𝑎 i ’ conditional on the signal 𝑡 i that player 𝑖 had received and his knowledge of the recommended action 𝑎 i . This is reflected in the following “ obedience ” condition. ’ 19
Definition 1 (obedience)
A BCE is defined to be a decision rule 𝜎 which is obedient: that is, one where no player has an incentive to deviate unilaterally from the recommended joint strategy, for any possible type they may be. In formal notation, decision rule 𝜎 is obedient (and a BCE) for game Γ = ( 𝐺 , 𝑆 ) if, for each 𝑖 = 1, … , 𝐼 , 𝑡 i ∈ 𝑇 i and 𝑎 i ∈ 𝐴 i , we have
for all 𝑎 i ’ ∈ 𝐴 i . The payoff of every player is higher should they follow the recommendation from the decision rule than by deviating unilaterally to any other possible actions. The definition of BCE requires obedience and nothing else.
Definition 2 (CE)
We note that if there is no uncertainty regarding the state of the world i.e. if Θ is a singleton, the definition reduces to Aumann ’s solution for a complete information game (Aumann 1987).
Proof 1
As discussed above, the BCE extends the concept of CE to scenarios of incomplete information, where private signals influence the strategic decisions of both senders and receivers. This aligns with how the CB could leverage private information to guide economic expectations and behaviours towards an equilibrium where no agent benefits from deviating unilaterally. For this to work, we require a mechanism, specifically, a mechanism whereby a sender can manipulate the flow of information to alter the receiver ’s actions beneficially. This was achieved by Kamenica & Gentzkow 2011.
19 Bergemann & Morris 2016.
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