# How could define the certainty equivalent in a Bayesian Persuasion model?

For once again I will start describing the Kamenica and Gentzkow Bayesian persuasion model.

Suppose that $$\Theta$$ is a finite set of states and $$\theta$$ is the element of the state set. To simplify the model, we assume that $$\Theta = \{\theta_1 = G, \theta_2 = B \}$$. A sender's signal is represented by $$(S,\pi)$$, where $$S$$ is the set of realized signals and $$\pi:\Theta\to\Delta(\Omega)$$ is the probability distribution over the set or realized signals and $$s$$ stands for the typical element of $$S$$.

A distribution of posteriors is denoted by $$\tau:\underbrace{\{\mu_s\quad\text{s.t. for all s\in S with \pi(s)>0, \mu_s is the induced posterior.}\}}_{\text{Supp}(\tau)} \to \Delta(\Delta(\Theta))$$ and has finite support. Given a signal $$\pi$$, any signal realization $$s$$ induces a posterior belief $$\mu_s(\theta) \triangleq \mu(\theta|s)$$. Each signal $$\pi$$ leads to a distribution over posterior beliefs, namely each $$\pi$$ induces $$\tau$$ if $$\text{Supp}(\tau)=\{\mu_s\}_{s\in \pi}$$ and we write $$\tau = <\pi>$$. Therefore, observing a signal realization $$s$$ with probability $$\mathbb{P}(s)>0$$ generates a unique posterior belief

$$$$\mu_s(\theta) = \frac{\pi(s|\theta)\mu_0(\theta)}{\sum_{\theta^{'}\in\Theta}\pi(s|\theta^{'})\mu_0(\theta^{'})},\quad\text{for all s and \theta} \tag{1}$$$$

where $$\pi(s) = \sum_{\theta^{'}\in\Theta}\pi(s|\theta^{'})\mu_0(\theta^{'})$$ is the marginal probability of $$s$$ and the distribution of posterior beliefs is

$$$$\tau(\mu) \triangleq \sum_{\{s\in \pi : \mu_s = \mu\}}\pi(s),\quad\text{for all \mu} \tag{2}$$$$

A distribution of posterior beliefs is $$\textit{Bayes plausible}$$ if the best projection about the posterior beliefs, given the prior distribution of beliefs, equals the prior beliefs, or in other words the beliefs satisfy the martingale property.

$$\mathbb{E}_{\tau}(\mu_s|\mu_0)=\sum_{\text{Supp}(\tau)}\mu_s\tau(\mu) =\mu_0 \tag{3}$$

The sender's utility is denoted by $$v_1(\alpha, \theta)$$ and the receiver's utility is denoted by $$v_2(\alpha, \theta)$$ where $$\alpha$$ denotes the action of the sender and $$\theta$$ the state of the world. Receiver forms the posterior belief $$\mu_s$$ using Bayes rule and then she takes an action that is $$\alpha^*(\mu_s)= argmax_{\alpha\in A}\mathbb{E}_{\mu_s}v_2(\alpha,\theta)$$.

They assume that they exist at least two actions and for every action $$\alpha$$, there exists a $$\mu$$ s.t. $$\alpha\in \alpha^*(\mu)$$ and the receiver's equilibrium outcome is denoted by $$\hat{\alpha}(\mu)$$. Any signal $$s$$ induces a posterior belief $$\mu_s$$, such that

$$\mathbb{E}_{\tau}(\mu_s(\theta)|\mu_0) = \sum_{\text{Supp}(\tau)}\tau(\mu)\mu_s(\theta)= \sum_{s\in \pi: \mu=\mu_s}\pi(s) \frac{\pi(s|\theta)\mu_0(\theta)}{\sum_{\theta^{'}\in\Theta}\pi(s|\theta^{'})\mu_0(\theta^{'})} =\mu_0(\theta)\underbrace{\sum_{s\in \pi: \mu=\mu_s} \pi(s|\theta)}_{=1}=\mu_0(\theta)$$

And hence

$$\mathbb{E}_{\tau}(\mu_s(\theta)|\mu_0) = \sum_{\text{Supp}(\tau)}\tau(\mu)\mu_s(\theta) = \mu_0(\theta) \tag{4}$$

Now by going to $$(1)$$, we have that

$$v_s(\mu_s)=\mu_s(G) u_s(\hat{\alpha}(\mu_s(G)), G)+\underbrace{(1-\mu_s(G))}_{\mu_s(B)}u_s(\hat{\alpha}(\mu_s(B)), B)\tag{5}$$

Therefore we end up with $$(5)$$ where the utility of the sender depends only on the posterior beliefs $$\mu_s=(\mu_s(G),\mu_s(B))$$ and the state $$\theta$$. However, by applying the expectations of the distribution of posterior beliefs on the latter equation, we do not longer have any concern about $$\theta$$ and thus

$$\mathbb{E}_\tau v_s(\mu_s) =\mathbb{E}_{\tau}\left(\sum_{\theta\in\Theta}\mu_s(\theta) u_s(\hat{\alpha}(\mu_s(\theta)), \theta)\right) = \sum_{\text{Supp}(\tau)}\tau(\mu)\sum_{\theta\in\Theta}\mu_s(\theta) u_s(\hat{\alpha}(\mu_s(\theta),\theta)$$

By setting $$\hat{V}(\mu_s)= \sum_{\theta\in\Theta}\mu_s(\theta) u_s(\hat{\alpha}(\mu_s(\theta),\theta)$$ the problem of the sender reduces to the following

$$\tau^*\in \text{argmax}\left(\mathbb{E}_{\tau}\left(\hat{V}(\mu_s)\right)\right) \tag{*}$$

$$\text{such that \sum_{\text{Supp}(\tau)}\tau(\mu)\mu_s(\theta) = \mu_0(\theta)}\tag{**}$$

and the sender's problem reduces to $$(*)$$ and $$(**)$$

My questions is the following

• Taking into account the above, how could someone define the certainty equivalent in such a setting in ordrer to measure the expected utility gain of the receiver and the sender expliciterly? Kamenica and Gentzkow show geometrically what are the gains of the sender, but could we show that with another qualitative criterio as the certainty equivalent?

• I believe that we could use the notion of gambles (or lotteries) as there is uncertaity in the model and define the V-NM utility and eventually the certainty equivalent of a gamble and hence we could somehow measure it. Does anyone have a clue about how to move towards this direction?