MINGZHAO

We consider one third-party certifier and one product/service buyer in the market.
Product/service quality type $i \in \{L,H\}$.

Certifier type $\theta \in [0,1]$ is the accurate rate when the certifier gives a certification saying that the product is with high quality, i.e. $\theta = P(i = H)$.

The buyer evaluates the expected profit of buying a certified product $\pi_C = \theta P_H + (1-\theta) P_L$ given a prior belief of $\theta$ and the profit of outside option $\pi_0 = \text{max} \{ P_H - s \ , \ P_L \}$.
For a critical value $\bar{\theta}$, the two options are indifferent, i.e. $\pi_C = \pi_0$. We get

$\bar{\theta} = \frac{\pi_0 - P_L}{P_H - P_L}.$

Now consider that the buyer does not know the type of the certifier, and it plays the game and considers whether to buy certified products repeatedly. The time starts from $t=1$ and continues until infinity.
We consider the buyer has a prior belief of the type of the certifier at the beginning of each period $t$, denoted by $\hat{\theta}_t$. Each prior belief follows a certain Beta distribution. At $t=1$, the buyer has an initial belief distribution $\hat{\theta}_1 \sim{\text{Beta}(\alpha , \beta)}$, with $E(\hat{\theta}_1) = \frac{\alpha}{\alpha + \beta}$. After playing the game at each period, the buyer updates its belief of the certifier type according to Bayes Rule, and the new prior belief follows the distribution below:

$\hat{\theta}_t \sim{\text{Beta}(\alpha + N_t(d=H), \beta + N_t(d=L))},$

where $N(d=H)$ and $N(d=L)$ are respectively the times of receiving high and low quality product from $t=1$ to $t-1$, and $N_t(d=H) + N_t(d=L) = t-1$. Following this we have

$E(\hat{\theta}_t) = \frac{\alpha + N_t(d=H)}{\alpha + \beta +t -1}.$

Then the buyer will take the prior belief $\hat{\theta}_t$ to evaluate the expected profit of buying a certified product, which is $E(\pi_{C,t}) = E(\hat{\theta}_t) P_H + (1-E(\hat{\theta}_t)) P_L$. Therefore, the buyer will choose to buy certified product only if

$E(\hat{\theta}_t) \geq \bar{\theta}.$