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Sorry in advance for asking too many questions. I am just seeking for some minor help and check if what I'm thinking is acceptable.

A customer R takes his car to the mechanic S. The mechanic can perfectly see what needs to be fixed; the customer only knows the prior probabilities. There are three possible states of the world: The car only needs new oil (probability 0.25), the car needs a new gear (probability 0.5), or the car needs a whole new motor (probability 0.25). There are three possible messages: $\{o, g, m\}$. After observing the state of the world, S chooses which message to send to R, then R observes the message and decides which part of the car should be replaced. The following table gives the payoffs of S and R for different states of the world and different actions. (Rows correspond to states of the world and columns to decisions of R. The first payoff in every cell is that of S.)

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  • Find a PBE in which no information is transmitted and write down the complete strategies of S and R in this equilibrium

Now in cheap talk we usually don't have fully separating equilibrium, as players tend to deviate. In this case for example, despite needing only oil, the mechanic will tend to deviate to changing gear as it has a payoff of 2. Thus, since the interests of the mechanic and customer are not in alignment, the mechanic has an incentive to lie, and the message will not be useful to the customer. So the customer will choose based on the highest expected utility:

$EU_2[Oil]=0.25*2=0.5$

$EU_2[Gear]=0.25*1+0.5*1=0.75$

$EU_2[Motor]=0.25*1=0.25$

which yields 'change gear' as the PBE of this game

  • Find a PBE in which R’s action depends on S’s message and describe it (just a statement). Are there any out-of-equilibrium beliefs? And what will be the posterior beliefs of R after receiving this message?

In this case, the customers best response is to choose 'change gear' upon receiving a signal of 'change gear'. When receiving a signal of change oil or change motor, the customer knows that it is not needed, and maximizes it's expected value

$EU_2[Oil]=0.25*2=0.5$

$EU_2[Gear]=0.25*1=0.25$

$EU_2[Motor]=0.25*1=0.25$

and thus, the customer is indifferent between changing gear and motor. But this is not making that much sense to me?

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    $\begingroup$ A PBE in this game consists of a strategy by S (i.e. a message $\mu$ as a function of the state) and a strategy by R (i.e. an action $a$ as a function of $\mu$) such that $a$ is a best response for R to a posterior belief induced by $\mu$ and $\mu$ is a best response for S to $a$. You need to be explicit about each of the three elements ($\mu$, $a$, and R's posterior belief) in your description of the PBE. $\endgroup$
    – Herr K.
    Sep 7 at 20:51
  • $\begingroup$ Isn't, in this case, the strategy for S just s={change oil, change gear, change motor}? $\endgroup$ Sep 7 at 21:12
  • $\begingroup$ No. S's strategy is a function of state, as it is conceivable that S does not convey the observed state, e.g. $\mu(\text{needs oil})=\text{needs gear}$. $\endgroup$
    – Herr K.
    Sep 7 at 21:48
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    $\begingroup$ After some more thinking I reached into this: for the first one we are in a babbling equilibrium where the sender sends messages randomly. For the second one we have {oil, gear}-> S send and R change gear. {Motor}-> S sends M and R changed motor. The o-o-e belief when S sends O, R believes it needs oil since if it needed gear S would have never sent oil. And after receiving g, R belief on needing oil is $\frac{0.25}{0.25+0.5}=1/3$. After m, R's belief that it needs motor is 1. $\endgroup$ Sep 8 at 8:29
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Responding to OP's comment

For the babbling equilibrium, it's important to note that how S randomizes over his messages matters. In particular, the messages should be randomized in the same way across all states. For example, if S sends messages $\{o, g, m\}$ with probabilities $(p,q,r)$ (where $p,q,r\in(0,1)$ and $p+q+r=1$) when the state is $o$, he needs to randomize over the messages using that same probability distribution in the other two states as well. Otherwise, the posterior will generally not be the same as the prior. Since $p,q,r\in(0,1)$, posterior beliefs are uniquely pinned down by Bayes' rule. In this case, they are the same as the prior. R acts based on her prior and chooses $g$. [I would also note that the distribution need not have full support (i.e. it's okay to let $p,q,r\in[0,1]$ s.t. $p+q+r=1$), as long as it is the same across states. But in this case, care needs to be taken to ensure that off-equilibrium beliefs are the same as the prior.]

For the partially informative equilibrium, your description is almost complete. You still need to specify an action after receiving message $o$ (which should be straightforward in this case). Otherwise there is no problem here.

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