I got very confused, so any help is appreciated. So, let's say there is a buyer that buys materials of two types - good quality and poor quality and 2 types of sellers (Poor quality and Good quality), both equally likely. The payoff matrix is a follows:
...........High Q job | Low Q job
High Q Seller 5, 2 | 0,0
Low Q Seller 3, -1 | 2,2
Sellers can send a message about their quality (P or G), So low Q sellers have an incentive to lie and send a message of G.
The question - what is the equilibrium? What I get is:
To make low Q seller indifferent:
f - how often should buyer trust the message
3*f+2*(1-f) = 2
f = 0,
so never trust the message! But with that payoffs for all (H seller, L seller and the buyer) are lower than if f = 2/3! So it shouldn't be an equilibrium.
You're right; the low-quality sellers will always send message "G" if there is no cost to sending that message. This is a cheap talk game, where all sellers have an incentive to report high-quality, regardless of their actual type. Because everyone sends the same message, the message has no informational content, and the buyer has no way of knowing the true type of the seller. Unfortunately, this leads to an inefficient outcome, as you have noticed - but it is still an equilibrium.
One way to resolve this problem is by assigning a cost to sending message "G", where that cost is higher for Low-Q sellers than for High-Q sellers. For some cost functions, there will be an equilibrium where High-Q sellers send message "G", Low-Q sellers send message "P", and buyers can trust the message.