There is a buyer and a seller. The seller wants to sell a used scooter. The scooter can either be of good or bad quality. The quality of the scooter is only observed by the seller. To the seller, the scooter is worth $v_i$ and to the buyer it is worth $\frac65v_i$ with $i$ either good or bad and $v_g>v_b$. Note, neither the seller nor the buyer has an outside option. Assume that the seller has all the bargaining power.
Now, I am trying to study this bayesian game. My problem is how to define the set of actions of the seller. Does it exist an obvious way to specify the set of possible actions of the seller?
I have tried a formulation in which the seller can either sell at a price $p=\frac65v_i$ or sell at a price $p>\frac65v_i$. But if the buyer can only accept or not accept the deal, I can draw a strategic form of the game with just two Nash Equilibria. The fact that puzzles me is that in these NE the trade does not take place.
First of all (and more a technicality), if the game you are describing is a static game, then the action of the buyer can not be accept or not accept the deal as they are playing simultaneously and he has therefore no idea about the price that is offered to him. Therefore, you have to either have to change the structure (or rather define, as currently the game has no structure) of the game (e.g. first the seller $S$ offers the buyer $B$ a car for a certain price $p$ (which leads to the seller having all the bargaining power)) or change the possible actions for $B$.
Which leads to the main question: What possible actions do the players have (in the static game):
And that is (in short) every price at which the buyer is willing to pay for a car and the prices $p_{b}$ and $p_{g}$ for which the seller is willing to sell a good and a bad car for.
Note, that in general, it is also possible that the buyer is willing to buy the car at some price $p_1$ but not willing to buy the car at $p_2 < p_1$ because he can infer something about the quality of the car from the price.
Of cause it is possible to restrict the action space more (e.g. restrict $p$ to $v_b \leq p \leq \frac65 v_g$) but not as much as you tried to do.
Now some more comments/hints:
The game you are trying to solve is an example of Akerlof's - Market for lemons.
The dynamic and static game I described above should be pretty much
the same but solving the dynamic game is (imho) much easier.
The result may depend on the prior belief of $B$ about the quality.
It is possible that trade does not occur sometimes (depending on the quality).