# What is the subgame perfect pricing policy for seller?

Suppose there is a seller S who is selling to 3 different potential customers H, M, L

The good is a durable good and it's utility in terms of dollar equivalent is as in the picture below: So if H buys in period 1 it's total payoff will be 1500+700-p1 where p1 is the price paid to seller in period 1.

If any buyer buys in 1st period they will be out of the game for the second period.

So if H buys in period 2 it's total payoff will be 700-p2 where p2 is the price paid to seller in period 2.

If any customer does not buy they get a payoff of 0.

The game starts with S announcing the price p1 the H, M and L simultaneously and independently decide whether to buy or not. Whoever buys in period 1 is left out in period 2 and all the players are observable to everyone involved. Then S announce price in p2.

Given this what will be the subgame perfect pricing policy for seller S and strategies for all players involved.

My attempt:

Below image shows the possible pay off matrix for each period. From all three possibilites we can see that H will always buy in period 1. Given that H will always buy in period 1 the best possible case is Case 2 for S which maximizes the payoff of S to 2697. Therefore the subgame perfect pricing policy is set p1 = 1199 and p2 = 299 To find the SPNE you want to consider all possible scenarios that the seller might face in the second period, and what his best strategy would be. This will help you study the seller's actions in the first period since they will lead to different scenarios in the second one. In fact, saying "the subgame perfect pricing policy is set p1 = 1199 and p2 = 299" is not a valid statement, the subgame perfect policy should be of the form: " the seller chooses p1=number in the first period and p2= a function of which players remain in the market"

Let me be more clear with what I mean:

The seller might find in t=2 that no one has bought the good, and must decide between choosing p2=300 and selling three units (revenue of 900), choosing p2=400 and selling 2 units (revenue of 800) or choosing p2=700 and selling one unit (revenue of 700). Clearly, in this case, the best pricing policy is 300.

Similarly, the seller might find himself in the situation that one player has bought the good in the first period. Say player M bought the good in period 1. In that case, it is better to set p2=700 and sell 1 unit than p2=300 and sell 2 units. (Notice that this case is not very relevant because if M finds it profitable to buy in period 1 so will H, but it is a possibility you must consider.)

It is not hard to see that unless buyers L and H are the only ones in the market in period 2, the seller will find optimal to choose p2 to be equal to the smallest valuation among the players still in the market, and every player that has not bought the good, buys it.

Given this strategy, you can now study the choice of p1. For example: If the seller sets p1=2200 no one will buy in period 1. This is because buyer M will rather wait to period 2, anticipating that no one will buy in period 1, thus the seller will drop the price to 300. Therefore, she will be able to make a surplus of 700-300=400. We learn that for buyer H to be willing to buy in the first period, she must get a surplus of at least 400. Therefore, the maximum that the seller can charge buyer H for the good in period 1 is 2200-400=1800. To sum up, if p1=1800, H buys in t=1 and the other two buy at t=2 at p2=300, giving a total revenue of 1800+600=2400.

Through a similar argument, you can find that if the seller wants buyer M to also buy at t=1, the largest price he could charge this buyer is p1=1500 (otherwise, buyer M would prefer to wait and have a surplus of 100 at t=2). Now let's compare the total revenue of p1=1800 vs p1=1500. If p1=1500, M and H buy at t=1 and L buys at t=2. The total revenues are 1500+1500+300=3,300, which is better than 2,400.

Lastly, would it make sense to try to sell to buyer L? in that case, the biggest price you can charge in t=1 so that L buys is p1=900. Since anyway L cannot guarantee himself any surplus. However if p1=900, all buyers buy at t=1 and you make a total revenue of 2,700.

We conclude that the subgame perfect pricing policy is to set p1=1500, and p2 equal to the smallest valuation among the buyers still in the market, except if only buyers L and H are in the market. In that case, choose p2=700. (Note that expressing the equilibrium strategy of the buyers will also be a bit wordy, but I think the question is interested only in the seller's pricing policy, so this is sufficient).

It might sound redundant to express p2 the way I did, but it is crucial in order to show that no player will have an incentive to deviate.