Is the Construction Price Method in Class the Best Method?

In one of the lectures this past week, Professor Immorlica went over a construction of prices method. In this method, the prices of an over demanded items were increased by one until there was a preferred seller graph. In one of the examples in class, the values were (12, 4, 2), (8, 7, 6), and (7, 5, 2) and the prices came out to be p1=3, p2=1, and p3=0. Though this might be a preferred seller graph, is this really that good equilibrium for the sellers? Person one's value for item one is 12 but is only paying 3; person two's value for item 2 is 7 but is only paying 1; and person three's value for item 3 is 2 but is only paying 0. This is a good scenario for the buyer because they are paying little but the sellers are not getting there full value.

One way sellers could get a better value is if everyone raises their value by 1. Since the preference on items is depended on the price of the item compared to the other items, if they all raise their price, none of the players will change their preference. Therefore, the sellers could get much larger amounts of money for their product.

So though the method we went over in class is good method for the buyer, the sellers are losing a lot of value.