# Deciding the winner in a Product-Mix auction

I am trying to understand how does Klemperer's Product-Mix auction work.

As I understand, given a supply I want to sell and a group of bidders, an auction system should determine who receives the products and at which price.

In a Product-Mix auction, there is a single auction to sell the entire supply and each bidder should bid for each bundle of articles simultaneously. Then if we consider the price space (the set of all possible prices of the articles, that is, the set $$\mathbb{R}^n$$ where $$n$$ is the number of kind of articles) each agent bid allow us to do some nice pictures in the price space and by analyzing this pictures we get the prices for which there is a distribution of the supply between the agents in such a way such that each agent its utility (a competitive equilibrium).

Nevertheless, there may be multiple different ways to distribute this articles maximizing the utility as above. So in practice, between all this different optimum ways to distribute the articles, how does we decide which one we pick? We just do it randomly?

As an example, suppose we have a house with three rooms, a big room, a medium room and a small room, and we want to rent the rooms between three agents $$A$$,$$B$$ and $$C$$. We then ask each agent to bid simultaneously for each room and they bid as follows:

$$\begin{array}{|c|c|c|c|c|} \hline \text{Agent} & \text{Big room} & \text{Medium room} & \text{Small room} & \text{No room} \\ \hline A & 400 & 300 & 200 & 0 \\ \hline B & 450 & 350 & 150 & 0 \\ \hline C & 400 & 350 & 250 & 0 \\ \hline \end{array}$$

Then for the price 400\$, 300\$ and 200\\$ there are two possible ways of distributing the supply. We can give the big room to $$A$$, the medium room to $$B$$ and the small room to $$C$$ or we can give the medium room to $$A$$, the big room to $$B$$ and the small room to $$C$$. How we decide between the two?