Notice that in a market s.t. all agents have unit demand valuation functions- if there is an allocation (not necessarily and a matching) that yields SW of value v, there is also a

I think one should assume that there are no outgoing edges in G_I from vertix j if j was not sold under OPT.

This makes the graph well defined and makes it possible to prove the theorem.

Is this an acceptable assumption ?

]]>"For each pair of items j, j', let the weight of the edge (j, j') in G_I be v_i({j}) − v_i({j'}),

where i is the agent that was matched by M to j."

What happens if j was not matched to any agent in M ?

This case isn't defined explictly.

Can one assume that it means j has no outgoing edges ?

Thanks.

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