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P −1 ) could not be further extended we know that nodes in the subpath (k, k − 1, . . , 1) were not the final node of some earlier path, while nodes in (k − 1, . . , 1, 0) were not the first node of some earlier path. This implies nk (˜s) ≤ nk (s), n0 (˜s) ≥ n0 (s), and ni (˜s) = ni (s) for all i ∈ {1, . . , k − 1}. We derive: ΔP (Φ) = f0 (n0 (˜s) + 1) − fk (nk (˜s)) ≤ f0 (n0 (s) + 1) − fk (nk (s)) = f1 (n1 (s)) + δ1 − fk (nk (s)) ≤ ΨF · Φ1,P (s) + f1 (n1 (s) + 1) + δ1 − fk (nk (s)) 2 = ΨF · Φ1,P (s) + f2 (n2 (s)) + δj − fk (nk (s)) j=1 36 M.

This completes the construction of G. Path-Cycle Decomposition. We now decompose G into directed cycles and paths as follows: – While there is a directed cycle C in G, remove C from G. } be the set of removed cycles. , one which cannot be augmented further, and remove it from G . Let P = {P1 , P2 , . } be the ordered set of removed paths. We call (C, P) the ordered path-cycle decomposition of G. By construction, reassigning all sub-players a ∈ A according to the path-cycle decomposition (C, P) has the same effect on the potential as reassigning all players i ∈ N from s to s∗ .

For each resource e ∈ E denote me the number of times e is a start node of some path in P. Observe, that me = max{0, ne (s) − ne (s∗ )}. ne fe (j) e −me +1 To simplify notation let ne := ne (s) and define αe = j=n , for me ·fe (ne ) each resource e ∈ E if me > 0 and αe = 1 if me = 0. If B ∈ P is a path, e is the start node of B then Φe,B (s) = αe · fe (ne ) − 1 − αe · fe (ne ) = ΨF − 1 ΨF · αe − 1 ΨF − 1 · fe (ne ) (4) In all other cases this contribution is Φe,B (s) = ne −me j=1 fe (j) n e − me + me 1 − αe · · fe (ne ).

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