  Policy makers focus on stable strategies as the ones adopted by rational
players. If there are many such solutions an important question is how to
select amongst them. We study this question for the Multicommodity Flow
Coalition Game, used to model cooperation between autonomous systems (AS) in
the Internet. In short, strategies are flows in a capacitated network. The
payoff to a node is the total flow which it terminates. Markakis-Saberi show
this game is balanced and hence has a non-empty core by Scarf's Theorem. In the
transferable utility (TU) version this gives a polynomial-time algorithm to
find core elements, but for ASs side payments are not natural. Finding core
elements in NTU games tends to be computationally more difficult. For this
game, the only previous result gives a procedure to find a core element when
the supply graph is a path. We extend their work with an algorithm,
Incorporate, which produces many different core elements.
  We use our algorithm to evaluate specific instances by generating many core
vectors. We call these the Empirical Core for a game. We find that sampled core
vectors are more consistent with respect to social welfare (SW) than for
fairness (minimum payout). For SW they tend to do as well as the optimal linear
program value $LP_{sw}$. In contrast, there is a larger range in fairness for
the empirical core; the fairness values tend to be worse than the optimal
fairness LP value $LP_{fair}$. We study this discrepancy in the setting of
general graphs with single-sink demands. In this setting we give an algorithm
which produces core vectors that simultaneously maximize SW and fairness. This
leads to the following bicriteria result for general games. Given any
core-producing algorithm and any $\lambda \in (0,1)$, one can produce an
approximate core vector with fairness (resp. social welfare) at least $\lambda
LP_{fair}$ (resp $(1-\lambda) LP_{sw}$).
