The Empirical Core of the Multicommodity Flow Game Without Side Payments

Abstract

Policy makers focus on stable strategies as the ones adopted by rational players. If there are many stable solutions, however, an important secondary question is how to select amongst them. We study this question for the multicommodity flow coalition game, introduced by Papadimitriou to model incentives and cooperation between autonomous systems in the Internet. In short, the strategies of the game are flows in a capacitated network (the supply graph). The payoff to any node is the total flow which it terminates. Markakis-Saberi show that this game is balanced and hence has a non-empty core by Scarf's Theorem. In the transferable utility (TU) version this also leads to a polynomial-time algorithm to find core elements but for the application to autonomous systems, side payments are not natural. Finding core elements in NTU games, however, tends to be computationally much more difficult, cf. [CS06]. Even for this multiflow game, the only previous result is due to Yamada and Karasawa who give a procedure to find a core element when the supply graph is a path. We extend their work by designing an algorithm, called incorporate, which produces many different core elements.We use our algorithm to evaluate several specific instances by running incorporate to generate multiple core vectors. We call these the empirical core of the 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 across fairness in the empirical core; the fairness values tend to be worse than the optimal fairness LP value LP f air . 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 λ ∈ (0, 1), one can produce an approximate core vector with fairness (resp. social welfare) at least λLP f air (resp. (1 − λ)LP sw ).