Equilibrium Computation in Resource Allocation Games

We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either games and, thus, leads -- via our first algorithm -- to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd [2016].Comment: This version contains some typo corrections onl

When Nash Meets Stackelberg

Motivated by international energy trade between countries with profit-maximizing domestic producers, we analyze Nash games played among leaders of Stackelberg games (\NASP). We prove it is both $\Sigma^p_2$-hard to decide if the game has a pure-strategy (\PNE) or a mixed-strategy Nash equilibrium (\MNE). We then provide a finite algorithm that computes exact \MNEs for \NASPs when there is at least one, or returns a certificate if no \MNE exists. To enhance computational speed, we introduce an inner approximation hierarchy that increasingly grows the description of each Stackelberg leader feasible region. Furthermore, we extend the algorithmic framework to specifically retrieve a \PNE if one exists. Finally, we provide computational tests on a range of \NASPs instances inspired by international energy trades.Comment: 40 Pages and a computational appendix. Code is available on gitHu

In Congestion Games, Taxes Achieve Optimal Approximation

In this work, we consider the problem of minimising the social cost in atomic congestion games. For this problem, we provide tight computational lower bounds along with taxation mechanisms yielding polynomial time algorithms with optimal approximation. Perhaps surprisingly, our results show that indirect interventions, in the form of efficiently computed taxation mechanisms, yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the agents' actions. It follows that no other tractable approach geared at incentivizing desirable system behavior can improve upon this result, regardless of whether it is based on taxations, coordination mechanisms, information provision, or any other principle. In short: Judiciously chosen taxes achieve optimal approximation. Three technical contributions underpin this conclusion. First, we show that computing the minimum social cost is NP-hard to approximate within a given factor depending solely on the admissible resource costs. Second, we design a tractable taxation mechanism whose efficiency (price of anarchy) matches this hardness factor, and thus is worst-case optimal. As these results extend to coarse correlated equilibria, any no-regret algorithm inherits the same performances, allowing us to devise polynomial time algorithms with optimal approximation

Potential function minimizers of combinatorial congestion games: Efficiency and computation

We study the inefficiency and computation of pure Nash equilibria in unweighted congestion games, where the strategies of each player i are given implicitly by the binary vectors of a polytope Pi . Given these polytopes, a strategy profile naturally corresponds to an integral vector in the aggregation polytope PN = Σ i Pi . We identify two general properties of the aggregation polytope PN that are sufficient for our results to go through, namely the integer decomposition property (IDP) and the box-Totally dual integrality property (box-TDI). Intuitively, the IDP is needed to decompose a load profile in PN into a respective strategy profile of the players, and box-TDI ensures that the intersection of a polytope with an arbitrary integer box is an integral polytope. Examples of polytopal congestion games which satisfy IDP and box-TDI include common source network congestion games, symmetric totally unimodular congestion games, non-symmetric matroid congestion games and symmetric matroid intersection congestion games (in particular, r -Arborescences and strongly base-orderable matroids). Our main contributions for polytopal congestion games satisfying IDP and box-TDI are as follows: (1) We derive tight bounds on the price of stability for these games. .is extends a result of Fotakis (2010) on the price of stability for symmetric network congestion games to the larger class of polytopal congestion games. Our bounds improve upon the ones for general polynomial congestion games obtained by Christodoulou and Gairing (2016). (2) We show that pure Nash equilibria can be computed in strongly polynomial time for these games. To this aim, we generalize a recent aggregation/decomposition framework by Del Pia et al. (2017) for symmetric totally unimodular and non-symmetric matroid congestion games, both being a special case of our polytopal congestion games. (3) Finally, we generalize and extend results on the computation of strong equilibria in bo.leneck congestion games studied by Harks, Hoefer, Klimm and Skopalik (2013). In particular, we show that strong equilibria can be computed efficiently for symmetric totally unimodular bottleneck congestion games. In general, our results reveal that the combination of IDP and box-TDI gives rise to an effiicient approach to compute a pure Nash equilibrium whose inefficiency is be.er than in general congestion games