Flows and Decompositions of Games: Harmonic and Potential Games

In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to "nearby" games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games

Differential information in large games with strategic complementarities

We study equilibrium in large games of strategic complementarities (GSC) with differential information. We define an appropriate notion of distributional Bayesian Nash equilibrium and prove its existence. Furthermore, we characterize order-theoretic properties of the equilibrium set, provide monotone comparative statics for ordered perturbations of the space of games, and provide explicit algorithms for computing extremal equilibria. We complement the paper with new results on the existence of Bayesian Nash equilibrium in the sense of Balder and Rustichini (J Econ Theory 62(2):385–393, 1994) or Kim and Yannelis (J Econ Theory 77(2):330–353, 1997) for large GSC and provide an analogous characterization of the equilibrium set as in the case of distributional Bayesian Nash equilibrium. Finally, we apply our results to riot games, beauty contests, and common value auctions. In all cases, standard existence and comparative statics tools in the theory of supermodular games for finite numbers of agents do not apply in general, and new constructions are required

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Food scares in an uncertain world

This is the accepted version of the following article: Food scares in an uncertain world. Journal of the European Economic Association, Volume 11, Issue 6, pages 1432–1456, December 2013, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/jeea.12057/abstrac

Food scares in an uncertain world

This is the accepted version of the following article: Food scares in an uncertain world. Journal of the European Economic Association, Volume 11, Issue 6, pages 1432–1456, December 2013, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/jeea.12057/abstrac

Supermodular Functions on Finite Lattices

The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43343/1/11083_2005_Article_9026.pd

Product–process matrix and complementarity approach

The relationship between different types of innovation is analysed from three different approaches. On the one hand, the distinctive view assumes that the determinants of each type of innovation are different and therefore there is no relationship between them. On the other hand, the integrative view considers that the different types of innovation are complementary. Finally, the product–process matrix framework suggests that the relationship between product innovation and process innovation is substitutive. Using data from Spain belonging to the Technological Innovation Panel (PITEC) for the years 2008, 2009, 2010, 2011 and 2012, we tested which of the three approaches is predominant. To perform the hypothesis test, we used the so-called complementarity approach. We find that there is no unique relation. The nature of the relationship depends on the types of innovation that interact. Our most significant finding is that the relationship between product innovation and process innovation is complementary. This finding contradicts the proposal of the product–process matrix framework. Consequently, the joint implementation of both types of innovation generates a greater impact on the performance of a company than the sum of their separate implementation