77 research outputs found
Abstract Interpretation of Supermodular Games
Supermodular games find significant applications in a variety of models,
especially in operations research and economic applications of noncooperative
game theory, and feature pure strategy Nash equilibria characterized as fixed
points of multivalued functions on complete lattices. Pure strategy Nash
equilibria of supermodular games are here approximated by resorting to the
theory of abstract interpretation, a well established and known framework used
for designing static analyses of programming languages. This is obtained by
extending the theory of abstract interpretation in order to handle
approximations of multivalued functions and by providing some methods for
abstracting supermodular games, in order to obtain approximate Nash equilibria
which are shown to be correct within the abstract interpretation framework
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):385â393, 1994) or Kim and Yannelis (J Econ Theory 77(2):330â353, 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
Productâprocess 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 productâprocess 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 productâprocess 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
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