In many games, players’ decisions consist of multiple subdecisions, and hence can give rise to an exponential number of pure strategies. However, this set of pure strategies is often structured, allowing it to be represented compactly, as in network congestion games, security games, and extensive form games. Reduction to the standard normal form generally introduces exponential blow-up in the s...

In this paper, we first extend the celebrated PageRank modification to a higher-order Markov chain. Although this system has attractive theoretical properties, it is computationally intractable for many interesting problems. We next study a computationally tractable approximation to the higher-order PageRank vector that involves a system of polynomial equations called multilinear PageRank. This...

Proposition 1.1. α1, . . . , αn forms a basis for V ∗ (called the dual basis). In particular, this shows that V and V ∗ are vector spaces of the same dimension. However, there is no natural way to choose an isomorphism between them, unless we pick some additional structure on V (such as a basis or a nondegenerate bilinear form). On the other hand, we can construct an isomorphism ψ from V to (V ...

This paper contains the material covered in a minicourse given by the author at the Centre de Recerca Matemàtica in Barcelona during the period May 4–9, 2009. The course was an expanded version of a series of three lectures delivered by the author two weeks earlier (April 23–25) at the New Mexico Analysis Seminar held at the University of New Mexico in Albuquerque. The author would like to than...

Let f : {−1, 1}n → [−1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L1 influence of f can also be bounded as a function of d. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of O(d) for general functions and O(d) for homogeneous functions. We improve on ...

This project consists of a rambling introduction to some basic notions in multilinear algebra. The central purpose will be to show that the div, grad, and curl operators from vector calculus can all be thought of as special cases of a single operator on differential forms. Not all of the material presented here is required to achieve that goal. However, all of these ideas are likely to be usefu...

Joux [10] presented a one round protocol for tripartitie key agreement and Al-Riyami et.al. [15] developed a number of tripartitie, one round, authenticated protocols related to MTI and MQV protocols. Recently, Boneh and Silverleg [4] studied multilinear forms, which provides a one round multi-party key agreement protocol. In this paper, we propose (n + 1) types of one round authenticated multi...

In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal leastsquares sense, by a tensor that has prespecified column rank value, row rank value, etc. For matrices, the solution is conceptually obtained by truncation of the singular value decomposition (SVD); however, this...