Three-player impartial games

Past efforts to classify impartial three-player combinatorial games (the theories of Li and Straffin) have made various restrictive assumptions about the rationality of one's opponents and the formation and behavior of coalitions. One may instead adopt an agnostic attitude towards such issues, and seek only to understand in what circumstances one player has a winning strategy against the combined forces of the other two. By limiting ourselves to this more modest theoretical objective, and by regarding two games as being equivalent if they are interchangeable in all disjunctive sums,as far as single-player winnability is concerned, we can obtain an interesting analogue of Grundy values for three-player impartial games.Comment: 8 pages, 10 tables; to appear in Theoretical Computer Scienc

Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding

Information Security as Strategic (In)effectivity

Security of information flow is commonly understood as preventing any information leakage, regardless of how grave or harmless consequences the leakage can have. In this work, we suggest that information security is not a goal in itself, but rather a means of preventing potential attackers from compromising the correct behavior of the system. To formalize this, we first show how two information flows can be compared by looking at the adversary's ability to harm the system. Then, we propose that the information flow in a system is effectively information-secure if it does not allow for more harm than its idealized variant based on the classical notion of noninterference