Lexicographic Composition of Simple Games

A two-house legislature can often be modelled as a proper simple game whose outcome depends on whether a coalition wins, blocks or loses in two smaller proper simple games. It is shown that there are exactly five ways to combine the smaller games into a larger one. This paper focuses on one of the rules, lexicographic composition, where a coalition wins G_1 => G_2 when it either wins in G_1, or blocks in G_1 and wins in G_2. It is the most decisive of the five. A lexicographically decomposable game is one that can be represented in this way using components whose player sets partition the whole set. Games with veto players are not decomposable, and anonymous games are decomposable if and only if they are decisive and have two or more players. If a player's benefit is assessed by any semi-value, then for two isomorphic games a player is better off from having a role in the first game than having the same role in the second. Lexicographic decomposability is sometimes compatible with equality of roles. A relaxation of it is suggested for its practical benefits.simple games, voting, game composition, game decomposition, semi-value, decisiveness, fairness

Voting Power in the EU Council of Ministers and Fair Decision Making in Distributive Politics

We analyze and evaluate the different decision rules describing the Council of Ministers of the EU starting from 1958 up to date. All the existing studies use the Banzhaf index (for binary voting) or the Shapley-Shubik index (for distributive politics). We argue that the nucleolus can be considered an appropriate power measure in distributive situations and an alternative to the Shapley-Shubik index. We then calculate the nucleolus and compare the results of our calculations with the conventional measures. In the second part, we analyze the power of the European citizens as measured by the nucleolus under the egalitarian criterion proposed by Felsenthal and Machover (1998), and characterize the first best situation. Based on these results we propose a methodology for the design of the optimal (fair) decision rules. We perform the optimization exercise for the earlier stages of the EU within a restricted domain of voting rules, and conclude that Germany should receive more than the other three large countries under the optimal voting rule.

Lexicographic Composition of Simple Games

A two-house legislature can often be modelled as a proper simple game whose outcome depends on whether a coalition wins, blocks or loses in two smaller proper simple games. It is shown that there are exactly five ways to combine the smaller games into a larger one. This paper focuses on one of the rules, lexicographic composition , where a coalition wins in G 1 =\u3e G 2 when it either wins in G 1 , or blocks in G 1 and wins in G 2 . It is the most decisive of the five. A lexicographically decomposable game is one that can be represented in this way using components whose player sets partition the whole set. Games with veto players are not decomposable, and anonymous games are decomposable if and only if they are decisive and have two or more players. If a player’s benefit is assessed by any semi-value, then for two isomorphic games a player is better off from having a role in the first game than having the same role in the second. Lexicographic decomposability is sometimes compatible with equality of roles. A relaxation of it is suggested for its practical benefits

Estimating topological entropy of multidimensional nonlinear cellular automata

Cellular automata are discrete dynamical systems whose configurations are determined by local rules acting on each cell in synchronous. Topological entoropy is a tool for measuring the complexity of these dynamical systems. In this paper we estimate topological entropy of a two-dimensional nonlinear cellular automaton. The method we use is that a one-dimensional cellular automaton with positive topological entoropy is “naturally” embedded into the twodimensional cellular automaton. Hence we obtain a multidimensional cellular automaton with infinite topological entoropy

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Mini-Workshop: Positional Games

Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions