Kaleidoscope of life: A 24-neighbourhood outer-totalistic cellular automaton

One of the challenges of cellular automaton research is finding models with a low complexity and at the same time a rich dynamics. A measure of low complexity is the number of states in the model and the number of transition rules to switch between those states. In this paper, we propose a 2-dimensional 2-state cellular automaton that-though governed by a single simple transition rule-has a sufficiently rich dynamics to be computationally universal. According to the transition rule, a cell’s state is determined by the sum of the states of the cells at orthogonal or diagonal distances one or two from the cell (distance-2 Moore neighbourhood), but not by the previous state of the cell itself. Notwithstanding its simplicity, this model is able to generate a great variety of patterns, including several types of stable configurations, oscillators and patterns that move over cellular space (gliders). We prove the computational universality of the model by constructing a universal set of logic gates (NOT and AND) from these patterns. A key element in this proof is the shifting of phases and positions of signals such that they meet the input requirements of the logic gates. Similarities of the model with classical spin systems are also discussed.     

临界情况下模糊矩阵对策的性质

讨论临界情况下模糊矩阵对策的性抽,对模型问题各种情况给出了临界值c*的定义,证明了当d(f1,f2)＜ε≤c*时,模糊矩阵对策模型问题关于价值函数f1(x)、f2(x)具有连续依赖性和可调和性。     

正则条件全竞赛空间与混合策略竞赛空间上的两个极值定理

在正则的全竞赛空间与混合策略竞赛空间推广并证明了两个新的最优决策存在定理．     

Uncomputability and undecidability in economic theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory[Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics[Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443-2463] are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.     

Opportunistic timing and manipulation in Australian Federal Elections

In many parliamentary systems, election timing is an important decision made by governments in order to maximize their expected remaining life in power. Governments can also introduce policy or economic actions to enhance their popular standing and thus their chance of being re-elected. On the other hand, an oppositions’ natural objective is to gain power, and they will also apply controls through their own policies to reduce the governments’ chance of being re-elected. In this paper we employ a dynamic programming approach to determine the optimal timing for governments and oppositions to best utilize their limited resources. At each decision branch, the optimal control is interpreted as a Nash–Cournot equilibrium of a zero-sum political game which, in certain states, admits mixed strategy solutions. We perform a case study on the Australian Federal Election for House of Representatives.     

Definitions of equilibrium in network formation games

We examine a variety of stability and equilibrium definitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of definitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible.Bloch is also affiliated with the University of Warwick.     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Social systems analysis

We present a model of social systems as special types of games in extensive form in which perception and learned strategies play an important role. We discuss personality, emotions, and the relation of utility to Maslow's hierarchy of values. We also formulate a new class of games, partially cooperative games, and attempt to define a solution concept for these.