385,728 research outputs found
Nonzero-sum Stochastic Games
No full textThis paper treats of stochastic games. We focus on nonzero-sum games and provide a detailed survey of selected recent results. In Section 1, we consider stochastic Markov games. A correlation of strategies of the players, involving ``public signals'', is described, and a correlated equilibrium theorem proved recently by Nowak and Raghavan for discounted stochastic games with general state space is presented. We also report an extension of this result to a class of undiscounted stochastic games, satisfying some uniform ergodicity condition. Stopping games are related to stochastic Markov games. In Section 2, we describe a version of Dynkin's game related to observation of a Markov process with random assignment mechanism of states to the players. Some recent contributions of the second author in this area are reported. The paper also contains a brief overview of the theory of nonzero-sum stochastic games and stopping games which is very far from being complete.average payoff stochastic games, correlated stationary equilibria, nonzero-sum games, stopping time, stopping games
Small maximal sum-free sets
No full textLet G be a group and S a non-empty subset of G. If ab∉S for any a,b∈S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates ⟨S⟩ then |S|≤2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a∈S such that a∉⟨S∖{a}⟩
Beyond sum-free sets in the natural numbers
No full textFor an interval [1,N]⊆N, sets S⊆[1,N] with the property that |{(x,y)∈S2:x+y∈S}|=0, known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering r(S)=|{(x,y)∈S2:x+y∈S}|, and analyze its behaviour as S ranges over the subsets of [1,N]. We obtain a comprehensive description of the spectrum of attainable r-values, constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.Peer reviewe
Weighted Sum of Correlated Lognormals: Convolution Integral Solution
No full textProbability density function (pdf) for sum of n
correlated lognormal variables is deducted as a special
convolution integral. Pdf for weighted sums (where weights can
be any real numbers) is also presented. The result for four
dimensions was checked by Monte Carlo simulation
A Hybrid Continuous Max-Sum Algorithm for Decentralised Coordination
No full textRecent advances in decentralised coordination of multiple agents have led to the proposal of the max-sum algorithm for solving distributed constraint optimisation problems (DCOPs). The max-sum algorithm is fully decentralised, converges to optimality for problems with acyclic constraint graphs and otherwise performs well in empirical studies. However, it requires agents to have discrete state spaces, which are of practical size to conduct repeated searches over. In contrast, there are decentralised non-linear optimisation methods that are capable of accurately finding local optima over multi-dimensional continuous state spaces, however these methods are not designed to navigate complex interactions between local constraints in order to find globally optimal solutions. Given this background, in this paper we tackle the problem of coordinating multiple decentralised agents with continuous state variables. Specifically we propose a hybrid approach, which combines the max-sum algorithm with continuous non-linear optimisation methods. We show that, for problems with acyclic factor graph representations, for suitable parameter choices, our proposed algorithm converges to a state with utility close to the global optimum. We empirically evaluate our approach for cyclic constraint graphs in a multi-sensor target classification problem, and compare its performance to the discrete max-sum algorithm, as well as a non-coordinated approach and the distributed stochastic algorithm (DSA). We show that our hybrid max-sum algorithm outperforms the non-coordinated algorithm, DSA and discrete max-sum considerably. Furthermore, the improvements in outcome over discrete max-sum come without significant increases in running time nor communication cost
Self-organising Sensors for Wide Area Surveillance Using the Max-sum Algorithm
No full textIn this paper, we consider the self-organisation of sensors within a network deployed for wide area surveillance. We present a decentralised coordination algorithm based upon the max-sum algorithm and demonstrate how self-organisation can be achieved within a setting where sensors are deployed with no a priori information regarding their local environment. These energy-constrained sensors first learn how their actions interact with those of neighbouring sensors, and then use the max-sum algorithm to coordinate their sense/sleep schedules in order to maximise the effectiveness of the sensor network as a whole. In a simulation we show that this approach yields a 30% reduction in the number of vehicles that the sensor network fails to detect (compared to an uncoordinated network), and this performance is close to that achieved by a benchmark centralised optimisation algorithm (simulated annealing)
Using or Hiding Private Information ? An Experimental Study of Zero-Sum Repeated Games with Incomplete Information
No full textThis paper studies experimentally the value of private information in strictly competitive interactions with asymmetric information. We implement in the laboratory three examples from the class of zero-sum repeated games with incomplete information on one side and perfect monitoring. The stage games share the same simple structure, but differ markedly on how information should be optimally used once they are repeated. Despite the complexity of the optimal strategies, the empirical value of information coincides with the theoretical prediction in most instances. In particular, it is never negative, it decreases with the number of repetitions, and it is nicely bounded below by the value of the infinitely repeated game and above by the value of the one-shot game. Subjects are unable to completely ignore their information when it is optimal to do so, but the use of information in the lab reacts qualitatively well to the type and length of the game being played.Concavification, laboratory experiments, incomplete information, value of information, zero-sum repeated games.
Continuous fictitious play in zero-sum games
No full textRobinson (1951) showed that the learning process of Discrete Fictitious Play converges from any initial condition to the set of Nash equilibria in two-player zero-sum games. In several earlier works, Brown (1949, 1951) makes some heuristic arguments for a similar convergence result for the case of Continuous Fictitious Play (CFP). The standard reference for a formal proof is Harris (1998); his argument requires several technical lemmas, and moreover, involves the advanced machinery of Lyapunov functions. In this note we present a simple alternative proof. In particular, we show that Brown''s convergence result follows easily from a result obtained by Monderer et al. (1997).mathematical economics;
Evaluating information in zero-sum games with incomplete information on both sides
No full textIn a Bayesian game some players might receive a noisy signal regarding the specific game actually being played before it starts. We study zero-sum games where each player receives a partial information about his own type and no information about that of the other player and analyze the impact the signals have on the payoffs. It turns out that the functions that evaluate the value of information share two property. The first is Blackwell monotonicity, which means that each player gains from knowing more. The second is concavity on the space of conditional probabilities.Value of information, Blackwell monotonicity, concavity.
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