On Chooser-Picker positional games

Two new versions of the so-called Maker-Breaker Positional Games are defined by József Beck. In these variants Picker takes unselected pair of elements and Chooser keeps one of these elements and gives back the other to Picker. In the Picker-Chooser version Picker is Maker and Chooser is Breaker, while the roles are swapped in the Chooser-Picker version. It seems that both the Picker-Chooser and Chooser-Picker versions are not worse for Picker than the original Maker-Breaker versions. Here we give winning conditions for Picker in some Chooser-Picker games that extend the results of Beck.     

On Ramsey‐type positional games

Let K denote the graph obtained from the complete graph K_s+t by deleting the edges of some K_t‐subgraph. We prove that for each fixed s and sufficiently large t, every graph with chromatic number s+t has a K minor. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 343–350, 2010     

On the stability property in positional differential games

Doklady Mathematics -     

Biased positional games and the phase transition

Let 𝔏(n, q) be the game in which two players, Maker and Breaker, alternately claim 1 and q edges of the complete graph K_n, respectively. Maker's goal is to maximize the number of vertices in the largest component of his graph; Breaker tries to make it as small as possible. Let L(n,q) denote the size of the largest component in Maker's graph when both players follow their optimal strategies. We study the behavior of L(n, q) for large n and q=q(n). In particular, we show that the value of L(n, q) abruptly changes for q∼n and discuss the differences between this phenomenon and a similar one, which occurs in the evolution of random graphs. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 141–152, 2001     

On stochastic games

In this paper, we consider the stochastic games of Shapley, when the state and action spaces are all infinite. We prove that, under certain conditions, the stochastic game has a value and that both players have optimal strategies.Part of this research was supported by NSF grant. The authors are indebted to L. S. Shapley for the useful discussions on this and related topics. The authors thank the referee for pointing out an ambiguity in the formulation of Lemma 2.4 in an earlier draft of this article.     

On mathematical games

In this paper we survey most of the games that, throughout history, can be classified as mathematical games. This paper is based on a talk given at the conference ‘Numeracy: historical, philosophical, and educational perspectives’ at St Anne's College, Oxford, December 2009.     

On Porter's pursuit-evasion games

A minimax problem for Porter's class of functionals is considered in the case when the two players' strategies are elements of convex and strongly or weakly compact subsets of real Hilbert spaces. Functional analysis is used to prove sufficient conditions ensuring the existence of saddle points. An example illustrates the results, and a necessary condition of optimality of strategies is given in a particular case.The author wishes to express his thanks to Professor L. D. Berkovitz for several remarks that have improved the exposition of this paper.     

On shortest path games

A class of cooperative TU-games arising from shortest path problems is introduced and analyzed. Some conditions under which a shortest path game is balanced are obtained. Also an axiomatic characterization of the Shapley value for this class of games is provided.     

On thek-stability ofm-quota games

We consider thek-stability ofm-quota games ofn players. We prove that anm-quota game (N, v), which satisfies the conditionv(S)=0 for allS, ¦S¦ ≤m ?1, is (m ?1)-stable if and only if there is no weak player. Further, some relationships between ak-stable pair and anm-quota are shown. Some ofLuce's results [1955] on Shapley quota games are generalized tom-quota games.     

A nested family of $$\varvec{}$$-total effective rewards for positional games

We consider Gillette’s two-person zero-sum stochastic games with perfect information. For each \(k \in \mathbb {N}=\{0,1,\ldots \}\) we introduce an effective reward function, called k-total. For \(k = 0\) and 1 this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all k, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that k-total reward games can be embedded into \((k+1)\)-total reward games.     

On fair coin-tossing games

Let Ω be a finite set with k elements and for each integer $\text{n ≧ 1}$ let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}\text{= {(a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) | (a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) ∈ Ω}{}_{\mathrm{n}}$ and a_j ≠ a_j+1 for some 1 ≦ j ≦ n ? 1}. Let {Y_m} be a sequence of independent and identically distributed random variables such that P(Y₁ = a) = k^?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in $\text{Ω}{}_{\mathrm{n}}$ and in Ω?_n with respect to the stochastic process {Y_m}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.     

On positional simulation in dynamic systems

The problem of simulating delays and controls in a dynamic system described by ordinary differential-difference equations is examined. The simulation is carried out in real time by the feedback principle on the basis of information and the current phase states of the system, measurable with a certain error. The simulation algorithm proposed —an algorithm for reconstructing the unknown delays and controls —is a regularizing one in the sense that the simulation results become better the less the measurement errors in the system's phase positions. The ideological source of the proposed method of solving the problem is Krasovskii's extremal aiming principle /1, 2/. The paper extends the investigation in /3, 4/ and touches on /1–5/.     

On stochastic games,II

In this paper, we consider positive stochastic games, when the state and action spaces are all infinite. We prove that, under certain conditions, the positive stochastic game has a value and that the maximizing player has an -optimal stationary strategy and the minimizing player has an optimal stationary strategy.The authors are grateful to Professor David Blackwell and the referee for some useful comments.     

On noncooperative hybrid stochastic games

The present article models and analyzes a noncooperative hybrid stochastic game of two players. The main phase (prime hybrid mode) of the game is preceded by “unprovoked” hostile actions by one of the players (during antecedent hybrid mode) that at some time transforms into a large scale conflict between two players. The game lasts until one of the players gets ruined. The latter occurs when the cumulative damage to the losing player exceeds a fixed threshold. Both hybrid modes are formalized by marked point stochastic processes and the theory of fluctuations is utilized as one of the chief techniques to arrive at a closed form functional describing the status of both players at the ruin time.     

On equilibria in finite games

We are concerned with Nash equilibrium points forn-person games. It is proved that, given any real algebraic numberα, there exists a 3-person game with rational data which has a unique equilibrium point andα is the equilibrium payoff for some player. We also present a method which allows us to reduce an arbitraryn-person game to a 3-person one, so that a number of questions about generaln-person games can be reduced to consideration of the special 3-person case. Finally, a completely mixed game, where the equilibrium set is a manifold of dimension one, is constructed.