In this paper we study some properties concerning the equilibrium point of a bimatrix game and describe a geometric method to obtain all the equilibria of a bimatrix game when one of the players has at most three pure strategies.

In this survey article we shall summarise some of the recent progress that has occurred in the study of topological games as well as their applications to abstract analysis. The topics given here do not necessarily represent the most important problems from the area of topological games, but rather, they represent a selection of problems that are of interest to the authors.

As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize $\sigma$-compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize $\sigma$-compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between $\sigma$-compact and Menger spaces.

Concepts, definitions, notions, and some facts concerning the Banach-Mazur game are customized to a more general setting of partial orderings. It is applied in the theory of Fraïssé limits and beyond, obtaining simple proofs of universality of certain objects and classes.

Some of the covering properties of spaces as defined in Parts I and II are here characterized by games. These results, applied to function spaces $C_p\left(X\right)$ of countable tightness, give new characterizations of countable fan tightness and countable strong fan tightness. In particular, each of these properties is characterized by a Ramseyan theorem.

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31/2}$-space. In [9] we showed that $C_p\left(X\right)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_p\left(X\right)$ has countable fan tightness and the Reznichenko property. 2....

We investigate some natural questions about the class of posets which can be embedded into ⟨ω,≤*⟩. Our main tool is a simple ccc forcing notion $H_E$ which generically embeds a given poset E into ⟨ω,≤*⟩ and does this in a “minimal” way (see Theorems 9.1, 10.1, 6.1 and 9.2).

We show that some classes of small sets are topological versions of some combinatorial properties. We also give a characterization of spaces for which White has a winning strategy in the point-open game. We show that every Lusin set is undetermined, which solves a problem of Galvin.

In this paper, which is an extension of [4], we first show the existence of solutions to a class of Min Sup problems with linked constraints, which satisfy a certain property. Then, we apply our result to a class of weak nonlinear bilevel problems. Furthermore, for such a class of bilevel problems, we give a relationship with appropriate d.c. problems concerning the existence of solutions.

1991 AMS Math. Subj. Class.:Primary 54C10; Secondary 54F65We provide both a spectral and an internal characterizations of arbitrary !-favorable spaces with respect to co-zero sets. As a corollary we establish that any product of compact !-favorable spaces with respect to co-zero sets is also !-favorable with respect to co-zero sets. We also prove that every C* -embedded !-favorable with respect to co-zero sets subspace of an extremally disconnected space is extremally disconnected.

We continue the efforts to characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorial or structural properties of the given filter. Previous results in the literature included those games where player II responded with natural numbers, or finite subsets of natural numbers. In this paper we concentrate on games where player II responds with members of the dual ideal. We also give a summary of known results on filter games.

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over ${\Sigma }_2^0$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over ${\Sigma }_2^0$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that ${\kappa }^{<\kappa }=\kappa$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^{\kappa }$ many such spaces up to homeomorphism. We also establish a Kuratowski-like...

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_n$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_n$ of X. The players must obey the rule that $O_n\subseteq O_{n+1}\subseteq T_{n+1}\subseteq T_n$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a $G_{\delta }$-set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets...