Dual selection games

Dual Selection Games

Abstract.

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if is coinitial in with respect to , where collects the choice functions on the set , then and are dual selection games.

Key words and phrases:
Selection principle, selection game, limited information strategies
2010 Mathematics Subject Classification:
54C30, 54D20, 54D45, 91A44

1. Introduction

Definition 1.

An \term-length game is a pair such that . The set is the \termmoveset of the game, and the set is the \termpayoff set for the second player.

In such a game , players and alternate making choices and during each round , and wins the game if and only if .

Often when defining games, and are restricted to choosing from different movesets . Of course, this can be modeled with by simply letting and adding/removing sequences from whenever player / makes the first “illegal” move.

A class of such games heavily studied in the literature (see [7] and its many sequels) are selection games.

Definition 2.

The \termselection game is an -length game involving Players and . During round , chooses , followed by choosing . Player wins in the case that , and Player wins otherwise.

For brevity, let

That is, wins in the case that , and wins otherwise.

Definition 3.

For a set , let be the collection of all choice functions on .

Definition 4.

Write if is coinitial in with respect to ; that is, , and for all , there exists such that .

In the context of selection games, we will say is a \termselection basis for when .

Definition 5.

The set is said to be a \termreflection of the set if

is a selection basis for .

Put another way, is a reflection of if for every , there exists such that and .

As we will see, reflections of selection sets are used frequently (but implicitly) throughout the literature to define dual selection games.

We use the following conventions to describe strategies for playing games.

Definition 6.

For and , let be the restrction of to . In particular, for and , describes the first terms of the sequence .

Definition 7.

A \termstrategy for the first player (resp. second player ) in a game with moveset is a function . This strategy is said to be \termwinning if for all possible \termattacks by their opponent, where is played by the opponent during round , the player wins the game by playing (resp. ) during round .

That is, a strategy is a rule that determines the moves of a player based upon all previous moves of the opponent. (It could also rely on all previous moves of the player using the strategy, since these can be reconstructed from the previous moves of the opponent and the strategy itself.)

Definition 8.

A \termpredetermined strategy for the first player in a game with moveset is a function . This strategy is said to be winning if for all possible attacks by their opponent, the first player wins the game by playing during round .

So a predetermined strategy ignores all moves of the opponent during the game (all moves were decided before the game began).

Definition 9.

A \termMarkov strategy for the second player in a game with moveset is a function . This strategy is said to be winning if for all possible attacks by their opponent, the first player wins the game by playing during round .

So a Markov strategy may only consider the most recent move of the opponent, and the current round number. Note that unlike perfect-information or predetermined strategies, a Markov strategy cannot use knowledge of moves used previously by the player (since they depend on previous moves of the opponent that have been “forgotten”).

Definition 10.

Write (resp. ) if player has a winning strategy (resp. winning predetermined strategy) for the game . Similarly, write (resp. ) if player has a winning strategy (resp. winning Markov strategy) for the game .

Of course, . In general, none of these implications (not even the second [4]) can be reversed.

It’s worth noting that is equivalent to the selection principle often denoted in the literature.

The goal of this paper is to characerize when two games are “dual” in the following senses.

Definition 11.

A pair of games defined for a topological space are \termMarkov information dual if both of the following hold.

  • if and only if .

  • if and only if .

Definition 12.

A pair of games defeind for a topological space are \termperfect information dual if both of the following hold.

  • if and only if .

  • if and only if .

2. Main Results

The following four theorems demonstrate that reflections characterize dual selection games for both perfect information strategies and certain limited information strategies.

The duality of the Rothberger game and the point-open game on for perfect information strategies was first noted by Galvin in [5], and for Markov-information strategies by Clontz and Holshouser in [3]. These proofs may be generalized as follows.

Theorem 13.

Let be a reflection of .

Then if and only if .

Proof.

Let witness . Since , for some . So let for all and . Suppose for all . Note that since is winning and , . Thus witnesses .

Now let witness . Let be defined by , and let . Suppose that for all . Choose such that . Since is winning, . Thus witnesses . ∎

Theorem 14.

Let be a reflection of .

Then if and only if .

Proof.

Let witness . Let . Suppose that for each , there was such that for all , . Then and , thus for all , a contradiction.

So choose such that for all there exists such that . It follows that when for , , so witnesses .

Now let witness . Then , so for , let satisfy , and let . Then if for , , so . Thus witnesses . ∎

Theorem 15.

Let be a reflection of .

Then if and only if .

Proof.

Let witness . Let . Suppose is defined for . Since , let satisfy , and let . Then let for , so

demonstrating that is a legal attack against .

Let . Consider the attack against . Then since is winning and , it follows that . Thus witnesses .

Now let witness . For , define by . Let , and for , choose such that (for other , choose arbitrarily as it won’t be used). Now let , and suppose has been defined for and . Then let and for choose such that (and again, choose arbitrarily for other as it won’t be used).

Then let attack , so and thus . Since is winning, . Thus witnesses . ∎

Theorem 16.

Let be a reflection of .

Then if and only if .

Proof.

Let witness . Let and assume is defined (of course, ). Suppose for all there existed such that for all , . Then and , and thus for all , a contradiction. So let satisfy for all there exists extending such that .

If is attacked by , then . So , and since is winning, . Therefore witnesses .

Now let witness . Let , and suppose is defined (again, ). For choose where , and let , and let extend by letting .

If is attacked by , then since and is winning, we conclude that is a legal strategy and . Therefore witnesses . ∎

Corollary 17.

If is a reflection of , then and are both perfect information dual and Markov information dual.

3. Applications of Reflections

Definition 18.

Let be a topological space and be a chosen basis of nonempty sets for its topology.

  • Let be the local point-base at .

  • Let be the fan at .

  • Let be the local finite-base at .

  • Let be the collection of basic open covers of .

  • Let be the collection of local point-bases of .

  • Let be the collection of basic -covers of .

  • Let be the collection of local finite-bases of .

  • Let be the collection of dense subsets of .

  • Let be the collection of converging fans at . (When intersected with , these are the non-trivial sequences of converging to .)

While these notions were defined in terms of a particular basis, the reader may verify the the following.

Proposition 19.

Let be a selection basis for .

  • .

  • .

  • .

  • .

Proposition 20.

Each selection set in Definition 18 is a selection basis for the set defined by replacing with the set of all nonempty open sets in .

As such, the choice of topological basis is irrelevant when playing selection games using these sets.

We may now establish (or re-establish) the following dual games.

Proposition 21.

is a reflection of .

Proof.

For every open cover , the corresponding choice function is simply the witness that . ∎

Corollary 22.

and are perfect-information and Markov-information dual.

In the case that , is the well-known Rothberger game, and is isomorphic to the point-open game : chooses points of , chooses an open neighborhood of each chosen point, and wins if ’s choices are a cover. So this was simply the classic result that the Rothberger game and point-open game are perfect-information dual [5], and the more recent result that these games are Markov-information dual [3].

Proposition 23.

is a reflection of .

Proof.

For every -cover , the corresponding choice function is simply the witness that . ∎

Corollary 24.

and are perfect-information and Markov-information dual.

Note that in the case that , is the Rothberger game played with -covers, and is isomorphic to the -finite-open game : chooses finite subsets of , chooses an open neighborhood of each chosen finite set, and wins if ’s choices are an -cover. These games were shown to be dual in [3].

Proposition 25.

is a reflection of .

Proof.

For every dense , the corresponding choice function is simply the witness that . ∎

Corollary 26.

and are perfect-information and Markov-information dual.

In the case that for some , is the strong countable dense fan-tightness game at , see e.g. [1]. is the game first studied by Tkachuk in [10]. Tkachuk showed in that paper that these games are perfect-information dual; Clontz and Holshouser previously showed these were Markov-information dual in the case that [3].

In the case that , then is the strong selective separability game introduced in [8], and is the point-picking game of Berner and Juhász defined in [2]. Scheepers showed that these were perfect-information dual in his paper.

Proposition 27.

is a reflection of .

Proof.

For every set with limit point , the corresponding choice function is simply the witness that . ∎

Corollary 28.

and are perfect-information and Markov-information dual.

In the case that for some , is Gruenhage’s game [6]. Its dual characterizes the strong Fréchet-Urysohn property at , which now seen to be equivalent to . This allows us to obtain the following result.

Corollary 29.

if and only if .

Proof.

As shown in [9], a space is at , that is, if and only if for all . ∎

For , is the variant of Gruenhage’s game for clustering. This game is now seen to be dual to the strong countable fan tightness game at .

4. Open Questions

Question 30.

Does there exist a natural reflection for or ?

Question 31.

Can these results be extended for ?

5. Acknowledgements

Thanks to Prof. Jared Holshouser for his input during the writing of these results.

References

  1. Doyel Barman and Alan Dow. Selective separability and . Topology Proc., 37:181–204, 2011.
  2. Andrew J. Berner and István Juhász. Point-picking games and HFDs. In Models and sets (Aachen, 1983), volume 1103 of Lecture Notes in Math., pages 53–66. Springer, Berlin, 1984.
  3. S. Clontz and J. Holshouser. Limited Information Strategies and Discrete Selectivity. ArXiv e-prints, June 2018.
  4. David Gale and F. M. Stewart. Infinite games with perfect information. In Contributions to the theory of games, vol. 2, Annals of Mathematics Studies, no. 28, pages 245–266. Princeton University Press, Princeton, N. J., 1953.
  5. Fred Galvin. Indeterminacy of point-open games. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26(5):445–449, 1978.
  6. Gary Gruenhage. Infinite games and generalizations of first-countable spaces. General Topology and Appl., 6(3):339–352, 1976.
  7. Marion Scheepers. Combinatorics of open covers. I. Ramsey theory. Topology Appl., 69(1):31–62, 1996.
  8. Marion Scheepers. Combinatorics of open covers. VI. Selectors for sequences of dense sets. Quaest. Math., 22(1):109–130, 1999.
  9. P. L. Sharma. Some characterizations of -spaces and -spaces. General Topology Appl., 9(3):289–293, 1978.
  10. Vladimir V. Tkachuk. Two point-picking games derived from a property of function spaces. Quaestiones Mathematicae, 0(0):1–15, 2017.
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