The difference between winning tactic and winning strategy

Question

In the topological game, what is the difference between winning tactic and winning strategy? Why the author (in this paper: LEFT SEPARATED SPACES WITH POINT-COUNTABLE BASES) said that, the first implies the second?

Answer 1

The usual definition is that a tactic only looks at the last move of the opponent, while a strategy looks at the whole prior sequence of moves. Marion Scheepers has several interesting results where he considers $k$-tactics for $k\in\mathbb N$. Here, one looks at the last $k$ moves of the opponent, so a tactic is a $1$-tactic.

Of course, if a tactic is winning, then it automatically gives us a winning strategy, where we have at our disposal all prior moves of the opponent but, in fact, only make use of the last one.

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An interesting example of the difference between strategies and $k$-tactics is the countable-finite game. Here, we start with a set $X$, player I plays countable subsets, player II plays finite subsets. Say that in move $n$, player I moves $C_n$ and player II moves $F_n$. We require that $C_n\subset C_{n+1}$. At the end, player II wins if they "catch up": $\bigcup_n F_n\supseteq \bigcup_n C_n$. For which $X$ does player II have a winning $2$-tactic? It is easy to see that II has a winning strategy.

Marion has several papers on the subject. See for example

Marion Scheepers. Concerning $n$-tactics in the countable-finite game. J. Symbolic Logic 56 (1991), no. 3, 786–794. MR1129143 (92m:03075).