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I'm a little confused about the strategy-stealing argument and how exactly it's supposed to work.

This question's answer tells us that strategy-stealing works for tic-tac-toe because 'having extra pieces on the board in Tic-Tac-Toe is never bad'. This makes sense (maybe!) in contrast with, say, chess, where moves can certainly be bad (not just 'failing to take your position forward compared to other available moves') (i.e., there are certainly moves it would have been better to pass than make, if you could pass in chess).

But how do we know a move in tic-tac-toe is never bad? (For example, it could leave fewer free squares for you to create a 3-in-a-row with.) I'd think one could easily wave their hands and make the same claim about Connect Four; yet provably Connect4 with certain board dimensions, e.g. 8x8, is a win for the second-player.

So why are tic-tac-toe or Hex addressed by the strategy-stealing argument, but not for example Connect Four? Gomoku may be the best example, since it's just Connect Five without 'gravity' (i.e. new stones on a given column must be placed on the lowest unoccupied space); why does the strategy-stealing argument apply there but not to Connect Four?

Mobeus Zoom
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    Note that in connect four (with "gravity"), a move you make can create a possible move for your opponent that wasn't possible before (= place on top of the disc you just placed). This breaks strategy stealing. – Noah Schweber Aug 03 '21 at 17:37
  • To add onto Noah's comment, to strategy steal in tic-tac-toe, you just drop the first move and swap the colors. You can't do that in connect 4 because some moves are only possible if certain other moves were played. – Rushabh Mehta Aug 03 '21 at 18:05
  • @NoahSchweber Why does that break strategy-stealing when, say, removing options (e.g. without "gravity", if I play on a certain position, my opponent can then no longer play at that position) does not break strategy-stealing? And what is the criterion for "move that wasn't possible before" that excludes strategy-stealing from applying to chess for example? – Mobeus Zoom Aug 14 '21 at 17:27

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