I was reading Andrew Bacon’s paper A Paradox for Infinite Decision Makers, and while I agreed with a lot of the conclusions, I didn’t agree with one of the arguments. The argument concerns this case,
For each n ∈ ω, at 1/n hours past 12 pm Alice and Bob will play a round of the game. A round involves two moves: firstly Alice chooses either 1 or 0, and then Bob makes a similar choice. The moves are made in that order, and both players hear each choice. Alice wins the round if Bob’s choice is the same as hers, and Bob wins if his choice is different. The game finishes at 1 pm, Alice wins the game if she wins at least one round, Bob wins the game if he wins every round.
As Andrew notes, it seems that Bob should win. At every stage, he waits for Alice’s move, and then makes a different move. But he claims Alice has a winning strategy, as follows.
There are various ways that Bob could play throughout a whole game, but any way he plays can be encoded as an ω-sequence of 1’s and 0’s, where the nth term in the sequence represents how he responds in the nth round. Before the game starts, Alice chooses her strategy as follows. Alice divides these sequences into equivalence classes according to whether they differ by finitely many moves at most. With the help of the Axiom of Choice, Alice then picks a representative from each equivalence class and memorises it. At any point after the game has started, Alice will know what moves Bob has made at infinitely many of the rounds, and will only be ignorant of the moves Bob is yet to play, of which there are only finitely many. Thus, at any point after 12 pm, Alice will know to which equivalence class the sequence of moves Bob will eventually make belongs. Her strategy at each round, then, is to play how the representative sequence for this equivalence class predicts Bob will play at that round. If the representative sequence is right about Bob’s move at that round, Alice will win that round. However, the representative sequence and the sequence that represents how Bob actually played, must be in the same equivalence class: they must be the same at all but finitely many rounds. If Alice played according to the representative sequence at every round, then she will have won all but finitely many of the rounds, meaning that she has won the game.
I think this can’t be right for two reasons.
First, I think we can make the intuitive argument that Bob will win slightly more rigorous. (This matters because intuitive arguments are more-or-less counter-indicative of truth when it comes to reasoning about infinities.) Assume that Bob plays the strategy “Wait and see what move Alice makes, then make the opposite move.” Let F(n) be the proposition that Bob wins the round after which there are n more rounds to play. Then F(0) is clearly true – Bob will win the last round. And for arbitrary k, we can prove F(k) → F(k+1). That’s because we can prove F(k+1), and then derive F(k) → F(k+1) by ∨-introduction! Then by mathematical induction, we can infer ∀x F(x).
Second, assuming that Alice’s strategy here works seems to lead to a reductio. Assume that Bob doesn’t play the “Wait and see what Alice does” strategy. Instead he plays a mirror image of Alice’s strategy. That is, he finds a representative strategy of each member of the set of equivalence classes as Andrew defines them. He then notes, as he can do at each stage, which equivalence class Alice’s play is in. He then plays the opposite move to the representative strategy. If Andrew’s reasoning is correct, then an exactly parallel argument should prove that Bob wins all but finitely many of the rounds. But it’s impossible for each of Alice and Bob to win all but finitely many of the rounds.
So something has gone wrong in the reasoning here. I think, though I’m not sure about this, that the strategy Andrew suggests for Alice will not have any distinctive advantages. Here’s the way I picture the situation. At any move, Alice will already have lost infinitely many games, or she will not have. If she has, the strategy can’t stop her losing infinitely many games, which is its main supposed advantage. If she has not, then it doesn’t matter what strategy she plays, she still won’t end up losing infinitely many games. So playing this strategy doesn’t help either way.
But I’m not at all sure about this diagnosis – it’s definitely a good case to think about, as are all the other points that are raised in the paper.