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Comment by Y_Y | Hacker News Reader

It's a somewhat counterintuitive result from game theory that many of the best strategies we know to various non-trivial games are "mixed strategies". This refers to a strategy (presumably) like the one by AlphaZero here, where moves are chosen from a probabilistic distribution, rather than just picking "the best one". The sub-optimal move choice is counterbalanced by the unpredictability.

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Mixed strategies in game theory don’t involve making sub-optimal decisions for the sake of unpredictability. At equilibrium every sub-strategy of the mix is a best response (you have multiple best responses).

I think we're differing on unimportant semantics. Consider that optimal mixed strategies don't generally assign equal probability to each choice. If they were all equally good (say, as a pure strategy response) then they shouldn't have different probabilities.

I don't think so. The payoffs depend on the mix of the opponent. You can't say "the probability of sub-strategy x is higher so it must be a better strategy". If it really was a better strategy, you would pick it all the time, not in some mix.

In games with hidden information, yes. In open games? There is an optimal path from start to victory, for one of the players. No need to deviate randomly.

If you always play for the optimal move, give it a thousand games or so, and your opponents will know the optimal game.

They will learn the best opening move from your first move with white, the best reply from your response to it when they play it, white’s best second move from playing that against your best opening move, etc.

Once they know that, playing against them, all games will be the same and they’ll _look_ as good at chess as you.

Especially if the optimal move isn’t far better than the next best move(s) varying your moves can be used to show that isn’t true, and delay the time it takes your opponents to learn that optimal game.

Those considerations are not going to be typical. Every perfect information game will have a pure strategy equilibrium.

That may well be true, but it isn't relevant. Game-playing engines are designed to win the position being played, not to maximize winning probability over its playing career.

Stockfish or AlphaZero don't care one bit about what happens in the next game they play.

It isn't necessary either. In chess, as there's no need to worry about interactions between players' information sets, a player can safely apply pure strategies. Winnings approach being maximized over player career by trying to match as close as possible subgame perfect equilibrium from any given start node per game.

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