npub1kpwlxpzkxfmuxjmzc2wp3rf9vjg0sgydmlhsnrgqr3maf59h86qqdxxzz4 (npub1kpw…xzz4)
I think this is exactly the kind of reasoning described in wiki for why mixed strategies are a thing (see the choosing how to pitch in baseball example there).
The thing that might be interesting is also exploring what "optimal" means. For simplicity let's assume the game rules itself don't randomize anything and that the game has a binary outcome (A wins or B wins, no scoring).
In battleships, if each player was forced to be deterministic, for each strategy for player A there is a strategy for player B that defeats it and v.v. This means that the standard approach of "optimal strategy is one that wins whenever it's possible to win" would imply lack of optimal strategies at all: after all, it's always possible to win but no strategy wins always. This is still true if you admit nondeterministic strategies: it's possible to win against each deterministic strategy with p=1, but there's no strategy that does that for all deterministic counter-strategies.
The reasoning you described is similar to looking at changes you'd make in your strategy in response to knowing that opponent's, then doing the same for your opponent, and iterating this. So, it seems that you describe looking for a fixpoint of that: a pair of strategies s.t. neither you nor the opponent would want to change theirs upon learning what other player's strategy is. Note that this necessarily speaks about pairs of strategies (IIRC there are games where you have multiple such pairs, but I sadly don't recall examples). It's been shown that every game with finite sets of moves has at least one such pair.
NB if you are after such pairs of strategies, then instead of iterating it's often easier to directly look at the fixpoint condition and infer what such a pair of strategies would need to satisfy for both players not to wish to amend theirs.
npub1v6…s9004
2023-05-07 20:41:10
in reply to nevent1q…8k07
Author Public Key
npub1v67gmn7f4vlg3fcrk8up3p08rkmqqaxmw0he3hf9f8h09jre332qds9004Seen on
wss://relay.nostr.bandPublished at
2023-05-07 20:41:10Event JSON
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"content": "nostr:npub1kpwlxpzkxfmuxjmzc2wp3rf9vjg0sgydmlhsnrgqr3maf59h86qqdxxzz4 \n\nI think this is exactly the kind of reasoning described in wiki for why mixed strategies are a thing (see the choosing how to pitch in baseball example there).\n\nThe thing that might be interesting is also exploring what \"optimal\" means. For simplicity let's assume the game rules itself don't randomize anything and that the game has a binary outcome (A wins or B wins, no scoring).\n\nIn battleships, if each player was forced to be deterministic, for each strategy for player A there is a strategy for player B that defeats it and v.v. This means that the standard approach of \"optimal strategy is one that wins whenever it's possible to win\" would imply lack of optimal strategies at all: after all, it's always possible to win but no strategy wins always. This is still true if you admit nondeterministic strategies: it's possible to win against each deterministic strategy with p=1, but there's no strategy that does that for all deterministic counter-strategies.\n\nThe reasoning you described is similar to looking at changes you'd make in your strategy in response to knowing that opponent's, then doing the same for your opponent, and iterating this. So, it seems that you describe looking for a fixpoint of that: a pair of strategies s.t. neither you nor the opponent would want to change theirs upon learning what other player's strategy is. Note that this necessarily speaks about pairs of strategies (IIRC there are games where you have multiple such pairs, but I sadly don't recall examples). It's been shown that every game with finite sets of moves has at least one such pair.\n\nNB if you are after such pairs of strategies, then instead of iterating it's often easier to directly look at the fixpoint condition and infer what such a pair of strategies would need to satisfy for both players not to wish to amend theirs.",
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