The Perfect Chess Game

[andretimpa]

Food for thought: what's the perfect game in Rock, Paper, Scissors?
For there to be a "perfect move" in any instance of the game, all other moves must undeniably lead to a loss.

There is a perfect strategy for RPS: play randomly (1/3 chance of each move). The perfect game would be both players choosing moves at random.

That's a Nash equilibrium iirc. It's not technically the same as a perfect strategy but it's the best you can do.

[OldTrees]

Food for thought: what's the perfect game in Rock, Paper, Scissors?
For there to be a "perfect move" in any instance of the game, all other moves must undeniably lead to a loss.

There is a perfect strategy for RPS: play randomly (1/3 chance of each move). The perfect game would be both players choosing moves at random.

That's a Nash equilibrium iirc. It's not technically the same as a perfect strategy but it's the best you can do.

While it is a Nash equilibrium, it is also the RPS perfect game. No individual play has any advantage over the others and patterns of play only reduce your win rate. Thus the optimal move each game is to pick randomly.

The same is true for the 4th move of tic tac toe's perfect game (previous moves being 1, 5, & 9 in that order). The 4th move is randomly selected from 2, 4, 6 or 8 to result in a tied game.

[CuCN]

Food for thought: what's the perfect game in Rock, Paper, Scissors?
For there to be a "perfect move" in any instance of the game, all other moves must undeniably lead to a loss.

There is a perfect strategy for RPS: play randomly (1/3 chance of each move). The perfect game would be both players choosing moves at random.

That's a Nash equilibrium iirc. It's not technically the same as a perfect strategy but it's the best you can do.

While it is a Nash equilibrium, it is also the RPS perfect game. No individual play has any advantage over the others and patterns of play only reduce your win rate. Thus the optimal move each game is to pick randomly.

The same is true for the 4th move of tic tac toe's perfect game (previous moves being 1, 5, & 9 in that order). The 4th move is randomly selected from 2, 4, 6 or 8 to result in a tied game.

But tic-tac-toe is not played simultaneously, so there is no need to use a mixed strategy. If after 1-5-9 the second player always chose 2, it would still guarantee a tie. Besides, the third move can actually be any unoccupied square (not just 9), because they all result in ties with optimal play.

[jawdirk]

Food for thought: what's the perfect game in Rock, Paper, Scissors?
For there to be a "perfect move" in any instance of the game, all other moves must undeniably lead to a loss.

There is a perfect strategy for RPS: play randomly (1/3 chance of each move). The perfect game would be both players choosing moves at random.

That's a Nash equilibrium iirc. It's not technically the same as a perfect strategy but it's the best you can do.

While it is a Nash equilibrium, it is also the RPS perfect game. No individual play has any advantage over the others and patterns of play only reduce your win rate. Thus the optimal move each game is to pick randomly.

Interestingly, among a field of competitors with variations in strategy, random play is not the best strategy for RPS. Random play only guarantees a 50% win rate, where as some other strategies can take advantage of non-random strategies and end up with > 50% win rate. So perfect play in RPS is not a competitive advantage.

[OldTrees]

Food for thought: what's the perfect game in Rock, Paper, Scissors?
For there to be a "perfect move" in any instance of the game, all other moves must undeniably lead to a loss.

There is a perfect strategy for RPS: play randomly (1/3 chance of each move). The perfect game would be both players choosing moves at random.

That's a Nash equilibrium iirc. It's not technically the same as a perfect strategy but it's the best you can do.

While it is a Nash equilibrium, it is also the RPS perfect game. No individual play has any advantage over the others and patterns of play only reduce your win rate. Thus the optimal move each game is to pick randomly.

The same is true for the 4th move of tic tac toe's perfect game (previous moves being 1, 5, & 9 in that order). The 4th move is randomly selected from 2, 4, 6 or 8 to result in a tied game.

But tic-tac-toe is not played simultaneously, so there is no need to use a mixed strategy. If after 1-5-9 the second player always chose 2, it would still guarantee a tie. Besides, the third move can actually be any unoccupied square (not just 9), because they all result in ties with optimal play.

Sequential play does remove the need for a mixed strategy. However since they are all equal there is no need for the perfect strategy to specify which.
The 3rd move is 9 because 9 is equally good against a perfect opponent and is optimal against more imperfect opponents. (Same reason the first move is 1 instead of 2 or 5).

Interestingly, among a field of competitors with variations in strategy, random play is not the best strategy for RPS. Random play only guarantees a 50% win rate, where as some other strategies can take advantage of non-random strategies and end up with > 50% win rate. So perfect play in RPS is not a competitive advantage.

True but as you progressively discard the least efficient strategy you will find the better strategies start to approximate random until eventually only the random strategy remains. The only reason this has not happened yet is that humans can't do random unassisted.