Puzzles with Rock-Paper-Scissors
Feb. 21st, 2013 05:43 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
blimixToday, I thought of some variants on an otherwise trivial puzzle.
This will be Rock-Paper-Scissors, as played by a mathematician and/or game theorist, rather than (as it is played in real life) a psychological contest of reading your opponent's intentions.
In these puzzles, we will posit that your opponent knows your strategy, and plays optimally for their own benefit. (Assume each player cares only about their own income. No generosity and no spite for the other player.) So if your strategy is to play Rock, your opponent will play Paper. Your saving grace is that you are allowed to use a random ("mixed") strategy.
I hope that even those among us who are not mathematically inclined can intuit that the ideal strategy is to pick randomly from Rock, Paper, and Scissors, with equal chances of each.
But what happens if we throw off the symmetry of the game, not by changing the system, but by introducing different rewards for different wins?
Problem 1: Good old Rock. The payout from the loser to the winner is as follows: $9 if the winner throws Rock; $3 if the winner throws Scissors; $1 if the winner throws Paper.
Now you will lose by keeping the probabilities equal: Your opponent would throw Rock and count on winning much more to Scissors than they would lose to Paper. And you couldn't just throw Paper in anticipation of them throwing Rock, because they get to know your strategy, and they would just throw Scissors instead.
What strategy can you employ, that your opponent can't beat even if they know what it is?
(If you said, "The only winning move is not to play," then congratulations on being a smart-ass. Now find a real answer.)
Hint: How do you make a strategy that your opponent cannot beat? A good path is to formulate your strategy such that all of your opponent's choices are equally good. If you leave them with one clearly better choice, then your strategy still has room to improve: You could make their best choice a little worse, at the cost of making their worst choice a little better, which is a good trade for you. Once you've given them no good reason to prefer one move over another, you've probably minimized their winnings — and your losses.
Problem 2: On the House. The payouts are the same as in problem 1, but instead of coming from the loser, they come from the bank. Does your strategy change, and if so, how?
I won't screen comments, so beware of spoilers if you read them. I'll post my answers in a day or two.
Some white space, in case you only clicked through to read the hint, but do not want to see answers in the comments:
This will be Rock-Paper-Scissors, as played by a mathematician and/or game theorist, rather than (as it is played in real life) a psychological contest of reading your opponent's intentions.
In these puzzles, we will posit that your opponent knows your strategy, and plays optimally for their own benefit. (Assume each player cares only about their own income. No generosity and no spite for the other player.) So if your strategy is to play Rock, your opponent will play Paper. Your saving grace is that you are allowed to use a random ("mixed") strategy.
I hope that even those among us who are not mathematically inclined can intuit that the ideal strategy is to pick randomly from Rock, Paper, and Scissors, with equal chances of each.
But what happens if we throw off the symmetry of the game, not by changing the system, but by introducing different rewards for different wins?
Problem 1: Good old Rock. The payout from the loser to the winner is as follows: $9 if the winner throws Rock; $3 if the winner throws Scissors; $1 if the winner throws Paper.
Now you will lose by keeping the probabilities equal: Your opponent would throw Rock and count on winning much more to Scissors than they would lose to Paper. And you couldn't just throw Paper in anticipation of them throwing Rock, because they get to know your strategy, and they would just throw Scissors instead.
What strategy can you employ, that your opponent can't beat even if they know what it is?
(If you said, "The only winning move is not to play," then congratulations on being a smart-ass. Now find a real answer.)
Hint: How do you make a strategy that your opponent cannot beat? A good path is to formulate your strategy such that all of your opponent's choices are equally good. If you leave them with one clearly better choice, then your strategy still has room to improve: You could make their best choice a little worse, at the cost of making their worst choice a little better, which is a good trade for you. Once you've given them no good reason to prefer one move over another, you've probably minimized their winnings — and your losses.
Problem 2: On the House. The payouts are the same as in problem 1, but instead of coming from the loser, they come from the bank. Does your strategy change, and if so, how?
I won't screen comments, so beware of spoilers if you read them. I'll post my answers in a day or two.
Some white space, in case you only clicked through to read the hint, but do not want to see answers in the comments:
(no subject)
Date: 2013-02-22 04:58 am (UTC)