Saturday, August 7, 2010

Solution to "Double Rock Paper Scissors"

See the original puzzle

Since Alice can't use paper, Bob has no reason to ever use scissors.  Rather than playing scissors, Bob can always achieve a better outcome with rock.  Given this knowledge, each player has four possible moves, and we can draw a table of the sixteen different outcomes.

From this table, it's clear that Alice has no reason to ever play RS or SR, because SS is always an improvement.  So we can cross out those columns.

Alice's best strategy is to pick RR or SS each with a 50% chance.  Bob's best strategy is to have a 50% chance of picking RR and a 50% chance of picking RP/PR/PP (it doesn't matter which).  The outcome is that both players have an even chance of winning.

I am making certain assumptions about what constitutes the best strategy.  I am assuming that Alice picks each choice with a certain probability.  She chooses those probabilities based on the assumption that Bob uses the best possible strategy to counter hers.  Likewise, Bob chooses probabilities based on the assumption that Alice uses the best possible strategy to counter his.

For example, if Alice picked RR with a 20% chance and SS with an 80% chance, then this strategy is suboptimal.  Bob can simply pick RR, and win 80% of the time.  Mind you, Bob doesn't necessarily know what Alice's strategy is, and thus doesn't know to pick RR.  But the fact remains that there exists some strategy which Bob could use to win 80% of the time.

Back in my high school days, I think I wrote a rock paper scissors type puzzle, except with six choices consisting of different "elements".  It was highly nontrivial.  I think I was pretty hardcore back then.


Larry Hamelin said...

I think I was pretty hardcore back then.

Back then!?!?!

miller said...

Haha, I mean that I constructed some really difficult puzzles. I used to submit puzzles to a website, and everyone there was really good at solving them. So I ramped the difficulty way up.

These days I try to construct much easier puzzles, because I have an audience that solves puzzles much more casually. But sometimes I have a really poor sense of what's too difficult.

Larry Hamelin said...

I'm just teasing you, mon frere. Your puzzles are always enjoyable, even when I can't even begin to try and solve them.