Reminds me of political system bugs where a candidate might win only by transitivity.
Nontransitive dice
31–40 of 49 posts
Re: Nontransitive dice
#32The non-transitivity of Rock-Scissors-Paper is easy to understand, partly because it's so simple, but mostly because you're likely never played outside the usual rules, even if adding Lizard-Spock. Non-transitive dice screw with the 'nature' of dice that most of us expect. To get to the mathematical intuition, one may have to get past a deeply-ingrained feeling that something about these dice just isn't right. That's…
I think the numeric and gambling aspect helps too. If you're used to probability, you probably start thinking about expected value automatically in these situations, but it isn't the case that the die with the highest expected value wins most of the time against a die with lower expected value. (e.g. 2,2,2,2,2,2s vs 1,1,1,1,1,100).
Seasoned gamblers count "outs" (winning outcomes), convert to "hand odds" (out:loss ratio), then weigh hand odds against "pot odds" (weigh expected profit vs investment).
Re: Nontransitive dice
#33This reminded me of Penney's Game ( https://en.wikipedia.org/wiki/Penney's_game ).
Freaky! I find that result even more surprising than the intransitive dice. Thanks for posting.
You might find the result less surprising after you solve a riddle by Martin Gardner:
> A young man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of the platform; to visit the girl in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally well, he simply takes the first train that comes along. In this way, he lets chance determine whether he rides to the Bronx or to Brooklyn. The young man reaches the subway platform at a random moment each Saturday afternoon. Brooklyn and Bronx trains arrive at the station equally often—every 10 minutes. Yet for some obscure reason he finds himself spending most of his time with the girl in Brooklyn: in fact on the average he goes there 9 times out of 10. Can you think of a good reason why the odds so heavily favor Brooklyn?
The idea shows up again in the Elevator paradox, which has a delightful article on Wikipedia: https://en.wikipedia.org/wiki/Elevator_paradox
Re: Nontransitive dice
#34OMG, I just finished working through an example of this with my student, like literally minutes ago, and look what Hacker News brings me ...
Re: Nontransitive dice
#35OMG, I just finished working through an example of this with my student, like literally minutes ago, and look what Hacker News brings me ...
Well what were the odds of that, eh?
Re: Nontransitive dice
#36Earlier quoted context omitted.
I think the numeric and gambling aspect helps too. If you're used to probability, you probably start thinking about expected value automatically in these situations, but it isn't the case that the die with the highest expected value wins most of the time against a die with lower expected value. (e.g. 2,2,2,2,2,2s vs 1,1,1,1,1,100).
Mean EV doesn't work because the dice have different variances? Trying to wrap my head around the rationale here
EV of what? The faces of dice cannot be added together, except in a very formal sense.
Re: Nontransitive dice
#37Earlier quoted context omitted.
Freaky! I find that result even more surprising than the intransitive dice. Thanks for posting.
You might find the result less surprising after you solve a riddle by Martin Gardner: > A young man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of the platform; to visit the girl in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally w…
I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?
Re: Nontransitive dice
#38Does anyone have any board game recommendations that take advantage of this dice configuration?
Good question. It seems this non-transitive games are suited to trick people, but can you make skill-based game out of the mechanics?
Re: Nontransitive dice
#39Earlier quoted context omitted.
In normal dice play expected value doesn't come into play. But if we were to do a long term sum of results (like amalcon proposed) then it expected value would come into play, and would determine the winner. This illustrates the difference between premature rounding (normal dice play) and non premature rounding (long term summing of results).
But a long term sum of results is a completely different game. It doesn't matter what's on the face of the dice at all; all that matters is the average value per roll. It's not dice anymore! And you've defined the winning condition completely differently! I also don't understand how normal dice play counts as "premature rounding". It's just how playing dice works -- you compare the numbers versus each other.
Maybe this description of the two games might help clarify the link.
Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b) dollars.
Game 2: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b > 0 ? 1 : -1) dollars.
You can kind of squint and see that game 2 is the same as game 1 just with (a - b) "rounded" to either 1 or -1.
Re: Nontransitive dice
#40It was a little difficult to believe that such a property (dice being transient) was possible. But a little simulation in python 2 (using the Wikipedia example) makes everything crystal:
https://gist.github.com/spapas/9b29e1dec752b85fc1d6ad660af7f...