Comment on The house always wins
thebestaquaman@lemmy.world 1 week agoThat’s not how standard deviations work though. The point is that if you are n players, the probability of any given player starting is 1/n. After an arbitrary number of dice throws, the probability that a given player is ahead remains 1/n, when you account for the throw that decided who would go first.
Let’s put it this way: Would it be “more random” who goes first if you throw ten dice to decide instead of one? Of course not. But that’s essentially what you’re doing when you go “warm up” rounds. You’re just throwing the dice more times, and letting whoever has the highest total go first. Clearly, the probability that any given player gets the highest total remains 1/n, regardless how many dice are thrown.
Amir@lemmy.ml 6 days ago
I didn’t mean dice rolls for who starts, but moving around the board.
If you go around the board 0 times, there’s a 100% chance the player who started will be ahead.
If you go around the board 1 times, there’s a less-than-100% chance the player who started will be ahead.
Every added round around the board increases the.standard deviation of spaces moved. While the expected amount of spaces moved will still be higher for the first mover after their turn, the significance of this difference goes down as the standard deviation goes up.
Therefore, running 100 rounds around the board before starting the game will change the first-mover advantage from being ahead 100% of the time to, likely slightly more than 25% of the time but very close to 25%.
thebestaquaman@lemmy.world 6 days ago
What you say is true. What you’re neglecting is that you need a random process to choose who will go first. Let’s use your own example: If four players go around the board 100 times, there’s a near 25% chance that a given player gets around first. As you correctly say (indirectly), you will asymptotically approach a 25% chance as you increase the number of rounds towards infinity.
What you seem to be forgetting is that there’s a very easy way to skip the infinite number of rounds, and get directly to the 25% chance: By choosing randomly who goes first. Of course, you need to do that anyway in order to start the warm-up rounds at all, so what you are effectively doing is
First: Give every player a 25 % chance to start. Then: Spend an arbitrary amount of “warm-up” rounds to randomly choose a different player that gets to start the real game.
Of course, these are not independent random processes, so the player that wins the first selection has an advantage in the second selection. The overall probability that a given player starts the “real” game first then becomes identical to the probability that they start the “warm-up” first. An infinite number of warmup rounds is literally identical to a single dice roll in terms of the probability that a given player goes first. So what you’re doing is one quick random selection, which you immediately throw out in favour of an infinitely time consuming random selection with the same distribution.
Amir@lemmy.ml 6 days ago
That does make sense, thanks for writing this out.
thebestaquaman@lemmy.world 6 days ago
Thank you so much for taking the time to read and understand what I mean! I honestly mean it, it means a lot to me.