Michael Kleber wrote
If the first player doesn't bust, then of course player 2 will adopt the strategy "beat player 1 or bust trying." But p2's probability of busting trying is just the same as was p1's of busting before reaching that point value in the first place.
Presumably p2 wants to win. So wouldn't the probability of p2 surpassing p1's score be less than the probability of p1 reaching it in the first place? On the other hand, as the number of players grows, the player who goes out first is at an increasing disadvantage. When there are six or eight people around the table the probability that no one will beat the "first out" score can be kind of crappy. If it is a close game with a lot of players, you don't want to go out with a mediocre roll. You'll lose. I have a large family. In our house we play a similar game, but the scoring is more volatile and we arbitrarily changed the ending condition to something that makes more sense given the number of kids usually playing. When one player goes out, the game play changes somewhat. When one player goes out their score becomes the high score, the score to beat. The next player in turn attempts to beat the high score. If they do, their score becomes the high score. If they don't, they are out. This cycles around the table eliminating players until only one player is left. The player who went out first is not treated any different than the others. When play returns to the player who went out first, they either have the high score and win or they get a chance to beat the current high score. Either way, the play continues around and around until there is a winner. Since the player who went out first gets another chance, it removes the almost certain loss associated with going out first. Maybe the biggest benefit with this ending condition is that it removes a good deal of pouting associated with children who are quick to perceive something that is "not fair". The scoring in our game is a little different than in Toss Up Komi, so it seldom takes long for a winner to emerge after someone goes out. I'm not sure how our ending conditions would work with Toss Up Komi. I can see a couple of conservative players going back and forth for hours... Mark -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber Sent: Friday, August 31, 2007 6:06 PM To: math-fun Subject: Re: [math-fun] Toss Up Komi Joshua Zucker wrote, about Toss Up at 99-to-99:
This argument (that going first is good) doesn't make sense.
The second player won't adopt the same strategy as the first: the second player adopts the strategy of stopping when they get at least one point past the first player (or maybe stopping on a tie, if the first player was ambitious enough Knowing how many points the first player earned is a huge advantage. If the first player busts, you have an almost automatic win by just rolling once.
If the first player busts, then the game isn't yet about to end! It becomes the second player's turn, and the score is still 99-to-99, and we're back in the original game state with the players' identities swapped. (Perhaps I wasn't clear that I'm analyzing the original rules, in which busting only loses you the tentative points you've accumulated so far this turn. Points you've "banked" by voluntarily passing the dice are yours forever.) If the first player doesn't bust, then of course player 2 will adopt the strategy "beat player 1 or bust trying." But p2's probability of busting trying is just the same as was p1's of busting before reaching that point value in the first place. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun