Comments on: Pulling the goalie http://bit-player.org/2007/pulling-the-goalie An amateur's outlook on computation and mathematics. Fri, 29 Aug 2008 05:30:44 +0000 http://wordpress.org/?v=1.5.2 Comment on Pulling the goalie by: markr http://bit-player.org/2007/pulling-the-goalie#comment-1552 Thu, 06 Dec 2007 05:33:22 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1552 Two suggestions for altering the model: 1) Players roll simultaneously after secretly choosing a die. 2) The game ends after a predetermined number of rolls. Two suggestions for altering the model:

1) Players roll simultaneously after secretly choosing a die.
2) The game ends after a predetermined number of rolls.

]]> Comment on Pulling the goalie by: Anonymous http://bit-player.org/2007/pulling-the-goalie#comment-1547 Sat, 01 Dec 2007 03:10:33 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1547 Barry is right. Dynamic programming is your friend on small problem instances like this. Barry is right. Dynamic programming is your friend on small problem instances like this.

]]> Comment on Pulling the goalie by: Barry Cipra http://bit-player.org/2007/pulling-the-goalie#comment-1546 Fri, 30 Nov 2007 15:51:24 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1546 Elgee's dice game might be better thought of in a take-away sense: Each player starts with a pile of N stones and, taking turns, removes the number equal to the roll of a chosen die, the winner being the first player to empty his pile. In this way one can inductively calculate P(h,k), the probability of winning when you've got h stones left, your opponent has k stones, and it's your turn to roll a die: P(h,k) = max{1-p1*P(k,h-1)-p2*P(k,h-2)-p3*P(k,h-3)-p4*P(k,h-4)-p5*P(k,h-5)-p6*P(k,h-6)} where the max is taken over the available choices of (p1,p2,p3,p4,p5,p6), representing the probabilities of rolling 1,2,3,4,5, or 6 with a given die (so that p1+p2+p3+p4+p5+p6=1). Note this is a kind of min-max probability, which assumes your opponent adopts the same P-maximizing strategy: If he doesn't, your probability of success can only go up. It shouldn't be terribly time-consuming to compute the 10,000 P(h,k)'s to reach P(100,100). Obviously one should also keep track of D(h,k), which tells you which die gives the max P from position (h,k). Elgee’s dice game might be better thought of in a take-away sense: Each player starts with a pile of N stones and, taking turns, removes the number equal to the roll of a chosen die, the winner being the first player to empty his pile. In this way one can inductively calculate P(h,k), the probability of winning when you’ve got h stones left, your opponent has k stones, and it’s your turn to roll a die:

P(h,k) = max{1-p1*P(k,h-1)-p2*P(k,h-2)-p3*P(k,h-3)-p4*P(k,h-4)-p5*P(k,h-5)-p6*P(k,h-6)}

where the max is taken over the available choices of (p1,p2,p3,p4,p5,p6), representing the probabilities of rolling 1,2,3,4,5, or 6 with a given die (so that p1+p2+p3+p4+p5+p6=1). Note this is a kind of min-max probability, which assumes your opponent adopts the same P-maximizing strategy: If he doesn’t, your probability of success can only go up.

It shouldn’t be terribly time-consuming to compute the 10,000 P(h,k)’s to reach P(100,100). Obviously one should also keep track of D(h,k), which tells you which die gives the max P from position (h,k).

]]> Comment on Pulling the goalie by: Mark http://bit-player.org/2007/pulling-the-goalie#comment-1545 Fri, 30 Nov 2007 14:08:59 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1545 In relation to the game Piggy, there has been work by Neller and Presser on the simpler game of Pig to determine when you should continue rolling the die on your turn, and the results point to the strategy that you should be more risky as the other player gets closer to winning. Todd W. Neller and Clifton G.M. Presser. Optimal Play of the Dice Game Pig, The UMAP Journal 25(1) (2004), pp. 25-47. In relation to the game Piggy, there has been work by Neller and Presser on the simpler game of Pig to determine when you should continue rolling the die on your turn, and the results point to the strategy that you should be more risky as the other player gets closer to winning. Todd W. Neller and Clifton G.M. Presser. Optimal Play of the Dice Game Pig, The UMAP Journal 25(1) (2004), pp. 25-47.

]]> Comment on Pulling the goalie by: brian http://bit-player.org/2007/pulling-the-goalie#comment-1544 Fri, 30 Nov 2007 13:25:17 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1544 And at the opposite pole from retirement savings we have the desperate-gambler problem—the one where you enter a casino with $1,000, and you have to parlay it into $10,000 or your bookie will have you rubbed out. It's well known that the optimal strategy is bold play. And at the opposite pole from retirement savings we have the desperate-gambler problem—the one where you enter a casino with $1,000, and you have to parlay it into $10,000 or your bookie will have you rubbed out. It’s well known that the optimal strategy is bold play.

]]> Comment on Pulling the goalie by: Josh http://bit-player.org/2007/pulling-the-goalie#comment-1543 Fri, 30 Nov 2007 07:44:57 +0000 http://bit-player.org/2007/pulling-the-goalie#comment-1543 Sounds like something that might have been studied by economists or financial analysts as a simple model of risk. A typical economic scenario is the comparison of, say, two mutual funds: one with higher risk and higher average payoff, and one with lower risk but lower average payoff. As in hockey, it's not uncommon for people to invest riskily when they're young and have little to lose, and invest more safely once they've retired and have little means of getting their money back if they lose it gambling on stocks (a bit like the winning team in hockey). Sounds like something that might have been studied by economists or financial analysts as a simple model of risk. A typical economic scenario is the comparison of, say, two mutual funds: one with higher risk and higher average payoff, and one with lower risk but lower average payoff.

As in hockey, it’s not uncommon for people to invest riskily when they’re young and have little to lose, and invest more safely once they’ve retired and have little means of getting their money back if they lose it gambling on stocks (a bit like the winning team in hockey).

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