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Here is a 'lunch break' problem from a rather old publication.

Devise one set of rules for a dice game, where any number of players and one representative of the bank (mandatory), with one die each, can be playing, and where the players and the representative can roll their respective dice any number of times. The rules have to be such that the game is attractive to all the players (i.e. the players feel like there is a good chance of them winning), but that the bank would generate a good profit in the long run. In your game, how much would it cost for a player to play the game, and how much would the bank pay out in the case of a win for one player? What are the odds of a player winning, and what is the ‘house edge’?

This seems rather open-ended to me. Can anyone think of some interesting and profitable rules?

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    Brian deleted the previous question after I commented "Yes this is open-ended and you are supposed to use your own imagination" and Brian replied with the same comment he made above.2012-03-01

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One thought would be to have a function $f(n,h)$ where $n$ is the number of rolls and $h$ is the highest roll achieved. You would like it increasing in $h$ and decreasing in $n$, thinking of it as a score for the result. Then pay off depending on who is higher, maybe depending on how much, with the house winning ties.

Added:I didn't have enough information worked out to define the odds. I imagined any number of players, each against the bank (like blackjack). Each player has one die, which he rolls. He can keep the roll or try again. The player score is based on the highest roll achieved and the number of rolls, so if you roll a 6 you quit; if you roll a 1 you try again; if you are in the middle you need to decide. Each player gets a score, then the bank rolls. Bets are settled based on who wins, or maybe the score difference. But that's as far as I went.

To calculate the odds, you define a strategy for the bank. One example would be to roll until you get 5 or 6, then quit. Based on that strategy, you can calculate the probability for the bank of each score. In this case P(f(5,1))=P(f(6,1))=1/6. P(f(6,2))=(4/6)*(1/6) and so on. Then each player can define his strategy, with the best approach to maximize the payoff subject to the known bank behavior. Per the problem, the payoff should be negative, but there should be a number of positive payoffs available.

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    @Brian: Given a player strategy, you would calculate a distribution of scores. For the simple strategy of "take the first roll", the distribution is f(1,1)/6+f(2,1)/6+... Then you can calculate the distribution of bank scores since you know the bank strategy. The expectation for the player is just sum(P(player score)P(bank score)(payoff)) over all pairs of scores. Then you have to find a payoff function that meets that requirement, which is a psychological problem, not a mathematical one. Since people overrate big plusses you should pay well if the player rolls a 6 on the first roll.2012-03-04