Contingencies - March/April 2017

The game is modified again so that if there are ever exactly two survivors, they split the pot.

3. What is the expected house take
for the modified two-round game
with five players?

$100( 44+ 20)/1024 = $6.25. See prior solution, except that the house no longer wins if there are two survivors on the first round.

Challenge: Suppose the game is modified yet again so that as many rounds as needed to get to only 0, one, or two survivors are played.

What is the expected house take per round for the modified multi-round game with 6, 7, 8, and 9 players?

The answers are $7.06, $10.01, $12.64, and $14.25 for 6, 7, 8, and 9 players, respectively. Two respondents solved it using different methods: John Snyder and Tomasz Serbinowski. Another, David Promislow, came within pennies of the correct answers.

Snyder’s solution is worth reading in its entirety for its generality, graph, and references. We have posted it at contingencies.org/JanFeb2017solution. pdf. My own solution is somewhat simpler but closer to that of Serbinowski.

The fundamental question is how many of the (n– 1)n possible sets of choices for a round with n players will have k survivors where 0 ≤ k ≤ n– 2. I call that number S(n,k) and compute it using the relationships:

4. S(n,k) = ( nk) T(n,k), where T(n,k) is the same as S(n,k) except that the survivors are preset. In particular S(n,0) = T(n,0) = the number of derangements of n items. 5. T(n,k) = (n–k)T(n– 1,k– 1) + T(n– 1,k))

From 2) we get the above table for T(n,k) from which all the above results can be calculated.

Finally, I asked for modifications or extensions to this problem. Al Spooner suggested considering the effect of collusion, and Andrew Dean suggested looking at the jai alai quiniela.

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Solvers

Robert Bartholomew, Doug Bass, Bob Byrne, William Carroll, Bob Conger, Bernie Erikson, Bill Feldman, Yan Fridman, Steve Gallancy, Michael Gordy, Rui Guo, Tom Koons, Chi Kwok, David Lovit, Jerry Miccolis, Daniel Nolan, David Promislow, Noam Segal, Tomasz Serbinowski, John Snyder, Al Spooner, and Daniel Wade.