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.
n\k01234567
21
322
49124
5 44 84 48 8
6 265 640 528 168 16
7 1,854 5,430 5,840 2,784 552 32
8 14,833 50,988 67,620 43,120 13,344 1,752 64
9 133,496 526,568 830,256 664,440 282,320 60,384 5,448 128