Poker Dice

In Poker Dice, the aim is to build the best possible poker hand, where straights and flushes don't count. The first person to roll has up to three throws of the dice and after each throw can put aside any dice they wish to use for their hand; also, they can stop after the first of second roll if they wish. After the first person establishes a number of rolls, each successive player may only throw the dice at most that many times. The player with the highest ranking hand after everyone has had a turn wins the game. Is there any way to gain an edge?


Here is a list of poker dice hand rankings and their probability of occurring with $N$ dice:

Hand         5Die 4Die 3Die 2Die

5 of a kind     6    -    -    -

4 of a kind   150    6    -    -

full house    300    -    -    -

3 of a kind  1200  120    6    -

2 pair       1800   90    -    -

1 pair       3600  720   90    6

Nothing       720  360  120   30

Total        7776 1296  216   36

Since the first player has a strategic decision to make on whether to keep his hand or continue rolling, we can analyze the game on each turn. For example, with what hands should the first player stop on the first roll?

Let's say we go first and roll a pair of 5s on the first roll with $N$ players. According to the above chart, the probability of another player rolling a worse hand on their first throw is 3120/7776 so $p=P(win\ or\ draw\ with\ 5s\ |\ N)=(\frac{3120}{7776})^{N-1}$. If $p$ is greater than 1/$N$, we should keep the hand. 


Following this logic, we can see what hands (or better) we should stick with on the first roll given the total number of players:

    2     3     4     5
2211X 3322X 4433X 5533X

Continuing on we see that we should not stick with the pair's and some two-pair combinations on the first roll given a certain amount of players. Considering we should roll again, the following table show us the updated percentage of making a certain hand when choosing keeping the sub-par hand from the first roll (hand\first roll):

            Nothing 1pair 2pair
5 of a kind       6     1     -
4 of a kind     150    15     -

full house      300     5     2

3 of a kind    1200    60     -

2 pair         1800    75     4

1 pair         3600    60     -

Nothing         720     -     -

Total          7776   216     6

In each case, players should keep the best hand from the first roll -- increasing their chances of a better hand and eliminating the possibility of not making anything. 

Now we have a rule set for the second roll: keep the first roll as indicated by the prior chart; otherwise, keep the best hand from the first roll and re-roll the remaining dice. Using this strategy favorably tweaks the ways certain hands can be made:

Hand         Ways

5 of a kind   209

4 of a kind  3725

full house   3700

3 of a kind 20800

2 pair      28950

1 pair      12000

Nothing       600

Total       69984

Using a similar analysis as above, we we can see what hands (or better) we should stick with on the second roll given the total number of players:

    2     3     4     5

6633X 111XX 222XX 333XX