Yahtzee

At lunch today the subject of Yahtzee came up.  I won a game last night against my girlfriend and eventually we drifted to score differentials.  It lead me to wonder. 

What’s the worst possible score a person could achieve playing a perfect strategy at Yahtzee

After a bit of thinking I settled on the score of 15 and here’s the logic behind it. 
Warning: This is not a mathematical proof or really a proof of any kind so please don’t read it as such (it lacks rigor).  It’s just a fun exercise I went through. 
The key to getting the lowest score is to get to a position where there is at least 1 roll of the dice which meets no category and forces a scratch of some category on the score sheet.  Once this the roller is in this position bad luck has them roll that set of dice N consecutive turns where N is the number of categories which have yet to receive a point or be scratched.
Ignore Chance for now as it’s a catch all and is unavoidable.  The perfect player would use Chance whenever they had either a very low or no scoring option and the sum of the dice was significant as compared to the max achievable Chance value.  The max achievable chance value is the highest sum of the dice which meets no other category (6,6,5,5,4=26). 
With chance out of the way the rest of the bottom half of a Yahtzee score sheet can be avoided by a roll of the dice which produces 2 pairs.  For example 1,1,2,2,3.  This matches none of the rest of the bottom and forces a scratch of a category, chance or use of the digit count category at the top.  The sequence 1,1,2,2,3 is chosen as the target roll because it has the lowest possible sum of any sequence which meets none of the bottom half of the yahtzee score card and hence will produce the lowest chance score. 
Now that we have a roll which doesn’t meet the bottom we need to eliminate it from the top with the lowest possible score.  Ideally we want a situation where 1,2 and 3 are filled up with the smallest values. 
Lets play our game.
Roll 1: 1,2,2,3,3
Here the player is at a loss.  In the top half of the score card you need to average 3 ticks for every digit category in order to hit your bonus.  If you only get one 1’s you are 2 point in debt, while if you get one 2 you are 4 points in debt (and one 3 has you 6 points in debt).  In this situation its best to take the one 1 because while all of the losses are equal the higher value dice have more upside with equal probability of hitting two or more in the same roll. 
Also the chance score here is 11 which is less than 1/2 of the maximum chance score of 26.  So the perfect player puts 1 tick in the 1 digit category.  
Roll 2: 1,1,2,3,3
Using the same logic as above, the perfect player puts 1 tick in the 2 digit category. 
Roll 3: 1,1,2,2,3
Using the same logic as above the perfect player puts 1 tick in the 3 digit category. 
Roll 4+: 1,1,2,2,3
This roll can go no other place than chance.  Eventually the perfect player is forced to take their chance of 9. 
Total Score:
1+2+3+9 = 15