There are 10 battlefields, numbered 1, 2, 3, ... , 10, and worth 1, 2, 3, ... , 10 points respectively. You have 100 fighters, which you can allocate between the battlefields however you wish. Your opponents also (independently) do the same.
3 strikes, you’re Out! Battlefields are fought in order from 1 to 10. The number of fighters on each battlefield is then compared, and for each battlefield, whoever has the most fighters on that battlefield wins its points (in the case of a tie, no one gets points). However, as soon as one player wins 3 consecutive battles, they automatically win all the remaining battles too.
For example, here is a match:
\begin{array} {|r|r|}\hline Battlefield & B1 & B2 & B3 & B4 & B5 & B6 & B7 & B8 & B9 & B10 \\ \hline Tom & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ \hline John & 5 & 10 & 15 & 17 & 8 & 5 & 5 & 15 & 18 & 2 \\ \hline \end{array}
In this match, Tom wins battlefields 1, 5, 6, 7, 8, 9 and 10 for a score of 46 points, and John wins battlefields 3 and 4 for a score of 7 points. No-one wins castle 2 (it’s a draw). Tom wins castles 8 and 9 because he has already won the 3 consecutive castles 5,6,7.
We're going to play a tournament. You get one entry and your final score is the average of your scores playing head-to-head against entries from several hundred people. You should try to maximise this average score. An entry should be submitted as a list of 10 non-negative integers, adding up to 100, where the nth element is the number of units of resources being sent to Battlefield n.
What's your entry? How did you go about coming up with it?
MY approach:
If you win B1, B2, B3 you automatically win all the other Bs, so the best thing to do would be to put 33,33,34 on these three Bs. However, the opponents are rational (I assume) and so they will do the same thing and at the end your score will not be maximized.
Any idea?
I brute-forced a simpler version of your game with only $10$ fighters in total and these were the top out of all possible entries where at least three battlefields have fighters in them:
$$\begin{array}{cc|cc|cc} 1 & (2, 3, 3, 2, 0, 0, 0, 0, 0, 0) & 6 & (1, 3, 3, 2, 1, 0, 0, 0, 0, 0) & 11 & (1, 4, 3, 2, 0, 0, 0, 0, 0, 0) \\ 2 & (3, 3, 3, 1, 0, 0, 0, 0, 0, 0) & 7 & (0, 3, 3, 3, 1, 0, 0, 0, 0, 0) & 12 & (0, 2, 3, 4, 1, 0, 0, 0, 0, 0) \\ 3 & (1, 3, 3, 3, 0, 0, 0, 0, 0, 0) & 8 & (2, 4, 3, 1, 0, 0, 0, 0, 0, 0) & 13 & (3, 3, 4, 0, 0, 0, 0, 0, 0, 0) \\ 4 & (0, 2, 3, 3, 2, 0, 0, 0, 0, 0) & 9 & (1, 3, 4, 2, 0, 0, 0, 0, 0, 0) & 14 & (0, 2, 4, 3, 1, 0, 0, 0, 0, 0) \\ 5 & (2, 3, 4, 1, 0, 0, 0, 0, 0, 0) & 10 & (1, 2, 3, 3, 1, 0, 0, 0, 0, 0) & 15 & (3, 4, 3, 0, 0, 0, 0, 0, 0, 0) \\ \end{array}$$
Using this as an inital population, I ran an evolutionary algorithm which returned the following:
$$\begin{array}{cc|cc} 1 & (1, 1, 40, 28, 29, 0, 1, 0, 0, 0) & 6 & (24, 35, 32, 0, 6, 0, 0, 3, 0, 0) \\ 2 & (23, 35, 37, 3, 0, 0, 2, 0, 0, 0) & 7 & (23, 35, 35, 2, 1, 0, 1, 2, 1, 0) \\ 3 & (1, 1, 37, 28, 28, 0, 1, 2, 1, 1) & 8 & (28, 37, 27, 2, 2, 2, 0, 2, 0, 0) \\ 4 & (25, 34, 33, 2, 2, 1, 2, 1, 0, 0) & 9 & (24, 37, 31, 1, 2, 1, 1, 1, 2, 0) \\ 5 & (25, 34, 32, 3, 5, 0, 0, 1, 0, 0) & 10 & (24, 37, 31, 0, 2, 0, 2, 3, 1, 0) \end{array}$$