# Optimal Number of Entries in Contest of Skill

Objective: I'm looking for the optimal number of unique entries in a contest of skill with monetary prizes.

Description: The contests vary from as little as 20 entries up to 10,000+ entries. You can enter an unlimited amount of entries. Within each entry, there are 9 slots that can be filled with variables. Each slot has a unique set of variables that cannot be used in any other slot. The unique variables for each slot may number from about 2-30 depending on the contest. Once you enter your chosen variables in the slots, the variables score points based upon real-life events. Some slots traditionally score many more points than other slots. The entry with the highest amount of points wins. Prize money is split in the event of a tie score.

Assumptions: I am an above average player. When facing similar competition, assume I can win 57% of the time (if playing in a fifty/fifty contest where half of the entrants win prize money). The contest has exactly 10,055 entries. The entry fee is $2. The prize distribution is detailed as follows (minus 10.5% entry fee going to contest organizers): Prizes: 1st:$2,000; 2nd: 1,200; 3rd: 800; 4th: 500; 5th: 350; 6th: 300; 7th: 250; 8th: 200; 9th: 180; 10th: 160; 11th: 150; 12th: 140; 13th: 130; 14th: 120; 15th: 110; 16th: 100; 17th - 18th: 90; 19th - 20th: 80; 21st - 22nd: 70; 23rd - 25th: 60; 26th - 28th: 50; 29th - 35th: 40; 36th - 50th: 30; 51st - 60th: 20; 61st - 75th: 18; 76th - 100th: 16; 101st - 200th: 14; 201st - 400th: 12; 401st - 600th: 10; 601st - 800th: 8; 801st - 1050th: 6

Discussion: I have an above average understanding of probability (but that's not really saying much), so I've gotten as far as running a simulation (the paper and pencil kine) with 10 entries and a 10 dollar entry fee (1st-$50, 2nd-30, 3rd-20). Internet research suggests, and my paper simulations concur, that using 2 or more of the exact same entries will always result in a negative expected value (EV). I then tried to run the simulation with two different entries. There seem to be 44 unique possibilities. I added up the prize money; then, I multiplied by 1/44. I took this answer then subtracted the$20 in entry fees. The result was an expected value of 45 cents. Is this correct?

Assuming my math is correct, I have no clue how to apply it to the contests with a much higher number of entries to find the optimal number of unique entries I should submit. I don't really know how to translate my assumed 57% success rate in one contest to another contest. I also don't know if it's best to have completely unique entries or just slightly unique entries (it seems that I should diversify my entries and make them as unique as possible while still maximizing points scored).

Any help, guidance, or tips are very much appreciated. Have a great day.

P.S. If you know the contests I'm talking about, please don't refer to them by name.

P.S.S. I wasn't sure whether to post this question here or on the math.stackexchange