Lottery with a twist

Thanks all for your feedback.

Here's a new idea.

We pick a number from 10000 to 99999.

There are 5 participants.
Player1: Buys a lottery ticket $10, picks 12000
Player2: Buys a lottery ticket $10, picks 23450
Player3: Buys a lottery ticket $10, picks 69990
Player4: Buys a lottery ticket $10, picks 52123
Player5: Buys a lottery ticket $10, picks 73234

We draw the number: 54321

Now, we will distribute the whole pot among all the players.

1. Rank the players based on how close their guess was to the actual number:

   Player4: 1 (closest)
   Player3: 2
   Player5: 3
   Player2: 4
   Player1: 5 (furthest)

2. Calculate the inverse of each player's rank:

   Player4: 1/1 = 1
   Player3: 1/2 = 0.5
   Player5: 1/3 = 0.33
   Player2: 1/4 = 0.25
   Player1: 1/5 = 0.2

3. Sum up all the inverse ranks: 1 + 0.5 + 0.33 + 0.25 + 0.2 = 2.28 (harmonic series)

4. Calculate each player's share of the pot:

   Player4: (1 / 2.28) * $50 = $21.93
   Player3: (0.5 / 2.28) * $50 = $10.96
   Player5: (0.33 / 2.28) * $50 = $7.24
   Player2: (0.25 / 2.28) * $50 = $5.48
   Player1: (0.2 / 2.28) * $50 = $4.39

In this system, we always distribute the pot, and the closer you are, the more you get. You even have the chance to break even.

Payouts with 100 players

Let's calculate the shares until we reach a share that is $10 or more: (to check break even)

Player1: (1 / 5.18738) * $1000 = $192.78
Player2: (0.5 / 5.18738) * $1000 = $96.39
Player3: (0.33 / 5.18738) * $1000 = $63.62
Player4: (0.25 / 5.18738) * $1000 = $48.19
Player5: (0.2 / 5.18738) * $1000 = $38.56
Player6: (0.166 / 5.18738) * $1000 = $32.01
Player7: (0.142 / 5.18738) * $1000 = $27.37
Player8: (0.125 / 5.18738) * $1000 = $24.11
Player9: (0.111 / 5.18738) * $1000 = $21.39
Player10: (0.1 / 5.18738) * $1000 = $19.28
Player11: (0.09 / 5.18738) * $1000 = $17.35
Player12: (0.083 / 5.18738) * $1000 = $16.00
Player13: (0.076 / 5.18738) * $1000 = $14.65
Player14: (0.071 / 5.18738) * $1000 = $13.68
Player15: (0.066 / 5.18738) * $1000 = $12.73
Player16: (0.0625 / 5.18738) * $1000 = $12.05
Player17: (0.058 / 5.18738) * $1000 = $11.18
Player18: (0.055 / 5.18738) * $1000 = $10.60
Player19: (0.052 / 5.18738) * $1000 = $10.02

This could be based on the The harmonic series. The pot distribution could be calculated on this function.

The harmonic series for 100 is:

H(100) = 1 + 1/2 + 1/3 + ... + 1/100 ≈ 5.18738...

In the harmonic series distribution the key factor is the accuracy of the guess, not the quantity of guesses. The distribution of the prize pool is still based on the rank of each player's guess, not the number of tickets they purchased. So, even if a player buys many more tickets than others, they would still need to guess closer to the drawn number to receive a larger portion of the prize pool. This aspect of the game could be seen as a form of fairness, as it rewards accuracy over quantity.

The point is, everyone will be a winner. But the winning amount will be based on how close you are to the actual drawn number. You can earn more, break even, or earn less than your initial guess. Thoughts?