Even reward to all the players in the race based on their power

Question

Here are the results of $1,000,000$ games between $7$ players

        1       2       7       10      16      17      47      

1       1.00    1.99    7.03    9.98    16.00   16.99   47.01   
2       1.48    2.99    9.92    13.88   20.89   22.05   28.78   
3       2.20    4.33    13.72   18.00   23.11   23.26   15.38   
4       3.56    6.86    19.24   22.72   20.83   20.20   6.58    
5       6.69    12.72   28.67   22.99   14.04   12.94   1.94    
6       22.18   41.03   17.35   10.54   4.56    4.06    0.29    
7       62.88   30.08   4.07    1.89    0.57    0.48    0.02

The first row represents the player power, the rows after it represents how many times that player won. For example, the player with the power of $47$ won the first place $47.01$ percent of the total games, he won the second place $28.78$ percent of the total games and so on.

The game is selecting a ball from a jar where each ball has a unique number between $0$ to $2^{256}$, during each game the players draw number of balls equal to their power, and never return it to the jar, then they keep the ball with the lowest number and throw away all the other balls, so each player only have one ball. The player with the lowest ball wins the first place, the player with the seconds lowest ball wins the second place an so on.

I have a reward of 100 coins and I would like to equally share it among the players based on their power. My first strategy was to only reward the first place, and it worked perfectly. However, it upset the rest of the players who didn't win.

Therefore I would like to come up with a new rewarding idea that will reward all the players after each game solely based on the game results. But still, after all the games are over, the total reward will be equally distributed among the players based on their power.

Answer 1

Sorry, but I think you've already found the only possible solution. Let $a_{ij}$ be the probability that player $i$ finishes in position $j$ on a given round. You want to award some fraction $f_j$ to the player who finishes in position $j$ on each round, and we may take $f_1+f_2+\dots+f_7=1.$ You also want the prizes to awarded in proportion to power, so the expectation of the players should be in proportion to their power. The expectation of player $i$ is $a_{i1}f_1+a_{i2}f_2+\dots+a_{i7}f_7$.

We can rescale the powers by dividing by the total power without changing the problem. In your example, we can make the powers $.01,.02, .07, .10, .16,.17,.47,$ and say that the players draw balls in proportion to their power; the problem doesn't change. Then the power of a player is simply the probability that he finishes in first place. We have $$ \mathbf{A}f = a,$$ where $\mathbf{A}$ is the matrix $(a_{ij})_{7\times 7},\ f=(f_1,\dots,f_7)^T,\ a=(a_{11},a_{21},\dots,a_{71})^T.$ Note that the right-hand side is the first column of $\mathbf{A}$ so that $$f=(1,0,0,0,0,0,0)^T$$ is a solution. Unless $\mathbf{A}$ is nonsingular, which seems highly unlikely (the set of singular matrices has measure zero) this is the only solution, and it's precisely what you are doing already.

Of course, there's nothing magic about $7$. The same result holds for any number of players.