Estimating the score of the 5th ranked player in an anonymous tournament I have the following problem that I want to solve using simulation. The context is the following. We have a tournament in which each game of the tournament is played by five players of different colors. Each player plays three games per round and only the Tournament director knows which player is playing each color (this information will be revealed at the end of the round). The players must play with three different colors in each of their three games in the round. The tournament round includes many players, for example, it could include 40 players playing a total of 16 games.
We have data of the current scores of each color in each game. Note that all games have the same colors.




Game
Blue
Red
Green
Purple
White




1
5.0
35.0
0.0
13.0
16.0


...
...
...
...
...
...




The final score of a player will be the sum of their scores in the three games of the round. We would like to estimate the current score (not final score) of the Nth (but really 5th) ranked player before the end of the round. When we are doing this estimation we don't know which player is in each game/color; so we don't know which three scores to add for each player.
My approach so far is "simulating" the score of each hypothetical player by picking scores from our data at random, being careful that each player gets three different colors for the round. I then sort the results and extract the 5th highest score and record that data.
My questions are:

*

*Is this a suitable approach?

*Is the mean the right summary statistics for the resulting values? Is there a different way to summarize such data?

*Is there a name for what I am doing that I can search for?

*Is there a way to estimate how many simulation trials I need to get an accurate number?

*How would I report appropriate intervals for my summary statistics?

Thanks for your attention.
Edit: I modified the table to provide more realistic scores
Edit 2: tried adding additional clarifying information (in italics)
 A: I suspect that your dataset may not be rich enough to determine what you want to calculate. To see why, it might be useful to see why your simulation approach will fail in a scaled down and stylized version of your setting.
Assume 3 instead of 5 players, and assume that only 2 rounds are played. Assume that player 1 always scores 1, player 2 always scores 2, and player 3 always scores 3. Finally, assume we are looking for the 3rd ranked player. In my above example, the correct answer should be that the third ranked player is player 1 who scores 2 total points.
Suppose you run your simulation now. The following situation happens with positive probability: your first hypothetical player takes player 1's score in game 1 and player 3's score in round 2. Your second hypothetical player takes player 2's score in both games. Your third hypothetical player takes player 3's score in game 1 while taking player 1's score in game 2. In that case, your three simulated players all scored 4, so you would conclude that the 3rd rank player sometimes scores 4, but this would be incorrect.
The point that the above example is meant to demonstrate is more general than the specifics of the example, so it does not seem like a more clever way of simulating things will help you. Specifically, the main problem is that given the information available in your dataset, multiple different processes could have generated the same observed outcome. For example, in my example above, your dataset will always have two rows with 1,2,3 shuffled around. Any alternative process where exactly one player scores 1, exactly one player scores 2, and exactly one player scores 3 each round would be equally consistent with the observed data, but the relative probabilities that each player scores 1, 2, or 3 will influence the distribution of how well the 3rd ranked player scores.
A: I do not think you need a simulation to accomplish what you're trying to do. We can take the simple example where there are only three colors (R, G, B) and two rounds. Let's assume that the scores in the two rounds were:
Round 1:
R:2    G:10   B:100
Round 2:
R:12   G:98   B:3
You do not know how the players switched colors. But since a player is not allowed to get the same color twice in a row, there are only two ways in how the color switching could have happened:
A) R->G, G->B, B->R
or
B) R->B, G->R, B->G
You could calculate for both scenarios what the final score of the 3rd best player would be. In this case it would be 13 for case A) and 5 for case B). These are the only two possibilities. How you want to choose your final guess is up to you and depends on how likely the different color switching scenarios are. One way is to assume all switching scenarios are equally likely, then the "best" guess (at least in terms of expected squared error) is the mean value of 5 and 13 which equals 9. I guess this is what your simulation is doing, generating different color switching scenarios with equal probability. In the example above, however, one scenario of switching is perhaps more probable then the other. If we just look at the scores it seems unlikely that one player would score only 2 in one round, and then suddenly 98 in the second, as it is the case in A). Assigning different probabilities to the switching scenarios would change the calculation of the expected value of the 3rd ranked player's final score.
If one extends the situation to 5 colors and 2 rounds, there are 44 possible switching scenarios (number of derangement permutations). If a player is not allowed to get the same color twice in a row then for 3 rounds this would result in $44*44 = 1936$ possible switching scenarios (if the rule is that in none of the three rounds a player can have the same color then there are even fewer valid ways of switching). Since that number is not that big, you could calculate the final score of the 5th ranked player for all scenarios and then for example take the mean as described above. This should give you the same result as running your simulation with infinite replications.
