I decided to do another mathematical analysis of a common event in pinball tournaments - ties in bracket matchplay. This is intended to be (marginally) informative for tournament directors concerned with tournament duration.
The following analysis assumes a group of 4 players who play N games (as is typical in a bracket format like single-elimination) and receive points based on each game result. A player’s points are summed across games to determine the winner, or who moves on to the next round. Each player is assumed to have an equal chance of getting 1st, 2nd, 3rd, or 4th on a game (i.e., all players are of equal skill).
If the players with the top 2 scores proceed to the next round, the relevant probability is the tie for 2rd/3rd. For the final round of a tournament, other probabilities may be of greater interest, such as the probability of a tie for 1st/2nd.
RECIPE
All permutations of game results are enumerated, and the points tallied. The number of outcomes which result in various kinds of ties are then counted, and these counts are divided by the total number of outcomes. The resulting fraction is then expressed as a percentage in the tables below.
RESULTS
First, the results for 4-2-1-0 game scoring:
| Type of Tie | 2 Games Played | 3 Games Played | 4 Games Played | 5 Games Played |
|---|---|---|---|---|
| Any Ties | 54.17% | 42.01% | 40.20% | 38.07% |
| 2nd/3rd | 20.83% | 19.62% | 17.27% | 15.52% |
| 2nd/3rd ONLY | 12.50% | 15.62% | 14.97% | 13.19% |
| 1st/2nd ONLY | 8.33% | 10.42% | 8.81% | 10.33% |
| 2nd/3rd/4th ONLY | 4.17% | 2.26% | 1.24% | 1.04% |
| 1st/2nd/3rd ONLY | 4.17% | 1.74% | 0.89% | 1.29% |
| All Tied | 0.00% | 0.00% | 0.17% | 0.00% |
Next the results for 3-2-1-0 game scoring (or, identically, any linear sequence of numbers such as 7-5-3-1, etc.):
| Type of Tie | 2 Games Played | 3 Games Played | 4 Games Played | 5 Games Played |
|---|---|---|---|---|
| Any Ties | 70.83% | 44.10% | 51.02% | 42.73% |
| 2nd/3rd | 37.50% | 20.14% | 23.73% | 19.14% |
| 2nd/3rd ONLY | 25.00% | 14.58% | 19.98% | 15.15% |
| 1st/2nd ONLY | 12.50% | 8.33% | 11.23% | 9.18% |
| 2nd/3rd/4th ONLY | 4.17% | 2.78% | 1.50% | 2.00% |
| 1st/2nd/3rd ONLY | 4.17% | 2.78% | 1.50% | 2.00% |
| All Tied | 4.17% | 0.00% | 0.76% | 0.00% |
And now the results for scoring akin to Fair Strikes: 2-1-1-0. I have not seen this scoring used anywhere, but there’s no reason why it couldn’t be used, so I’ve included it for completeness:
| Type of Tie | 2 Games Played | 3 Games Played | 4 Games Played | 5 Games Played |
|---|---|---|---|---|
| Any Ties | 100.00% | 79.17% | 74.07% | 66.97% |
| 2nd/3rd | 50.00% | 45.83% | 37.62% | 33.98% |
| 2nd/3rd ONLY | 41.67% | 40.28% | 32.23% | 29.16% |
| 1st/2nd ONLY | 41.67% | 12.50% | 13.54% | 12.73% |
| 2nd/3rd/4th ONLY | 0.00% | 1.39% | 1.39% | 1.54% |
| 1st/2nd/3rd ONLY | 0.00% | 1.39% | 1.39% | 1.54% |
| All Tied | 8.33% | 2.78% | 2.60% | 1.74% |
By request, here are the results for Marburg scoring: 7-4-2-0. I was not familiar with this scoring method. It’s sort of halfway between 4-2-1-0 and 3-2-1-0, allowing it to dodge a bunch of tie situations.
| Type of Tie | 2 Games Played | 3 Games Played | 4 Games Played | 5 Games Played |
|---|---|---|---|---|
| Any Ties | 45.83% | 24.31% | 25.72% | 23.15% |
| 2nd/3rd | 12.50% | 9.72% | 12.13% | 9.42% |
| 2nd/3rd ONLY | 8.33% | 7.29% | 11.33% | 8.70% |
| 1st/2nd ONLY | 8.33% | 7.29% | 4.99% | 6.69% |
| 2nd/3rd/4th ONLY | 4.16% | 0.69% | 0.28% | 0.58% |
| 1st/2nd/3rd ONLY | 0.00% | 1.74% | 0.35% | 0.14% |
| All Tied | 0.00% | 0.00% | 0.17% | 0.00% |
And now some more that nobody asked for. Here are the results for the ‘OG’ 10-5-1-0 scoring:
| Type of Tie | 2 Games Played | 3 Games Played | 4 Games Played | 5 Games Played |
|---|---|---|---|---|
| Any Ties | 45.83% | 24.31% | 23.98% | 20.12% |
| 2nd/3rd | 20.83% | 12.85% | 10.44% | 8.34% |
| 2nd/3rd ONLY | 16.67% | 10.42% | 9.81% | 7.78% |
| 1st/2nd ONLY | 8.33% | 7.29% | 6.55% | 6.15% |
| 2nd/3rd/4th ONLY | 4.16% | 0.69% | 0.28% | 0.49% |
| 1st/2nd/3rd ONLY | 0.00% | 1.74% | 0.17% | 0.07% |
| All Tied | 0.00% | 0.00% | 0.17% | 0.00% |
CODE
The code used to populate these tables is a Mathematica script:
tieProbabilities[nGames_, scoring_] :=
Module[{nPlayers, perms, results, nResults, tiedResults,
tied23PlusResults, tied12Results, tied23Results, tied123Results,
tied234Results, tiedAllResults},
nPlayers = Length@scoring;
perms = Permutations@scoring;
results =
Map[Total[#] + scoring &, Tuples[perms, {nGames - 1}], {1}];
results = Sort[#, Greater] & /@ results;
nResults = Length@results;
tiedResults = Select[results, Length@Union@# != nPlayers &];
tied23PlusResults = Select[results, #[[2]] == #[[3]] &];
tied12Results =
Select[results, #[[1]] == #[[2]] &&
Length@Union@# == nPlayers - 1 &];
tied23Results =
Select[results, #[[2]] == #[[3]] &&
Length@Union@# == nPlayers - 1 &];
tied123Results =
Select[results, #[[1]] == #[[2]] == #[[3]] &&
Length@Union@# == nPlayers - 2 &];
tied234Results =
Select[results, #[[2]] == #[[3]] == #[[4]] &&
Length@Union@# == nPlayers - 2 &];
tiedAllResults = Select[results, Length@Union@# == nPlayers - 3 &];
Return[N[100 Length[#]/nResults] & /@ {tiedResults,
tied23PlusResults, tied23Results, tied12Results, tied123Results,
tied234Results, tiedAllResults}]
]
