### Ranked Draws & Rank-Based Scoring: Fibonacci-Diplo

Posted:

**05 Jun 2018, 04:39**The following is representative of my personal opinions as a player and GM, not as a moderator.

Background

Here on PlayDiplomacy, there are roughly two layers of scoring that occur with Rated Games:

In this graph and others in this thread, raw DSS score is shown on a 420-point scale. Elo changes are estimated as the average change in a match with seven longtime players of equal rating, and the Elo bonus for a solo is estimated as 20%. These numbers are probably incorrect, but the actual numbers matter less than the overall proportions of points awarded/lost.

We don't see the actual DSS points, but we do see our own Elo Rating changes (#2) after a game is finished. There are a number of modifications that occur on top of PD's Elo system, but roughly the way it works is that you have an expected performance value based on your current rating as compared to your opponents' ratings. If all opponents start with the same rating, then their expected value is 14.29% of the points (one-seventh). If you score higher-than-expected in a game, you earn Elo rating. If you score lower-than-expected, you lose points. If your expected value was higher than the average, you can potentially lose points even if you are part of a draw.

So What?

So, here's the thing: I really like our Elo system. I think it's one of the smartest decisions this site made, and it's especially nice that everyone has varied categories to judge their ratings in. Earning points based on the relative skill levels of your opponents is a good long-term strategy for making ratings meaningful and reward players for exceptional results, particularly against better-skilled competitors.

What I'm not a huge fan of, however, is the Draw-Sized Scoring. Or rather, I'm not a fan of the fact that it is the only option for scoring a game. There are many different scoring systems used for Diplomacy tournaments, particularly in the face-to-face competitive scene, and jay65536 has provided some lengthy discussion of their relative merits in this thread, which I recommend. I have a short summary of some of the scoring system rules in another thread.

So Where is this Going?

I would like to see PlayDiplomacy provide an option for an alternative rank-based scoring system, to be set on a per-game basis, that will still play nicely with our existing Elo rating system.

Background: Carnage

One popular rank-based system is Carnage; it is used in a number of US tournaments. In a solo, Carnage still awards 100% of points to the winner, just like our current DSS system (in an older form, it actually awarded a solo one million points, but that's been adjusted to make Carnage a fixed-sum system). In a draw, however, not every player is equal. Instead, players receive score based on their rank relative to their opponents, as calculated by comparing SC counts for surviving players and years-of-elimination for the rest. The draw size does not matter, and draws include all survivors. Getting points in a Carnage game comes from preventing the solo, surviving as long as possible, and finding ways to one-up your peers (without having to fully remove them from the map in order to gain an edge on them).

(I again highly suggest reading jay65536's thread to understood what benefits that rank-based scoring can bring, and why so few tournaments use DSS.)

The trouble with simply implementing Carnage here on PlayDip is that it isn't designed to scale with different maps, variants, and SC counts. It's designed to work with only Classic Diplomacy, which is all they play at F2F tournaments. Now, you can hypothetically drop Carnage's SC count points, which are really just a tiebreaker, and provide points that scale linearly based only on rank:

By itself, it's not a bad system, but it doesn't scale consistently with different player counts. As seen above, in a 7-player match, the board-topping player in a draw receives 25% of the points (7/28).

In a 5-player match (like Ancient Med), the board-topping player gets 33.3% of the points (5/15). In a 10-player match (which we don't have yet, but one can dream), an equivalent system would grant the board-topper only 18% of the points (10/55).

If one really wants a rank-based system to work across the vast spectrum of games that PD does support or could support in the future, and one wants the system to play nicely alongside other games that are DSS scored, Carnage isn't the way to go. The best that a game could ever offer a 1st-rank-in-draw player would be the equivalent of a four-way draw in DSS.

(Incidentally, Sum-of-Squares, another popular scoring system, has the opposite problem. When compared alongside DSS, potential Sum-of-Squares results are too good, with single-player results possible far over 50% of points in a draw.)

So standard Carnage scoring is out. So is a related rank-based system, C-Diplo, since it also can't scale with different map sizes or player counts. I've been playing around with the mathematics, however, and after chatting with Dave Maletsky (the originator of Carnage) I think I have a potential solution...

Introducing: Fibonacci-Diplo

A rank-based scoring system that operates like Carnage, but scales with varying player counts

While Carnage points scale linearly with rank, Fibonacci-Diplo uses Fibonacci numbers as a basis for score.

One useful mathematical property of Fibonacci numbers is that the ratio of the highest term in a finite Fibonacci sequence, as compared to the total of all terms in that sequence, quickly approaches a constant 38% (which is related to the Golden Ratio and the Golden Spiral).

In Fibonacci-Diplo, players in a draw are still ranked by SC counts and years-of-elimination at game end, just like Carnage. The number of points awarded, however, scales with the Fibonacci Sequence instead of linearly.

In a typical Fibonacci draw for a game of 7, the board-topper would receive 40% of total points. This is nicely wedged in-between the 33.3% awarded in DSS for a three-way and the 50% for a two-way. The second-place player would net the equivalent of a four-way. The third-place would earn 15%, between a six-way and seven-way. Further ranks would earn less than the equivalent 7-way draw in DSS; once converted to Elo, this would amount on average to a loss in rating, but it would not always be as harsh as a loss in DSS (particularly for the fourth-place finisher, who depending on their current rating, might earn a few points after all).

Meanwhile, as mentioned previously, a solo would still work exactly the same as before.

What Would this Look Like on the Site?

Under the "Draw Proposals" setting during game creation, there would be a new option in the dropdown: "Rank all players in a draw". When a game using this option ends in a draw (DIAS), the results would show each player's rank (1st-7th, or 1st-5th for Ancient Med).

If that game is also rated, then points would be awarded according to the Fibonacci-Diplo system described above prior to converting those values into Elo ratings adjustments. Below are several potential end results possible (since scores shift around based on ties in rank, actual draws can vary in points awarded).

Support this Proposal!

Thanks for taking the time to read or skim through this proposal. I really believe that having the option of rank-based scoring on PlayDiplomacy, rather than solely draw-sized scoring, would add to the overall variety of games and to the enjoyment of players. Rank-based games would shift the emphasis on draw-whittling to jockeying for position, providing more incentive to attack equal-or-better-positioned peers rather than knock out weaker powers. With Fibonacci-Diplo, I believe I've come up with a fixed-sum rank-based scoring formula that will scale well for any game of five or more players and will provide endgame scores comparable to those provided in standard DSS games.

Background

Here on PlayDiplomacy, there are roughly two layers of scoring that occur with Rated Games:

- Draw-Sized Scoring (DSS) on a per-game basis
- Zero-sum Elo Rating modifications

In this graph and others in this thread, raw DSS score is shown on a 420-point scale. Elo changes are estimated as the average change in a match with seven longtime players of equal rating, and the Elo bonus for a solo is estimated as 20%. These numbers are probably incorrect, but the actual numbers matter less than the overall proportions of points awarded/lost.

We don't see the actual DSS points, but we do see our own Elo Rating changes (#2) after a game is finished. There are a number of modifications that occur on top of PD's Elo system, but roughly the way it works is that you have an expected performance value based on your current rating as compared to your opponents' ratings. If all opponents start with the same rating, then their expected value is 14.29% of the points (one-seventh). If you score higher-than-expected in a game, you earn Elo rating. If you score lower-than-expected, you lose points. If your expected value was higher than the average, you can potentially lose points even if you are part of a draw.

So What?

So, here's the thing: I really like our Elo system. I think it's one of the smartest decisions this site made, and it's especially nice that everyone has varied categories to judge their ratings in. Earning points based on the relative skill levels of your opponents is a good long-term strategy for making ratings meaningful and reward players for exceptional results, particularly against better-skilled competitors.

What I'm not a huge fan of, however, is the Draw-Sized Scoring. Or rather, I'm not a fan of the fact that it is the only option for scoring a game. There are many different scoring systems used for Diplomacy tournaments, particularly in the face-to-face competitive scene, and jay65536 has provided some lengthy discussion of their relative merits in this thread, which I recommend. I have a short summary of some of the scoring system rules in another thread.

So Where is this Going?

I would like to see PlayDiplomacy provide an option for an alternative rank-based scoring system, to be set on a per-game basis, that will still play nicely with our existing Elo rating system.

Background: Carnage

One popular rank-based system is Carnage; it is used in a number of US tournaments. In a solo, Carnage still awards 100% of points to the winner, just like our current DSS system (in an older form, it actually awarded a solo one million points, but that's been adjusted to make Carnage a fixed-sum system). In a draw, however, not every player is equal. Instead, players receive score based on their rank relative to their opponents, as calculated by comparing SC counts for surviving players and years-of-elimination for the rest. The draw size does not matter, and draws include all survivors. Getting points in a Carnage game comes from preventing the solo, surviving as long as possible, and finding ways to one-up your peers (without having to fully remove them from the map in order to gain an edge on them).

(I again highly suggest reading jay65536's thread to understood what benefits that rank-based scoring can bring, and why so few tournaments use DSS.)

The trouble with simply implementing Carnage here on PlayDip is that it isn't designed to scale with different maps, variants, and SC counts. It's designed to work with only Classic Diplomacy, which is all they play at F2F tournaments. Now, you can hypothetically drop Carnage's SC count points, which are really just a tiebreaker, and provide points that scale linearly based only on rank:

Super-Simplified Carnage: 7 player game

- 1st - 7000 points
- 2nd - 6000 points
- 3rd - 5000 points
- 4th - 4000 points
- 5th - 3000 points
- 6th - 2000 points
- 7th - 1000 points

By itself, it's not a bad system, but it doesn't scale consistently with different player counts. As seen above, in a 7-player match, the board-topping player in a draw receives 25% of the points (7/28).

Super-Simplified Carnage: 5 player game

- 1st - 5000 points
- 2nd - 4000 points
- 3rd - 3000 points
- 4th - 2000 points
- 5th - 1000 points

In a 5-player match (like Ancient Med), the board-topping player gets 33.3% of the points (5/15). In a 10-player match (which we don't have yet, but one can dream), an equivalent system would grant the board-topper only 18% of the points (10/55).

If one really wants a rank-based system to work across the vast spectrum of games that PD does support or could support in the future, and one wants the system to play nicely alongside other games that are DSS scored, Carnage isn't the way to go. The best that a game could ever offer a 1st-rank-in-draw player would be the equivalent of a four-way draw in DSS.

(Incidentally, Sum-of-Squares, another popular scoring system, has the opposite problem. When compared alongside DSS, potential Sum-of-Squares results are too good, with single-player results possible far over 50% of points in a draw.)

So standard Carnage scoring is out. So is a related rank-based system, C-Diplo, since it also can't scale with different map sizes or player counts. I've been playing around with the mathematics, however, and after chatting with Dave Maletsky (the originator of Carnage) I think I have a potential solution...

Introducing: Fibonacci-Diplo

A rank-based scoring system that operates like Carnage, but scales with varying player counts

While Carnage points scale linearly with rank, Fibonacci-Diplo uses Fibonacci numbers as a basis for score.

Example Fibonacci Number Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...

One useful mathematical property of Fibonacci numbers is that the ratio of the highest term in a finite Fibonacci sequence, as compared to the total of all terms in that sequence, quickly approaches a constant 38% (which is related to the Golden Ratio and the Golden Spiral).

In Fibonacci-Diplo, players in a draw are still ranked by SC counts and years-of-elimination at game end, just like Carnage. The number of points awarded, however, scales with the Fibonacci Sequence instead of linearly.

Fibonacci-Diplo: 7 player game (points in brackets represent scaling to 420 fixed-sum basis)

Solo: 20 points (420) to the winner, 0 to all others.

In a draw, rank all players by SC counts (eliminated players by years-of-elimination):Players that tie split the points awarded to all applicable ranks.

- 1st - 8 points (168)
- 2nd - 5 points (105)
- 3rd - 3 points (63)
- 4th - 2 points (42)
- 5th - 1 point (21)
- 6th - 1 point (21)
- 7th - 0 points (0)

In a typical Fibonacci draw for a game of 7, the board-topper would receive 40% of total points. This is nicely wedged in-between the 33.3% awarded in DSS for a three-way and the 50% for a two-way. The second-place player would net the equivalent of a four-way. The third-place would earn 15%, between a six-way and seven-way. Further ranks would earn less than the equivalent 7-way draw in DSS; once converted to Elo, this would amount on average to a loss in rating, but it would not always be as harsh as a loss in DSS (particularly for the fourth-place finisher, who depending on their current rating, might earn a few points after all).

Meanwhile, as mentioned previously, a solo would still work exactly the same as before.

What Would this Look Like on the Site?

Under the "Draw Proposals" setting during game creation, there would be a new option in the dropdown: "Rank all players in a draw". When a game using this option ends in a draw (DIAS), the results would show each player's rank (1st-7th, or 1st-5th for Ancient Med).

If that game is also rated, then points would be awarded according to the Fibonacci-Diplo system described above prior to converting those values into Elo ratings adjustments. Below are several potential end results possible (since scores shift around based on ties in rank, actual draws can vary in points awarded).

Support this Proposal!

Thanks for taking the time to read or skim through this proposal. I really believe that having the option of rank-based scoring on PlayDiplomacy, rather than solely draw-sized scoring, would add to the overall variety of games and to the enjoyment of players. Rank-based games would shift the emphasis on draw-whittling to jockeying for position, providing more incentive to attack equal-or-better-positioned peers rather than knock out weaker powers. With Fibonacci-Diplo, I believe I've come up with a fixed-sum rank-based scoring formula that will scale well for any game of five or more players and will provide endgame scores comparable to those provided in standard DSS games.