Ranked Draws & Rank-Based Scoring: Fibonacci-Diplo

[NoPunIn10Did]

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:

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

We as players don't actually see the draw-sized scores (#1), but they're certainly a part of the system. Every game has some finite amount of points available. In a solo, the winner gets 100% of the points. In a draw, the points are divided up equally between the players who are part of the draw: 50% each for a two-way, 33.3% for a three-way, 25% for a four-way, et cetera.


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

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

1. 1st - 5000 points
2. 2nd - 4000 points
3. 3rd - 3000 points
4. 4th - 2000 points
5. 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):

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

Players that tie split the points awarded to all applicable ranks.

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.

[Custer]

I've had that Fibonacci-Diplo before......and Its no joke! I wouldn't wish that on anybody!

The SHIV

[Nanook]

Speaking purely as a player...I LOVE this idea. Carnage is an awesome scoring system, and I'm glad that somebody much better at math than me found a way to adapt it to online play.

[rd45]

I love it. That's a very subtle & achievable way of enabling a potentially huge impact in terms of game play. End games can be such a grind, and large parts of the grind are based around the current focus on draw size, with all the drama around who has or hasn't "earned" their place in a draw. All of that could be gone. I love it.

[The Red Sphinx]

I’m not a numbers person so forgive me for not comprehending anything after you started discussing carnage.

But perhaps I can get an understanding of exactly what your proposed change wants to achieve. You mention it will prevent “draw whittling”? Would this be defined as a number of stronger powers in a game, agreeing to remove one or more smaller powers to then secure more points in the draw?

If so what exactly about the carnage system (or the variation your propose) would discourage it? Are you saying, as an example, that should you have three players, on 13, 11 and 4 SCs respectively... rather than the players on 13/11 SCs ganging up on the third to remove them prior to the draw agreement, the player on 11 will have an incentive to attack the top positioned player in an effort gain a higher ranking? If this is the case I can see some merit.

However, if it is simply saying that in the event these three players agree to a draw, the player with 13 SCs deserves more points than the player with 11, or 4... than I don’t think I agree at all. I don’t proscribe to the theory of those who believe a draw is practically a loss, however I don’t think it constitutes a win either. A player who shares in a draw should not be rewarded just because they finished with more SCs - If they want extra points they should play well and attain a solo.

I think you also risk discouraging solos, as some players know they will get a points boost, and decide to take an easy draw option rather than pursuing a slightly more difficult solo option. I’m neither a carebear nor solo only adherent, but solos should not be discouraged in anyway either.

Finally I disagree with the premise that by attaining 13 SCs, you have played a better game than the person on 11. I think most middling to experienced players, would have a very good idea of how to go about ensuring they reach 9-12 SCs, at the expense of ever having a solo opportunity. I think most players get to a point after playing this game where they can secure themselves territory into the mid-game, but learning how to progress and solo is still the difficult part, this proposed change would see that extra progression discouraged. What about an alliance on one side of the board who work brilliantly to secure their sphere, but before they can rush across the stalemate lines they run into a wall that is one player from across the other side who has conquered all and the game is drawn? Should they be punished because they worked well together rather than as an individual?

The game recognises 18 SCs as the tipping point at which a single person controls the board. The draw therefore recognised, by its very definition, that no one player controls the board, but collectively a number of players do. This is the important part - no matter the disparity in SC count, without all players contributing their SC count to the draw, then that draw would not be viable, therefore those players do not control the board.

[Nanook]

In general, carnage scoring tends to reward solos more than draw based scoring, because of the increased incentive to solo. In this system nopun has devised, a solo is worth 2.5 times what a board top is worth. That’s a pretty substantial increase, and incentive to go for a solo rather than settle for a board top (which is comparable in this system to a “good” draw in the current system). It’s also worth noting that in our current system, there are still many people who forsake a solo chance to secure a safe three way. No viable scoring system is going to eliminate that completely, but I doubt that this modified carnage will encourage it any more than our current system does.


Your criticism is a bit misguided where it starts talking about 13 SCs vs. 11, and one player having a bunch vs. two players having a bunch, at least as it pertains to carnage. Part of the idea behind carnage is that the more SCs you have, the closer you are to a solo—so one player with, say, 17 centers, is going to be rewarded more than two players with 17 between them (let’s say one at 10, one at 7 for the sake of argument), because that player with 17 is closer to a solo than a player at 10 or at 7. It kinda seems like your criticism here is getting a bit crossed—you say the goal is to solo, but you’re also saying that 17 for one or 17 between two powers is indistinguishable in regards to who is closer to soloing, which seems...not right. Maybe I’m misreading you here?

[The Red Sphinx]

In general, carnage scoring tends to reward solos more than draw based scoring, because of the increased incentive to solo. In this system nopun has devised, a solo is worth 2.5 times what a board top is worth. That’s a pretty substantial increase, and incentive to go for a solo rather than settle for a board top (which is comparable in this system to a “good” draw in the current system). It’s also worth noting that in our current system, there are still many people who forsake a solo chance to secure a safe three way. No viable scoring system is going to eliminate that completely, but I doubt that this modified carnage will encourage it any more than our current system does.


Your criticism is a bit misguided where it starts talking about 13 SCs vs. 11, and one player having a bunch vs. two players having a bunch, at least as it pertains to carnage. Part of the idea behind carnage is that the more SCs you have, the closer you are to a solo—so one player with, say, 17 centers, is going to be rewarded more than two players with 17 between them (let’s say one at 10, one at 7 for the sake of argument), because that player with 17 is closer to a solo than a player at 10 or at 7. It kinda seems like your criticism here is getting a bit crossed—you say the goal is to solo, but you’re also saying that 17 for one or 17 between two powers is indistinguishable in regards to who is closer to soloing, which seems...not right. Maybe I’m misreading you here?

Hmm, I guess my main point was that i don’t think it is true to say just because someone has more SCs, then they are closer to a solo... I understand however, that this is a matter of perspective. Definitely if this encourages more solos, I’d be for it, but at its basest level (as I understand it) is punishes those who scrap for a draw and make themselves so indespensible, that a draw cannot happen without them. This is diplomacy after all, and if a player with few SCs is able to convince others that they can’t possibly eliminate him, then I see that equally as a person forced to share in a draw despite being on 17.

There are too many other variables beyond SC count, including the stalemate line. Both in the east or the west, it is relatively easy to get 17 SCs - getting that final one is often hardest, requires careful alliance building on both sides of the board, and crucially, stabbing at the RIGHT time to ensure total victory. To me having more SCs is not immediately worth rewarding. Good alliance building and diplomacy is worth rewarding however and that is measurable in three ways I think - 1) solo victory, which shows you dominated the board, 2) draw sharing, which shows you were good enough to separate the board between you and X number of players, with less players requiring more diplomacy skill; and 3) eliminated from the game... which shows you failed completely.

If this scoring doesn’t adversely affect the smaller SC count in a draw then I’d be all for it.

[super_dipsy]

It's a very interesting and well thought out idea.

Could you factor in the big bugbear for online Diplomoacy of Surrenders? How would you handle surrenders in your proposed system, both in terms of the person surrendering and the person who may have picked up a surrendered country x turns in?

[NoPunIn10Did]

It's a very interesting and well thought out idea.

Could you factor in the big bugbear for online Diplomoacy of Surrenders? How would you handle surrenders in your proposed system, both in terms of the person surrendering and the person who may have picked up a surrendered country x turns in?

It would operate as closely as possible to how we currently handle surrenders and replacements.

1. Any surrendered player receives an immediate total loss (like losing to a solo). That player is no longer included in calculations
   (except possibly generating comparative Elo expected values for everyone else; I’m not sure how you currently handle that math, but I wouldn’t alter it)
2. Any surrendered power(s) that stayed surrendered until game end would tie for Last Place, prior to ranking remaining powers / players by SC counts and elimination years.
3. Replacement players would be ranked normally at end-of-game. Their Elo rating change would be modified in the same manner it is currently. The delta would be multiplied by the percentage of turns taken by that person for that power, with a Rating Shield amending a negative Elo delta to zero.

[NJLonghorn]

Hmm, I guess my main point was that i don’t think it is true to say just because someone has more SCs, then they are closer to a solo... I understand however, that this is a matter of perspective. Definitely if this encourages more solos, I’d be for it, but at its basest level (as I understand it) is punishes those who scrap for a draw and make themselves so indespensible, that a draw cannot happen without them. This is diplomacy after all, and if a player with few SCs is able to convince others that they can’t possibly eliminate him, then I see that equally as a person forced to share in a draw despite being on 17.

This. When a game ends 15-15-4, the player who managed to cling to life deserves full credit for the draw. My favorite game of all time was when I survived as the 1 (Tunis) in a 17-16-1 draw. It took all I was worth to convince both other players that they couldn't trust the other to try to get rid of me. Downgrading my credit for that game would be a HUGE step in the wrong direction.

Plus, I don't buy the idea that the OP proposal would encourage solos. Giving the 15s/16s/17s extra points would give them an extra incentive to accept a draw rather than try to finish off weaker players.

If you want to incentivize solos, award extra points for solos. The game is too dynamic to do it any other way.