Contents

• What is a Diplomacy Rating?
• Provisional Ratings
• Established Ratings
• Win-Loss Change
• Survival Change
• Performance Change
• Replacement Positions
• Balance & Implications
• Variables

What is a Diplomacy Rating?

The Diplomacy Rating System is a method for calculating the relative skill levels of players in Diplomacy games. It is based on the Elo Rating System for 2-player games, and expanded to accommodate the varying number of players in a game of Diplomacy.

Elo was originally invented as a chess rating system although it is used in many other games today. It is used as a rating system for competitive multi-player play in a number of computer games, and has been adapted to team sports including international soccer / football, Major League Baseball, and American college football and basketball.

One's Diplomacy Rating is a number, typically between 1000 and 2000, with new ratings starting around 1200-1300. A player's rating is changed any time a game is completed in which the person played. With the completion of each game, points are reassigned from losers to winners.

Points are awarded or removed on a scale that has solo wins at one end, followed by increasingly larger draws, down to survivals and the various types of eliminations. These point totals are affected to a lesser degree by the player's existing rating, the number and quality of his opponents, how long he survived, and with how many centers.

The Diplomacy Rating is a closed system: No points enter or leave the system via a game completion. New points are introduced only when new players join, and points leave the system only if a player retires.

Provisional Ratings

One's initial rating is the average of his ratings calculated from the result of his first 3 games:

Established Ratings

After the completion of 3 games, a rating becomes established.

One's established rating change for any subsequent game is calculated as the sum of 3 components:

Ra_Aft = Ra_End + WLC + SC + PC

WLC = Win-Loss Change

The measure of one's actual result in a game: Win, Lose, or Draw.

To calculate the Win-Loss change, one needs only to know the number of players in the game, and the result achieved by the player. A solo is the best possible result, followed by a small draw, then larger draws, and last, a loss.

Solo:   WLC = BP

Base Points (BP) are calculated, as 13 + P
  A 5-player game is worth a base of 18 points.
  A 7-player game is worth a base of 20 points.
  A 10-player map is worth a base of 23 points.
  A 17-player map is worth a base of 30 points.
The Base Points were chosen so that a Standard game is played for 20 points to a solo victor, and a World game is played for 30 points -- with all others falling in a linear progression.

Draw:   WLC = PA / R

The Points Awarded (PA) are reduced by multiplying the Base Points by the draw factor: PA = BP * DF
Better results pay out a larger portion of the base points:
  A 2-Way Draw has a DF of .833, and pays only 16.67 points to the victors.
  A 3-Way Draw has a DF of .667, and pays only 13.33 points to the victors.
  A 4-Way Draw has a DF of .500, and pays only 10.00 points to the victors.
  A 5-Way Draw has a DF of .333, and pays only 6.67 points to the victors.
  A 6-Way Draw has a DF of .167, and pays only 3.33 points to the victors.
  A 7-Way Draw has a DF of 0, and pays no points to the victors.

Next, The Points Won (WLC) are calculated by dividing the Actual Points Awarded, evenly among the victors:
  A 2-Way Draw is split between 2 players (16.67 / 2 = 8.33 points won each.)
  A 3-Way Draw is split between 3 players (13.33 / 3 = 4.44 points won each.)
  A 4-Way Draw is split between 4 players (10.00 / 4 = 2.50 points won each.)
  A 5-Way Draw is split between 5 players ( 6.67 / 5 = 1.33 points won each.)
  A 6-Way Draw is split between 6 players ( 3.33 / 6 = 0.56 points won each.)
  A 7-Way Draw is split between 7 players ( 0 / 7 = no points won.)

Large draws including most of the players will result in very few points being won, whereas small draws or solos have the greatest gains.

Loss:   WLC = BP / (P - 1)

Any result other than a Win or a Draw is a Loss. The points each player loses is simply the Base Points for the game divided by (the number of players minus 1). Each player loses the same amount:

Zero Sum: In all the scenarios above, the points won will always sum up to equal the points lost:
  7-player game, Solo: winner wins 20 points, 6 players lose 3.33. 3.33 x 6 = 20.
  7-player game, 2-Way: 2 winners score 8.33 (8.33 x 2 = 16.67). 5 players lose 3.33 (3.33 * 5 = 16.67).
  7-player game, 3-Way: 3 winners score 4.44 (4.44 x 3 = 13.33). 4 players lose 3.33 (3.33 * 4 = 13.33).
  7-player game, 4-Way: 4 winners score 2.50 (2.50 x 4 = 10.00). 3 players lose 3.33 (3.33 * 3 = 10.00).
  7-player game, 5-Way: 5 winners score 1.33 (1.33 x 5 = 6.67). 2 players lose 3.33 (3.33 * 2 = 6.67).
  7-player game, 6-Way: 6 winners score 0.56 (0.56 x 6 = 3.33). 1 player loses 3.33 (3.33 * 1 = 3.33).
  7-player game, 7-Way: 7 winners score 0 (Total of 0). 0 players lose anything (Total of 0).

SC = Survival Change:

The measure of one's ability to stay alive and control centers in a game.

To calculate the Survival Base, one needs to understand two components: The Duration Base and Center Base. Survival scores only take effect for players that score a loss in a game. They act as a consolation score, to indicate a player has played a certain portion of the game, or even survived to the end controlling what might be a substantial number of centers (often referred to as Seconds). Players who survived longer will get a better score than those who were eliminated sooner, and players who end the game in control of more supply centers will score better than those who control fewer centers or were eliminated.

SB = DB + CB

DB = Duration Base = 2 * (Player_Seasons - Player_NMRs) / (Total_Seasons)
  DB amounts to a number between 0 and 2 points, making a bonus for turns played without NMRs.

CB = Center Base = 4 * (Player_Centers) / (Total_Centers)
  CB amounts to a number between 0 and 2 points, making a bonus for centers controlled.

Combined Survival Base amounts to a number between 0 and 4 points. A player that is eliminated may score up to 2 points as their Center Base will be 0. A player that survives to the end will score between 2 and 4 points, as their Survival Base will be 2 (for survival to the end) plus up to another 2 based on the number of centers they control. A Second will score close to 4 points.

Any country that goes into Civil Disorder scores 0 for their Survival Score, irrelevant of how long they played or centers controlled - a player must play the game out to conclusion to qualify for their Survival Score.

Zero Sum: After calculation of all the Survival Bases, one has a list of numbers between 0 and 4. These scores are shifted linearly about 0 to create a set that will sum to zero. The Survival Base scores are averaged, then that average is subtracted from each base to create the Survival Change, or SC.

SC = SB - S_Avg

A sample Survival calculation: 7-Player Game, 3-Way Draw, game time 5 years (10 turns):

Player	Result	Centers	Turn Elim	DB	CB	SB	SC
1	Draw	12	N/A	N/A	N/A	N/A	N/A
2	Draw	8	N/A	N/A	N/A	N/A	N/A
3	Surv	3	N/A	2.0	0.35	2.35	0.82
4	Surv	1	N/A	2.0	0.12	2.12	0.59
5	Elim	0	10	2.0	0	2.00	0.47
6	Elim	0	6	1.2	0	1.20	-0.33
7	CivD	0	8	0	0	0.00	-1.53
S_Avg = (2.35 + 2.12 + 2.00 + 1.20 + 0.00) / 5 = 1.53

PC = Performance Change:

The measure of one's actual result in a game relative to how well he should theoretically perform.

To calculate the Performance Base, one needs only know his rating, the average rating of his opposition, and his actual result.

PB = K * (SA - EA)

A player's Performace Factor is equal to his Actual Score (SA) minus his Expected Score (EA). This will always produce a number between -1 and +1, indicating the degree to which a player exceeded (positive) or underperformed (negative) his expected result. The Performance Factor is multiplied by K to determine the Performance Base for points gained / lost.

K = A constant, 10.

SA = Actual Score, a fixed number based on result:

Solo	SA = 1
Any Draw	SA = .5
Survival	SA = .1
Elimination / Civil Disorder	SA = 0

EA = Expectation, a number between 0 and 1, calculated based on the players' ratings

EA = 1/{1+10^[(R_Opp - R_Eff)/400]}

Some sample EA calculations:

Player's rating (Ra_Eff) = 1200
Opponents' average rating (Ra_Opp) = 1200
Ra_Opp - Ra_Eff = 0 point differential
EA = .50

Player's rating (Ra_Eff) = 1290
Opponents' average rating (Ra_Opp) = 1220
Ra_Opp - Ra_Eff = -70 point differential
EA = .60

Zero Sum: After calculation of all the Performance Bases, one has a list of numbers between -10 and +10. These scores are shifted linearly about 0 to create a set that will sum to zero. The Performance Base scores are averaged, then that average is subtracted from each base to create the Performance Change, or PC.

PC = PB - P_Avg

Replacement Positions

Some GMs allow the use of Replacement players to take over a country for which the original player is unable to continue playing. The Diplomacy Rating system is designed to accommodate this replacement phenomenon. No implication is made as to the intention of replaced players, whether their replacement is for good reason or otherwise - the rating system is intent-neutral, and only considers the statistics as of the time the one player left the game, and the final result that the replacement player obtains.

Replacement positions with a Provisional rating:

• Solo: Ra_Avg + 99 + P
  (Same as above - a replacement that manages to score a Solo is awarded full credit for the Solo)
• Draw: Ra_Avg + 36 * DF
  (Same as above - a replacementr that manages to score a Draw is awarded full credit for the Draw)
• Survival: Ra_Avg - 8 + DB + CB
  (A replacement Survival is pegged -8 below his opposition instead of -18: 10 point bonus.)
• Elimination: Ra_Avg - 12 + DB
  (A replacement Elimination is pegged -12 below his opposition instead of -24: 12 point bonus.)
• Civil Disorder: Ra_Avg - 30
  (Same as above - a replacement that goes into Civil Disorder forfeits all bonus)

Replacement positions with an Established rating:

• Solo: If a replacement player scores a Solo, the replacement player gets the bulk of the point gains. The WLC point gain is divided between the original player and the replacement player based on the number of supply centers under the control of the original player at the time he departed the game. If a player was in control of 5 centers, and his replacement ends up scoring a Solo, the original player would win 5/34 of the WLC points, and the replacement player would win the balance of 29/34 of the WLC points.
• Draw: If a replacement player scores a Draw, the replacement player gets the bulk of the point gains. The WLC point gain is divided between the original player and the replacement player based on the number of supply centers under the control of the original player at the time he departed the game. If a player was in control of 5 centers, and his replacement ends up scoring a Draw, the original player would win 5/34 of the WLC points, and the replacement player would win the balance of 29/34 of the WLC points.
• Loss: If a replacement player scores a Loss, the original player bears the bulk of the point loss. The WLC point loss is divided between the original player and the replacement player based on the number of supply centers under the control of the original player at the time he departed the game. If a player was in control of 5 centers, and his replacement ends up scoring a Loss, the original player would lose 29/34 of the WLC points, and the replacement player would lose only 5/34 of the WLC points.

There is a strong incentive for replacement players to take on games, for the worse the position taken over (by center count), the less liability the new player takes on, while still retaining the high potential for gain. There is also an incentive for a player that needs to leave a game - to find a good substitute to take over his position. The original player is still liable for the bulk of the loss, but could possibly minimize that loss by finding a player that can play the position to hopefully a good result.

Balance & Implications

The relative impact of the 3 components WLC, SC, and PC varies with differing game results:

• WLC: For a solo, the largest impact to the winner comes from the WLC, for the base of 20 points won. As games end in progressively larger draws, the impact of the WLC is reduced, and with very large draws, hardly any WLC points are won at all. This is because as there are fewer and fewer losers, there are no players from which the lost points can be taken to reassign to the winners.
• SC: Survival score has a relatively large impact for the losers of the game, and no impact for the draw / winners of a game. For a loss, a player can expect to lose 3.33 WLC points in a standard game, but can win up to about 2 SC points back just by trying to survive. This can mean that one player may take a 5 point loss while another may only take a 2 point loss, even tho both players have lost the same game. If one of them outlived the other, or had a substantial number of centers at game end (i.e. Seconds) their Survival Score will reflect the difference.
• PC: Performance scores can be in the range of +10 to -10, depending on how far one exceeded or failed to meet his theoretical expectations. PC tends to have the largest impact when there is a large spread of ratings in a game, and has a smaller spread when there is a relatively equal matchup. With a large rating gap, it becomes progressively more difficult for the higher rated players to gain further rating points at all by scoring Draws, for one's EA will tend to be .60, .70, and up as the differential increases. With a .70 EA, even a Draw will net a PC loss of about 2 points (0.2 * 10), making a 5-Way Draw in which one only wins 1.33 WLC points, a net loss. Conversely, players that got off to a low rating start should find that even a simple draw would net them substantial gains - for instead of a 5-way draw winning 1.33, it may win 3-4 points if drawn by an underdog.

Performance can't be measured absolutely; it can only be inferred from wins and losses. Ratings therefore have meaning only relative to other ratings. Both the average and the spread of ratings are arbitrarily chosen. The average Diplomacy rating was chosen to be around 1200-1300, with a low end of about 1100 and an upper end near 2000. The extreme values have yet to be seen as insufficient games have been run to create such diversity.

Variables

P = Number of Players

R = Result (# of players sharing in the solo / draw)

DF = Draw_Factor = (P-R)/(P-1)

Ra_Avg = The average of the effective ratings for all countries in a game.

Ra_Eff = The effective rating of any country. If a country was played by one player from start to finish, then Ra_Eff = Ra_Sta. If a country was played by multiple players, R_Eff is the average of all the players that played the country, weighted by the number of turns they played.

Ra_Opp = The average of the effective ratings of one's opponents.

Ra_Sta = A player's rating of record for a game. A player's rating is marked as of the date the game started (or as of the date joined if a replacement). Even tho the player's rating may change during the course of the game, his rating for the purpose of all calculations related to the game, is that rating he had when the game began.

DB = Duration Base = 2 * (Player_Seasons - Player_NMRs) / (Total_Seasons)
  DB amounts to a number between 0 and 2 points, making a bonus for turns played without NMRs.

CB = Center Base = 4 * (Player_Centers) / (Total_Centers)
  CB amounts to a number between 0 and 2 points, making a bonus for centers controlled.