echidna wrote: ↑Fri Nov 10, 2023 4:43 am
A strange difference between the ghost rating adjustments of the scoring schemes:
In a classic game where 6 of the players have the same Ghost Rating (G, say), the seventh, player P, has a rating of 2G, and the game ends in a 2-way draw with one of the 6 and the other player P, with exactly 17 centers each, then it seems with Draw Size Scoring player P will get a larger ghost rating adjustment than with Sum of Squares scoring:
Based on how I THINK the code in https://github.com/kestasjk/webDiplomacy/blob/master/ghostratings/calculations.php works:
In DSS: P's ratingAdjustment = 8G/17.5 * (1/2 - 2G/8G) = 8G/17.5 * (1/2 - 1/4) = 2G/17.5
In SoS: P's ratingAdjustment = 8G/17.5 * (17*17/(17*17 + 17*17) - 2G*2G/(6*G*G + 2G*2G)) = 8G/17.5 * (1/2 - 4/10) = 8G/17.5 * (1/2 - 2/5) = G/17.5
I have the intuition that it's harder to achieve this outcome in a DSS game than an SoS game, so it should provide a bigger GR uplift. Of course, a more experienced player / someone with access to game outcome data might be able to prove me wrong on this point.
I played around with the code in GPT and I think I have an accurate analysis of some of illustrative GR change examples, which are pasted in detail below the line.
In short, the GR consequences of a game vary a lot based on the scoring system used:
- To your earlier point, two-way draws in SoS are always better than in DSS.
- It is much better to just survive with just a few centres in a DSS game than in an SoS game.
- It can be a bit better to solo under SoS than DSS, but only if you're the underdog.
I think each of these disparities can be justified by the difficulty of achieving these outcomes given the incentives of each scoring system, but of course more experienced folks / real player data could definitely change my mind.
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Variables Explanation:
P_GR: The initial Ghost Rating of the player in question.
O_GR: The initial Ghost Rating of each other player.
n: Total number of players in the game (7 in classic Diplomacy).
d: Number of players surviving to the draw, including the player in question.
s: Number of supply centers controlled by the player in question at the end of the game.
S: Total supply centers in the game (34 in classic Diplomacy).
General Formulas for GR Adjustments:
Draw-Size Scoring (DSS)
GR Adjustment_DSS = (8 * (P_GR + (n - 1) * O_GR) / 17.5) * (1/d - P_GR / (P_GR + (n - 1) * O_GR))
Sum-of-Squares (SoS)
GR Adjustment_SoS = (8 * (P_GR + (n - 1) * O_GR) / 17.5) * (s^2 / (s^2 + (S - s)^2 * (d - 1) / (n - 1)) - P_GR^2 / (P_GR^2 + (n - 1) * O_GR^2))
Illustrative Scenarios:
Four-way Draw, Player Has Fewer Centers (Player controls 4 centers)
Assumption: Other players' centers are evenly distributed.
P_GR | O_GR | DSS Adjustment | SoS Adjustment
200 | 100 | 0.00 | -133.73
100 | 100 | 34.29 | -34.73
50 | 100 | 51.43 | -1.68
Solo Win by the Player (Player controls 18 centers for a solo win)
Assumption: Remaining centers are evenly distributed among other players.
P_GR | O_GR | DSS Adjustment | SoS Adjustment
200 | 100 | 274.29 | 219.43
100 | 100 | 274.29 | 274.29
50 | 100 | 274.29 | 285.26
Two-way Draw, Player Has Half the Centers (Player controls 17 centers)
Assumption: Player controls exactly half of the total centers.
P_GR | O_GR | DSS Adjustment | SoS Adjustment
200 | 100 | 91.43 | 167.18
100 | 100 | 114.29 | 228.57
50 | 100 | 125.71 | 242.81