How is the ELO rating recalculated after a match?

[nyczar]

Community:

I have had some matches recently that make me want to better understand how the ELO gets recalculated. For example, how much is a change dependent on the relative ELO of the opponents? Does the score of a game get factored in so that a large disparity will lead to a larger ELO increase/decrease or make no difference? What happens when a game gets timed out in a tournament or a picked up ranked game gets stalled do to inactivity?

I tried searching the forum for help on this but didn't find anything.

Thanks for whomever can share the inner workings.

[Karvon]

Community:

I had some matches recently that make me want to better understand how the ELO get recalculated. For example, how much is a change dependent on the relative ELO of the opponents? Does the score of a game get factored in so that a large disparity will lead to a larger ELO increase/decrease or make no difference? What happens when a game gets timed out in a tournament or a picked up ranked game gets stalled do to inactivity.

I tried searching the forum for help on this but didn't find anything.

Thanks for whomever can share the inner workings.

For Slitherine ELO at least, the relative score has no bearing on the ELO adjustment; it's simply a matter of whomever has the highest point total is the winner. There is no regard for the normal winning definition of 25+ or over 60 in this determination. Thus, results of 1-0, 40-0, or 60-59 would all result in the same result of a win for the higher scorer.

The difference between players' ELO does impact on the adjustment. I'm fairly certain this is a standard formula which you can find online.

However, I'm not sure what the base value Slitherine uses for their adjustment, and you'd need to know that to play around with ELO in a spreadsheet. Perhaps Richard could tell you this? That base value is what players are competing for in a game and the division of that prize is relative to the players' ELO.

For TDC, we have adjustments to that base value based on the degree of victory or a draw. You can find a quite extensive explanation of our system of calculation provided by Mike who's our spreadsheet/math guy.

Regards,

Karvon

[tyronec]

I have had some matches recently that make me want to better understand how the ELO gets recalculated.

In this thread: https://www.slitherine.com/forum/viewtopic.php?t=113421


Here is the formula:

-----------------------

$K = 30;
$Probability1 = 1 / (1 + (pow(10, ($Rating2 - $Rating1) / 400)));
$Probability2 = 1 / (1 + (pow(10, ($Rating1 - $Rating2) / 400)));

if ($Player1HasWon)
{
$S1 = 1;
$S2 = 0;
}
else if ($Player2HasWon)
{
$S1 = 0;
$S2 = 1;
}
else // draw, it never happens but better to have it
{
$S1 = 0.5;
$S2 = 0.5;
}
$Rating1 += round($K * ($S1 - $Probability1));
$Rating2 += round($K * ($S2 - $Probability2));

---------------------

Pow(x, y) equals to x^y (Ex. Pow(10, 2) => 10^2 => 100)
Round(x) returns the rounded value (an integer)

[nyczar]

Thanks Gentlemen.

[nyczar]

Well, the formula was not intuitively clear to me so I decided to use this new hot thing called AI. I asked ChatGPT to write the formula more clearly and to provide any commentary that helps with understanding. Here is what ChatGPT provided:

Screenshot 2024-09-29 210011.jpg (217.04 KiB) Viewed 420 times

With this, I was able to make an excel file that helped me to see the relationships and to form a descriptive narrative of the FOG ELO methodology. Here we go:

Every player has a rating. The difference in player rating provides a probability of victory for each player. Players of equal rating have a 50%/50% chance of victory. For every difference of 50 in rating, the win probability increases/decreases by a little over 7%. For example, if player 1 has a rating of 1050 and player 2 has a rating of 1000, Player 1's win probability is roughly 57% and Player 2's is 43%.

The maximum points earned in a ranked game is 30. To determine how much your rating might go up in victory, multiply 30 by your OPPONENTS win probability. In the example above, if Player 1's opponent win probability is 43%, the maximum a victory is worth to Player 1 is 13, (30*43%). However, because Player 2 is a lower rated player, his rating would change 17 if he won (30*57%).

In a ranked game, the loser's rating is lowered by the opposite of the winners increase. Continuing the example, if Player 1 wins, his rating increases 13 and Player 2's decrease 13. If Player 2 wins, his rating increases 17 and Player 1's rating decreases 17. Per the included ChatGPT commentary, "This formula ensures that a player gains more points if they win against a higher-rated opponent and loses fewer points if they lose to a higher-rated opponent. Conversely, beating a much lower-rated opponent gains fewer points, while losing to them costs more."


I welcome vetting of my findings and edits to my narrative, if needed.

[Karvon]

Mike discusses how we calculate ELO for TDC in this thread.
https://www.slitherine.com/forum/viewtopic.php?t=113776