# Absolute strength and stability of Elo

Not sure how to phrase this...

I am aware that Elo measures the relative strength of players.

Let's assume for this question that there is such thing as an absolute strength in chess and that players stay at their level without improving or getting worse over time.

If there are only two people in the pool playing many games against each other their Elo ratings will settle on some value and these values are a measure of their absolute strength.

Now, if there are more than two players, I was wondering, whether under the assumptions above (absolute strength exists, no change over time), the Elo ratings would also settle on constant values or would keep on changing constantly?

For the answer, I would like to exclude chess arguments like: player A cannot deal with player B's aggressive style and scores worse than against other players of the same rating (but who play more calmly).

Basically I am wondering whether there is some kind of analogue to frustration (physics term) in the Elo system, namely that, if you have >2 players in a pool they would not happily settle on a rating.

To put it concretely, I am wondering whether the system of players ends up in a steady state of constant player ratings such that if at that point in time any two players play against each other a million games, their outcome reflects exactly their rating difference.

It is assumed that players are of comparable strength so that they have a non-zero probability to draw (or win).

No. In the general case, the ELO ratings of three people will not converge to some stable equilibrium, and here's why.

Suppose you have two players, A and B. B on average scores 66% vs A, so if they played enough games, B's ELO rating would converge to about 100 more than A's.

Now suppose we introduce player C, who scores 66% vs B on average, but never plays against player A. If C and B play many games, their ELO difference will stabilize to about 100 as well. If B kept playing games with A during this time, A would converge to about 100 less than B, with C about 100 more than B.

So, what happens when A finally starts playing C? For there to be stability in the ratings, we MUST have C score 76% vs A, because C is 200 ELO points higher than A. If A scores better than that, A would gain points when playing C, causing the gap between A and B to decrease. But then A would lose the points A gained if they started another long series with B. In this case, the three ELO ratings are not only a function of how well they play against eachother, but also how often they play each opponent - clearly not an absolute rating in any sense.

So this begs the question, if B beats A 66% of the time, and C beats B 66% of the time, should we automatically assume that C beats A 76% of the time, as the ELO functions imply? That doesn't sound too crazy, but for the 3 body ELO problem to be stable, this implied score would have to be accurate for ANY pair of ELO gaps between A-B and B-C. There aren't any assumptions in the ELO model that this transitivity should work.

• Isn't there an implicit assumption that this will work? Otherwise why was 76% chosen?
– D M
Commented Jun 21, 2020 at 5:13
• Given a ratings difference of D points, the ELO formulas predict a score of 1/(1+10^(D/400)) for the weaker player. For a D of 100, this is about 34% for the weaker player, and 24% for a D of 200. Commented Jun 21, 2020 at 6:40
• Yes, that's the formula.By the way, I'm not saying that it does work. I just think that this formula was chosen in part because it was, in fact, assumed to work - or at least be close - when it was created.
– D M
Commented Jun 21, 2020 at 13:05
• Historically, yes 200 ELO difference was chosen to be about a 75% score, and from that our single degree of freedom was set, namely the 10^.0025 as the base. Some choice HAD to be made to define how “wide” a point is. So perhaps that historical baggage makes the concrete example muddier. But for the ELO ratings to be stable, this must hold for all differences. So if B beats A 75% of the time and C beats B 75% of the time, we would also require C to beat A 90% of the time. There isn’t any reason a model with only one degree of freedom should have this hold in all cases. Commented Jun 21, 2020 at 13:51
• After further research I stand corrected. One quirk of ELO is that win ratios work like gear ratios; eg if A wins 1 game for every x games B wins and B wins 1 game for every y games that C wins, then if A played C we should expect A to win once for every x*y win of C. If we accept this assumption as true, then the 3-body ELO problem (and the general n-body ELO problem) has stable solutions. I think the truth of this assumption is field specific; it may apply to some games that use an ELO system, but not all. Commented Jun 21, 2020 at 17:43

Let's assume for this question that there is such thing as an absolute strength in chess and that players stay at their level without improving or getting worse over time.

If there are only two people in the pool playing many games against each other their Elo ratings will settle on some value and these values are a measure of their absolute strength

Now let's add in another false assumption, although it is less immediately obvious than that a player's playing strength remains absolutely constant.

To understand why let's look at an example. Suppose I, FIDE rated 1718, play a long series of standard time control games against Magnus Carlsen, FIDE rated 2863.

Carlsen's k factor is 10 and mine is 20 and our rating difference is over 400 so every time Carlsen beats me he will gain 0.8 points and I will lose 1.6 points. I am never going to beat Carlsen. I am never going to draw with him. My rating is going to keep on going down and Carlsen's keep on going up. Our ratings are never going to "settle on some value".

Of course if Carlsen repeatedly plays his clone then with unvarying playing strength all games are going to be draws and neither player's rating will change. So at that extreme the model works.

Now, if there are more than two players, I was wondering, whether under the assumptions above (absolute strength exists, no change over time), the Elo ratings would also settle on constant values or would keep on changing constantly?

We've already established that the two player system wouldn't necessarily result in "settled" ratings so clearly that will also apply in the multi-player system.

In the more general case where a large number of players with widely differing strengths play each other without the restriction of the 400 point rule the system is still not going to reach a steady state because the underlying formula (P = 1/(1 + 10^(-D/400)) behind the tables used for calculating rating does not reflect the real world probabilities when two players differ in playing strength by several hundred points play each other. Weaker players get better than expected results in those circumstances. Hence the 400 point rule encourages stronger players to play in those circumstances.

Here is what Wikipedia says on the subject:

Subsequent statistical tests have suggested that chess performance is almost certainly not distributed as a normal distribution, as weaker players have greater winning chances than Elo's model predicts. Therefore, the USCF and some chess sites use a formula based on the logistic distribution. Significant statistical anomalies have also been found when using the logistic distribution in chess

they will all oscillate around the true strength. people have off days. some they play are improving and so the WL ratio changes.

the biggest problem with ratings is that we do not have good starting ratings and that many new players rapidly improve distorting the ratings because there is not sufficient competition amongst all the players and the initial ratings never get properly set nor do the later ones.

another problem is that old players do get worse with time. their brain slows down. their attention span lowers.

ratings are just an interesting guide to about how good a player might be but are not the magical answer that so many players think.

pursuing a rating is a misuse of your life and has no value at all except to some people and their ego.

you should see the amount of money that bridge players spend playing weekly as well as frequently in tournaments monthly if not travelling to tournaments every week. all in pursuit of a meaningless title of 'master' with some adjective associated with their level they paid for with time and money.

They wouldn't stay totally still because ratings vary after every game but in the long term they'd tend to converge to the same value

• If this is so, why can't we say that ratings also measure the absolute strength? Commented Jun 20, 2020 at 8:59
• Because there is no such thing as quantifiable absolute strength. Two separate player pools will converge to uncomparable ratings Commented Jun 21, 2020 at 9:48