One way to objectively judge the relative strength of game players is the ELO rating system. We will use a specific example to directly compare the relative strength of two players and thus deduce their *Relative Mind Power*.

The expected probability of winning for a player with rating R against a player with rating x, is:

Let's pick two specific examples then, with corresponding ratings 2500 and 1000:

> AR:=2500; #player A rating

> IR:=1000; #player B rating

> AEF:=x->(1+10^((x-AR)/400))^(-1); #expected probability of player with rating AR beating a player with rating x

> IEF:=x->(1+10^((x-IR)/400))^(-1); #expected probability of player with rating IR beating a player with rating x

> with(plots):

> plot(AEF(x),x=0..AR+1000,color=red);

The function depicted above is a probability distribution. Hence a player with rating AR=2500 has a probability pA of winning a player with rating IR=1000:

> pA:=evalf(AEF(IR));#probability of player with rating AR=2500 beating a player with rating IR=1000

pA := .9998222036

while a player with rating IR=1000 has a probability pI of winning a player with rating AR=2500:

> pI:=evalf(IEF(AR));#probability of player with rating IR=1000 beating a player with rating AR=2500

pI := .1777963238e-3

Note that pA+pI=1.

Now let's follow an example of how the ratings are updated when the two players above play each other. A's rating is AR=2500, so according to FIDE, we use AK=10, and I's rating is 1000, so we use IK=15.

> NR:=proc(AR,IR,AS,IS)

>#Calculate new rating after a game

> local AE,IE,AK,IK,ARN,IRN;

> AK:=10;IK:=15;

> AE:=(1+10^((IR-AR)/400))^(-1);#expected A

> IE:=(1+10^((AR-IR)/400))^(-1);#expected I

> ARN:=AR+AK*(AS-AE);#new A

> IRN:=IR+IK*(IS-IE);#new I

> print(`ELO A`, evalf(ARN));

> print(`ELO I`,evalf(IRN));

> end:

The parameters now determine the outcome. There are three possible outcomes for the scores AS, IS:

AS=1 and IS=0, AS=1/2 and IS=1/2 and AS=0 and IS=1. Let's see the results:

>AS:=1;IS:=0;#stronger player A wins

>NR(AR,IR,AS,IS);

ELO A, 2500.001778

ELO I, 999.9973331

>AS:=1/2;IS:=1/2;#draw

>NR(AR,IR,AS,IS);

ELO A, 2495.001778

ELO I, 1007.497333

>AS:=0;IS:=1;#weaker player I wins

>NR(AR,IR,AS,IS);

ELO A, 2490.001778

ELO I, 1014.997333

We see that the only cases which really matter for I are the draw and win cases, because these are the cases which raise his rating and lower the rating of A.

__Relative Mind Power in Chess__

We can now define the *Relative Mind Power* (RMP) to be the ratio: pA/pI, or formally, for two players with ratings AR and IR, and AR>IR:

Using Maple to calculate:

> AR:=2500;IR:=1000;

> RMP:=(x,y)->AEF(x)/IEF(y);

> evalf(RMP(IR,AR));

5623.413252

In other words, the mind of a player with a rating AR=2500 is roughly 5623 times (!) more powerful than the mind of the player with a rating IR=1000. Now you understand why it is so difficult to beat stronger rated players.

You may ask, in what *sense* is A's mind *more powerful* than I's? In other words, *how* does this difference/ratio in power manifest? Exactly in the sense of influencing future events in chess in terms of statistics.

Let's see some examples: Suppose there are three people, A, I and O, an observer. Then if A's and I's ratings are the same, O can make the probabilistic prediction that I needs to play roughly 1 game(s) to see a draw. That's exactly what RMP is in this case.

When A and I have different ratings, say AR=2500 and IR=1000, the expected probabilities of winning are pA=.9998222036 and pI=1-pA, so O can make a fairly safe prediction that I needs to play roughly pA/pI=pA/(1-pA)=5623 games to win one. That's again RMP.

Now suppose A and I have the ratings AR=2400 and AI=2000. In this case, O can again make a fairly safe prediction that I must play roughly pA/(1-pA) games to win one. That's exactly RMP=10, in this case.