Code:

`CRE = Rating - 4 * Deviation`

- Rating changes are too slow. You'll need to beat quite a lot of players in order to see your rating change acceptably. This makes players use more alts.
- The higher the rating deviation of the player, the more the player's true skill is underestimated.
- It provides horribly incorrect ratings for people whose rating deviation is very high. For an example, just visit this page.

- It is simple to calculate.

Because of this, I set out to try to find a better way of finding a player's overall rating given his Rating R and Deviation RD... and I managed to do this yesterday.

I read Glickman's paper (the inventor of the Glicko and Glicko-2 rating systems) and he provides an equation that essentially calculates the probability that a player with rating R_1 and deviation RD_1 beats another player with rating R_2 and deviation RD_2. It is written below:

Code:

```
Probability = 1 / (1 + 10^(((R_2 - R_1) / (400 * sqrt(1 + C * (RD_1^2 + RD_2^2))))))
where C = 3 * ln(10)^2 / (400 * pi)^2 (approximately 0.0000100724)
pi = 3.14159265359
sqrt(x) is the square root of x
ln(x) is the natural logarithm of x
```

However, this is a strenuous effort to do, and hence I wanted to approximate this probability for every player using just his R and RD (not everyone else's as well). After considering various possibilities, it dawned on me that the probability of the player beating a 1500 rating, 350 deviation player (the rating and deviation of a player that has just joined the ladder) would provide a good approximation. When testing it out, it did provide a good approximation of the true rating... a very, very good approximation actually!

The only time it didn't provide a good approximation was when the deviation of the player was high. This confirmed yet again that players that have a rating deviation that is too high (meaning that his rating is too uncertain) shouldn't even be listed on the leaderboard. And this is what I propose for the estimated rating to be done.

After consulting a bit with the community, it was decided that this system's rating should represent the estimated percentage that that player has of winning a battle against a random opponent.

So, finally, here is what I propose to be a much better estimate of the player's rating. I'm calling it GLIXARE, short for '

**Gli**cko -

**X**-

**A**ct

**R**ating

**E**stimate':

Code:

```
Given a player rating R and a rating deviation RD:
GLIXARE Rating = 0, if RD > 100
GLIXARE Rating = round(10000 / (1 + 10^(((1500 - R) * pi / sqrt(3 * ln(10)^2 * RD^2 + 2500 * (64 * pi^2 + 147 * ln(10)^2)))))) / 100, otherwise
```

Code:

```
Rank Rating Deviation True Rating CRE Rank For CRE GLIXARE Rank For GLIXARE
1 1991.408347 52.4913089 86.25% 1781.443112 1 (=) 86.77% 1 (=)
2 1992.528854 68.34131414 86.21% 1719.163597 13 (+11) 86.73% 2 (=)
3 1989.461831 80.60840798 85.95% 1667.028199 18 (+15) 86.51% 3 (=)
4 1969.23615 50.08961382 85.08% 1768.877695 2 (-2) 85.78% 4 (=)
5 1968.509612 50.00035476 85.04% 1768.508193 3 (-2) 85.75% 5 (=)
6 1972.675135 99.88585936 84.88% 1573.131698 34 (+28) 85.58% 6 (=)
7 1963.494349 50.60336877 84.76% 1761.080874 5 (-2) 85.51% 7 (=)
8 1962.47472 50.03981759 84.71% 1762.31545 4 (-4) 85.46% 8 (=)
9 1953.22279 52.20270923 84.18% 1744.411953 6 (-3) 85.00% 9 (=)
10 1958.445485 96.63628542 84.13% 1571.900343 36 (+26) 84.94% 10 (=)
11 1943.00504 50.10101356 83.60% 1742.600986 7 (-4) 84.51% 11 (=)
12 1941.259317 52.64803719 83.48% 1730.667168 10 (-2) 84.41% 12 (=)
13 1938.159617 50.92018226 83.31% 1734.478888 8 (-5) 84.26% 13 (=)
14 1936.183665 51.14032287 83.19% 1731.622373 9 (-5) 84.16% 14 (=)
15 1930.057618 50.16740178 82.84% 1729.388011 11 (-4) 83.85% 15 (=)
16 1927.779647 51.0262639 82.70% 1723.674591 12 (-4) 83.73% 16 (=)
17 1922.220146 57.65787498 82.32% 1691.588646 14 (-3) 83.40% 17 (=)
18 1918.558883 74.43201955 81.99% 1620.830805 28 (+10) 83.09% 18 (=)
19 1908.558873 59.43764926 81.47% 1670.808276 15 (-4) 82.65% 19 (=)
20 1898.317729 81.4379487 80.68% 1572.565934 35 (+15) 81.93% 20 (=)
21 1876.359618 52.21151457 79.44% 1667.513559 17 (-4) 80.86% 21 (=)
22 1870.366607 50.00313579 79.06% 1670.354064 16 (-6) 80.51% 22 (=)
23 1867.964646 51.3491295 78.89% 1662.568128 21 (-2) 80.36% 23 (=)
24 1867.766236 50.56176086 78.88% 1665.519192 19 (-5) 80.35% 24 (=)
25 1863.669023 50.11759802 78.61% 1663.198631 20 (-5) 80.10% 25 (=)
26 1866.541991 95.77888912 78.50% 1483.426435 48 (+22) 79.95% 26 (=)
27 1859.313589 55.68806985 78.28% 1636.561309 25 (-2) 79.81% 27 (=)
28 1854.389046 51.55994467 77.97% 1648.149268 22 (-6) 79.52% 28 (=)
29 1855.299562 72.27265208 77.92% 1566.208954 37 (+8) 79.46% 29 (=)
30 1853.503073 52.54886057 77.90% 1643.307631 23 (-7) 79.46% 30 (=)
31 1841.225405 50.60827664 77.06% 1638.792298 31 (=) 78.70% 31 (=)
32 1834.513128 50.14590546 76.59% 1633.929506 26 (-6) 78.26% 32 (=)
33 1834.145367 52.8071478 76.55% 1622.916775 27 (-6) 78.23% 33 (=)
34 1801.474785 50.04016895 74.21% 1601.314109 29 (-5) 76.04% 34 (=)
35 1804.267316 94.72400253 74.15% 1425.371306 55 (+20) 75.93% 35 (=)
36 1795.795989 50.01171801 73.78% 1595.749117 30 (-6) 75.64% 36 (=)
37 1793.496594 50.25237181 73.61% 1592.487107 31 (-6) 75.47% 37 (=)
38 1782.059218 51.06317634 72.75% 1577.806512 32 (-6) 74.65% 38 (=)
39 1775.843559 50.18930449 72.28% 1575.086341 33 (-6) 74.20% 39 (=)
40 1748.052482 56.82618848 70.11% 1520.747729 42 (+2) 72.08% 40 (=)
41 1748.543861 92.78899389 69.97% 1377.387886 67 (+26) 71.89% 41 (=)
42 1744.455748 50.99377091 69.85% 1540.480664 39 (-3) 71.83% 42 (=)
43 1743.084558 60.21302067 69.71% 1502.232476 44 (+1) 71.68% 43 (=)
44 1742.067022 50.19645562 69.66% 1541.2812 38 (-6) 71.65% 44 (=)
45 1740.620853 84.04522098 69.40% 1404.439969 58 (+13) 71.37% 46 (+1)
46 1738.536514 51.44593668 69.38% 1532.752767 40 (+6) 71.35% 45 (-1)
47 1728.468483 59.51691032 68.55% 1490.400841 45 (-2) 70.54% 47 (=)
48 1727.948725 50.16195168 68.54% 1527.300918 41 (-7) 70.54% 48 (=)
49 1708.596637 59.16778978 66.96% 1471.925478 50 (+1) 68.94% 49 (=)
50 1705.528967 50.05894259 66.73% 1505.293197 43 (-7) 68.72% 50 (=)
```