Conquer Club

Concise description:

• Show (the logarithm of) the ratio of the number of opponents beaten to the number of games lost in the profile

Specifics:

• This could be instead of or in addition to the percentage of games won. It's similar to that percentage in that it's a simple statistic that reflects the proportion of games a player wins without trying to take into account different player strengths, but whereas the percentage depends strongly on how many players a player typically plays in a game, this ratio is independent of that and thus a more direct measure of a player's strength.

This will improve the following aspects of the site:

• relevance of statistics in profile for gauging player strength

This was originally a suggestion by Itrade in a General Discussion thread; I replied to it and someone liked the ideas and suggested to put the entire post in the suggestion box. For context, here's the URL for Itrade's original post:
Subject: Win Percentage Seems Cool, but is Actually Pretty Useless

Here are my comments on that thread:

It seems the discussion in this thread mixed a couple of objectives, and I want to try to disentangle it somewhat.

The objective of Itrade's original post, as I understood it, was to fix a specific problem with the win percentage, namely that it doesn't take into account the number of players per game. I found this post searching this forum because I wanted to make the same comment and a similar suggestion.

A lot of the ensuing discussion was about things like points per game and how best to reflect the skill levels of the opponents. This is an interesting discussion in itself, but it goes way beyond the original remark and proposal. The win percentage is a very simple stat whose attraction (it's clear from this and other threads that many people like it) lies in its simplicity and obviousness. A scheme for taking into account skill levels is inevitably more complicated and contains more free and somewhat arbitrary parameters, and no matter how nifty it is, there will inevitably be people who disagree with some detail of it.

So while I personally find the point system more interesting and would like to see a points per game stat (perhaps as a rolling average), this is a somewhat different topic from Itrade's proposal, which was about fixing a simple problem in a simple stat that people seem to value even though, or perhaps precisely because, it doesn't try to take into account player skills.

The solution Itrade proposed doesn't introduce any arbitrary parameters; it just ensures in a straightforward way that the stat tells you the most basic thing that one might want it to reflect, namely whether a player is doing better or worse than if the winners had been determined randomly.

So I second Itrade's proposal to introduce a comparison between # of players beaten and # of games lost. This could be instead of the win percentage, or if people really want to keep that, in addition to it.

One post that did comment on Itrade's original proposal was this:

lackattack wrote:

Itrade wrote:
I personally like amount of players beaten divided by amount of games lost as a way to tell how good a player is. It doesn't take into account the skill level of the other players, though.

Not bad, but how about something simpler: instead of wins divided by games we use wins divided by # of opponents?

A subsequent post seconded this, and Itrade asked what the difference is.

In #wins/#opponents, the correction for the opponents is the wrong way around -- the more opponents you have, the less your wins count. So if you want to use the number of wins, the opponents would need to be in the numerator, with each factor normalized by the number of games: #wins/#games * #opponents/#games. That's not quite right yet, though, either: a player winning an average number of two-player games would get a score of 1/2 on this, whereas a player winning an average number of eight-player games would get a score of 7/8. What we need is actually #wins/#games * #players/#games. That would yield a score of 1 for an average player, no matter what sorts of games they play. It would have the advantage of looking like a slightly corrected version of the current stat #wins/#games. However, it still has a major disadvantage -- among good players, it favours those who tend to play more opponents. In the extreme case of a player who always wins, their score would be 2 if they only play two-player games, but 8 if they only play 8-player games.

Itrade's original proposal, #opponents beaten/#games lost, on the other hand, has none of these disadvantages: It's always 1 for an average player, it's always zero for a player who always loses, and it's always infinity for a player who always wins, irrespective of the number of opponents per game.

It also has the nice feature that it's similar to the point system, just without depending on the other players' skills. You earn points from players you beat and lose points to players you lose to, just instead of the points depending on your own and your opponents' current points, it's always exactly one point, and instead of starting out with 1000 points, you just take the ratio between the number of points you've earned and the number of points you've lost.

One more thing that I personally would like to see but that might perhaps seem too "mathematical" to others is to take the logarithm of the ratio -- that would remove the arbitrary asymmetry of putting one of the numbers in the numerator and the other in the denominator, and would transform the asymmetric range of 0 to 1 for below-average and 1 to infinity for above-average into the symmetric and more "natural" range of -infinity to 0 below average and 0 to infinity above average.

I think this could be the most accurate way to gauge a player's skill. Great suggestion.

I love it.

Love the post.
Love the idea.

I can drop this straight into Map Rank for the time being if you like to see the numbers in advance.

What's more this proposal is free from points so it would be accurate regardless of missing logs from Map Rank's point of view.
Also you could also get a ratio for each map or even game type etc.

Nice.

chipv wrote:I can drop this straight into Map Rank for the time being if you like to see the numbers in advance.

Great idea.

Trouble is, now I'm thinking more about it, is that the logarithm or ratio is not an obvious frame of reference.

Win/loss ratio (= wins/ wins + losses) is a fraction so can be turned into a percentage - so the denominator is always 100.
Easy to see if you're good or bad.
90% is good, 10% is bad.

The ratio of defeats/losses is not necessarily a fraction so is just a number. So is the log of this number.
On its own meaningless unless you compare against everyone else's number.

So if you look at your own number it tells you nothing about how good you are as a player aside from the sign of the number (in the case of log) so you must compare with other people's numbers to see.

So, yes a negative number is bad, and a large positive number is very good... but how bad and how good?

Now if instead you had opponents beaten/ opponents played - this would be fraction and can be turned into a percentage.

It is similar to the win/loss calculation but only deals in number of players not number of games.

So ratio = opponents defeated / opponents defeated + opponents not defeated

If you win 1 2 player and lose 3 2 player games this ratio is 1/4 = 25%
If you win 1 8 player and lose 3 2 player games this ratio is 7/10 = 70% showing bias for larger number of players.

Note that the win/loss games ratio is exactly the same for both. (1 win out of 4 games played = 25%).

Now it is arguable which is more difficult 8p or 2p so I'm not commenting on that.

Not sure if this was rejected already in the other thread but I guess someone may have mentioned it already so apologies for repetition.

chipv wrote:So ratio = opponents defeated / opponents defeated + opponents not defeated

Make it so then...

C.

chipv wrote:Trouble is, now I'm thinking more about it, is that the logarithm or ratio is not an obvious frame of reference.

Good point. Although I think the current percentage merely generates the illusion of being an obvious frame of reference, whereas you can't actually tell whether a given value is good or bad without looking at the numbers of opponents, I agree that a measure with a natural scale, ideally 0 to 1 or equivalently 0% to 100%, would be preferable to a logarithm of a ratio, whose scale is inherently arbitrary. (The ratio does have a nice interpretation -- it's the factor by which you've expanded or reduced the average number of games between two of your losses, compared to what it would be if winners were chosen randomly, but I admit that isn't quite as straightforward as a percentage.)

I think your proposal for such a measure is flawed, but it can be remedied and then links up nicely with Itrade's proposal.

chipv wrote:So ratio = opponents defeated / opponents defeated + opponents not defeated

If you win 1 2 player and lose 3 2 player games this ratio is 1/4 = 25%
If you win 1 8 player and lose 3 2 player games this ratio is 7/10 = 70% showing bias for larger number of players.

Note that the win/loss games ratio is exactly the same for both. (1 win out of 4 games played = 25%).

OK, but this comparison doesn't work out:

If you win 1 2-player and lose 3 2 player-games the ratio is 1/4 = 25%
If you win 1 8-player and lose 3 8 player-games the ratio is also 7/28 = 1/4 = 25%

But the first is below average and the second is above average, so they shouldn't lead to the same score.

What does the trick, though, is #opponents defeated / (#opponents defeated + #games lost).

This is always between 0 and 1 (or 0% and 100%). It encodes the same information as Itrade's ratio; in fact it's simply c / (c + 1), where c is Itrade's ratio, #opponents defeated / #games lost. This has a neat interpretation. As I showed in the original thread (Subject: Win Percentage Seems Cool, but is Actually Pretty Useless), a player who would achieve the same Itrade ratio c independent of the number of opponents would have a probability c / (c + k) of winning against k opponents. So c / (c + 1) is the probability of such a player winning a two-player game.

That is, the score #opponents defeated / (#opponents defeated + #games lost) is a number between 0 and 1 (or equivalently 0% and 100%) that says how many percent of your games you would have won if you had achieved the same Itrade ratio playing only two-player games -- the winning percentage has been "remapped" to a two-player equivalent to make it comparable among all players with different numbers of opponents. A score of 0 (0%) means you always lose, a score of 1 (100%) means you always win, and a score of 1/2 (50%) means you win as many games as would be expected if the winners were chosen randomly. That seems like a nice and easily interpretable scale.

Thanks for your criticism and ideas; I think they've improved the proposal (and its chances of being taken up) considerably.

Skimmed the idea, didn't want to read it all. Sounds like a good idea though.

joriki wrote:I think your proposal for such a measure is flawed, but it can be remedied and then links up nicely with Itrade's proposal.

chipv wrote:So ratio = opponents defeated / opponents defeated + opponents not defeated

If you win 1 2 player and lose 3 2 player games this ratio is 1/4 = 25%
If you win 1 8 player and lose 3 2 player games this ratio is 7/10 = 70% showing bias for larger number of players.

Note that the win/loss games ratio is exactly the same for both. (1 win out of 4 games played = 25%).

OK, but this comparison doesn't work out:

If you win 1 2-player and lose 3 2 player-games the ratio is 1/4 = 25%
If you win 1 8-player and lose 3 8 player-games the ratio is also 7/28 = 1/4 = 25%

But the first is below average and the second is above average, so they shouldn't lead to the same score.

Why not? Unless you're generalising all of the types of games you can play here,
these 2 players above in your example (one winning 1 out of 4 2p and one winning 1 out of 4 8p) are equally skilled... at their
chosen game type.

It's where a player has a mixed bag of game types played that the formula is trying to bias for.
A player is surely more skillful having played different game types so there ought to be bias for that and against
players who play the same game type all of the time.

Your logic is not in question , I agree with (setting c=#defeated/#lost) that #defeated/#defeated + #games lost = c/(c+1), that's just basic maths.

The probabilities of winning a general n-player game are not as simple if you consider all of the game types.

Special cases are Assassin and Team games where the probabilities of winning are different than a Standard winner take-all game.
Different criteria define winning in each of the games and therefore the probabilities of winning them.
That means that trying to generalise a player's strength not only has to be independent of number of players, it should also be
independent of game type.

So, like a number of posters mentioned in the other thread, it is probably better to have a separate measure of skill per game type... and that leads to a whole host of numbers.

All is not lost though, we are due in-depth statistics which will split stats along number of players, maps and game types, we shall see.
At that time, there will be no need to find a general formula for skill because the win/loss ratio per game type/number of players will tell us that.

chipv wrote:

joriki wrote:OK, but this comparison doesn't work out:

If you win 1 2-player and lose 3 2 player-games the ratio is 1/4 = 25%
If you win 1 8-player and lose 3 8 player-games the ratio is also 7/28 = 1/4 = 25%

But the first is below average and the second is above average, so they shouldn't lead to the same score.

Why not? Unless you're generalising all of the types of games you can play here,
these 2 players above in your example (one winning 1 out of 4 2p and one winning 1 out of 4 8p) are equally skilled... at their
chosen game type.

It's where a player has a mixed bag of game types played that the formula is trying to bias for.
A player is surely more skillful having played different game types so there ought to be bias for that and against
players who play the same game type all of the time.

It looks like we're trying to solve different problems. You seem to be saying that the problem I (and I believe Itrade) was trying to solve isn't a problem -- that it's OK that winning 1 out 4 two-player games and winning 1 out of 4 8-player games leads to the same score, as it currently does with the winning percentage. I disagree. The player winning 1 out of 4 two-player games is doing rather poorly, since all players on average win 1 out of 2 two-player games, whereas the player winning 1 out of 4 eight-player games is doing rather well, since all players on average win 1 out of 8 eight-player games. The two-player player is less skilled than the people she's been playing, and the eight-player player is more skilled than the people he's been playing, so unless there's a systematic bias in the people they've been playing (which this sort of score can't take into account; that's what the point system is for), the two-player player is less skilled than the eight-player player. The whole point of Itrade's proposal was to reflect this; if you disagree with this premise, we're unlikely to come to an agreement about what score to use.

It's great that there will be statistics specific to the types of games and the numbers of player. However, it would still be nice to have a simple overall score one can take in and compare at a glance, and the two are not mutually exclusive, so I'd appreciate further feedback on the opponents beaten/games lost idea.

joriki wrote:
It looks like we're trying to solve different problems. You seem to be saying that the problem I (and I believe Itrade) was trying to solve isn't a problem -- that it's OK that winning 1 out 4 two-player games and winning 1 out of 4 8-player games leads to the same score, as it currently does with the winning percentage. I disagree. The player winning 1 out of 4 two-player games is doing rather poorly, since all players on average win 1 out of 2 two-player games, whereas the player winning 1 out of 4 eight-player games is doing rather well, since all players on average win 1 out of 8 eight-player games. The two-player player is less skilled than the people she's been playing, and the eight-player player is more skilled than the people he's been playing, so unless there's a systematic bias in the people they've been playing (which this sort of score can't take into account; that's what the point system is for), the two-player player is less skilled than the eight-player player. The whole point of Itrade's proposal was to reflect this; if you disagree with this premise, we're unlikely to come to an agreement about what score to use.

I do not agree with that premise, you're probably right on agreement.

An 8 player assassin game vs a 2 player 'assasin game' is a good example why.
A 2 player 'assassin game' is exactly the same as a Standard one so same degree of difficulty to win.
But the degree of difficulty in winning an 8 player assassin game is different than that of an 8 player Standard game.
Because the degree of difficulty varies across game types, it is not necessarily the case that a player on average wins
1 out of 8 eight-player games and 1 out of 2 player games because of that variation in game type - it is not the same type of game.

Where we would agree is if your proposal was split across game types - that would then be comparing like for like. (Same probability of winning).
I would also completely agree with you if we were only talking about Standard Type games where you have one winner defeating the rest of the players.

It's great that there will be statistics specific to the types of games and the numbers of player. However, it would still be nice to have a simple overall score one can take in and compare at a glance, and the two are not mutually exclusive, so I'd appreciate further feedback on the opponents beaten/games lost idea.

If we can agree that this is a generalised formula, I can probably live with that assumption. As far as a single indicator goes, your last suggestion is the best one yet as it most closely fits the model ltrade was looking for.

chipv wrote:If we can agree that this is a generalised formula, I can probably live with that assumption. As far as a single indicator goes, your last suggestion is the best one yet as it most closely fits the model ltrade was looking for.

Yes, I agree that it's a generalised formula; there are lots of things it won't tell you and it'll lump together lots of things that would be treated separately in a more detailed analysis. I'm glad we agree that as far as such single indicators go, the last version was the best.

Since that's buried somewhere up in the thread, here it is again for future reference: The score would be #opponents beaten / (#opponents beaten + #games lost), the value would be between 0 and 1 (or equivalently 0% and 100%); 0 would mean always losing; 1 would mean always winning; 1/2 would mean winning as many games as one would expect if winners were chosen randomly; so > 1/2 would be good, < 1/2 would be bad.

I do want to reply to your argument about assassin games, though, too:

chipv wrote:An 8 player assassin game vs a 2 player 'assasin game' is a good example why.
A 2 player 'assassin game' is exactly the same as a Standard one so same degree of difficulty to win.
But the degree of difficulty in winning an 8 player assassin game is different than that of an 8 player Standard game.
Because the degree of difficulty varies across game types, it is not necessarily the case that a player on average wins
1 out of 8 eight-player games and 1 out of 2 player games because of that variation in game type - it is not the same type of game.

I still disagree. The fact that averaged over all players a player on average wins 1 out of 8 eight-player games has nothing to do with the type or difficulty of the game. Each eight-player game has eight players and one winner, so if someone is winning more than 1 out of 8 games someone else is necessarily winning less than 1 out of 8 games. The game may be more difficult in the sense that there's more to learn about it, so it'll take longer before you can beat the best players at it, but it can't be more difficult in the sense that it's harder to win and hence you win fewer than 1 out of 8 games on average. It's just as hard for everyone else to win, and the 1/8 ratio of winners to players is independent of the difficulty.

Team games are different, since there's more than one winner, but that could be taken into account by counting each team as one player.

For terminator games, there'd be a choice how to treat them. They could be treated just like standard games, with the final winner being treated as the winner and all intermediate eliminations that win points being ignored for the score, or alternatively one could reinterpret "#opponents beaten" and "#games lost" as "#opponents eliminated" and "#times eliminated oneself" -- then the score would take the intermediate terminations into account and the final win would just count as a single elimination; this would all fit in well with the interpretation of the score as above.

I think that covers all the game types.

I'm not trying to argue that this simple score would capture all the subtleties of measuring skill at all the various game types. For instance, I think that if a certain type of game is more difficult, that will lead to a greater spread in skill levels with respect to that type, and hence probably a greater spread in scores for people who play that type of game a lot, so perhaps a score of 20% at a difficult game would underestimate how often a player would win at a simpler game. All I'm trying to argue is that independent of the type and difficulty of the game, the score is set up to yield 50% on average, because the number of opponents beaten and the number of games lost will be the same on average, simply because someone's beaten opponent is someone else's lost game, so independent of game type and player count you'll be able to tell whether a player is doing well (> 50%) or not so well (< 50%).

Just read your post.

We'll agree to disagree on the game types logic, I think this could well continue for a while and detract from the fact that
we agree with your final formula.

I can put this in Map Rank to generate a separate number per map,game type whatever combination and that should keep both of us happy.

As for this appearing on the site, you may have to wait a little longer, but I still think it's a good idea and you have presented it well and conducted your debating in a manner that has made it very pleasing to talk to.

Game type only comes into consideration if its a team game. Terminator and assassin should count the same as standard.

I'm gonna mention team games here...

Winning "odds" of a Quads game is 1/2 - whereas "odds" of a standard game are 1/8 (All I'm talking here are total winners per game)

That needs to be addressed into the formula...

Perhaps something like...

(#Opponents defeated / (#Players Played + #Teammates))

(Where Players played = Teammates & Opponents)

And then you need to add in the #games lost thing that Joriki mentioned.

C.

Having slept on this, I think joriki is right on every single count including teams, so I agree in its entirety.

So count each team as one player (so #opponents defeated = #teams defeated).

(You now get #defeated = #total teams - 1. Mirrors non-team games where #defeated = #total players - 1)

This way you keep the addition #total = #winners + #losers which in the team games case is the number of teams not players.

Also fits with the number of games you expect to win on average.
As yeti_c mentioned take Quads (2 team game) as an example - you expect to win the same proportion of games as a 2 player game.

So concluding we still have joriki's formula

#opponents defeated / #opponents defeated + #games lost

For team games #opponents defeated = #teams defeated

But #teams defeated = #defeated players / #players per team
(e.g. You win an 8p doubles game (4 teams) so defeating 3 teams.
#defeated players = 6, #players per team = 2 so 3 = 6/2 as required)

You can wrap this up in the same formula now by replacing #opponents defeated by #defeated players / #players per team:

Works for non teams games , #players per team =1 so this boils down to joriki's original formula

So now we have

(#defeated players / #players per team) / (#defeated players / #players per team) + #games lost

Should cover it.

Awesome...

C.

Why not just display the 'Average No of Opponents' next to the Win Percentage?

MrBenn wrote:Why not just display the 'Average No of Opponents' next to the Win Percentage?

Or the "expected %" of winning. Say for example, I play only 4 player standard games and win 30% of them. I would have 30%/25% (actual win%/expected win%) next to my score. This would indicate how well someone fared, compared to the "average".

Expected win = 1 / Average No of Players

MrBenn wrote:Expected win = 1 / Average No of Players

exactly, as long as you count teams as a single opponent.

MrBenn wrote:Why not just display the 'Average No of Opponents' next to the Win Percentage?

I'm not sure whether you meant opponents or players (the latter being one more), but neither would give the right results -- if you wanted to go in that direction, you'd need the average of the reciprocal of the number of players:

If someone plays an equal number of two-player games and eight-player games, they'd be expected to win 4 out of 8 two-player games and 1 out of 8 eight-player games, for a total of 5 out of 16 games, or roughly 1 out of 3. The average number of players (or opponents), which is 5 (or 4), would lead you to expect 1 out of 5 (or 4). The average of the reciprocals of the numbers of players, 1/2 and 1/8, on the other hand, is 5/16; this is just another way of expressing Timminz' proposal: the expected win percentage.

MrBenn wrote:Expected win = 1 / Average No of Players

The expected win percentage is not the reciprocal of the average number of opponents; it's the average of the reciprocal of the number of opponents. That difference is important, as the above example shows.

The expected win percentage is certainly a simple and useful alternative to my proposal. They both have advantages. In itself, the expected win percentage is a more direct measure; everyone will immediately understand what it means, whereas a "two-player equivalent" percentage is a bit more indirect. On the other hand, the two-player equivalent percentage gives you one number you can use to compare players, whereas with the expected win percentage, you need to first compare that number to the actual win percentage, and then compare that comparison against the corresponding comparison for the other player, which is a bit more indirect. So once one has understood what the two-player equivalent percentage means, it offers a more direct comparison.

Considering this, I would suggest that all three percentages be displayed: actual win percentage (#games won/#games played), expected win percentage (sum over reciprocals of player numbers/#games played), two-player equivalent win percentage (#opponents beaten/(#opponents beaten + #games lost)) (appropriately taking team games into account in each case)

chipv wrote:Having slept on this, I think joriki is right on every single count including teams, so I agree in its entirety.

I'm glad we agree. Sleep is a wonderful thing

what about this: In a profile,

OVERALL: wins * #players/games

Win ratio (straight wins/ # games) for 1v1

Win ratio for teams (all teams) (wins * team players defeated) /games

Win ratio for all multiple player games. (wins * players)/games
OR for standard, assassin and Terminator seperately.


To make OVERALL ratio: (wins* players)/#games (teams would be multiplied by players)


I realize those are multiple statistics, but if in the player's personal profile, I don't think extra lines matter that much. Some people would really like to see this.

You really don't have to, in fact should not use logarythmic equations. It is just a straight ratio, but with the number of players. This will inherently be "weighted" because the number of players will vary. So, someone who plays a lot of 1v1 (like I) will have a different statistic than someone who plays a lot of other games.

ALSO, you really don't need to subtract wins from losses. It is more intuitive and actually better to just show wins. (or losses, but I believe most folks would just prefer a winning statistic : )....)

The only real issue is whether you JUST want the overall totals or want to see things broken out.

PLAYER57832 -- I'm having a bit of trouble following you. I don't think anyone in this thread ever suggested subtracting wins from losses -- which proposal are you referring to there? We had dropped the idea of taking a logarithm a while ago, since it was pointed out that it's a disadvantage that neither the ratio I had originally proposed nor its logarithm has a natural scale to compare it to, so we agreed that #opponents beaten/(#opponents beaten + #games lost) would be better (suitably modified for team games) since it yields an easily interpretable number between 0 and 1 (or 0% and 100%) without requiring a logarithm.

Regarding your new formulas, I'm not sure how to interpret them. If by "#players" you mean the total number of players, you need another division by #games to get a ratio that doesn't grow with the number of games played. If you mean the average number of players, please see my post above with an example that shows that that doesn't solve the problem we were trying to solve. I can't think of any other interpretation of "#players" in your formulas.

PLAYER57832 wrote:So, someone who plays a lot of 1v1 (like I) will have a different statistic than someone who plays a lot of other games.

That's precisely what the proposal was trying to avoid. I agree that there's enough space on the profile page to have various statistics, and I'm not arguing against whatever statistics you'd like to see there, but this thread was about a proposal for a statistic that as far as possible does not depend on the number of players one tends to play.

PLAYER57832 wrote:The only real issue is whether you JUST want the overall totals or want to see things broken out.

I disagree. That may be the only issue you're interested in, but we've discussed some other issues in this thread, and we've come up with interesting ideas that I'd like to see realized.