# Is there a long-term scoring system for 4 or 5 players who have played the same game repeatedly?

We are a group of 4, sometimes 5 players who have played Power Grid every week for nearly 2 years. For every game we have played, I have kept track of game dates and rankings (1st place, 2nd,..., 5th).

Is there a scoring system which would allow me to award points for each rank in each game and possibly calculate the probability of any one player winning the next game?

• There's some suggestions for ELO/Ranking options on ChessExchange. Some seem to take multiplayer games as well, or could be combined with the ideas in the answer below. – Jontia Jan 17 '20 at 10:35

There's a relatively direct translation of the Elo rating system to multi-players: just treat each game as a set of games between each pair of players, with each pair having a winner and a loser based on their relative final rank in the game.

See http://www.tckerrigan.com/Misc/Multiplayer_Elo/ for an example of an even simpler method: pairing each player only with the players directly ahead of and behind him or her in the final standing. That page also includes performance of this method.

Doing a full pairwise calculation is more work (more calculations), of course, but it seems warranted in my opinion. Restricting the score comparison to one's immediate neighbors in finish position seems arbitrarily restricted when attempting to measure skill.

One thing to consider when using either approach is how much the game score is (or should be considered) a measure of relative performance. For pure footrace kinds of games (multi-player solitaire, where each player's actions do not affect the scores of the other players), this system is absolutely fine.

At the other end of the spectrum, king-of-the-hill games where there's really just two levels of finishing: win or don't, you may be better off (i.e., get a better measure of skill) by treating every pair of non-winners as tied, even if there is some score kept to determine who wins that you might obviously use to rank two non-winners relative to each other. You might choose to treat them as tied to reflect the fact that a player, by employing the best strategy to win, may have to risk enough of his or her score so that, if the gambit fails, their score is reduced below other non-winners.

If you rank the non-winners according to score, you may find that the fact of keeping score affects the way they play (i.e., a player may play for second rather than risk a play for the win).

This is already true in the base 2p Elo system, where a player might, by virtue of the score being kept, play for a sure draw over a possible win (and possible loss). But it's more of an issue in multi-player games. It's just a matter of what you want your score to emphasize / what you want the players to play for (max score vs. max chance to win)

(The linked page also mentions another algorithm: TrueSkill by Ralf Herbrich et al. at Microsoft, which I hadn't heard of before now).

I don't know where you keep the tracks, but BG Stats is great for this functionality (and the app syncs automatically with BGG). They have a lot of statistics and charts, based on the players and on the games. You can make some predictions based on the information displayed, although it doesn't show the chances of someone winning.