Probability of winning races/multi participant events

I am working on a project in which I am keeping track of results and analysing them in a game myself and 5 friends play, because I am a bit of a spreadsheet nerd. I want to calculate probabilities of each player winning the next game, each player finishing 2nd, each player finishing 3rd etc. However, because I don't want to just use the win/2nd/3rd rate, I am struggling. I want to have the probability factor in multiple things (for the winning probability):
  • Number of wins total
  • Number of 2nd and 3rd places (maybe lower down if possible too) (Person with 1 "lucky" win but loads of poor results will have a lower probability than someone consistently 2nd or 3rd with no win)
  • Recent form, last 5 or maybe 10 games

From this, I want to be able to put a rough estimate on it so I can say that player x has a 23% chance of finishing 2nd in the next game, or player y has a 47% chance of finishing last because they are so awful at the game.

If it helps, the game does give you a certain number of points, and the winner is the one with the most points, so I am thinking I could use that an model each players graph against each other, but there is a maximum amount of points that can be achieved, and that'll affect any distributions like normal or Poisson. I'm sorry if the way I have worded this question is really stupid or vague and I have missed something obvious, but I have tried so many different ways, and my lack of knowledge of the entire world of statistics is plaguing me here.

Thanks all for your responses!