# Probability of winning multi participant events

I will just start off with context about myself. I am currently studying further mathematics at A-Level and have a keen passion in statistics. I am sure most don't care about that, but just so you know my knowledge is probably fairly limited on a wide range of topics that all of you know!

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. 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.

### Simple model

You could model the score from each of the players as some function (in the example image below a beta-binomial, but you may choose whatever is appropriate to you) that is independent of the others. Based on those curves you can compute the probability of the ranking of a player. (The order statistic could be computed, but with multiple players, it may be better to simulate it)

For instance when we simulate the above 10 000 times then the outcome frequencies are:

          3rd     2nd    1st
Alice     7150    2078   772
Bob       1909    5518   2573
Carol      941    2404   6655


You could actually also compute that table simply from the observed frequencies of places, but with this method you can handle situations when the group of competitors are not always of the same composition.

### Estimating the simple model

• If you have insight into the scores then you will be able to use those data to estimate the curves for each player.

• Otherwise you will need to estimate the curves based on the outcomes. This is not easy. The likelihood function is not easy to express and optimize. One way that you could tackle this is by using an approximate method for instance Approximate Bayesian Computation

### More complex model

The above method represents the scores from each player as independent from the opponents. But there might be correlations. Especially when interactions in the game occur.

For instance, in a game of soccer, the probability of the score is dependent not only on the player quality but also on the opponent's quality. Even in a relatively individual game like darts you get influences, not just psychologically, but also because of the layered scoring system with legs and sets.

In these circumstances, you might still think of some underlying player score function, and match it against others, but it will not be so easy to match the scores to an outcome. Possibly you may also think of more complex functions (e.g. a score for defense and a score for offense).

Now the player score function will be an input for another model. You can see this as a latent variable model the reasoning of the above player score functions is very similar to the probit model which models a probability that determines some binary outcome as the probability that a latent variable that is Gaussian distributed, surpasses some level.

You could base a model on something like that (a categorical probit model), but at this point, you might need to explain the mechanics of your game better.

• Thank you so much! That is a huge help. I will give you some info on the game mechanics: 1. In the first round, players are given 2 questions, and people vote for their favourite answer. The players are then given 1000 points for each question multiplied by the percentage of the vote they got in each question. If they win, they also get a bonus 100 points, and 250 points if they get 100%. In round 2 points are doubled but the same. In round 3, every answer is ranked best to worst and winner gets 3000. 2nd gets 2500 and so on. (1/2) Nov 12, 2020 at 21:37
• ...So reading this, I do like the beta binomial model, however as the points values are very specific, I imagine this may be difficult to achieve. I am also collecting data on the winner of each round, and how many 100%'s they get and the score in each round and final score, but unfortunately I am unable to collect data for each player head to head when their answers "face off" because then this would be very easy and I would establish a probability A beats B, B beats C and E beats A etc which could lead me to a final answer. (2/2) Nov 12, 2020 at 21:40
• And also, the other issue with beta binomial as you alluded too, is that the outcomes of points are not independent events since player A getting points means player B loses points, but obviously I hadn't told you that. I am going to delve deeper into some of the more complex models you mentioned and let you know how I got on. Thanks again Nov 12, 2020 at 21:42