How to devide some fixed amount of reward points to players of a racing game in a fair way depending on their finishing time

Question

I'm in need of some kind of algorithm I can't figure out on my own sadly.

My biggest problem is that I have no good way to describe the problem... :/

I will try like this:

Imagine you have a racing game where everyone can try to be the fastest on a track or map. Every Map is worth 100 Points in total. If someone finished a map in some amount of time he gets a record in a database. If the player is the first and only player to finish this map he earns all the 100 points of this map.

Now, that's easy ;) but...

Now another player finishes the map. Let's imagine the first player finishes in 50 Seconds and the 2nd player finishes in 55 seconds, so a bit slower. I now need a calculation depending on both records in the database. Each of both players now earn a part of the 100 points. The faster player a bit more then the slower player. Let's say they finished the exact same time they both would get 50 points from 100, but as the first one is slightly faster, he now earns something around 53 of the points and the slower player just 47.

I started to calculate this like this:

Sum of both records is 105 seconds, the faster player took 50/105 in percent of this, so he earns 100-(50/105*100) points and the slower player 100-(55/105*100) points. The key to this is, that all points distributed among the players always equals to 100 in total. This works for 2 players, but it breaks at 3 and more.

For example:

Player 1 : 20 seconds

Player 2 : 20 seconds

Player 3 : 25 seconds

Calculation would be:

Player 1: 100-(20/65*100) = 69 points

Player 2: 100-(20/65*100) = 69 points

Player 3: 100-(25/65*100) = 61 points

This would no longer add up to 100 points in total.

Fair would be something around values of:

Player 1 & 2 (same time) = 35 points

Player 3 = 30 points

My problem is i can't figure out a algorithm which solves this.

And I need the same algorithm for any amount of players. Can someone help with an idea? I don't need a complete finished algorithm, maybe just an idea at which step i used the wrong idea, maybe the sum of all times is already a bad start.

Thx in advance :)

Answer 1

Simple observation: Let the sum of times for all players be S. A person with lower time t would have a higher value of S-t. So you can reward points proportional to S-t for each player.

Formula:
Let the scores for N players be a,b,c...,m,n. Total sum S = a+b+c...+m+n. Then score for a given player would be

score = [S-(player's score)]/[(N-1)*S] * 100

You can easily see that using this formula, the sum of scores of all players will be always be 100.

Example 1:

S = 50 + 55 = 105, N-1 = 2-1 = 1
Player 1 : 50 seconds => score = ((105-50)/[1*105])*100 = 52.38

Player 2 : 55 seconds => score = ((105-55)/[1*105])*100 = 47.62

Similarly, for your second example,

S = 20 + 20 + 25 = 65
N - 1 = 3 - 1 = 2
For Player 1, (S-t) = 65-20 = 45
Player 1's score => (45/(2*65))*100 = 34.6 
Player 2 => same as Player 1  
For Player 3, (S-t) = 65-25 = 40
Player 3's score => (40/(2*65))*100 = 30.8

This method avoids any division in the intermediate states, so there will be no floating point issues for the calculations.

Answer 2

We can give each player points proportional to the reciprocal of their time.

One player with t seconds gets 100 × (1/t) / (1/t) = 100 points.

Of the two players, the one with 50 seconds gets 100 × (1/50) / (1/50 + 1/55) ≈ 52.4, and the one with 55 gets 100 × (1/55) / (1/50 + 1/55) ≈ 47.6.

Of the three players, the ones with 20 seconds get 100 × (1/20) / (1/20 + 1/20 + 1/25) ≈ 35.7, and the one with 25 seconds gets 100 × (1/25) / (1/20 + 1/20 + 1/25) ≈ 28.6.