4-Player ELO Simulation

Scope

I wanted to create a game that allows a user to pick their favorite choice out of four options. By picking their favorite, the winning competitor should go up in rank, while the losing competitors propotionately decrease in rank.

This simulation takes in 2 parameters: # of competitors & # of rounds

ELO Algorithm

I based this 4-player ELO off of the standard 2-player ELO Algorithm used on most chess websites:

E_a = probability of player a winning

R_a = Rating of player a; R_b = Rating of player b

R_A = Rating before; R^'_A = Rating After

S_A = 1 if player is winner; S_A = 0 if player is loser

K = constant

The resulting 4-player version of this alogirthm is as follows:

E_a1 = (1/(1+10^{(r₂-r₁)/400}) + 1/(1+10^{(r₃-r₁)/400}) + 1/(1+10^{(r₄-r₁)/400}))/6

E_a2 = (1/(1+10^{(r₁-r₂)/400}) + 1/(1+10^{(r₃-r₂)/400}) + 1/(1+10^{(r₄-r₂)/400}))/6

E_a3 = (1/(1+10^{(r₁-r₃)/400}) + 1/(1+10^{(r₂-r₃)/400}) + 1/(1+10^{r₄-r₃)/400}))/6

E_a4 = (1/(1+10^{(r₁-r₄)/400}) + 1/(1+10^{(r₂-r₄)/400}) + 1/(1+10^{(r₃-r₄)/400}))/6

R^'_A1 = round(R_A1 + k(S_A - E_a1))

R^'_A2 = round(R_A2 + k(S_A - E_a2))

R^'_A3 = round(R_A3 + k(S_A - E_a3))

R^'_A4 = round(R_A4 + k(S_A - E_a4))

Instructions

1. Clone Repo: git clone git@github.com:P-Carth/Multiplayer-ELO-Simulation.git

2. Open ELO_Simulation.ipynb in Jupyter Lab

3. Run the first 4 code blocks

4. The 5th code block begins the simulation, where it will ask you to input # of rounds and # of competitors

5. Finally, run the last code block to view the simulation's analysis

Results & Discussion

While I was successful in creating a normally distributed 4-Player ELO System, it does seem that there is a consistent skew to the right. Fortunately, for my purposes this is not an issue.

This algorithm can be applied to a number of scenarios to compare competitors in an adjusted and fair manner. For example, I applied this algorithm to an NFT game that I made with Flask that allows users to vote on their favorite NFT (shown below).