In [1] a simple problem was posted:

- We have a population of \(N=24\) players, each with an ELO rating \(r_i\)
- We need to select \(2 \times 6\) players for 2 teams (each team has \(n=6\) members).
- We want to minimize the difference in average ELO ratings of the teams.

The poster asked for an algorithm. But, of course, this looks like a problem we can solve as a mathematical programming model.

First I generated some random data:

---- 11 PARAMETER r ELO rating player1 1275, player2 1531, player3 1585, player4 668, player5 1107, player6 1011 player7 1242, player8 1774, player9 1096, player10 1400, player11 1036, player12 1538 player13 1135, player14 1206, player15 2153, player16 1112, player17 880, player18 850 player19 1528, player20 1875, player21 939, player22 1684, player23 1807, player24 1110

I have no idea if these numbers are realistic or not.

It make sense to look at this from an

**assignment problem**point of view. It is amazing how often this concept is encountered in modeling. So we define:\[x_{i,j}=\begin{cases} 1 & \text{if player $i$ is assigned to team $j$}\\ 0 & \text{otherwise}\end{cases}\]

A high-level model can look like:

High-level Model |
---|

\[\begin{align}\min\>& | \color{DarkRed}{\mathit{avg}}_2 - \color{DarkRed}{\mathit{avg}}_1 | \\ & \sum_j \color{DarkRed} x_{i,j} \le 1 && \forall i\\ & \sum_i \color{DarkRed} x_{i,j} = \color{DarkBlue} n && \forall j \\ & \color{DarkRed}{\mathit{avg}}_j = \frac{\displaystyle \sum_i \color{DarkBlue} r_i \color{DarkRed} x_{i,j}}{\color{DarkBlue} n} \\ & \color{DarkRed}x_{i,j} \in \{0,1\}\end{align} \] |

As you can see, this model is largely an assignment problem plus some average calculations.

Notes:

- The absolute value is easily linearized in different ways:
- Bounding \[\begin{align} \min\> & z \\ & -z \le \mathit{avg}_2-\mathit{avg}_1\le z\end{align}\]
- Variable splitting: \[\begin{align} \min\> & z^{+}+z^{-} \\ & z^{+}-z^{-} = \mathit{avg}_2-\mathit{avg}_1 \\ & z^{+},z^{-}\ge 0 \end{align}\]
- Removing symmetry. We can require that the average rating of team 1 is not lower than of team 2. Now we just can minimize difference: \[\begin{align} \min\> & \mathit{avg}_1 - \mathit{avg}_2\\ & \mathit{avg}_1 \ge \mathit{avg}_2 \end{align}\]
- We can use the sum instead of the average

Surprisingly, the solution looks like:

---- 43 VARIABLE x.L assignment team1 team2 player1 1.000 player2 1.000 player4 1.000 player5 1.000 player6 1.000 player7 1.000 player8 1.000 player9 1.000 player10 1.000 player11 1.000 player17 1.000 player18 1.000 ---- 43 VARIABLE avg.L average rating of team team1 1155.833, team2 1155.833 ---- 43 PARAMETER report solution report team1 team2 player1 1275.000 player2 1531.000 player4 668.000 player5 1107.000 player6 1011.000 player7 1242.000 player8 1774.000 player9 1096.000 player10 1400.000 player11 1036.000 player17 880.000 player18 850.000 sum 6935.000 6935.000 avg 1155.833 1155.833

We achieved a perfect match!

The probability of this must be somewhat low. Well, no. If we try this 10 times with different random \(r_i\), we get 10 times a perfect match.

---- 50 PARAMETER trace results for each solve avg1 avg2 k1 1155.833 1155.833 k2 1583.333 1583.333 k3 1385.333 1385.333 k4 1258.500 1258.500 k5 1423.167 1423.167 k6 1491.833 1491.833 k7 1262.167 1262.167 k8 1736.167 1736.167 k9 1514.167 1514.167 k10 1483.667 1483.667

Using a solve loop of length 100 gives the same result. It looks that we almost always can form two teams with equal average ELO rating. I suspect it will be very difficult to derive anything analytically about the probability of not being able to find a perfect match. I was surprised by these results.

#### Discussion

Often, people with a Computer Science/programming background immediately think about algorithms instead of mathematical models. They miss out on a paradigm that can be very powerful and efficient (in terms of time needed to design, implement and maintain an algorithm or model). This model was thought out, implemented and tested in less than 20 minutes (including finding out what this ELO is). I am sure developing some special purpose algorithm will take much more time. In addition, for this model, the solver will find proven optimal solutions while developing an algorithm yourself will likely result in some heuristic without any concept of optimality.

#### References

- Team matchmaking algorithm based on ELO, https://stackoverflow.com/questions/53119389/team-matchmaking-algorithm-based-on-elo