[Math] How to calculate Team Strength for future prediction

combinatoricsgame theorylogicpuzzlestatistics

You are given with $4$ players name, namely Player $A$, Player $B$, Player $C$ and Player $D$. These players are grouped into two teams with two players each. A Game is played between the two team.For example,

Game 1: $AB$ vs $CD$
Game 2: $AC$ vs $BD$
Game 3: $AD$ vs $BC$

We also know the outcome of each game, let say

Game 1: Team $AB$ wins
Game 2: Team $BD$ wins [Bcoz, $A$ and $C$ hate each other]
Game 3: Team $AD$ wins

  • All i wanted is to come up with a measure with this data to find each player strength and their team strength so that i can predict the outcome of the game.
  • For four players we can enumerate all possible team composition and come up with the measure. How do we generalize if you are given with $N$ players and each team containing $M$ players.

[Number of Ways to choose Team 1 = $\frac{{N}\choose{M}}{M}$]

Best Answer

We can model this problem as a network as follows. We can describe the outcomes for teams of size 2 as a weighted graph, where an edge between two players indicates the number of wins that pair has had. The toy example in the question results in the graph:

Team graph

In this graph, members A, B and D are equivalent (formally, there are automorphisms of the graph that can map $x$ to $y$ for all $x,y \in \{A,B,D\}$). Thus, there is insufficient information to distinguish whether team $AB$ is better than $AD$ or $BD$. The graph does suggest, however, that any team involving $C$ will be poor.

As for a general method for deciding which pair might make a good team, we will need to make some choice as to what data would indicate a good team. I.e., we will need to decide on some kind of network measure for the edges in the network. A lot of network theory follows along these lines: there can be numerous intuitively good choices of measures, but can give results that contradict one another. Moreover, practical constraints, such as the ability to compute the measures, also plays a large role in what measures to use.

Here's one simple possibility (there's probably much more sophisticated measures in the literature):

  • For each player $p$, let the player weight $W(p)$ be the sum of the weights of the edges it is an endpoint of.
  • We assign a team $pq$ the team weight $W(pq)=W(p)+W(q)$. The larger the weight, the better.

To illustrate, let's suppose players $A,B,C,D$ paired up and played some more games, and the resulting graph looks like this:

Another team graph

We can calculate the weights of the players

  • $W(A)=4$,
  • $W(B)=6$,
  • $W(C)=1$, and
  • $W(D)=3$.

and the weights of the teams

  • $W(AB)=4+6=10$,
  • $W(AC)=4+1=5$,
  • $W(AD)=4+3=7$,
  • $W(BC)=6+1=7$,
  • $W(BD)=6+3=9$, and
  • $W(CD)=1+3=4$.

By this measure, we would conclude that $AB$ is the best team.

One might argue that this measure does not capture some important property of real-world data. This is to be expected of such a basic model. The next step is to develop a better model that incorporates the missing property (which, in turn, will have its own deficiencies). Then we improve that model, and repeat until we're at a point where we're generally satisfied.

The above will extend to $k$-player teams by using $k$-uniform hypergraphs.

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