Copula to ensure one team wins and the other loses (Bernoulli margins)

Question

Team $1$ has a historical win percentage of $p_1$.

Team $2$ has a historical win percentage of $p_2$.

The upcoming game features team $1$ against team $2$ and cannot end in a tie (one team wins, and the other loses).

While this isn't strictly true, assume the time series of wins for each team to be $iid$ (so no player injuries, improvements, etc).

I want to know the joint distribution of the game outcome. It seems like I can use margins of Bernoulli$(p_1)$ and Bernoulli$(p_2)$. For the copula, I figured I could use a Gaussian copula with a "correlation" parameter of $-1$ so that the Bernoulli margins always have opposite outcomes. However, when I simulated this, I did not observe such behavior.

library(copula)
set.seed(2023)

# Gaussian copula with -1 as the "correlation" parameter
#
cop <- copula::normalCopula(-1)

# Define the joint distribution with Bernoulli (binomial) margins and
# "cop" as the copula
# This features two good teams that win 90% and 80% of their games
#
joint_dis <- copula::mvdc(
  cop,
  c("binom", "binom"),
  list(
    list(size = 1, prob = 0.9),
    list(size = 1, prob = 0.8)
  )
)

# Simulate ten games
#
X <- copula::rMvdc(10, joint_dis)

# Nine of ten are (1, 1) outcomes where both teams win
# Huh?
#
X

Nine of the ten simulated games gave (1, 1) outcomes that I interpret as ties (yet, curiously, there are not any (0, 0) outcomes, even in a million simulated games).

Therefore, using a Gaussian copula with the "correlation" parameter set to $-1$ does not force anyone to lose (Bernoulli outcome of $0$).

If this Gaussian copula doesn't do the trick, what would?

Answer 1

A copula is just a way to construct a joining, or coupling, of two distributions.

Let $X$ and $Y$ be two random variables. There is a notion of maximal coupling of (the distributions of) $X$ and $Y$. This is a coupling $(X', Y')$ for which the probability $\Pr(X' = Y')$ has the highest possible value.

Now let's come to your context. Denote $B_1 \sim \text{Ber}(p_1)$ and $B_2 \sim \text{Ber}(p_2)$. In your case you don't want the maximal coupling of $X_1$ and $X_2$ but, on the contrary, you want to maximize $\Pr(X'_1 \neq X'_2)$ where $X'_1 \sim X_1$ and $X'_2 \sim X_2$ are jointly defined. Then apply the maximal coupling theorem to $X=X_1 \sim \text{Ber}(p_1)$ and $Y=1-X_2 \sim \text{Ber}(1-p_2)$. Indeed the equality $X'_1 = 1-X'_2$ is equivalent to $X'_1 \neq X'_2$ here. We find that the joint law of $X'_1$ and $1-X'_2$ given by $$ \begin{array}{c|cc} & 0 & 1 \\ \hline 0 & 0.1 & 0 \\ 1 & 0.7 & 0.2 \end{array} $$ is a maximal coupling, and for this coupling the joint law of $X'_1$ and $X'_2$ is given by $$ \begin{array}{c|cc} & 0 & 1 \\ \hline 0 & 0 & 0.1 \\ 1 & 0.2 & 0.7 \end{array}. $$ The probability of tie is $0.7$, and it is not possible to have a smaller value.

Answer 2

You've set up the problem to preserve the teams' historical win probabilities in the final result, but with $p_1 \not = 1 - p_2$ this is impossible. If two teams with such high win probabilities play a series with each other, one or both of them must do worse than their historical average.

The thing to remember is that the choice of copula does not affect the margins at all. If you specify margins of Bernoulli(0.9) and Bernoulli(0.8), then you're going to get (on average) 90% and 80% wins, respectively. All the copula is going to do is determine how they get paired up. Because you chose a countermonotone copula, the losses, when they occur, will always get paired up with wins for the other team (that's why you never see a (0,0) result), but you haven't allowed enough losses for either team to avoid some wins getting paired up with other wins.

More importantly, copulas aren't the right tool to use for a problem like this because you really only have a single random variable. That is, $x_1$ and $x_2$ aren't merely correlated; they are one and the same, with one of them simply in disguise. Copulas aren't meant to handle scenarios like that. It turns out that it sort of works for continuous variables, but as you can see, it falls apart for discrete variables.