battle of the sexes; `home_away()` data format

[RobinHankin]

I was looking at Battle of the Sexes as implemented in sagemath, where I found the following problem (which I have slightly rephrased):

Two players, Amy and Bob, are modeled. Amy prefers to play video games and Bob prefers to watch a movie. They both however want to spend their evening together. This can be modeled using the following two matrices:

$$ A= \begin{pmatrix}3 & 1\\0&2 \end{pmatrix},\qquad B=\begin{pmatrix}2&1\\0&3\end{pmatrix} $$

Matrix A represents the utilities of Amy and matrix B represents the utility of Bob. The choices of Amy correspond to the rows of the matrices: • The first row corresponds to video games. • The second row corresponds to movies. Similarly Bob’s choices are represented by the columns: • The first column corresponds to video games. • The second column corresponds to movies. Thus, if both Amy and Bob choose to play video games: Amy receives a utility of 3 and Bob a utility of 2. If Amy is indeed going to stick with video games Bob has no incentive to deviate (and vice versa).

The reason that this problem appears here as a hyper2 github issue is that the above system looks very similar to the data format needed by function home_away(), and if there is common ground it might be a good idea to exploit it.

[RobinHankin]

Amy <- matrix(c(3,0,1,2),2,2)
dimnames(Amy) <- list(Amy=c("video","movie"),Bob=c("video","movie"))
Bob <- matrix(c(2,0,1,3),2,2)
dimnames(Bob) <- dimnames(Amy)
Amy
#>        Bob
#> Amy     video movie
#>   video     3     1
#>   movie     0     2
Bob
#>        Bob
#> Amy     video movie
#>   video     2     1
#>   movie     0     3

^{Created on 2023-11-22 with reprex v2.0.2}