Leo, Greg and Lou, Jian and Van der Linden, Martin and Vorobeychik, Yevgeniy and Wooders, Myrna H., “Matching Soulmates.” Journal of Public Economic Theory. *forthcoming*

In Matching Soulmates, my co-authors and I study a a recursive process in matching that
forms coalitions that are mutually most-preferred by members of that coalition. We call this process the iterated matching of soulmates: *IMS*. We show that mechanisms that implement IMS have strong properties among those who are matched by IMS.

How many people can “usually” be matched by IMS? This depends a lot on the environment and structure of preferences. But what about in totally unstructured environments?

Here is some R code that takes advantage of R's array-centric functions and some custom operators to count the number of people that can be matched as soulmates in 10000 random 10-person stable roommates problems.

The core of this code is contained in the *whos_a_soulmate* function. This function is somewhat non-standard R code. It is inspired by the type of programming normally done in one of R's predecessors APL. Here is the function:

```
whos_a_soulmate <- function(p){which((p%r%rank%>%hadamard%r%min)==1)}
```

To see how this works, let's first setup an example preference matrix.

```
p <- matrix(c(3,1,2,1,3,2,2,1,3),3,3,byrow=TRUE)
p
```

```
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 1 3 2
## [3,] 2 1 3
```

Plyer 1 likes 2 best, player 2 likes 1 best, player 3 likes 2 best. Note that 1 and 2 are soulmates. We should find that 1 and 2 are soulmates in the first round of IMS.

The first step of the function ensures the matrix “p” is a ranking matrix. We apply the “rank” function to the “p” matrix by row using the by-row operator created here. Since “p” is already a ranking matrix, this does not do anything. It is more important once some players have been removed and we need to re-rank.

```
p %r% rank
```

```
## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 1 3 2
## [3,] 2 1 3
```

Now we take take Hadamard product of the preference matrix with its own transpose by piping it to our “hadamard” function with the built-in R pipe.

```
p %r% rank %>% hadamard
```

```
## [,1] [,2] [,3]
## [1,] 9 1 4
## [2,] 1 9 2
## [3,] 4 2 9
```

Note how off diagonal 1's in this matrix represent positions where both players prefer eachother most. We want to pull these players who have a 1 in their row. To do this, we pipe what we have to the “min” function using our by-row operator. If a 1 is present, that player has a soulmate.

```
p %r% rank %>% hadamard %r% min
```

```
## [,1] [,2] [,3]
## [1,] 1 1 2
```

Note that 1 and 2 have a “1” here, indicating that players 1 and 2 have a soulmate from this group. Select just these players, we compare this vector to 1. The indicies that map to “TRUE” are players who have a soulmate.

```
p %r% rank %>% hadamard %r% min == 1
```

```
## [,1] [,2] [,3]
## [1,] TRUE TRUE FALSE
```

We see now that players 1 and 2 have a soulmate. We can now remove these players by using our remove operator “%rm%”.

```
p <- p %rm% whos_a_soulmate(p) %>% as.matrix
p
```

```
## [,1]
## [1,] 3
```

What we get is the remainder of the preference matrix “p” after removing soulmates. We only have player 3 left. Note that player 3 originally ranked being “alone” as third-best. That's why a 3 shows up here. Let's pipe this to rank again to have the player(s) rerank their potential partners.

```
p %r% rank
```

```
## [,1]
## [1,] 1
```

Now we can run the process again.

```
p <- p %rm% whos_a_soulmate(p)
p
```

```
## <0 x 0 matrix>
```

Since being alone is the only outcome left, player 3 is a solemate-group-of-one and is removed. This demonstrates how the “IMS” function proceeds. It calls “remove_soulmates”. If anyone is removed, “IMS” calls itself. If there is no one to remove or no one further can be removed, the function returns the remaining preference matrix.

```
#Operator "m %r% f" applies function "f" to matrix "m" by rows.
`%r%` <- function(m,f){t(apply(m,1,match.fun(f)))}
#Operator "m %rm% i" removes rows and column indicies "i" from matrix "m".
`%rm%` <- function(m,i){if(length(i)>0){m[-i,-i]}else{m}}
#Operator applies function "f" across list "l" with sapply.
`%s%` <- function(l,f){sapply(l,match.fun(f))}
#Returns Hadamard product of matrix m with its transpose.
hadamard <- function(m){m * t(m)}
#Given preference matrix, returns vector of player indicies who are first-order soulmates.
whos_a_soulmate <- function(p){which((p%r%rank%>%hadamard%r%min)==1)}
#Removes first order soulmates from preference matrix p.
remove_soulmates <- function(p){p %rm% whos_a_soulmate(p) %>% as.matrix}
#Recursively applies remove_soulmates until there's none left to remove.
ims <- function(p){if(dim(p)[1]==0 || identical(remove_soulmates(p),p)){p}else{ims(remove_soulmates(p))}}
#Create a random roommates preference matrix with "dim" players.
create_preference <- function(dim){
p <- matrix(runif(dim*dim),dim,dim,byrow=TRUE)
diag(p)<- 1
p %r% rank
}
#Set up parameters. "dim" is the number of players. "n" is the number of random trials.
dim <- 10
n <- 100
#Create List of "n" Preference Matricies with "dim" players.
preference_list <- lapply(1:n,function(x){create_preference(dim)})
#Apply IMS to the list of preferences then count the number of players remaining.
remaining <- preference_list %s% ims %s% dim
#Get the number of players removed by IMS.
removed <- dim - remaining[1,]
#Make a table of frequencies of the number of removed players.
removed %>% table/n
```

```
## .
## 0 2 4 6 10
## 0.57 0.25 0.13 0.01 0.04
```