Is this a good method for deterministically randomising the order of two names?

Question

In English speaking culture, when a man and a woman get married, the woman traditionally takes the man's surname. My brother took his wife's surname, but he is the exception, not the rule.

Some people have said that this system is unfair because it prioritises the man over the woman. Benefits of this system:

1. It is simple
2. It keeps surnames short
3. It is somewhat random, in that for any pair of surnames there's likely no gender bias
4. It always gives the same answer, so it can't be gamed

Alternatives I have heard suggested:

1. Use both surnames - surnames become long and unwieldy after only a few generations
2. Use the surname that comes first alphabetically - eventually all surnames start with "A"

My (non-serious hypothetical) idea:

1. Concatenate both names into two strings, one in alphabetical order, one in reverse alphabetical order
2. Apply a hash to each
3. The hash that sorts first determines which name "wins"
4. The system is not allowed to be gamed by using uncommon spelling, unusual capitalisation, alternate encodings, one party changing their name before the marriage, etc. Assume each party is acting sincerely and agrees to abide by these rules without attempting to gain an advantage

It satisfies all but the first of the benefits of the traditional system while at the same time ignoring gender:

1. It is simple It requires a computer to work out
2. It keeps surnames short
3. It is somewhat random, in that for any pair of surnames there's likely no gender bias
4. It always gives the same answer, so it can't be gamed

Using MD5 as the hash, here is a table of which names would "win":

         | Brown    | Jones    | Smith    | Taylor   | Williams | Wilson
---------+----------+----------+----------+----------+----------+---------
Brown    |          | Brown    | Smith    | Taylor   | Williams | Wilson
Jones    | Brown    |          | Jones    | Jones    | Williams | Jones
Smith    | Smith    | Jones    |          | Smith    | Williams | Smith
Taylor   | Taylor   | Jones    | Smith    |          | Taylor   | Taylor
Williams | Williams | Williams | Williams | Taylor   |          | Wilson
Wilson   | Wilson   | Jones    | Smith    | Taylor   | Wilson   |

If the two names are the same there's no need to choose one since the answer is the same either way, so I haven't included those in the table for clarity.

As you can see, there are many "rock-paper-scissors" triads, such as "Brown/Jones/Smith" where each name beats only one of the other names:

• Brown/Jones > Brown
• Brown/Smith > Smith
• Jones/Smith > Jones

Is this scheme mathematically sound? With enough names would this give each name a 50% chance of surviving after a couple was married? Would there be a 50% chance of the surname coming from the husband's side and a 50% chance of the surname coming from the wife's side?