Website security breaches seem to be a common occurrence, giving the attacker password hashes that he can conduct a brute force attack against, often given him a list of passwords that will work on other sites since users commonly reuse passwords.
Would it improve overall security by making the hashing algorithm slightly weaker and more prone to collisions, so even if the attacker does brute force a password that works on the hacked site, it's not likely to work at any other site?
So if, for example, some hash algorithm were 1000 times more likely to have a collision than some other hash algorithm, it makes the attacker's work 1000 times easier on that site, but any password he brute forces has only a 1 in 1000 chance of working on another site. He could continue to brute force and come up with all 1000 passwords that collide with the hash, but if he tried to cycle through all 1000 passwords on other sites, hopefully that site would shut him down before he got through all passwords.
Would this lead to better overall security by reducing the ability of a hacker to exploit passwords shared with other sites, or would it just weaken security by deliberately making it much easier to brute force a single site's password?