Two suggestions...
If you can do it for two, then start with AxB, BxC, and AxC. Then feed the results of that to another layer of the same logic. That should give you a simple result.
I'd do it with silicon rather than relays. I's use a PIC or AVR, and do it in software. First run each input signal through a pair of FFs to avoid metastability. (That's probably included in the PIC/AVR front end.) Then in software, spin until at least one bit comes on. If there is only one, your job is done. If two or three, then you have more work to do.
One approach is to change the rules of the game. Say everybody gets an extra $100 in the case of a tie. (Or only the teams that tied.)
Or you could run a counter in the background, and use the counter to decide ties. Say divide the counter by 2 or 3 and use the remainder to decide.