It's my thing

Yup, they can do that unless you can show that the approach is detrimental in some fashion to a 'protected group'. Management prerogative. Seems pretty bizarre though. Got a Union?

Let me take a punt.

Example: Employer says some people need to work night shift others day shift, no one wants to work the night shift. A ballot is abitrary but fair the alternative is something like length of service or buggins turn which is grossly unfair. The post office operated this which meant the longer serving staff had all the cushy jobs and didn't put in a shift whilst the new staff took the brunt.

I agree it's not nice if you are stuck on the bad shift but I think you would need to show the employer a valid reason why you need to work that new shift? Family commitments or religious reasons something like that. I'm afraid "I just fancy the new shift" probably won't get it done.

Anyway I hope you resolve it satisfactorily, maybe you could trade with one of the winners?

One more point

In this day and age it Should not be down to a sweepstake where future work schedules are involved .

Is it the type of company that does an employee engagement survey and has a strong view of its culture and behaviours? If so maybe you could use that as a lever.

Then that suggests that the ballot was rigged in which case you have a case.

One more point

Did you get an interview? Were these better positions? If you got an interview, how was it scored? Were positions advertised correctly? Who oversaw such a sweepstake? Was it an open process?
Have you written asking for reasons on paper and would the company be able to offer a reasonable response?
I'm thinking in any legal case a company is not going to look good just randomly selecting people. However, from my questions, you can probably tell that this could be quite a complicated case.
My advice is this: write to the company formally requesting how you have been overlooked. This may give them a push in the right direction.
Simultaneously, write a much stronger letter complaining to the union. At the least, you will see if they are in cahoots. Be very suspicious if neither reply.
Particularly, if you are unhappy with responses, find out which solicitors have had recent dealings with employment hearings against the company. You may know someone who has recently had a settlement. Often, they are not allowed to say anything but you can guarantee they will tell you in private - as they will still actively dislike the organisation.
There are certain solicitors that companies are afraid of and will generally cave into. Sometimes just a consultation and a letter from them can have the desired effect.
These are just some things from the top of my head but I believe you have nothing to lose by writing a couple of well-worded letters. Could put the cat amongst the pigeons, so to speak.

Red and Blue Army!