Thanks for engaging on this. I hope it is helpful and we each further our own understanding of these concepts.  I see what you are saying, and you very well might be correct. 

Still, it feels like there is probably some way to do it with oblivious signing that would satisfy your objection. 

VzxPLnHqr

I added a test [demonstrating 1 of n oblivious signing](https://github.com/VzxPLnHqr/secp256k1-scala/blob/main/Oblivious.test.scala#L77). It does not quite solve the problem you point out, but it gets closer (notice how Alice can privately shuffle the messages and yet Bob still receives one and only one valid signature, while Alice still remains oblivious as to which one he receives).

bitcoin

A Provably Fair Off-Chain Lottery (and DLC) Protocol

conduition

The issue is that none of your Bobs can trust each other not to collude with Alice. If even one player colludes with Alice, they can steal everyone else's money by simply telling Alice which `Pick` transaction to publish. 