No, there is an unexpected twist. Any block of height t containing N txns maps to exactly the same satoshis. The lottery is deterministic rather than random. The only random element is who happens to have their computer online. We could predict who is going to mine in advance. This isn't necessary to think about, but in theory the whole mining process could be scheduled because of this.

However, once the block hash is signed by the first winner, then the block contents cannot be modified by the other winners. By default, the group is not supposed to sign more than one version of block t. They could cheat and sign a very large number of versions of block t with N txns.

Three cases if they cheat:
1) If one version has already been built upon and the other blocks are orphans, then keep building on the longest version.
2) You are aware of more than one version, but haven't seen any that have been built on. In this case, combine any two duplicate blocks t and mine a 'null block'. The record of the cheating is put in the blockchain. All txn for time t are ignored and you move on to block t+1.
3) You are only aware of one version. In this case try to build on it until you hear about another. (You don't know that cheating has occurred yet.)