It seems like this is a moot point (since it's apparently accepted now), but it's worth pointing out that your translation has a totally different meaning than the offered sentence. Saying, "Even I didn't win," seems to place an emphasis on it being strange that you didn't win, whereas, "I didn't win either," is simply commiseration with another contestant.
In il passato prossimo, it can be translated as EITHER:
"I have not won"
"I did not win"
It is an idiocyncrasy of English that we put the work "did" in the sentence when there is a negative. It's not actually an issue with the Italian.
If it were not negative, (ho vinto), it could be translated as either:
"I have won"
It can mean both. In Italian, the present perfect tense is also used to indicate the past tense. You'll learn the passato remoto later, but that's only used in literature (and, I believe, in some southern regions). In this case, "did not" probably makes more sense, given the content of the sentence, but it can go either way without a surrounding conversation/paragraph to put it all in context.
When I questioned which of the two translations was correct(I didn't win either of the two prizes or Too bad you lost, but I didn't win either), what I was really asking was, can that same construction apply for both meanings, since the context would usually be provided in a previous sentence. After all, the word "neither" would always require an antecedent context.
Thank you, Jordy for your patience, but I still didn't get the answer to my basic question, which is: how would we say in Italian, after establishing that we're talking about either of two prizes, rather than either of two winners? Would the Italian sentence be written in the same way as with the two winners?