Ok, so sometimes "nem van" is "nincs" and sometimes it's just "nem" omitting "van"...anyone who could explain, please ? :)
Nincs is "there is no.." or "doesn't exist". Nem is just simply no. Nincs játék = there is no game (or toy).
As far as I know, játék can mean "toy" as well. I would certainly use it instead of játékszer in that meaning too. (Though in that case the sentence would be more natural with the indefinite article included).