Is expiation obligatory for an oath I took concerning a forbidden electronic game, given that I did not have it on my device but only paid for it, and am I sinful if my brother plays it after taking it from me by force?

Your oath to your brother not to take the toy is considered a binding oath if it was sworn by one of the names of Allah Almighty or one of His attributes. If your brother took it, you have broken your oath, and the expiation for an oath becomes binding upon you.

The obligation of expiation does not fall away from you unless you were compelled under duress that is considered valid by Sharia (definition). This has conditions, among them: that the one compelling you must have authority or dominance, that you strongly believe the threatened harm will occur if you do not comply, and that the harm you would suffer is significant.

The fact that you swore the oath based on a mistaken assumption does not change the ruling; expiation is still binding upon you.

As for the sin of your brother playing with a forbidden toy, there is no sin upon you if you did not permit him or consent to it. However, you must prevent him if you are able. Otherwise, Allah does not burden a soul beyond its capacity.