Potentially ambiguous sentence/understanding

It is ambiguous.

The offer is of the following form:

• You pick one, and I’ll do something to another.

What happens to the one you picked is unspecified. It is left to the context to supply the missing information.

Consider a host saying to his guest:

• You stay seated. Pick a drink and I’ll get another.

There is a strong expectation that the host will get the drink the guest picked (as well as another drink for himself). It would be odd for the host to return with a single serve of a drink the guest didn’t pick. Another situation with similar semantics is at sales: “You choose any product and I’ll give you another.”

Now consider two friends choosing costumes for a ball.

• You pick one and I’ll wear another.

Now there is a strong expectation that the speaker is not going to wear the friend’s costume as well as her own.

Why does the speaker ‘do’ the verb with the picker’s choice in one case and not in the other? It’s because the context dictates different expectations in each case.

For games such as the one you describe, the context is not defined by broader social customs. If the organisers wanted to, they could change the rules so that they open both boxes - the one the contestant picked, plus another one. That would change the game, but they could still legitimately say, “You pick one and I’ll open another.”

This reliance on unspoken context makes the offer ambiguous when one is not familiar with the context.