I'm almost convinced my definition is faulty, but not completely yet. In this case, if the offer was sure to be rejected, Alice (A) may move out, or evict Bob (B), or react in a different way that minimizes Bob's utility, or Alice may just decide to stop chores anyway because she just prefers a messy and just household than a clean and injust one.

So precommitment to reject the offer doesn't necessarily help Bob. But I need to think about this. Upvoting both examples.