That example helps clarify. In the A/C situation, you and your brother aren't really starting with a game. There isn't a natural set of strategies you are each independently choosing from; instead you are selecting one temperature together. You could construct a game to help you two along in that joint decision, though. To solve the overall problem, there are two questions to be answered:

1. Given a set of outcomes and everyone's preferences over the outcomes, which outcome should be chosen? This is studied in social choice theory, cake-cutting/fair division, and bargaining solutions.
2. Given an answer to the first question, how do you construct a game that implements the outcome that should be chosen? This is studied in mechanism design.

One possible solution: If everything is symmetric, the result should split the resource equally, either by setting the temperature halfway between your ideal and his ideal or alternating nights where you choose your ideals. With this as a starting point, flip a coin. The winner can either accept the equal split or make a new proposal of a temperature and a payment to the other person. The second person can accept the new proposal or make a new one. Alternate proposals until one is accepted. This is roughly the Rubinstein bargaining game implementing the Nash bargaining solution with transfers.

Another possible solution: Both submit bids between 0 and 1. Suppose the high bid is p. The person with the high bid proposes a temperature. The second person can either accept that outcome or make a new proposal. If the first player doesn't accept the new proposal, the final outcome is the second player's proposal with probability p and the status quo (say alternating nights) with probability 1-p. This is Moulin's implementation of the Kalai-Smorodinsky bargaining solution.