I agree with Nancy that this doesn’t look like a prisoner’s dilemma.
You could think about this as a dynamic game, but it seems simplest to model it as a static game with two strategies: eat heavily and eat lightly. Both have to choose eat heavily to actually eat enough to gain weight, since it sounds like both have to agree every time they eat. The payoffs then look something like:
with the bodybuilder as the row player and the model as the column player. Then (Heavily, Lightly) and (Lightly, Lightly) are both Nash Equilibria. Do those payoffs seem to capture the situation you were thinking of?
I don’t know enough about Game Theory to expect what my Nash Equilibria would look like, but what I’m trying to find out in general is how to split resources when people have to use the same pot at the same time when they have competing or contradictory goals.
So for a real life example that I think captures what I’m trying to illustrate: My brother likes to turn the air conditioner as cold as possible during the summer so that he can bundle up in a lot of blankets when he goes to sleep. I on the other hand prefer to sleep with the a/c at room temperature so that I don’t have to bundle up with blankets. Sleeping without bundling up makes my brother uncomfortable, and having to sleep under a lot of blankets so I don’t freeze makes me uncomfortable. We both have to use the a/c, but we have contradictory goals even though we’re using the same resource at the same time. And the situation is repeated every night during the summer (thankfully I don’t live with my brother, but my current new roommate seems to have the same tendency with the a/c).
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:
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.
I agree with Nancy that this doesn’t look like a prisoner’s dilemma.
You could think about this as a dynamic game, but it seems simplest to model it as a static game with two strategies: eat heavily and eat lightly. Both have to choose eat heavily to actually eat enough to gain weight, since it sounds like both have to agree every time they eat. The payoffs then look something like:
with the bodybuilder as the row player and the model as the column player. Then (Heavily, Lightly) and (Lightly, Lightly) are both Nash Equilibria. Do those payoffs seem to capture the situation you were thinking of?
I don’t know enough about Game Theory to expect what my Nash Equilibria would look like, but what I’m trying to find out in general is how to split resources when people have to use the same pot at the same time when they have competing or contradictory goals.
So for a real life example that I think captures what I’m trying to illustrate: My brother likes to turn the air conditioner as cold as possible during the summer so that he can bundle up in a lot of blankets when he goes to sleep. I on the other hand prefer to sleep with the a/c at room temperature so that I don’t have to bundle up with blankets. Sleeping without bundling up makes my brother uncomfortable, and having to sleep under a lot of blankets so I don’t freeze makes me uncomfortable. We both have to use the a/c, but we have contradictory goals even though we’re using the same resource at the same time. And the situation is repeated every night during the summer (thankfully I don’t live with my brother, but my current new roommate seems to have the same tendency with the a/c).
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:
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.
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.
Thanks! This gives me more resources to study directly instead of hoping to land on what I was looking for randomly.