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Reverse Game Theory

30 Jan

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So on Less Wrong I proposed what I thought was an example of the Prisoner’s Dilemma:

I am trying to formalize what I think should be solvable by some game theory, but I don’t know enough about decision theory to come up with a solution.

Let’s say there are twins who live together. For some reason they can only eat when they both are hungry. This would work as long as they are both actually hungry at the same time, but let’s say that one twin wants to gain weight since that twin wants to be a body builder, or one twin wants to lose weight since that twin wants to look better in a tuxedo.

At this point it seems like they have conflicting goals, so this seems like an iterated prisoner’s dilemma. And it seems like if this is an iterated prisoner’s dilemma, then the best strategy over the long run would be to cooperate. Is that correct, or am I wrong about something in this hypothetical?

It was pointed out that this isn’t a PD since the twins have competing goals. So instead of going with a confusing hypothetical, I decided to use an example from my own life:

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).

User badger was able to use this real life example to explain that this indeed isn’t a PD since this isn’t a pre-made game, so I would have to design a game that could be solved. This is mechanism design, or reverse game theory:

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.

According to my own moral intuitions (which admittedly aren’t universal moral intuitions) alternating weeks over who has control over the A/C seems the most fair. But someone else might prefer to place bids. However, introducing money also introduces a power system where whoever has or earns more money gets their wants achieved asymmetrically.

Of course there could also be a power differential due to knowledge, such as a game that is designed to violate the laws of probability to favor the game designer. But that’s why it’s always in your favor to learn rationality!

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