I had to google for the nash bargaining game and I still don’t entirely understand your analogy there. If you could expand on that bit, I’d be interested.
There have been two recentpostings on bargaining that are worth looking at. The important thing from my point of view is this:
In any situation in which two individuals are playing an iterated game with common knowledge of each other’s payoffs, rational cooperative play calls for both players to pretend that they are playing a modified game with the same choices, but different payoffs. The payoff matrix they both should use will be some linear combination of the payoff matrices of the two players. The second linked posting above expresses this combination as “U1+µU2”. The factor µ acts as a kind of exchange rate, converting one players utility units into the “equivalent amount” of the other player’s utility units. Thus, the Nash bargaining solution is in some sense a utilitarian solution. Both players agree to try for the same objective, which is to maximize total utility.
But wait! Those were scare quotes around “equivalent amount” in the previous paragraph. But I have not yet justified my claim that µ is in any sense a ‘fair’ (more scare quotes) exchange rate. Or, to make this point in a different way, I have not yet said what is ‘fair’ about the particular µ that arises in Nash bargaining. So here, without proof, is why this particular µ is fair. It is fair because it is a “market exchange rate”. It is the rate at which player1 utility gets changed into player2 utility if the bargained solution is shifted a little bit along the Pareto frontier.
Hmmm. That didn’t come out as clear as I had hoped it would. Sorry about that. But my main point in the grandparent—the thing I thought was ‘cool’ - is that Nash bargaining, discounting of future utility, and discounting of simulations relative to reality are all handled by doing a linear combination of utilities which initially have different units (are not comparable without conversion).
There have been two recent postings on bargaining that are worth looking at. The important thing from my point of view is this:
In any situation in which two individuals are playing an iterated game with common knowledge of each other’s payoffs, rational cooperative play calls for both players to pretend that they are playing a modified game with the same choices, but different payoffs. The payoff matrix they both should use will be some linear combination of the payoff matrices of the two players. The second linked posting above expresses this combination as “U1+µU2”. The factor µ acts as a kind of exchange rate, converting one players utility units into the “equivalent amount” of the other player’s utility units. Thus, the Nash bargaining solution is in some sense a utilitarian solution. Both players agree to try for the same objective, which is to maximize total utility.
But wait! Those were scare quotes around “equivalent amount” in the previous paragraph. But I have not yet justified my claim that µ is in any sense a ‘fair’ (more scare quotes) exchange rate. Or, to make this point in a different way, I have not yet said what is ‘fair’ about the particular µ that arises in Nash bargaining. So here, without proof, is why this particular µ is fair. It is fair because it is a “market exchange rate”. It is the rate at which player1 utility gets changed into player2 utility if the bargained solution is shifted a little bit along the Pareto frontier.
Hmmm. That didn’t come out as clear as I had hoped it would. Sorry about that. But my main point in the grandparent—the thing I thought was ‘cool’ - is that Nash bargaining, discounting of future utility, and discounting of simulations relative to reality are all handled by doing a linear combination of utilities which initially have different units (are not comparable without conversion).