### Thinking About Coase

Isaac made a comment last Fall in Micro regarding the Coase Theorem. Sans transactions costs, the Coase Theorem basically states that if you have two agents, A and B, and B is taking an action that has a positive or negative effect on A (an externality), A and B have the incentive to privately come to an agreement about the externality as long as property rights are well-defined. For example, if A are a group of fisherfolk (thanks, Larry) and B is a creek polluter the efficient outcome will result if you give A the right to have a non-polluted creek or if you give B the right to pollute the creek. This comes about through bargaining. Isaac's point was that the situation essentially gives A and B a pie of surplus to split up and that the bargaining process will eat up the whole pie. The logic behind this is that the loser in the bargain will always want to hold out for a little bit more. (If I'm not explaining this well please clarify!)

But if we put some structure behind this, the argument doesn't hold. I modeled this as a chain of Proposer-Decider games. In one iteration of the game, one player proposes a division of the pie and the other player can either accept or reject the division. If rejected, there is another iteration of the game, but with a smaller pie. The game continues until a division has been accepted or the pie reaches 0. We can determine the unique subgame perfect equilibrium by unraveling from the back. The result depends on who you give the power to. If you let one player be the Proposer in every iteration of the game then in the subgame perfect equilibrium you have that player initially propose to take the whole pie himself and the other player accepts that plan. If you let the players alternate in the roles of proposer and decider then in the subgame perfect equilibrium the pie is divided nearly equally by the first player to propose and the other player accepts. But in both situations none of the pie is wasted on costly extra rounds of bargaining.

I suppose the intuition here is that when the bargaining is set up in this way, each player wants to prevent the game from going on to the next stage in which they do not get to propose. In order to do this, A has to give B as much as B would get if the game did move to the next stage and then A gets the remainder of the pie. So the first player to propose divides the whole pie up in such a way that this occurs in the first round. The result where one player proposes the whole time is simple, they don't have to give the other player anything so they just take it all in the first round (like the one-stage Proposer-Decider game.) Of course there are other ways you could set up the bargaining...

But if we put some structure behind this, the argument doesn't hold. I modeled this as a chain of Proposer-Decider games. In one iteration of the game, one player proposes a division of the pie and the other player can either accept or reject the division. If rejected, there is another iteration of the game, but with a smaller pie. The game continues until a division has been accepted or the pie reaches 0. We can determine the unique subgame perfect equilibrium by unraveling from the back. The result depends on who you give the power to. If you let one player be the Proposer in every iteration of the game then in the subgame perfect equilibrium you have that player initially propose to take the whole pie himself and the other player accepts that plan. If you let the players alternate in the roles of proposer and decider then in the subgame perfect equilibrium the pie is divided nearly equally by the first player to propose and the other player accepts. But in both situations none of the pie is wasted on costly extra rounds of bargaining.

I suppose the intuition here is that when the bargaining is set up in this way, each player wants to prevent the game from going on to the next stage in which they do not get to propose. In order to do this, A has to give B as much as B would get if the game did move to the next stage and then A gets the remainder of the pie. So the first player to propose divides the whole pie up in such a way that this occurs in the first round. The result where one player proposes the whole time is simple, they don't have to give the other player anything so they just take it all in the first round (like the one-stage Proposer-Decider game.) Of course there are other ways you could set up the bargaining...

## 2 Comments:

Ah yes, that whole proposer-decider thing. I've never quite gotten used to the idea that rationally you ought to unravel games from the back. It's elegant and correct (insofar as that is the game you are playing), but I haven't yet retrained my instinct to try to solve the game directly.

But isn't bargaining in Coase more equivalent to simultaneous proposals? To assume that the structure is proposer-decider assumes that one party somehow has more claim to the surplus than the other party (because that party will obviously come out better in the proposer-decider contest). If you implicitely assign the surplus to one party, the proposer, you'll get a solution without eating up the surplus, true. Yet this glosses over the big idea in the Coase Theorem: the surplus isn't properly "held" by anyone, it is interdependent. This interdependence means that there is no natural way to decide who should be the proposer (that presupposes a solution): maybe you flip a coin, but that coin flip is a rather expensive bet. And so the classic critique of game theory that "how do we know, or decide, what game we're playing" hits again.

If both people have the power to to propose, then you get that expensive war of wills, which has to be the case in the Coase Theorem.

Backwards induction is elegant and correct but it may also have no relation to how people actually play, as in the centipede game. Most solution concepts in game theory don't correspond well to real life, which I think is a more important criticism in general than "which game are we playing".

But in this case, "which game are we playing" is crucial because it is totally unclear what game might actually represent real-world bargaining. In the real world, time is continuous, people make offers whenever they want, there is cheap talk, etc. In the Coase situation, though, in order for bargaining to be possible, someone is given the right to whatever externality is in question. The fisherfolk are given the right to have a pollution-free stream. Then the coal company can make an offer to pollute a certain amount, or the fisherfolk can go to the coal company and make an offer themselves. But the fisherfolk are the ones in control of the surplus.

I don't think its clear that there are simultaneous proposals any more than it is clear that there are alternating proposals. If anything I would bet against simultaneous proposals because very rarely do bargainers do something like write down offers separately and then compare notes. Or yell out their offers at the same time. It seems to me that it is usually in an offer/counter-offer framework, at least loosely, which is the model I used. Although someone is always the proposer in each round, the other party gets to propose in the next round. Who starts? It doesn't really matter since the result is that they split it evenly.

Even if you modeled it with simultaneous offers I think you'd expect that they'd reach a deal in the first round or no deal at all. But I'm starting to think that dividing a pie is not the way to look at it. At least, dividing a pie with constant returns to the size of your slice is not the way to look at it. There should be diminishing returns, otherwise there's no unique efficient allocation in the first place.

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