Each of two roommates simultaneously decides whether to contribute $100 to buying a common dining table. (Roommates contribute ei- ther the full $100 or nothing; they cannot contribute an intermediate amount. Also, they cannot change their minds after making the deci- sion.) If both roommates contribute, so that they buy a $200 table, they both get an enjoyment benefit out of it that is worth $150. If one of them contributes, so that they buy a $100 table, they both get a benefit worth $75. If neither contributes, they do not buy a table and do not get the benefits.

(a) (8pts) Draw the payoffs of the game in a 2 by 2 matrix and solve for the Nash equilibirum.

(b) (8pts) Argue informally but carefully that with the distributional preferences models we have considered in class, there can be at least two equilibria: one in which both roommates contribute and one in which neither does.

(c) (8pts) Argue informally but carefully that the same is true in Rabin's intentions-based model of fairness.

(d) (8pts) Suppose roommates are playing an equilibrium in which they are both contributing to the public good. As you have ar- gued in the previous two parts, this is consistent with distribu- tional preferences models as well as intentions-based models. But suppose you want to know which model is motivating the room- mates' behaviour. What is a simple modification of the game in part (a) such that by comparing play in the original game to that in the modification, we can tell whether intentions-based motiva- tions are contributing to the roommates' behaviour?

1. Each of two roommates simultaneously decides whether to contribute $100 to buying a common dining table. (Roommates contribute ei- ther the full $100 or nothing; they cannot contribute an intermediate amount. Also, they cannot change their minds after making the deci- sion.) If both roommates contribute, so that they buy a $200 table, they both get an enjoyment benefit out of it that is worth $150. If one of them contributes, so that they buy a $100 table, they both get a benefit worth $75. If neither contributes, they do not buy a table and do not get the benefits. (a) (b) (8pts) Draw the payoffs of the game in a 2 by 2 matrix and solve for the Nash equilibirum. (8pts) Argue informally but carefully that with the distributional preferences models we have considered in class, there can be at least two equilibria: one in which both roommates contribute and one in which neither does. (8pts) Argue informally but carefully that the same is true in Rabin's intentions-based model of fairness. (8pts) Suppose roommates are playing an equilibrium in which they are both contributing to the public good. As you have ar- gued in the previous two parts, this is consistent with distribu- tional preferences models as well as intentions-based models. But suppose you want to know which model is motivating the room- mates' behaviour. What is a simple modification of the game in part (a) such that by comparing play in the original game to that 1