Each of two roommates simultaneously decides whether to contribute $100 to buying a common stereo. (Roommates
contribute either the full $100 or nothing; they cannot contribute an intermediate amount. Also, they cannot change their minds after making the decision.) If both roommates contribute, so that they buy a $200 stereo, they both get an enjoyment benefit out of it that is worth $150. If one of them contributes, so that they buy a $100 stereo, they both get a benefit worth $75. If neither contributes, they do not buy a stereo and do not get the benefits.
(a) Write out the payoff matrix for this simultaneous game. (Note: in the set up above are both benefits and costs listed. To determine the payoffs (in dollar terms) for each possible outcome you will need to calculate the net benefit for each player given their strategy of contributing or not in each case.) (6 points)
(b) Argue carefully that with the distributional models we have considered in class, there will often be at least two equilibrium: one in which both roommates contribute and one in which neither does. Take each equilibrium one at the time and state the necessary assumptions, if any, you need to make for each equilibrium to hold. (these will be assumptions with respect to the roommates σ or ρ) (14 points)
(c) Argue informally but carefully that the same is true in Rabin's intentions-based model of fairness, that there are two equilibrium: one in which both roommates contribute and one in which neither does. State the necessary assumptions, if any, you need to make for each equilibrium to hold. (14 points)
(d) Suppose roommates are playing an equilibrium in which they are both contributing to the public good. As you have argued in the previous two parts, this is consistent with distributional models as well as intentions-based models. But suppose you want to know which one is motivating the roommates. What is a simple modification of the game such that by comparing play in the original game to that in the modification, we can tell whether intentions-based motivations are contributing to the roommates' behavior? (7 Points)
(e) Could face-safing concerns play a role in this simultaneous decision by the two roommates? Argue informally and if so, give a brief hypothetical example of how this could influence the decision reasoning of a roommate. (6 points)
(f) Referring back to the setup in part (b) (so no intentions-based preference). Let's assume the distributional preference of both players are such that σ = ρ = 0.1 . Use the idea of a utility function for face saving concerns (in addition to distributional preferences) as seen in the lecture. For the case in which both roommates contribute how big would the face saving concern need to be to sustain this outcome as an equilibrium.
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