12. Preferences and Condorcet winners

Gilberto, Lorenzo, and Sam are roommates who are trying to decide how much money to spend on heat in their apartment each month.

Sam does not care about heat and is on a tight budget; therefore, he prefers to contribute no money to heat, and his utility is constantly declining as they increase the amount of money they spend. Sam's preferences for spending on heat_____ single peaked.

Gilberto hates being cold; therefore, he would ideally like each to contribute $50 to heat. If they decide to contribute less than $30 each, he will end up spending most nights sleeping at his parents' house. Thus, he prefers contributing no money to heat to contributing any amount between $0 and $30 (because in going from $0 to $30, the amount he pays increases, but he doesn't spend time at the apartment and thus gets no benefit from the heat). Gilberto's preferences_____ single peaked.

Lorenzo would ideally contribute $20, but he would prefer contributing $50 to $0 because he does not want to be freezing. His utility, therefore, increases from $0 up until they are each contributing $20, at which point his utility from additional spending on heat begins to decline. Lorenzo's preferences____ single peaked.

Suppose the three roommates vote on purchasing either $0, $20, or $50 on heat. Among these proposals there____ a Condorcet winner. Answer choices -Are and are not