Which of the following properties of vNM utilities most particularly prevent us from using them in a utilitarian Bergson-Samuelson social welfare function?

Q 1 Which of the following properties of vNM utilities most particularly prevent us from using them in a utilitarian Bergson-Samuelson social welfare function?

1. We can re-scale vNM utilities by adding or subtracting any constant.

2. A vNM utility of zero has no particular significance.

3. The distinction between “positive” and “negative” vNM utilities has no particular significance

4. We can re-scale vNM utilities by multiplying by any positive constant.

Q2 On the sixtieth anniversary of the formation of The Beatles, a newspaper has decided to find out “the nation’s favourite Beatle”. Having surveyed its readers, asking them for a preference order over John, Paul, George and Ringo, it combines the preferences in the following way. Each Beatle is awarded 3 points for every reader that has ranked him first, 2 points for every reader that has ranked him second, and 1 point for every reader that has ranked him third. The newspaper will then suppose that Beatle x is weakly socially preferred to Beatle y if and only if Beatle x was awarded at least as many points as Beatle y.

Exactly which of the axioms in Arrow’s “Impossibility Theorem” is fulfilled by the proposed social preference function?

1. Universal Domain 2. Pareto Efficiency 3. Non-Dictatorship 4. Independence of Irrelevant Alternatives

Q3 A rival newspaper conducts a similar survey, but combines the preferences in a different way. In Round 1 (using pairwise, simple majority rule) they decide on a winner between Paul and Ringo. In Round 2 they decide similarly on a winner between George and John. Then, they use pairwise simple majority rule again, to determine first-place and second-place (between the winners of Rounds 1 and 2), and to determine third-place and fourth-place (between the losers of Rounds 1 and 2).

Exactly which of the axioms in Arrow’s “Impossibility Theorem” is fulfilled by the second newspaper’s social preference function?

1. Universal Domain 2. Pareto Efficiency 3. Non-Dictatorship 4. Independence of Irrelevant Alternatives

Q4 Three individuals (dining together in a restaurant) are trying to choose a bottle of wine to share. There are four types of wine on the wine-list, and each individual’s weak preference relation over the four alternatives is an “ordering”. The individuals’ preference orderings must be somehow combined to make a group decision.

Suppose the group uses the following social preference function to determine their joint (“social” or “group”) preferences over the four alternatives. In the social preference relation, any one alternative (x) will be weakly preferred to any other (y) if and only if at least two individuals weakly prefer x to y.

Exactly which of the axioms in Arrow’s “Impossibility Theorem” is fulfilled by the social preference function?

1. Universal Domain 2. Pareto Efficiency 3. Non-Dictatorship 4. Independence of Irrelevant Alternatives

Q5 In the situation above, individuals 2 and 3 then discover it is individual 1’s birthday. So the group adopts a new social preference function. They now use the following rule. In the social preference relation, any one alternative (x) will be weakly preferred to any other (y) if and only if individual 1 weakly prefers x to y. Exactly which of the axioms in Arrow’s “Impossibility Theorem” is fulfilled by the new social preference function?

1. Universal Domain 2. Pareto Efficiency 3. Non-Dictatorship 4. Independence of Irrelevant Alternatives

Q6 Later on, individual 1 leaves, but individuals 2 and 3 decide to stay on and order another bottle. They now decide to use the following rule. (In this case we shall assume that the wines are all different prices.) If x is cheaper than y then x will be weakly socially preferred to y if and only if: either at least one individual strictly prefers x to y, or both individuals weakly prefer x to y. If x is more expensive than y then x will be weakly socially preferred to y if and only if y is not weakly socially preferred to x.

Which one of the axioms in Arrow’s “Impossibility Theorem” is not fulfilled by this social preference function?

1. Universal Domain 2. Pareto Efficiency 3. Non-Dictatorship 4. Independence of Irrelevant Alternatives

Q7 In which of the following situations could we most plausibly suppose that preferences are “single peaked”. (Suppose a group of individuals is choosing between four different bottles of white wine.)

1. THE MOST EXPENSIVE WINE IS WIDELY KNOWN TO TASTE TERRIBLE, SO IT IS LIKELY TO BE EVERYBODY’S LEAST-PREFERRED.

2. The cheapest wine is widely known to be also the tastiest, so it is likely to be everybody’s most-preferred.

3. The wines are all the same price, and the only information on the menu is that one is sweet, one is medium, one is dry and the fourth is very dry.

4. All individuals have very clear preferences. No individual is indifferent between any two different wines.

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