1.(a) Let G be the set of gambles over a set of basic rewards R, and ≥∗ a preference order on G. State the conditions that must be satisfified by a function U : G → R for it to be a utility function for G and ≥∗.
(b) Explain the relevance of these conditions for decision making under uncertainty.
(c) Describe (without proof) a procedure to construct a utility function for the set of gambles and a preference order over a set of rewards.
(d) Give one condition that the preference order must satisfy for this procedure to work.
(e) Explain to what extent a utility function representing a given set of preferences on a set of rewards is unique.
2.(a) Individuals A and B have utilities for non-negative amounts of money of the form UA(£x) = log(x + a) and UB(£x) = log(x + b), where a < b. Discuss and compare the attitudes to risk of A and B. (Any results about risk attitudes that you require should be stated clearly, but need not be proved.)
(b) Suppose that individuals A and B each have a utility for money of the form U(£x) = log(x + 1). Individual A currently has no money and B has £7. Individual A has a rafflfflffle ticket that, with probability 1/2 will pay £8 and with probability 1/2 will pay nothing. Show that there is an amount £t that B would be prepared to pay for the ticket and which A would be prepared to accept.
3.Defifine what it means for attributes X and Y to be mutually utility independent with respect to a set of preferences or a utility function.
Suppose that you may receive two amounts of money. You receive M1 immediately and you receive M2 in three years’ time. Suppose that you consider M1 and M2 to be mutually utility independent. Your marginal utilities for M1 and M2 are both of the form U(£m) = √m for non-negative m.
Suppose that you are indifffferent between the three choices:
(i) (m1, m2) = (16, 0);
(ii) (m1, m2) = (0, 64);
(iii) (m1, m2) = (9, 36).
With origin (m1, m2) = (0, 0), evaluate your utility as a function of m1 and m2.Comment on the interpretation of the constant that is specifified in your utility function.
4.In a particular game, R chooses strategy R1, R2 or R3, C chooses strategy C1, C2,
C3, C4 or C5. The payoffffs to R are as follows
C1 C2 C3 C4 C5
R1 2 0 3 -1 -2
R2 -2 3 0 0 4
R3 1 -2 4 -1 -2
The payoffff to C is minus the payoffff to R.
(a) Reduce this game to a game where R has only two possible strategies. Explain carefully why this can be done.
(b) Use a graphical method to identify the minimax strategies for R and for C, and the value of the game.
5.(a) Consider a bargaining problem with 4 options: A, B, C, D. The utilities for these options to John and David are given in the table below, together with their utilities for the status quo (SQ).
A B C D SQ
John -2 2 4 8 0
David 9 5 3 -1 1
i.Without optimising the function in the defifinition of the Nash point, show that at the unique solution to the Nash axioms John has utility 3 and David has utility 4. Explain brieflfly how this solution is derived and which Nash axioms are used. (Hint: Use a transformation to get a symmetric bargaining problem.)
ii.Specify all bargains over the options that correspond to the solution.
iii. Discuss whether or not a player may end up, at the end of the whole process of solving this decision problem, with an option for which he has lower utility than for the status quo; include the Individual Rationality axiom in your discussion.
(b) Consider a group decision problem with fifive voters and three alternatives, A,B and C. Each voter’s preference ordering is transitive. Three voters prefer A over B and B over C. The other two voters prefer B over C and C over A.Combine these preferences into a group preference order using the Borda count procedure, assigning scores 2, 1 and 0, to a person’s fifirst, second and third preferred alternative, respectively. Discuss the resulting group preference order by considering all pairwise preferences based on the use of the simple majority rule.
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