这是一篇英国math决策论限时测试**数学代写**

**SECTION A**

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 *U*A(*£**x*) = log(*x *+ *a*) and *U*B(*£**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 *M*1 immediately and you receive *M*2 in three years’ time. Suppose that you consider *M*1 and *M*2 to be mutually utility independent. Your marginal utilities for *M*1 and *M*2 are both of the form *U*(*£**m*) = *√**m *for non-negative *m*.

Suppose that you are indifffferent between the three choices:

(i) (*m*1*, m*2) = (16*, *0);

(ii) (*m*1*, m*2) = (0*, *64);

(iii) (*m*1*, m*2) = (9*, *36).

With origin (*m*1*, m*2) = (0*, *0), evaluate your utility as a function of *m*1 and *m*2.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.