程序代写案例-MATH 331
时间:2022-05-02
PAPER CODE NO.
MATH 331



MAY 2020 EXAMINATIONS




NON-PHYSICAL APPLICATIONS I (MATHEMATICAL ECONOMICS)

TIME ALLOWED : Two Hours and a Half

INSTRUCTIONS TO CANDIDATES

This paper contains SEVEN questions, all carrying equal weight. Full marks will be
awarded for complete solutions to FIVE questions. Candidates may attempt all
questions, but only the best FIVE solutions will be taken into account.

Candidates are only permitted to use calculators deemed acceptable, and affixed with
an official holographic sticker, by the Department of Mathematical Sciences.




















PAPER CODE …Math331…… PAGE 1 OF 6 CONTINUED
Examiner: Dr D M Lewis
Tel: ext 4014 (Maths Sci.)





1. A rustic poker game for two players, and , begins with both players
staking £1 into the kitty. In this game there are two hats, each containing two
cards. Hat No. 1, used in the first round of bidding, contains a card marked 1
and a card marked 5. Hat No. 2 used in the second round of bidding, contains
a card marked 3 and another card marked 5. When selecting a card from a hat
both players are careful not to show it to their opponent.
Player goes first and has the choice of either calling “Raise £3”, putting £3
into the kitty and selecting a card from hat No. 1, or calling “Stick”, in which
case he does nothing. Now it is Player ’s turn. He too can choose to call
“Raise £3” pay and select a card from Hat No. 1, or just call “Stick”. That
concludes the first round of bidding.
If (and only if) both players raised in round 1, then a second round of bidding
commences, with Hat No.1 replaced by Hat No. 2. The procedure is the same.
Player goes first, then Player . Each player can “Raise £3” or “Stick” and
any selections are made from Hat No. 2.
Once all the bidding has concluded the game ends. The players turn over
their cards and compute their scores. A player’s score is given by the total on
their cards (if any) minus the amount of money they contributed to the kitty
(including their initial stake) to get them. The player with the highest score
wins the kitty, if the scores are level the kitty is shared.

a) Draw a game tree for this game including the information sets, payoffs, and
the relevant probabilities of the possible wining/losing scenarios.
[10 marks]

b) Write out all of Player ’s playing strategies in full. Explain carefully why
Player has a total 34 possible playing strategies and write out any four of
these in full.
[6 marks]

c) Suppose Player decides to adopt his most aggressive strategy of raising in
all circumstances. Compute the expected payoff if:

i) Player also raises in all circumstances, [2 marks]
ii) Player raises in round 1, but only raises in round 2 if he possesses
the 5 card. [2 marks]







PAPER CODE …Math331…… PAGE 2 OF 6 CONTINUED





2. a) In a bi-matrix game, the payoff matrices of two players A (row) and B
(column) are

(
(3, 1) (−2, −1)
(1, −2) (2, 0)
).

Using the swastika method, or otherwise, find the Nash equilibria of the game.
[8 marks]

b) A generalised version of the above bi-matrix game is summarised by the
following payoff matrices

(
(3, 1) (, −1)
(, ) (2, 0)
),

with constants ∈ ℝ and ∈ ℝ. Establish the necessary conditions on and
for both players to possess mixed Nash equilibrium strategies for this
generalised version of the game.
[8 marks]

c) Show that if both players play their respective mixed Nash equilibrium
strategies 〈〉 = ( , 1 − ) and 〈〉 = ( , 1 − )
established in part
b), then player can expect to receive a payoff of the form

2 +
2( − 2)
[2 + ( − 1)( − 5)]
,

for ∈ (0, 1) and < 3.
[4 marks]


3. The following are the payoffs for the row player in two zero-sum games:

a) (
−1
3
6
4

5
−2
−4
−1
) b) (
7 3 12
4 5 2
8 6 1
)

Find the maximin solutions and the values of both the games. In game b) for
full marks you must compute the optimum strategies of both players.
[9 and 11 marks]


PAPER CODE …Math331…… PAGE 3 OF 6 CONTINUED





4. a) State the Expected Utility Proposition (EUP).
[2 marks]
[Assume EUP holds in both parts b) and c) of this question.]

b) Let () denote Bob’s utility regarding the sure prospect of winning £x.
Suppose Bob’s utility function satisfies (−10) = −10, (100) = 100 and
(1000) = 1000.
Bob is considering paying £10 to enter an online draw (denoted by ) in
which the first prize is a new car worth £20,000. If the chance of winning the
car were 0.01%, then Bob’s utility value for the draw would be () = −8.
In fact his chance of winning the car is 1 ⁄ , where is the number of
entrants (assuming he competes). Bob is actually indifferent to entering this
draw and playing an online raffle (denoted by ), which also costs £10 to
enter. This pays out much lower prizes of £100 and £1000 and is defined by

~ [
1
10
[
1
30
1000, 100] , [
3
100
100,
139
150
−10]].

Compute the actual values of and (). Hence establish how many rivals
Bob would be up against (to the nearest integer) if he entered the online draw
to win the car.
[8 marks]

c) Bob and his friend Dave are very much into music and are planning to
establish their own band. Bob has £300 to invest for new musical equipment
for the band. He is wondering what to buy. His has three musical prospects to
consider: = a new electric guitar for himself: = a new stereo turntable
system: = a new drum kit for Dave. Consider the following three musical
lotteries 1, 2 and 3 of music prospects

1 = [
1
7
,
6
7
] , 2 = [
4
11
,
7
11
] and 3 = [
1
5
,
4
5
] .
In the past Bob has expressed a 10% preference for lottery 3 over 2 and he
prefers lottery 1 over 3 by a factor of 15 14⁄ . His utility values for the
prospects also satisfy 24() = () + 8(), where > 0 is an
integer value. If Bob’s utility values are all positive and the price of each item
£ = () for = , , & , then find the price of a guitar if Bob spends all
his money on the item he values most.
[10 marks]


PAPER CODE …Math331…… PAGE 4 OF 6 CONTINUED




5. a) A two-player co-operative game has a payoff bi-matrix given by

(
(5, 1) (4, 3)
(4, 4) (0, 1)
(4, 0) (1, 7)
).

Draw the attainment set for this game, indicating the Pareto-Optimal set. Calculate
the status quo point for the game and the Nash bargaining payoffs.
[11 marks]

b) Find the threat bargaining solution of the game (
(0, 1) (4, 4)
(4, 0) (1, 7)
).
[9 marks]


6. Four TV companies, ATV, BTV, CTV and DTV are bidding for the rights to
broadcast live football matches from the Leftovers League. Broadcasting rights
cost £1 million per match. ATV has a budget of £3.1 million to spend on matches.
BTV can spend up to £5.7 million, whilst CTV has a budget of £6.1 million.
Finally DTV can spend up to £4.2 million. The Board of the Leftovers will accept
bids from coalitions of the TV companies, but only the highest bidder will be
awarded any matches to broadcast.

a) Calculate the coalition function for the various coalitions defined as the number
of matches a coalition can be certain of broadcasting at a price of £1 million per
match (matches are indivisible). Hence derive the conditions satisfied by
imputations lying in the core. Solve the game and show there are four possible
solutions which satisfy the conditions of the core.
[12 marks]

b) To secure its respective broadcasting rights each TV company must pay the
Board of Leftovers the exact amount of £1 million for each match it wants to
cover. If they don’t have the requisite amount, the individual TV companies can
borrow their respective short fall from the Bank. However, the Bank must be re-
paid once the match has been broadcast. The TV companies expect to make a
profit of £100,000 (after payment of the rights) through advertising on each match
they broadcast. Determine which of your four solutions found in part a) gives the
best collective profit, whilst still allowing the individual TV companies to pay off
their Bank debts in full.
[8 marks]




PAPER CODE …Math331…… PAGE 5 OF 6 CONTINUED




7. a) In the early Middle Ages, one the most important and profitable trading
relationships centred on the port cities of Venice and Constantinople, capital of
the Byzantine Empire. Venetian merchants were eager to obtain high quality
Indian silks, imported via Constantinople, for their garment industries. In return
Byzantine traders required Venetian timber, which was used to maintain their
merchant fleet. At the beginning of the year 1082CE, wharfs in Venice held
approximately 400 tonnes of timber and crates containing 20 tonnes of silk. By
contrast Constantinople’s wharfs held 100 tonnes of timber and crates containing
280 tonnes of silk. Venice’s utility function for the commodities of timber and
silk was given by

(, ) =
8
6 + 3
,

whilst that of Constantinople (timber ′ and silk ′) was given by

(′, ′) =
7′′
2′ + 4′
.

Find the contract curve for the trading relationship between the two cities in terms
of (, ). [8 marks]

b) In 1082CE ships sailing between the two cities came under attack from Norman
pirates, fresh from their recent conquest of England. In order to raise money to
ward off these attacks and keep trade going, the authorities in the two cities
propose that exchanges this year should be regulated by maximising a mutual
welfare function given by

= 6 +
1296
343
.

Derive the two equations that maximise and hence show that any maximum
must also lie on the contract curve. Find how much timber and silk must be
transported between the two cities (using the much slower overland route) in order
for trade to continue in the future.
[12 marks]









PAPER CODE …Math331…….PAGE 6 OF 6 END


essay、essay代写