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CSYS5020 Calculation Exercise

创建日期：2024-05-06 16:35:31

所属分类：计算机computer science

标签： CSYS5020

Calculation Exercise

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CSYS5020: Interdependent Civil Systems – Assignment

Calculation Exercise

Q.1

Pagerank algorithm can be used as an effective centrality measure particularly for directed
networks.

i. Let us compute some ‘pagerank’ values for this network. Assume alpha = 0.8, and all
nodes have a value of (1/N) to begin with, where N is the size of the network. Prepare
an Excel spreadsheet, which dynamically updates the pagerank value of each node
based on the values of relevant nodes in the previous timestep. Based on your
spreadsheet, prepare a sorted list of pagerank values for all nodes after 100 iterations.
(15 marks)

You may assume that pagerank scores are governed by the expression:

Where Pit is the pagerank of node i at time t, A is the adjacency matrix of the network, and
is the damping factor.
ii. Verify that the pagerank values still add up to 1. (2 marks)
iii. How does the pagerank of Node 6 and Node 9 compare? Do you think this makes
sense? Please explain how. (2 marks)
iv. If all the edges are now considered bidirectional (except the ones between Node 6 and
Node 7, which together already make a bidirectional edge), how would it affect the
relative rank of Node 6? Predict and justify your answer without re-computing the
pagerank values of the whole network. (6 marks)

Q.2

Two competing Super Luxury bus services, Alpha Travels and BuzzMe, are operating a daily
service between the Indian cities of Chennai and Bangalore. Each service runs only once a
day, and starts from the same buss stand in each city at 6PM. Travellers do not pre-book
tickets, but come to the buss stand and simply get on board one or the other buss. Dinner is
served on board each bus for all passengers free of charge. The ticket prices are the same,
therefore the travellers will simply choose the bus based on the choice of meal available in
each luxury buss. Usually, each service has a market share of 50%.
Alpha travels has only two meal choices, A1 and A2, however on a given day, it can offer
only one to its customers. Similarly, BuzzMe has four choices of meals, but on a given day
will offer only one. Given the choice of meals on a particular day, a certain percentage of
customers will switch from one service to the other, as given in the following matrix.

B1 B2 B3 B4
A1 2 2 3 -1
A2 4 3 2 6

The numbers given are percentage gains, for Alpha Travels.

(a) What strategy must Alpha Travels choose, to minimise its maximum losses? (to minimise
the damage in their worst-case scenario?) (4 marks)
(b) What strategy BuzzMe must choose, to minimise its maximum losses? (to minimise the
damage in their worst-case scenario?) (4 marks)
(c) Is there a pure saddle point (a pure strategy Nash equilibrium)? If not, what does the
absence of such equilibrium signify? (4 marks)
Suppose that, on a given day, Alpha travel will choose meal A1 with probability x (and meal
A2, therefore, with probability 1-x).
(d) Calculate the expected payoff for Alpha Travels for each of BuzzMe’s pure strategies
(meal choices), in terms of x. Based on this, fill the following table. (8 marks)

B’s pure Strategy A’s expected payoff
B1
B2
B3
B4

(e) Given that x is a probability and ranges from 0 to 1.0, plot A’s expected pay-off against x
as a graph for each of B’s pure strategies, on the same plot. (8 marks)
(f) Using your plot, for each of the following values of x, identify what is the worst pay-off
for A, and which pure strategy of B it corresponds to. (7 marks)
x = 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1.0
(g) Using the plot, identify what is the best (highest) value of the minimum payoff of A, and
what is the value of x this corresponds to. (5 marks)
(i) If A wants to optimise its outcome of the worst-case scenario, how often it should serve
meal A1 and how often it should serve meal A2? (4 marks)
(j) A similar approach can be used to prescribe a ‘mixed strategy’ for BuzzMe. Comment on
why this task is relatively more difficult. (6 marks)
Q.3
Consider an Iterated Stag Hunt (ISH) played in a networked system. The networked system
on which the game is played is given by Fig.2. The pay-off metric for stag-hunt is given by
Fig. 3. The nodes at each iteration play a round of SH with each of their nearest neighbours.
Thus, M games are played per iteration, where M is the number of edges. The nodes use the
simple strategies of coordination C or defection D. Your overall objective is to maximise the
system utility (total payoff for the system).

i. What is the pay-off for the system from a D-C link, a C-C link, and a D-D link,
respectively? Consequently, which types of links do you need to maximise to
increase system pay-off? Which is the second-best type of link? (7 marks)
ii. If there is a constraint that the number of players playing C must equal the number
of players playing D, come up with a scheme which ensures highest public utility per
iteration. In your answer, denote what pure strategy each node must play, and the
corresponding total pay-off per iteration. If the number of nodes is odd, you must
choose a configuration which gives you the highest utility where the difference
between the number of Coordinators and number of Defectors is exactly one. (10
marks)
iii. Now assume that each node plays tit-for-tat in an iterated game, and the pure
strategies that you have assigned are simply the initial pure strategies for iteration
one. Will the system pay-off change in the second iteration? By how much? Justify
the answer. Assume that a given node could play C against one neighbour and D
against another in the same iteration, if the tit-for-tat strategy prescribes that. (4
marks)
iv. If there is no constraint on how many nodes must play C or D, assign the strategy for
each node which will maximise the system pay-off. (4 marks)

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COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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