CSYS 5020 - Interdependent Civil Systems

Assignment Task

Question 

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.

ii. Verify that the pagerank values still add up to 1.

iii. How does the pagerank of Node 6 and Node 9 compare? Do you think this makes sense? Please explain how.

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.

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.

(a) What strategy must Alpha Travels choose, to minimise its maximum losses? (to minimise the damage in their worst-case scenario?

(b) What strategy BuzzMe must choose, to minimise its maximum losses? (to minimise the damage in their worst-case scenario?

(c) Is there a pure saddle point (a pure strategy Nash equilibrium)? If not, what does the absence of such equilibrium signify? 

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.

(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.

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