MATH 240: Problem Set 4

Transcription

1 MATH 240: Problem Set 4 You may either complete this problem set alone or in partners. If you complete it in partners, only one copy needs to be submitted. Show all your work. This problem set is due on: Monday, November 7 th. 1. Consider the graph below. Suppose that each node represents a web page and each direct edge represents a link. a) This graph can be viewed as a strongly connected component, an upstream component and a downstream component. Label each vertex with S, U, or D to indicate whether it is in the strongly connected, upstream, or downstream component respectively. b) Suppose that you are able to add one link to this system. It is your goal to expand the strongly connected component as much as possible. Draw in an edge on the graph above that would accomplish this.

2 2. Consider the graph of web pages below. You decide to use the hub and authority viewpoint in order to rank the pages. Start by assigning each hub a value of 1. a) Apply the authority update rule. What are the new values for each authority? b) Apply the hub update rule. What are the new value for each hub?

3 c) Apply the authority and hub update rules again. What are the new values for each page? d) Recall that in practice we scale the values such that the sum of the authority is 1 and the sum of the hub is 1. What is the scaled authority value of J?

4 3. Consider the web page and link structure below. Now suppose you are going to use basic page rank to rank the results. a) Assign each page equal page rank. Apply the basic page rank update rule once. What are the new page rank values for each page?

5 b) Recall that any strongly connected graph has a unique set of equilibrium page rank values. What are these equilibrium page rank values for the network above? Show your work. c) Recall that we know that we realized we needed to update the basic page rank rule to include a scaling factor. Delete an edge (place an X on the edge to delete) to create a network that illustrates the need for this scaled update rule. Explain why this new graph needs the scaled page rank rule.

6 4. Recall that factors other than the core page ranking algorithm effect what search results we see. Suppose that we consider the search for basketball. Give an example of how each of the following factors may influence the search results for this term: time, geography/location, and personal profile. [For example, if the search term is flowers we may get information about planting flowers in the early Spring but get information about buying flowers near February.]

7 5. Suppose a search engine has three slots (a, b, and c) that it can sell. Slot a has a clickthrough rate of 5, slot b has a click-through rate of 3, and slot c has a click-through rate of 2. There are three advertisers (x, y, and z) who are interested in these slots. Advertiser x values clicks at 8 per click, advertiser y values clicks at 6 per click and advertiser z values clicks at 5 per click. a) Viewing the problem as a matching market problem, fill in the pay off matrix below. x y z a b c b) Who should get assigned which slot in order to maximize total value? What is the total value of this matching?

8 6. Suppose a search engine is running a generalized second-price auction on its web page, and has three ad slots (a, b, and c) that it can sell. Slot a has a click-through rate of 5, slot b has a click-through rate of 3, and slot c has a click-through rate of 1. There are three advertisers (x, y, and z) who are interested in these slots. Advertiser x values clicks at 8 per click, advertiser y values clicks at 5 per click, and advertiser z values clicks at 2 per click. Suppose all advertisers bid their true values for the clicks. a) Which advertiser should be assigned which slot? b) How much should each advertiser be charged (per click) for their slot? c) What is the total net value for each advertiser for their slot (not per click, but total)? d) Supposes that advertiser x learns that the other two advertisers will bid their true values. Assuming that x wants to maximize his net value, should advertiser x bid his true value or not? Justify.

9 7. A search engine has three ad slots, 1, 2, and 3, that it can sell. Slot 1 has a click-through rate of 6, slot 2 has a click-through rate of 4, and slot 3 has a click-through rate of 1. There are three advertisers, A, B, and C, who are interested in these slots. Each advertiser wants at most one slot. Advertiser A values clicks at 5 per click, advertiser B values clicks at 3 per click, and advertiser C values clicks at 1 per click. a) Which advertiser should be assigned which slot? b) What is the VCG price that each advertiser should pay? Show your work.

10 8. Currently, most if not all search engines use either generalized second price auctions or VCG priced auctions to sell their ads. However, it is unclear which method, if either, is best in terms of profit generation for the search engine. If you were running a search engine, which of the two methods would you use and why?