Networks, Crowds, and Markets Homework 1 Solution

Description

Part 0: Slack

Join the slack channel for the course, via the following link: SLACK

Part 1: Game Theory

1. For each of the following three two-player games, nd (i) all strictly dominant strategies, (ii) the action pro les which survive iterative removal of strictly dominated strategies, and (iii) all pure-strategy Nash equilibria. Give a brief justi cation for each part.

		( ;L)	( ;R)
(a)	(U; )	(5, 4)	(4, 5)
	(D; )	(4, 4)	(0, 0)
		( ;L)	( ;R)
(b)	(U; )	(2, 2)	(2, 1)
	(D; )	(3, 2)	(0, 3)
		( ;L)	( ;R)
(c)	(U; )	(6, 5)	(4, 5)
	(D; )	(5, 4)	(2, 2)

2. Consider the two-player game given by the following payo matrix:

	( ;L)		( ;M)		( ;R)
(t; )	(-1, 2)		(5,	1)	(0, 0)
(m; )	(1,	2)	(-1, 0)		(6,	2)
(b; )	(4,	1)	(3,	1)	(2,	0)

1. 1. Does either player have a strictly dominant strategy? If so, which player, what strategy, and why? If not, what is the smallest number of entries in the payo matrix which would need to be changed so that some player did have a strictly dominant strategy? Justify why this is the minimum, i.e. there is no smaller value that works.

1. 2. What are player 1’s and player 2’s best-response sets given the action pro le (m; L)?

1. 3. Find all pure-strategy Nash equilibria for this game. (Argue why all that you wrote are PNEs and why there are no others.) Describe how best-response dynamics might converge to each pure-strategy Nash equilibria.

3. (a) Prove the following: If player 1 in a two-person game has a dominant strategy s₁, then there is a pure-strategy Nash equilibrium in which player 1 plays s₁ and player 2 plays a best response to s₁.

3. 2. Is the equilibrium from part (a) necessarily a unique pure-strategy Nash equilibrium? Justify your answer.

3. 2. In particular, can there also exist a pure-strategy Nash equilibrium where player 1 does not play s₁? Justify your answer.

3. 2. If s₁ is instead a strictly dominant strategy for player 1, how do the answers to (a)-(c) change? Provide proper justi cations for each part.

4. Formulate a normal-form game (as a payo matrix) that has a unique pure-strategy Nash equilibrium, but for which best-response dynamics does not always converge (i.e. there are possible starting states for which BRD will not converge). Justify your answer. (Hint: Rock-paper- scissors has no equilibrium, and thus BRD will not converge. Can you combine this with a game that does have an equilibrium?)

Part 2: Graph Theory

Note: Unless stated otherwise, please assume for any problem involving graphs that we refer to undirected and unweighted graphs.

5. Given a graph, we call a node x in this graph pivotal for some pair of nodes y and z if x (not equal to y or z) lies on every shortest path between y and z.

5. 1. Give an example of a graph in which every node is pivotal for at least one pair of nodes. Explain your answer.

5. 2. For any integer c 1, construct a graph where every node is pivotal for at least c di erent pairs of nodes. That is, if I give you any value for c 1, you should be able to give me such a graph. Explain your answer.

5. 3. Give an example of a graph having at least four nodes in which there is a single node x which is pivotal for every pair of nodes not including x. Explain your answer.

5. Given some connected graph, let the diameter of a graph be the maximum distance (i.e. shortest path length) between any two nodes. Let the average distance be the expected shortest path length between a randomly selected pair of distinct nodes.

5. 1. Let G be a graph with average distance A. What is the smallest diameter possible for such a graph? Provide a graph G that attains this minimum and prove that any smaller is impossible.

5. 2. Give a graph G with diameter at least 3 A.

5. 3. Repeat (b) for a diameter of at least 100 A. (You don’t need to draw the graph, just describe it and brie y justify why the diameter is at least 100 times larger than than the average distance.) Describe how you could extend this to an arbitrarily large factor C A.

5. 4. Discuss what the diameter and average distance of a social network (given as a graph) might represent. What might it mean if the diameter is very similar to the average distance? What might it mean if the diameter is much greater?

5. Consider a graph G on n nodes.

5. 1. What is the fewest number of edges such that G is connected? Give an example with that many edges, and argue why any fewer edges must result in a graph G which is disconnected.

5. 2. What is the fewest number of edges such that any two nodes in G have a shortest path length of 1? Again, prove that this is the minimum by arguing that no fewer is possible and that the number you give is attainable.

(c) Repeat part (b) for a shortest path length of at most 2.

Part 3: Coding: Shortest Paths

8. Submit the following:

8. 1. In graph.py, implement (and turn in) a function create graph(n,p) that produces an undirected graph with n nodes where each pair of nodes is connected by an edge with probability p.

8. 2. Implement a general shortest-path algorithm for graphs, as described in lecture, that works on your graph. In graph.py, include a function shortest path(G,i,j) that outputs the length of the shortest path from node i to j in your graph G. Make sure to handle the case where the graph is disconnected (i.e. no shortest path exists) by outputting \in nity”.

8. 3. Construct a graph for n = 1000 and p = 0:1. Estimate the average shortest path between a random pair of two (connected) nodes in the graph. For accuracy, repeat for 1000 random pairs of nodes in your graph. Output an execution trace in avg shortest path.txt containing all path lengths written as (i, j, length).

8. 4. For n = 1000, run the shortest-path algorithm on data sets for many values of p (for instance, 0.01 to 0.04 using .01 increments, and then 0.05 to 0.5 using .05 increments). Turn in your numerical data as varying p.txt, and plot the average shortest path as a function of p and submit as an image le varying p.(image extension) or include in your main .pdf le.

Note: For p = 0:01 there is actually a small but reasonable chance (around 4%) to produce a disconnected graph. If this occurs, resample and produce a connected graph for the purposes of gathering data.

8. 5. Intuitively explain the behavior of the data you found; speci cally, as p increases (in particular, look at the larger values, e.g. 0.3 and above), what function does the average shortest path length seem to asymptotically approach and why?

8. Now run your code on the Facebook social network data available at: http://snap.stanford.edu/ data/egonets-Facebook.html

(In particular, please refer to the le \facebook combined.txt.gz”; the data is formatted as a list of undirected edges between 4,039 nodes, numbered 0 through 4038. You will need to parse this data as part of your code; knowing how to do this will be useful for subsequent assignments!)

8. 1. Repeat the same analysis as in part 8(c) (i.e. run your algorithm on 1000 random pairs of nodes and determine the average shortest path length). Include your code in graph.py and include an execution trace in fb shortest path.txt

8. 2. For the Facebook data, estimate the probability p that two random nodes are connected by an edge. Explain how you computed p.

8. 3. Is the average shortest path length of the Facebook data greater than, equal to, or less than you would expect it to be if it were a random graph with the same number of nodes and value of p? (To answer this, you may wish to run your code from question (8c) using

the p you determined in part (9b) and 4039 nodes.) Explain why you think this is the case.

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