1. Games for Crowds & Networks
EXERCISES
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2. Exercise Session
Week 1

3. Wrong Nash (from A Beautiful Mind)
This is a game with four players (men) and five actions (women to dance with). The assumption is that if one man asks to dance with a woman she says yes, but if two or more men ask to dance with her she says no to all. All want to dance with the blonde woman, and are indifferent between the rest, but would prefer to dance with someone rather than none.
Questions
Why is the solution described in the movie not a Nash equilibrium?
Does a Nash equilibrium exist?

4. Badminton Tournament (from 2012 London Olympics)
Top two teams from each group (A, B, C, D) proceed to the quarter-finals. This is followed by a standard “knockout” phase.
#1 from group A/B plays against #2 from group C/D
#2 from group A/B plays against #1 from group C/D
In the 2012 Olympic games the following happened:
A feared Chinese team (C1) in an upset came in 2nd place in group D.
The South Korean team (SK) and another Chinese team (C2) had a 2-0 record in group B and both guaranteed to enter the knockout stage, but had to play each other to determine who got 1st place in group B.
SK and C2 both purposely tried to lose their game.
Questions
Why would they do this? Try to model formally.
How can we change the rules so that this does not occur?

5. Exercise Session
Week 2

6. Selling Multiple Items
Assume there are k identical items and n > k bidders, each of whom is interested in purchasing exactly one item.
Consider the following proposed mechanisms. Are these mechanisms DSIC?
Hold k simultaneous Vickrey auctions.
Hold k Vickrey auctions sequentially.
Design one sealed-bid auction that will simultaneously sell all items. Ensure that it is DSIC.
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.

7. Depedent/Unknown Values
Is the Vickrey auction DSIC in the following scenarios?
Assume all buyers have an independent Bernoulli value v for the item: vi is 1 with probability pi and 0 with probability 1-pi, and the true value is only discovered after the purchase. (You may have to think about what it means to bid truthfully in a probabilistic setting).
Now, assume all buyers have the same dependent Bernoulli value v for the item: v is 1 with probability ½ and 0 with probability ½; the true value is only discovered after the purchase and is the same for all bidders.
Does this change if some buyers are informed and know the true value v before they bid?
Assume all buyers have the same value v for the item, but their estimates of this value are noisy; e.g., the true value is v but each buyer believes her value to be v + X_i where X_i is some Gaussian noise N(0, .01). The true value (without noise) is discovered after the purchase.

8. Exercise Session
Week 3

9. Combinatorial Auctions
Assume there is a combinatorial auction where bidders may want more than one item. Consider the following setting: There are two goods (A, B) and three bidders (1,2,3). Bidder 1 only has value for both items, so v1(AB) = 1 and 0 otherwise. Bidder 2 is interested in A, so v2(A) = v2(AB) = 1 and 0 otherwise. Similarly, Bidder 3 is only interested in item B.
Compute the VCG allocation and prices when only bidders 1 and 2 are present (apply the same VCG principle to this setting – first maximize social welfare in the allocation, then charge the “harm” to others as the price).
Compute the VCG allocation and prices when all three bidders are present.
Look at the two revenues above. Can the same thing happen in the Vickrey auction?
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.

10. More Combinatorial Auctions
Consider a combinatorial auction in which a bidder can secretly submit multiple bids under different pseudonyms. The value and payment are the sum of all values and payments for the different pseudonyms.
Show the following is possible: A bidder can increase her utility by doing the above.
Can the same thing happen in a Vickrey auction?

11. Exercise Session
Week 4

12. Local Connection Game
Show that forα< 1 the complete graph is the unique equilibrium.
Does this imply anything about the price of stability?
About the price of anarchy?
Construct a Nash equilibrium that is not a star for some α> 2.
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.

13. Global Connection Game
Consider a congestion version of the global connection game where the cost of an edge increases linearly with the number of players ce = ½ ke. Show that this game has a potential function.
(Bonus question) Consider a weighted version where each player has a weight wi > 0 which corresponds to the amount of traffic she will send, and costs are shared proportionately to the weights. Specifically, if We is the sum of weights of all players using edge e, then player i pays cewi/We. Show that this game does not have a potential function.

14. Exercise Session
Week 5

15. Homophily & Cascades Does this network exhibit homophily?
For which range of p = b / (a+b) does a complete cascade occur from white to pink?
In the social network below, which edge(s) are the next most likely to form?
If you wanted to initiate a cascade, which vertex would you want to as an initial adopter?
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.

16. Selection & Influence
Consider Schelling’s tipping model with 10 people.
What is the largest ravg such that everybody still riots?
Assume that the people are on a social network, and their willingness to riot depends only on their own neighbors. Construct a few connected networks with 8 people and ri = 0 for 2 people, ri = 2 for 2 people, ri = 4 for everyone else.
Explore the tradeoff between maximizing the number of edges and minimizing the number of rioting nodes.

17. Exercise Session
Week 6

18. Information Cascades
Consider the information cascade model, with the following modifications. For the red/blue urn example, can a cascade to the wrong guess form in this model?
Individuals see the private signal (red/blue draw) in addition to the guesses of other players.
One out of every 10 players is “stubborn” and only takes into account their private information.
Individuals only see the guess of the person before who came immediately before them (but no signals and no guesses from anyone else).
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.

19. Market for Lemons
Suppose that there are three types of used cars: good ones, medium ones and lemons, and that sellers know which type of car they have, but buyers do not. The fraction of used cars of each type is 1/3. A seller values a good car at $8,000, a medium car at $5,000 and a lemon at $1,000. Buyers values are, $9,000, $8,000 and $4,000, respectively. Buyers are willing to pay at most their expected value for a car.
Is there an equilibrium in the used-car market in which all types of cars are sold?
Is there an equilibrium in the used-car market in which only medium quality cars and lemons are sold?
Is there an equilibrium in the used-car market in which only lemons are sold?

20. Exercise Session
Week 7

21. Multi-Player Games
Suppose there are three players with pure strategies (U, D), (L, R) and (l, r) respectively. The payoff are given by the following 3D matrix (written as two 2D matrices):
Assume the players make simultaneous decisions. What are the pure Nash equilibria?
Assume player 3 moves first, and then 1 and 2 move simultaneously. What are the subgame perfect equilibria?
Assume player 3 moves, then player 2, then player 1. What are the subgame perfect equilibria?

22. Repeated Games Consider the following game.
What are the Nash equilibria in a single round of this game?
If the game is repeated twice, is there a subgame perfect equilibrium where the players use the strategy (A, A) in at least one round?
In 1990 new zeland sold tv licenses. One license had high bid of 10,000 and second-high of 6. another high bid of 7 million and second high of 5000.
In 2000 switzerland auctioned 3 blocks of spectrum in sequential vickrey auctions. First, 121 mil, second 134 mil, third 55 mil.