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Game Theory and Strategy

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Game Theory and Strategy
Spring 2012, Assignment 2
The assignment must be done by 6-00 p.m. May 1st. You should submit your assignment assignments before the class begins. Good luck!

Part 1 (30%)

For multiple choice questions choose a letter corresponding to your answer. For true- false questions the answer is “TRUE” if you agree with the sentence and “FALSE” otherwise.

1. Mixed strategy is
A) When a player plays different actions in response to different actions taken by other players
B) When a player uses advises from different people
C) When a player chooses each action with some probability 2. “The game where players undertake actions with uncertain outcomes” is a definition of the game with asymmetric information. Falls

3. In a separating equilibrium with two types of informed player and two possible actions each type of informed player chooses different action.. True 4. Auction is a public sale in which property or merchandise are sold to the highest bidder. True

5. Private Value Auction is one where True * Each bidder knows his or her value for the object * Bidders differ in their values for the object

6. Common Value Auction is one where the item has a single though unknown value. True

7. English auction is one where: Falls * Bidders call out prices (outcry) * Auctioneer calls out prices (silent) * Bidders hold down button (Japanese) * Second-Highest bidder gets the object

8. Second-price auction is one where: True * All buyers submit bids * Buyer submitting the highest bid wins and pays the second highest bid

9. In a second-price auction bidding your true valuation is a dominated strategy. Falls

10. More bidders usually lead to higher prices. True Part 2 (50%)
Explanations and/or calculations are required in all exercises!!! Without it you can get maximum only 25% of possible points.
1. Consider the following game in the normal form

Player 2 A B A 100,50 20,30
Player 1 B 20,40 50,100 a. Identify all Nash Equilibria in pure strategies.
The Nash Equilibrium is AABB because 100 > 20 and 50 >20 and 50>40 and 100 >30.
b. find all equilibria in mixed strategies.

Let's assume p is probability that Player 1(P1) chooses A and 1-p if Pl chooses B. q is probability that Player2 (P2) chooses A and 1-q is P2 chooses B.
If P1 chooses A the expectation for P2 to choose A are: 50p+30(1-p) = 10p+40
If P2 chooses B the expectation for P2 to choose B are: 30p + 100(1-p) = 70p+100
The critical value is: 10p+40 = -70p + 100 = 3/4
If p< 3/4 P2 will choose A
If p > 3/4 P2 will choose B
If p = 3/4 A or B is the same for P2
As a result, for P1 only if choosing p = 3/4 can P1 make P2 unknown to P1's strategy and do not use pure strategy. So 3/4, 1/4 is the mixed strategy for P1.

2. In the market for old cars there are two types of cars: good quality and bad quality cars. The fraction of good quality cars is q. The Sellers’ reservation prices are 1000 and 2000 (for bad and good quality cars respectively). The Byuers’ reservation prices are 1220 and 2400. Describe equilibrium in this market. Explain “market for lemons” term. What is adverse selection?

To describe equilibrium in this market we should take a look at several cases:

1. When prices (p) are: p<1000

3. Present the “Prisoner’s dilemma” game as an extensive form game. Solve the game by using Bayesian equilibrium approach.

4. Consider the following second-price auction: * There are 3 bidders and they have following valuations:. Homer: $5, Bart $3, Lisa $2
Who wins the auction and how much will he/she pay?
Bart $3

5. Suppose that Bombardier had organized its sale to Air Canada or WestJet as a first-best price sealed-bid auction. The Value of wining for Air Canada is 80 and that for WestJet is 40. Possible bids are 20, 40, 60 or 80. Alternatively, each firm can decide not to participate in the auction. If bids are equal, Air Canada wins. The utility from wining is the Value of winning minus the bid. (a) (6%) Model this as a two-by-two game and (b) (6%) Find all Nash equilibria (in pure strategies) (c) (6%) It is known in theory that a situation where all players bid the valation of player with second-high valuation is an equilibrium. Is this game consistent with this result?

Part 3 (20%)

1. Explain the concept of mixed strategies? Why do we need this concept in addition to Standard Nash equilibrium concept?

A mixed strategy is an assignment of a probability to each pure strategy. A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face. A mixed strategy allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even if their strategy set is finite.
We need mixed strategies concept in addition to Standard Nash equilibrium concept because many games don't have Nash equilibrium in pure strategy. 2. Explain the revenue equivalence theorem. Explain why it can break down when bidders are risk-averse.

The Revenue Equivalence Theorem is generalized to the case of asymmetric auctions in which each player's valuation is drawn independently from a common support according to his/her distribution function.

The revenue equivalence theorem does apply to auctions in which the value that bidders attach to the auctioned item is unknown to them and related, but only if the signals they receive about the value are independent, and they come from the same distribution. The theorem does not apply when bidders receive signals about the value of the object to them that are correlated with each other.

Attitude towards risk
The revenue equivalence theorem implies that in private value auctions, the bidders and the auctioneer are indifferent between a wide range of auctions if they are risk neutral. For example we showed that the all pay auction, the first price sealed bid auction (which is strategically equivalent to the descending price auction in this case) and the second price sealed bid auction (strategically equivalent to an ascending auction) are all revenue equivalent. But what if bidders are risk averse?

In the case of second price sealed bid auctions with private values, the arguments we used before apply to risk averse bidders in this case. It remains a weakly dominant strategy for each player to bid his or her valuation. Therefore the optimal bidding strategy for the second price sealed bid auction (and also the Japanese and English auctions) is independent of a bidder’s attitude towards risk and uncertainty when private values are drawn from a common probability distribution.
The same claims cannot be made for first price sealed bid auctions with private values. In this case, note first that a strategy of bidding your valuation guarantees exactly zero surplus. If you place a lower bid than your valuation your expected surplus initially increases until it reaches the maximum for a risk neutral bidder, and then falls, but the variance of the surplus increases as well. Since a risk averse gambler is willing to trade a lower expected value to reduce the amount of uncertainty, he accordingly bids higher than a risk neutral bidder.
So when comparing first and second price sealed bid auctions, the revenue generated by a second price auction is independent of the bidders’ preferences over uncertainty, since bidding is unaffected. Yet the revenue generated by the first price auction is the same as the revenue generated by a second price auction when bidders are risk neutral. Therefore risk averse bidders generate more revenue in a first price auction than they would in a second price auction, and they generate more revenue in a first price auction than do risk neutral bidders.

Risk-averse bidders tend to be more aggressive at the first-bid auction.
The revenue equivalence theorem implies that in private value auctions, the bidders and the auctioneer are indifferent between a wide range of auctions if they are risk neutral.…...

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