ECON 212 Lecture Notes - Lecture 10: Strategic Dominance, Oligopoly, Nash Equilibrium

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There are two kinds of games: simultaneous game (cournot oligopoly) where the firms make decisions at the same time, sequential game (stackelberg oligopoly) where one firm makes a decision first, the other follows. To solve the games, label all possible outcomes with a number representing the payoff each player. If new airline enters: existing airlines will: business>price war. New airlines will enter the industry because 5>0. The players decide on a course of action at the same time, their game is depicted in a matrix, where each row is one of player 1"s. Existing airlines will: business>price war choices and each column is one of player 2"s choices. If player 1 is going to withdraw then player 2 would like to as well. If player 1 is not going to withdraw then player 2 will also be content not to withdraw.

Related Questions

1- Two friends on a trip, A and B, meet by chance and want to dine together, although both recognize that A prefers to eat at Chili's and B prefers to eat at Steak-n-Shake. Each wonders what to say and accept in negotiating a joint decision. Let O₁ = both eat at Stake-n-Shake, O₂ = A eats at Stake-n-Shake and B eats at Chili's, O₃ = B eats at Stake-n-Shake and A at Chili's, and O₄ = both eat at Chili.

Write an ordinal utility function for each of the friends in respect of the possible outcomes:

Friend A: u_A(O₁) = Friend B: u_B(O₁) =

u_A(O₂) = u_B(O₂) =

u_A(O₃) = u_B(O₃) =

u_A(O₄) = u_B(O₄) =

Construct a 2 x 2 matrix that models this situation, including players, strategies, and the ordinal utilities of each player over the possible outcomes. Make sure you label everything (i.e., strategies, players, etc.).

2- Using the situation described above:

Construct an interval utility function for each player, assuming the following: A is indifferent between (i) its second most preferred outcome and a lottery composed of a .6 chance of its most preferred outcome and a .4 chance of its least preferred outcome, and (ii) A is indifferent between its third-most preferred outcome and a lottery with a .9 chance of its second most preferred outcome and a .1 chance of its least preferred outcome. However, B is indifferent between (i) its second most preferred outcome and a lottery composed of a .7 chance of its most preferred outcome and a .3 chance of its least preferred outcome, and (ii) B is indifferent between its third-most preferred outcome and a lottery with a .8 chance of its second most preferred outcome and a .2 chance of its least preferred outcome.

Construct a 2 x 2 matrix that models the situation, as described above, including players, strategies, and the interval utilities of each player for the possible outcomes.

3- An angry faction, A, is considering whether or not to challenge a status quo power, B, by highjacking B's plane and threatening to kill the passengers if B does not release some terrorists previously imprisoned by B, and whether to carry out its threat or release the passengers if B does not concede to its demands, were the demands made. Let the possible outcomes be: O₁ = A does not highjack the plane, O₂ = The passengers are saved after B releases the imprisoned terrorists in response to the threat made by A, O₃ = the passengers are killed by A after B refuses to submit to A's demand for the release of prisoners, O₄ = the passengers are released by A after B refuses to submit to A's demands. Assume (a) that the best outcome for B, but the worst for A, is for the passengers on the plane to be released after a highjacking, given the reputation for toughness B would gain at A's expense, (b) that the best outcome for A would be for B to release the prisoners after A makes its threat, (c) that A prefers to kill the passengers after a failed attempt to obtain the release of the terrorists over doing nothing at all to obtain the release of the terrorists, and (d), of course, B prefers having to do nothing to have to concede to A's demands, if A decides to challenge B.

Write an ordinal utility function for the angry faction and the status quo power in respect of those outcomes:

Angry A: u_A(O₁) = Power B: u_B(O₁) =

u_A(O₂) = u_B(O₂) =

u_A(O₃) = u_B(O₃) =

u_A(O₄) = u_B(O₄) =

b. Construct a 2 x 2 matrix that models this situation, including players, strategies, and the ordinal utilities of each player over the possible outcomes. Make sure you label everything (i.e., strategies, players, etc.).

4- Why, as game theorists, do we want to obtain an interval measurement of preferences rather than an ordinal measurement? In particular, what does an interval measurement allow us to do that an ordinal measurement doesn't? Are there any limitations on interval measurements? Be specific.

5- What is the difference between decisions under risk and decisions under certainty? What significance does this have for the relevant decision principles for a given game? If a decision principle used for games under uncertainty is applied to a game under risk and it contradicts the decision principle normally used for games under risk, which decision principle should be used, and why? Give an example.

6- Name the two types of irrational actions and explain what they are. Give an example of each. What relevance do irrationalities have for game theory?

7- Josiah, a young man, is considering whether or not to ask Lydia to go to prom with him. Lydia is either willing to go with him or not. Josiah surmises that there is a .6 chance that Lydia would say yes. If Josiah does not ask Lydia out, he will find out whether or not she would have accepted his offer afterward. Let O₁ = Josiah asks and Lydia accepts, O₂ = Josiah asks and Lydia denies, O₃ = Josiah doesn't ask and Lydia would have accepted, and O₄ = Josiah doesn't ask and Lydia wouldn't have accepted. Josiah prefers O₁ to O₄, O₄ to O₃, and O₃ to O₂. That is, O₁ > O₄ > O₃ > O₂.

a) Assume that Josiah is indifferent between O₄ and a lottery with a .9 chance of O₁ and a .1 chance of O₂, and likewise assume that he is indifferent between O₃ and a lottery with a .4 chance of O₁ and a .6 chance of O₂. Construct an interval utility function for Josiah over the respective outcomes:

Josiah: u(O₁) =

u(O₂) =

u(O₃) =

u(O₄) =

b) Use these utilities to construct a 2 X 2 matrix representing the situation. Use the Bayesian decision principle to determine which strategy Josiah would take, were he rational.

c) We treated this game as a game against nature, or a parametric game, rather than a strategic game. Which of these two types of games would be a more realistic model of the described situation? Why or why not?

8- It's Saturday night and Josiah is trying to decide where to take his new girlfriend, Lydia, out to dinner. He has two options: Taco Bell and Macaroni Grill. Lydia prefers Macaroni Grill to Taco Bell, and Josiah prefers what Lydia prefers. There is a problem, however. Martin works at both Taco Bell and Macaroni Grill, and he is practically Josiah's stalker. If Josiah is anywhere near Martin, Martin will find Josiah and ruin dinner for Lydia, which is all that Josiah cares about. Luckily, Martin is either at Macaroni Grill or Taco Bell, not both. Unluckily, neither Josiah nor Lydia has any idea which restaurant Martin is working at or any way of finding out. Lydia prefers eating at Taco Bell without putting up with Martin to eating at Macaroni Grill while putting up with Martin. If Lydia has to put up with Martin, she would still rather eat at Macaroni Grill than Taco Bell.

a) What are the relevant outcomes?

O₁ =

O₂ =

O₃ =

O₄ =

b) Construct an ordinal utility function for these outcomes. Remember, Josiah only cares about what Lydia prefers.

u(O₁) =

u(O₂) =

u(O₃) =

u(O₄) =

c) Construct a 2 X 2 matrix using the utilities from above. Consider two decision principles: maximin and the Bayesian decision principle + the principle of insufficient reason. Determine if each principle can actually be applied to this game. For any principle which does apply, use it to determine Josiah's rational strategy, if the principle recommends one.

9- Consider the following game between play S and player T.

	T1	T2	T3	T4
S1	1, -1	4, -4	10, -10	1, -1
S2	9, -9	5, -5	7, -7	6, -6
S3	3, -3	1, -1	8, -8	5, -5
S4	10, -10	3, -3	1, -1	2, -2

a) Determine the saddlepoint strategies of both players. Show your work.

b) Remove T4 and S4 from the game and determine the outcome using dominance reasoning. Show your work.

whiteherring176

(Please show as much work as possible)

1. You are bidding in a second-price auction for a painting that you value at $800. You estimate that other bidders are most likely to value the painting at between $200 and $600. Which of these is likely to be your best bid?

a. $1,000

b. $800

c. $600

d. $400

2. Which of the following is true about different ways of conducting a private-value auction?

a. A first-price auction is strategically equivalent to a second-price auction.

b. A first-price auction is strategically equivalent to an English auction.

c. A second-price auction is strategically equivalent to an English auction.

d. None of the above

3. Suppose that five bidders with values of $500, $400, $300, $200, and $100 attend an oral auction. Which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

4. In the above auction, if the bidders with the first- and third-highest values ($500 and

$300) collude, which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

5. If a seller is concerned about collusion among bidders, which of the following changes to the auction, should the seller make?

a. Hold frequent, small auctions instead of infrequent large auctions.

b. Conceal the amount of winning bids.

c. Publically announce the name of each auction's winner.

d. Hold a second-price instead of a first-price auction.

6. You're holding an auction to license a new technology that your company has developed. One of your assistants raises a concern that bidders' fear of the winner's curse may encourage them to shade their bids. How might you address this concern?

a. Release your analyst's positive scenario for the technology's future profitability.

b. Release your analyst's negative scenario for the technology's future profitability.

c. Use an oral auction.

d. All of the above

7. In a first-price auction, you bid ________ your value, and in a second-price auction you bid _________ your value.

a. at; above

b. below; above

c. below; at

d. below; below

8. You hold an auction among three bidders. You estimate that each bidder has a value of either $16 or $20 for the item, and you attach probabilities to each value of 50%. What is the expected price? If two of the three bidders collude, what is the price?

9. In Sweden, firms that fail to meet their debt obligations are immediately auctioned off to the highest bidder. (There is no reorganization through Chapter 11 bankruptcy.) The current managers are often high bidders for the company. Why?

10. When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, representatives of other museums that have no chance of winning are actively wooed by the auctioneer to attend anyway. Why?

11. The deities Mars and Venus often do battle to create the weather conditions on Earth. Venus prefers extreme temperatures (especially heat), while Mars prefers temperate conditions. The payoffs (expressed in Points of Wrath) are given below.

		Venus
		Warm	Chill
Mars	Warm	20 , 0	0 , 10
Chill	0 , 90	20 , 0

What is the unique mixed-strategy equilibrium of the above game?

(Let p be the probability of "Warm" for Mars, and q the probability of "Warm" for Venus.)

a) p=9/10, q=1/2

b) p=1/2, q=1/10

c) p=1/2, q=1/2

d) p=1/10, q=1/10

Player 2

		H	D
Player 1	H	0 , 0	4 , 1
D	1 , 4	2 , 2

12. The above game is the title of the hawk-dove game and used by evolutionary biologists to describe evolutionary processes. It is also used to model how a business should grow. In the above game, what is the Nash equilibrium in pure strategies and mixed strategies.?

Assume the cost of producing the goods is zero and that each consumer will purchase each good as long as the price is less than or equal to value. Consumer values are the entries in the table.

	Good 1	Good 2
Consumer A	$2,300	$1,700
Consumer B	$2,800	$1,200

13. Suppose the monopolist only sold the goods separately. What price will the monopolist charge for good 1 to maximize revenues for good 1?

a. $2,300

b. $2,800

c. $1,200

d. $1,700

14. What is the total profit to the monopolist from selling the goods separately?

a. $4,500

b. $6,300

c. $7,000

d. $6,000

15. What is a better pricing strategy for the monopolist? At this price, what are the total profits to the monopolist?

a. Bundle the goods at $2,800; Profits = $5,600

b. Bundle the goods at $4,000; Profits = $8,000

c. Charge $2,800 for good 1 and charge $1,700 for good 2; Profits = $4,500

d. Charging the lowest price for each good individually is the best pricing strategy; Profits = $7,000

16. The prisoners' dilemma is an example of

a. a sequential game.

b. a simultaneous game.

c.a shirking game.

d. a dating game

17. Nash equilibrium

a. is where one player maximizes his payoff, and the other doesn't.

b. is where each player maximizes his own payoff given the action of the other player.

c.is where both players are maximizing their total payoff.

d. is a unique prediction of the likely outcome of a game.

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ECON 212 Lecture Notes - Lecture 10: Strategic Dominance, Oligopoly, Nash Equilibrium

ECON 212 Full Course Notes

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