ECN 3667 – Strategic Game Theory
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ECN 3667 – Strategic Game Theory
QUESTION 1
Is it always beneficial to keep private information hidden? Explain why or why not.
QUESTION 2
There are three people living on a dead end street. Two are home (Jack and Jill) and the third (Henry) is away. Jack and Jill think Henry is being robbed. If a robbery is stopped they both get a benefit of 10 regardless of who stops the robbery. If the robbery happens they both get a payoff of zero. However, they both think that there is some risk from walking over to the neighbor's house and possibly confronting the thief. The risk subtracts 3 from the payoffs of only those that intervene.
2A. Model this as a sequential game. Draw the extensive form (game tree) of this game. Assume Jack is closer and acts first. Jack and Jill can see each other through the windows and can observe the actions of the other.
2B. In the sequential game identify all Rollback Equilibria and identify the equilibrium payoff(s). 2C. Is there a first/second mover advantage? Explain.
2D. What if they live on opposite sides of Henry and cannot observe what the other has chosen. Model this as a simultaneous game. Write out the strategic form (game table/payoff matrix).
2E. Identify all Nash Equilibria in pure strategies and the respective payoffs.
2F. Is there a mixed strategy Nash Equilibrium in the simultaneous game? If you found any Nash Equilibria in part E compare the expected payoffs to Jack and Jill of the mixed strategy to the pure strategy Nash Equilibrium.
G. In the mixed strategy Nash Equilibrium what is the probability no one intervenes? Discuss what this outcome means and any social implications this might have for public policy
QUESTION 3
A three person Academic Review Board decides the punishment on cases of plagiarism. The bylaws of the committee state that by secret ballot they first determine if the offense warrants a strong or weak punishment. After that they then determine whether the accused should receive the punishment.
First Vote: strong versus weak punishment
Second vote: winner of the first vote versus no punishment.
The three members have different preferences. The first prefers strong sanctions but would rather let the person go free rather than lower the reputation of the school by giving a weak punishment. The second person dislikes sanctions and prefers the lesser of any two options and the third person prefers weak but any punishment is better than none:
Person 1: strong > none > weak
Person 2: none > weak > strong
Person 3: weak > strong > none
3A. If we model this as two simultaneous games played in sequence list the strategies available to player 1 (think carefully about the second round described above)?
3B. Construct the strategic form for the second round one table if strong wins the first vote and another if weak wins the first vote. Use player 1 as the row, player 2 as the column and player 3 as the page.
3C. Based on your answer in part B identify all Nash Equilibria in these two subgames. Are all of these rationalizable? Are there any focal points?
3D. If the judges vote myopically (meaning they pick their favorite from the choices available in each round) what will the outcome of each round of voting be?
3E. Now assume that they have all had game theory and will think ahead and use rollback. How will strategic thinking change the voting in each round and the final outcome? In your discussion use the information from parts B and C to help illustrate your points.
QUESTION 4
Fung Wah is considering adding a new route to Buffalo but they would then be competing with a Greyhound route. Fung Wah estimates that it will cost $200,000 to buy additional equipment for the new route. Greyhound is currently running the route and earning $1,000,000 from the route. Greyhound has the option of fighting off the entry by offering special pricing or they can let Fung Wah in and share the market. If Greyhound fights they earn 500,000 and Fung Wah earns - 100,000. If Greyhound does not fight they earn 600,000 and Fung Wah earns 200,000.
4A. Write out the extensive form (game tree) for this game and identify all Nash Equilibria.
4B. Now assume the game changes and Fung Wah can observe Greyhound's decision to fight or accommodate and Fung Wah can then decide to stay in or exit the market. If they decide to leave the market and Greyhound decided to fight then Greyhound earns 700,000 and Fung Wah loses their 200,000 investment. If Fung Wah leaves and Greyhound had decided to accommodate then Greyhound makes the original $1,000,000 and Fung Wah loses 200,000.
4C. Write out the strategic form (game table) for this game and identify all Nash Equilibria. If there are multiple Nash Equilibria identify the subgame perfect Nash Equilibrium and explain why the others are not subgame perfect.
QUESTION 5
There are only two firms producing natural spring water, Stillwater and Blue Bubbles. The firms draw their water from different springs (Stillwater's water is "still" and Bubbles' is carbonated), so each firm has some "market power." Specifically, the demand functions for the two firms' waters are given by
qS = 30 - 2pS + pB and qB = 15 - 2pB + pS,
where p denotes the price per gallon charged q denotes the resulting number of gallons that will be purchased from the firm. Each firm can produce as much water as it chooses, costlessly. The firms compete with one another via price: each firm chooses a price to charge, taking as given the price the other firm is charging.
5A. Find the best response functions and draw them in a diagram with pS and pB on the axes.
5B. Determine the Bertrand equilibrium prices and quantities -- i.e., the Nash equilibrium when each firm, in making its own pricing decision, takes the other firm's price as given.
5C. How would the outcome in part B change if the two firms created the NSWC (Natural Spring Water Cartel) and they operated as a monopoly?
5D. Would you expect the cartel outcome in part C could be maintained long term? Explain.
2023-02-24