Reference no: EM132311719

Assignment

1. Chicken: In the classic game, “Chicken,” two angry youths drive their cars at one another in a foolish attempt to prove who is tougher. The two players have a choice of either staying or swerving. If both stay, they will collide head- on, in the worst possible outcome. If they both swerve, they will be seen as quitters, as lose the esteem of their peers. However, quitting is not as bad as losing, which happens when one player swerves while the other (the winner) stays.

Let’s examine a general version of Chicken with the payoff matrix below:
The payoffs are C < L < Q < W (collision is worse than losing, which is worse than quitting, which is worse than winning).

What are the pure strategy Nash equilibria to this game, if any exist? If one does not exist, show that instead.

What is the mixed strategy Nash equilibrium to this game?

What are the players’ expected payoffs in the mixed strategy Nash equilibrium?

How does the probability of a head-on collision -- outcome (C, C) – change as

W increases?

L decreases?

C decreases?

Are any of the results from part (d) interesting or counterintuitive? Discuss them.

2. The Battle of the Bismarck Sea is named for a part of the Southwestern Pacific Ocean separating the Bismarck Archipelago from Papua-New Guinea. The following scenario describes an actual World War II naval engagement between the United States and Japan.

In 1943, a Japanese admiral was ordered to move a convoy of ships to New Guinea. He had to choose between a rainy northern route and a sunnier southern route, both of which required 3 days’ sailing time. The Americans knew the convoy would sail, and wanted to send bombers to attack the convoy, but they did not know in advance which route it would take.

The Americans had scout planes to look for the convoy, but had only enough planes to scout one route (northern or southern) at a time. Both the Americans and Japanese had to make their decisions over which route to take and which to scout without any knowledge of the others’ decision. In other words, they had to movesimultaneously.

If the convoy was on the route selected by the Americans, they could begin bombing right away. If not, they lost a day of bombing. Poor weather on the northern route would also hamper bombing. If the Americans explored the northern route and found the Japanese right away, they could expect to have only 2 days to bomb the Japanese (out of 3).

If they explored the northern route, and then found that the Japanese had chosen the southern route, they could also only expect 2 days of bombing. If the Americans chose to explore the southern route, and found the Japanese there, they could bomb for three days, but only 1 day if the Japanese had actually taken the northern route.

a. Express this scenario as a normal form game. In other words, specify the players, their actions, and the payoffs associated with any outcomes. Hints: Making a table is the way to go here. You should characterize the Americans’ utility as the number of days they get to bomb the convoy. You should characterize the Japanese admiral’s utility as the number of days the convoy isn’t being bombed.

b. Find the pure strategy Nash equilibrium, or equilibria if there are more than one, to this game.

c. Show that this game does not have a valid mixed strategy Nash equilibrium. One way you can do this is by solving for the mixed strategy equilibrium and then explaining why the solution you find does not fit with the definition of a mixed strategy equilibrium.

3. Private Beach Problem: N property owners (where N is greater than or equal to 2) hold the rights to N parcels of beachfront land. The local government has recently allowed the property owners to privatize access to their beachfront land. All of the owners simultaneously decide how much of their land to privatize. More formally, owners can choose the level of p, which can take on any value between 0 and 1 (inclusive), on privatization efforts.

For example, when pi = 0, that means the owner is not attempting to privatize their land, when pi = 1, the owner is privatizing everything they can privatize, and when p = 0.25, the owner is privatizing a quarter of what they can privatize, and so on. The utility functions (payoffs) for owners are as follows:

??" =1−3??¯+2??"

Where ??¯ is the average privatization level of the population of all N players

and pi is player i’s individual privatization effort.

Find the pure strategy Nash equilibrium of this game when there are just 2

property owners (N = 2).

Show that the Nash equilibrium you found in part a remains the same if N is instead a fairly large number of property owners (like 100 or more).

How do the payoffs that the property owners receive at equilibrium differ from the payoffs they would get if the government did not open the beachfront to privatization?

4. The Terrorist Hunt: The CIA and the FBI must often work together in order to catch suspected terrorists. Both agencies can pursue either the terrorist Leader or a terroristOperative. They can only capture the terrorist Leader if they work together and share intelligence, but either agency can capture an Operative independently. If one agency pursues the Leader without the other’s help, they will fail to make an arrest. The CIA prefers capturing the terrorist Leader to capturing an Operative, and prefers capturing an Operative to making no arrests at all.

However, the FBI director has recently been dismissed by the President, and three deputy directors are vying for control of the Bureau. Each of the three deputies has a different vision for how the Bureau should proceed.

Deputy Director Smith has the same priorities as the CIA. Smith prefers capturing the terrorist Leader to taking down an Operative, but is would prefer to arrest anOperative to making no arrest.

Deputy Director Jones first and only priority is to capture a terrorist Leader. Jones receives the same amount of satisfaction from capturing an Operative as from making no arrest at all – there is no middle option.

Deputy Director Wilson believes that terrorist bosses fail to provide as much valuable intelligence as their underlings. Wilson prefers to capture an Operativethe most, but prefers capturing a Leader to making no arrest at all.

The CIA and the FBI must choose to pursue either a Leader or an Operative at the same time. However, the CIA does not know which deputy will be supervising the mission for the FBI. The CIA only knows that each deputy has a 1/3 probability of calling the shots on the day of the mission.

Express this as a game with incomplete information. Hints: You should start by making one or more tables. For each player, you should assign a utility of 2 to their most preferred outcome, a utility of 1 to their next most preferred outcome (if one exists), and a utility of 0 to their least preferred outcome or to making no arrest at all.

Find the Bayesian Nash equilibrium (or equilibria if there exists more than one) of this Terrorist Hunt Game.

5. Lord Whitney’s Estate: Lord Whitney, a renowned art collector, is on his deathbed. Since he knows his end is near, he summons his two rivalrous daughters, Abigail and Bethany, to his side in order to tell them his last wishes. His instructions are as follows:

“My beloved daughters, the time has come for you to inherit the art I have spent my life gathering and curating. My collection stands at 100 pieces. Each individual piece is worth the same amount of money and no piece can be destroyed or split.

Abigail, as my first-born, I know you are capable of making difficult decisions. It is your responsibility to divide my entire collection into two galleries. You may divide the art into two galleries however you see it, placing as many as 100 or as few as 0 into one gallery, and placing the remaining pieces in the other gallery.

Bethany, you have a keen eye for beauty, so you will then face a choice of your own. You will decide which gallery you want as your own, and which gallery will belong to your sister.”

In short, first Abigail will divide the collection between the two galleries (every piece must be allocated), and then Bethany will decide which allocation goes to herself and which to give to her sister.

Model this interaction between Lord Whitney’s daughters, Abigail and Bethany, as an extensive-form game. Drawing a tree-like diagram may be helpful here.

Solve for the subgame perfect Nash equilibrium of this game. Be sure to show that there is just one equilibrium.

Lord Whitney’s dying wish was to see his daughters put aside their rivalry and cooperate. Does this help accomplish that? Why or why not?