Game theory Problem set

Game theory Problem set

Problem set 1

 

Answer the following questions, and upload your answers to Canvas.  You can write your answers in word, or write them by hand and upload a picture or scanned copy as long as it is clearly readable.  You may collaborate with any of your classmates, but make sure that you fully understand how to answer each problem so that you can do a similar problem on the exams.

 

Part 1:  Rationality

For each of the following situations, determine whether the actor is behaving rationally or not, according to the definition of rationality we are using in this course.  Briefly explain your determination.  If the actor is or could be rational, explain the value system that would make their decision rational.  If the actor is not rational, state the rationality criterion that is violated.

  1. Athens wanted to incorporate Melos (an ally of Sparta) into the Athenian empire. The Athenians demanded that Melos surrender and pay annual tribute, while still retaining control of their internal affairs. However, if Melos refused to surrender, Athens promised to kill all the men and enslave the women and children once they had conquered Melos.  Athens was much more powerful than Melos, so an Athenian victory was essentially certain.  However, the Melians did refuse to surrender, were conquered and killed or enslaved accordingly.  Did the Melians behave irrationally?  Why or why not?
  2. Some have argued that many Republican voters are tricked or irrational, because they would benefit from many of the economic policies of the Democratic party (e.g. Thomas Frank, “What’s the Matter with Kansas?”). According to this argument, many Republican voters do not benefit significantly from the tax cuts favored by the Republican party, and would benefit from additional social spending on health care and welfare. However, they vote for the Repbulican party anyway based on cultural issues, such as opposition to LGBTQ rights and abortion, and support for gun rights.  Assess this argument.  Leaving aside any personal value judgements on the best policies, are these voters necessarily irrational?  Why or why not?
  3. Caledonia is considering three proposals for health care reform. The first option is a completely private health care system, where everyone is responsible for buying their own insurance and if they can’t afford it they are out of luck. The second option is a completely public health care system, such that the government pays for health care for everyone, and directly runs the hospitals and pays the doctors.  The third option is a mixed system, where everyone gets private insurance but the government provides sufficient subsidies that everyone can afford insurance.  A Member of Parliament has the following preferences.  She prefers the private health care system (option 1) to universal health care (option 2).  She prefers the mixed system (option 3) to a completely private system (option 1).  She prefers universal health care (option 2) to the mixed system (option 3).  Is this Member of Parliament irrational?  Why or why not?
  4. The country of Comoros is considering entering a trade agreement. They have fully analyzed the situation, and believe that the trade agreement is in their best interest. In other words, they prefer to enter the trade agreement over staying out of the trade agreement.  In the two months that it takes to negotiate their entry (which they expected), Comoros decides to stay out of the trade agreement.  The situation remains unchanged, and no new information came to light.  In other words, the situation is exactly as it was when Comoros first considered entering the trade agreement.  Is Comoros’ behavior irrational?  Why or why not?

 

Part 2:  Expected utility

In the following problems, you will be required to calculate the expected utility of various actions.  Make sure to show your work, so that partial credit can be awarded if you make minor mistakes

  1. Claire is thinking about whether to run for a seat in the Scottish Parliament. She believes that she has a 60% chance of winning the election. She believes that gaining a seat would gain her 100 utility relative to not holding the seat.  However, the effort and costs of campaigning would cost her 20 utility, which she must pay regardless if she wins or loses.  What is Claire’s expected utility for running in the election?  Should Claire run for election?
  2. The country of Brittany is considering going to war against Aquitaine. They believe they have a 25% chance of winning, a 45% chance of a draw and 30% chance of losing. Winning would gain them 50 utility relative to the status quo, a draw would retain the status quo (0 utility), and losing would lead to a loss of 60 utility.  Regardless of the outcome, they would also have to pay 20 utility for the costs of the war.  What is Brittany’s expected utility of going to war?  Should Brittany go to war?
  3. Assume that there is some deadly virus that has been going around, which kills 0.5% of those it infects. A vaccine against this virus has recently been developed. The vaccine will not change the chances of getting infected, but has 80% efficacy at preventing serious illness and death.  Efficacy is defined as , so the risk of dying of a vaccinated individual is the (1-efficacy) multiplied by the risk of dying of an unvaccinated.  However, the vaccine kills 1 out of 1,000,000 people that take it.  Further, assume that your chances of encountering the virus sufficient to contract it are 50%.  What is the expected utility of taking the vaccine?  What is the expected utility of not taking the vaccine?  (Assume that living gives you 1 utility and dying 0).  Should you take the vaccine?
  4. Assume all the facts from the previous problem. Except, it is unclear how dangerous the vaccine is. Assume the vaccine kills with probability p.  What would the value of p have to be for a rational person to be indifferent between taking the vaccine and not taking the vaccine?  (Another way to think of this is: what would the minimum value of p be such that it would no longer make sense to take the vaccine?)

 

Part 3:  Spatial preferences

  1. Samuel and Deborah are running for provincial governor in a developing country that does not yet have widespread access to COVID vaccines. Obviously, everyone’s main concern is what precautions to take against COVID. For simplicities sake, assume there are 100 voters, with the following positions
    • 5 favor no restrictions of any type
    • 12 favor mandating masks in public, but no other restrictions
    • 18 favor mask mandates + limiting the number of people that can be in businesses or events at once
    • 13 favor mask mandates, limited occupancy, and restricting travel in and out of the province
    • 35 favor a partial shutdown, with restaurants and bars closed but all other businesses allowed to open with limited occupancy
    • 17 favor a complete shutdown until the virus is under control, such that only grocery stores and medical facilities are open

Both candidates have perfect information about the preferences of the electorate.  The candidate to get the most votes wins, and Samuel and Deborah are the only candidates in the election.  To maximize their chances of getting elected, which option should Samuel and Deborah support?  Explain your reasoning.

 

  1. Bajor is considering how many refugees to accept. The procedure is that the Prime Minister makes a proposal for how many refugees to accept, and then the Parliament (60 members) approves or disapproves this number. Currently, Bajor accepts 20,000 refugees annually.  If no agreement is reached (i.e. Parliament rejects the Minister’s proposal), this number will remain.  The actors have the following preferences:
    • The Prime Minister wants to accept 35,000 refugees
    • 8 Members of Parliament want to accept 0 refugees
    • 14 Members of Parliament want to accept 10,000 refugees
    • 11 Members of Parliament want to accept 15,000 refugees
    • 18 Members of Parliament want to accept 25,000 refugees
    • 7 Members of Parliament want to accept 30,000 refugees
    • 2 Members of Parliament want to accept 50,000 refugees

Assume that all actors have single peaked and symmetric preferences, such that they prefer any outcome closer to their ideal point to one further away.  How many refugees will the Prime Minister propose accepting?  Will Parliament accept or reject this proposal?  How many refugees will Bajor accept?

 

  1. Consider the same problem, but if Parliament rejects the Prime Minister’s proposal, assume that no refugees will be accepted. How many refugees will the Prime Minister propose accepting? Will Parliament accept or reject this proposal? How many refugees will Bajor accept?
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