Problem Set #3

PlSc 349
Due: Tuesday, 21 September 2010

  1. [12 points] Using your strategy from the $10 Ultimatum Game in class, write out your payoffs to the extensive form game. (Note: No student had the same strategy, so it is unlikely that any students have the same payoffs.) Show that your strategy is an equilibrium given those payoffs. Propose a general utility function for yourself in which the Ultimatum Game utilities are a special case. Note what social preferences (if any) you are including (e.g. altruism), and show how you incorporate those preferences into the utility function.
  2. [4] Problem U4.1(c) and U4.2(c). Convert the payoffs in the table to the usual format.
  3. [5] Problem U4.3. Use dominance for at least one table, best response for at least one table, and brute force for at least one table.
  4. [8] Problem U4.10.
  5. [10] [loosely from Osborne 2004] Nations face a "security dilemma" whether to arm themselves. This has been framed in a variety of ways. Here are two (payoffs: Nation 1, Nation 2):

    Table A
    Nation 2
    Refrain Arm
    Nation 1 Refrain 3, 3 0, 2
    Arm 2, 0 1, 1

    Table B
    Nation 2
    Refrain Arm
    Nation 1 Refrain 2, 2 0, 3
    Arm 3, 0 1, 1

    Find the (pure-strategy) Nash equilibria of both tables. What type of game is each table? Which table do you think more accurately reflects the "security dilemma"? Why? (Do the Nash equilibria affect your choice?) Find the Nash equilibria if one country has payoffs from Table A and the other country has payoffs from Table B.

  6. [11] [from Geddes 1991 via Morrow 1994, 101-104] "Reform of the civil service has been a recurrent problem in democracies. Patronage is the traditional source for government employees. Victorious parties reward their workers and followers with government jobs. Patronage often leads to inefficiency and corruption, and the demand for reform soon follows. But civil service reform is difficult to achieve after the demand for it arises. The fight to establish merit-based hiring over patronage is often blocked by established political parties. Why is the move to meritocracy difficult to achieve?"

    Table C "displays the incentives of politicians to use patronage, that is, promise their workers jobs, during a campaign." "Each candidate can choose to use patronage in the campaign or not. If neither candidate uses patronage, then Politician 1's chance of winning is p and Politician 2's chance is 1-p. Patronage provides an advantage to the candidate using it. Politician 1's chance of winning rises by v1 if he uses patronage, and Politician 2's chance rises by v2 if she uses it in her campaign. Each candidate's advantage from patronage reduces its opponent's chance of winning." (Payoffs: Politician 1, Politician 2)

    Table C
    Politician 2
    Do Not Use Patronage Use Patronage
    Politician 1 Do Not Use Patronage p, 1-p p-v2, 1-p+v2
    Use Patronage p+v1, 1-p-v1 p+v1-v2, 1-p-v1+v2

    What type of game is this? Find the Nash equilibria. Explain what the result means in layperson's terms.

    "If patronage helps a candidate's chances of winning, why would parties support reforms that would deny them the exercise of patronage? Supporting reform can provide an electoral benefit." Table D "has two actors, a majority party and a minority party. Without the support of the former, civil service reform cannot pass. With that support, it passes. Each party can now support or oppose reform. If the majority party supports reform, reform laws pass, and neither party can use patronage in its campaigns. Each side's chance of winning depends on its underlying chance of winning, p for the majority party and 1-p for the minority party. If the majority party opposes reforms, both parties can and will use patronage in their campaigns, even if they support reform. (Why?) However, there is a benefit to supporting reform when the other party does not, given by e. This benefit arises from advancing reform even if the party goes on to use patronage in its campaign. If both parties support reform, neither gains an electoral advantage from doing so."

    Table D
    Minority Party
    Support Reform Oppose Reform
    Majority party Support Reform p, 1-p p+e, 1-p-e
    Oppose Reform p+v1-v2-e, 1-p-v1+v2+e p+v1-v2, 1-p-v1+v2

    What type of game is this? Find the Nash equilibria. (Examine two cases: v1-v2 < e, and v1-v2 > e.) Explain what the results mean in layperson's terms (including what v1-v2 means).

  7. [for future planning] Which problem was most useful in learning the concepts? Which problem was least useful?

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