1
Political Economy Qualifying Exam
Summer 2019
There are three questions, worth a total of 10 for this section. Do not consult any books or notes. Partial
credit will be given for incomplete answers, but no credit will be given for purely intuitive answers.
Consider a game between a political leader and citizens.
(A, 2 points) Find all Nash equilibria of the following normal-form game. (This requires to indicate some
strategy profiles, to prove that these are Nash equilibria, and to prove that there are no other Nash
equilibria.)
Citizens
Innovative
production
Subsistence
Leader
No corruption
4, 9
3, 4
Corruption
5, 0
4, 2
Now, suppose that the game is played repeatedly, and the two players have the common discount factor ๐ฟ๐ฟ.
The one-stage payoffs of the repeated game are given in the table above.
Solution: For Leader, No corruption is a strictly dominated strategy, which cannot be a part of any Nash
equilibrium. With its exclusion, Innovative production is a strictly dominated strategy of Citizens. The only
Nash equilibrium is (Corruption, Subsistence).
(B, 4 points) Describe two subgame perfect Nash equilibria of the game that have different life-time payoffs
for the players; you can choose the discount factor yourself. In each case, you need to fully describe
strategy profiles that constitute an equilibrium and then prove that this is indeed a subgame perfect Nash
equilibrium. Does one of the two equilibria Pareto-dominates the other?
Solution: (Corruption, Subsistence) is the first SPNE for any discount factor. For the second example,
consider the following strategy profile. Leader alternates No corruption and Corruption, while Citizens play
Innovative production. If any of the players deviates, they play (Corruption, Subsistence) in the resulting
subgame. Denoting the discount factor ๐ฟ๐ฟ, Leaderโs life-time payoff is 4+5๐ฟ๐ฟ
1โ๐ฟ๐ฟ2, and the Citizensโ - 9
1โ๐ฟ๐ฟ2. The
Leader deviates in period 1 unless ๐ฟ๐ฟ โฅ โ5โ1
2; Citizens deviate in period 2 unless ๐ฟ๐ฟ โฅ โ301โ7
14 . Thus, for any ๐ฟ๐ฟ
that satisfies the second inequality, the described profile is an SPNE.
Finally, suppose that there are two possible states of the world, โnormal timesโ and โrevolutionโ. The
normal times are described by the table above; the payoffs for โrevolutionโ are in the table below. In
normal times, if the leader chooses corruption, there is some probability ๐๐ that the next state will be
โrevolutionโ, which is an absorbing state.
(C, 4 points) Find all (pure-strategy) Markov perfect equilibria for all possible values of ๐ฟ๐ฟ and ๐๐.
