(10%). Find and write down all Nash equilibria in the following game (payoffs on
the left correspond to Player 1 and payoffs in the right correspond to Player 2):
Player 1,Player 2 E F G
A 5,6 3,7 0,4
B 8,3 3,1 5,2
C 7,5 4,4 5,6
D 3,5 7,5 3,3
2 (20%). Player 1 faces two options: A or B. If it chooses A, it obtains a payoff of 2.5.
If it chooses B, player 1 will play the following game in pure strategies with player 2
(payoffs on the left correspond to Player 1 and payoffs in the right correspond to
Player 2):
Player 1,Player 2 L R
U 3,4 1,1
D 2,3 2,2
With this information in hand, what is Player’s 1 best strategy: A or B and why?
3. Consider 2 identical players (i.e. i = 1, 2) with utility function:
πi = b(qi + q-i) – cqi.
Where qi is equal to one if player i contributes to the provision of a public good and
zero if she does not, q-i is the sum of the contributions by all other players, b is the
constant marginal benefit of contributing to the public good, and c is the cost of
contributing to the good. Assume that c = 1.
The players have two possible actions: to cooperate to the public good (C), or not to
cooperate to the public good (NC). The players choose actions simultaneously and
only one time.
a) Write down the game in strategic form (20%).
b) Suppose that player 2 will always play C (ie, will always cooperate). For what
values of “b” will player 1 play C? Show your calculations (20%).
4 (30%). Jesse (2009) considers an ideal point model where the probability of a ‘yea’
vote by actor I on proposal j is p(yij=1|γ, α, x)=Φ(γj xi- αj), where xi is a voter’s ideal
point and αj and γj are vote-specific difficulty and discrimination parameters. Here,
Φ(.) is the cumulative density function (CDF) of the normal distribution, with
associated probability density function (PDF) ϕ(.). Jesse extends this ideal point
model to consider the effect of political information where a factor φi is a function of
a voter’s information, denoted PolInfoi.
In lecture we explored the effect of a change in a voter’s ideal point on the
probability of a ‘yea’ vote; we mentioned that this effect is also known as a marginal
effect. Now assume that xi, αj, and γj are constants. Write down the marginal effect
of PolInfoi on the extended ideal point model, also known as a heteroskedastic ideal
point model.