April 10th , 2015
ECON 451 • Spring 2015
Econ 451 - Homework #4
Due: April 21st , 2015
1. For the following game:
C
D
C
3 3
0 5
D
5 0
1 1
Consider the following set of strategies:
• ALL C - Play C always, regardless of other player’s actions.
• ALL D - Play D always, regardless of other player’s actions.
• Alternate CD (Alt) - Plays C in odd periods and D in even periods, regardless of other player’s actions.
• Grim Trigger (GT) - Play C until other plays D, then D forever.
• Tit-for-tat (TFT) - Start with C, then copy other player’s action from previous period.
• Win Stay, Lose Shift (WSLS) - Start with C, then copy my action from previous period if the other person
plays C, and switch actions if the other person plays D.
• Suspicious Tit-for-tat (STFT) - Start with D, then copy other player’s action from previous period.
• Suspicious Win Stay, Lose Shift (SWSLS) - Start with D, then copy my action from previous period if the
other person plays C, and switch actions if the other person plays D.
a) Make an 8x8 table that displays the (non-discounted) payoff for player 1 in every combination of the above strategies.
For example, if both players play All C, then the payoff will be 600. Another example, if player 1 plays WSLS and
player 2 plays SWSLS, then player 1 receives 595 and player 2 receives 600.
b) Determine the rankings in a tournament like that run by Axelrod, where
• Each strategy plays every other strategy (including itself) one time for 200 periods.
• The winner of the tournament is the strategy that has the highest payoff summed across all eight matches (when
matched against the 7 other strategies and when matched against itself).
c) Determine the rankings in a variation (this variation is like sports league where you either win, lose or tie) of the
tournament like that run by Axelrod, where
• Each strategy plays every other strategy (including itself) one time for 200 periods.
• In an interaction between two strategies, the one that has the higher payoff gets a score of 1 (because they win),
the strategy that has the lower payoff gets a score of 0 (because they lose), and if they have the same payoff,
then the two strategies each get a payoff of 0.5.
• The winner of the tournament is the strategy that has the highest combined score in all eight matches (most
number of wins).
2. Problem removed, will discuss in class.
3. Consider the following duopoly where firms choose their quantities (qi ∈ [0, ∞)) simultaneously. Firm 1 has a cost of
L
c1 (q1 ) = 20q1 and firm 2 knows this cost. Firm 2 has either a cost of cH
2 (q2 ) = 30q2 or a low cost of c2 (q2 ) = 15q2 .
Firm 2 knows their cost, but firm 1 doesn’t know firm 2’s cost, but believes that firm 2 has a high cost with probability
1/3 and a low cost with probability 2/3. The market demand curve is,
(
120 − 4Q Q < 30
P (Q) =
0
otherwise
Page 1/2
April 10th , 2015
ECON 451 • Spring 2015
a) Find the Nash equilibrium of the Bayesian game.
b) Determine the profits in equilibrium.
4. Consider the independent private value auction setting that we talked about in class. Assume that there are a total of
n bidders (including yourself). For each of the following bidding functions answer the following:
• Assuming that everyone else plays the proposed bidding function, what do you want to do?
• Is the proposed bidding function a Nash equilibrium?
a) First price auction - highest bidder gets the item and pays their bid, everyone else pays nothing and gets nothing.
•
•
•
•
•
b(v) = v
b(v) = 2v
b(v) = v2
b(v) = n−1
n v
n
b(v) = n−1
n v
b) All pay auction - highest bidder gets the item and everyone pays their bid.
•
•
•
•
•
b(v) = v
b(v) = 2v
b(v) = v2
b(v) = n−1
n v
n
b(v) = n−1
n v
Page 2/2