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Microeconomics FHS Problem Set 4 Sketch Solutions
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Official sketch solutions for select questions in Problem Set 4: Static & Dynamic Games
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UNIVERSITY OF OXFORD Last updated: September 16, 2013Department of Economics
Undergraduate - Microeconomics
Tutorial/Class 4: Static & Dynamic Games
Sketch solutions
FOR USE BY TUTORS ONLY
DO NOT CIRCULATE, ESPECIALLY TO STUDENTS
, Problems and Multipart Questions
1. 2x2 Cournot game; Dominant Strategies
The two firms in an industry have identical cost functions c(qk ) = 2qk , k = 1, 2, and
the inverse demand function in the industry is p(q) = 14 q, where q = q1 + q2 . The
joint-monopoly quantity is q M = 6 and the Cournot quantity is q C = 4. (Verify this if
you wish.) Each firm can choose to produce either 21 q M = 3 (half the joint-monopoly
quantity) or q C = 4 (the Cournot quantity). Construct the one-shot simultaneous-
move game corresponding to this set-up and show that each firm has a dominant
strategy.
The strategic form of the game is:
1 M
2q qC
1 M 18 20
2q 18 15
15 16
qC
20 16
This is a ‘‘Prisoner’s Dilemma’’ where q C is dominant for each player.
2. Pure and Mixed Strategies
Identify the Nash Equilibria in pure strategies of the following game:
Left Right
3 1
Up
5 1
0 5
Down
0 3
Is there a Nash Equilibrium in mixed strategies, where Column plays Right w.p. p
and Row plays Up w.p. q? If so, what is each player’s expected payo↵?
Is this better or worse than a player’s average payo↵ across the two pure-strategy Nash
Equilibria? Why?
This is the ‘‘Battle of the Sexes’’.
The pure-strategy Nash Equilibria are (U, L) and (D, R).
Each player’s average payoff across these two equilibria is 4.
The mixed-strategy Nash Equilibrium is (U w.p. q, R w.p. p), where p = q = 57 .
Each player’s expected payoff in this equilibrium is 157 .
15
Clearly, 7 < 4. This is because when players mix, there is a sizable chance of
mis-coordination, e.g. they play (U, R) w.p. 25
49 , i.e. just over half the time.
1
1
Maybe draw a diagram showing best responses/reaction curves: with 0 p 1 on the horizontal
axis and 0 q 1 on the vertical axis, Row’s reaction curve, for example, joins (0, 1) to ( 57 , 1) to ( 57 , 0)
to (1, 0) with straight lines. The reaction curves intersect at ( 57 , 57 ), and also at (0, 1) and (1, 0) – the
pure-strategy equilibria.
1
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