Download EXERCISE 9: GAME THEORY AND OLIGOPOLY

MICROECONOMICS 2
TUTORIAL 9: GAME THEORY AND DUOPOLY
Timing of Tutorial This chapter reinforces the material in chapters 30 and 31 of the
text.
Purpose of Tutorial To learn through experience - by allowing you to play a game
(and its economic application) and hence understand the power (and limitations) of
the theory.
Prior Preparation Before the tutorial, each of you individually should look at the two
halves of this tutorial (Game Theory and Duopoly) and decide what you would do as
either of the Teams. In the tutorial proper your tutor will divide the tutorial group into
two Teams; and will give both teams 5 minutes at the start of each half to decide on
what they are going to do.
Written Work after Tutorial None.
Relevance to Examination The examination will ask you to analyse, different kinds
of economic games, all being 2 person games. This tutorial examines a particularly
important class of games, referred to as prisoner’s dilemma games. You should
understand the basic structure of this type of game and the relevance to the economic
theory of duopoly. There will also be a question on duopoly.
In this tutorial, like Tutorial 2, the tutorial group will be divided into two subgroups, or Teams. These Teams will play each other in two contests - one a
straightforward Game, the second a Duopoly Problem. Your tutor will act as the
Umpire.
Game Theory
Here is a payoff matrix. The amounts are in (hypothetical) pounds. The first
number is the payoff to Team 1; the second number is the payoff to Team 2. So, for
example, if Team 1 plays 2 and Team 2 plays 1, then Team 1 will get a payoff of -£2
(yes, will lose £2) and Team 2 will get a payoff of £20.
Team 2's choice
Team 1's choice
Payoffs
1
2
1
£1,£1
£20,-£2
2
-£2,£20
£6,£6
YOU SHOULD TRY AND MAXIMISE YOUR PAYOFF FROM PLAYING THIS
GAME.
There are various variants that you can play on this game:
 change the payoffs while keeping the structure of a Prisoner’s Dilemma;
 play it once without or with communication;
 play it a fixed number of times without or with communication;
 play it a random number of times without or with communication.
Depending upon how your tutorial is going your tutor may introduce some of these
variations – as well as similar variations in the Duopoly problem below. As far as the
repeated variations are concerned, an obvious tool is backward induction. This is
another tool used by economists. You might like to discuss this tool in the context of
three different ‘games’:

The Unexpected Hanging Paradox:
follows:
Wikipedia describes this paradox as
“A judge tells a condemned prisoner that he will be hanged at noon on one weekday
in the following week but that the execution will be a surprise to the prisoner. He will
not know the day of the hanging until the executioner knocks on his cell door at noon
that day. Having reflected on his sentence, the prisoner draws the conclusion that he
will escape from the hanging. His reasoning is in several parts. He begins by
concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged
by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on
Friday. Since the judge's sentence stipulated that the hanging would be a surprise to
him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging
cannot be on Thursday either, because Friday has already been eliminated and if he
hasn't been hanged by Wednesday night, the hanging must occur on Thursday,
making a Thursday hanging not a surprise either. By similar reasoning he concludes
that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he
retires to his cell confident that the hanging will not occur at all. The next week, the
executioner knocks on the prisoner's door at noon on Wednesday — which, despite all
the above, was an utter surprise to him. Everything the judge said came true.”

The Centipede Game: Here is an example lifted from Wikipedia

The Beauty Contest: Here is the statement of a related game described by
Nagel in a paper in the American Economic Review in 1995:
“Consider the following game: a large number of players have to state simultaneously
a number in the closed interval [0,100]. The winner is the person whose chosen
number is closest to the mean of all the numbers chosen multiplied by a parameter p,
where p is a predetermined positive parameter of the game: p is common knowledge.
The payoff to the winner is a fixed amount, which is independent of the chosen
number and p. If there is a tie, the prize is divided equally among the winners. The
other players whose chosen numbers are further away receive nothing.”
Dominic Spengler also suggests that you may find amusing the following game, both
played once and several times:
Player 1
T
C
B
L
£3,£3
£4,£0
£0,£0
Player 2
M
£0,£4
£1,£1
£0,£0
R
£0,£0
£0,£0
50p,50p
He writes “Played once, the NEq is {C,M}. Played a finite amount of times with t > 3
the social optimum {T,L} is enforceable, because players can punish by playing {B}
or {R} respectively. The amount of rounds needed for enforcement can obviously be
varied by varying the respective payoffs.”
Duopoly
Each Team is Duopolist. Each Duopolist has constant marginal and average
costs of 10p per unit. The aggregate demand curve for the duopolists’ product is
p = 100 - (q1 + q2 )
where p is the market price (in pence) and q1 and q2 are the outputs of Teams 1 and 2
respectively. Teams must decide on their q’s; the Umpire works out p using the
formula above.
So, for example, if q1 = 20 and q2 = 30 then p = 50 and revenues to Teams 1
and 2 are respectively 50 x 20 = 1000 (= £10) and 50 x 30 = 1500 (= £15). Costs are
respectively 10 x 20 = 200 (= £2) and 10 x 30 = 300 (= £3), so profits are £8 and £12
respectively.
YOUR TEAM SHOULD TRY TO MAXIMISE ITS PROFITS.