Transcript Game Theory and Its Application in Managerial Economics

MANAGERIAL ECONOMICS
An Analysis of Business Issues
Howard Davies
and Pun-Lee Lam
Published by FT Prentice Hall
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Chapter 13:
Game Theory and Its Application
in Managerial Economics
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Objectives

On completion of this chapter you should:
– understand the place of game theory in
Economics
– be able to represent and solve simple games
– apply game theory to the issue of collusion
– model Cournot, Bertrand and von Stackelberg
competition
– be able to take a game-theoretic approach to entry
deterrence
– appreciate the limits of game theory
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A ‘Paradigm Shift’ Bringing Industrial
Economics and Managerial Economics
Together
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The ‘old’ Industrial Economics used the
Structure-Conduct-Performance paradigm
dependent variable is the performance/
profitability of a sector
– performance determined by structure; with high entry barriers, high
concentration and high product differentiation profits will be high
– cross- sector multiple regressions the dominant empirical technique

behaviour of individual firms (conduct) largely
implicit
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A ‘Paradigm Shift’ Bringing Industrial
Economics and Managerial Economics
Together

Industrial Organization now dominated by
game theory
– focus is on what happens within an oligopolistic
industry, not on differences across industries
– decisions taken by individual firms have become
the centrepiece of analysis
– case studies of firms’ conduct have become the
dominant empirical method
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Basic Concepts in Game Theory
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Warning! Game theory is difficult and can involve highly complex
chains of reasoning
A game is any situation involving interdependence amongst
‘players’
Many different types of game:
–
–
–
–
co-operative versus non-co-operative (which is the main focus)
zero-sum, non-zero-sum
simultaneous or sequential
one-off versus repeated
• repeated a known number of times, infinite number of times or an
unknown but finite number of times
– continuous versus discrete pay-offs
– complete versus asymmetric information
– Prisoners’ Dilemma, assurance games, chicken games,
evolutionary games
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Representing and Solving Games
A Simultaneous Game in Strategic Form:
The Payoff Matrix
Company B’s
Actions
High Price
Low Price
Company A’s Actions
High Price
Low Price
100A,100B
120A, -20B
-20A,120B
50A,50B
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Representing and Solving Games

The same game in sequential form
High Price
100A, 100B
Company A
High Price
120A, -20B
Low Price
Company B
High Price
-20A, 120B
Low Price
Company A
50A, 50B
Low Price
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Representing and Solving Games

Use rollback or backward induction to solve sequential games
– if Company B has set a high price then A chooses a low price: the
‘B high/A high’ branch can be pruned
– if B set a low price then A will choose a low price: the ‘B low/ A high’
branch can be pruned
– B is therefore choosing between a high price (-20) and a low price
(50): it chooses low price
– A will also choose a low price

For simultaneous games search for dominant strategies (ones
which will be preferred whatever the rival does)
– A prefers a low price if B sets a high price and a low price if B sets
a low price: low price is a dominant strategy
– low price is also dominant for B
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This Simple Example Illustrates a
Number of Key Ideas
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Nash Equilibrium - ‘a set of strategies such that each
is best for each player, given that the others are
playing their own equilibrium strategies’
Rollback and Dominant strategies
The Prisoners’ Dilemma
– an important class of game where players can choose
between co-operating or cheating/defecting
– the best outcome for both is co-operation but the ‘natural’
result is cheating
– collusion is a key example
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How to Find Nash Equilibria?
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1.Note a distinction between pure strategies - the players
choose one or other ‘move’ with certainty - and mixed strategies
- the players choose ‘high price with a 60% probability and low
price with a 40% probability’. Mixed strategies are too complex
to deal with here: see Dixit and Sneath (1999) for a nontechnical introduction
For pure strategies
– look for dominant strategies
– if dominant strategies are not to be found, look for dominated
strategies and delete them
– for zero sum games use the minimax criterion: pick the strategy for
each player for which the worst outcome is the least worst
– cell-by-cell inspection: examine every cell and for each one ask
(does either player wish to move from this cell, given what the other
has done?)
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For example
No Dominant Strategies: Eliminate Dominated Strategies
Company B’s
Actions
High Price
Medium Price
Low Price
Company A’s Actions
High Price
Medium Price
100A,100B
120A, 65B
65 A, 120B
80A,80B
65A,60B
55A, 60B
Low Price
60A, 65B
60A, 55B
50A,50B
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How Many Nash Equilibria?
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There may be no Nash equilibria in pure
strategies
There may be more than one Nash
equilibrium
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Collusion
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The Prisoner’s Dilemma game, leading to low
price/low price illustrates a problem for firms trying to
collude. But managers will recognise the problem and
try to find ways round it. How can the dilemma be
escaped?
– Repetition
– Punishments and rewards
– Leadership
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Repetition and the Prisoners’ Dilemma
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If the game is repeated an infinite number of times there are
very clearly advantages in co-operating and players can
observe each others’ behaviour, hence co-operation may be
established. THE DILEMMA HAS BEEN ESCAPED
BUT the logic of this depends on the number of rounds of the
game being infinite. If the number of rounds is finite, in the last
round there is no longer any incentive to co-operate. Hence
cheating will take place in the last round and therefore in the
round before….and so on. THE DILEMMA RE-APPEARS
COMMONSENSE suggests that solutions are found to this
problem, and a good deal of evidence supports that view,but the
basic insight remains.
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Contingent Strategies in Repeated Games

Contingent strategies are strategies whereby the
actions taken in repeated games depend upon
actions taken by rivals in the last round
– grim strategy: co-operate until the rival defects and then
defect forever; this is ‘too unforgiving’ - one mistake and the
prospect of collusion is gone forever
– tit-for-tat: co-operate when the rival co-operated in the last
round, defect if they defected; this strategy makes
permanent cheating unprofitable and out-performs most
others in simulations and experiments
• in terms of the original game, defecting in every round under titfor-tat leads to profit of 120,50,50,50,50, whereas not defecting
leads to 100,100,100,100. Choice of defect is only rational at a
very high discount rate
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Contingent Strategies in Repeated Games
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If the game is not repeated an infinite number of
times, but there is a probability ‘p’ of another round
the analysis can follow the same logic but adjust the
discount rate so that $1 accruing two periods in the
future is discounted by p/(1+r)2 instead of 1/(1+r)2 .
If ‘p’ is relatively low, cheating becomes relatively
more profitable, because the future gains from cooperation become less.
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Penalties and Rewards to Support
Collusion
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Introducing penalties and rewards changes the payoff structure
Firms may ‘punish’ themselves in order to set up
structures which assist collusion
– e.g.’I will match any lower price set by my competitor’ or
looks like a highly competitive move, but it produces a payoff structure in which collusion is the outcome
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Penalties and rewards might come from other
sources - consumers or the law might punish
collusion - trade associations might punish those who
defect from the collusion
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Leadership to Support Collusion
Leadership as a Solution
Company B’s
Actions
High Price
Low Price
Company A’s Actions
High Price
Low Price
100A,300B
120A, 120B
-20A,280B
50A,50B
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Leadership to Support Collusion
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If one firm has much larger pay-offs than the other it
may suit it to charge the higher price even if the rival
charges a lower price - see the example
Furthermore, the large firm may increase overall
profits by making side-payments to rivals
Saudi Arabia in the oil market?
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Cournot and Bertrand Competition
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From historical curiosities to ‘analytical workhorses’
Cournot competition
– two firms, identical products, firms choose output levels
– each firm’s profit-maximising output depends on the other
firm’s output; hence each firm has a reaction function
– as each firm will operate on its reaction function the point
where they cross is the Cournot Nash equilibrium
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Bertrand competition
– two firms, identical products, firms choose price levels
– price is forced down to marginal cost
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Von Stackelberg equilibrium
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A ‘Leader’ and a ‘Follower’
The Leader chooses a price and the Follower then
chooses the price that suits it best, given what the
Leader has done
A sequential game and the solution is found by
working backwards. The Leader maximises his profit
(which depends on what the Follower does in
response to what the Leader does) by taking into
account what the Follower will do
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Entry Deterrence
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A major area of application for game
theory
“Firms may deter entry by threatening
retaliation”
– in particular firms may build excess
capacity in order to deter entry by
indicating that they will cut price and
increase output if entry takes place
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Entry Deterrence
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But is the threat credible?
If it is more profitable for the incumbent to allow entry
and share the market the threat is not credible
However, if the incumbent can make commitments
which effectively force him to fight the entrant. For
instance, if the incumbent installs excess capacity so
that his cost per unit rises if market share is given up
to an entrant, he is forced to fight: see Figure 13.7
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Airbus and Boeing: No
Government Intervention
Boeing and Airbus Produce a Super-Jumbo:
No Government Intervention
Boeing
Develop SuperJumbo
Stay Out
Airbus
Develop SuperJumbo
-100A,-100B
500A,0B
Stay Out
0A, 500B
0A,0B
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Airbus and Boeing: No
Government Intervention
Boeing and Airbus Produce a Super-Jumbo:
Boeing is Subsidized by the US Government
Boeing
Develop SuperJumbo
Stay Out
Airbus
Develop SuperJumbo
-100A, 10B
500A,0B
Stay Out
0A, 610B
0A,0B
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How Useful Is Game Theory?
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A powerful tool, BUT
– outcomes are very sensitive to the protocols
– there may be many equilibria
– when using it to model real life cases there may be many different
options - “ a model may be devised to fit almost any fact” Saloner
1991
– analysis works backwards - instead of theory to hypotheses to
empirical data - empirical data to theory
– sometimes players’ commonsense tells them what to do despite
multiple equilibria
– the requirement that firms do as expected (in rollback, for instance)
– where do the protocols come from, how do they change? GE and
Westinghouse found new protocols which helped them to collude why then and not before?
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How Useful Is Game Theory?
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An overall judgment?
Some very useful insights
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Nash equilibrium concept
collusion; the price matching result
entry deterrence; the importance of credibility
Cournot and Bertrand provide determinate solutions for
oligopoly
The degree of complexity involved may limit its
usefulness as a predictive tool
The degree of rationality which has to be assumed on
the part of players is uncomfortable
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