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MANAGERIAL ECONOMICSAn Analysis of Business Issues

Howard Davies

and Pun-Lee LamPublished by FT Prentice Hall

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Chapter 13:

Game Theory and Its Application in Managerial Economics

INTRODUCTION TO GAME THEORY: Was developed in 1950’s by JOHN

VON NEUMANN and OSKAR MORGENSTERN.

Design to evaluate situations where individuals and organizations have conflicting objectives.

It can be used to analyze the bargaining process between two parties

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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

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 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 A’s ActionsHigh Price Low Price

High Price 100A,100B 120A, -20BCompany B’sActions Low Price -20A,120B 50A,50B

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Representing and Solving Games The same game in sequential form

Company B

Company A

Company A

High Price

Low Price

100A, 100B

120A, -20B

-20A, 120B

50A, 50B

High Price

Low Price

High Price

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

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? 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 non-technical 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 A’s ActionsHigh Price Medium Price Low Price

High Price 100A,100B 120A, 65B 60A, 65B

Medium Price 65 A, 120B 80A,80B 60A, 55B

Company B’sActions

Low Price 65A,60B 55A, 60B 50A,50B

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How Many Nash Equilibria?

There may be no Nash equilibria in pure strategies

There may be more than one Nash equilibrium

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Collusion

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 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 tit-for-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 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 co-operation become less.

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Penalties and Rewards to Support Collusion

Introducing penalties and rewards changes the pay-off 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 pay-off structure in which collusion is the outcome

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 A’s ActionsHigh Price Low Price

High Price 100A,300B 120A, 120BCompany B’sActions Low Price -20A,280B 50A,50B

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Leadership to Support Collusion

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

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

Bertrand competition– two firms, identical products, firms choose price levels– price is forced down to marginal cost

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Von Stackelberg equilibrium

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

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

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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How Useful Is Game Theory? 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? An overall judgment? Some very useful insights

– 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