MANAGERIAL ECONOMICSAn Analysis of Business Issues Howard Daviesand Pun-Lee LamPublished by FT Prentice Hall

Chapter 13:

Game Theory and Its Application in Managerial Economics

ObjectivesOn completion of this chapter you should:understand the place of game theory in Economicsbe able to represent and solve simple gamesapply game theory to the issue of collusionmodel Cournot, Bertrand and von Stackelberg competitionbe able to take a game-theoretic approach to entry deterrenceappreciate the limits of game theory

A Paradigm Shift Bringing Industrial Economics and Managerial Economics TogetherThe old Industrial Economics used the Structure-Conduct-Performance paradigmdependent variable is the performance/ profitability of a sectorperformance determined by structure; with high entry barriers, high concentration and high product differentiation profits will be highcross- sector multiple regressions the dominant empirical techniquebehaviour of individual firms (conduct) largely implicit

A Paradigm Shift Bringing Industrial Economics and Managerial Economics TogetherIndustrial Organization now dominated by game theoryfocus is on what happens within an oligopolistic industry, not on differences across industriesdecisions taken by individual firms have become the centrepiece of analysiscase studies of firms conduct have become the dominant empirical method

Basic Concepts in Game TheoryWarning! Game theory is difficult and can involve highly complex chains of reasoningA game is any situation involving interdependence amongst playersMany different types of game:co-operative versus non-co-operative (which is the main focus)zero-sum, non-zero-sumsimultaneous or sequentialone-off versus repeatedrepeated a known number of times, infinite number of times or an unknown but finite number of timescontinuous versus discrete pay-offscomplete versus asymmetric informationPrisoners Dilemma, assurance games, chicken games, evolutionary games

Representing and Solving Games

A Simultaneous Game in Strategic Form:

The Payoff Matrix

Company As Actions

High Price

Low Price

Company Bs Actions

High Price

100A,100B

120A, -20B

Low Price

-20A,120B

50A,50B

Representing and Solving GamesThe same game in sequential formCompany BCompany ACompany AHigh PriceLow Price100A, 100B120A, -20B-20A, 120B50A, 50BHigh PriceLow PriceHigh PriceLow Price

Representing and Solving GamesUse rollback or backward induction to solve sequential gamesif Company B has set a high price then A chooses a low price: the B high/A high branch can be prunedif B set a low price then A will choose a low price: the B low/ A high branch can be prunedB is therefore choosing between a high price (-20) and a low price (50): it chooses low priceA will also choose a low priceFor 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 strategylow price is also dominant for B

This Simple Example Illustrates a Number of Key IdeasNash Equilibrium - a set of strategies such that each is best for each player, given that the others are playing their own equilibrium strategiesRollback and Dominant strategiesThe Prisoners Dilemmaan important class of game where players can choose between co-operating or cheating/defectingthe best outcome for both is co-operation but the natural result is cheatingcollusion is a key example

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 introductionFor pure strategieslook for dominant strategiesif dominant strategies are not to be found, look for dominated strategies and delete themfor zero sum games use the minimax criterion: pick the strategy for each player for which the worst outcome is the least worstcell-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?)

For example

No Dominant Strategies: Eliminate Dominated Strategies

Company As Actions

High Price

Medium Price

Low Price

Company Bs Actions

High Price

100A,100B

120A, 65B

60A, 65B

Medium Price

65 A, 120B

80A,80B

60A, 55B

Low Price

65A,60B

55A, 60B

50A,50B

How Many Nash Equilibria?There may be no Nash equilibria in pure strategiesThere may be more than one Nash equilibrium

CollusionThe Prisoners 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?RepetitionPunishments and rewardsLeadership

Repetition and the Prisoners DilemmaIf 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 ESCAPEDBUT 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-APPEARSCOMMONSENSE suggests that solutions are found to this problem, and a good deal of evidence supports that view,but the basic insight remains.

Contingent Strategies in Repeated GamesContingent strategies are strategies whereby the actions taken in repeated games depend upon actions taken by rivals in the last roundgrim strategy: co-operate until the rival defects and then defect forever; this is too unforgiving - one mistake and the prospect of collusion is gone forevertit-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 experimentsin 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

Contingent Strategies in Repeated GamesIf 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.

Penalties and Rewards to Support CollusionIntroducing penalties and rewards changes the pay-off structureFirms may punish themselves in order to set up structures which assist collusione.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 outcomePenalties and rewards might come from other sources - consumers or the law might punish collusion - trade associations might punish those who defect from the collusion

Leadership to Support Collusion

Leadership as a Solution

Company As Actions

High Price

Low Price

Company Bs Actions

High Price

100A,300B

120A, 120B

Low Price

-20A,280B

50A,50B

Leadership to Support CollusionIf 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 exampleFurthermore, the large firm may increase overall profits by making side-payments to rivalsSaudi Arabia in the oil market?

Cournot and Bertrand CompetitionFrom historical curiosities to analytical workhorsesCournot competitiontwo firms, identical products, firms choose output levelseach firms profit-maximising output depends on the other firms output; hence each firm has a reaction functionas each firm will operate on its reaction function the point where they cross is the Cournot Nash equilibriumBertrand competitiontwo firms, identical products, firms choose price levelsprice is forced down to marginal cost

Von Stackelberg equilibriumA Leader and a FollowerThe Leader chooses a price and the Follower then chooses the price that suits it best, given what the Leader has doneA 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

Entry DeterrenceA major area of application for game theoryFirms may deter entry by threatening retaliationin 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

Entry DeterrenceBut is the threat credible?If it is more profitable for the incumbent to allow entry and share the market the threat is not credibleHowever, 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

Airbus and Boeing: No Government Intervention

Boeing and Airbus Produce a Super-Jumbo:

No Government Intervention

Airbus

Develop Super-Jumbo

Stay Out

Boeing

Develop Super-Jumbo

-100A,-100B

0A, 500B

Stay Out

500A,0B

0A,0B

Airbus and Boeing: No Government Intervention

Boeing and Airbus Produce a Super-Jumbo:

Boeing is Subsidized by the US Government

Airbus

Develop Super-Jumbo

Stay Out

Boeing

Develop Super-Jumbo

-100A, 10B

0A, 610B

Stay Out

500A,0B

0A,0B

How Useful Is Game Theory?A powerful tool, BUToutcomes are very sensitive to the protocolsthere may be many equilibriawhen using it to model real life cases there may be many different options - a model may be devised to fit almost any fact Saloner 1991analysis works backwards - instead of theory to hypotheses to empirical data - empirical data to theorysometimes players commonsense tells them what to do despite multiple equilibriathe 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?

How Useful Is Game Theory?An overall judgment?Some very useful insightsNash equilibrium conceptcollusion; the price matching resultentry deterrence; the importance of credibilityCournot and Bertrand provide determinate solutions for oligopolyThe degree of complexity involved may limit its usefulness as a predictive toolThe degree of rationality which has to be assumed on the part of players is uncomfortable