Chapter 12 Strategy and Game Theory © 2004 Thomson Learning/South-Western.

Chapter 12

Strategy and Game Theory

© 2004 Thomson Learning/South-Western

2

Basic Concepts

Any situation in which individuals must make strategic choices and in which the final outcome will depend on what each person chooses to do can be viewed as a game.

Game theory models seek to portray complex strategic situations in a highly simplified setting.

3

Basic Concepts

All games have three basic elements:– Players– Strategies– Payoffs

Players can make binding agreements in cooperative games, but can not in noncooperative games, which are studied in this chapter.

4

Players

A player is a decision maker and can be anything from individuals to entire nations.

Players have the ability to choose among a set of possible actions.

Games are often characterized by the fixed number of players.

Generally, the specific identity of a play is not important to the game.

5

Strategies

A strategy is a course of action available to a player.

Strategies may be simple or complex. In noncooperative games each player is

uncertain about what the other will do since players can not reach agreements among themselves.

6

Payoffs

Payoffs are the final returns to the players at the conclusion of the game.

Payoffs are usually measure in utility although sometimes measure monetarily.

In general, players are able to rank the payoffs from most preferred to least preferred.

Players seek the highest payoff available.

7

Equilibrium Concepts

In the theory of markets an equilibrium occurred when all parties to the market had no incentive to change his or her behavior.

When strategies are chosen, an equilibrium would also provide no incentives for the players to alter their behavior further.

The most frequently used equilibrium concept is a Nash equilibrium.

8

Nash Equilibrium

A Nash equilibrium is a pair of strategies (a*,b*) in a two-player game such that a* is an optimal strategy for A against b* and b* is an optimal strategy for B against A*.– Players can not benefit from knowing the equilibrium

strategy of their opponents.

Not every game has a Nash equilibrium, and some games may have several.

9

An Illustrative Advertising Game

Two firms (A and B) must decide how much to spend on advertising

Each firm may adopt either a higher (H) budget or a low (L) budget.

The game is shown in extensive (tree) form in Figure 12.1

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A makes the first move by choosing either H or L at the first decision “node.”

Next, B chooses either H or L, but the large oval surrounding B’s two decision nodes indicates that B does not know what choice A made.

11

7,5

L

LH

L

H

HB

B

A

5,4

6,4

6,3

FIGURE 12.1: The Advertising Game in Extensive Form

12


The numbers at the end of each branch, measured in thousand or millions of dollars, are the payoffs.– For example, if A chooses H and B chooses L,

profits will be 6 for firm A and 4 for firm B.

The game in normal (tabular) form is shown in Table 12.1 where A’s strategies are the rows and B’s strategies are the columns.

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Table 12.1: The Advertising Game in Normal Form

B’s Strategies L H

L 7, 5 5, 4 A’s Strategies H 6, 4 6, 3

14

Dominant Strategies and Nash Equilibria

A dominant strategy is optimal regardless of the strategy adopted by an opponent.– As shown in Table 12.1 or Figure 12.1, the

dominant strategy for B is L since this yields a larger payoff regardless of A’s choice.

If A chooses H, B’s choice of L yields 5, one better than if the choice of H was made.

If A chooses L, B’s choice of L yields 4 which is also one better than the choice of H.

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A will recognize that B has a dominant strategy and choose the strategy which will yield the highest payoff, given B’s choice of L.– A will also choose L since the payoff of 7 is one

better than the payoff from choosing H.

The strategy choice will be (A: L, B: L) with payoffs of 7 to A and 5 to B.

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Since A knows B will play L, A’s best play is also L.

If B knows A will play L, B’s best play is also L.

Thus, the (A: L, B: L) strategy is a Nash equilibrium: it meets the symmetry required of the Nash criterion.

No other strategy is a Nash equilibrium.

17

Two Simple Games

Table 12.2 (a) illustrates the children’s finger game, “Rock, Scissors, Paper.”– The zero payoffs along the diagonal show that if

players adopt the same strategy, no payments are made.

– In other cases, the payoffs indicate a $1 payment from the loser to winner under the usual hierarchy (Rock breaks Scissors, Scissors cut Paper, Paper covers Rock).

18

TABLE 12.2 (a): Rock, Scissors, Paper--No Nash Equilibria

B’s Strategies Rock Scissors Paper

Rock 0, 0 1, -1 -1, 1 Scissors -1, 1 0, 0 1, -1 A’ Strategies

Paper 1, -1 -1, 1 0, 0

19

Two Simple Games

This game has no equilibrium. Any strategy pair is unstable since it offers at

least one of the players an incentive to adopt another strategy.– For example, (A: Scissors, B: Scissors) provides

and incentive for either A or B to choose Rock.– Also, (A: Paper, B: Rock) encourages B to choose

Scissors.

20

Two Simple Games

Table 12.2 (b) shows a game where a husband (A) and wife (B) have different preferences for a vacation (A prefers mountains, B prefers the seaside)

However, both players prefer a vacation together (where both players receive positive utility) than one spent apart (where neither players receives positive utility).

21

TABLE 12.2 (b): Battle of the Sexes--Two Nash Equilibria

B’s StrategiesMountain Seaside

Mountain 7, 5 5, 4A’s Strategies

Seaside 6, 4 6, 3

22

Two Simple Games

At the strategy (A: Mountain, B: Mountain), neither player can gain by knowing the other’s strategy.

The same is true with the strategy (A: Seaside, B: Seaside).

Thus, this game has two Nash equilibria.

23

APPLICATION 12.1: Nash Equilibrium on the Beach

Applications of the Nash equilibrium concept have been used to analyze where firms choose to operate.

The concept can be used to analyze where firm’s locate geographically.

The concept can also be used to analyze where firm’s locate in the spectrum of specific types of products.

24


Hotelling’s Beach– Hotelling looked at the pricing of ice cream sellers

along a linear beach.– If people are evenly spread over the length of the

beach, he showed that each seller had an advantage selling to nearby consumers who incur lower (walking) costs.

– The Nash equilibrium concept can be used to show the optimal location for each seller.

25


Milk Marketing in Japan– In southern Japan, four local marketing boards

regulate the sale of milk.– It appears that each must take into account what the

other boards are doing, since milk can be shipped between regions.

– A Nash equilibrium similar to the Cournot model found prices about 30 percent above competitive levels.

26


Television Scheduling– Firms can also choose where to locate along the

spectrum that represents consumers’ preferences for characteristics of a product.

– Firms must take into account what other firms are doing, so game theory applies.

– In television, viewers’ preferences are defined along two dimensions--program content and broadcast timing.

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– In general, the Nash equilibrium tended to focus on central locations

There is much duplication of both program types and schedule timing

– This has left “room” for specialized cable channels to attract viewers with special preferences for content or viewing times.

Sometimes the equilibria tend to be stable (soap operas and sitcoms) and sometimes unstable (local news programming).

28

The Prisoner’s Dilemma

The Prisoner’s Dilemma is a game in which the optimal outcome for the players is unstable.

The name comes from the following situation.– Two people are arrested for a crime.– The district attorney has little evidence but is

anxious to extract a confession.

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– The DA separates the suspects and tells each, “If you confess and your companion doesn’t, I can promise you a six-month sentence, whereas your companion will get ten years. If you both confess, you will each get a three year sentence.”

– Each suspect knows that if neither confess, they will be tried for a lesser crime and will receive two-year sentences.

30


The normal form of the game is shown in Table 12.3.– The confess strategy dominates for both players so

it is a Nash equilibria.– However, an agreement not to confess would

reduce their prison terms by one year each.– This agreement would appear to be the rational

solution.

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TABLE 12.3: The Prisoner’s Dilemma

BConfess Not confess

ConfessA: 3 yearsB: 3 years

A: 6 monthsB: 10 yearsA

Not confess A: 10 years B: 6 months

A: 2 yearsB: 2 years

32


The “rational” solution is not stable, however, since each player has an incentive to cheat.

Hence the dilemma:– Outcomes that appear to be optimal are not stable

and cheating will usually prevail.

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Prisoner’s Dilemma Applications

Table 12.4 contains an illustration in the advertising context.– The Nash equilibria (A: H, B: H) is unstable since

greater profits could be earned if they mutually agreed to low advertising.

– Similar situations include airlines giving “bonus mileage” or farmers unwilling to restrict output.

The inability of cartels to enforce agreements can result in competitive like outcomes.

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Table 12.4: An Advertising Game with a Desirable Outcome That is Unstable


L 7, 7 3, 10 A’s Strategies H 10, 3 5, 5

35

Cooperation and Repetition

In the version of the advertising game shown in Table 12.5, the adoption of strategy H by firm A has disastrous consequences for B (-50 if L is chosen, -25 if H is chosen).

Without communication, the Nash equilibrium is (A: H, B: H) which results in profits of +15 for A and +10 for B.

36

TABLE 12.5: A Threat Game in Advertising


L 20, 5 15, 10 A’s Strategies H 10, -50 5, -25

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Cooperation and Repetition

However, A might threaten to use strategy H unless B plays L to increase profits by 5.

If a game is replayed many times, cooperative behavior my be fostered.– Some market are thought to be characterized by

“tacit collusion” although firms never meet.

Repetition of the threat game might provide A with the opportunity to punish B for failing to choose L.

38

Many-Period Games

Figure 12.2 repeats the advertising game except that B knows which advertising spending level A has chosen.– The oral around B’s nodes has been eliminated.

B’s strategic choices now must be phrased in a way that takes the added information into account.

39

7,5

L

L

H

L

H

H B

B

A

5,4

6,4

6,3

FIGURE 12.2: The Advertising Game in Sequential Form

40

Many-Period Games

The four strategies for B are shown in Table 12.6.– For example, the strategy (H, L) indicates that B

chooses L if A first chooses H.

The explicit considerations of contingent strategy choices enables the exploration of equilibrium notions in dynamic games.

41

TABLE 12.6: Contingent Strategies in the Advertising Game

B’s Strategies L, L L, H H, L H, H

L 7, 5 7, 5 5, 4 5, 4 A’s Strategies H 6, 4 6, 3 6, 4 6, 3

42

Credible Threat

The three Nash equilibria in the game shown in Table 12.6 are:– (1) A: L, B: (L, L);– (2) A: L, B: (L, H); and– (3) A: H, B: (H,L).

Pairs (2) and (3) are implausible, however, because they incorporate a noncredible threat that firm B would never carry out.

43

Credible Threat

Consider, for example, A: L, B: (L, H) where B promises to play H if A plays H.– This threat is not credible (empty threats) since, if A

has chosen H, B would receive profits of 3 if it chooses H but profits of 4 if it chooses L.

By eliminating strategies that involve noncredible threats, A can conclude that, as before, B would always play L.

44

Credible Threat

The equilibrium A: L, B: (L, L) is the only one that does not involve noncredible threats.

A perfect equilibrium is a Nash equilibrium in which the strategy choices of each player avoid noncredible threats.– That is, no strategy in such an equilibrium requires a

player to carry out an action that would not be in its interest at the time.

45

Models of Pricing Behavior: The Bertrand Equilibrium

Assume two firms (A and B) each producing a homogeneous good at constant marginal cost, c.

The demand is such that all sales go to the firm with the lowest price, and sales are evenly split if PA = PB.

All prices where profits are nonnegative, (P c) are in each firm’s pricing strategy.

46

The Bertrand Equilibrium

The only Nash equilibrium is PA = PB = c.– Even with only two firms, the Nash equilibrium is

the competitive equilibrium where price equals marginal cost.

To see why, suppose A chooses PA > c.– B can choose PB < PA and capture the market.

– But, A would have an incentive to set PA < PB.

This would only stop when PA = PB = c.

47

Two-Stage Price Games and Cournot Equilibrium

If firms do not have equal costs or they do not produce goods that are perfect substitutes, the competitive equilibrium is not obtained.

Assume that each firm first choose a certain capacity output level for which marginal costs are constant up to that level and infinite thereafter.

48


A two-stage game where the firms choose capacity first and then price is formally identical to the Cournot analysis.– The quantities chosen in the Cournot equilibrium

represent a Nash equilibrium, and the only price that can prevail is that for which total quantity demanded equals the combined capacities of the two firms.

49


Suppose Cournot capacities are given by

A situation in which is not a Nash equilibrium since total quantity demanded exceeds capacity.– Firm A could increase profits by slightly raising price

and still selling its total output.

price.capacity full the is P that and q and q BA

PPP BA

50


PPP BA

Similarly,

is not a Nash equilibrium because at least one firm is selling less than its capacity.

The only Nash equilibrium is which is indistinguishable from the Cournot result.

This price will be less than the monopoly price, but will exceed marginal cost.

,PPP BA

51

Comparing the Bertrand and Cournot Results

The Bertrand model predicts competitive outcomes in a duopoly situation.

The Cournot model predict monopolylike inefficiencies in which price exceed marginal cost.

The two-stage model suggests that decisions made prior to the final (price setting) stage can have important market impact.

52

APPLICATION 12.2: How is the Price Game Played?

Many factors influence how the pricing “game” is played in imperfectly competitive industries.

Two such factors that have been examined are – Product Availability– Information Sharing

53


Product availability is an important component of competition in many retail industries.

The impact of movie availability in the video-rental industry was examined in 2001 by James Dana.

His data showed that Blockbuster’s prices were 40% higher than at other stores.

He argued that Blockbuster’s higher price in part stems from its reputation for having movies available and that those prices act as a signal.

54


Firms in the same industry often share information with each other at many levels.

A 2000 study of cross-shareholding in the Dutch financial sector showed clear evidence that competition was reduced when firms had financial interests in each other’s profits.

A famous 1914 antitrust case found that a price list published by lumber retailers facilitated higher prices by discouraging wholesalers from selling at retail.

55

Tacit Collusion: Finite Time Horizon

Would the single-period Nash equilibrium in the Bertrand model, PA = PB = c, change if the game were repeated during many periods?– With a finite period, any strategy in which firm A,

say, chooses, PA > c in the last period offers B the possibility of earning profits by setting PA > PB > c.

56

Tacit Collusion: Finite Time Horizon

– The threat of charging PA > c in the last period is not credible.

– A similar argument is applicable for any period before the last period.

The only perfect equilibrium requires firms charge the competitive price in all periods.

Tacit collusion is impossible over a finite period.

57

Tacit Collusion: Infinite Time Horizon

Without a “final” period, there may exist collusive strategies.– One possibility is a “trigger” strategy where each

firm sets its price at the monopoly price so long as the other firm adopts a similar price.

If one firm sets a lower price in any period, the other firm sets its price equal to marginal cost in the subsequent period.

58


Suppose the firms collude for a time and firm A considers cheating in this period.– Firm B will set PB = PM (the cartel price)

– A can set its price slightly lower and capture the entire market.

– Firm A will earn (almost) the entire monopoly profit (M) in this period.

59


Since the present value of the lost profits is given by (where r is the per period interest rate)

This condition holds for values of r < ½. Trigger strategies constitute a perfect equilibrium for

sufficiently low interest rates.

.1

2

if profitable be willcheating

,1

2

r

r

MM

M

60

Generalizations and Limitations

Assumptions of the tacit collusion model:– Firm B can easily detect whether firm A has cheated– Firm B responds to cheating by adopting a harsh

response that punishes firm A, and condemns itself to zero profit forever.

More general models relax one or both of these assumptions with varying results.

61

APPLICATION 12.3: The Great Electrical Equipment Conspiracy

Manufacturing of electric turbine generators and high voltage switching units provided a very lucrative business to such major producers and General Electric, Westinghouse, and Federal Pacific Corporations after World War II.

However, the prospect of possible monopoly profits proved enticing.

62


To collude they had to create a method to coordinate their sealed bids.– This was accomplished through dividing the country

into bidding regions and using the lunar calendar to decide who would “win” a bid.

The conspiracy became more difficult as its leaders had to give greater shares to other firms toward the end of the 1950s.

63


The conspiracy was exposed when a newspaper reporter discovered that some of the bids on Tennessee Valley Authority projects were similar.

Federal indictments of 52 executives lead to jail time for some and resulted in a chilling effect on the future establishment of other cartels of this type.

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Entry, Exit, and Strategy

Sunk Costs– Expenditures that once made cannot be recovered

include expenditures on unique types of equipment or job-specific training.

– These costs are incurred only once as part of the entry process.

– Such entry investments mean the firm has a commitment to the market.

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First-Mover Advantages

The commitment of the first firm into a market may limit the kinds of actions rivals find profitable.

Using the Cournot model of water springs, suppose firm A can move first.– It will take into consideration what firm B will do to

maximize profits given what firm A has already done.

66


Firm A knows fir B’s reaction function which it can use to find its profit maximizing level of output.

Using the previously discussed functions.

.2120

gives qfor Solving

.2

60

2

)120(120120

2120

A

Pq

Pq

Pq

Pqq

qq

A

A

ABA

AB

67


Marginal revenue equals zero (revenue andprofits are maximized) when qA = 60.

With firm A’s choice, firm B chooses to produce

Market output equals 90 so spring water sells for $30 increasing A’s revenue by $200 to $1800.

Firm B’s revenue falls by $700 to $900. This is often called a “Stakelberg equilibrium.”

.302

)60120(

2

120

A

B

qq

68

Entry Deterrence

In the previous model, firm A could only deter firm B from entering the market if it produces the full market output of 120 units yielding zero revenue (since P = $0).

With economies of scale, however, it may be possible for a first-mover to limit the scale of operation of a potential entrant and deter all entry into the market.

69

A Numerical Example

One simple way to incorporate economies of scale is to have fixed costs.

Using the previous model, assume each firm has to pay fixed cost of $784.– If firm A produced 60, firm B would earn profits of

$116 (= $900 - $784) per period.– If firm A produced 64, firm B would choose to

produce 28 [ = (120-64) 2].

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A Numerical Example

– Total output would equal 92 with P = $28.– Firm B’s profits equal zero [profits = TR - TC =

($28·28) - $784 = 0] so it would not enter.– Firm A would choose a price of $56 (= 120 - 64)

and earn profits of $2,800 [= ($56·64) - $784].

Economies of scale along with the chance to be the first mover yield a profitable entry deterrence.

71

APPLICATION 12.4: First-Mover Advantages for Alcoa, DuPont, Procter and Gamble, and Wal-Mart

Consider two types of first-mover advantages– Advantages that stem from economies of scale in

production.– Advantages that arise in connection with the

introduction of pioneering brands.

72


Economies of Scale for Alcoa and DuPont.– The first firm in the market may “overbuild” its initial

plant to realize economies of scale when the demand for the product expands.

– Antitrust action against the Aluminum Company of America (Alcoa) claimed that it built larger plants than justified by current demand.

73


– In the 1970s, DuPont expanded its capacity to produce titanium dioxide which is a primary coloring agent in white paint.

– Studies suggest that this strategy was successful in forestalling new investment by others into the titanium dioxide market.

74


Pioneering Brands for Proctor and Gamble– Introducing the first brand of a new product

appears to provide considerable advantage over later-arriving rivals.

– Proctor and Gamble was successful in this with Tide laundry detergent in the 1940s and Crest toothpaste in the 1950s.

– New products are a risk for consumers, and if the first one works, consumers may stick with it.

75


The Wal-Mart Advantage– Its success stems from its first mover advantage in

economies of scale and its initial “small town” strategy.

– Started in the 1960s, it started serving smaller, mostly Southern markets.

– This profitable near monopoly situation allowed it to grow and gain economies of scale in distribution and in buying power.

76

Limit Pricing

A limit price is a situation where a monopoly might purposely choose a low (“limit”) price policy with a goal of deterring entry into its market.– If an incumbent monopoly chooses a price

PL < PM (the profit-maximizing price) it is hurting its current profits.

– PL will deter entry only if it falls short of the average cost of a potential entrant.

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Limit Pricing

– If the monopoly and potential entrant have the same costs (and there are no capacity constraints), the only limit price is PL = AC, which results in zero economic profits.

Hence, the basic monopoly model does not provide a mechanism for limit pricing to work.

Thus, a limit price model must depart from traditional assumptions.

78

Incomplete Information

If an incumbent monopoly knows more about the market than a potential entrant, it may be able to use this knowledge to deter entry.

Consider Figure 12.3.– Firm A, the incumbent monopolist, may have “high”

or “low” production costs based on past decisions which are unknown to firm B.

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1,3Entry

High costNo entry

Entry

No entry

Low costB

B

A

4,0

3, -1

6,0

FIGURE 12.3: An Entry Game

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Incomplete Information

– Firm B, the potential entrant, must consider both possibilities since this affects its profitability.

If A’s costs are high, B’s entry is profitable (B = 3).

If A’s costs are low, B’s entry is unprofitable (B = -1).

– Firm A is clearly better off if B does not enter.– A low-price policy might signal that firm A is low cost

which could forestall B’s entry.

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Predatory Pricing

The structure of many predatory pricing models also stress asymmetric information.

An incumbent firm wishes its rival would exit the market so it takes actions to affect the rival’s view of future profitability.

As with limit pricing, the success depends on the ability of the monopoly to take advantage of its better information.

82

Predatory Pricing

Possible strategies include:– Signal low costs with a low-price policy.– Adopt extensive production differentiation to indicate

the existence of economies of scale.

Once a rival is convinced the incumbent firm possess an advantage, it may exit the market, and the incumbent gains monopoly profits.

83

APPLICATION 12.4: The Standard Oil Legend

The Standard Oil case of 1911 was one of the landmarks of U.S. antitrust law.

In that case, Standard Oil Company was found to have “attempted to monopolize” the production, refining, and distribution of petroleum in the U.S., violating the Sherman Act.

The government claimed that the company would cut prices dramatically to drive rivals out of a particular market and then raise prices back to monopoly levels.

84


Unfortunately, the notion that Standard Oil practiced predatory pricing policies in order to discourage entry and encourage exit by its rivals makes little sense in terms of economic theory.

Actually, the predator would have to operate with relatively large losses for some time in the hope that the smaller losses this may cause rivals will eventually prompt them to give it up.

This strategy is clearly inferior to the strategy of simply buying smaller rivals in the marketplace.

85


In a famous 1958 article, J.S. McGee concluded that Standard Oil neither trieds to use predatory policies nor did its actual price policies have the effect of driving rivals from the oil business.

McGee examined over 100 refineries bought by Standard Oil and found no evidence that predatory behavior by Standard Oil caused these firms to sell out.

Indeed, in many cases Standard Oil paid quite good prices for these refineries.

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N-Player Game Theory

The most important additional element added when the game goes beyond two players is the possibility for the formation of subsets of players.

Coalitions are combinations of two or more players in a game who adopt coordinated strategies.– A two-person game example is a cartel.

87

N-Player Game Theory

The formation of successful coalitions in n-player games if influenced by organizational costs.– Information costs associated with determining

coalition strategies.– Enforcement costs associated with ensuring that a

coalition’s chosen strategy is actually followed by its members.

Chapter 12 Strategy and Game Theory © 2004 Thomson Learning/South-Western.

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Transcript of Chapter 12 Strategy and Game Theory © 2004 Thomson Learning/South-Western.