game theory models of pricing -- ucla microeconomic analysis (lecture)

8/3/2019 Game Theory Models of Pricing -- UCLA Microeconomic Analysis (Lecture)

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Chapter 8: Game Theory Models of Pricing Chapter 8: Game Theory Models of Pricing

• Unlike monopoly or perfect competition, mostfirms must consider the likely responses ofcompetitors when they make decisions aboutprice, advertising, and investments in newproduct lines

• Premise of game theory is that players(decisionmakers) are rational and act tomaximize profits

• Key element of strategy is understanding youropponents actions and how he/she will respondto your actions

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Two types of games: cooperative and Two types of games: cooperative and noncooperative noncooperative

• Cooperative games are based on binding contractsbetween parties

− firms enter agreement to make a joint investmentin a new technology

− firms agree to divide profit from joint venture

• General focus of game theory is on noncooperativegames

− firms make no formal agreements amongthemselves

− alternatively, agreements may exist but they arenot fully observed



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Advertising Game in Duopoly: How Does Advertising Game in Duopoly: How Does a Firm Develop an Advertising Strategy? a Firm Develop an Advertising Strategy?

• This is called the payoff matrix, because itsummaries the outcome of the game for differentchoices of Firm A and B

• The first number is each cell is the payoff to A andthe second is the payoff to B

• For example, if both firms advertise, then A make aprofit of 10 and B makes a profit of 5

10,5

6,8

15,0

10,2

Advertise Don’t Advertise

Advertise

Don’t Advertise

Firm B

Firm A

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What Strategy Should Firm A Choose? What Strategy Should Firm A Choose?

• Consider A’s strategy− If B advertises, then A makes more profit by advertising

(10 instead of 6)− If B does not advertise, A also makes more profit by

advertising (15 instead of 10)

• In this case, A is better off advertising irrespective of what Bdoes

• This is a special case where Firm A has a dominant strategy –i.e., a strategy that is optimal no matter what his opponentchooses

10,5

6,8

15,0

10,2


Advertise

Don’t Advertise

Firm B

Firm A



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What Strategy Should Firm B Choose? What Strategy Should Firm B Choose?

• Consider B’s strategy− If A advertises, then B makes more profit by advertising

(5 instead of 0)− If A does not advertise, B also makes more profit by

advertising (8 instead of 2)

•In this case, B is also better off advertising irrespective ofwhat A does – advertising is a dominant strategy for B as well

• Since both firms are rational, we predict that each will pursueits dominant strategy and the outcome will be that both firmswill advertise

10,5

6,8

15,0

10,2


Advertise

Don’t Advertise

Firm B

Firm A

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Modified Advertising Game Modified Advertising Game

• Advertising is still dominant for Firm B, but Firm A has nodominant strategy

− If B advertises, then A does best by advertising− If B does not advertise, the A does best by not advertising

• What should A do?− B will always advertise, since this is B’s dominant strategy− Therefore, A will choose its best outcome, given that B will

advertise, and will choose to advertise

• Solution is that both firms advertise

10,5

6,8

15,0

20,2


Advertise

Don’t Advertise

Firm B

Firm A



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Nash equilibrium: a set of strategies such that each player Nash equilibrium: a set of strategies such that each player is doing the best it can given the actions of its opponents is doing the best it can given the actions of its opponents

• Each player has no incentive to deviate from theNash equilibrium, so the strategy is stable andneither firm has an incentive to change its decisionsin response to the others actions

• Technical definition: A pair of strategies (a*, b*)represents a Nash equilibrium solution to a two-player game if a* is an optimal strategy for A againstb* and b* is an optimal strategy for B against a*

• In the two examples, the Nash equilibrium solution isthat both firms advertise. Given the decision of itscompetitor, each firm is satisfied that it has made thebest possible solution and has no incentive tochange

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Dominant Strategies Versus Nash Equilibrium Dominant Strategies Versus Nash Equilibrium

• In dominant strategy equilibrium, each party is doingthe best it can irrespective of what the other partydoes

• In Nash equilibrium, each party is doing the best itcan given what the other party is doing

• Hence, dominant strategies equilibrium is a specialsubset of Nash equilibrium

• Conditions are much weaker for Nash equilibriumthan for dominant strategies



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Example of Nash Equilibrium: Battle of the Sexes Example of Nash Equilibrium: Battle of the Sexes

• Husband prefers wrestling match to opera, but wife prefers opera towrestling match. If they cannot agree, they will stay home and get noentertainment

• No dominant strategies for either, since the best choice of each depends onthe choice of the other

• Two Nash equilibrium: either they both go to wrestling match or both go toopera

− If husband choose wrestling, then wife is better off by choosingwrestling

− If wife chooses opera, then husband is better off by choosing opera− Bottom right is also a Nash equilibrium by the same logic

• We cannot predict which Nash solution will occur

2,1

0,0

0,0

1,2

Wrestling OperaWife

HusbandWrestling

Opera

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Nash Nash

equlibriums equlibriums

are not necessarily Pareto efficient are not necessarily Pareto efficient

Non-equilibrium strategies will sometimes make both partiesbetter off

Prisoner’s Dilemma: Two suspect are captured for a crime andquestioned separately. Prosecution has a weak case so thesuspects are offered a plea bargain if they confess

Payoffs

• If both confess, they are both held for 3 months

• If both deny involvement, they are both held for 1 monthon a lesser charge

• If one confesses and the other denies, then the confessoris set free and the other prisoner serves 6 months



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Prisoner Prisoner ’ ’ s Dilemma Continued s Dilemma Continued

• The dominant strategy for both players is to confess.The outcome will be upper, left where they bothserve 3 months

• Both players are better off if they can agree tocooperate and both deny. This solution is not stable,however, because each has an incentive to deal andconfess

• The dilemma is that the Pareto optimal solution isnot a stable equilibrium

-3,-3

-6,0

0,-6

-1,-1

Confess DenyPlayer B

Player A ConfessDeny

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First First - - Mover Advantage Mover Advantage • In some games, a player gains an advantage by

being the first mover – It makes a choice that greatlylimits the alternatives of the other player

• Example: Stackelberg equilibrium in natural springwater

− Two identical firms− P=30-Q, homogeneous product so one price

− MC1=MC2=0, no marginal cost− Q=Q 1+Q2, consumers cannot distinguish the

product of firm 1 from that of firm 2



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Example continued Example continued

1221

21

121

1

1

111

5.15similarlyand5.15

0230

))(30(

))(30(

0

1

QQQQ

QQ

QQQ

QQ

PQ

C R

Q

−=−=

=−−=−−=

−=−=

−=

∂∂π

π

• These two equations show how the output decision of eachfirm is related to the output of the other – the equations arecalled reaction curves

• In equilibrium, these equations must be mutually consistent:each firm produces what the other thinks that it will

• Therefore, reactions curves are solved simultaneously forequilibrium values

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Example continued Example continued • Solution is called Cournot equilibrium

− Q1=Q2=10− Q=20, P=10, and ππππ=200(100 for each firm)− Firms make production decision simultaneously,

but they anticipate the other firm’s decision

• Now change strategy somewhat and assume thatone party acts first (Stackelberg assumption)

−Suppose Firm 1 goes first (the leader) and Firm 2goes second (the follower)

− Firm 2 has the same reaction curve as beforesince it takes Firm 1’s output as fixed



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Example continued Example continued Firm 1 anticipates Firm 2’s output by using Firm 2’sreaction curve

This changes Firm 1’s decision calculus

15

015 /

5.15

)])(5.15[30(

))(30(

1

11

211

111

1211

=

=−=∂∂−=

−−−=

−−=

Q

QQ

QQ

QQQ

QQQ

π

π

• After Firm 1 choose 15, Firm 2 chooses 7.5 (itsbest output given Firm 1’s output)

• Q=22.5, P=7.5, ππππ1=112.5, and ππππ2=56.25

• Corresponding results if Firm 2 goes first

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Example continued: Payoff Matrix Showing Firm Example continued: Payoff Matrix Showing Firm Profit Under Different Output Strategies Profit Under Different Output Strategies

• If both sides act simultaneous, then Cournot solution is Nashequilibrium: only solution where each firm is doing best given whatthe other firm is doing

• If Firm 1 acts first this constrains Firm 2’s choices− If #1 picks 7.5, then #2 picks 10 & #1 earns 94− If #1 picks 10, then #2 picks 10 & #1 earns 100− If #1 picks 15, then #2 picks 7.5 & #1 earns 112.5

• #1 picks 15 with first-mover advantage: leader is much better offand follower much worse off than at Cournot solution

112.5, 56.25

125, 93.75

112.5, 112.5

7.5

75, 50

100, 100

93.75, 125

10

0, 0

50, 75

56.25, 112.5

15

7.5

10

15

Firm 2

Firm 1

Outputs



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Example continued Example continued

• Firms must find a way to agree on leader and followerstatus, i.e., who gets to go first

• If both players act like leader, then the competitivesolution results

• The two leader-follower pairs are Nash equilibriumstrategies: If Firm 1 knows that Firm 2 has acted like aleader, its best alternative is to act like a follower

112.5, 56.25

125, 93.75

112.5, 112.5

7.5

75, 50

100, 100

93.75, 125

10

0, 0

50, 75

56.25, 112.5

15

7.5

10

15

Firm 2

Firm 1

Outputs

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Game Theory and Competitive Strategy: Part 2 Game Theory and Competitive Strategy: Part 2

Special type of game →→→→ zero-sum game where what Firm 1gains always equals Firm 2’s loss

Payoff matrix shows profit for Firm 1 and loss for Firm 2

• What strategy should Firm 1 choose?− Strategy A is dominant for Firm 1

• What strategy should Firm 2 choose?− Firm 2 has no dominant strategy: #1 is best if Firm 1

chooses B, but #2 is best if Firm 1 chooses A− Firm 2 recognizes that Firm 1 will always choose A,

however, so Firm 2 does best by choosing #2

• The Nash equilibrium solution is A2

8101

8.592

10113

BAFirm 1

Firm 2Example 1



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Example 2 Example 2

• What strategy will Firm 1 choose?− Strategy B is dominant for Firm 1, so it

chooses B irrespective of Firm 2’s strategy

• What strategy will Firm 2 choose?− Neither #1 nor #2 is a dominant strategy for

Firm 2

− Firm 1 will always choose #B, however, soFirm 2 chooses #2

• What is the Nash equilibrium?− B, 2

8, 77, 10

1

7, 86, 7

2

BAFirm 1

Firm 2

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Semi Semi - - Dominance Dominance

These examples show semi-dominance

• A strategy is dominant for one of the players, but theother player has no dominant strategy

• The strategy for the player with a dominant strategyis predictable, so the other player need only considerthe best choice from a subset of the alternatives

In semi-dominance, one players choice is independent ofthe other’s choice, but the second players choicedepends on that of the first



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Example 3: Marking choices to find Nash Equilibrium Example 3: Marking choices to find Nash Equilibrium

• As you inspect a payoff matrix, it is useful to markpreferred strategies by underlining them

• After considering each choice, the Nash equilibriumsare represent by cells where both entries areunderlined

• In this case, the Nash equilibriums are bottom left(BL) and top right (TR)

3,3

4,5

5,4

2,2

Produce Don’t ProduceFirm B

Firm A ProduceDon’t Produce

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Pure and Mixed Strategies Pure and Mixed Strategies • All games so far have involved specific choices like

advertise or not advertise, produce a certain amount, orinvest in a certain product. These types of strategies arecalled pure strategies

• More sophisticated games are based on mixed strategieswhere players make random choices among possiblealternatives, based on a set of probabilities

− In some cases, a game has no Nash equilibrium inpure strategies, but the game has a Nash equilibrium

in mixed strategies− The idea behind mixed strategies is that players donot react to the choices of their opponents, but ratherto what they expect their opponents to do



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Example 4: Nash Equilibrium in Mixed Strategy Example 4: Nash Equilibrium in Mixed Strategy

• First, see that the game does not have a dominantstrategy for either player

• Next notice that the game does not have a Nashequilibrium in pure strategies

− When one player choose a strategy and the otherresponds, the first player will always revise theirstrategy

− Outcomes follow arrows around payoff matrix

1,2

2,1

3,0

0,3

Top

Bottom

Left Right

Player 1

Player 2

↓↓↓↓ →→→→ ↑↑↑↑←←←←

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Example 4 continued Example 4 continued

• Now consider mixed strategies− Let P 1 be the probability that player 1 chooses T, so (1- P 1) is

the probability that player 1 chooses B− Similarly, let P 2 be the probability that player 2 chooses L and

(1- P 2) be the probability that player 2 chooses R

• Now consider the new choice variables to be P 1 and P 2 instead ofactual discrete outcomes

• Consider how player 1’s profit varies with P 2

− If player 1 plays T, then E ππππT= P 2(1)+(1- P 2)3=3-2P 2

− If player 1 plays B, then E ππππB= P 2(2)+(1- P 2)0=2P 2

1,2

2,1

3,0

0,3

Top (P 1)

Bottom (1-P 1)

Left (P 2) Right (1-P 2)

Player 1

Player 2

↓↓↓↓ →→→→ ↑↑↑↑←←←←



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• Now suppose P 2=.5, then the beststrategy for Player 1 is to choose T,because it has a higher expectedpayoff than B (2 instead of 1)

• Why is this not an equilibrium?− If player 2 chooses P 2=.5, then

player 1 would always play T (apure strategy)

− If player 1 were choosing T,however, player 2 would preferL, so P 2 would increase

• Equilibrium occurs where thecurves cross: Player 1 is indifferentbetween T & B when E ππππT= EππππB, so3-2P 2=2P 2 and this implies P 2=0.75

0 1

1

2

3

Player 1Eπ

P2

Expected ProfitCurves for Player 1

EπT

EπB

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Now consider Player 2

• EππππL= P 1(2)+(1- P 1)1=P 1+1, andEππππR= P 1(0)+(1- P 1)3=3-3P 1

• As before, the equilibriumoccurs where the curvescross: Player 2 is indifferentbetween L & R when E ππππL= E ππππR,so P 1+1=3-3P 1 and P 1=0.5

0 1

1

2

3

Player 2Eπ

P1

Expected ProfitCurves for Player 2

At P 1=0.5 and P 2=0.75, eachplayer is doing the best that itcan, given what the otherplayer is doing. This is theNash equilibrium.

EπR

EπL



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2,14,24,3Bottom (B)

4,22,42,5Center (C) Player 1

3,75,41,2Top (T)

Right (R)Middle (M)Left (L)

Player 2

• Suppose that the game in part a is played sequentially.Discuss what is the best strategy for Player 1.− If player 1 goes first, he/she should pick B and earn 4

as compared with 2 for C and 3 for T− If player 2 goes first, he/she will pick middle (as

compared with 3 for L and 2 for R). When player 2plays first and picks M, player 1 will earn 5.

− Therefore, player 1 should insist that player 2 playsfirst, so the equilibrium outcome is TM.

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Example 6: Mixed Strategy Example 6: Mixed Strategy

Find the mixed strategy solution for this game

• Expected profits for Player 1− EππππT =7P L+3(1-P L)− EππππB =2P L+6(1-P L)− Profits for both are equal at equilibrium, so

P L=3/8=0.375

• Expected profits for Player 2− EππππL =3P T+9(1-P T)− EππππR =5P T+6(1-P T)− Profits for both are equal at equilibrium, so

P L=3/8=0.6

6,62,9Bottom (B)

3,57,3Top (T) Player 1

Right (R)Left (L)

Player 2



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• Now suppose that the game is played sequentially.Discuss whether the players would agree on who shouldplay first and who should play second.

− If player 1 goes plays first then TR is the outcome,where player 1 earns 3 and player 2 earns 5.

− If player 2 goes first, then BR is the outcome and bothplayers are better off than at TR.

− If they play sequentially, they will agree that #2 shouldplay first, so they are both better off than letting #1play first.


6,62,9Bottom (B)

3,57,3Top (T) Player 1

Right (R)Left (L)

Player 2

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Price Competition with Homogeneous Price Competition with Homogeneous Products: Bertrand Model Products: Bertrand Model Consider a duopoly again

• P=30-Q and MC 1=MC2=3 (different costs than before)

• Cournot solution is Q 1=Q 2=9, P=12, and ππππ1=ππππ2=81

Bertrand suggested that firms would compete bychoosing price and not quantities as in the Cournotmodel

• Product is identical, so if two firms charge differentprices, then the lower-priced firm supplies the wholemarket, i.e., high-priced product does not sell



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Bertrand Model Bertrand Model

• What is the Nash equilibrium in this case?− P 1=P 2=3=MC, Q=27, and ππππ1=ππππ2=0− Firms behave as if the industry is perfectly competitive

• Why is this the solution?− If either firm raised its price it would lose all sales and

be no better off− If either lowered price, it would capture all sales but

lose money and be worse off− Hence, neither firm has an incentive to deviate from

the competitive solution

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Bertrand Model (continued) Bertrand Model (continued) • Why isn’t there a solution with positive profits?

− Consider some higher price than 3, say 6. At 6,both firms earn a profit.

− Each firm could nearly double its profit bycutting the price slightly and capturing the entiremarket

− Thus, each firm undercuts its competitor until wereach the competitive solution

Implication of Bertrand Model: Even with two firms in amarket, price competition leads to an efficient solution



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Threats and Entry Deterrence Threats and Entry Deterrence • Consider an incumbent monopolist that faces a

potential competitor

• The incumbent want to convince the new firm thatentry would be unprofitable

• Suppose that the new firm has a sunk cost of $40million to build a plant

30, 30-40=-1050, 50-40=10

Enter

60, 0100, 0

Not Enter

Low PriceHigh Price

Potential Entrant

IncumbentMonopolist

Monopoly profits are split evenly with entrant. High priceis accommodation strategy, and low price is warfare strategy.

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Threats continued Threats continued

Suppose the incumbent threatens warfare if the new firmenters the market. How successful is the threat ofwarfare (a low price) from the incumbent?

• A high price is a dominant strategy for the

incumbent, so the new firm knows that it will fact ahigh price if it enters

• Therefore, the threat is empty because it would notbe rational for the incumbent to enforce it. There isno credible threat.

30, 30-40=-1050, 50-40=10

Enter

60, 0100, 0

Not Enter

Low PriceHigh Price

Potential Entrant

IncumbentMonopolist



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Threats continued Threats continued • Now suppose the the incumbent makes an irrevocable commitment

that changes its incentives to engage in warfare

• The firm purchases new equipment that will allow it to increaseoutput at the same MC as pre-entry output

− Equipment cost is $30 million− Extra capacity is wasted if the firm maintains the high price, but

cost is recovered at the higher output levels with low price

30, -1020, 10Enter

60, 070, 0

Not Enter

Low PriceHigh Price

Potential Entrant

IncumbentMonopolist

New Matrix

• Now incumbent profits are reduced at high price, but not at low price• Incumbent threat of low price with entry is now credible

− the incumbent is better off at low price if new firm enters− The new firm is deterred, because it loses money at a low price

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Strategic Trade Policy Strategic Trade Policy

• If both firms produce the new airplane, they will both losemoney

• If one firm produces and the other does not, then the producerwill earn large profits: two Nash equilibrium are (0,100) and(100,0)

• Suppose that Boeing has a head start in the process ofdevelopment, then Boeing will produce and Airbus will not

Consider the commercial aircraft industry andthe decision to build a new airplane

0, 100-10, -10Produce

0, 0100, 0

Not Produce

Not ProduceProduce

Airbus

Boeing



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Strategic Trade Policy Continued Strategic Trade Policy Continued

• Now “Produce” is a dominant strategy for Airbus

• Boeing will recognize that Airbus will produce, so they willchoose not to produce

• Under the strategic trade policy, Airbus profits go from 0 to 120− $100 million is transferred from US to Europe− For a subsidy of $20 million, European profits increase by

$100 million

Now suppose that European governments make commitment toAirbus of $20 million to produce the plane before Boeing decidesand agree to give the subsidy irrespective of Boeing's decision

0, 120-10, -10+20

Produce

0, 0100, 0

Not Produce

Not ProduceProduce

Airbus

Boeing

game theory models of pricing -- ucla microeconomic analysis (lecture)

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