1. BUS 525: Managerial Economics Game Theory: Inside Oligopoly
Lecture 10
Game Theory: Inside Oligopoly

2. Overview I. Introduction to Game Theory
10-2
Overview
I. Introduction to Game Theory
II. Simultaneous-Move, One-Shot Games
III. Infinitely Repeated Games
IV. Finitely Repeated Games
V. Multistage Games

3. Game Environments Players are individuals who make decisions
10-3
Players are individuals who make decisions
Players’ planned decisions are called strategies.
Payoffs to players are the profits or losses resulting from strategies.
Order of play is important:
Simultaneous-move game: each player makes decisions without knowledge of other players’ decisions.
Sequential-move game: one player observes its rival’s move prior to selecting a strategy.
Frequency of rival interaction
One-shot game: game is played once.
Repeated game: game is played more than once; either a finite or infinite number of interactions.
Simultaneous-move game: Boxing
Sequential-move game: Chess
Managerial decisions in presence of interdependence, Game theory applications: price change, advertising, introduction of new products, entry into new markets.
Also used in decision making within the firm: monitoring and bargaining with workers

4. Simultaneous-Move, One-Shot Games: Normal Form Game
10-4
Simultaneous-Move, One-Shot Games: Normal Form Game
A Normal Form Game consists of:
Set of players i = {1, 2, … n} where n is a finite number.
Each players strategy set or feasible actions consist of a finite number of strategies.
Player 1’s strategies are S1 = {a, b, c, …}.
Player 2’s strategies are S2 = {A, B, C, …}.
Payoffs, to the players are the profits or losses that results from the strategies.
Due to the interdependence, payoff to a player depends not only the player’s strategy but also on the strategies employed by other players
Player 1’s payoff: π1(a,B) = ?
Player 2’s payoff: π2(b,C) = ?

5. Normal Form Game Player 2 12,11 11,12 14,13 Player 1 11,10 10,11 12,12
10-5
Normal Form Game
Normal form game: A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies
In game theory, Strategy, is a decision rule that describes the actions a player will take at each decision point
Player 2
12,11
11,12
14,13
Two players: 1 and 2; three strategies each: abc and ABC
First entry (in red) refers to player 1, second entry (in blue) refers to player 2
Payoff of player 1 critically depends on strategy of player 2 and vice-versa
Payoffs.
Player 1’s payoff: π1(a,B) = 11.
Player 2’s payoff: π2(b,A) = 10.
Player 1
11,10
10,11
12,12
10,15
10,13
13,14

6. A Normal Form Game Player B Player A 10-6 Price up or down
Advertising up or down
If A chooses up B chooses left, the resulting payoffs are 10 for A and 20 for B
One-shot game cannot make conditional decision

7. 10-7
Dominant Strategy
A strategy that results in highest payoff to a player regardless of opponents action
Player B
The dominant strategy for player A is Up
B chooses Left, A should choose Up; 10>-10
B chooses Right, A should choose Up; 15>10
Regardless of whether player B’s strategy is left or right, the best choice for player A is Up
Player A
Principle: Check to see if you have a dominant strategy. If you have one, play it.

8. Dominant Strategy in a Simultaneous-Move, One-Shot Game
10-8
A dominant strategy is a strategy resulting in the highest payoff regardless of the opponent’s action.
If “Up” is a dominant strategy for Player A in the previous game, then:
πA(Up,Left) ≥ πA(Down,Left);
πA(Up,Right) ≥ πA(Down,Right).

9. Secure Strategy Player B Player A Does B have a dominant strategy?
10-9
Does B have a dominant strategy?
Play secure strategy, i.e. A strategy that guarantees the highest payoff under the worst possible scenario. i.e choose Right; 8>7
Could you do better? Yes, choose left. Because A will choose Up, A’s dominant strategy; 20>8
Player B
A chooses Up, B should choose Left; 20>8
A chooses Down, B should choose Right; 10>7
Player B does not have a dominant strategy
Player A
Principle: Put yourself in rivals shoes. If your rival has a dominant strategy, anticipate that he or she will play it.

10. Nash Equilibrium Nash Equilibrium Player B Player A
10-10
The Nash equilibrium is a condition in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategy
Suppose A chooses Up, B chooses Left. Either player will have no incentive to change his or her strategy. Given that A’s strategy is Up, the best player B can do is to choose Left. Given that B’s strategy is Left, the best player A can do is to choose Up.
The Nash equilibrium strategy for player A is Up, and for player B it is Left.
Nash Equilibrium
Player B
Suppose A chooses Down, B chooses Left. Given that A’s strategy is Down, the best player B can do is to choose Right. Given that B’s strategy is Right, the best player A can do is to choose Up. Therefore, (-10,7) it is not a Nash equilibrium
Compare dominant strategy with Nash equilibrium
A beautiful mind John Nash
Dominant strategy from individual viewpoint, NASH equilibrium both the players.
Player A

11. Key Insights Look for dominant strategies.
10-11
Key Insights
Look for dominant strategies.
Put yourself in your rival’s shoes.

12. A Pricing Game Two managers want to maximize market share: i = {1,2}.
10-12
Two managers want to maximize market share: i = {1,2}.
Strategies are pricing decisions
SA = {Low Price, High Price}.
SB = {Low Price, High Price}.
Simultaneous moves.
One-shot game.

13. A Pricing Game Firm B Firm A Is there Nash equilibrium?
10-13
A Pricing Game
Firm B
Firm A
Is there Nash equilibrium?
Is Nash equilibrium the best?
Cheating
The Nash equilibrium is a condition in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategy
If firm B charges a high price, firm A’s best choice is to charge the low price, since 50 units of profits are better than 10 units if A charged the high price. Similarly, if B charges the low price, firm A’s best choice is to charge the low price, since 0 unit of profit is preferred to the 10 units of losses that would result if A charged the high price. Nash equilibrium strategy for both firms is to charge low price. Similar arguments hold from firm B’s perspective
This is a classic dilemma because Nash equilibrium is inferior to collusive outcome, both charging high prices..
Collusion is illegal.
. Cheating: If A cheated and charged low prices it could increase its profit from 10 to 50. B knows this and does not agree to collude.
Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated the prisoners, visit each of them to offer the same deal. If one testifies for the prosecution against the other (defects) and the other remains silent (cooperates), the defector goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent.
B: Silent
B: Defects
A: Silent
6 months prison for both
A:10 years
B: Free
A: Defects
A: Free
B: 10 years
Each serves 5 years

14. An Advertising game Firm B Firm A
10-14
An Advertising game
Firm B
Firm A
Dominant strategy for each firm is to advertise. Nash equilibrium for the game is to advertise
Collusion would not work because this is a one shot game; Reach agreement not to advertise to get $10 million each. Each firm has an incentive to cheat.

15. Key Insight:
10-15
Game theory can also be used to analyze situations where “payoffs” are non monetary!
We will, without loss of generality, focus on environments where businesses want to maximize profits.
Hence, payoffs are measured in monetary units.

16. Coordination Games
10-16
In many games, players have competing objectives: One firm gains at the expense of its rivals.
However, some games result in higher profits by each firm when they “coordinate” decisions.
Coordination is different from collusion

17. A Coordination Game Firm B Firm A
10-17
A Coordination Game
Firm B
Firm A
The game has two Nash equilibria. Better outcome by coordination. If firms produce different outlets, lower demand as consumers needs to spend more money in wiring their houses .
One Nash equilibrium is for each firm to produce 220 volt appliances; the other is for each firm to produce 110 volt appliances.
How to achieve the result?
Firms agree, Govt. regulation

18. Key Insights: Not all games are games of conflict.
10-18
Key Insights:
Not all games are games of conflict.
Communication can help solve coordination problems.
Sequential moves can help solve coordination problems.

19. Games With No Pure Strategy Nash Equilibrium
10-19
Worker
Manager
Suppose Managers strategy is to monitor the worker. Then the best choice for the worker is to work. But if the worker works, the best strategy for the manager not to monitor. Therefore, monitoring is not a Nash equilibrium strategy. Not monitoring is also not a Nash equilibrium strategy. Because, if manager’s strategy is “don’t monitor;” The worker would choose shirking. Manager should monitor.
Will every game has a Nash equilibrium? No.

20. Strategies for Games With No Pure Strategy Nash Equilibrium
10-20
In games where no pure strategy Nash equilibrium exists, players find it in there interest to engage in mixed (randomized) strategies.
This means players will “randomly” select strategies from all available strategies.
Mixed (randomized A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her actions
Surprise visit
VC or Dean will come
Announced or unannounced visits?

21. A Nash Bargaining Game Union Management 10-21
How much of a $100 surplus is to be given to the union?. The parties write the amount (either 0,50 or100). If the sum exceeds 100, the bargaining ends in a stalemate. Cost of delay -$1 both for the union and the management. Three Nash equilibrium to this bargaining game(100,0), (50,50), and (0,100).
Six of the nine potential payoffs are inefficient as they result in total payoff lesser than amount to be divided. Three negative payoffs. Settle for split which is fair, look at history

22. Lessons in Simultaneous Bargaining
10-22
Lessons in Simultaneous Bargaining
Simultaneous-move bargaining results in a coordination problem.
Experiments suggests that, in the absence of any “history,” real players typically coordinate on the “fair outcome.”
When there is a “bargaining history,” other outcomes may prevail.

23. Infinitely Repeated Games
10-23
Infinitely Repeated Games
Infinitely Repeated Games
A game that is played over and over again forever and in which players receive payoffs during each play of the game
Recall key aspects of present value analysis
Value of the firm for T period with different payoffs at different periods
Value of the firm for ∞ period with same payoffs at different periods
Trigger strategy
A strategy that is contingent on the past play of a game and in which some particular past action “triggers” a different action by a player
PV Firm = π0 + { π1/(1+i)} + { π2/(1+i)2} + …..+ { πT/(1+i)T}
= Σ1i-1{ πt/(1+i)t}
PV Firm = (1+i)π/i

24. A Pricing Game that is Repeated
10-24
A Pricing Game that is Repeated
Firm B
Firm A
Nash equilibrium in one shot play of this game is for each firm to charge low prices and earn zero profits
Impact of repeated play on the equilibrium outcome of the game
Trigger strategy: ”Cooperate provided no player has ever cheated in the past. If any players cheats, punish the player by choosing one shot Nash equilibrium strategy forever after.”
.Suppose firm A and B meet and agree that “we will each charge the high price, provided neither of us have ever cheated (i.e. charged low price in the previous period). If one of us cheats and charges low price, the other player will punish the deviator by charging the low price in every period thereafter.”
Pv Cheat Firm A = $ ….
Pv coop = $ 10 + {$10/(1+i)} + {$10/(1+i)}2………= 10(1+i)/i

25. A Pricing Game that is Repeated
10-25
A Pricing Game that is Repeated
The benefit of cheating today on the collusive agreement is earning $50 instead of $10 today. The cost of cheating today is earning $0 instead of $10 in each future period. If the present value of cost of cheating exceeds one time benefit of cheating, it does not pay for a firm to cheat, and high prices can be sustained.
In the example, if the interest rate is less than 25 percent both firms A and B will loose more (in present value) from cheating. That’s why we see collusion in oligopoly.
Condition for sustaining cooperative outcome with trigger strategies
(ΠCheat- πCoop)/(πCoop- πN)≤ 1/i or
ΠCheat- πCoop ≤ 1/i(πCoop- πN)
The LHS represents the one time gain from cheating. The RHS
represents the present value of what is given up in the future by cheating today.
In one-shot games there is no tomorrow, and threats have no bite.
When interest rate is low firms may find it in their interest to collude and
charge high prices
Pv Cheat Firm A = 50 ≤ 10(1+i)/I
= Pv Coop Firm A
10(1+0.25)/0.25
= 12.5/i
i = 0.25
πN = the one shot Nash equilibrium payoff
If interest rate is less than 25% both firm will loose due to cheating

26. Factor Affecting Collusion in Pricing Games
10-26
To sustain collusive arrangements via punishment strategies firms must know: (i) who the rivals are; (ii) who the rivals customers are; (iii) When the rivals deviate from collusive arrangement; and (iv) must be able to successfully punish rivals for cheating.
These factors are related to:
(1) Number of firms; (2) Firm size; (3) History of the market; (4) Punishment mechanism

27. Finitely Repeated Games
10-27
Finitely Repeated Games
Finitely Repeated Games
Games in which players do not know when the game will end
Games in which players know when the game will end
Games with an uncertain final period
When the game is repeated a finite but uncertain times, the benefits of cooperating look exactly the same as benefits from cooperating in an infinitely repeated game except that 1-θ plays the role of 1/(1+i); PLAYERS DISCOUNT THEIR FUTURE NOT BECAUSE OF THE INTEREST RATE BUT BECAUSE THEY ARE NOT CERTAIN FUTURE PLAYS WILL OCCUR
In our example Firm A has no incentive to cheat if
ΠFirm ACheat=50 ≤10/θ = ΠFirm ACoperate, which is true if θ ≤1/5, probability the game will end is less than 20 percent, If θ=1, one shot game, dominant strategy for both firm is to charge the low price

28. Repeated Games with a Known Final period
10-28
Repeated Games with a Known Final period
Provided the players know when the game will end, the game has only one Nash equilibrium.
Same outcome as one shot game
Trigger strategies would not work
Rival cannot be punished in the last period
Trigger strategies wont work
Backward unraveling.
End of period problem
Workers work hard due to threat of firing
Announces plans to quit, the cost of shirking is reduced
Manager’s response, fire immediately after announcements
Worker responds by telling you at the very end
What to do? Incentives, gratuity

29. 10-29
Multistage Games
Multistage framework permits players to make sequential rather than simultaneous decisions
Differs from one-shot and infinitely repeated games
Extensive-form game
A representation of a game that summarizes the players,
the information available to them at each stage, the
strategies available to them, the sequence of moves, and
the payoff resulting from alternative strategies

30. Pricing to Prevent Entry: An Application of Game Theory
10-30
Pricing to Prevent Entry: An Application of Game Theory
Two firms: an incumbent and potential entrant.
Potential entrant’s strategies:
Enter.
Stay Out.
Incumbent’s strategies:
{if enter, play hard}.
{if enter, play soft}.
{if stay out, play hard}.
{if stay out, play soft}.
Move Sequence:
Entrant moves first. Incumbent observes entrant’s action and selects an action.

31. The Pricing to Prevent Entry Game in Extensive Form
10-31
The Pricing to Prevent Entry Game in Extensive Form
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A(Entrant)
Out
B is an existing firm; A potential entrant
A stays out B earns 10 million, A zero
A decides to enter B must decide whether to start a price war (hard) or share the market
0, 10

32. Identify Nash and Subgame Perfect Equilibria
10-32
Identify Nash and Subgame Perfect Equilibria
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
Subgame Perfect Equilibria: A condition describing a set of strategies that constitutes a Nash equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies
Two Nash equilibria
The first occurs where firm B threatens to choose price war if A enters the market, thus A stays out of the market (0,10)
Is the threat credible? No
What happens if A enters? Will B engage in price war
Share the market (5,5)
0, 10

33. Two Nash Equilibria -1, 1 Enter 5, 5 Out 0, 10 Price war Share
10-33
Two Nash Equilibria
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
Firm B threatens price war. If A enters the market thus A stays out of market. Given B’s strategy price war A’s best strategy is not to enter. Given, A does not enter B’s best strategy is to threaten price war A earns 0. B earns 10.
Is the threat credible? No. If A enters B will chase share 5>1. Given, B plays safe, A should enter.
0, 10

34. One Subgame Perfect Equilibrium
10-34
One Subgame Perfect Equilibrium
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
A condition describing a set of strategies that constitutes a Nash equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.
Involves only credible threats.
0, 10
Subgame Perfect Equilibrium Strategy:
{enter; If enter, Share market}

35. Insights
10-35
Establishing a reputation for being unkind to entrants can enhance long-term profits.
It is costly to do so in the short-term, so much so that it isn’t optimal to do so in a one-shot game.

36. An Innovation Game: An Application of Game Theory
10-36
An Innovation Game: An Application of Game Theory
Class Exercise
Two firms: Innovator and Rival
Innovator’s strategies:
Introduce.
Don’t introduce.
Rival’s strategies:
{if introduced, clone}.
{if introduced, don’t clone}.
Move Sequence:
Innovator moves first. Rival observes innovator’s action and selects an action.

37. Class Exercise
If you do not introduce the new product, you and your rival will earn $1 million each
If you introduce the new product
Rival clones : you will loose $ 5 million and your rival will earn $20million
Rival does not clone: you will make $100 million and your rival will make $0 million
Set up the extensive form of the game
Should you introduce the new product?
How would your answer change if your rival has promised not to clone the product?
What would you do if patent law prevented your rival from cloning the product?

38. An Innovation Game: An Application of Game Theory
10-38
-5,20
Clone
B (Rival)
Introduce
Don’t clone
100, 0
A (Innovator)
Don’t
Introduce
If you introduce the product, B’s best strategy is to clone: you loose $5 million and B earns 20 million. If you don’t introduce you earn $1 million. Thus your profit maximizing strategy is not to introduce
Should not take promise seriously
Introduce, if patent law is there
1, 1

39. Simultaneous-Move Bargaining
10-39
Simultaneous-Move Bargaining
Management and a union are negotiating a wage increase.
Strategies are wage offers & wage demands.
Successful negotiations lead to $600 million in surplus, which must be split among the parties.
Failure to reach an agreement results in a loss to the firm of $100 million and a union loss of $3 million.
Simultaneous moves, and time permits only one-shot at making a deal.

40. Fairness: The “Natural” Focal Point
10-40
Fairness: The “Natural” Focal Point
Union
Management
Three Nash equilibria

41. Single Offer Bargaining
10-41
Single Offer Bargaining
Now suppose the game is sequential in nature, and management gets to make the union a “take-it-or-leave-it” offer.
Analysis Tool: Write the game in extensive form
Summarize the players.
Their potential actions.
Their information at each decision point.
Sequence of moves.
Each player’s payoff.
Am extensive form game. A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them at each stage, the sequence of moves, and the payoffs resulting from alternative strategies.

42. Step 1: Management’s Move
10-42
Step 1: Management’s Move 10 5
Firm 1

43. Step 2: Add the Union’s Move
10-43
Step 2: Add the Union’s Move
Accept
Union
Reject 10
Accept 5
Firm
Union
Reject 1
Accept
Union
Reject

44. Step 3: Add the Payoffs Accept 100, 500 Union Reject -100, -3 10
10-44
Step 3: Add the Payoffs
Accept
100, 500
Union
Reject
-100, -3 10
Accept
300, 300 5
Firm
Union
Reject
-100, -3 1
Accept
500, 100
Union
Reject
-100, -3

45. The Game in Extensive Form
10-45
The Game in Extensive Form
Accept
100, 500
Union
Reject
-100, -3 10
Accept
300, 300 5
Firm
Union
Reject
-100, -3 1
A simple example:
Circles are called “decision nodes” and each circle indicates that at that stage of the game the particular player must chose a strategy.
Decision node branches at payoffs.
Player A moves first
up (10,15) Both up = A: 10 B:15
A up B down = A:5 B:5
A down B up = A:0 B:0
up down (5,5) Both down = A:6 B:20
(0,0)
down up
down (6,20)
Accept
500, 100
Union
Reject
-100, -3 B A B

46. Step 4: Identify the Firm’s Feasible Strategies
10-46
Step 4: Identify the Firm’s Feasible Strategies
Management has one information set and thus three feasible strategies:
Offer $10.
Offer $5.
Offer $1.
Two Nash equilibria:
Suppose player B’s strategy is “chose down if player A chooses up, and down if player A chooses down”
The strategies are:
Player A: down
Player B: down if player a chooses up and down if player A chooses down
A chooses up, she earns 5, since B will chose down, she earns 6, since B will chose down. A will choose down.
Higher payoff for A, results when A chooses up and B chooses up why up not chosen. Because B threats to choose down.

47. Step 5: Identify the Union’s Feasible Strategies
10-47
Step 5: Identify the Union’s Feasible Strategies
The Union has three information set and thus eight feasible strategies:
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Accept $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
Is the threat credible? No. if A chooses up B’s best payoff is up (payoff 15). But if B chooses up, A earns 10. (higher than 6)
Another Nash equilibrium:
Suppose player B’s strategy is choose up if player A chooses up, choose down if player A chooses down.
Player A:up
Player B: choose up if player A choose

48. Step 6: Identify Nash Equilibrium Outcomes
10-48
Step 6: Identify Nash Equilibrium Outcomes
Outcomes such that neither the firm nor the union has an incentive to change its strategy, given the strategy of the other.

49. Finding Nash Equilibrium Outcomes
10-49
Finding Nash Equilibrium Outcomes $1
Yes $5
Yes $1
Yes $1
Yes
$10
Yes $5
Yes $1
Yes
$10, $5, $1 No

50. Step 7: Find the Subgame Perfect Nash Equilibrium Outcomes
10-50
Step 7: Find the Subgame Perfect Nash Equilibrium Outcomes
Outcomes where no player has an incentive to change its strategy, given the strategy of the rival, and
The outcomes are based on “credible actions;” that is, they are not the result of “empty threats” by the rival.
A condition describing a set of strategies that constitute a Nash equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.

51. Checking for Credible Actions
10-51
Checking for Credible Actions
Yes No No No No No No No

52. The “Credible” Union Strategy
10-52
The “Credible” Union Strategy
Yes No No No No No No No

53. Finding Subgame Perfect Nash Equilibrium Strategies
10-53
Finding Subgame Perfect Nash Equilibrium Strategies $1
Yes $5
$10 No
$10, $5, $1
Nash and Credible
Nash Only
Neither Nash Nor Credible

54. 10-54
To Summarize:
We have identified many combinations of Nash equilibrium strategies.
In all but one the union does something that isn’t in its self interest (and thus entail threats that are not credible).
Graphically:

55. There are 3 Nash Equilibrium Outcomes!
10-55
There are 3 Nash Equilibrium Outcomes!
Accept
100, 500
Union
Reject
-100, -3 10
Accept
300, 300 5
Firm
Union
Reject
-100, -3 1
Accept
500, 100
Union
Reject
-100, -3

56. Only 1 Subgame-Perfect Nash Equilibrium Outcome!
10-56
Only 1 Subgame-Perfect Nash Equilibrium Outcome!
Accept
100, 500
Union
Reject
-100, -3 10
Accept
300, 300 5
Firm
Union
Reject
-100, -3 1
Accept
500, 100
Union
Reject
-100, -3

57. 10-57
Bargaining Re-Cap
In take-it-or-leave-it bargaining, there is a first-mover advantage.
Management can gain by making a take-it-or-leave-it offer to the union. But...
Management should be careful; real world evidence suggests that people sometimes reject offers on the the basis of “principle” instead of cash considerations.

58. An Advertising Game
10-58
Two firms (Kellogg’s & General Mills) managers want to maximize profits.
Strategies consist of advertising campaigns.
Simultaneous moves.
One-shot interaction.
Repeated interaction.

59. A One-Shot Advertising Game
10-59
A One-Shot Advertising Game
General Mills
Kellogg’s

60. Equilibrium to the One-Shot Advertising Game
10-60
Equilibrium to the One-Shot Advertising Game
General Mills
Kellogg’s
Nash Equilibrium

61. Can collusion work if the game is repeated 2 times?
10-61
Can collusion work if the game is repeated 2 times?
General Mills
Kellogg’s

62. No (by backwards induction).
10-62
No (by backwards induction).
In period 2, the game is a one-shot game, so equilibrium entails High Advertising in the last period.
This means period 1 is “really” the last period, since everyone knows what will happen in period 2.
Equilibrium entails High Advertising by each firm in both periods.
The same holds true if we repeat the game any known, finite number of times.

63. Can collusion work if firms play the game each year, forever?
10-63
Can collusion work if firms play the game each year, forever?
Consider the following “trigger strategy” by each firm:
“Don’t advertise, provided the rival has not advertised in the past. If the rival ever advertises, “punish” it by engaging in a high level of advertising forever after.”
In effect, each firm agrees to “cooperate” so long as the rival hasn’t “cheated” in the past. “Cheating” triggers punishment in all future periods.

64. Suppose General Mills adopts this trigger strategy. Kellogg’s profits?
10-64
Suppose General Mills adopts this trigger strategy. Kellogg’s profits?
Cooperate = /(1+i) + 12/(1+i)2 + 12/(1+i)3 + …
= /i
Value of a perpetuity of $12 paid
at the end of every year
Cheat = 20 +2/(1+i) + 2/(1+i)2 + 2/(1+i)3 + …
= /i
General Mills
Kellogg’s

65. Kellogg’s Gain to Cheating:
10-65
Kellogg’s Gain to Cheating:
Cheat - Cooperate = /i - ( /i) = /i
Suppose i = .05
Cheat - Cooperate = /.05 = = -192
It doesn’t pay to deviate.
Collusion is a Nash equilibrium in the infinitely repeated game!
General Mills
Kellogg’s

66. Benefits & Costs of Cheating
10-66
Benefits & Costs of Cheating
Cheat - Cooperate = /i
8 = Immediate Benefit ( today)
10/i = PV of Future Cost ( forever after)
If Immediate Benefit - PV of Future Cost > 0
Pays to “cheat”.
If Immediate Benefit - PV of Future Cost  0
Doesn’t pay to “cheat”.
General Mills
Kellogg’s

67. Key Insight
10-67
Collusion can be sustained as a Nash equilibrium when there is no certain “end” to a game.
Doing so requires:
Ability to monitor actions of rivals.
Ability (and reputation for) punishing defectors.
Low interest rate.
High probability of future interaction.