Game Theory

1
Game Theory
2
Learning Objectives

• Define game theory, and explain how it helps to
  better understand mutually interdependent
  management decisions
• Explain the essential dilemma faced by
  participants in the game called Prisoners
  Dilemma
• Explain the concept of a dominant strategy and
  its role in understanding how auctions can help
  improve the price for sellers, while still
  benefiting buyers

3
Overview

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

4
Game Theory

• Optimization has two shortcomings when applied to
  actual business situations
• Assumes factors such as reaction of competitors
  or tastes and preferences of consumers remain
  constant.
• Managers sometimes make decisions when other
  parties have more information about market
  conditions.
• Game theory is concerned with how individuals
  make decisions when they are aware that their
  actions affect each other and when each
  individual takes this into account.
• Game Theory is a useful tool for managers

5

• In the analysis of games, the order in which
  players make decisions is important
• Simultaneous-move game- Each player makes
  decision without knowledge of other players
  decision
• Sequential-move game player makes a move after
  observing other players move

6

• One shot game underlying game is played only
  once
• Repeated game underlying game is played more
  than once

7

• How managers use game theory
• Betrand Duopoly game
• 2 gas stations no location advantage.
  Consumers view product as perfect substitutes and
  will purchase from station that sells at lower
  price.
• First thing manager must do in the morning is to
  tell attendant to put up price without knowledge
  of rivals price.
• This is a simultaneous move game.
• If Manager of station A calls in price higher
  than B ? will lose sales that day

8
Normal Form Game

• A Normal Form Game consists of
• Players.
• Strategies or feasible actions.
• Payoffs.

9
A Normal Form Game
Player 2
12,11
11,12
14,13
Player 1
10
Simultaneous-move, One shot game

• Important to managers making decisions in an
  environment of interdependence. E.g. profits of
  firm A depends not only on firms A actions but
  on the actions of rival firm B as well.

11
Normal Form GameScenario Analysis
Player 2
10,20
15,8

Player 1
12

• Whats the optimal strategy?
• Complex question. Depends on the nature game
  being played.
• The game above is easy to characterize the
  optimal decision a situation that involves a
  dominant strategy.
• A strategy is dominant if it results in the
  highest payoff regardless of the action of the
  opponent

13

• For player 1, the dominant strategy is UP.
  Regardless of what player 2 chooses, if A chooses
  UP, shell earn more.
• Principle
• Check to see if you have a dominant strategy. If
  you have one, play it.

14

• What should a player do in the absence of a
  dominant strategy (e.g. Player 2)?
• Play a SECURE STRATEGY
• -- A strategy that guarantees the highest payoff
  given the worst possible scenario.
• Find the worse payoff that could arise for each
  action and choose the action that has the highest
  of the worse payoffs.

15

• Secure strategy for player 2 is RIGHT.
  Guarantees a payment of 8 rather than 7 from LEFT
• 2 shortcomings
• Very conservative strategy
• Does not take into account the optimal decision
  of your rival and thus may prevent you from
  earning a significantly higher payoff.
• Player 2 should actually choose LEFT, knowing
  that player 1 will play UP

16

• Principle Put yourself in your rivals shoes
• If you do not have a dominant strategy, look at
  the game from your rivals perspective. If your
  rival has a dominant strategy, anticipate that
  she will play it.

17
Putting Yourself in your Rivals Shoes

• What should player 2 do?
• 2 has no dominant strategy!
• But 2 should reason that 1 will play a.
• Therefore 2 should choose C.

Player 2
12,11
11,12
14,13
Player 1
18
The Outcome
12,11
11,12
14,13

• This outcome is called a Nash equilibrium
• a is player 1s best response to C.
• C is player 2s best response to a.

19
Nash Equilibrium

• Given the strategies of other players, no player
  can improve her payoff by unilaterally changing
  her own strategy.
• Every player is doing the best she can given what
  other players are doing.
• In original example, Nash equilibrium is when A
  chooses UP and B chooses LEFT.

20
Application of One shot games

• Two managers want to maximize market share.
• Strategies are pricing decisions. (charge high or
  low prices)
• Simultaneous moves.
• One-shot game. (firms meet once and only once in
  the market)

21
The Market-Share Game in Normal Form
Manager 2
Manager 1
22
Market Share game Equilibrium

• Each managers best decision is to charge a low
  price regardless of the others decision.
  Outcome of game is that both firms charge a low
  price and earn 0 profits
• Low prices for both managers is the Nash
  Equilibrium

23

• If firms collude to charge high prices, profits
  will be higher for both
• ? Classic case in Economics called dilemma
  because the Nash equilibrium outcome is inferior
  (from the firms viewpoint) to the situation where
  they both agree to charge high prices
• Even if firms meet secretly to collude, is there
  an incentive to cheat on the agreement?

24
To advertise or Not?

• Your firm competes against another firm for
  customers
• You and your rivals know your product will be
  obsolete at the end of the year (one shot game)
  and must simultaneously determine whether or not
  to advertise.
• In your industry, advertising does not increase
  industry demand but induces consumers to switch
  among the products of the different firms

25
An Advertising Game
Manager 2
Manager 1
26
To advertise or Not?

• Dominant strategy of each firm is to advertise. ?
  unique Nash equilibrium.
• Collusion will not work because this is a
  one-shot game and if theres agreement not to
  advertise, each firm will have an incentve to
  cheat.

27
Key Insight

• Game theory can 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.

28
Examples of Coordination Games

• Industry standards
• size of floppy disks.
• size of CDs.
• National standards
• electric current.
• traffic laws.

29

• Coordination Decisions
• Firms dont have competing objectives but
  coordinating their decisions will lead to higher
  profits
• e.g. Producing appliances that require either
  90-volt or 120-volt outlets

30
A Coordination Game in Normal Form
Firm B
Firm A
31
Coordination Game 2 Nash Equilibria

• What would you do if you manage Firm A?
• If you do not know what firm B is going to do,
  youll have to guess what B will do.
• Effectively, both you and firm B will do better
  by coordinating your actions.
• 2 Nash equilibria. If the firms can talk to
  each other, they can agree on what to produce.
• Notice, theres no incentive to cheat here
• This is a game of coordination rather than game
  of conflicting interest

32
Simultaneous-Move Bargaining

• Management and a union are negotiating a wage
  increase.
• Strategies are wage offers wage demands.
• Players have one chance to reach an agreement and
  offer is made simultaneously.
• Parties are bargaining over how much of 100 in
  surplus must go to the union

33

• Assume the surplus can be split only into 50
  increments
• One shot to reach agreement
• Parties simultaneously write the amount they
  desire on a piece of paper.
• If the sum of the amounts does not exceed 100,
  players get the specified amount
• If sum exceeds 100, stalemate, costing each
  player 1

34
The Bargaining Game in Normal Form
Union
Management
35
Simultaneous-Move Bargaining

• 3 Nash equilibria outcomes.
• Multiplicity of equilbria leads to inefficiency
  if parties fail to co-odinate on an equilibrium
• 6 of 9 outcomes are inefficient because they
  dont sum up to 100
• Clearly, in this game management must ask for 50
  if they

36
Key Insights

• Not all games are games of conflict.
• Communication can help solve coordination
  problems.
• Sequential moves can help solve coordination
  problems.

37
Infinitely Repeated Games

• Game played over and over again. Players receive
  payoff during each repetition of game
• Firms compete week after week, year after year ?
  game is repeated over time
• To evaluate profits earned during this game,
  consider the PV of all payoffs.
• If payoffs are the same in each period, then for
  an infinitely played game
• PV (1i)/i constant profit

38
An Advertising Game

• Two firms (Kelloggs General Mills) managers
  want to maximize profits.
• Strategies consist of pricing actions.
• Simultaneous moves.
• Repeated interaction.

39
Equilibrium to the One-Shot Pricing Game
General Mills
Kelloggs
40

• When firms repeatedly face this type of matrix,
  they use trigger strategy
• Trigger Strategy is a strategy that is
  contingent on the past plays of players in a game
• A player who adopts a trigger strategy continues
  to choose the same action until some other player
  takes an action that triggers a different
  action by the first player

41
Can collusion work if firms play the game each
year, forever?

• Consider the following trigger strategy by each
  firm
• We will each charge the high price, provided
  neither of us has ever cheated in the past. If
  one of us cheats and charges a low price, the
  other player will punish the deviator by
  charging low price in ever period thereafter
• In effect, each firm agrees to cooperate so
  long as the rival hasnt cheated in the past.
  Cheating triggers punishment in all future
  periods.

42
Kelloggs profits?

• ?Cooperate 10 10/(1i) 10/(1i)2 10/(1i)3
  
• 10 10/i

Value of a perpetuity of 12 paid at the end of
every year
?Cheat 500 0 0 0
Theres no incentive to cheat if the PV from
cheating is less than the PV from not cheating
43
Kelloggs Gain to Cheating

• ?Cheat - ?Cooperate 50 - (10 10/i) 40 -
  10/i
• Suppose i .05
• ?Cheat - ?Cooperate 40- 10/.05 40 - 200
  -160
• It doesnt pay to deviate.
• As long as i is less than 25, it pays not cheat.
• Collusion is a Nash equilibrium in the infinitely
  repeated game!

44
Benefits Costs of Cheating

• ?Cheat - ?Cooperate 40 - 10/i
• 40 Immediate Benefit (50 - 10 today)
• 10/i PV of Future Cost (10 - 0 forever after)
• If Immediate Benefit - PV of Future Cost gt 0
• Pays to cheat.
• If Immediate Benefit - PV of Future Cost ? 0
• Doesnt pay to cheat.

45
Application of Infinitely repeated games (product
quality)
Firm
Consumers
46

• If one shot game, Nash equilibrium low quality
  product and dont buy
• If infinitely repeated and consumers tell firm
  Ill buy your product and will continue to buy
  if it is of good quality. But if it turns out to
  be shoddy, Ill tell my friends not to buy
  anything from you again.
• Given this strategy of consumers, what should the
  firm do?
• If the interest rate is not too high, the best
  alternative is to sell a high product quality

47

• If firm cheats and sells shoddy product, it will
  earn 10 now but 0 forever thereafter.
• It will not pay for the firm to cheat if the
  interest rate is low.

48

• FINITE REPEATED GAMES
• Games that eventually end
• Games in which players do not know when the game
  will end
• Games in which players know when it will end.

49

• Suppose two duopolists repeatedly play the
  pricing game until their product become obsolete.
  Suppose the firms dont know when the game will
  end but theres a probability p that the game
  will end after every given play
• Probability the game will be played tomorrow if
  played today is (1-p). If the game is played
  tomorrow, the probability it will be played the
  next day is (1-p)2 etc.

50
Pricing Game that is infinitely repeated
General Mills
Kelloggs
51

• Suppose firms adopt trigger strategies, whereby
  each agrees to charge a high price but if a firm
  deviates and charges a low price, the other firm
  will punish it by charging low price until the
  game ends.
• Assume interest rate is zero
• Does Kelloggs have an incentive to cheat?

52
Kelloggs profits?

• ?Cooperate 10 10/(1-p) 10/(1-p)2 10/(1-p)3
  
• 10/p

?Cheat 500 0 0 0
Theres no incentive to cheat if the profit from
cheating is less than the profit from not
cheating. If there is a 10 that the government
will ban the sale of the item, then profit from
not cheating is 100 ? It pays not to cheat
53
Key Insight

• 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.

54
End of Period Problem

• When players know precisely when a repeated game
  will end, end-of-period problem arises
• In the final period, theres no tomorrow and
  theres no way to punish a player for doing
  something wrong in the last period.
• Consequently, players will behave as if it was a
  one shot game

55
Resignations, Quits Snake Oil salesmen

• Workers work hard if threatened with being fired
  if benefits of shirking are less than cost of
  being fired
• When worker announces that she wants to quit, say
  tomorrow, the cost of shirking is low so threat
  of firing has no effect
• What can managers do to overcome problem?
• Fire the worker as soon as she announces plan to
  quit? Problems
• Snake Oil Salesmen move about so no punishments

56
Factors affecting collusion in pricing games

• Number of firms Collusion is easier when there
  are few firms rather than many.
• Firm Size Economies of scale exists in
  monitoring. Easier for large firms to monitor
  small ones than other way round
• History of the Market Explicit meeting to
  collude or tacit collusion?
• Punishment Mechanism How do we punish our
  rivals when they cheat?

57
Real World Examples of Collusion

• Garbage Collection Industry
• OPEC
• NASDAQ
• Airlines