Repeated Games I
Week 2
Repeated Games I Finite Repetition
Repeated Games Interactions / games between competitors A and B take place not only once but repeatedly
Repeated Games Interactions / games between competitors A and B take place not only once but repeatedly A can threaten B: If A does not behave cooperatively this time, B will retaliate next time – and vice versa Can this effectively enforce cooperation?
Types of Repeated Games Finite Repetition It is clear from the beginning how often the game is repeated and when it ends Infinite Repetition There is no defined end of the game and the number of repetitions is not clear
Street Lights for the Olympics
Street Lights for the Olympics The organizing committee orders 500 street lights from a contractor The contractor manufactures the street lights and installs them in the Olympic Park • Every month only 100 street lights can be manufactured and installed
• The whole process takes 5 months • One month before the Olympics starts all street lights have to be installed
Actions Every month… …the contractor has to decide about delivery • High-quality street lights (worth £45,000 / costs £15,000) or • Low-quality street lights (worth £30,000 / costs £12,000) …the organizing committee has to decide about
• Paying the agreed price of £30,000 or • Renegotiating the price down to £20,000
Matrix Form Every month, the game looks like this:
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Threats The contractor threatens the organizing committee: If the organizing committee renegotiates the price in one month, they will deliver low quality in all subsequent months The organizing committee threatens the contractor: If the contractor delivers low quality in one month, they will renegotiate the price in all subsequent months Does this help enforcing cooperative behaviour?
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games II Backward Induction
Backward Induction Can be used to analyse repeated games with finite repetitions Process of reasoning backwards in time: • First consider the last stage of a game and determine the best action at that time • With this information, determine what to do in the penultimate stage • Continue until the best strategy for every stage of the game is found
Street Lights Recap Installation takes 5 months, every month 100 street lights can be manufactured and installed The contractor threatens the organizing committee:
If the organizing committee renegotiates the price in one month, they will deliver low quality in all subsequent months The organizing committee threatens the contractor:
If the contractor delivers low quality in one month, they will renegotiate the price in all subsequent months
Payoffs in Month 5 Last month of interaction: No threat of retaliation in subsequent period
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 4 No cooperation in month 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 3 No cooperation in months 4 and 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 1 No cooperation in months 2 to 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Summary Following backward induction • The contractor will deliver low quality in all months • The organizing committee will always renegotiate a lower price The outcome is triggered by the fact that in the last stage of the game there is no further threat of retaliation
Endgame effect This holds for any prisoners‘ dilemma with finite repetitions
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games III Infinite Repetition
Types of Repeated Games Finite Repetition It is clear from the beginning how often the game is repeated and when it ends Infinite Repetition There is no defined end of the game and the number of repetitions is not clear
Diamond Cartel South Africa and Australia control the market for diamonds as luxury good Each country has enough resources and production capacity to supply the diamonds that are sold every year Every January, the countries decide about the prices they charge This game is repeated every year, with probability p it goes on in the next year
Actions Both countries charge the monopoly price • The market is shared equally • Overall profit is $50mn
One country charges a slightly lower price • This country serves the whole market • Its profit equals $49mn
Both countries charge lower prices • Ends in fierce competition • Overall profit is $0mn
One Stage Game South Africa Australia
Monopoly Price
Lower Price
Monopoly Price
$50mn 2
/
$50mn 2
$0mn
/
$49mn
Lower Price
$49mn
/
$0mn
$0mn
/
$0mn
Agreement The countries agree on the following strategy: They charge the monopoly price As soon as one country charges a low price in one year, the other one will charge low prices for all periods in the future Does this ensure cooperation?
Payoffs For each country: This year
Next year
Ever following year
Cooperate this year (Charge monopoly price)
$50mn 2
$50mn 2
$50mn 2
Deviate this year (Charge lower price)
$49mn
$0mn
$0mn
Payoffs are unsecure (Game continues with probability p)
Expected Payoffs For each country:
Cooperate this year (Charge monopoly price)
$50mn 2
Deviate this year (Charge lower price)
$49mn
x
1 (1-p)
p is between zero and one
Cooperation Each country will cooperate this year if Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price) $49mn
Summary In infinitely repeated games, cooperation is possible If one country deviates it sacrifices long-term profits for short-term gains Determinants • Likelihood of future payoffs • Relative value of payoffs
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games IV Factors Influencing Cooperation
Diamond Cartel Recap Each country will cooperate this year if Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
p is probability that game goes on
Payoff from deviating (Charge lower price)
>
$49mn
Number of Competitors If there are more competitors, the share from the monopoly profit gets smaller: Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price)
>
$49mn
Importance of the Future If the future becomes more important, the interest rate i decreases: Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price) 1
x 1-
(
1 1+i
)
>
$49mn
Degree of Punishment Change in the agreement: As soon as one country charges a low price in one year, the other one will charge low prices for the next five years Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price)
>
$49mn
+
p5 (1-p)
x
$50mn 2
Summary Factors enhancing cooperation • High importance of future payoffs
Factors hindering cooperation • Large number of competitors • Low degree of punishment
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Repeated Games I Finite Repetition
Repeated Games Interactions / games between competitors A and B take place not only once but repeatedly
Repeated Games Interactions / games between competitors A and B take place not only once but repeatedly A can threaten B: If A does not behave cooperatively this time, B will retaliate next time – and vice versa Can this effectively enforce cooperation?
Types of Repeated Games Finite Repetition It is clear from the beginning how often the game is repeated and when it ends Infinite Repetition There is no defined end of the game and the number of repetitions is not clear
Street Lights for the Olympics
Street Lights for the Olympics The organizing committee orders 500 street lights from a contractor The contractor manufactures the street lights and installs them in the Olympic Park • Every month only 100 street lights can be manufactured and installed
• The whole process takes 5 months • One month before the Olympics starts all street lights have to be installed
Actions Every month… …the contractor has to decide about delivery • High-quality street lights (worth £45,000 / costs £15,000) or • Low-quality street lights (worth £30,000 / costs £12,000) …the organizing committee has to decide about
• Paying the agreed price of £30,000 or • Renegotiating the price down to £20,000
Matrix Form Every month, the game looks like this:
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Threats The contractor threatens the organizing committee: If the organizing committee renegotiates the price in one month, they will deliver low quality in all subsequent months The organizing committee threatens the contractor: If the contractor delivers low quality in one month, they will renegotiate the price in all subsequent months Does this help enforcing cooperative behaviour?
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games II Backward Induction
Backward Induction Can be used to analyse repeated games with finite repetitions Process of reasoning backwards in time: • First consider the last stage of a game and determine the best action at that time • With this information, determine what to do in the penultimate stage • Continue until the best strategy for every stage of the game is found
Street Lights Recap Installation takes 5 months, every month 100 street lights can be manufactured and installed The contractor threatens the organizing committee:
If the organizing committee renegotiates the price in one month, they will deliver low quality in all subsequent months The organizing committee threatens the contractor:
If the contractor delivers low quality in one month, they will renegotiate the price in all subsequent months
Payoffs in Month 5 Last month of interaction: No threat of retaliation in subsequent period
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 4 No cooperation in month 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 3 No cooperation in months 4 and 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Payoffs in Month 1 No cooperation in months 2 to 5 in any case: Threat of retaliation not credible
Contractror
Organizing Committee Accept price
Renegotiate price
High quality
£15,000 / £15,000
£5,000 / £25,000
Low quality
£18,000 / £0
£8,000 / £10,000
Summary Following backward induction • The contractor will deliver low quality in all months • The organizing committee will always renegotiate a lower price The outcome is triggered by the fact that in the last stage of the game there is no further threat of retaliation
Endgame effect This holds for any prisoners‘ dilemma with finite repetitions
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games III Infinite Repetition
Types of Repeated Games Finite Repetition It is clear from the beginning how often the game is repeated and when it ends Infinite Repetition There is no defined end of the game and the number of repetitions is not clear
Diamond Cartel South Africa and Australia control the market for diamonds as luxury good Each country has enough resources and production capacity to supply the diamonds that are sold every year Every January, the countries decide about the prices they charge This game is repeated every year, with probability p it goes on in the next year
Actions Both countries charge the monopoly price • The market is shared equally • Overall profit is $50mn
One country charges a slightly lower price • This country serves the whole market • Its profit equals $49mn
Both countries charge lower prices • Ends in fierce competition • Overall profit is $0mn
One Stage Game South Africa Australia
Monopoly Price
Lower Price
Monopoly Price
$50mn 2
/
$50mn 2
$0mn
/
$49mn
Lower Price
$49mn
/
$0mn
$0mn
/
$0mn
Agreement The countries agree on the following strategy: They charge the monopoly price As soon as one country charges a low price in one year, the other one will charge low prices for all periods in the future Does this ensure cooperation?
Payoffs For each country: This year
Next year
Ever following year
Cooperate this year (Charge monopoly price)
$50mn 2
$50mn 2
$50mn 2
Deviate this year (Charge lower price)
$49mn
$0mn
$0mn
Payoffs are unsecure (Game continues with probability p)
Expected Payoffs For each country:
Cooperate this year (Charge monopoly price)
$50mn 2
Deviate this year (Charge lower price)
$49mn
x
1 (1-p)
p is between zero and one
Cooperation Each country will cooperate this year if Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price) $49mn
Summary In infinitely repeated games, cooperation is possible If one country deviates it sacrifices long-term profits for short-term gains Determinants • Likelihood of future payoffs • Relative value of payoffs
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
Week 2
Repeated Games IV Factors Influencing Cooperation
Diamond Cartel Recap Each country will cooperate this year if Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
p is probability that game goes on
Payoff from deviating (Charge lower price)
>
$49mn
Number of Competitors If there are more competitors, the share from the monopoly profit gets smaller: Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price)
>
$49mn
Importance of the Future If the future becomes more important, the interest rate i decreases: Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price) 1
x 1-
(
1 1+i
)
>
$49mn
Degree of Punishment Change in the agreement: As soon as one country charges a low price in one year, the other one will charge low prices for the next five years Payoff from cooperating (Charge monopoly price) $50mn 2
x
1 (1-p)
Payoff from deviating (Charge lower price)
>
$49mn
+
p5 (1-p)
x
$50mn 2
Summary Factors enhancing cooperation • High importance of future payoffs
Factors hindering cooperation • Large number of competitors • Low degree of punishment
Competitive Strategy Tobias Kretschmer Professor of Management, LMU Munich
© 2013 LMU Munich
