H4: Mergers and collusions
=> Firms attempt to jointly reduce competition, might also benefit consumers through
cost efficiencies or customer service
=> We mostly looked at equilibria where they chose the best outcome for themself
Jointly escaping competition
Homogeneous goods Bertrand duopoly
- marginal cost c = 60, demand q = 100 - p
- All firms set the lowest price share demand equally
- Bertrand paradox: p - c - 60 = 0
- If they both choose to set pm = 80, the split profits
- They can each time undercut the other by deviating from the agreement
=> Not a NE
How to make sure all firms don’t undercut each other
=> Cartels: “carrot and stick” to deter undercutting
- Firms (illegally) agree to increase prices
- Firms must find a way to punish those that do not stick to the agreement
- To punish in the future, there needs to be a future => repeated games
=> Mergers: integrate ownership
- If the two firms merge, then they benefit from setting the monopoly price
- Mergers, especially those leading to a monopoly, can be blocked by the
antitrust
Repeated games
=> Special type of extensive game
- We first assumed that players only meet once
- No incentive to stick to promises that are not the best responses
- In real life, agents often interact more than once (e.g. competing firms,
colleagues at work, political parties, husband and wife…)
- This gives players the opportunity to punish bad behaviour, or reward
cooperation
=> These situations are better described using repeated games
Categorization
- Extensive game
- Agents meet T times and play the same strategic game, but they remember
what was played before = perfect recall
- If T is finite, the game is an T-repeated game
- If T is infinite, the game is an infinitely-repeated game
=> Players don’t know when they will meet food the last time
1
,How to create a repeated game (sl14)
- Choose if finite or infinite => if finite, choose T
- Start from strategic game such as prisoners dilemma
- Each action profile of the “first round” creates a new decision node/history at
each node, the game is played again, and each action profile generates a
new history
- Repeat T times
- A strategy is a rule (function) prescribing one action for each possible
decision node
- Payoffs for the repeated are the present value of the payoffs obtained in
each node that is visited (each iteration of the game)
Present value and discounting
- Money tomorrow is not worth as much as today
- Waiting might involve risks
- Resources today can be fruitfully invested
- Might provide higher benefit today
Discounting example
- Plan A: Invest 60 euro today for a return of 200 euro in 10 years
- Plan B : Invest 100 euro today for a row of profits of 20 euro a year for 10
years
- Plan C: Invest 100 euro today to receive 5 euro a year forever
Interest calculations
- vt+k = vt(1 + i)k
- vt = vt+k / (1 + i)k
- Discount factor: δ = 1/(1+r)
Discounting in game theory
- Many things can go wrong in 10 years, might have to value even less
- Discount factor is always 0<δ<1/(1+i*)k, where i* is the interest rate for a risk-
free investment
- Lower discount factor: risk, lack of liquidity, uncertainty, impatience
- Increase discount factor: patience, trust, probability of future interactions
=> In repeated games, the discount factor can be interpreted as the probability of
meeting again
2
, Comparing investment projects (sl22)
=> The firm should plan B
Equilibrium in repeated games
- If T is finite, then it is a standard extensive game
- Backward induction and find the SPNE
- In each period a NE of the strategic game must be played
- If T is infinite, you cannot use backward induction => no end
- Guess-and-verify: We conjecture a strategy/plan of action for each player,
then we verify that no player can do better by switching
- We rely on the one-deviation-property of SPNE: “If players cannot improve
their payoff by deviating in any specific round (action), then they cannot
improve by changing multiple actions simultaneously”
- Makes it very easy to think of deviations
3
=> Firms attempt to jointly reduce competition, might also benefit consumers through
cost efficiencies or customer service
=> We mostly looked at equilibria where they chose the best outcome for themself
Jointly escaping competition
Homogeneous goods Bertrand duopoly
- marginal cost c = 60, demand q = 100 - p
- All firms set the lowest price share demand equally
- Bertrand paradox: p - c - 60 = 0
- If they both choose to set pm = 80, the split profits
- They can each time undercut the other by deviating from the agreement
=> Not a NE
How to make sure all firms don’t undercut each other
=> Cartels: “carrot and stick” to deter undercutting
- Firms (illegally) agree to increase prices
- Firms must find a way to punish those that do not stick to the agreement
- To punish in the future, there needs to be a future => repeated games
=> Mergers: integrate ownership
- If the two firms merge, then they benefit from setting the monopoly price
- Mergers, especially those leading to a monopoly, can be blocked by the
antitrust
Repeated games
=> Special type of extensive game
- We first assumed that players only meet once
- No incentive to stick to promises that are not the best responses
- In real life, agents often interact more than once (e.g. competing firms,
colleagues at work, political parties, husband and wife…)
- This gives players the opportunity to punish bad behaviour, or reward
cooperation
=> These situations are better described using repeated games
Categorization
- Extensive game
- Agents meet T times and play the same strategic game, but they remember
what was played before = perfect recall
- If T is finite, the game is an T-repeated game
- If T is infinite, the game is an infinitely-repeated game
=> Players don’t know when they will meet food the last time
1
,How to create a repeated game (sl14)
- Choose if finite or infinite => if finite, choose T
- Start from strategic game such as prisoners dilemma
- Each action profile of the “first round” creates a new decision node/history at
each node, the game is played again, and each action profile generates a
new history
- Repeat T times
- A strategy is a rule (function) prescribing one action for each possible
decision node
- Payoffs for the repeated are the present value of the payoffs obtained in
each node that is visited (each iteration of the game)
Present value and discounting
- Money tomorrow is not worth as much as today
- Waiting might involve risks
- Resources today can be fruitfully invested
- Might provide higher benefit today
Discounting example
- Plan A: Invest 60 euro today for a return of 200 euro in 10 years
- Plan B : Invest 100 euro today for a row of profits of 20 euro a year for 10
years
- Plan C: Invest 100 euro today to receive 5 euro a year forever
Interest calculations
- vt+k = vt(1 + i)k
- vt = vt+k / (1 + i)k
- Discount factor: δ = 1/(1+r)
Discounting in game theory
- Many things can go wrong in 10 years, might have to value even less
- Discount factor is always 0<δ<1/(1+i*)k, where i* is the interest rate for a risk-
free investment
- Lower discount factor: risk, lack of liquidity, uncertainty, impatience
- Increase discount factor: patience, trust, probability of future interactions
=> In repeated games, the discount factor can be interpreted as the probability of
meeting again
2
, Comparing investment projects (sl22)
=> The firm should plan B
Equilibrium in repeated games
- If T is finite, then it is a standard extensive game
- Backward induction and find the SPNE
- In each period a NE of the strategic game must be played
- If T is infinite, you cannot use backward induction => no end
- Guess-and-verify: We conjecture a strategy/plan of action for each player,
then we verify that no player can do better by switching
- We rely on the one-deviation-property of SPNE: “If players cannot improve
their payoff by deviating in any specific round (action), then they cannot
improve by changing multiple actions simultaneously”
- Makes it very easy to think of deviations
3
