3.1 — Dynamic Games & Collusion

# 3.1 — Repeated Games & Collusion

## ECON 326 • Industrial Organization • Spring 2023

### Ryan Safner Associate Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/ioS23">ryansafner/ioS23</a> <a href="https://ioS23.classes.ryansafner.com"> ioS23.classes.ryansafner.com</a>

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# Outline

### [Reframing Oligopoly Theory](#3)
### [More Game Theory](#13)
### [Introduction to Sequential Games](#26)
### [Repeated Games](#39)
### [Finitely Repeated Games](#43)
### [Infinitely Repeated Games](#49)
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# Reframing Oligopoly Theory

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# Reframing Oligopoly Theory

![:scale 50%](../images/bertrand.jpg)
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.right-column[
- The “classic” models of oligopoly (Cournot & Bertrand) have significant flaws
  - Primarily: *static* games where players only interact once
  - In reality, players *continue* interacting and .hi-turquoise[change behavior in response to previous observed behavior]!

- Cournot *ignored* the possibility of collusion (we considered it alongside Cournot's model)
  - Introduced by Edward Chamberlin (1933) — *The Theory of Monopolistic Competition*
  - .hi-turquoise[It's in the interest of the firms to find a way to collude!]

]

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# Reaching a Collusive Bargain

.pull-left[
.smaller[
- Consider a .hi[profit possibilities frontier] between Coke and Pepsi should they collude
  - Using our conditions from Cournot (2.2)

- Point C: Cournot-Nash equilibrium
  - Each firm produces 30, gets $45 in profit

- Points `$M_c$`, `$M_p$`, if Coke or Pepsi were a monopolist, respectively
 - One firm produces 45 (other 0), gets $101.25 in profit
]
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.pull-right[
<img src="3.1-slides_files/figure-html/unnamed-chunk-1-1.png" width="432" style="display: block; margin: auto;" />

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# Reaching a Collusive Bargain

- Anything northeast of C is a Pareto improvement (for the firms)

- A bargaining problem between .red[Coke] and .blue[Pepsi]

- Coke would prefer point E, Pepsi point D, point F is a 50:50 split

- But in any case, a lot of room for a mutually-beneficial agreement to cooperate instead of (Cournot) competing
]

.pull-right[
<img src="3.1-slides_files/figure-html/unnamed-chunk-2-1.png" width="432" style="display: block; margin: auto;" />

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
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.smaller[
> “The present paper accepts the hypothesis that oligopolists wish to collude to maximize joint profits. It seeks to reconcile this wish with the facts, such as that collusion is impossible for many firms and collusion is much more effective in some circumstances than in others. The reconciliation is found in the problem of policing a collusive agreement, which proves to be a problem in the theory of information,” (44).

]

.source[Stigler, George J, 1964, “A Theory of Oligopoly,” *Journal of Political Economy* 72(1): 44-61]
]

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
]
]
]

.right-column[
.smaller[
> “We shall show that collusion normally involves much more than ‘the’ price...The colluding firms must agree upon the price structure appropriate to the transaction classes which they are prepared to recognize. A complete profit-maximizing price structure may have almost infinitely numerous price classes: the firms will have to decide upon the number of price classes in the light of the costs and returns from tailoring prices to the diversity of transactions,” (44-46).

]
.source[Stigler, George J, 1964, “A Theory of Oligopoly,” *Journal of Political Economy* 72(1): 44-61]
]

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
]
]
]

> “Let us assume that the collusion has been effected, and a price structure agreed upon. It is a well-established proposition that if any member of the agreement can secretly violate it, he will gain larger profits than by conforming to it. It is, moreover, surely one of the axioms of human behavior that all agreements whose violation would be profitable to the violator must be enforced. The literature of collusive agreements...is replete with instances of the collapse of conspiracies because of ‘secret’ price-cutting. This literature is biased: conspiracies that are successful in avoiding an amount of price-cutting which leads to collapse of the agreement are less likely to be reported or detected. But no conspiracy can neglect the problem of enforcement,” (46)

]
.source[Stigler, George J, 1964, “A Theory of Oligopoly,” *Journal of Political Economy* 72(1): 44-61]
]

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
]
]
]

.smallest[
> “Enforcement consists basically of detecting significant deviations from the agreed-upon prices. Once detected, the deviations will tend to disappear because they are no longer secret and will be matched by fellow conspirators if they are not withdrawn. If the enforcement is weak, however — if price-cutting is detected only slowly and incompetently — the conspiracy must recognize its weakness: it must set prices not much above the competitive level so the inducements to price-cutting are small...” (46).

]
.source[Stigler, George J, 1964, “A Theory of Oligopoly,” *Journal of Political Economy* 72(1): 44-61]
]

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
]
]
]

.smallest[
> “Policing the collusion sounds very much like the subtle and complex problem presented in a good detective story. [But] there is a difference: In our case the man who murders the collusive price will recieve the bequest of patronage. The basic method of detection of a price-cutter must be the fact that he is getting business he would not otherwise obtain,” (47).

]
.source[Stigler, George J, 1964, “A Theory of Oligopoly,” *Journal of Political Economy* 72(1): 44-61]

]

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# Reframing Oligopoly Theory

1911—1991

Economics Nobel 1982
]
]
]

- We should focus oligopoly theory less on static models of Cournot/Bertrand/etc competition

- Focus more on examining the types of conditions where firms can effectively form and maintain collusive agreements, and conditions where agreements break down into competition
  - Stigler (1964) specifically focused on the problem of a cartel policing against “secret price cutting”

- Consider more of a .hi-purple[dynamic game] of cooperation and/or competition between firms
]

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# More Game Theory

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# Game Theory: Some Generalizations

There's a *lot* more to game theory than a one-shot prisoners' dilemma:

- one shot vs. repeated game

- discrete vs. continuous strategies

- perfect vs. imperfect vs. incomplete/asymmetric information

- simultaneous vs. sequential games

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# Solution Concepts

.smallest[
- We use various .hi-purple[“solution concepts”] to allow us to predict an .hi-purple[equilibrium] of a game

- .hi[Nash Equilibrium] is the primary solution concept
  - Note it has *many* variants depending on type of game!

- Recall, .hi[Nash Equilibrium]: no players want to change their strategy given what everyone else is playing
  - All players are playing a best response to each other
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# Solution Concepts: Nash Equilibrium

- Important about Nash equilibrium:

1. N.E. `$\neq$` the “*best*” or *optimal* outcome
  - Recall the Prisoners' Dilemma!

2. Game may have *multiple* N.E.

3. Game may have *no* N.E. (in “pure” strategies)

]

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# Example: Coordination Game

- A .hi-purple[Coordination Game]
  - No dominant strategies

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# Example: Coordination Game

- .hi-purple[Two Nash equilibria]: (.red[A],.blue[A]) and (.red[B],.blue[B])
  - Either just as good
  - Coordination is most important

]

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# Example: Coordination Game

- Two general methods to solve for Nash equilibria:

.hi-purple[1) Cell-by-Cell Inspection]: look in each cell, does either player want to deviate?
  - .hi-purple[If no: a Nash equilibrium]
  - If yes: *not a Nash equilibrium*

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# Example: Coordination Game

- Two general methods to solve for Nash equilibria:

.hi-purple[2) Best-Response Analysis]: take the perspective of each player. If the other player plays a particular strategy, what is your strategy(s) that gets you the highest payoff?
  - Ties are allowed
  - .hi-purple[Any cell where both players are playing a best response is a Nash Equilibrium]

]

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# Example: Coordination Game

- Two general methods to solve for Nash equilibria:

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# Example: Coordination Game

- Two general methods to solve for Nash equilibria:

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# Example: Coordination Game

- Two general methods to solve for Nash equilibria:

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# A Change in the Game

- Two Nash equilibria again: (.red[A],.blue[A]) and (.red[B],.blue[B])

- But here (.red[A],.blue[A]) `$\succ$` (.red[B],.blue[B])!

]

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# A Change in the Game

.pull-right[
- .hi-purple[Path Dependence]: early choices may affect later ability to choose or switch

- .hi-purple[Lock-in]: the switching cost of moving from one equilibrium to another becomes prohibitive

- Suppose we are currently in equilibrium (B,B)

- .hi-purple[Inefficient lock-in]:
  - Standard A is superior to B
  - But too costly to switch from B to A

]

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# Introduction to Sequential Games

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# Sequential Games

- Consider a sequential game of Cournot competition between .red[Coke] and .blue[Pepsi]

- .red[Coke] moves first, then .blue[Pepsi], then the game ends

- Each player can:
  - **Cooperate**: produce cartel quantity (22.5)
  - **Defect**: produce Cournot quantity (30)
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# Sequential Games

- .hi-purple[Decision nodes]: decision point for each player
  - Solid nodes, I've labeled and color-coded by player (.red[C.1], .blue[P.1, P.2])

- .hi-purple[Terminal nodes]: outcome of game, with payoff for each player (profits)
  - Hollow nodes, no further choices
]

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# Sequential Games

1. (.red[Cooperate], .blue[Cooperate]): .red[50], .blue[50]
2. (.red[Cooperate], .blue[Defect]): .red[38], .blue[57] 
3. (.red[Defect], .blue[Cooperate]): .red[57], .blue[38]
4. (.red[Defect], .blue[Defect]): .red[45], .blue[45]
]

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# Strategies

.pull-left[
- .hi[(“Pure”) strategy].magenta[†]: a player’s **complete** plan of action for every possible contingency
 - i.e. what player will choose at **every possible decision node**, _even if it’s never reached_

- Think of a strategy like an .hi-turquoise[algorithm]:

> If we reach node 1, then I will play X; if we reach node 2, then I will play Y; if...
]

.tiny[.magenta[†] “Pure” is meant to contrast against **“mixed” strategies**, where players take a range of actions according to a probability distribution. That's beyond the scope of this class.]
]

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# Strategies

.pull-left[
- .hi-red[Coke] has `$2^1=2$` possible strategies:
  1. .red[Cooperate] at .red[C.1]
  2. .red[Defect] at .red[C.1]
]

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# Strategies

.pull-left[
- .hi-red[Coke] has `$2^1=2$` possible strategies:
  1. .red[Cooperate] at .red[C.1]
  2. .red[Defect] at .red[C.1]

- .hi-blue[Pepsi] has `$2^2=4$` possible strategies:
  1. (.blue[Cooperate] at .blue[P.1], .blue[Cooperate] at .blue[P.2])
  2. (.blue[Cooperate] at .blue[P.1], .blue[Defect] at .blue[P.2])
  3. (.blue[Defect] at .blue[P.1], .blue[Cooperate] at .blue[P.2])
  4. (.blue[Defect] at .blue[P.1], .blue[Defect] at .blue[P.2])

]

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# Solving the Game: Backward Induction

- To determine the outcome of the game, start with the *last-mover* (i.e. decision nodes just before terminal nodes) and work to the beginning

- A process of considering .hi-purple[“sequential rationality”]:

> “If I play X, my opponent will respond with Y; given their response, do I really want to play X?”

- What is that mover's best choice to maximize their payoff?
]

]

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# Solving the Game: Backward Induction

- We start at .blue[P.1] where .hi-blue[Pepsi] can:
  - .blue[Cooperate] to yield outcome (.red[50], .blue[50])
  - .blue[Defect] to yield outcome (.red[38], .blue[57])

- And .blue[P.2] where .hi-blue[Pepsi] can:
  - .blue[Cooperate] to yield outcome (.red[57], .blue[38])
  - .blue[Defect] to yield outcome (.red[45], .blue[45])
]

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# Solving the Game: Backward Induction

- .hi-blue[Pepsi] will .blue[Defect] if the game reaches node .blue[P.1] and .blue[Defect] if the game reaches node .blue[P.2]

- Recognizing this, what will .hi-red[Coke] do?
]

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# Solving the Game: Backward Induction

- Work our way up to .red[C.1] where .hi-red[Coke] can:
  - .red[Cooperate], knowing .hi-blue[Pepsi] will .blue[Defect], to yield outcome (.red[38], .blue[57])
  - .red[Defect], knowing .hi-blue[Pepsi] will .blue[Defect], to yield outcome (.red[45], .blue[45])

]

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# Solving the Game: Backward Induction

- .hi.purple[Nash Equilibrium]:

(.red[Defect], .blue[(Defect, Defect)])

]

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# Solving the Game: Pruning the Tree

- As we work backwards, we can .hi-turquoise[prune] the branches of the game tree
  - Highlight branches that players **will** choose
  - Cross out branches that players will **not** choose

- .hi-purple[Equilibrium path of play] is highlighted from the root to one terminal node
  - All other paths are not taken

]

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# Repeated Games

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# Prisoners' Dilemma, Reprise

- Each player's preferences: 
 - 1st best: you Defect, they Coop. ("temptation payoff")
 - 2nd best: you both Coop.
 - 3rd best: you both Defect
 - 4th best: you Coop., they Defect ("sucker's payoff")

- Nash equilibrium: (.red[Defect], .blue[Defect])
  - (.red[Coop.], .blue[Coop.]) an unstable Pareto improvement

]
]

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# Prisoners' Dilemma: How to Sustain Cooperation?

- .hi-purple[How can we sustain cooperation in Prisoners' Dilemma?]

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# Repeated Games: Finite and Infinite

.pull-left[
.smallest[
- Analysis of games can change when players encounter each other *more than once*

- .hi[Repeated games]: the same players play the same game multiple times, two types:

- Players know the *history* of the game with each other

- .hi-purple[Finitely-repeated game]: has a known final round

- .hi-purple[Infinitely-repeated game]: has no (or an unknown) final round

]
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.pull-right[
.center[
![](../images/infinitelyrepeated.JPG)
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# Finitely-Repeated Games

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