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Principles of Game Theory. Repeated Games. Administrative. Midterm exam results soon. Let me know if you want to cover any problems in class. Final exam 3 weeks from today (last day of class). Mid-mini feedback Slight change in the schedule:

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administrative
Administrative
  • Midterm exam results soon.
    • Let me know if you want to cover any problems in class.
  • Final exam 3 weeks from today (last day of class).
  • Mid-mini feedback
  • Slight change in the schedule:
    • I’m moving repeated games up because I feel they fit better with extensive-form and stage games: Chapter 11 in Dixit and part of Chapter 2 in Dixit.
wisdom in proverbs
Wisdom in Proverbs

“Доверяй, НоПроверяй”

“Trust, but Verify”

prisoners dilemma
Prisoners’ Dilemma
  • You can also think of the PD as a price war between firms:

Equilibrium: $54 K

Cooperation: $60 K

price wars as pds
Price wars as PDs
  • Why does the dilemma occur?
    • Firms:
      • Lack of monopoly power
      • Homogeneity in products and costs
      • Overcapacity
      • Incentives for profit or market share
    • Consumers
      • Price sensitive
      • Price aware
      • Low switching costs
    • Interaction
      • No fear of punishment
      • Short term or myopic play
irrational cooperative behavior
Irrational Cooperative Behavior

Is it irrational to cooperate in the Prisoners’ Dilemma?

    • Private rationality  collective irrationality
      • The equilibrium that arises from using dominant strategies is worse for every player than the outcome that would arise if every player used her dominated strategy instead
    • Goal:
      • To sustain mutually beneficial cooperative outcome overcoming incentives to cheat

(A note about tacit collusion)

  • Firm interactions:
    • No fear of punishment  exploit repeated play
    • Short term play  introduce uncertainty.
repeated games
Repeated Games
  • Repeated Games are just stage games
    • We’ve already covered, so why are we still talking about them? Well…
      • Two main types:
      • Finitely repeated and
      • Infinitely repeated
      • Why infinitely repeated – none of us live forever?
finitely repeated games
Finitely Repeated Games

We’ve already seen these…

  • Each repetition is a stage in the multi-stage game.
  • How do solve them?
    • Subgame perfection: Work backwards. If it ends at stage T, find the equilibrium in stage T and rollback.
    • The strategy space can be really big in general
repeated price wars
Repeated Price Wars
  • Think about the PD repeated for 10 periods. What would happen in the 10th round?
  • If there is a unique equilibrium in every stage of the game, then there is a unique SPNE.

 Cooperation is not an equilibrium with a finite and known length of time.

finite repetition take ii
Finite Repetition take II
  • Unraveling prevents cooperation if the number of periods is fixed and known.
  • What if ends probabilistically?
    • The game continues to the next period with some probability p
      • Mathematical trickery: it’s equivalent to an infinitely repeated game with discounting…
discounting
Discounting
  • Easier to work with discount factors δ rather than an interest rate r. δ = 1/ (1+r); 0 < δ < 1.
    • e.g.: interest rate of 0.25, implies δ=1/(1.25) = 0.8. So a $100 invested today would be δ*100 = 80.
  • So why is a probabilistic end the same?
    • Invest $50.
    • Expected Value of future = 50δ * p
    • Since p is the same every period, δ*p = δ’
infinitely repeated games
Infinitely repeated games
  • So now we can call a probabilistic end an infinitely repeated game.
  • Infinitely repeated
    • No last period  no rollback.
    • Can only use history-dependent strategies.
      • What would you use?
trigger strategies
Trigger strategies
  • Trigger strategies (more common than you think)
    • Begin cooperating and cooperate as long as other player does
    • Upon observing a defection, punish for some period of time.
  • Grim Trigger
    • Cooperate but defect forever if there is a deviation
    • ∞ memory
  • Tit-for-tat
    • Do what your opponent does: cooperate if she cooperated in the previous period. Defect if she defected.
      • 1 period of memory.
trigger strategy extremes
Trigger Strategy Extremes
  • Grim trigger is
    • least forgiving
    • longest memory
    • ≈ nuclear option
    • deterrence but lacks credibility

Grim trigger answers:

“Is cooperation possible?”

  • Tit-for-Tat is
    • most forgiving
    • shortest memory
    • proportional
    • credible but lacks deterrence

Tit-for-tat answers:

“Is cooperation easy?”

why cooperate against grimm trigger
Why Cooperate against Grimm Trigger?
  • Cooperate if the PV of cooperation is is higher
  • Cooperate: 60 today, 60 next year, 60 … 60
  • Defect: 72 today, 54 next year, 54 … 54
payoff stream
Payoff Stream

profit

72

cooperate

60

defect

54

t

t+1

t+2

t+3

time

why is it an equilibrium
Why is it an equilibrium?

How do we calculate if Grimm Trigger (or TFT) is an equilibrium?

  • It’s an finite game with discounting: must calculate infinite sums.
  • So the present discounted value of receiving k in every round is just: kδ / (1+δ)
grimm trigger eq
Grimm Trigger Eq
  • To verify it’s an equilibrium must show that there is no incentive to deviate. So check the PV(cooperate) ≥ PV(defect):
  • So PV(cooperate) > PV(defect): only if δ > 2/3, or an interest rate r < 0.5.

PV(cooperation)

60 + 60+60+…

  • 60 + 60δ / (1-δ)

PV(defection)

72 + 54+54+…

  • 72 + 54δ / (1-δ)
is tit for tat an equilibrium
Is Tit for Tat an equilibrium?
  • You’ll find out for yourselves on the homework.
  • What about other strategies?
    • Yes. Lots.
  • δ will depend on the size of the payoffs (i.e., not the ordinal ranking)
    • Surprising?
folk theorem
“Folk” Theorem

Welcome to one of the most obvious and most confusing theorems in Game Theory: the “folk” theorem.

  • Theorem: For any infinitely repeated game G, for any feasible set of payoffs (π1, π2, …) that are ≥ to the payoffs of an equilibrium in the game G, there exists a δ that achieves (π1, π2, …) as an equilibrium.
folk theorem1
Folk Theorem
  • In the context of the PD: any outcome that on average yields the mutual defection outcome payoffs, or higher, can be sustained as a subgame perfect Nash equilibrium, for a large enough discount factor.