Principles of Game Theory

1. Principles ofGame Theory Repeated Games

2. 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.

3. Wisdom in Proverbs “Доверяй, НоПроверяй” “Trust, but Verify”

4. Prisoners’ Dilemma • You can also think of the PD as a price war between firms: Equilibrium: $54 K Cooperation: $60 K

5. 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

6. 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.

7. 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?

8. 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

9. 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.

10. 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…

11. 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’

12. 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?

13. 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.

14. 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?”

15. 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

16. Payoff Stream profit 72 cooperate 60 defect 54 t t+1 t+2 t+3 time

17. 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)

18. 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)

19. 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?

20. “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.

21. 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.