Lecture 2: What is game theory?*

1. Lecture 2: What is game theory?* • Formal analysis of strategic behaviour, that is, relation between inter-dependent agents. The interplay of competition and cooperation. • Informal definition: game theory and analysis is the art of: • Seeing through the future. Decisions taken today may have an impact in future decisions, both by myself and by my competitors. • Seeing through your competitors’ eyes. Your results depend not only on your decisions but on your competitors’ decisions as well. • (*I am grateful to Professor L. Cabral for some of these notes)

2. Model Strategic Interactions • To create an awareness for the importance of strategic considerations in many circumstances of business life (and, in fact, of everyday life). I think it is instructive to use game theory analysis... Game theory forces you to see a business situation over many periods from two perspectives: yours and your competitor’s. — Judy Lewent (in “Scientific Management at Merck: An interview with CFO Judy Lewent,” Harvard Business Review, January-February 1994).

3. Game theory and strategy:“content” vs “process” Today's most successful managers craft their strategies on the basis of knowledge of their own companies … At their worst, game theorists represent a throw back to the days of such whizz kids as Robert McNamara … who thought that rigorous analytical skills were the key to success. Managers have much to learn from game theory -- provided they use it to clarify their thinking, not as a substitute for business experience. -- The Economist

4. Historical notes [JvN, also co-inventor of the computer/ operating system.] The 1995 U.S. spectrum auction was partly designed by game theorist Paul Milgrom. John von Neuman, percursor of game theory. John Nash, 1994 Nobel laureate, the first game theorist to receive the prize. Game theory is now commonly used by various consulting companies such as McKinsey.

5. Basic concepts • What is a game? • Players (e.g., Vodaphone and Cellnet) • Rules (e.g., simultaneously choose prices) • Strategies (e.g., a price between 10 and 30) • Payoffs (e.g., sales minus production costs) • What can I do with it? • Determine how good each of my strategies is • Figure out what my rival is probably going to do

6. How to represent a game • Extensive form or game-tree form: • useful when decisions are sequential. • Normal form: • useful when decisions are simultaneously taken. • Important note: the meaning of “simultaneously” • Order of moves VERY important to outcome

7. L C R 5 6 7 T 9 8 1 3 5 6 M 2 0 1 7 6 8 B 2 3 3 Normal form: example Player B Player A

8. Dominant and dominated strat’s • Dominant strategy: payoff is greater than any other strategy regardless of rival’s choice. • Rule 1: if there is one, choose it and that’s the end of it. • Dominated strategy: payoff is lower than some other strategy regardless of rival’s choice. • Rule 2: do not choose dominated strategies. • Check whether there are dominant and/or dominated strategies in the example above. What can we say based on this?

9. Equilibrium • Sometimes a game can be “solved” just by looking at dominant and dominated strategies (e.g., example above). • However, there are many games for which this does not work. • Concept of equilibrium: a rest point of the system. • Nash equilibrium: Situation such that, given what other players are doing, no player would want to change strategy unilaterally.

10. The prisoner’s dilemma Rest of Cartel’s Output 2 4 Example: output setting (million barrels a day) by OPEC members 450 500 2 Saudi Arabia’s Output 450 375 375 400 4 500 400

11. The prisoner’s dilemma (cont) • Dominant strategies: high output. • Equilibrium payoffs are (400,400), worse than those attained by low output, (450,450). • Conflict between individual incentives and joint incentives. • Typical of many business situations.

12. Ballet Soccer 5 2 Ballet 10 2 Husband 0 10 Soccer 5 Battle of the Sexes • Battle of Standards Wife 0 Two Nash Equilibria (B,B), (S,S) – Illustrates Problems with NE

13. Dynamic Games & Subgame Perfection • In Nash Equilibrium, players take opponents’ strategies as given & do not consider the possibility of influencing them • In games in which a player chooses some actions after observing some of his opponents’ actions (dynamic games), the above conjecture is naïve and leads to some absurd Nash equilibria (see example next slide) • Subgame perfection is a refinement that mitigates some of the deficiencies of Nash equilibrium • An outcome is said to be subgame perfect if it induces a Nash equilibrium in every subgame of the original game. • Subgame perfect equilibria can be found by backwards induction.

14. Example: Subgame Perfection Incumbent dioxide industry: Monopoly Capacity (M) Excess Capacity (C) Entrant N Enter E E Not Enter N 2 1 4 0 1 -1 3 0 Nash Equilibria: {M, [ E,E ] } , {C, ( E,N ) } Subgame Perfect Equilibrium: {C, ( E,N ) }

15. One-shot vs repeated games • A repeated game is simply a game made up of a finite or indefinite repetition of a one-shot game. • The equilibrium of a repeated game may be very different from the repetition of the equilibrium of the one-shot game. Reasons: • Learning about competitors • Influencing their learning/expectations • achieving a “co-operative solution” • How repetition can make co-operation an equilibrium: tit-for-tat, grim strategies, etc.

16. Repeated Games Example: Cartel Rest of Cartel’s Output 2 4 450 500 2 Saudi Arabia’s Output 450 375 375 400 4 500 400 Unique Nash equilibrium of one shot game is (4,4)

17. Repeated Games: Example Continued Suppose that each firm adopts the following Tit for Tat “trigger” strategy: I will produce at the “cooperative” level as long as my competitor did so if the previous period. If, however, my competitor deviates from that level, I will produce 20 million barrels forever. (Punishment threats must be credible to be effective.) Each player must determine whether it is worthwhile to deviate from the cooperative output level. Such a deviation results in a short term gain. But there are long term losses. Hence there is a tradeoff which depends, in part, on the discount rate.

18. Cooperation yields the following payoff: 450+ 450/(1+r) + 450/(1+r)2 +…. = 450(r +1)/r Deviation yields the following payoff 500+ 400/(1+r) + 400/(1+r)2 +….= 500+ 400/r Cooperation can be sustained if: r<(450-400)/(500-450)=1, that is if the discount rate is less than 100%. In general cooperation can be sustained if r<(cartel profit – one shot eq. Profit)/(deviation profit - cartel profit).

19. Summary of Repeated Games Without repetition of play, players are less likely to cooperate Repetition can create much stronger incentives to cooperate Trading off the gains from being non-cooperative today with the last future cooperation Tradeoff of SR gains vs. LR losses means that the discount rate matters