Principles of Game Theory

1. Principles of Game Theory Lecture 12: Cheap Talk

2. Administrative • Homework due tomorrow • Quiz Thursday • Repeated Games – look at chapter 11 in Dixit and the end of Chp 2 in Gibbons • No class Sunday, Sept 29th ; next quiz Oct 6 (week from Sunday)

3. Rational Irrationality • U.S. / U.S.S.R. nuclear deterrence • Mutually Assured Destruction (MAD) • like Grim Trigger Strategy • Proportional Response • like Tit-for-Tat • Dr Strangelove.

4. Doomsday device Is it credible? • Severity “Create fear in the mind of the enemy” • Irreversibility “It is essential” • Irrationality “Not something a sane man would do” • Practicality “It wasn’t a practical deterrent” • Clarity “Tell the world”

5. Commitment Under Uncertainty For you seniors: • An offer you can’t refuse: • After a seemingly successful interview, the interviewer asks where the firm ranks on your list of potential employees • BUT Before answering, you are told: • The firm only hires applicants who rank it first • If the firm is in fact your first choice, then you must accept a job offer in advance, should one be made

6. Commitment Under Uncertainty Why make such offers? • Take advantage of your uncertainty • Take advantage of your risk-aversion • Make you commit before they do!

7. Games in real life • Last time I asked you to write down an interaction as a game. So let’s hear about them

8. Communication • So far we’ve assumed that the players can’t talk before they play the game • At most players could make moves that “signal” their intentions. • But people talk all the time before they actually interact. How does that change things? • Behaviorally: lots of implications (not necessarily consistent) • Theoretically: it depends.

9. Back to simple games • Split into two groups and have a sheet of paper ready • Now we’ll allow communication…

10. Game 1 • Group 1: you may, but are not compelled to, write a message to Group 2 before selecting your action

11. Game 2 • Group 2: you may, but are not compelled to, write a message to Group 1 before selecting your action

12. Game 3 • Group 1: you may, but are not compelled to, write a message to Group 2 before selecting your action

13. Any strategic moves? • Can anyone think of a strategic move group 2 – the receiver – could have made in the last game?

14. Talk is cheap… • These games are examples of what we call “Cheap Talk” games • Sending the message is costless (hence cheap) • How do we know that a cheap (costless) message is informative? • Short answer is that we don’t… • At best it changes the beliefs of the players (note: why both players and not just the receiver?)

15. Equilibria in Cheap Talk Games • We’ll analyze the strategy space and equilibria in more detail next time, but: • With perfectly aligned interests (pure coordination – game 1) cheap talk can help with equilibrium selection • Similarly with partially aligned interests (game 3) • With opposing interests, why should a player believe what the other is saying?

16. Equilibria in Cheap Talk Games • With communication, there are lots, and LOTS, of equilibria • Often “nonsensical” ones are referred to as babbling equilibria • It’s clearest to see these in zero-sum games • But…they actually still exist in games of perfectly aligned interests as well • Much in the same way non-subgame perfect equilibria exist in extensive form games.

17. Bayes Theorem In preparation for next time, I want to make sure you are familiar with Bayes Theorem: But it’s often useful to use the law of total probability:

18. Bayes Theorem • When those A’s and B’s (types of other players and messages that can be sent) are continuous it requires integration over a densities fAand fB:

19. Bayes Tables But when there are only a couple of events (types of players, etc), we can construct a table: where a and b are often described as the accuracy of the test: • a = P(“A”|A) or using the notation in the table P(X|A) • b = P(“B”|B) or using the notation in the table P(Y|B)

20. An Example • H = Patient HIV+ is either {0,1} • Patient is actually either {negative, positive} • T = Patient tests positive for HIV+ is either {0,1} • Or the test says {“negative, “positive”} • What could happen? • “True Positive:”P(H = 1 | T = 1) is the probability of being HIV+ given testing positive for HIV+ • “False Positive:” P(H = 0 | T = 1) is the probability of not being HIV+ given testing positive for HIV+ • “False Negative”: P(H =1 | T = 0) is the probability of being HIV+ given testing negative for HIV+ • “True Negative”: P(H=0 | T=0) is the probability of not being HIV+ given testing negative for HIV

21. An Example Assume • P(HIV=1) = 0.0076 • P(“Positive” | HIV = 1 ) = 0.976 • P(“Negative” | HIV = 0 ) = 0.995 • What’s the probability of having HIV given the test came back positive? P( HIV | “Positive”)

22. The Probability Matrix 1 – Rate of True Positives (“False Negatives”) Rate of True Positives (“Accuracy”) Joint Probability 0.024 × 0.0076 1 – Rate of True Negatives (“False Positives”) Probability Test Says NOT HIV+ 0.0002 + 0.9874 P(HIV|“Positive”) = 0.0074/0.0124 = 0.5992