SIGNALING GAMES: Dynamics and Learning

1. SIGNALING GAMES:Dynamics and Learning NASSLI 2016 Monday

2. Course Plan • Monday Signaling Games • Tuesday Information in Signals • Wednesday Evolutionary Dynamics • Thursday Learning Dynamics • Friday Networks

3. Oxford University Press

5. Conventional Meaning in Lewis Signaling Games

6. W. V. O Quine 1908-2000 “Truth by Convention” 1936

7. Bertrand Russell 1872-1970 “We can hardly presuppose a parliament of hitherto speechless elders, meeting together and agreeing to call a cow a cow and a wolf a wolf.”

8. Democritus The Laughing Philosopher 400BC Proclus states that Pythagoras and Epicurus agree with Cratylus, but Democritus and Aristotle agree with Hermogenes, the former that names arise by nature, the latter that they arise by chance. Democritus Laughing, by Hendrick ter Brugghen, 1628

9. 1. Lewis on Convention

10. Overview A convention is a certain kind of equilibrium in a coordination game. … where everyone plays their part in this equilibrium … and all the foregoing is common knowledge.

11. Equilibrium Nash No gain from unilateral deviation. Strict Nash Loss from unilateral deviation. Lewis Loss for everyone from unilateral deviation.

12. Coordination Games The Winding Road

13. Coordination Game? Stag Hunt

14. Coordination Game Stag Hunt 2

15. Common Knowledge Now: We all know that p. We all know that we all know that p. We all know that we all know that we all know that p. etc. Lewis We all have reason to believe that p. We all have reason to believe that we all have reason to believe that p. How important?

16. 2. Lewis Signaling Games

17. Lewis Sender-Receiver Games • States, Signals, Acts (equal numbers) • Nature chooses a state according to some probability. • Senderobserves the state and sends a signal. • Receiverobserves the signal and chooses an act. • Pure Common Interest • Both get 1 if act matches the state, 0 otherwise

18. For example

19. How is this different? Extensive form Sender’s strategy - map from states to signal sent Receiver’s strategy - map from states to signal sent

20. How is this different? Extensive form Sender’s strategy - map from states to signal sent Receiver’s strategy - map from states to signal sent Incomplete Information Probabilities of the states

21. Equilibrium Analysis(states equiprobable) 2 Conventions -- Signaling Systems 1 => A =>1 2 => B => 2 1 => B =>1 2 => A => 2 But also there are also Pooling Equilibria: Sender’s probabilities independent of state Receiver’s probabilities independent of signal

22. Equilibrium Analysis(state 2 more probable) 2 Conventions -- Signaling Systems 1 => A =>1 2 => B => 2 1 => B =>1 2 => A => 2 Pooling Equilibria are now: Sender’s probabilities independent of state Receiver always guesses state 2

23. Partial Pooling Equilibrium (states equiprobable)

24. N states M signals N acts? To few signals create information bottlenecks. Too many signals create possibility of synonyms. Lewis signaling systems?

25. Too Many Signals • Example: • 2 states, 3 signals, 2 acts

27. Too FewsSignals • Example: • 3 states, 2 signals, 3 acts

28. Now we must have bottlenecks. For example Sender Receiver s1 => t1 t1 => a1 s2 => t2 t2 => a2 s3 => t2 12 efficient equilibria

29. From Donaldson et. al. JTB 2007

30. In light of these, does Lewis need to be revised? • The notion of a signaling system? • The notion of a convention?

31. Back to the SimplestLewis Signaling Games Remember Quine’s skeptical challenge. Explain existence of signaling systems without circularity. Does Lewis do this?

32. How do we get from pooling tosignaling?

33. Thank you