Innuendos and Common Knowledge

1. Innuendos and Common Knowledge

2. Consider the following examples of innuendos: "I hear you're the foreman of the jury in the Soprano trial. It's an important civic responsibility. You have a wife and kids. We know you'll do the right thing." [a threat] "Gee, officer. I was thinking that maybe the best thing would be to take care of the ticket here, without going through a lot of paperwork." [a bribe] "Would you like to come up and see my etchings?" [a sexual advance] "We're counting on you to show leadership in our Campaign for the Future." [a solicitation for a donation] Notice that, in each example, the speaker obfuscates their message Source: Lee and Pinker 2010

3. “The puzzle for social psychology and psycholinguistics is why people so often communicate in ways that seem inefficient and error-prone rather than stating their intentions succinctly, clearly, and unambiguously.” Source: Lee and Pinker 2010

4. Today, we’ll provide one potential explanation (Next Class we will provide evidence for this explanation) (In your next pset, you will prove some of the details)

5. Here’s the intuition: Innuendo ensures that the recipient gets the message while at the same time preventing the message from being common p-believed We will use an information structure to formally represent innuendos Thereby allowing the speaker to: 1) Prevent switching equilibria in a game where common p-beliefs matter 2) Influence the listener’s behavior in a game where only first-order beliefs matter We will combine these individual games into a large game in which it is a Nash equilibrium to use innuendo

6. We’ll use the same information structure as we did in for “intermediaries,” but applied to innuendos…

7. Innuendos aren’t always understood Ω= {silent, innuendo not understood, innuendo understood} Player 1 can’t tell whether player 2 understood the innuendo When player 2 doesn’t understand the innuendo, it’s as though it were not stated π1= {{silent}, {innuendo not understood, innuendo understood} π2= {{silent, innuendo not understood}, {innuendo understood}} For now, assume innuendo happens randomly. Also assume that when it happens, player 2 understands half the time μ(silent)=1/2 μ(innuendo not understood)=1/4 μ(innuendo understood)=1/4

8. Now we’re ready to see the game in which p-beliefs matter

9. A B Ω = {silent, innuendo not understood, innuendo understood}π1 = {{silent}, {innuendo not understood, innuendo understood}π2 = {{silent, innuendo not understood}, {innuendo understood}}μ(silent)=1/2μ(innuendo not understood)=1/4μ(innuendo understood)=1/4 2, 2 0, 0 A 0, 0 1, 1 B S1({silent}) = A S1 (innuendo) =B

10. Now, here is a game where common p-beliefs don’t matter, just first order beliefs…

11. The “Identification Game”: player 2 tries to identify 1’s type, and they each prefer that 2 succeeds 1 is Type B 0,0 1,1 1 is Type A A Player 2 0, 0 1,1 B

12. Ω = {silent, innuendo not understood, innuendo understood}π1 = {{silent}, {innuendo not understood, innuendo understood}π2 = {{silent, innuendo not understood}, {innuendo understood}}μ(silent)=1/2μ(innuendo not understood)=1/4μ(innuendo understood)=1/4 1 is Type B 0,0 1,1 1 is Type A A Player 2 0, 0 1,1 B S2 ({silent, innuendo not understood}) =A S2({innuendo understood}) = B

13. Until now we have assumed the information structure is “exogenous” But now we will let player 1 choose whether to use an innuendo Afterwards the players will play the guessing game or the coordination game (randomly determined)

14. The “Innuendo Game” 1 is Type B B A 1 is Type A 1,1 2,2 0,0 0,0 A A Type A Innuendo .5 .5 Silent 1 1 Player 2 0,0 0,0 Explicit 1,1 1,1 B B Type B

15. There exists a NE of the Innuendo game where: Player 1 Type A: remains silent, plays A in coordination game Type B: uses innuendo, plays A in coordination game Player 2 1 is silent: plays A in the coordination game and A in the guessing game 1 uses innuendo and 2 understands: plays A in the coordination game and guesses B 1 speaks explicitly: plays B in both games

16. The “Innuendo Game” 1 is Type B B A 1 is Type A 1,1 2,2 0,0 0,0 A A Type A Innuendo .5 .5 Silent 1 1 Player 2 0,0 0,0 Explicit 1,1 1,1 B B Type B

17. The “Innuendo Game” 1 is Type B B This is the situation we observe that we’re puzzled by! A 1 is Type A 0,0 2,2 1,1 0,0 A A Type A Innuendo .5 .5 Silent 1 1 Player 2 0,0 0,0 Explicit 1,1 1,1 B B Type B If player 2 understands she plays B, If she doesn’t understand, she plays A

18. The “Innuendo Game” 1 is Type B B If player 1 were to be explicit… A 1 is Type A 1,1 2,2 0,0 0,0 A A Type A Innuendo .5 .5 Silent 1 1 Player 2 0,0 0,0 Explicit 1,1 1,1 B B Type B

19. How does this explanation fit our motivating examples? E.g., why would a man say to a woman “would you like to come up to see my etchings”?

20. Innuendos aren’t always understood Ω= {silent, he says “etching” & she is naive, he says etchings & she isn’t naïve} He can’t tell whether she is naive When she is naive, it’s as though it wasn’t stated π1= {{silent}, {etchings & naive, etchings & sophisticated}} π2= {{silent, etchings & naive}, {etchings & sophisticated}} For now, assume innuendo happens randomly. Also assume that when it happens, player 2 understands half the time μ(silent)=1/2 μ(innuendo not understood)=1/4 μ(innuendo understood)=1/4

21. StayFriends If she is NOT interested, the next day at work they play the following coordination game Do not Stay Friends 2, 2 0, 0 Stay Friends 0, 0 1, 1 Do not Stay Friends Their friendship cannot be affected by his innuendo

22. If she IS interested, she chooses whether or not to go upstairs with him 1 is interested 1 isn’t interested 0,0 1,1 She wants to go up, if she thinks he is also interested (otherwise, she would prefer to get home early) Don’t Go 0, 0 1,1 He too would only want her to come up if he is in fact Interested Go Up She will go up if she thinks he is interested, regardless of what she thinks he thinks she thinks…

23. The “Innuendo Game” He is interested Not Friends Friends He isn’t interested 2,2 1,1 0,0 0,0 Friends A He isn’t interested “Etchings” .5 .5 “Have a good night” 1 1 Player 2 0,0 0,0 “Do you want…” 1,1 1,1 Not Friends B He is interested She is interested

24. So we have a game-theoretic explanation for innuendos! Aside from a post-hoc explanation…what have we gained from this formalism?

25. What property does the speech act have to have to work like an innuendo? Our theorem suggests the answer: The speech act has to create common p-beliefs Notes: Intermediaries are expected to work the same way. But “explicit speech while a bus goes by” won’t

26. What property do the games have to have for innuendos to play this role? Our theorem suggests the answer… There needs to be a “mix” of the following two games: • one which has multiple equilibria • one where the optimal choice depends on the private information of the other player but does not depend on the choice of the other player

27. 2 more insights…

28. When should one use an innuendo? Our model tells us the costs/benefits of an innuendo relative to explicit statement: Cost: reduces the chance of recipient “getting the message” Benefit: increases the chance that you don’t switch to bad equilibria So use when: Care a lot about the game with multiple equilibria Recipient is likely to “get the message” when use innuendo Etc.

29. Suppose there are two possible innuendos you could use in a particular situation The theory teaches us that it’s better to choose the one the recipient is most likely to “get”, provided he doesn’t know you know he will get it This is counterintuitive. Without the theory, one might have thought the point of an innuendo were to obscure first-order beliefs

30. Next time, we’ll show you some studies that provide evidence for this explanation