innuendos and common knowledge

Innuendos and Common Knowledge

Consider the following examples of innuendos:

"I hear you're the foreman of the jury in the Soprano trial. It's animportant civic responsibility. You have a wife and kids. We knowyou'll do the right thing." [a threat]

"Gee, officer. I was thinking that maybe the best thing would be totake care of the ticket here, without going through a lot ofpaperwork." [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 theFuture." [a solicitation for a donation]

Notice that, in each example, the speaker obfuscates their message

Source: Lee and Pinker 2010

“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

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)

Here’s the intuition:

Innuendo ensures that the recipient gets the message while at thesame 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

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

Innuendos aren’t always understood

Ω = {silent, innuendo not understood, innuendo understood}

Player 1 can’t tell whether player 2 understood the innuendoWhen 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)=1/2μ(innuendo not understood)=1/4μ(innuendo understood)=1/4

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

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

S1({silent}) = A S1 (innuendo) =B

2, 2 0, 0

0, 0 1, 1

A

B

A B

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

The “Identification Game”: player 2 tries to identify 1’s type, and they each prefer that 2 succeeds

1,1 0,0

0, 0 1,1

A

B

1 is Type B1 is Type A

Player 2

1,1 0,0

0, 0 1,1

A

B

1 is Type B1 is Type A

Player 2

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

S2 ({silent, innuendo not understood}) =A S2({innuendo understood}) = B

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)

2,20,00,01,1

A

B

BA

1,1 0,00,01,1

A

B

1 is Type B1 is Type A

Player 2

1

Type A

Type B

.5

1

Innuendo

Silent

Explicit

.5

The “Innuendo Game”

There exists a NE of the Innuendo game where:

Player 1Type A: remains silent, plays A in coordination gameType B: uses innuendo, plays A in coordination game

Player 21 is silent: plays A in the coordination game and A in the guessing

game1 uses innuendo and 2 understands: plays A in the coordination

game and guesses B1 speaks explicitly: plays B in both games

2,20,00,01,1

A

B

BA

1,10,00,01,1

A

B

1 is Type B1 is Type A

Player 2

1

Type A

Type B

.5

1

Innuendo

Silent

Explicit

.5

The “Innuendo Game”

2,20,00,01,1

A

B

BA

1,1 0,00,0 1,1

A

B

1 is Type B1 is Type A

Player 2

1

Type A

Type B

.5

1

Innuendo

Silent

Explicit

.5

The “Innuendo Game”

If player 2 understands she plays B,If she doesn’t understand, she plays A

This is the situation we observe that we’re puzzled by!

2,20,00,0 1,1

A

B

BA

1,1 0,00,0 1,1

A

B

1 is Type B1 is Type A

Player 2

1

Type A

Type B

.5

1

Innuendo

Silent

Explicit

.5

The “Innuendo Game”If player 1 were to be explicit…

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”?

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 naiveWhen 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)=1/2μ(innuendo not understood)=1/4μ(innuendo understood)=1/4

2, 2 0, 0

0, 0 1, 1

Stay Friends

Do notStay Friends

Stay FriendsDo notStay Friends

If she is NOT interested, the next day at work they play the following coordination game

Their friendship cannot be affected by his innuendo

If she IS interested, she chooses whether or not to go upstairs with him

1,1 0,0

0, 0 1,1

Don’t Go

Go Up

1 is interested1 isn’t interestedShe wants to go up, if she thinks he is also interested(otherwise, she would prefer to get home early)

He too would only want herto come up if he is in fact Interested

She will go up if she thinks he is interested,regardless of what she thinks he thinks she thinks…

2,20,00,01,1

Friends

Not Friends

Not FriendsFriends

1,1 0,00,01,1

A

B

He is interestedHe isn’t interested

Player 2

1

He isn’t interested

He is interested

.5

1

“Etchings”

“Have a good night”

“Do you want…”

.5

The “Innuendo Game”

She is interested

So we have a game-theoretic explanation for innuendos!

Aside from a post-hoc explanation…what have we gained from this formalism?

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

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:

1) one which has multiple equilibria 2) 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

2 more insights…

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 equilibriaRecipient is likely to “get the message” when use innuendoEtc.

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

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