Symbolic Gestures and Common Knowledge

Reall…

Salala Pakistan, Nov 26 2011…

US gunships responding to request for air-support accidentally killed 24 Pakistani soldiers

In response…

Pakistan closed supply routes

Leading to ~$1 billion in extra shipping fees

Until the US apologized…

“We are sorry for the losses suffered by the Pakistani military”

-Hillary Clinton, July 3 2012

Over half a year later!

Then, immediately…

“…the ground supply lines into Afghanistan are opening”

Why wouldn’t US just say sorry? -Mere words? -Worth a billion dollars?

Why would Pakistan care?-Again, mere words?-US could just “fake it”; why “believe”?

We make a big deal out of other words too like…

The main features of the treaty were mutual recognition, cessation of the state of war that had existed since the 1948 Arab–Israeli War, normalization of relations and the complete withdrawal by Israel of its armed forces and civilians from the Sinai Peninsula which Israel had captured during the Six-Day War in 1967.

-Source: http://en.wikipedia.org/wiki/Israel-Egypt_Peace_Treaty

And not just words…

Who goes through the door first? Barak and Arafat at Camp David

More generally…

Why do symbolic gestures matter?

Is this just silliness, and our laws and militaries and relationships should focus on the “tangibles”?

Or if symbolism does matter,

WHEN does it matter?

WHAT makes it work or not work?

Our answer:

Ω = {symbolic gesture, no symbolic gesture}

π1 = { {symbolic gesture}, {no symbolic gesture} }π2 = { {symbolic gesture}, {no symbolic gesture} }

a, a b, c

c, b d, d

A

B

A B

a > c , d > bS1({symbolic gesture}) = A S2({symbolic gesture}) = AS1({no symbolic gesture}) = B S2({no symbolic gesture}) = B

Symbolic gesture can influence behavior, because can act as a coordination device!

In what kinds of situations is this possible?

Our model tells us…

Subsequent interaction needs to have multiple equilibria!

Ω = {symbolic gesture, no symbolic gesture}

π1 = { {symbolic gesture}, {no symbolic gesture} }π2 = { {symbolic gesture}, {no symbolic gesture} }

v/2-c v

0 v/2

HD

H D

S1({symbolic gesture}) = H S2({symbolic gesture}) = DS1({no symbolic gesture}) = D S2({no symbolic gesture}) = H

Ω = {symbolic gesture, no symbolic gesture}

π1 = { {symbolic gesture}, {no symbolic gesture} }π2 = { {symbolic gesture}, {no symbolic gesture} }

S1({symbolic gesture}) = Accepts S1({no symbolic gesture}) =Rejects

2, 2 -2, 2

0, 0 0, 0

1Accepts

1Rejects

2 is low Type2 is high Type

(supposing gesture probability Independent of type)

So symbolic can influence whether the U.S. has the “right” to use Pakistani airspace without permission (this is a hawk-dove game!)

Or can influence how much we invest in a romantic relationship, if we both really like each other (this is a coordination game, presuming I don’t want to be invested when other isn’t but do

What property need symbolic gestures have?

Our model tells us…

Gesture needs to create common p-beliefs

What kinds of gestures create common p-beliefs?

Eye contact, explicit statements, handshakes, public displays

(Intuition: when I believe they happened, I believe that you believe they happened, and I believe that you believe that…)

What kind of gestures do not create comon p-belief?

Innuendos, Rumors, Intermediaries

Ω = {no gesture, told intermediary, but wasn’t relayed, message relayed}

π1 = {{no gesture}, {told intermediary but wasn’t relayed, message relayed }π2 = {{no gesture, told intermediary but wasn’t relayed}, {message relayed}}

μ(no gesture)=1/2μ(told intermediary but wasn’t relayed)=1/4μ(message relayed)=1/4

S1({no gesture}) = A S1 ({told intermediary but wasn’t relayed, message relayed }) =B

2, 2 0, 0

0, 0 1, 1

A

B

A B

Why?

Because whenever tell intermediary don’t think other will receive?

But what if know intermediary is really reliable?

Well, even then, if other doesn’t know that intermediary is reliable…

Ω = {no gesture, told reliable intermediary and relayed, told unreliable intermediary and wasn’t relayed, told reliable intermediary and was relayed}

π1 = …π2 = …

μ(no gesture)=1/2μ(told intermediary but wasn’t relayed)=1/5μ(message relayed)=1/5

S1({no gesture}) = A S1 ({told reliable intermediaty}) =B

2, 2 0, 0

0, 0 1, 1

A

B

A B

Even when you know message got delivered, you suspect that other suspects that you suspect message didn’t get delivered…

I.e. with intermediaries, even if p-belief, isn’t isn’t common p-belief…

Note: we do seem to require that symbolic gestures are explicit and public

What evidence can you think of that CK is behind symbolic gestures?

How would you test?

Supposing symbolic gestures are explained by CK, what advice would this lead to?

To make the model fit these applications better, what other math would you want to work out?