Convention

Recap

What We’re Doing

The relation between a word and its meaning is conventional.

To understand this we want to know what a convention is. To do that we are using certain ideas from game theory:• Equilibrium point• Coordination problem

The Disarmament Game

Country 2

Country 1

Disarm Don’t Disarm

Disarm

10

10

15

-15

Don’t Disarm

-5

5

0

0

Equilibria

An equilibrium point is a square on the grid where no player can improve his position through unilateral deviation.

Unilateral deviation is when one player changes strategy and all the other players do not.

The Disarmament Game

Country 2

Country 1

Disarm Don’t Disarm

Disarm

10

10

15

-15

Don’t Disarm

-5

5

0

0Equilibrium

Coordination Problems

A coordination problem is a game where there are multiple equilibrium points. The problem is to “coordinate”: make sure we play strategies that will put us at the equilibrium points.

The Meeting Game

Person 2

Person 1

Place 1 Place 2

Place 1

1

1

0

0

Place 2

0

0

1

1

The Meaning Game

Person 2

Person 1

“Dog” “Chien”

“Dog”

1

1

0

0

“Chien”

0

0

1

1

Conventions

Solving Coordination Problems

How do you solve a coordination problem? Two ways are salience and precedent.

Example: Meeting

Suppose two people want to meet, but they have no way of communicating with each other.

It does not matter where they go, as long as they go to the same place.

Salience

Suppose you and a friend are in a strange city and you get split up. It doesn’t matter where you go, as long as your friend goes there to. Where should you go?

Your goal is to guess where your friend will go and go there. The salience of a location can suggest it as a good meeting place.

Precedent

If you’ve successfully navigated this situation before you might choose to meet at wherever you met before.

Convention

Finally, if you get in situations like this a lot, you might have a conventional rule that you follow for what to do.

Example: my wife and I have the rule: always meet at Exit A of the closest MTR station.

Convention: First Definition

It is a convention for a group to follow a rule in a certain situation, if whenever they are in that situation:1. Everyone follows the rule.2. Everyone expects everyone else to follow the rule.3. The situation is a coordination problem.4. Everyone following the rule is a solution to the coordination

problem.5. The reason why they follow the rule is (3) and (4).

Example: Money

Throughout history, people have used different things as money: gold, silver, sea shells, salt (whence ‘salary’), goats, cigarettes (in prison), coins and paper currency.

It doesn’t matter to me what I accept in exchange for my goods and labor as long as it’s what everyone else accepts (as long as I can spend it).

The Currency Game

Person 2

Person 1

Accepts HKD Accepts AUD

Accepts HKD

100

100

-500

-500

Accepts AUD

-500

-500

100

100

Accepting HKD is Conventional

It is a convention for Hong Kongers to follow the rule “accept HKD” in exchange for goods and services, because:1. Everyone gets paid in HKD and makes purchases with HKD.2. Everyone expects everyone else to accept payment and make

purchases with HKD.3. Which currency to accept is a coordination problem.4. Everyone accepting HKD is an equilibrium point in that problem.5. The reason why we follow the rule is (3) and (4).

Not All Rules Are Conventions

• Laws contain lots of rules. Many of these are not conventions: the reason we obey the law is not because other people do, it’s because we will be punished (however, compare driving on the right).• Mathematics textbooks contain rules for calculating the values of

certain functions. Again, when we follow these rules it is not because others do– it’s because they get us the right answer!• Moral rules?

Language and Convention

Many aspects of language are also conventional:• The meaning of each word.• Some facts about the grammar.• How the words get pronounced.

What the Word Means

Person 2

Person 1

“dog” “dog”

“dog”

1

1

0

0

“dog”

0

0

1

1

Word Order

Person 2

Person 1

SOV SVO

SOV

1

1

0

0

SVO

0

0

1

1

What Word We Use

Person 2

Person 1

“Dog” “Chien”

“Dog”

1

1

0

0

“Chien”

0

0

1

1

Not Everything Conventional

“Certainly not every feature of a language is conventional. No humanly possible language relies on ultrasonic whistles, so it is not by convention that Welshmen do not… but so long as even two languages are humanly possible, it must be by convention that a population chooses to use one or the other.” – David Lewis

Common Knowledge

City of Gold

Jungle of Danger

War Games

General 2

General 1

Attacks at Dawn Does Not Attack

Attacks at Dawn

Victory!

Victory!

Escape

Death

Does Not Attack

Death

Escape

Victory!

Victory!

A Messenger

General 1 wants to let General 2 know that he (G1) wants to attack at dawn.

So he sends a messenger through the Jungle of Danger, with the hopes that he will get through the snipers and snakes.

A Worry

General 1 is still worried: Did the messenger survive?

I know that I want to attack at dawn, But I do not know that General 2 knows that I want to attack at dawn.If he doesn’t know this then he might not attack at dawn.

A New Messenger

Luckily a new messenger arrives from the Jungle of Danger. The messenger says, “General 2 got your message!”

General 1 feels relieved.

A New Worry

But something is still wrong. General 1 now has a new worry:

I know that I want to attack at dawn.And I know that General 2 knows that I want to attack at dawn.But General 2 doesn’t know that I know that General 2 knows that I want to attack at dawn.If he doesn’t know this he might not attack at dawn.

Another Messenger

So General 1 sends another messenger into the Jungle of Danger to tell General 2 that General 1 got G2’s message that G2 got G1’s original message.

Another message comes back from General 2 saying he got the latest message. General 1 feels relieved.

Another Worry

But something is still wrong. General 1 now has a new worry:

I know that I want to attack at dawn.And I know that General 2 knows that I want to attack at dawn.And I know that General 2 knows that I know that General 2 knows that I want to attack at dawn.But General 2 doesn’t know that I know that G2 knows that I know that G2 knows that I want to attack at dawn.If he doesn’t know this he might not attack at dawn.

Common Knowledge

What the generals need is what Lewis calls common knowledge. It’s common knowledge among a group that P:

• Everyone knows that P• Everyone knows that everyone knows that P• Everyone knows that everyone knows that everyone knows that P• Etc.

Common Knowledge

Common knowledge might seem hard to get, but that’s not true. Imagine that there is no Jungle of Danger and the generals just meet the night before the attack.

City of Gold

Attack at dawn!

General 1 Reasons…

I know that G2 will attack at dawn, because he saw me say “Attack at dawn!”And I know G2 knows that I know that G2 will attack at dawn, because he saw me see him see me say “Attack at dawn!”And I know that G2 knows that I know G2 knows that I know that G2 will attack at dawn, because he saw me see him seeing me seeing him see me say “Attack at dawn!”

Lewis argues that conventions require not just following rules, but common knowledge.

Convention: Second Definition

It is a convention for a group to follow a rule in a certain situation, if whenever they are in that situation:1. It’s common knowledge that everyone follows the rule.2. It’s common knowledge that everyone expects everyone else to

follow the rule.3. It’s common knowledge that it is a coordination problem.4. Everyone following the rule is a solution to the coordination problem.5. It’s common knowledge that the reason why they follow the rule is (3)

and (4).