Amparo Urbano (with P. Hernandez and J. Vila)

University of Valencia. ERI-CES

Pragmatic Languages with Universal Grammars:

An Equilibrium Approach

Motivation

Economic agents communicate to reduce uncertainty and achieve coordination in either complete or incomplete information frameworks.

Language is a central tool in the process of making decisions

Most of the times, communication is noisy. Information transmission may involve different sources of misunderstanding:

Cultural, different mother tongues, Different specialization field (marketing, finance,….) Non verbal (unconscious) communication

However, the equilibrium approach to communication misunderstandings is not too widespread.

Common dictionary or corpus

A CB

Speaker/sender

Hearer/receiver

Noiseless communication

B B

A CB

Speaker/sender

Hearer/receiver

Noisy communication

A CB

B C

P( | C)

P( | C)

P( | C)

Hearer/receiverSpeaker/sender

P( A | B)

P( B | B)

P( C | B)

Inference of meaning

AAA

CCC

BBB

BBB

AAB

{A,B,C}3AAB

Speaker/sender Hearer/receiver

P( A | B)

P( B | B)

P( C | B)

P( A | B)

P( B | B)

P( C | B)

P( A | B)

P( B | B)

P( C | B)

Pragmatic inference of meaning

{A,B,C}3

The partition of the message space does not only depend on the

transition probabilities but also on the context of the communication

episode

Our context is a Sender-Receiver

game

The message space is

partitioned by a BEST RESPONSE

criterion

Agenda

We construct pure strategies in a Sender-Receiver game with noisy information transmission, based on:

Coding and inference of meaning rules (pragmatic Language).

The coding has a universal grammar and the meaning inference model is a partition of the message space.

We characterize the hearer/receiver best response in terms of some vicinity bounds in a pragmatic way.

We measure how much the communicative agents depart from noiseless information transmission equilibrium payoffs.

We calculate the minimum length of the communicative episode to guarantee any efficiency payoff approximation.

The basic model: The sender-receiver game Γ

Let be a set of states of nature. We have a

game defined by:

A set of two players:

A set of actions for player R:

A payoff function for both players, given by:

},...,{ ||1

},{ RS

RAu :

},...,{ ||1 AaaA

ASSUMPTION: is an aligned interest game

For each state we have an action such that jja

otherwise

aaifMau jtj

jt 0

ˆ 0),(

Noisy channel

Input basic signals

}1,0{X

Output basic signals

}1,0{Y)/( xyp

X Y0

1

0

1

Players communicate with noise. We follow an unifying approach and

consider a discrete noisy channel to model general

misunderstandings that may appear in information transmission.

nx nyInput

sequence

Output

sequence

n-timeCom.

),( 10

xy

The Extended communication game

Natures chooses a state wj with probability qj

S is informed of the actual state.

S utters an input sequence of length n to R, through the

noisy channel.

R hears an output sequence of length n, and chooses an

action accordingly (infers a meaning).

Payoffs are realized

nGAME : communication length n. Messages are i.i.d. variables

Strategies of the extended game

SENDER: )(wwhere

nX:

RECEIVER: )(yd

where AYd n :

Our construction: Corpus and pragmatic variations

We construct pure strategies based on a pragmatic Language.

This language consists of:

A Corpus or set of standard prototypes (sequences of basic signals which are one-to one with the set of sender's meanings=actions)

• The specific structure of the prototypes is defined by a grammar

Pragmatic variations of each standard prototype: output sequences from which the receiver will infer the meaning associated to the corresponding prototype

• Each sequence is assigned to a particular pragmatic variation in terms of its “vicinity” to the standard prototypes.

i-th block of the i-th standard

prototype is formed with 0’s

Block coding grammar: the corpus

0...0...1...11...1ˆ

......

1...1...0...01...1ˆ

1...1...1...10...0ˆ

||||

22

11

mn ||

),...,(ˆ 1 njjjj xxx

otherwise

jmsmjx s

j 1

1)1(0

m

Why this specific corpus?

Universal: It does not depend on the parameters of game Γ (initial probabilities and payoffs)

• It can be applied to any sender-receiver game

• It enables an easy characterization of the receiver’s pragmatic variations in terms of the Hamming distance, only depending of both the game and noise parameters of any sender-receiver game.

• (We have also characterized the pragmatic variations for any feasible corpus grammar, but it depends on some features of the specific coding rule).

EXAMPLE: the sender-receiver game

)7,7(

)43,43()0,0(

1

2

1a 2a

)1,1( )0,0(

)0,0(

)0,0(

)0,0(

)0,0(

3a

3

5.01 q

25.02 q

25.03 q

R

)7,7(

)43,43()0,0(

1

2

1a 2a

)1,1( )0,0(

)0,0(

)0,0(

)0,0(

)0,0(

3a

3

5.01 q

25.02 q

25.03 q

R

)43,43()0,0(

1

2

1a 2a

)1,1( )0,0(

)0,0(

)0,0(

)0,0(

)0,0(

3a

3

5.01 q

25.02 q

25.03 q

R

EXAMPLE: the noisy channel

0

1

0

1

0.9

0.4

0.1

0.6

654321 xxxxxxx

654321 yyyyyyy

Communication length: n = 6

)6.0,1.0(

EXAMPLE: the corpus

111100ˆ

110011ˆ

001111ˆ

33

22

11

23

6

||

nm

Vicinity measure: Hamming distance

To characterize the pragmatic variation sets, we need a

measure of distance.

Linguistics uses Levenshtein distance as a measure of

phonological distance between two corpora of phonetic

data.

Given two n-strings x=x1,x2,…,xn and y=y1,y2,…,yn , the

Hamming Distance between them is given by:

Let hb(x,y) be the Hamming distance between b-th blocks of

sequences x and y

tt

n

ttt yxwhereplacesyxyxh

#1:),(1

The receiver’s problem

d(y) is the solution of the maximization problem:

lkkl

k

k

ll M

M

yp

ypayd

ˆˆ,ˆ)|ˆ(

)|ˆ(ˆ)(

j

jja

auypyd ),()|ˆ(maxarg)(

myhyh

k

l

k

l

lkll

q

q

yp

yp

where

),ˆ(),ˆ(

1

1

0

0

11)|ˆ(

)|ˆ(

Noise level

Relative expected

payoff loss of

playing action l

instead of action k

Vicinity bound: largest number of errors permitted in blocks l and k to

play action l instead of action k

The Receiver: Pragmatic variations. The vicinity bounds

} ,),ˆ(),ˆ(/{ lkkCyhyhYyY lklklln

l

llY prototype of variationpragmatic :

mLn

qMqM

Ln

C ll

kk

lk

1

1

0

0

11

An interpretation of the vicinity bounds

mLn

qMqM

Ln

C ll

kk

lk

1

1

0

0

11

mCqMqMif lkllkk mCqMqMif lkllkk

mCqMqMif lkllkk

The minimum is associated to the maximum relative expected

payoff loss of playing action l instead of action k:

lkC

ll

kkqM

qM

kkllkllk

kkllkllk

lk

MqMqmCC

MqMqmCC

nm withC

whenever ,2

whenever ,12

with hence and

The Receiver’s best response.

}' ,~

'),ˆ(),ˆ(

,' ,~

'),ˆ(),ˆ(

,~

),ˆ(),ˆ(/{

''

''

lkkCyhyh

lkkCyhyh

kCyhyhYyY

llklkll

llklkll

llklklln

l

then,}such that 1{~

Let kklll MqMq}Ω,...,{kΩ

.ˆ

Then,

11 Yyad(y) l

Vicinity bounds

increase with relative

expected payoffs

EXAMPLE: vicinity bounds

*00

3*1

32*

*

*

*

2313

3212

3121

CC

CC

CC

50.011 Mq 75.122 Mq 75.1033 Mq

)7,7(

)43,43()0,0(

1

2

1a 2a

)1,1( )0,0(

)0,0(

)0,0(

)0,0(

)0,0(

3a

3

5.01 q

25.02 q

25.03 q

R

EXAMPLE: pragmatic variations

*00

3*1

32*

*

*

*

2313

3212

3121

CC

CC

CCVICINITY

BOUNDS

}0),ˆ(),ˆ(

2),ˆ(),ˆ(/{

232322

2121226

2

Cyhyh

CyhyhYyY

}3),ˆ(),ˆ(

3),ˆ(),ˆ(/{

323233

3131336

3

Cyhyh

CyhyhYyY

}0),ˆ(),ˆ(

1),ˆ(),ˆ(/{

131311

1212116

1

Cyhyh

CyhyhYyY

PRAGMATIC

VARIATIONS

EXAMPLE: pragmatic variations

}0),ˆ(),ˆ(

1),ˆ(),ˆ(/{

131311

1212116

1

Cyhyh

CyhyhYyY

} 000111 ,001011 ,001111 {1 Y

216

3 YYYY

Utterances with meaning ‘action 1’

} 000011 ,010011 ,100011 ,110011 {2 YUtterances with meaning ‘action 2’

Utterances with meaning ‘action 3’

6Y

1Y2Y

3Y

1a

2a

3a

Main result

Give an aligned interest sender-receiver game, a noisy

channel and a finite communication length n, the

strategies given by

ii ˆ)(

ii Yyayd )(

are a pure strategy Bayesian Nash equilibrium of the

extended noisy communication game .n

The sender’s truth-telling problem

We must check that sender has no incentive to send a message different from when she knows that actual state of nature is

nn Yy

lYy

ll yduxypyduyp )),(()|()),(()ˆ|(

ll Yy

llYy

l MxypMyp )|()ˆ|(

1)|(

)ˆ|(

l

l

Yy

Yyl

xyp

yp

l l

Probability of a correct meaning

inference

Efficiency of meaning inference

Given a channel with , a length n of the

communication episode, and game , then for all

we have that:

where and is a polynomial on the

channel parameters such that

1),(0 10 l

,...,1}min{ klkl CC ),( 10

,...,1l

),( 10 n

10

The vicinity bound depends on both n and the relative expected payoff loss

),(1)ˆ|( 101

1 lc

lllYp

Ex-ante expected payoff

without noise

Ex-ante payoffs efficiency

Given , then for any length of the communication

episode , we have that

n

iiiMq

1

where are the ex-ante expected payoffs of the

extended communication game , and

0[*,[ nn

n

)1)(1(ln

}max{ln

lnln

1*

10

10

,...,1,

,...,11

klll

kk

iii

Mq

Mq

Mqn

n

We have constructed a pragmatic Language with a universal

grammar in noisy information transmission situations.

We have shown that such a Language is an equilibrium language.

We have also shown that such a Language is an efficient inference of

“meaning” model: in spite of initial misunderstandings, the hearer is

able to infer with a high probability the speaker’s meaning

Therefore: Pragmatic languages with a short number of basic signals

support coordination, even when misunderstandings may appear

Our analysis can be extended to explain the role of communication in

specific situations such as communication in organizations, some

types of advertisement, market research and sub-cultural languages

among others

Conclusions

Thank you