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 ACB Speaker/senderHearer/receiver

Noiseless communication B B ACB Speaker/senderHearer/receiver

Noisy communication ACB B C P( | C) Hearer/receiverSpeaker/sender P( A | B) P( B | B) P( C | B)

Inference of meaning A AAAAAA CCCCCC BBBBBB B BBBBBB AABAAB {A,B,C} 3 AABAAB 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: ASSUMPTION: is an aligned interest game For each state we have an action such that

Noisy channel Input basic signalsOutput basic signals  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. Input sequence Output sequence n-time Com. x y

The Extended communication game  Natures chooses a state w j with probability q j  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 GAME : communication length n. Messages are i.i.d. variables

Strategies of the extended game SENDER: where RECEIVER: where

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

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

EXAMPLE: the noisy channel Communication length: n = 6

EXAMPLE: the corpus

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=x 1,x 2,…,x n and y=y 1,y 2,…,y n, the Hamming Distance between them is given by:  Let h b (x,y) be the Hamming distance between b-th blocks of sequences x and y

The receiver’s problem d(y) is the solution of the maximization problem:

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

An interpretation of the vicinity bounds The minimum is associated to the maximum relative expected payoff loss of playing action l instead of action k:

The Receiver’s best response.

Vicinity bounds increase with relative expected payoffs EXAMPLE: vicinity bounds

EXAMPLE: pragmatic variations VICINITY BOUNDS PRAGMATIC VARIATIONS

EXAMPLE: pragmatic variations Utterances with meaning ‘action 1’ Utterances with meaning ‘action 2’ Utterances with meaning ‘action 3’

Main result Give an aligned interest sender-receiver game, a noisy channel and a finite communication length n, the strategies given by are a pure strategy Bayesian Nash equilibrium of the extended noisy communication game.

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

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

Ex-ante expected payoff without noise Ex-ante payoffs efficiency Given, then for any length of the communication episode, we have that where are the ex-ante expected payoffs of the extended communication game, and

 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

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