Amparo Urbano (with P. Hernandez and J. Vila) University of Valencia. ERI-CES

1. Pragmatic Languages with Universal Grammars: An Equilibrium Approach Amparo Urbano (with P. Hernandez and J. Vila) University of Valencia. ERI-CES

2. 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, communicationis 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.

3. Common dictionary or corpus A B C Speaker/sender Hearer/receiver

4. Noiseless communication A B C B B Speaker/sender Hearer/receiver

5. Noisy communication P( A | B) P( B | B) P( C | B) A B C Speaker/sender Hearer/receiver B C P( | C) P( | C) P( | C)

6. Inference of meaning 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) AA A B B B C C C Hearer/receiver Speaker/sender BB B A A B A A B {A,B,C}3

7. Pragmatic inference of meaning Our context is a Sender-Receiver game The partition of the message space does not only depend on the transition probabilities but also on the context of the communication episode The message space is partitioned by a BEST RESPONSE criterion {A,B,C}3

8. 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.

9. 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

10. 0 1 0 1 n-time Com. Noisy channel • 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. y x Input basic signals Output basic signals Input sequence Output sequence

11. The Extended communication game GAME : communication length n. Messages are i.i.d. variables • 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

12. Strategies of the extended game SENDER: where RECEIVER: where

13. 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.

14. Block coding grammar: the corpus i-th block of the i-th standard prototype is formed with 0’s

15. 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).

16. EXAMPLE: the sender-receiver game

17. EXAMPLE: the noisy channel Communication length: n = 6 0.9 0 1 0 1 0.1 0.6 0.4

18. EXAMPLE: the corpus

19. 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

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

21. The Receiver: Pragmatic variations. The vicinity bounds 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 Noise level

22. 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:

23. The Receiver’s best response.

24. EXAMPLE: vicinity bounds Vicinity bounds increase with relative expected payoffs

25. EXAMPLE: pragmatic variations VICINITY BOUNDS PRAGMATIC VARIATIONS

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

27. 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 .

28. 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

29. The vicinity bound depends on both n and the relative expected payoff loss Efficiency of meaning inference Given a channel with , a length n of the communication episode, and game , then for all we have that: Probability of a correct meaning inference where and is a polynomial on the channel parameters such that

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

31. Conclusions • 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

32. Thank you