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THE LANGUAGE GAME:
A GAME-THEORETIC APPROACH TO
LANGUAGE CONTACT
Part I: Static Analysis
by
Nagore Iriberri and José R. Uriarte
2006
Working Paper Series: IL. 24/06
Departamento de Fundamentos del Análisis Económico I
Ekonomi Analisiaren Oinarriak I Saila
University of the Basque Country
THE LANGUAGE GAME:
A Game-Theoretic Approach to Language Contact
Part I: Static Analysis
Nagore Iriberri
a
and José Ramón Uriarte
b
Universidad del Pais Vasco-Euskal Herriko Unibertsitatea.
Departamento de Fundamentos del Análisis Económico I
Ekonomi Analisiaren Oinarriak I Saila
Avenida Lehendakari Aguirre, 83
48015 Bilbao, Basque Country-Spain
June 2006
We are grateful to Ehud Kalai, Karl Schlag and Reinhard Selten for their
comments and criticisms to an earlier version of this work.
a. University of California at San Diego. Department of Economics.
b. Please, send any comment to jr.uriarte@ehu.es
ABSTRACT
We study a society inside which two o¢ cial languages, the majority language A and
the minority language B, are in contact and compete for the same social functions. We
propose a non-cooperative game to capture some features of this competitive situation.
In the game, there are two type s of players: the bilingual one who speaks both A and
B and the monolingual one who speaks only A. The information about which type is
each player is private.
A real life situation captured by the game is that in many interactions bilingual
players must decide under incomplete information about which language to use. One
implication of this information structure is that while A satis…es the main properties
of a public good, B does not. Another implication is that it may have dangerous
consequences on the language diversity of the society. We show that in many equilibria
bilingual players fail to coordinate in their preferred language and end up using the
majority language A.
2
1. Introduction
Sociolinguists remind us that (i) economic and/or political power is always
exerted in a speci…c language and (ii) since language is such a central ingredient
in human culture and cultural identity, we tend to convert language into a very
important issue, more so when it is felt that the society’s perceived “own”language
is under threat : “The western world has had a love a¤air with its own languages
for over two centuries. Because of this love a¤air (the seeds of which the power and
prestige of the West has also sown throughout much, if not most, of the world),
it is easy to overstate the importance of language in human social and cultural
a¤airs”(Fishman(2001)).
An issue frequently studied in the sociology of language is the minority lan-
guage maintenance in the context of powerful forces operating in modern societies
that push in the direction of minority language shift. The outcome of this process
is that there are many languages under threat, many thousands are dying and
many more will die in the near future. “Unfortunately, democratic regimes can
be just as blind to cultural pluralism and to the needs of minorities for cultural
recognition and support, as are autocratic regimes”. (Fishman (1991)). The,
approximately 6,000, existing languages of the world (about the di¢ culties to
quantify the number of existing languages, see Crystal(2001)) were classi…ed by
Krauss(1992) into three categories: “(1) 20-50% of the total are “moribund lan-
guages”that children have stopped using as mother tongues, (2) 40-75% of them
are “endangered languages”that children are still learning as native speakers but
that are likely to become moribund by the end of the 21st century, and (3) 5-10%
of them are “safe languages” that seem to be assured of their future existence.
So, in the worst case, 95% of the languages spoken in the world today could be
extinct or be on the way of extinction during the next century”. (Nara (2000);
see also Skutnabb-Kangas(2000)). India is an exception of this trend, a nation
with many languages and a Constitution that designates no one language as the
national language. The linguist Annamalai (2000) says that “the linguistic history
of India has been one of addition of languages, and not their reduction”.
Needless to say that the perception that your own language is under threat
may have profound political implications. The Mysore Document (2000) stresses
3
the idea that “the preservation of linguistic diversity is necessary not only for
ethical reasons but also as an input for human progress and development. Every
language codi…es a way of cognizing, experiencing and organizing knowledge of
the world. Each way has the potential to correct any wrong step taken by “suc-
cessful languages and cultures” which may be detrimental to humanity”. The
…rst recommendation of the Mysore Document says that every State “shall en-
sure equal rights and opportunities to all linguistic communities for survival and
development”.
We shall concentrate on societies inside which two o¢ cial languages are in con-
tact and compete for the same social functions (in education, research, government
-executive and legislative wings-, administration of justice, business, banking and
insurance regulations, information technologies, mass media, entertainment etc.).
It is very common to observe in that type of societies that one of the languages
has a relatively smaller social support than the other; one language is spoken by
some, while the other one is known by all the members of the society and domi-
nates both the private and the public domain. In those societies we would have a
minority language community who is able to speak both languages and a majority
who speaks only one language. It is easy to predict what would be the outcome of
language competition under those circumstances. The weaker language will loose
social functionality, unless its speech community takes some actions. Examples of
this type of languages and societies abound. To name just a few: Irish in Ireland,
Welsh in Wales and Gaelic in Scotland are minority languages in contact with
English; Basque in the Basque Country is a minority language in contact with
“big languages”, French and Spanish; Galician spoken in Galicia is in contact
with Spanish; Frisian spoken in the province of Friesland in The Netherlands is in
contact with Dutch; the Aboriginal languages of New Zealand and Australia are
in contact with English, etc.
Hence, we have a relevant social con‡ict a¤ecting many societies, the contact
of languages which compete inside a society for the same social functions, with
some languages and their related cultures and collective identities under threat
or …ghting for survival. The present paper models certain central features of this
competitive situation, the contact of languages, by using non-cooperative game
tools.
Language, and particularly the economic analysis of bilingualism and language
policy, has been central in the works of Canadian economists, such as Breton (1998,
1999) and Vaillancourt (1996). Most of these works are a mixture of theory and
applied economics, one of the purpose being the evaluation of minority language
4
policies, as in Grin and Vaillancourt (1999). Along this line, Grin (1996, 2003 )
shows how fruitful can be the economic approach to language; his work is an initial
step in trying to introduce basic microeconomic tools in the study of language
and facing the di¢ culties of de…ning the notions of supply, demand and price of
a language (speci…c good).
Even though it is a fact that linguistic diversity and language contact is ac-
companied by con‡ict (some authors like Nelde (1997) emphasize that there is no
language contact without social con‡ict, and others, like Grin (2003), emphasize
that linguistic diversity tends towards con‡ict) it is surprising how little or null
attention has received tis issue from economic theory. By this we do not mean that
language is not considered relevant to economic theory. Language appears mainly
in relation with cheap-talk games, (see, for instance, Crawford and Sobel (1983)
and Farrell (1993)) where what matters is that players have a unique system of
communication or a unique labeling -and it does not matter which one- so that
information is transmitted and messages are understood. But, as stressed by Pei-
yu Lo (2005), the restriction imposed by the assumption that players speak the
same natural language is not taken into account in cheap-talk games. Neverthe-
less, even when it is introduced the notion of language, the analysis usually deals
with the corpus of the language (that is, the internal structure and functioning of
a language) or, in a di¤erent setting, with pragmatics ("the study of factors that
govern the choice of language -here language means certain words and sentences-
in social interaction and the e¤ects of our choice on others", Crystal (1987)), as
it happens in the approach of Rubinstein (2000).
The present paper deals instead with the status of the language (the relative
prestige and power -measured in terms of the size of the speech community, the
degree of use in education, mass media, information technologies, etc...- of a lan-
guage in a language contact situation), and it is in this sense that it is more related
to the work of Pool (1986) and Selten and Pool (1991), although the issue studied
here is of a di¤erent nature.
The Language Game models the interaction between two players with incom-
plete information. There are two types of players, the bilingual (i.e. the one who
can speak both the majority language A and the minority language B) and the
monolingual ( i.e. the one who only speaks the majority language A). We assume
that A and B are in contact; they could be the o¢ cial languages of the society,
although this is not needed for A and B be in contact. The competitive situation
between A and B, that we mentioned above, is embedded in apparently innocuous
interactions, such as in a simple conversation. Suppose the interaction is between
5
a seller and a potential buyer and that it is located in a shop. Suppose too that
none of the two players know the type of the other and that there are no signals
(accents, speech style, etc.) that could help to infer the type of a player; suppose
too that it is common knowledge that both know the majority language A and
that only a small proportion is bilingual (there are surveys that quantify the pro-
portion of bilingual individuals equal to ). The initial exchange of customary
words and phrases, -such as, “good morning”, “how may I help you”, etc.-, will
determine the language, A or B, that will be used in the subsequent conversation.
Note that only when the two players happen to be bilingual (an event that occurs
with probability
2
) there would be a possibility of using B in the conversation
and it is only in this case that both players make a choice of language. Obviously,
in the rest of the cases, that is when both players are monolingual or at least one
is monolingual, A will be the language used. When only one of the players is
bilingual, that player may choose B, but then he will be forced to use A.
But why is the study of the choice of language, if there is any choice at all, in a
conversation important? Because the minority language B could avoid the danger
of being culturally marginal or wiped out only if it is socially used by bilingual
individuals in no matter what kind of interaction. The Language Game is a model
that describes that initial stage of a conversation as a non-cooperative game in
extensive form with incomplete information. The goal is to study the language
conventions that the bilingual agents may build in that natural setting, since that
will be an indicator of the degree of social use of language B.
The game stresses the fact that monolingual players have (trivially) perfect
information during the course of the game because they do not make language
choices. But the presence of monolingual players introduce uncertainties upon
those who talk both languages. Bilingual players must make choices under incom-
plete information, because they do not know the type of player they are interacting
with. Both the bilingual sender and receiver are uninformed. Given this informa-
tion structure and being common knowledge that everybody speaks A and only
a minority speaks B, it is not strange that in the interaction among bilingual
players the majority language A is more frequently used than the minority B.
Even though, and this is important, bilingual players share the common interest
of using B, this is a phenomenon that happens frequently in real life situations.
A corollary of this paper, to our knowledge not stressed in sociolinguistic
studies, is that since bilingual players are uninformed and since it is common
knowledge that everybody speaks A and only a minority speaks B, the language
contact situation produces a negative externality upon the use of the minority
6
language B. This e¤ect clearly happens when only one of the players is bilingual;
in that event the talk must be in the majority language A. But even when both
players are bilingual they frequently fail to coordinate in the minority language B,
probably because they are not informed about the bilingual nature of each other.
Hence, in language contact situation, only the majority language A satis…es the
main properties of a public good (non-rivalrous consumption –the consumption
of the good by one individual does not detract from that of another–and non-
excludability–it is di¢ cult if not impossible to exclude an individual from enjoying
the good).
The paper is organized as follows. In section 2 we present a detailed description
of the game in its extensive form. We introduce two assumptions: …rst, that
bilingual players prefer B to A, and therefore, they would get a higher (von
Neumann and Morgenstern) utility when they interact in the language they prefer
most than in the case when they choose to use A (see Pool (1986) and Crystal
(1987)). Second, when a bilingual player, who wishes to use his preferred language,
is forced to speak in his less preferred language he will su¤er a frustration cost,
represented by a payo¤ loss (see Fishman (1991)). In section 3 we carry out
the equilibrium analysis of the game assuming that the frustration cost could be
greater, smaller or equal to the bene…t derived from using B. In each case we
would have multiple Nash equilibria and in each equilibrium a speci…c language
will be used among the bilingual players. We conclude that the most interesting
case of equilibrium selection problem occurs when the cost equals the bene…t.
Since speech conventions are a social construct, the equilibrium selection prob-
lem will be studied in a follow-up paper where an evolutionary setup will be used.
2. Description of the Language Game
Supp ose a society in which there are two o¢ cial languages. One of them
is spoken by every individual in the society and the other by a relatively small
prop ortion of individuals. Thus, even though, by law, they both have the same
legal status, one is a majority language and the other is a minority language. To
distinguish among the two o¢ cial languages, let us proceed as sociolinguists do,
by denoting the majority language as A and the minority language as B:
We shall assume that there are two types of individuals in the society. The
bilingual type, who speaks both A and B, and the monolingual type, who only
speak the majority language A. The bilingual type of individuals are a minority in
this society. Let and (1 ) denote the proportion of bilingual and monolingual
agents, respectively. We assume that < (1 ), (but we should note that we
7
assume that is not only smaller than (1 ), it is much smaller). Since there are
sociolinguistic studies in that so ciety, the value of is known by every individual;
in other words, we shall assume that the value of is common knowledge.
Let us imagine a simple social relation in which a potential buyer enters a shop
and interacts with the shopkeeper. Suppose they do not know each other and that
there are no signals showing that the buyer or the seller, or both, is monolingual
or bilingual. What they do know is that both speak the majority language A and
that only a minority speaks B. In a society such as the one described above,
bilingual players must make a decision about which language they will use in the
interaction, and, very likely, that decision will be conditioned by the (speech)
conventions of that society.
The actions are taken sequentially, as in an ordinary conversation. For exam-
ple, a bilingual shopkeeper starts the interaction using one of the two languages
and the buyer replies. If the shopkeeper chooses A and the bilingual buyer replies
using B, the shopkeeper has a second chance to change or not his initial choice
of language. In some cases, this decision is made out of necessity: the monolin-
gual player dictates that the language A has to be used in the interaction. In
other cases, when it happens that both agents are bilingual, they have to agree
on the language they will use. In any case, as stressed above, what matters is
that that both the buyer and the shopkeeper do not know in advance whether
the person they have to interact with is bilingual or not. This type of interaction
could be represented by a non-cooperative game in extensive form with incomplete
information, a Bayesian game, shown in Figure 1.
We shall assume that player I is the shopkeeper and player II the buyer. The
presence of Nature represents the ignorance of each player about the other’s type.
Nature chooses bilingual players with probability and monolingual players 1
: Hence, in Figure 1 it is shown how Nature begins by deciding whether player
I is bilingual or monolingual. After player I’s move, Nature intervenes again by
choosing whether player II is bilingual or not. Recall that there are studies about
the knowledge of B in the society, so that we may assume that the exact values
of and 1 is common knowledge among the two players.
The action set for the bilingual player is therefore A
bi
= fA; Bg and the
corresponding set for the monolingual player is the single action set A
mono
=
fAg. As a consequence, when a bilingual player is matched to a monolingual
player, independently of the choice of language made by the bilingual player, the
interaction will necessarily take place in language A.
Players can only observe (hear) the action (language) used by the other player.
8
In some cases this observation reveals the type of player, but in other cases do es
not, as it will be seen.
A bilingual player I has two information sets:
Bilingual Ia: player I knows that he is a bilingual type of player, and thus in
this information set he must choose the language A or B to start the conversation
with player II, without knowing which type of player is II. Hence, player I is
uninformed in this information set.
Bilingual Ib: this information set starts a proper subgame and player I chooses
after player II has revealed that he is bilingual. Thus, player I is informed in this
information set.
The bilingual player II has two information sets:
Bilingual II : this information set starts another proper subgame. Here, player
II may choose after player I has revealed his bilingual nature because he decided
at Bilingual Ia to choose B. Hence player II is informed in this information set.
Majority Language (ML): a bilingual player II makes choices in this set after
having heard player I start the conversation using the majority language A. Hence,
in this set player II does not know which type of player is I, whether bilingual
or monolingual. Thus, this information set contains two nodes, x and y. When
player II moves in this set he must interpret the signal sent by player I and select
accordingly the best outcome. For instance, if player I happens to be bilingual,
both players could get the highest payo¤ m depending on the right choice of
actions. If player I happens to be monolingual player II may get the lowest payo¤
n c if he responds with B.
Monolingual players do not choose, because they can only speak A. The
monolingual player I is located at the top of the game tree of …gure 1 where Nature
chooses the type of player I. Since the bilingual player I may start the conversation
with either A or B and a monolingual one with A, there are three nodes where
Nature chooses again to determine which type of player is II. Monolingual player
II’s action is located after each of these three nodes of Nature. Hence, in this game
each bilingual player has in some information set a lack of information about the
type of player he is interacting with: player I at Bilingual Ia and player II at
Majority Language. After choosing an action in these two sets, both player I,
the sender, and player II, the receiver, are uninformed. In the interaction they
may speak A and therefore they do not reveal their type or they may reveal their
nature by speaking B.
9
NATURE
Nature
Nature
Nature
BILINGUAL Ia
BILINGUAL II
n
n
n
n
m
m
n-c
n
n-c
n
m
m
n
n-c
n
n
n
n
n
n-c
{s}, B
{1-s}, A
1-
{r}, B {1-r}, A
{q}, B
{1-q}, A
1-
1-
BILINGUAL Ib
II
II II
1-
{q}, B
{1-q}, A
{p}, B A, {1-p}
MonolingualBilingual
MAJORITY LANGUAGE
I
x
y
I
Figure 1. The Language Game, where A denotes the majority language, B the
minority language, c=c() the "frustration" cost and and 1 the proportion
of bilingual and monolingual individuals, respectively in the society. The
probability with which an action is chosen is shown by {.}.
2.1. Payo¤s
The culturally speci…c language of any society “is more than just a tool of
communication for its culture. (...). Such a language is often viewed as a very
speci…c gift, a marker of identity and a sp eci…c responsibility vis-à-vis future
generations.”(Fishman (1991)). This is a statement valid for any language.
Pool (1986) introduced a game with perfect information and only bilingual
players,-each player’s native language is the other’s second language,- and assumed
that both players prefer to speak in their own native language. Here we shall make
an assumption ab out the preferences of the bilingual players along the same lines.
10
B could be a bilingual player’s mother tongue or not. But in any case, we shall
assume that a bilingual player prefers to speak B rather than A. Formally, let
bi
denote the preference relation of a bilingual agent then, we assume
B
bi
A
Several reasons could justify that preference. Let the motivation behind that
ordering be just that bilingual players are aware tat B is an endangered language
and that they consider that the only way to avoid its disappearance is by using
it. Monolingual players do not have those concerns about the only language they
speak because they know that A is not under threat. Monolingual players do not
care who they are interacting with because it is common knowledge that everybody
in the society speaks A and so they know that the interaction in which they are
involved will take place in that language. Thus, the payo¤s to a monolingual player
are not a¤ected by which language is intended to be used by the other player. Let
us assume that the monolingual player gets the payo¤ n in the interaction. This
will also be the payo¤ obtained by a bilingual agent who, for whatever reasons,
chooses to speak A at some information set.
The e¤orts made by the bilingual population to reverse language shift are
an indication of dissatisfaction with the cultural life which is dominated by the
majority language (see Fishman (1991) p.17). Thus, one might think, that when
a bilingual player, who wishes to use his preferred language, is forced to speak in
his less preferred language, he will su¤er a frustration that will be represented by
a payo¤ loss of c():Hence, we shall make the following assumption
Assumption 1
For all 2 (0; 1)
m() > n > c() > 0
The …rst inequality, m() > n; means that, for a given value of , since bilin-
gual players prefer B to A they will get a higher von Neumann and Morgenstern
utility m() when they interact in the language they prefer most than in the case
when they choose the use of A: This assumption seems to be observed by those
specialized in di¤erent areas of linguistics.
“Switching to a minority language is very common as a mean of expressing
solidarity with a so cial group. The change signals to the listener that the speaker
11
is from a certain background; and if the listener responds with a similar switch, a
degree of rapport is established”. (Crystal (1987), Chapter 60).
We may think that as the prop ortion of bilingual population increases and
the normalization of the minority language B is being achieved, the payo¤, m();
obtained from speaking in B would get normalized too. Thus, we might assume
that the von Neumann and Morgenstern utility or payo¤ to a bilingual player
interacting in the language he prefers, m(); is a constant function for 2 (0; d];
say, m() = n + k() (where k() = k > 0 is a constant and d < 1 should have a
value of, at least, 1=2): For values of > d; that is, when the minority language
B is gradually achieving a normal status, we might assume that m() is strictly
decreasing, i.e.
@m()
@
< 0. In the limit, m(1) = n ( and in the other extreme,
when no B speakers exist at all , m(0) = n ).
The cost c(), which we assume to be smaller than n, is intended to capture
the dissatisfaction, mentioned by Fishman (1991), felt by the bilingual player who
must face the fact that, in many interactions, he is forced to use the majority
language. Notice that, since n is the payo¤ that a bilingual player can obtain for
sure had he chosen to speak A; the frustration cost c( ) should be subtracted
from n: Thus, n c() is the payo¤ to a bilingual player who, having chosen
B; is matched to someone, monolingual (or bilingual), who uses language A and,
therefore, must end up speaking in A:We assume that the graph of c() is similar
to that of m():See Figure 2.
12
m() = n + k
c()
von Neumann
and Morgenstern
utility
% of bilingual population
1
0
n
d
m()
Figure 2. The payo¤s
As it was mentioned above, the present work assumes a given which is much
below 50% and, in …gure 2, with a value in the range where the graph of m()
has reached its ‡at zone.
2.2. Pure Strategies
Player I has four pure strategies, S
I
= fBBA; BAA; ABA; AAAg : In each
strategy, from left to right, the …rst two components are the actions taken by a
bilingual player I; the …rst component is the decision taken at the left information
set, named Bilingual Ia, and the second component is the decision taken at his
second information set, named Bilingual Ib. The last component A corresponds
to the action of monolingual player I. Even though it is redundant, we shall keep
recalling in each strategy the action of a monolingual player I. Hence, a strategy
of a player describes the actions chosen when that player is bilingual and the
action A, on the right side of each strategy, when the player is monolingual. The
interpretation of player I’s pure strategies is the following.
- BBA : If I am bilingual I will start speaking B and if I reach the Bilingual
Ib information set, I would speak B. If I am monolingual I speak A. We may
call this strategy “I Speak always B”.
13
- BAA : If I am bilingual I will start speaking B and if I reach the Bilingual
Ib information set, I would speak A: If I am monolingual I speak A: This strategy
could be called “I Use B …rst and then switch to the language not spoken by
player II ”.
- ABA : If I am bilingual I will start speaking A and if I reach the Bilingual
Ib information set, I would sp eak B after hearing player II using B: If I am
monolingual I speak A: We may call this strategy “I Use A …rst and then switch
to the language spoken by player II ”.
- AAA : If I am bilingual I will start speaking A and if I reach the Bilingual
Ib information set, I would speak A: If I am monolingual I speak A: We may call
this strategy “I Speak always A ”.
Notice that Player I’s BBA and BAA strategies are equivalent because the
bilingual player I’s information set Bilingual Ib is o¤ the path of play and
therefore he will not have the opportunity to reply when player II uses B at
ML information set.
Player II’s pure strategy set is S
II
= fBBA; BAA; ABA; AAAg : In each strat-
egy, from left to right, the …rst two components are the actions taken when player
II happens to be bilingual. The …rst component is the decision taken at his left
information set, named Bilingual II; the second component is the action taken
at the information set Majority Language (ML), after he has heard player I
speaking A: The third component is the action taken when he is monolingual.
- BBA : If I am bilingual I will always answer using B: If I am monolingual I
speak A: We may call this strategy “I Speak always B ”.
- BAA : If I am bilingual and I hear B, I will answer using B; but if I hear A;
then I will answer in A: If I am monolingual I speak A: We may call this strategy
“I Speak always the language spoken by player I ”.
- ABA : If I am bilingual and I hear B, I will answer in A; but if I hear A;
then I will answer in B: If I am monolingual I speak A: We may call this strategy
“I Speak always the language not spoken by player I ”.
14
- AAA : If I am bilingual and I hear B, I will answer sp eaking A; but if I hear
A; then I will answer in A: If I am monolingual I will speak A: We may call this
strategy “I Speak always A ”.
2.3. The Language Matrix
Each pure strategy pro…le can be thought of as the initial phase of a conver-
sation b etween two bilingual players in which it is determined the language that
could be used during the interaction. Figure 3 shows a matrix in which, given
a pair of pure strategies, the corresponding entry is the language which will be
used in the event that the conversation takes place between a bilingual player I
and a bilingual player II (note that in each pure strategy we keep the action of a
monolingual player, -the third component from left to right-, even though it has
no consequences)
P layerII
Player I
BBA BAA ABA AAA
BBA B B A A
BAA B B A A
ABA B A B A
AAA A A A A
Figure 3. The Language Matrix. For each pair of pure strategies, one for each
player, the corresponding entry shows the language which will be spoken when the
occurring event is the matching of two bilingual players.
Four possible events or combination of player types may occur: with prob-
ability
2
; b oth players I and II are bilingual, with probability (1 ) player
I is bilingual and player II is monolingual, with probability (1 ) player I is
monolingual and player II is bilingual and, …nally, with probability (1 )
2
both
players are monolingual. Therefore, the minority language B could only be spoken
when the realized event is the matching of two bilingual players and if the chosen
strategies by both players allow it. The Language Matrix, in Figure 3, shows the
language associated to each pair of strategies when played by bilingual players
(note that in this case the far right action A of each strategy is redundant). In
the rest of the three possible events, since at least one player is monolingual, the
spoken language will be A: For instance, let us suppose that the strategy pro…le
15
is (ABA; ABA): In this situation, the language that will be used by the bilin-
gual players is determined as follows: player I starts choosing A at Bilingual Ia
information set and player II, after hearing language A;replies by choosing B at
his Majority Language information set. Then, player I reaches his Bilingual Ib
information set and there he switches to B:Hence, under this strategy pro…le, the
minority language B will be actually spoken if the realized event is the matching
of two bilingual players. The pro…le (ABA; BAA) shows that bilingual player I
chooses A at Bilingual Ia set and then bilingual player II chooses A as well at Ma-
jority Language information set and hence, the conversation will be in language
A in any of the four possible events. As a consequence, in the above matrix, 6
strategy pro…les, out of 16, would allow the use of B and therefore attach the
probability
2
to the use of B. The other 10 strategy combinations attach zero
probability to the use of B:
At …rst sight one would be tempted to say that a language satis…es the main
prop erties of a public good. That is, non-rivalrous consumption–the consumption
of the good by one individual does not detract from that of another–and non-
excludability–it is di¢ cult if not impossible to exclude an individual from enjoying
the good. Suppose L is the o¢ cial language of a society. Then, it is easy to see
that if I belong to the speech community of L, I continue to enjoy the use of L at
the same time that you do. By the same token, anyone in that society (i.e. the
language community of L ) can enjoy the use of L. Hence,we would deduce that
a language, from an economic point of view, can be thought of as a public good.
But, clearly, this is not true for any language. When two languages are in contact
-and one is spoken by every member of the society and the other by a minority-,
the Language Game shows that even inside the speech community of the minority
language B there could exist coordination failures which will block the use of B.
For instance, let us consider the strategy pro…le (ABA; BAA). Note that in this
pro…le both bilingual players do wish to speak B whenever one is addressed by
the other in language B. So let us assume that the event of the matching of two
bilingual players is realized and that they play the above pro…le (excluding the
A on the right of each strategy). Since player I starts the conversation using A,
-but is ready to switch to B if II responds with B-, Player II is deterred from
enjoying the use of B and thus he answers using A:This is because Player II does
not know whether Player I is bilingual or monolingual. That is, Player II does
not know in which node of the Majority Language information set is at. But,
the event which would occur with a higher frequency would be the matching of
monolingual players or players of di¤erent type. Then in the latter event the
16
conversation takes place in language A and the bilingual player would be deprived
from the use of B.
Therefore, since bilingual players are uninformed and it is common knowledge
that everybody speaks A and only a minority speaks B, the language contact
produces a negative externality upon the use of minority language B. Hence, we
should say that it is only language A, the majority language, that satis…es the
main properties of a public good. This, of course, does not mean that a minority
language B is a private good. We would rather say that a minority language, in
a language contact situation such as the one described here, has a public good
nature that su¤ers negative externalities from the majority language and for that
reason it needs a relatively higher governmental intervention (in the form of a
minority language policy) to ensure its provision and maintain diversity.
Proposition 1
When two languages are in contact, being one of them spoken by every member
of the society (i.e. the majority language A) and the other language is spoken
by a minority group (i.e., the minority language B), then it is language A that
might be viewed as a (strict) public good. Language B has not the same status
as A because, …rst when a bilingual player is matched with a monolingual one the
prop erty of non-excludability may not be satis…ed and second, when two bilingual
players are matched they may mutually exclude each other from enjoying the
language B that they prefer most because they are uniformed about their true
type.
3. Equilibrium Analysis
Note …rst that the Language Game has two proper subgames. One subgame
starts at player I’s Bilingual Ib information set and the other at player II’s Bilin-
gual II information set. In both subgames, B is the best choice for both players
( i.e. r = 1 for player I and s = 1 for player II) because the player who moved
previously, by choosing B, has revealed that he is bilingual; thus, both players
may select the outcome they prefer most.
Player I’s expected payo¤ at Bilingual Ia information set is (recall that is
given and, as it was mentioned above, has a small value; thus, from now on we
denote c() and m() as c and m , respectively):
17
E
I
(Bilingual Ia) = (1 p)fq[rm + (1 r)n ] + (1 q)ng + (1 )(1 p)n +
p[sm + (1 s)(n c)] + (1 )p(n c)
Therefore
E
I
(Bilingual Ia) =
qr(m n) + n
s(m n) + n c(1 s)
if p = 0
if p = 1
::::::::::::::(1)
The probability of reaching player II’s Majority Language (ML) information
set is
2
(1p)+(1). Hence, player II’s Bayes consistent beliefs at nodes x and
y of ML information set are
II
(x) = (1p)=1p and
II
(y) = (1)=1p,
respectively. Thus, the expected payo¤ of player II at this information set is
E
II
(ML) =
(1 p)
1 p
fq[(1 r)(n c) + rm] + (1 q)ng
+
1
1 p
(n qc)
Therefore
E
II
(ML) =
8
>
>
<
>
>
:
n
1
1p
[r(m n) c(1 r)
pr(m n)
prc + p(c n) + n]
if q = 0
if q = 1
:::::::::::::(2)
Recall that player I’s pure strategy set is S
I
= fBBA; BAA; ABA; AAAg.
Since BBA and BAA are equivalent, we shall refer to them as BBA, hence S
I
=
fBBA; ABA; AAAg, and fx
1
; x
2
; x
3
g will denote the probability distribution over
S
I
. Let S
II
= fBBA; BAA; ABA; AAAg and fy
1
; y
2
; y
3
; y
4
g the corresponding
sets for player I I.
We shall consider three cases that derive from the assumption that could be
made ab out the possible value of the frustration cost, c; felt by bilingual players,
vis-a-vis the bene…t, (m n), this type of players might obtain when they are able
to use his preferred language.
Case 1. Suppose that Assumption 1 is satis…ed.
Assumption 2: For all 0 < < 1 ;
18
c = (m n)
(1 )
This means that, for a given ; a bilingual player feels that the cost c equals
the bene…t (m n) weighted by the ratio
(1)
.
In societies in which the proportion of bilingual people, , is small, a bilingual
person (in the role of player I), who has no information whatsoever about the
person who is about to interact with, will usually start the conversation speaking
language A. The reason for this convention is simple: because it is common
knowledge that everybody speaks A. But the Language Game gives additional
reasons. Player I, by choosing A at his Bilingual Ia information set, may avoid
the lowest payo¤ n c.
Formally, when r = 1 (thus, player I chooses B in the subgame starting at
Bilingual Ib information set) (2) becomes
E
II
(ML) =
n
1
1p
[n pm]
if q = 0
if q = 1
If p = 0 and r = 1, and this means that player I’s strategy is ABA, then
E
II
(ML) = n, for all q, s = [0; 1]. In other words, if player I starts the conversation
using language A and is planning to switch to B at Bilingual Ib, then player II is
made to be indi¤erent between A and B in his two information sets, which is to
say that he is indi¤erent between all his pure strategies.
When s = 1, meaning that player II chooses B in the subgame starting at
Bilingual II informations set, then (1) becomes
E
I
(Bilingual Ia) =
qr(m n) + n
n
if p = 0
if p = 1
We can see that if s = 1 and q = 0, -that is, if player II’s strategy is BAA-,
then E
I
(Bilingual Ia) = n for all p, r = [0; 1]. Hence, player I is indi¤erent
between the actions available at Bilingual Ia information set as well as those
available at Bilingual Ib information set. This means that Player I is indi¤erent
between all his pure strategies when player II chooses s = 1 and q = 0.
Therefore, we have found the mixed strategy equilibrium (f0; 1; 0g ; f0; 1; 0; 0g) =
(ABA; BAA), which is Bayesian perfect (here, "sequential equilibrium" and "per-
fect Bayesian equilibrium" are equivalent): the behaviour strategy pro…le =
(
I
;
II
) = (p = 0; r = 1; s = 1; q = 0) and player II’s b elief system = (
II
(x) = and
II
(y) = 1 ) is a Bayesian perfect equilibrium for the Lan-
guage Game. In this equilibrium the spoken language between bilingual players
19
is A:
If we keep p = 0 and q = 0, then the Bilingual Ib and Bilingual II information
sets are not reached with positive probability. Hence, combining the values 0 and
1 assigned to r and s; we would get Nash equilibria that are not Bayesian perfect,
such as (ABA; AAA) (AAA; BAA) and the pooling equilibrium (AAA; AAA).
Note that in all these equilibria, the language spoken among the bilingual players
is A.
The bayesian perfect equilibrium (ABA; BAA) might be viewed as a language
coordination failure because bilingual players do intend to use the language they
prefer most, but they fail and end up using A. Hence, any departure from p = 0,
r = 1, q = 0, s = 1 that permits the realization of the desired coordination in
language B would be an equilibrium too because no player would get a lower
payo¤ and at least one player would get a higher payo¤. These are the following
cases:
(i) If p = 1, r = 1, then E
II
(ML) =
n
n c
if q = 0
if q = 1
; therefore q = 0 is
the best choice and at Bilingual II, and since player I has revealed his bilingual
nature by choosing p = 1, then player II must choose s = 1. Against s = 1, q = 0
(which means, against player II’s BAA), we have seen that player I is indi¤erent
between any p, r = [0; 1]; thus p = 1, r = 1 is a best reply. Hence, (BBA; BAA)
is a Bayesian p erfect equilibrium (with = (p = 1; r = 1; s = 1; q = 0) and = (
II
(x) = 0 and
II
(y) = 1)), in which the spoken language among the bilingual
players is B.
(ii) If q = 1, s = 0, then E
I
(Bilingual Ia) =
r(m n) + n
n c
if p = 0
if p = 1
.
Hence p = 0 is the best choice at Bilingual Ia and, since player II has revealed
his type by choosing q = 1, it must b e r = 1 at Bilingual Ib. Again, if p = 0 and
r = 1 player II is indi¤erent among all his pure strategies, so q = 1, s = 0 is a
best choice. Thus, we get the non Bayesian p erfect equilibrium (ABA; ABA) in
which the spoken language among the bilingual players is B.
(iii) If q = 1, s = 1, then E
I
(Bilingual Ia) =
r(m n) + n
n
if p = 0
if p = 1
.
Hence, the best choices are p = 0 and r = 1. Hence, (ABA; BBA) is a Bayesian
perfect equilibrium (with = (p = 0; r = 1; s = 1; q = 1 and = (
II
(x) =
and
II
(y) = 1 )) in which the spoken language between bilingual players is
B.
Thus, the set of equilibria for Case 1 is:
20
N =
(ABA; BAA); (ABA; AAA); (AAA; BAA); (AAA; AAA);
(BBA; BAA); (ABA; ABA); (ABA; BBA)
Note that player I’s ABA strategy is a best response against all player II’s
strategies (against some strategy it is not the only one though). Hence, ABA is a
weakly dominant strategy for player I. This is so because a bilingual player I who
uses this strategy starts the conversation with A and with this choice he will avoid
to su¤er the frustration cost, n c; moreover, if it is matched to a bilingual player
II who reveals his bilingual nature by choosing B at ML information set (i.e. a
player II using either BBA or ABA) player I then switches to B at Bilingual Ib
information set and gets the maximum payo¤ m. For player II, BAA is a best
response against all player I’s strategies (against some strategies it is not the only
one though). Thus, BAA is a weakly dominant strategy for player II, because to
a bilingual player I who reveals his bilingual nature by beginning the conversation
using B (when using BBA or the equivalent strategy BAA) he will answer with
B and if player I starts the conversation using A he replies with A. Therefore,
the strategy pro…le ABA for I and BAA for II allows both players to avoid the
minimum payo¤ n c and reach the maximum payo¤ m.
We have seen that (ABA; BAA) is a Bayesian perfect equilibrium. An im-
portant feature of this equilibrium is that the linguistic convention among the
bilingual players is to speak the majority language A. The equilibria in which
the spoken language among the bilingual players is B, such as (BBA; BAA),
(ABA; ABA) and (ABA; BBA) involve the play of weakly dominated strategies.
Case 2. Suppose that Assumption 1 is satis…ed.
Assumption 3: For all 0 < < 1 ;
c > (m n)
(1 )
Assumption 3 says that, for a given ; the cost c that a bilingual player su¤ers
when he chooses the minority language B but is forced to switch to the majority
language A; is greater than the bene…t (m n) (weighted by the ratio of bilingual
population over the monolingual population,
(1)
) that he would obtain when
he is able to interact using B:
Let us assume that p = 0, then from (2)
21
E
II
(ML) =
n
r(m n) c(1 r) + n
if q = 0
if q = 1
Hence, under assumption 3, given p = 0 and any r 2 [0; 1],- which means that
we are considering player I’s pure strategies ABA and AAA-, q = 0 maximizes
E
II
(M L). Note that this result is independent of any s 2 [0; 1], therefore, player
II will play the (strictly undominated) BAA and AAA strategies.
Let us assume now that q = 0, then from (1)
E
I
(Bilingual Ia) =
n
s(m n) c(1 s) + n
if p = 0
if p = 1
Therefore, given q = 0 and any s 2 [0; 1], - which means that we are considering
player II’s pure strategies BAA and AAA-, p = 0 maximizes E
I
(Bilingual Ia).
This result is independent of the value of r 2 [0; 1]. Therefore, player I will play
the (strictly undominated) ABA and AAA strategies.
Hence, we have shown that the set of equilibria is,
N
0
= f(ABA; BAA); (ABA; AAA); (AAA; BAA); (AAA; AAA)g
where (ABA; BAA) is the only Bayesian perfect equilibrium.
Note that, as the intuition would tell us, when the frustration cost is higher
than the bene…ts of speaking B, the language spoken among the bilingual players
in all the equilibria is A:
Case 3. Suppose that Assumption 1 is satis…ed.
Assumption 4 : For all 0 < < 1
c < (m n)
(1 )
Assumption 4 says that, for a given ; the cost c that a bilingual player su¤ers
when he chooses the minority language B but is forced to switch to the majority
language A; is smaller than the bene…t (m n) (weighted by the ratio of bilingual
population over the monolingual population,
(1)
) that he would obtain when
he is able to interact using B:
Let us assume that p = 0 and r = 1 (in other words, assume player I’s ABA
strategy), then from (2)
22
E
II
(ML) =
n if q = 0
(m n) + n c(1 ) if q = 1
By assumption 4, q = 1 maximizes E
II
(M L) given any s 2 [0; 1] (thus against
player I’s ABA, player II’s best responses are BBA and BAA.
Now suppose that q = 1, then
E
I
(Bilingual Ia) =
r(m n) + n
s(m n) c(1 s) + n
if p = 0
if p = 1
When p = 0 and r = 1 the expected payo¤ for player I at Bilingual Ia is
maximized when q = 0 and any s 2 [0; 1]: Hence, against player I I’s ABA and
BBA strategies, player Is best response is ABA. When b oth player I and II choose
B at Bilingual Ib and Bilingual II informations sets, the strategy pro…le (ABA,
BBA) would be a Bayesian perfect equilibrium (that is, the behaviour strategy
pro…le = (
I
;
II
) = (p = 0; r = 1; s = 1; q = 1) and player II’s belief system
= (
II
(x) = and
II
(y) = 1 ) is a Bayesian perfect equilibrium of the
Language Game). (ABA; ABA) is a Nash equilibrium, but not Bayesian perfect.
In both equilibria the language spoken by the bilingual players is B.
Supp ose now that bilingual player I chooses p = 1 and r = 1( that is, player
I chooses BBA). Then, E
II
(ML) = n cq. Hence, E
II
(ML) is maximized
when q = 0 and any s 2 [0; 1]. Now, if q = 0 and s = 1, then from (1) and
Assumption 4 we have E
I
(Bilingual Ia) = (m n) + n c(1 ) if p = 1
and any r 2 [0; 1]. Then (BBA, BAA) is a Bayesian perfect equilibrium (where
now = (
I
;
II
) = (p = 1; r = 1; s = 1; q = 0) and = (
II
(x) = 0 and
II
(y) = 1)):In this equilibrium the language spoken by the bilingual players is B
as well.
It can be seen that (AAA, AAA) is an equilibrium too in this case, but not
Bayesian perfect. Hence, the set of pure equilibria is
N
00
= f(ABA; BBA); (ABA; ABA); (BBA; BAA); (AAA; AAA)g
We can conclude that when the bene…ts of speaking the minority language B
are bigger than the frustration cost, we get the obvious consequence that in all
the equilibria but (AAA; AAA) the conversation among bilingual players is in B.
We have seen that the risk that a bilingual player faces is the loss of utility, c.
The higher is this loss of utility relative to the bene…t of speaking B; (mn)
(1)
;
the higher is the risk a bilingual player would face. As a consequence, it becomes
safer for a bilingual player to use the majority language A instead of B:Therefore,
in equilibrium, the probability that the interaction might happen using language
23
B decreases as the frustration cost c increases and in the limit, represented by
Case 2, the minority language B is not used at all.
From the above analysis we have obtained the following result
Proposition 2
Given assumption 1, there exists Nash equilibria in which B could be the
language used by bilingual players if, for all 0 < < 1 ,
c 5 (m n)
(1 )
A will be the spoken language in equilibrium, otherwise.
Remark
Case II above has no interest at all because it is only A the language spoken in
all the resulting equilibria and that would be the convention among the bilingual
players. Case III has no theoretical interest either; we found that all the equilibria
in which both bilingual players decide, at certain information set, to use B they
end up speaking B. Hence, it is very likely that in the long run the convention
that would be built among the bilingual players will, certainly, be to speak B.
In Case I we have a di¤erent situation. For instance, the sequential equilibrium
(ABA; BAA) describes a situation in which both players intend to use B at some
information set, but there is a communication failure among them and they end
up speaking A. Further, this is the only equilibrium of the set N in which both
players use undominated strategies. In other words, the rest of equilibria of the
set of Nash equilibria N in which the spoken language is B, at least one player is
using a weakly dominated strategy. Therefore, the interesting case is the situation
described by Case I.
4. Conclusion
The Language Game highlights the incomplete information structure that
bilingual players face. This may have dangerous consequences on the language
diversity of the society. First, because the minority language B su¤ers negative
externalities from the majority language A; and second, because under some pay-
o¤ stucture, it is the main reason of language B coordination failures among the
24
bilingual players. As a consequence, the probability that B, an already “endan-
gered minority language”, became a “moribund language”may easily increase in
the medium term. Therefore, a political intervention, in the form of language pol-
icy, is called for to ensure the provision of B and mantain the linguistic diversity
of the society.
5. Future Work ( Part II: Dynamic Analysis)
This part studies Case I in a di¤erent setting. In real-life situations, the
Language Game described previously is not an interaction that happens once. On
the contrary, this game is played many times by many di¤erent people and its
solution is not an individual one but a social construct. Therefore, a di¤erent
analytical approach is needed. This part studies the issue of building a speech
convention among the bilingual agents (which is equivalent to equilibrium selection
in the set of equilibria N) using an evolutionary approach.
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