Mar 19, 2018 - At the Relational Crossroads: Narrative Selection, Contamination, Biodiversity in Trans-Local. Contexts. A. Antocia, N. Bellancab, G. Galdib,â.
At the Relational Crossroads: Narrative Selection, Contamination, Biodiversity in Trans-Local Contexts
a
A. Antoci , N.
a Dipartimento b Dipartimento
b
Bellanca , G.
b,∗
Galdi
di Scienze Economiche e Aziendali, University of Sassari, Via Francesco Muroni 25, 07100 Sassari, Italy
di Scienze dell'Economia e dell'Impresa, University of Florence, Via delle Pandette 32, 50127 Firenze, Italy
Abstract
Agents act according to their knowledge of the state of the world and the relevant consequences which they may foresee, i.e. the payos corresponding to their choice of action. Such a pervasive representation of the world and of the position of the agents in the succession of cause-eect links is more than an aseptic information, but is rather culturally and emotionally dense. These overarching maps have been introduced in the literature as narratives, which include both the comprehension of the mechanisms of reality and the role covered in the latter by the self. Contrary to strategies and actions, narratives cannot be swiftly changed to better t dierent situations and to achieve higher payos. We thus study what is the resulting narrative dynamics when two social groups with dierent "favoured" narratives interact in a Trans-Local Context. Indeed, we present the outcomes, framing them into three categories according to their interpretation in terms of narratives diusion:
Selection, Contamination, Biodiversity .
Keywords:
Trans-local context; Narrative; Social identity; Cultural change; Behavioural
economics; Beliefs
∗
Corresponding author. Current email address is: giulio.galdi@uni.it
Preprint submitted to Elsevier
March 19, 2018
2
1.
Introduction
1. Introduction
Social identities and narratives have been the subject of some key contributions since the 1990s (e.g. Sen, 1985; McCloskey, 1990; Strassmann, 1993; McAdams, 1995), and of numerous contributions in recent years (e.g. Akerlof and Kranton, 2000; Darity et al., 2006; Davis, 2010; Shiller, 2017). In June 2016, a special issue of this journal was largely devoted to these topics (Snower, 2016). The relationship between social identities and narratives is particularly important for analysing how identities change. Indeed, Collier (2016) highlights that a certain story of the mechanisms of the world and the causal relations with an agent's own actions, i.e. a narrative, may only be maintained or adopted by an agent if it does not contrast with an individual self-representation into her society, i.e. her social identity. In addition, the narrative embraced by an agent may also inuence the social identity she adopts, also by redening the social group within which such narrative is more widely accepted. In other words, since identity is a socially constructed preference-ordering, sharing the same narrative with others may lead an agent to co-shape her social identity with them, ultimately aecting the agent's preferences. Since there are many contributions in the literature dening this conceptual couplet, it is useful to specify how we understand narratives and social identities in this work.
Narratives are compelling stories about the interpretation of reality and the position in the world of the members of a social group. A narrative tells where we come from, who we are and where we go; it is a set of stories that, ringing events along a plot, gives sequential and causal coherence to the world and/or to the experience of our group in the world. A narrative does not necessarily have to convey new information: if members of the group, for example, recite a daily prayer, or evoke a proverb for the thousandth time, they are not providing (new) informative content; rather, they are reproducing their meaning as a group. Moreover, there can be much "ction" or "imagination" in a narrative, since its task is not to gauge events, but to constitute an instrument for making meaning that dominates much of life in culture (Bruner, 1990, p. 97). This is the case, for instance, of ctitious examples which provide an interpretative illustration of events. Collier (2016) reinforces this point by underlining that the informative content, the causal link, contained in the narrative does not need to be correct for the narrative to be believed. The recent surge of "fake news" in our world and their capacity to diuse rapidly makes us witnesses of such independence of narratives' spread from the truthfulness of their informative content. In game theoretic terms, narratives can be thought of as
xed
strategies, which are to be played in multiple games simultaneously. Since
they have deep cultural roots, narratives cannot swiftly be changed from one game to another.
As concerns social identity, the literature presents it as the process through which subjects recognize - and are recognized by - other subjects as part of a group and through which, on the basis of such aliations, they attribute meaning to their own experiences over time (Tajfel, 1981;
3
1.
Introduction
Turner et al., 1987). In the denition we adopt here, social identity is characterised by two main factors: the prevalence and the relative advantage of a narrative with respect to the others that may be available in the social group of reference. On the one hand, hence, we see social identity as being characterised by the main vision of the world that is shared by the members of the social group. As we detail later in the modelisation, this is the most volatile part of social identity and may be subject to change in a shorter time horizon.
On the other hand, we see social identity
as inuencing the relative performance of narratives, yielding dierent rewards to the individuals adopting them. As narratives are representations of the world, they may be more coherent with the social context of one group with respect to another, thus increasing the performance of agents who adopt the
right
narrative in the
right
social group.
In this work, we rely on these two key concepts to tackle our research question: what happens when two separate social groups interact in a purely relational space?
In order to answer this
question, we build a game theoretic environment in which two sub-populations or social groups interact, each with its own social identity and preferred narrative. In this respect, we agree with Bates et al. (2000) in seeing such theoretical framework as among the most suited to investigate narratives. In two simultaneous games, all agents interact with one co-member and a member of the other group. In within-group interaction, each group has a dominant narrative which would always yield higher payos. However, in the cross-group interaction no narrative gives agents an advantage over the other. Indeed, the second game takes place in a purely relational space, which we dene as Trans-Local Context (TLC, hereafter), in which no narrative is dominant. This is the only place in which agents belonging to dierent social groups may interact. Indeed, in this work we do not deal with migration, understood as the stable change in the membership of an agent, from one group to the other. Obviously, we all know that migrations are a relevant and persistent phenomenon in human history. The reason why we exclude migrations from our analysis is that they are hardly sucient to explain the change in social identities of agents. In particular, we sustain that there is a wide divergence in human societies between the individual desire to change the social group membership and the actual ability to do so. This means that a migrant, an agent leaving her own social group and entering another, may not be able to automatically become a member of the new social group just by migrating into the latter's territory. In order to understand what it takes to be recognised by the members of the new group and being attributed with the group's social identity we need to look at other phenomena.
Indeed, there are other relevant social components of life that we all have before our own eyes and which take place on a larger scale than the migratory phenomenon. In particular, there are many occasions in which members of a social group exchange ideas, information, goods, resources with members of another group. Also, there are occasions in which they live with members of the other group life experiences, whether aective or competitive, cooperative or conicting. There is
4
2.
The model
a wide spectrum of occurrences in which a member of a social group can meet and exchange with a member of another group: from commercial and nancial transactions to tourist trips, from religious pilgrimages to military clashes, from health journeys to extended family reunions, from scientic conferences to technology exhibitions. We call mixed meetings the whole set of these exchanges: they are frequent, have the most diverse content, but are often episodic for the agent, and never come to the denitive change of the subject's belonging to a social group. In other words, we could say that in this model we disable any entry or exit mechanisms for the social groups, with all agents sticking to their initial membership, while interactions between the social groups occur on a regular basis.
Agents are localised (territorially rooted) in a population without migration (intended as
social group shift). This restriction seems plausible, since even in a globalised world, the localisation of social groups remains strong and pervasive and migration involves only a minority of the world population (estimated in a constant 3% by Berg and Besharov, 2015b,a).
The aim of this work is to study the role of such frequent interactions occurring within TLCs in the diusion of a narrative. We dene and characterise three main functions a TLC may serve according to the narrative dynamics that takes place:
enhancement.
Selection, Contamination, Biodiversity
Indeed, a rst possibility is for one of the two narratives to disappear from both social
groups, with all agents belonging to one group completely forsaking the narrative that is withingroup dominant. The second function describes the case in which all agents in a social group stick to their within-group dominant narrative, whereas agents in the other group are divided into adopters of one or the other narrative. Finally, the TLC may enhance the biodiversity of the narratives if at the end of the dynamic process of interaction both the existing narratives are present in some measure within both social groups. We develop a formal model based on replicator dynamics to highlight what are the conditions under which one of the three functions is served by the TLC.
2. The model
The model we propose in this work studies the evolutionary dynamics of the narratives in two dierent social groups.
In order to illustrate such model, we develop an example based on the
evocative work by Greene (2000), representing the institutional and cultural clash of civilisations that occurred since late Middle Age in Crete. from the domination of the
Indeed, the author describes how the isle passed
Repubblica Serenissima di Venezia
to the rule of the Ottoman Empire.
Venice and the Ottomans were very close commercial partners, to the point that even the occasional territorial conicts in the Balkans and over the Mediterranean Isles have never resulted in a lasting break of diplomatic and trading exchanges between the two countries.
In particular, Venetian
merchants strived to maintain the brokerage role between Europe and the luxury goods coming from the East, which required them to be in good terms with the Ottomans. The Ottoman economy, in turn, relied substantially on the taxes they could levy on Venetian trade. This reciprocal dependence
5
2.
The model
laid the ground for an enduring diplomatic relationship that lasted through centuries, with the countries overlooking occasional conicts in the name of economic cooperation. This co-existence also brought their respective religions to confront and to peacefully cope with each other, in order to maintain a favourable environment for both the Christian Venetians and the Muslim Ottomans to conduct their trade. In particular, the inuence of these competing powers over the domination of Crete had religious fragmentation as a side eect, with both Christians and Muslims living on the isle.
Religions are superb examples of narratives: they are visions of the world with a strong cultural connotation, overarching inspiring messages on the mechanics of life. For this reason, we adopt the Christian-Muslim divide as the narrative confrontation of the following illustrative example, which features Venetians and Ottomans as the two social groups interacting with each other in a neutral, Trans-Local Context:
the market.
Let us think of Venetian and Ottoman traders conducting
businesses in their home countries, where religious congruence with the fellow countrymen allows for smoother economic interactions. Indeed, Ottomans would not provide food during Ramadan (the fasting month for Muslims), whereas Venetians would deem unholy to eat meat on Fridays during Lent. However, their strong trading bonds persisted, which led them to seek coordination in the Cretan food market, for instance.
Analytically, in our model we assume that every trader from the two countries interacts with two other traders: one coming from the same country and another coming from the historically rival country. On the one hand, interactions with fellow countrymen are localised and embedded in the social and cultural context of the country.
On the other hand, interactions with traders
from the other country occur in the market, that is a neutral context in which traders would extract the highest benets by interacting with fellow traders following the same religion. However, although Christianity is more rewarding than Islam in Venice (e.g. because it is more culturally homogeneous with Venetian culture) and the reverse applies in the Ottoman Empire, none of them is advantageous
per se
in the Trans-Local Context, i.e. the market.
Since religious homogeneity increases the economic benets of trades from both countries, traders would prefer to adopt the same religion of their partner. Beyond a coercive imposition of one religion (which is not the subject of our study), the only possibility is then for the trader to switch her own religion to the one of her partner.
However, religions are not changed so easily and the convert
trader would still need to interact with home partners.
A Muslim Venetian could obtain higher
benets with muslim Ottomans in cross-country transactions, but she would incur into a loss (or reduced benets) when interacting with Christian Venetian traders. It should be recalled that such trader does not consider to (or cannot) move to the Ottoman Empire, since we excluded migration from our analysis.
6
2.
The model
According to the preferences of the traders over the possible matches in both the home country and in the Trans-Local Context, dierent outcomes may realise. Venetian traders may value interactions in the Trans-Local Context more than business at home, whereas Ottomans may be willing to face harsh negotiations in the Trans-Local Context as long as traders can be most eective in the domestic market. We consider all possible preferences of traders in the two social groups and study the correspondent outcomes. Is the co-existence in Crete of Christianity and Islam a result of a common preference toward market relations by Venetians and Ottomans? What if only Venetian traders prefer to coordinate whereas Ottomans are less concerned, or else if market cooperation is more convenient to the former with respect to the latter? We propose to outline some answers to these questions, investigating the conditions allowing for one scenario rather than another.
2.1. A formal modelisation The economic interactions of Venetians and Ottomans are a powerful exemplication of subpopulations interacting in a TLC, with the narrative dynamics represented by the confrontation of the two religions in Crete. However, we do not aim to analyse this historical occurrence here, but rather to study the more general phenomenon underlying this example. Indeed, in this paper we provide a formal analysis of the dynamics of interaction between two identity groups, each with its own worldview encapsulated in a narrative. In essence, our study is based on two simple premises: 1) there are mutual gains from trade between actors and 2) trade is easier if two actors have the same worldview, and hence accept the same narrative. The initial position of our analysis is that actors within each group trade with each other, but not with members from the other group. A Trans-Local Context then opens up, in which actors from each group can trade with those from the other group. This creates some payo for those actors who want to trade with the other group to adopt the narrative of the other group. We then formalise how these pressures to switch narratives play out, deriving the dynamics of adjustment to dierent possible equilibria. In one, one narrative wipes out the other; in a second,
Contamination,
Selection,
one narrative remains dominant in
one group and attracts a proportion of adherents in the other group; in a third,
Biodiversity ,
both
narratives have adherents in both groups. The formalisation enables the derivation of the key values of particular variables that determine which of these states occurs. This provides considerable scope for applications to many real contexts, but we leave this to scholars with the requisite contextual knowledge.
Let us now lay down a more formal denition of the problem. The social groups shall now be referred to as North, or
N,
and South, or
S.
This geographical discrimination has no roots in any
real case dierence and merely serves the purpose of distinguishing the two groups. The narrative are then dened as follows: Narrative 1 is dominant in group
N , whereas Narrative 2 is dominant in
7
2.
group
S.
The model
A dominant narrative always grants higher payos to the agents adopting it, with respect
to the other narrative.
At each point in time, agents from both social groups interact with their co-members in a strategic game in which a narrative is dominant. Simultaneously, they also interact with members from the other group in a Battle of the Sexes, in which all agents are better o when coordinating on a narrative, although none is dominant. The payo structure of the combined game does not lead to an overall dominant narrative, so that the shares of agents playing Narrative 1 or 2 in the two social groups is not predetermined. Indeed, we identify the initial conditions and the specic payo structure driving to the several dierent scenarios. We categorise these scenarios according to whether they result in both groups adopting the same narrative and forsaking the other one (Selection); in only one group being divided between the two narratives (Contamination); or else in both narratives being adopted in both groups (Biodiversity ).
The assumptions we pose on the payo structure are limited to what has been introduced discursively until now: within-group dominance of diering narratives and preference for coordination when interacting with agents from the other group.
In order to keep the model as general as
possible, we do not take any further assumption on the payo structure.
In Table 1 we give an
illustration of the payos for agents residing in the North, allowing for a more analytical statement of our assumptions.
Table 1: Payos structure for group N
N1 a1 a2
N1 N2
In this representation, either group
N
or
N2 b1 b2
N1
N1 N2
and
S, respectively.
S1
S1 c1 c2
S2 d1 d2
represent the adoption of Narrative 1 by an agent belonging to
Analogously,
N2
and
S2
represent the adoption of Narrative 2 by
the agents. The outcomes presented in the payo matrices refer to what the row agent (belonging to group
N, here) is rewarded depending on which narrative she adopts and which one her co-members
and agents from the other group adopt.
The payos structure for intra-group interactions in the North is congured as to make favourable, so that
and
b1 > b 2 .
We can thus easily conclude that Narrative 1 would be
N, if it could be played in a separate game. from N interact with members of S, we assume
dominant in group in which agents
a 1 > a2
N1
As regards the mixed meeting, that
c1 > c2
and that
d1 < d 2 .
This eectively congures the mixed meeting as a Battle of the Sexes in which both agents wish
8
2.
The model
to coordinate with each other, but each might prefer a dierent narrative to coordinate on. The payos for agents from group existing with group
S
are analogously represented in Table 2, to highlight the symmetry
N. Table 2: Payos structure for group S
S1 S2
S1 α1 α2
S2 β1 β2
S1 S2
N1 γ1 γ2
N2 δ1 δ2
Also in this case group members are assumed to favour one of the two narratives, and in particular Narrative 2. The payos are ordered accordingly:
N,
agents from group
δ1 < δ2 .
S
α2 > α1
and
β2 > β1 .
also prefer to coordinate in the trans-local context, so that
As for group
γ1 > γ 2
and
We do not make further assumptions regarding the ordering of the other payos.
Without loss of generality, we can replace the payo matrices in Tables 1 and 2 with the matrices in Tables 3 and 4, where the second row has been set to zero.
Table 3: Normalised payos structure for group N
N1 N2
N1 a 0
N2 b 0
S1 c 0
N1 N2
S2 d 0
Table 4: Normalised payos structure for group S
S1 S2
S1 α 0
S2 β 0
S1 S2
N1 γ 0
N2 δ 0
These matrices were obtained by applying:
a − a = a > 0 1 2 c − c = c > 0 1
2
α − α = α < 0 1 2 γ 1 − γ 2 = γ > 0
b1 − b2 = b > 0
(1)
d1 − d2 = d < 0 β1 − β2 = β < 0 δ1 − δ2 = δ < 0
(2)
9
2.
The model
2.2. The evolutionary dynamics To study how the diusion of the two narratives evolves, we study how the shares of agents adopting Narrative 1 varies in time in both populations. We dene adopting Narrative 1 who belong to group
1 − x and 1 − y
N
and group
x
S , respectively.
and
y
the shares of agents
The complementary shares
thus represent the shares of agents adopting Narrative 2 in the two groups. In order
to understand what drives such shares to change, we lay out the expected payos for all narratives available to agents:
PN1 (x, y) = ax + b(1 − x) + cy + d(1 − y)
(3a)
PS1 (x, y) = αy + β(1 − y) + γx + δ(1 − x)
(3b)
PN2 (x, y) = PS2 (x, y) = 0
(3c)
The expected payos in formulas
(3a) - (3c) are the sums of the expected payos of each
strategy in the two games, where the shares an agent adopting Narrative 1 from group
N
x
and
or
S,
y
measure the probability to be matched with
respectively. We then assume that the diusion
process of the two narratives follows the well-known replicator dynamics (Weibull, 1995):
In the above equations,
x˙ = x(1 − x) [PN1 (x, y) − PN2 (x, y)]
(4a)
y˙ = y(1 − y) [PS1 (x, y) − PS2 (x, y)]
(4b)
x˙ and y˙ indicate the time derivatives of x(t) and y(t), respectively, which
depend on the payo dierences:
PN1 (x, y) − PN2 (x, y) = b + d + (a − b)x + (c − d)y
(5a)
PS1 (x, y) − PS2 (x, y) = β + δ + (γ − δ)x + (α − β)y From equations (1) and (2), we derive that be noted from (4) that since of playing
N1
x(1 − x) > 0 ∀ x ∈ (0, 1), x˙ N2 ,
is higher than the one of
This means that in
N
c − d > 0, β + δ < 0
and
γ − δ > 0.
(5b) Moreover, it can
takes on positive values only if the payo
whereas it takes on negative values for
PN2 > PN1 .
the better rewarding narrative diuses at the expense of the other one. The
same argument holds in group
S,
so that
PS1 > PS2
implies
y˙ > 0
and vice versa. Finally, for both
groups there is no variation in time of the share playing Narrative 1, i.e. dierence between playing Narratives
1
or
2
x˙ = y˙ = 0,
if the payo
is null. What plays a crucial role in the dynamics of
the model are the parameters aecting the payo dierences (5a) and (5b):
10
2.
a − b = a1 + b2 − a2 − b1
(6a)
α − β = α1 + β2 − α2 − β1
(6b)
Indeed, (6a) denes the eect that the current share and
x plays on the payo dierence between PN1
PN2 , whereas (6b) measures the eect of the current value of y
relevant to note that for with the share
x
The model
a − b < 0 (a − b > 0)
of agents adopting
N1 .
the payo dierence
This means that for
on
PS1 − PS2 .
PN1 − PN2 b > a
In particular, it is
decreases (increases)
the benets of adopting
Narrative 1 are negatively related with the number of agents who adopt the same narrative in the same social group. The more
exclusive
is the action of adopting Narrative 1, the higher are the
related rewards. Furthermore, by checking Table 3 we may also add that the condition equivalently,
a1 + b2 − a2 − b1 < 0)
describes a payo ordering for group
N
b>a
(or,
which favours the
adoption of dierent narratives over coordination. Specically, it requires the payos lying on the main diagonal of the rst matrix in Table 1, i.e.
a1 + b2 ,
corresponding to coordination on one
of the two narratives, to be lower than the ones on the minor diagonal miscoordination. On the contrary, when
b 0)
the benet
in adopting Narrative 1 increases with the number of fellow adopters, incentivising coordination. The sign of equation (6b) is interpreted analogously for narratives in social group
Moreover, since
c−d
and
γ−δ
S.
are always positive, from (5a) and (5b) we derive that in both
social groups the relative benet of adopting Narrative 1 depends positively on the proportion of agents adopting the same narrative in the other social group. This feature of the model represents the intent of agents from both social groups to coordinate in the TLC.
2.3. Steady states of the dynamics A more in-depth analysis of the system of equations and its steady states is discussed in the appendix of the paper. We here develop only the main points which are functional to the description of the model. First of all, it is useful to point out that the dynamics of the model develop in the two-dimensional square region Q:
Q
According to (4) and (5),
= {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}
x˙ = 0
holds for
y=−
x = 0, x = 1
and along the line:
b+d a−b − x c−d c−d
(7)
11
3.
where we recall that
c − d > 0.
Analogously,
y=−
with
γ − δ > 0.
The functions of the TLC
y˙ = 0
for
x˙ > 0
above line (7) and
x˙ < 0
2.
y˙ > 0
to the right of (8) and
1
or along the line :
γ−δ β+δ − x α−β α−β
We can characterise the regions of the
1.
y = 0, y = 1
(8)
(x, y)
plane as follows:
below it;
y˙ < 0
to its left.
We then proceed to identify the points in the plane where
x˙ = 0, y˙ = 0,
i.e. the steady states of
system (4): (a) All vertices of Q
= (0, 0); (1, 1); (0, 1); (1, 0),
describing a scenario in which each social group
adopts a single narrative; (b) the intersection points, when they exist, between line (7) and the side of Q having between line (8) and the side of Q having
y = 0,
and
x = 1;
(c) the points, when they exist, internal to the square region Q and lying at the intersections of (7) and (8). In most cases (see the Mathematical Appendix), these lines intersect at most in one point, representing a situation in which both narratives grant the same payos and thus co-exist in both social groups.
These three possible sets of steady states are also functional in classifying the three possible functions of TLCs, respectively:
Selection, Contamination, Biodiversity .
3. The functions of the TLC
When two social groups interact, with their members holding dierent representations of the world, three main outcomes may be intuitively derived. Firstly, both social groups may settle to rely on the same narrative, with agents from one of the groups changing the narrative prevalence part of their social identity. In this case one of the two narratives is forsaken and discarded by all agents. Secondly, a social group may keep its grip on its own narrative, whereas some members from the other group may be inuenced into adopting a dierent narrative with respect to the one previously dominating. No narrative is selected out in this case, with two narratives co-existing in
1 For the sake of simplicity, we do not analyse the cases in which α − β = 0 or a − b = 0 here, although such cases are discussed in the Mathematical Appendix of this paper.
12
3.
The functions of the TLC
only one social group. Thirdly, it could happen that agents from both social groups are divided between the two possible narratives. We see it as an increase in the availability of representations of the world for the agents of both groups.
Finally, it could be noted that the agents from both social groups may decide to stick to the narratives that are dominant among their co-members, a scenario represented by the steady state
(1, 0).
In addition, agents could do the exact opposite, each of them adopting the narrative that
is dominant in the other group. This scenario is represented by the steady state
(0, 1).
Although
these two extreme scenarios are included in the rst set (a) of steady states, they are not relevant for this analysis. Indeed, we deem the former to be a trivial case and the latter to be unlikely (as shown in the Mathematical Appendix, the steady state
(0, 1)
is always repulsive).
3.1. Selection Beyond these two cases we discarded from our analysis (the trivial one and the unrealistic one), referring to steady states
(1, 0)
and
(0, 1),
the steady states set (a) also includes the two points
(0, 0) and (1, 1) which are indeed noteworthy.
On the one hand,
(0, 0) identies a situation in which
no agent adopts Narrative 1, although it would be dominant in within-group interactions in the other hand, the steady state
(1, 1) identies the case in which agents from S
Narrative 2 to adopt the one that is dominant and adopted in
N. On
completely forsake
N. In both instances, one of the two
representations of the world dies out, leaving in the system a single narrative adopted by all agents.
As concerns the trivial scenario, from the stability analysis performed (and shown in the Mathematical Appendix), we derive that its steady state
(1, 0) is locally attractive if the following condition
holds:
β+γ 0
and
As concerns the attractiveness of
Selection
steady states
(0, 0)
(9)
and
(1, 1),
it is determined by
the sign of the following inequalities:
Indeed, steady state
(0, 0)
b+d0
(10b)
is attractive if (10a) holds, whereas steady state
(1, 1)
(10b) holds. In other words, condition (10a) describes a context in which agents in from sticking to Narrative 1 when agents in
S
N
is attractive if
are discouraged
uphold Narrative 2. Indeed, the benet
b they receive
13
3.
Figure 1: In this scenario, only Selection steady states can be reached. According to whether the system is currently in the blue area or the pink one, it will converge to either (0, 0) or (1, 1), respectively.
The functions of the TLC
Figure 2: In this scenario, the system may converge to either a Contamination or a Selection steady state, according to whether it currently lies in the yellow or the pink area, respectively.
from within-group interactions is more than oset from loss
d resulting from miscoordination in the
TLC. Analogously, condition (10b) describes a symmetrical situation in which agents in higher payo by adopting the within-group dominated Narrative 1, as the loss with fellow southerners are more than oset from coordination gains
γ
α
S
obtain a
from encounters
obtained in the TLC. It
should be noted that the two conditions may hold simultaneously, so that both steady states and
(1, 1)
(0, 0)
may be attractive at the same time. In this case, the system converges to one steady
state or the other according to the distribution of the two alternative narratives at the initial time
t = 0,
i.e.
x(0)
and
y(0).
An example of phase diagram in which only the steady states
(0, 0)
and
(1, 1)
are attractors is
represented in Fig. 1. The basins of attraction of the two steady states are separated by the stable manifold of the internal steady state, which is a saddle point.
Moreover, in Fig.
2 we show an
example of how the TLC may serve dierent functions according to its initial conditions. Indeed, in Fig. 2 we show how the system may reach alternatively either one of the a border solution, which we classify as a
Contamination
Selection steady states
or
scenario. In the gures provided, the two
basins of attraction are separated by the stable manifold of the interior saddle point and represented by a dierent colouring of the phase diagram.
3.2. Contamination In this case one of the two social groups sticks to its within-group dominant narrative, while the other is divided between agents adopting either Narrative 1 or 2: the TLC has a
Contamination
function in allowing one of the two narratives to partly diuse in the other group.
The steady
states corresponding to this function of the TLC are the ones described in (b) and lie on the sides
14
3.
The functions of the TLC
of the region Q. Indeed, steady states lying on the sides identify those cases in which one of the two social groups embraces entirely the within-group dominant narrative, whereas in the other group a share of agents in the interval
(0, 1)
adopts Narrative 1 and its complement adopts Narrative 2.
In particular, it should be noted that no steady states are found on the sides of Q with
y=1
x=0
and
(see the Mathematical Appendix). This would represent a case in which either group adopts
a within-group dominated narrative, even at the cost of miscoordination in the TLC.
Analytically, as can be read in the appendix,
Contamination
equilibria may occur and be
attractive according to the sign of coecients (6a) and (6b): 1. if (6a) is negative, whereas (6b) is positive, then only the steady state lying on the side of Q with
y =0
can be attractive, which occurs when it lies on the left of line (8). This steady
state exists when the following conditions hold:
a+d0
and
(11)
In other terms, conditions (11) describe a situation in which it is preferable to adopt Narrative 1 in group
N
only if at least some of the fellow northerners are adopting the other Narrative.
This means that a tive 2, i.e.
y = 0,
Contamination
equilibrium in which everybody in group
whereas some northerners do not adopt Narrative 1, i.e.
only if narratives have an "elitist" nature in group
N.
S
adopts Narra-
x < 1,
Moreover, by looking at the relative
payo dierence of Narrative 1 (5a), we may see that since conditions (11) imply as
x
increases the relative payo of Narrative 1 decreases in
as Narrative 2 becomes more popular in
N,
feedback on the diusion of narratives in
N,
S
a − b < 0,
and vice versa. Equivalently,
Narrative 1 yields better payos. This negative
N
makes it possible for an equilibrium to occur
in which northerners are divided between the two narratives. positive, in
is possible
By contrast, since
α−β
is
we have self-reinforcing dynamics, so that as the share of southerners adopting
either narrative increases, its relative payo increases as well, making it even more benecial to adopt. 2. if (6b) is negative, whereas (6a) is positive, then only the steady state lying on the side of with
x = 1 can be attractive, which occurs when it lies above line (7).
Q
In this case, the steady
state exists if:
α+γ 0
(12)
This situation is specular to the previous one, with narratives showing an "elitist" dynamic in group
S.
We now have that
α − β < 0,
and the relative payo dierence in
S
(5b)
15
3.
The functions of the TLC
tells us that this translates into decreasing relative payos for Narrative 1 as
y
increases.
Equivalently, as the share of southerners adopting either narrative increases, it yields lower payos.
Analogously to the previous case, this negative feedback dynamics allows for an
equilibrium in
S
in which both narratives co-exist, whereas in
N
a−b > 0
the coecient
allows self-reinforcing dynamics that make a mixed equilibrium unstable, when it exists. 3. if both (6a) and (6b) are negative, then the steady states lying on the side of and on the one with
y = 0
Q
with
x=1
are simultaneously attractive if an internal steady state exists
2
and is a saddle . In this nal case, the negative feedback operates in both groups, leading to possible mixed stable equilibria in
S
and in
N.
Which equilibrium is eventually reached
depends on the initial conditions of the dynamical system. No narrative is self-enforcing in either group. 4. if both (6a) and (6b) are positive, then neither of the
Contamination
equilibria can be
attractive (although they may exist).
In the following gures, we show some instances of
Contamination
equilibria referring to case
3, with dierent parameter specications to allow for variety in the scenarios. When the line (7) is above line (8) in the phase diagram (as in Fig. 3), the steady state on the side with global attractor, so that all agents in in
N
S
y=0
is a
adopt their within-group dominant narrative while agents
are divided between Narrative 1 and Narrative 2. Conversely, in Fig. 4 line (8) is above line
(7) and the global attractor is the steady state on the side with to their within-group dominant narrative while agents in
S
x = 1,
so that all agents in
N
stick
split between the two. In both Figs. 3
and 4 only one colour has been adopted to represent the presence of a single basin of attraction, as there is a global attractor.
2 Check
equations (A.16) - (A.19) in the appendix for more details.
16
3.
Figure 3: In this scenario, only one global attractor exists, so that the system always converges to the side of Q with y = 0.
The functions of the TLC
Figure 4: In this scenario, only one global attractor exists, so that the system always converges to the side of Q with x = 1.
In the following phase diagram we show an instance in which two
Contamination
equilibria
are attractive, while an internal steady state, which is a saddle, exists where lines (6a) and (6b) intersect.
This scenario is shown in Fig.
5, in which there are two attractors with their related
basins of attraction separated by the stable manifold of the interior saddle point.
Figure 5: In this scenario, the system may reach one of the two Contamination steady states, according to its initial position in the phase diagram.
3.3. Biodiversity Finally, the internal steady state, where both narratives coexist in both groups, represents the case in which the variety of narratives increases in both groups. The conditions for the existence and
17
3.
The functions of the TLC
3
stability of this internal steady state are detailed in the appendix . For the sake of this discussion, it suces to point out that in order for the internal steady state to be attractive, both coecients (6a) and (6b) need to be negative, i.e. narratives have an "elitist" dynamic in both social groups. By looking at (7) and (8), we see that the negative coecients imply that both lines have positive slope. The lines represent the locus of points in which Narratives 1 and 2 have the same payo in the two groups, so that a positive slope implies that for this equality to hold, changes in and
y
must have the same sign.
North (i.e.
x
1
1
increases its diusion in the
increases), its "elitist nature" will reduce its relative payo for Northerners with
respect to Narrative Narrative
In other words, when Narrative
x
2.
In order for the two narratives to yield the same payos, the diusion of
must increase in the South as well (i.e.
y
needs to increase). This holds symmetrically
for both narratives in both groups: in order for the two narratives to be just as good, within-group externalities and between-group externalities need to oset each other.
Beyond the negative sign of coecients (6a) and (6b), the existence and stability of the internal steady states requires further conditions to hold, which are presented below. Firstly, we note that it must hold that:
a+d0
(14)
Analogously, if (14) did not hold, line (8) (along which payos in the South are equal for both narratives) would lie below square
Q,
so that
y˙
would always be negative inside the square. The
interpretation is analogous to the one of condition (13). Indeed, condition (14) describes the least
3 Check
section Appendix A.2
18
3.
favourable case for a southerner to adopt Narrative always outperform Narrative
1,
The functions of the TLC
2.
If it did not hold, then Narrative
2
would
thus making an internal steady state impossible.
Finally, in order for the internal steady state to exist and be stable, the following two conditions must be satised:
a+d β+γ < c−d α−β
(15)
b+d β+δ < a−b γ−δ
(16)
Condition (15) requires northerners to obtain lower payos by adopting Narrative to Narrative
(1, y˜),
2
when the system is on the
Narrative
in group
S.
1
Contamination
steady state
is adopted by all the members of group
N,
(1, y˜),
with
1 with respect 1 > y˜ > 0.
In
whereas the two narratives coexist
In such a context, condition (15) requires that the benets due to the coordination
with southerners adopting Narrative the "elitist" nature of Narrative
1
1
are more than compensated by the negative eects due to
in group
N.
In geometrical terms, it requires such steady state
to lie under line (7).
Analogously, condition (16) requires southerners to obtain lower payos by adopting Narrative
2
with respect to Narrative
1 > x b > 0.
In
(b x, 0),
1
when the system is on the
Narrative
narratives coexist in group
N.
1
Contamination
steady state
is adopted by all the members of group
S,
(b x, 0),
whereas the two
In such a context, condition (16) requires that the benets due to
the coordination with northerners adopting Narrative eects due to the "elitist" nature of Narrative steady state to lie under line (8).
2
2 are more than compensated by the negative
in group
S.
In geometrical terms, it requires such
Conditions (14) and (15) also imply (see the Mathematical
Contamination steady state (1, y˜), whereas condition (16) Contamination steady state (b x, 0). The existence of (1, y˜), in its turn,
Appendix) the existence of the the existence of the
that when Narrative
1
exists a threshold level
is adopted by the whole social group
y˜
of narrative diusion in group
same payos for members of group the
Contamination In Fig.
with
steady state
S.
S
N
implies implies
in which it is favoured, then there
such that the two narratives yield the
Analogous comments can be made about the existence of
(b x, 0).
6, we show the case in which the internal steady state exists and is stable.
In such
a context, the internal steady state is globally attractive (see the Mathematical Appendix), as suggested by the presence of a single colour representing a single basin of attraction.
19
4.
Discussion and further research directions
Figure 6: in this scenario, the internal steady state is a global attractor. 4. Discussion and further research directions
We argued that Trans-Local Contexts, in which agents from two dierent social groups interact, allow for the narratives held within such social groups to change.
Indeed, we consider dierent
narrative diusion scenarios that may prompt agents not to adopt the narrative that is withingroup dominant. We categorised the possible outcomes of the system according to three dierent functions that may be covered by TLCs:
Selection, Contamination, Biodiversity enhancement.
In all the three cases analysed, we found that the existence of TLCs is crucial to narrative dynamics in a population, leading to very dierent outcomes with respect to within-group narrative dynamics. However, the model we presented in this paper is not representative of all possible characterisations of the interactions, and others may be envisaged. For instance, it is relevant to remark that in the model we presented no narrative has intrinsic value to the agents, as such value actually relies on the share of group and non-group members adopting the same narrative. This assumption is taken in order to describe the most general case possible and also resonates with the illustration provided, in which being either christian or muslim does not increase payos
per se.
However, for other
instances of narratives there might be one narrative which eectively enhances the performance of its adopters. Let us consider one of the examples provided by Collier (2016), where "Together we stand, divided we fall" and "It's every man for himself" are two opposed narratives. Depending on the situation under study (working team output, personal wage, number of goals scored by a soccer team Vs a specic footballer), one of these two narratives could enhance payos, aecting the incentive structure. Moreover, insisting on the same couple of narratives, we can see how the assumption that coordination on a narrative increases the payo from inter-group exchanges does not need to hold either. Indeed, members of a social group believing that "It's every man for himself" could benet from interacting with agents who uphold that "Together we stand, divided we fall".
20
4.
Discussion and further research directions
As concerns the historical illustration we proposed, religions can hardly be framed according to the latter relationship and let us focus on coordination benets. Furthermore, although a narrative does not incur into changes throughout the diusion process, it could also develop an inner mutation, even to the point that two alternative versions of the narrative arise.
It could be argued, for
instance, that once a narrative diuses where it was once dominated in within group interactions, it is adapted in such a way by the recipient group that it actually constitutes a variation of the original narrative. Syncretic faiths may be one example of such narrative variations, although the eects on coordination would need to be further investigated. In for
authenticity
Selection
scenarios, the urgency
may also lead one group (perhaps the one where the narrative is dominant) to
develop a "purer" form of the narrative, to strengthen social identity within that group. Religion may serve as a good analogy once again. Finally, we characterise social identity as constituted from both the current diusion of the existing narratives within the corresponding social group and the payo structure of the group for each narrative. In this perspective, the variation of the relative diusion of a narrative, i.e.
x or y , aects social identity in one of its components.
If "It's every man
for himself" becomes more widespread in a group where its opposite was once dominant, we can safely conclude that the social identity of the group is changing, just as if incentives switch in favour of miscoordination of narratives. On this respect, we do not suggest what the interplay between these two components is and further research could shed light on their interdependence. Indeed, it would allow for a more comprehensive theory of the evolution of social identity in sub-populations and the role of TLCs in it.
From an empirical point of view, scholars might investigate narrative dynamics in countries after international trade between them opens or is fostered by specic policies. Beyond the market, other TLCs for countries may be represented by supra-national organisations such as the EU or the WTO and narrative change may thus be sought in converging national policies. On the same line, rms or organisations working on a common project expose their respective members to the narrative that is prevalent in the partner rm or organisation, possibly prompting a pull to coordination on one of the two narratives. Besides the tools and interpretative frameworks already employed by economic anthropologists and ethnographers, new analytical tools are being developed with the potential to make meaningful estimates of narrative diusion. A remarkable study on the subject is oered by Houghton et al. (2013), in which the authors also propose a theoretical model of the interaction between alternative narratives which includes congruence feedbacks on such narratives from the real world. As the world becomes increasingly interconnected, TLCs are likely to emerge or increase in relevance. We thus call on new research to investigate the way in which such interconnection in non-identitarian contexts may inuence narrative dynamics within social groups.
21
4.
Discussion and further research directions
Disclosure statement
The authors are not aware of any aliations, memberships, funding, or nancial holdings that might be perceived as aecting the objectivity of this article.
22
Appendix A.
Appendix A.
In this appendix, we shall analyse the system:
x˙ = x(1 − x) [PN1 (x, y) − PN2 (x, y)]
(A.1a)
y˙ = y(1 − y) [PS1 (x, y) − PS2 (x, y)]
(A.1b)
PN1 (x, y) − PN2 (x, y) = b + d + (a − b)x + (c − d)y
(A.2a)
where:
PS1 (x, y) − PS2 (x, y) = β + δ + (γ − δ)x + (α − β)y
(A.2b)
under the assumptions:
a, b, c α, β, δ
> 0
and
d 0, b + c > 0, c − d > 0, α + δ < 0, β + δ < 0, γ − δ > 0
(A.5)
Appendix A.1. Steady States The dynamics of the model develop in the two-dimensional square
Q:
Q = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} According to equations (A.1a) and (A.1b),
y=−
By the condition
c−d > 0
below it. Analogously,
y˙ = 0
for
x˙ = 0
holds for
x = 0, x = 1
and along the line:
b+d a−b − x c−d c−d
(see (A.5)), it follows that
y = 0, y = 1
x˙ > 0
or along the line:
(A.6)
above the line (A.6) and
x˙ < 0
23
Appendix A.
By the condition
y˙ < 0
β+δ γ−δ − x α−β α−β β+δ δ−γ
y
= −
x
=
γ−δ > 0
y˙ > 0
(see (A.5)),
α − β 6= 0
if
(A.7)
α−β =0
if
(A.8)
holds to the right of lines (A.7) and (A.8) while
to the left.
The steady states of system (A.1a)(A.1b) are: 1. All vertices
(0, 0), (1, 1), (0, 1), (1, 0)
of
Q;
each of them represents a scenario in which each
social group adopts a single strategy. 2. The intersection point (when it exists) between the line (A.6) and the side of if of
a − b 6= 0. Q
having
In the non robust case in which
y=0
are steady states if
a − b = 0,
b+d=0
(see (A.6)).
3. The intersection point between the line (A.7) and the side of In the non robust case in which
x=1
are steady states if
α − β = 0,
Q having y = 0,
we have that all the points of the side
Q
having
x = 1,
if
α − β 6= 0.
we have that all the points of the side of
(β + δ) / (δ − γ) = 1
Q
having
(see (A.8)).
4. The points, when they exist, internal to the square region between the line (A.6) and the line (A.7) ((A.8), if
Q
α − β = 0).
and lying at the intersection
Note that at most one internal
steady state exists if the lines (A.6) and (A.7) have dierent slopes, that is, if
(a − b) (α − β) 6=
(c − d) (γ − δ). It is easy to check that the line (A.6) cannot intersect the side of holds for every
x ∈ (0, 1),
when
y = 1:
if all agents in group
S
Q
with
y=1
in that
x˙ > 0
adopt Narrative 1, then the latter
is more rewarding than Narrative 2 for the agents belonging to group N, whatever the share adoption of Narrative 1 in group of
N.
x
of
Analogously, the lines (A.7) and (A.8) cannot meet the side
Q with x = 0 in that y˙ < 0 holds for
every
y ∈ (0, 1),
when
x = 0:
if all agents in group
N
adopt
Narrative 2, then the latter is more rewarding than Narrative 1 for the agents belonging to group
S,
whatever the share
y
of adoption of Narrative 2 in group
S.
Appendix A.2. The internal steady state The Jacobian matrix of system (A.1a)(A.1b), evaluated at an internal steady state
0 0
and
and therefore
α−β > 0
(x, y)
hold, then
is a saddle point.
Det J < 0
(and consequently
(x, y)
is a saddle) if:
γ−δ a−b < c−d α−β
(A.10)
According to condition (A.10), the slope of the line (A.6) must be higher than that of the
a−b > 0 b+d − 0,
and
−
(A.6) and (A.7) have negative slopes and intersect the
β+δ > 1, α−β
y axis
at
respectively.
3. If both the conditions
a−b0
if
x=1
x=x b := −(b + d)/(a − b)
(A.12)
y = yb := −(a + d)/(c − d)
(A.13)
at:
b + d > 0; yb > 0
if
a + d < 0,
and
yb < 1
always. The line (A.7) meets the
x-axis
at:
and the vertical line
where:
a−b0 and
x=1
always;
α − β < 0,
x=x ˜ := −(β + δ)/(γ − δ)
(A.14)
y = y˜ := −(β + γ)/(α − β)
(A.15)
at:
y˜ > 0
if
β + γ > 0,
and
y˜ < 1
a unique internal steady state
if
α + γ < 0.
(x, y)
Therefore, in the context
exists if and only if:
a + d < 0 (b y>0
holds)
(A.16)
β + γ > 0 (˜ y>0
holds)
(A.17)
hold, and either:
a+d β+γ > c−d α−β b+d β+δ > a−b γ−δ 4 Only
(y b
< y˜ holds)
(A.18)
(x b
x ˜
(A.21)
holds)
hold. When (A.18) and (A.19) hold, then the internal steady state is a saddle point, while it is attractive when (A.20) and (A.21) are satised. It is easy to check that if hold, and that in which in Fig.
6).
(x, y)
x˙ < 0
and
is attractive, then the region in
y˙ < 0,
Q
x˙ > 0
in which
and
are both positively invariant (see the arrow diagram
This implies that no closed trajectory can exist around
Poincaré-Bendixson Theorem, the steady state
(x, y)
(x, y)
and therefore, by the
saddle, then all the trajectories in the interior of
Q,
Q. By (x, y) is a
is globally attractive in the interior of
the Index Theory (see e.g. Lefschetz, 1963) and the Poincaré-Bendixson Theorem, when
approach a steady state in the boundary of
y˙ > 0
not belonging to the stable manifold of
(x, y),
Q.
When no internal steady state exists, then by the Poincaré-Bendixson Theorem, all the trajectories in the interior of
Q
approach a steady state in the boundary of
Q.
Appendix A.3. The steady states (0, 0), (1, 1), (0, 1), (1, 0) Appendix A.3.1. The steady state (0, 0) The Jacobian matrix of system (A.1a)(A.1b), evaluated at the steady state
J(0, 0) =
with eigenvalues
b+d
0
0
β+δ
(0, 0),
is:
! (A.22)
b + d in direction of the side of Q with y = 0 and β + δ (< 0 always, see (A.5))
in direction of the side with
x = 0.
Appendix A.3.2. The steady state (1, 1) The Jacobian matrix of system (A.1a)(A.1b), evaluated at the steady state
J(1, 1) =
with eigenvalues
−α − γ
−a − c (< 0
−a − c
0
0
−α − γ
x = 1.
is:
!
always, see (A.5)) in direction of the side of
in direction of the side with
(1, 1),
(A.23)
Q
with
y = 1
and
27
Appendix A.
Appendix A.3.3. The steady state (0, 1) (0, 1),
The Jacobian matrix of system (A.1a)(A.1b), evaluated at the steady state
J(0, 1) =
b + c (> 0
with eigenvalues
−α − δ (> 0
!
b+c
0
0
−α − δ
(A.24)
always, see (A.5))in direction of the side of
always, see (A.5)) in direction of the side with
is:
Q
with
y = 1
and
x = 0.
Appendix A.3.4. The steady state (1, 0) The Jacobian matrix of system (A.1a)(A.1b), evaluated at the steady state
J(1, 0) =
−a − d
with eigenvalues side with
−a − d 0 0 β+γ
in direction of the side of
Q
with
(1, 0)
is:
! (A.25)
y = 0
and
β+γ
in direction of the
x = 1.
Appendix A.4. The steady states on the sides of Q with x = 1 and with y = 0 Appendix A.4.1. The steady state on the side of Q with x = 1 In the non robust case
α − β = 0,
points belonging to the side of
x=1
Q
if
with
(β + δ) / (δ − γ) = 1
x=1
β + γ = 0)
are steady states. In the context
holds, then all the
α − β 6= 0,
posing
in (A.7) we obtain (see (A.15)):
y = y˜ := −
The point
y˜ < 1
(i.e.
(i.e.
(1, y˜)
β+γ α−β
is a steady state of system (A.1a)(A.1b) if
(α + γ) (α − β) > 0)
y˜ > 0
(i.e.
(β + γ)(α − β) < 0)
and
hold.
The Jacobian matrix of system (A.1a)(A.1b), evaluated at a steady state
(1, y) with 0 < y < 1,
is:
J(1, yb) =
with eigenvalues
y)(α − β) (< 0
for
−a − d − (c − d)y
0
y(1 − y)(γ − δ)
y(1 − y)(α − β)
−a − d − (c − d)y
α − β < 0)
! (A.26)
in direction of the interior of the square
in direction of the side
x = 1.
It holds
Q
and
y(1 −
−a − d − (c − d)y < 0
if
28
Appendix A.
PN1 (1, y) − PN2 (1, y) > 0
(see (A.2a)), that is, if the steady state
Note that the steady state
(1, y˜)
can be attractive only if
(1, y)
5
lies above the line (A.6) .
α − β < 0.
Appendix A.4.2. The steady state on the side of Q with y = 0 In the non robust case of
Q
with
y=0
a − b = 0,
if
b+d = 0
are steady states. In the context
holds, then all the points belonging to the side
a − b 6= 0,
posing
y=0
in (A.6) we obtain (see
(A.12):
x = x b := − The point
x b 0)
x b>0
(i.e.
(b + d)(a − b) < 0)
and
hold.
The Jacobian matrix of system (A.1a)(A.1b), evaluated at a steady state
(x, 0), with 0 < x < 1,
is:
J(x, 0) = with eigenvalues for
x(1 − x)(a − b)
x(1 − x)(c − d)
0
β + δ + (γ − δ)x
(see (5b)), that is, if the steady state
If both the steady states
(b x, 0)
(x, y) exists and is a saddle point. (1, y˜)
(A.27)
β+δ+(γ−δ)x in direction of the interior of the square Q, and x(1−x)(a−b) (< 0
a − b < 0) in direction of the side y = 0.
and
!
It holds
β + δ + (γ − δ)x < 0 if PS1 (x, 0) − PS2 (x, 0) < 0
(x, 0) lies to the left of line (A.7) (line (A.8), when α−β = 0)6 .
and
(1, y˜)
exist and are attractive, then the internal steady state
In such a case, the basins of attraction of the steady states
are separated by the stable manifold of
(b x, 0)
(x, y).
5 In the context α − β = 0, one eigenvalue is always equal to zero. The other eigenvalue −a − d − (c − d)y is dierent from zero if the steady state (1, y) does not coincide with the intersection point between the line (A.6) and the side of Q with x = 1. When −a − d − (c − d)y < 0 (i.e. the point (1, y) lies below the line (A.6)), only one trajectory approaches (1, y) in the interior of Q (the point (1, y) has a stable manifold of dimension 1). When −a−d−(c−d)y > 0, no trajectory converses to it in the interior of Q ((1, y) has an unstable manifold of dimension 1). Finally, if −a − d − (c − d)y = 0 (i.e. the point (1, y) belongs to the line (A.6)), the point (1, y) has a two-dimensional centre manifold. In such a case, it can be checked that there exists a trajectory tangent to the line x = 1 at the point (1, y). 6 In the context a − b = 0, one eigenvalue is always equal to zero. The other eigenvalue β + δ + (γ − δ)x is dierent from zero if the steady state (x, 0) does not coincide with the intersection point between line (A.7) (line (A.8), when α − β = 0) and the side of Q with y = 0. When β + δ + (γ − δ)x < 0 (i.e. the point (x, 0) lies to the left of line (A.7) (line (A.8), when α − β = 0), only one trajectory approaches (x, 0) in the interior of Q (the point (x, 0) has a stable manifold of dimension 1). When β + δ + (γ − δ)x > 0, then no trajectory converges to it in the interior of Q ((x, 0) has an unstable manifold of dimension 1). Finally, if β + δ + (γ − δ)x = 0, then the point (x, 0) has a two-dimensional centre manifold. Note that the steady state (x, 0) can be attractive only if a − b < 0
29
Appendix A.
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