A two population growing model: exogamic or endogamic Текст научной статьи по специальности «Биологические науки»

A Two Population Growing Model: Exogamic or Endogamic

Elvio Accinelli1, Juan Gabriel Brida2, Edgar J.S. Carrera2

1 Universidad Autonoma de San Luis, Mexico e-mail address: [email protected] 2 University of Siena, Dept. of Economics,

Piazza S. Francesco, 7, 53100 Siena (Italy),

Tel.: (+39)0577232780; fax (+39)0577232661 e-mail address: [email protected]

Abstract. We show an analytic model for a situation in which two populations are confronted in an exogamic or endogamic way. Our approach is based on Evolutionary Game Theory for Non-Symmetric Games but considering a new rule of imitation: evolutive regret when the probability of selecting the best strategy is included. The rule states to choose the actions with the best results, with a probability proportional to the expected gains. In particular, we show the relation between Dynamic Strategy and Nash equilibrium in an asymmetric game of imitation strategies.

JEL classification: C70, C72, C69.

Keywords: Imitation, replicator dynamic, stable population, stability and Nash

equilibrium.

Introduction

In Evolutionary Game Theory (EGT) we deal with populations of players that are programmed for a certain strategy. Players replicate and pass on their strategy to their offsprings. The number of offsprings is directly related to the average payoff of the parent strategy. EGT can be thinking as a theory that combines the dynamics of animal behavior with population (It is a predatorprey relationship, see [Schlag, 1998]). This article focuses on the relation between dynamical equilibria and Nash equilibria by appealing to an evolutionary process (as defined by Samuelson, and J. Zhang [Samuelson, 1992]) from two populations in the long run, with evolutionary imitation strategies, to give conditions under which dynamic selection processes will

yield stability requirement ensures that outcomes will be Nash equilibria. Since, imitation dynamics is more appropriate here than the replicator dynamics that is used in applications of EGT to theoretical biology. According to the imitation dynamics, players are not mortal and have no offsprings. However, every so often, a player is offered the opportunity to pick out some other player and to change his own strategy against the strategy of the other player. The probability that a certain strategy is adopted for imitation is positively correlated to the gain in average utility that is to be expected by this strategy change. So here as well as in the standard model, successful strategies will tend to spread while unsuccessful strategies die out. Moreover, exactly the same strategies are evolutionary stable under the replicator dynamics and under the imitation dynamics (see [Basar, 1982]).

An imitation game is characterized when the players show imitative behavior (see [Hines, 1980]): never imitate an individual that is performing worse than oneself, and imitate individuals doing better with a probability proportional to how much better they perform. It is shown that this behavioral rule results in an adjustment process that can be approximated by the replicator dynamics. The model that we present has a significant difference with the basic models; the asymmetric context with different populations appearing, the exogamic or endogamic situation. Hence, we delete the assumption of a single symmetric population, since interaction takes place between individuals from distinct populations. In economics, we may think of contests between the owner of a territory and an intruder, that is, the problems of migration; the analogy is a tourism economic activity in which resident attitudes toward and perceptions of tourism development efforts. Moreover, our model permits to obtain an evolutionary dynamics coinciding with the Replicator Dynamics. Hence, the replicator dynamics that we show is adequate for modeling social interactions in a population dynamics.

The paper is organized as follows. Experimentation is governed by one initial law of motion, this is explained in section 2. In section 3 we discuss our game where the friendship strategy is implemented by imitating your neighbor where players are nearby located. Subsection 3.1 discuss the dynamic and Nash equilibrium. In section 4 both populations choose their bids on the basis of their friendspast successful experiences in the same population: a successful imitation strategy.

1. A situation description

Let us assume that people at a given time on a given territory can be distinguished into two populations, residents R and migrants M. Each population can be in its turn split into two sets, or clubs, depending upon the strategy being played against the opponent. Suppose that these strategies are: to admit or not the marriage with a member of the other population. We identify these strategies respectively as m and nm. Let xT G R+, where the vector xT = (x1m,x1mn), t G {R,M} is normalized so that x'm + x1mn = 1 and each entry is the share of individuals in the respective club over the total population. Each period an individual of a given population t playing the i-th strategy i G {m, mn} faces the decision whether remaining unchanged or

change it with probability rT. Let pj j be the probability of changing from the i-th to the j-th strategy, hence of moving across clubs. Clearly, rjpj is the probability of changing from the i-th to j-th club. In the sequel, em = (1, 0) and enm = (0,1) will indicate vectors of pure strategies, m or nm independently from the population t.

Now let us assume that the probability to change the strategy is exclusively a function of the proportion of members belonging to each club.

For example, migrants who are considering whether to change or not their strategy, will take into account only the behaviours of those other fellows they are meeting and will totally disregard the residents decisions. Let it be the same for the latter population. Thus, the expected change in the share of players type i in the population t they belong to, will be given on average by the probability of a j-club member moving into the i-th club multiplied by the existing share of members in their initial club less than the probability of members of the i-th to move over to the j-th club weighted in the same way. For any given population t this hypothesis yields the following equation of motion:

¿I = rjpTjx — ri pTjxi, vi,j G {m,nm},j =i,T e {R,M}. (1)

Since (1) is a representation of the interaction of two clubs, or species, in a two-patch environment assuming that individuals behave adaptatively, hence, they maximize Darwanian fitness.

2. Imitating your neighbor

We introduce hereafter the first evolutionary exercise. This is called imitation: between choices each individual may observe the performance of one other individual (see [Hines, 1980], [Samuelson, 1992]). We make an individuals decision as to whether stick to a strategy/club or change over a function of the type of individuals in their own population they encounters. We consider the following assumptions:

(i) let us assume that the decision of an individual depends upon the utility deriving from it uT(ei,x-T), where (—t G {R, M}, —t = t) and the share of individuals in the same club:

ri = fi (U (ei,x-T)).

The function fj(uT(ei,x-T)) is reasonably interpreted as the propensity of a member of the i-th club to consider switching membership as a function of the expected utility gains from such a choice.

(ii) next, let us assume that once opted for a change, he/she will adopt the strategy chosen by the first population fellow to be encountered, her neighbor, i.e. for any t G {R, M}

p(i ^ j \ he/she consider to change strategy) = p\j = xj, i,j G m, nm.

With the above considerations and by (1), we have:

x = xTfT(uT(ej,x-T))xT - xTfT(uT(ei,x-T))xT, (2)

xi = (1 - xT)xT[fj (uT(ej,x T)) - fi (uT(ei,x T))]. (3)

This is the general form of the dynamical system representing the evolution of a two-population four clubs structure. It is a system of four simultaneous equations with four state variables (a state variable being the share of the club members over their respective population). However, given the normalization rule xm + xmn = 1 for each t e {R, M}, equation (3) can be reduced to two equations with two independent state variables. Taking advantage of this property, from now onwards we choose variables xR and xM with their respective equations.

For a first grasp of the problem, let us assume fj to be population specific, but the same across all its components independently from club membership, and furthermore to be linear in the utility levels. Thus, it makes sense to think that the propensity to switch behaviour will be decreasing in the level of the utility:

fT(uT(ei, x-T)) = aT - (3tuT(ei, x-T) e [0, 1] with aT,3t > 0. To get a full linear form we assume:

uT(ei, x-T) = eiATx-T, i e {m, nm};

in other words, utility is a linear function of both variables, through a population-specific matrix of weights or constant coefficients, AT e M2x2, (t e {R,M}). This latter assumption implies that utility levels reflect population specific (and therefore in principle different) properties, i.e. broadly speaking preference structures over their outcomes. This reduces the previous model to a much simplified version:

or in full

xi, = 3t xm(1 - xm)[(1, -1)at xcT ],t e {r,m }, (4)

I xm = 3rxR(1 - xm)(aRxM + bR));

\xRm = 3m xM (1 - xM )(aM xm + bM))., ( )

whose coefficients aM and bR depend of course upon the entries of the two population-specific matrices AM and AR, respectively.

2.1. Dynamic stability and Nash properties

System (5) admits five stationary states or dynamical equilibria, i.e.

(0,0), (0,1), (1,0), (1,1) and a positive interior equilibrium (xm,xM), where

xR = xM = - —

m R m t '

aR aT

or

In fact, the interesting case is when P = (xm, xM) is an equilibrium lying in the interior of the square C = [0,1] x [0,1], this happening when

bR bM

0 <------n < 1 and 0 < ——tf < 1,

aR bM

while of course the other four equilibria are the corners of the square itself. We can proceed to inquire about the stability of the five equilibria. These equilibria can be interpreted as follows:

• Clearly the trivial equilibrium is one where none of the residents is inclined to admit migrants, nor any migrant is willing to mix up with residents.

• On the other hand, there is another equilibrium at the opposite corner (where the sharing clubs involve all of their respective population): this is the case where reciprocal integration of the two populations is complete. The two remaining border equilibria show a different club dominating the two populations and in a sense a mismatch between strategies.

• Finally, of course we have the possibly interior equilibrium, this situation implies that certain percentage of each population has a good disposition to the other population, and the rest only accept marriage between persons of his own population.

Stability analysis looks at the properties of the Jacobian of the system (5):

3R(1 - 2xM )(aT xm + bM ) 3RaT xTm (1 - xM )

j (xm ,xM)

3taRxm(1 - xm) 3t(1 - 2xm)(mRxC + bR)

(6)

whose value is of course dependent on the population specific matrices, among other things.

• If bR < 0 and bM < 0, the interior equilibrium P is the unique attractor in C.

• If bR > 0 and bT > 0, on the contrary, P is the unique repulsor, all other four being local attractors in C. The latter is partitioned in four regions, the respective basins of attractions of the corner attractors.

• Otherwise, P is a saddle point, with the outset going through C from southwest to north-east, or the contrary. Accordingly the corner equilibria may be attractors or repulsors: in the former case (0, 0) and (1, 1) will turn up to be repulsors and (1, 0) and (0, 1) - attractors; the opposite in the latter case.

The dynamic described above has a ready interpretation in terms of game theory, as matrices AM and AR are the pay-off matrices of a bi-matrix game with two players, the population of migrants M and the population R of residents.

The analysis of the relation between dynamical equilibria and Nash equilibria provides useful additional information. The following properties can be easily derived (see, e.g. [Schlag, 1998]).

• Stability is a sufficient condition for a dynamic equilibrium to be a Nash equilibrium.

• Being an interior point is sufficient for an equilibrium to be a Nash equilibrium.

Therefore, for bR < 0 and bM < 0, at the interior equilibrium P agents have no incentives to switch away from the chosen strategy, moreover if they happen to be shocked away from it, they will return. Thus, the relative sizes of m and nm communities within either population will be evolutionary stable. On the contrary, i.e. when bR > 0 and bM > 0, the repulsor P is such that any slight shock or move away from it, will initialize a path systematically diverging towards one of the corner equilibria where we have homogeneous, or single community, populations. These combined properties lend themselves to interpret some of the population development scenarios we know of. First of all, it shows how fragile can be the cooperative equilibrium associated with the population interaction.

3. Picking up the most successful strategy

Now a migrant or a resident considers changing strategy imitating the most successful strategy played by members of the same population. The rule states to choose the actions with the best results, with a probability proportional to the expected gains. In other words, a migrant (or a resident, respectively) will change over to a different strategy played by another member of the population if, and only if, the latter brings a greater benefit in terms of expected utility.

Let us assume an individual of population t e {R, M} in the community j e {m, nm} encountering somebody of the alternative community, i = j, and that the former will change over to the latter membership/strategy if uT(ei,x-T) > uT (ej , x-T).

Note that the utility of each individual depends on his/her own strategy and on the characteristics of the individuals of the other population. Further, we assume that there is some uncertainty in the estimated return of the alternative strategies, so that must estimate the value uT(ei,x-T). Thus, the probability of a j-individual to change strategy is given by the probability of encountering a i-individual multiplied

by the probability that the estimate of the return:

D = uT (ei, x-T) - uT(ej, x-T)

is positive. By such way, the probability pTj of changing from the j-th to the i-th community/strategy will be xi times PT (jD > 0), the probability of the observing3 D = uT(ei, x-T) - uT(ej, x-T) > 0. Finally, pjj = xjPT(5 > 0).

Let us assume that PT (5 > 0) depends upon the true value of the difference

uT(ei, x-T) - uT(ej, x-T) which is unknown to the i-th individual. That is to say

PT (5 > 0) = 4>T (uT (ej , x-T) - uT (ei, x-T)).

3 I.e. the probability of the event where the estimator of the utility associate to the i-th community/strategy, given the characteristics of the other population, be greater than her own strategy given the characteristics of the other population.

Therefore, the probability of a j-th individual in the population t to observe a positive value of D increases with the true value of the difference uT(ej,x-T) - uT(ei,x-T). For the sake of simplicity (by considering an existence of an expected utility), let uT(ei,x-T) be linear, i.e. uT(ei,x-T) = eiATxT.

Thus, the probability of a j-th individual to change over to the i-th strategy is

PTj(uT(ei - ej,x)) = V(u(ei - ej,x-T))xT, (7)

the change in the share of the i-players will be given by the probability of a j-

player to become an i-player weighted by the relative number of j-players in its population, minus the probability of an i to become a j -player likewise weighted:

xT -- [x TPT _ PT x ] xT

xj \-xj pji pij xi -lxi .

In our case the equation becomes

xj = xTjxT[4’T(uT(ej - ei,x-T)) - 4>T(uT(ei - ej,x-T))], (8)

and its first order approximation is

xT = xT 4>T (0,xT )[uT (ej - ei,xT) - uT (ej - ei,x-T)] =

= 24>T (0,xT)uT(ei - xT,e-T)xT. (9)

Then, in a neighborhood of an interior stationary point the dynamics is approximately represented by a replicator dynamics multiplied by a constant. Stability analysis of the local type can, therefore, be carried out using the linear part of the nonlinear system4.

In the special case where 4>T is linear: 4>T = ^T + pTuT(ej - ei, xT) with AT and /jt:

0 < AT + /jTu(xT, x-T) < 1, x e {z e R+ : max zi < 1; i = 1, 2}

we get the equation xT = 2^TuT(ei - xT,x-T)xT. In this case stability analysis is similar to the one of the model of simple imitation.

4. Conclusion

Although the model is different from the one of simple imitation, as long as the linear approximation is mathematically valid, it permits to obtain an evolutionary dynamics of the same type. Similar conclusions apply to the issue of the relation between stable and Nash equilibria. For further characterization results on the set of Evolutionary Stable Strategies we refer the reader to [Cover, 1991], [Schlag, 1998]. The real justification for this extension of the simpler model lies in the description of an observed behaviour. Here people choose for an expected maximization of benefits, since they do what the others do.

4 If the equilibrium is non hyperbolic.

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