Документ взят из кэша поисковой машины. Адрес оригинального документа : http://io.cmc.msu.ru/translation2.pdf Дата изменения: Wed Nov 9 18:12:17 2011 Дата индексирования: Sun Feb 3 21:50:17 2013 Кодировка:

EGT and economics: I. Optimality principles and models of behaviour dynamics Alexander A. Vasin

1


Introduction Classical game theory
Game theory is widely used for description and analysis of economic players' behavior in microeconomics, public sector economics, political economics and other fields of economic theory. General ideas of classical game-theoretic analysis: Game in normal form (as a model of players interaction) -Several participants (players) - Several possible strategies for each player -Payoff functions Principle of Nash equilibrium (as a method to define agents strategies during their interaction) -Nash equilibrium is a basic concept of game theory. According to this optimal outcome of a game is one where no player has an incentive to deviate from his or her chosen strategy after considering the choices of other players. Principle of elimination of dominated strategies -Strategy is called dominated if there exists an alternative strategy which provides a greater gain no matter what strategies are chosen by the other players -Domination principle means that rational players will not use dominated strategies -Dominance elimination can be made iteratively 2


Evolutionary game theory: considered problems


1. Correspondence of real behavior of economic agents to Nash equilibrium and dominance elimination principles


Typically the search of Nash equilibrium and sets of nondominated strategies deliveres rather sophisticated math tasks. It is necessary to know all sets of strategies and payoff functions for their solution ( see models of Cournout and Bertrand economical competition) . A usual participant of such interaction has precise information only about his own strategies and payoff function and often doesn't know about mentioned decision-making principles Why should we expect that his behavior will be relevant to the principles? Justification by means of models of adaptive and imitative behavior (MAIB)








These models show that convergence to Nash equilibrium and dominance elimination proceed from general properties of evolutionary, adaptative and imitative mechanisms of behavior formation In this case complete information awarness and rationality in choosing strategies are not required. It suffices to compare the payoffs for current behavior strategy and chosen alternative. Definition of evolutionary stable strategy and its relation to Nash equilibrium.





3




2.How to determine players utility functions for particular interactions? Standard approach: use concepts of particular sciences.


In economics: 'homo economicus` concept (P.Samuelson)


In the role of a producer: profit maximisation In the role of a consumer: maximisation of the demand utility Discrepancy to real behavior: russian labour market.

Alternative: consider evolution of preferences (L. Samuelson)


Model of evolutionary mechanisms natural selection


In this model a society of interacting populations with different evolutionary mechanisms is considered Analysis of this model shows that if replication is in the set of competing mechanisms then behavior dynamics in the society is coordinated with maximization of individual fitness (Ch. Darwin). What about human populations?





3.Do individuals adjust to other interests or extensively influence payoff functions of other players? (agressive advertising, drug distribution)
4


Population game Population game is a static model of interaction in a large homogenuous group of individuals. This concept is an analog of a normal form game in classic game theory, so general noncooperative optimality principles are summarized: Nash equilibrium Dominance solution Idea of evolutionary stable strategy is added Formally a population game G is a set of parametres G = < J , f ( , ), j J , , >, where J is the set of players strategies , 0- 1 standart simplex f ( , ) is the payoff function for players that use strategy j under strategy distribution and other parametres of the model (e.g. total population size and environmental conditions). For social populations, the payoff function usually corresponds to consumer utility, income or profit. In this paragraph the function is exogenous.
j j j j J j

5


Example of population game (M.Smith) Consider pair competitive contests for one resource. Individuals of a population try to find the desired object (food, lodging or a female). Some of them get it without collision, others conflict in pairs, where one of them is the owner and another one is invader J , J - the sets of strategies in each role j , j j - profits of individuals if choses j J and - j J j ( N ) - probability of collision. It doesn't depend on startegies and is determined by the size of population N 0 - profit of individuals avoided collision Strategy of individual is a pair j ( j , j ) , where j J and j J . It is a rule of behavior choice according to the role. Function f j ( , N ) shows the average payoff of individuals with strategy j



6


Let p ( ) and p ( ) denote distributions over alternatives for the both roles and which correspond to strategy distribution. Then
p j ( )
j J







j j



,

p j ( )

j J











j j



And for strategy

i (i , i )

f

i i



1 ( , N ) (1 ( N )) 0 ( N )( 2

j J







i j



p j ( ))

Consider also the situation where players don't distinguish their roles. Then the set J J J , ( , j J ), , of strategies is the set of alternatives:
j ij ij ij ij

f i ( )


jJ

jij , f i ( , N ) (1 ( N )) 0 ( N ) f i ( )

In this case game G is equivalent to the game G J , f i ( ), i J , The set of behavior alternatives and individual payoffs may not depend on the role in the previous model too. However, behavior models for those similar situations are completely different.

7


Main static optimality principles Nash equilibrium for the populational game G is a distribution * such that any startegy which is used with positive frequence is an optimal reply to the given distribution under any parameter , i.e. (2.1) , j J ( * 0) j Arg max f i ( * , ) j
iJ

Let payoff functions in game G be expansible, i.e. f j ( , ) a( , ) f j ( ) b( , ) 0 as in the model of pair contests. Note that a part of the payoff function that depends on the strategy chosen by a player is independent of the parameter . Then (2.1) is equivalent to the following condition, which doesn't include the parameter :
j J : * 0 j Arg max f i ( * ) j
iJ

The concept of Nash equilibrium is the best-known optimality criterion used in strategic behavior modeling. However, it is known from analysis of dynamic models, that among Nash equilibrium there can exist unstable states, that are not realized in practice. For this reason we also consider stronger optimality criteria.

8


Evolutionary stable strategy (ESS) for the populational game G is a distribution * such that * ( ) (0,1) : (0, ( ))
f * ( (1 ) * , ) f ( (1 ) * , )

where f ( ' , ) j f j ( ' , ) is the average payoff for mixed strategy or distribution j J when individuals in the population are distributed in pure strategies according to '. The concept of ESS can be interpreted in the following way. Let a small group of 'mutants' with strategy distribution be implemented in a population *. If distribution * is evolutionary stable then the implemented group can't survive in the population, because its average fitness is less then the fitness of initial strategy *.

9


Any ESS is a Nash equilibrium. If is not an equilibrium then mutants with pure strategy that is a best reply to have grater payoff than the average payoff for the main population. This statement is correct if the share of one individual in the population is negligible, i.e. any change of his strategy doesn't influnce payoff ordering. Otherwise it is necessary to revise the ESS concept (Schaffer,1989).

For a symmetric function f j ( s j , sJ \ strategy change players with the

al game in normal form with n players, set of strategies S and payoff S j S such that any j ), ESS is defined as a symmetric situation by any player doesn't make his profit more than the profit of other former strategy. I.e. a mutant doesn't have any profit benefits.

Such ESS can be not a Nash equilibrium. In particular, in the game which correspond to symmetric Cournout oligopoly, players use 'market power' at the Nash equilibrium and realise lower production volumes in comparison with the competitive equilibrium, while the ESS corresponds to the competitive equilibrium.
10


Strict equilibrium of population game G is distribution * such that all players use one strategy, which is the best reply to itself:
0, j J : * 1, i j , j

f j ( * , ) f i ( * , )

Note, that any strict equilibrium is an ESS, even for groups with sufficiently large finite size. Selten (1988) showed that there are no ESS except for strict equilibria for random contests with role asymmetry of players ('owner' -- 'invader'). For general payoff functions f i ( , ) Nash equilibrium may not exist. In other classes of games there are a lot of equilibria, some of them are unstable. In this case another optimality principle is appealing -- domination that is also relative to the Darwinian principle of natural selection.

11


Strategy j dominates strategy i j i on the set of distributions ' if for any distribution over strategies ' the strategy j provides a greater gain than strategy i ( 0 : , ' f j ( , ) f i ( , ) ) Set J ' J is called a dominating set if it can be obtained by iterative exclusion of dominated strategies, i.e. there exists integer T >1 such as
J ' JT J
T 1

... J 1 J , wherek ,..., T 1, i J k \ J 1

k 1

j J

k 1

:

j i on k , j 0, j J k The described procedure for iterative elimination of dominated strategies can be considered as a quasi-dynamic model of behavior microevolution within a population. Indeed, this procedure describes a sequential reduction of the set of strategies used by players: at each stage, more efficient (better fitted) strategies are substituted for less efficient ones.
If >0 then strategy j strictly dominates strategy i ( j i ), J' is a strictly dominating set. Concepts of domination by a mixed strategy and a set domination in mixed strategies are similar
12


Search of Nash equilibrium and dominating sets of a population game is generally a sophisticated optimization problem. For random pair contests it is possible to reduce them to the known computational tasks for appropriate bimatrix games . Proposition2.1 Distribution * is a Nash equilibrium of game G such that f i ( ) jij and competitors don't differ conditions if and only if (* ,*) is Nash equilibrium in mixed strategies of symmetrical bimatrix game (ij )i , jJ , ( j,i )i , jJ i.e
j J ( * 0) j Arg max f i ( ) j
iJ

Proposition2.2 Distribution * such that s* 1 is strict equilibrium of game G if and only if for any i s 22 i 2 ,i.e. (s,s) is strict symmetric Nash equilibrium of game in pure

strategies.
Proposition2.3 Strategy s dominates strategy r ( r ) in game G if and only if s r in s game , i.e sj rj for any j J Proposition2.4 Distribution (
j j


)

contests if and only if = ( j ), ( j j ) j


( p ( ), p ( ))





is Nash equilibrium of game G for assymetric pair is Nash equilibrium in mixed strategies of game =
13




So, for any random pair contests Nash equilibrium of population game correspond to Nash equilibrium of bimatrix game that describes pair interaction Analogue relation exists for random interactions with larger number of participants when a separate local interaction is characterized by the game of n players. The results are easily generalized for the case of interpopulation collisions when individuals from different populations or social groups ('predatorprey', 'employer-employees' etc) play different roles. The main condition of the correspondence is non-correlation of distribution in interacting groups with players' strategies



14


Model of adaptive-imitative behavior (MAIB)
In which cases adeptive-imitative mechanisms form population behavior that corresponds to Nash principle and principle of dominated strategy elimination? Let population game G J , f j ( ) describe interaction of population individuals that happens continiously, at every moment of time. The number of individuals and the external factors are fixed( f j do not depend on ) With intensityrj rj ( f ( ), ) which depends on current player distribution over strategies and current payoff vector f ( ) ( f j ( ), j J ) a player with strategy j turns into 'adaptive' status where reconsiders his behavior In adaptive status player with startegy j chooses i as alternative with probability
q ji q ji ( f ( ), )

Current and alternative strategies are compared. If alternative strategy i is better then initial strategy j (i.e. gives individual the greater distribution under this distribution over strategies) then the player changes his strategy to i with probability
ji ji ( f ( ), )
15


Then rj

j

i: f i f



q ji
j

ji

is the average share of players who change their strategy j to strategy at time t,
j

from set i f i f

j



from set i f i f



to j.

i: f i f



ri i q ji ji - the share of players who change their strategy
j



So, equations of behavior dynamics look like Functions r j , ji , q
ji

j r j

j

i: f i f



q ji ji
j

i: f i f



ri i q ji ji (2.2)
j

are such that j J

rj 0; i, j J
0

ji 0, q ji 0; j J


iJ

q ji 1

Mentioned conditions guarantee that path (t, ) doesn't come out set at any moment of time t and with any initial distribution
0

16


MAIB. Examples.
Example 1. Let the intensity of status change to adaptive be constant. Alternative strategy is chosen by means of random imitation. And the probability of changing current strategy to alternative is proportional to the difference between the corresponding payoff functions. So,

rj ( f ( ), ) r , q ji ( f ( ), ) i , ji ( f ( ), ) ( f i ( ) f j ( ))
And (2.2) looks like


j r j ( f j ( )



i f i ( ), j J

J This system is an analogue of autonomus continuious model of replicator dynamics (see part 3)

Example 2. Alternative strategy is chosen with equal probabilities from the set of possible strategies, i.e.

q ji ( f ( ), ) 1 / J
This example illustrates the mechanism of individual adaptation when each player knows the whole set of possible strategies, and adaptation happens according to current payoff values. Adaptation doesn't depend on behavior of other population individuals. It is evident that there are many different MAIBs. The following theorems reveal relation MAIB stable states and solutions of the corresponding population game. Note that any Nash equilibrium of population game G is a steady state of MAIB.

17


Relation between MAIB stable states and population game solutions
Denote J ( ) Arg max f k ( ) as a set of best replies to distribution .
kJ

Theorem 2.1: Let MAIB meet the following conditions 1), 2) and 3) or 3') : 1)For any j J and any r j 0 (intensity of changing status to adaptive is positive for all strategies) 2) For any i, j J functions ji look like ( f i ( ) f j ( )) where for any x 0, ( x) 0 (probability of strategy choice as alternative is function of corresponding payoff difference and is positive funder the positive argument ) 3)For any j J , i J ( ) q ji 0 (probability of strategy choice as alternative is positive for any strategy that gives maximum payoff under the current distribution over strategies) 3')For any j J , i J ( ) q ji q i and q>0 (for any pure strategy with maximum payoff, probability of this strategy choice as alternative is not less than its share in the population multiplied by some constant) Then Any Lyapunov stable point * of system (2.2) is Nash equilibrium of population game G J , f j ( ), j J , 0 * 0 b) if initial distribution 0 and for path (t , 0 ) there exists lim (t , ) then * is Nash t equilibrium, for game G c) if * is a of strict equilibrium for population game G then * is an assymptoticaly stable point of system (2.2)

18


Theorem 2.2: Let MAIB (2.2) meet conditions 1 and 2 of theorem 2.1 and moreover 1.for any i, j J q ji i (alternative strategy is chosen by random imitation) 2. if f j f i then r j ri (intensity of changing status to adaptive decreases with the raise of payoff function) 3. ( x) increases monotonically in x (probability of strategy choice as alternative rises monotonically for payoff remainder) If J is a strictly dominating set of strategies in populational game G J , f j ( ), j J , then, lim j (t , 0 ) 0 for any j J and initial distribution 0 0 t Notes ·Other variants of such consistency conditions of MAIB dynamics with Nash and dominance decisions (see Samuelson L., Zhang J.(1992) and Weibull (1995)) relate to the concept of monotonous dynamics j j g j ( ), j J s.t. g j ( ) g i ( ) f j ( ) f i ( ), i, j J , At the same time there exist adaptation models that don't meet theorems 2.1 and 2.2. Models of evolutional mechanisms' natural selection considered in the next part explain why we nevertheless should expect coherence of real behavior dynamics with the mentioned optimality principles. Moreover, payoff functions of players are endogeneously defined in the frames of these models.

19


Replicator Dynamics Model (RDM) Population is characterized with a set S of possible strategies Strategy distribution of individuals at the current moment is set by vector ( s , s S ) Individuals differ only in behavior strategies, they don't change it during their life, strategy is inherited In case of populations with both males an females, each of them should be considered as a separate population Genetical mechanism of inheritance: the strategy is determined by genes, connected with sex. Mechanism of imitation: the strategy is defined by imitating behavior of the parent of the same sex. The result of the interaction in population during a period of time is characterized for players with strategy s by fertility function fers ( , N) that determines the average number of offsprings. It is also characterized by survival function s ( , N ) that determines the probability to survive under distribution and population size N.

20


Denote N s s N - the number of those who use strategy s . Then popultion dynamics N s (t ), s S , meets the following system: (3.1) N s (t 1) N s (t ) f s ( (t ), N (t )) Where f s ( , N ) fers ( , N ) s ( , N ) is a strategy fitness function. It formalizes the Darwinian concept of individual fitness. At first sight a concept of payoff function is inapplicable to this model. Players' strategies are fixed, the are not approaching to smt and don't choose anything. However, the picture changes if we look at strategy distribution dynamics. The following theorem shows that behavior assymptotics in such population corresponds to fitness as a payoff function for this population. Particularly, if for t strategy distribution approaches to stationary, then there are only those strategies in population that maximize fitness (corresponds to Darwinian principle of natural selection: only most fitted survive). If in any distribution one strategy fits better than another, then a part of the worst strategy in distribution (t) approaches to 0 while t .In this case fitness is at endogenuous utility function of this model.

21


Relation of Nash equilibria with stable MRD points Asymptotic stability of ESS Relation of dominating sets of strategies with behavior dynamics Theorem 3.1 (on relation of Nash equilibrium with stable points of RDM): Assume that the fitness function f s is reperesentable in additive form f s ( , ) a( , ) b( , ), a( , ) 0 Then 1) any stable (Lyapunov) distribution * of system (3.1) is a Nash equilibrium in population game G S , f s ( ), s S , 2) if for a certain path N (t ) 0 the initial distribution N (0) 0 and lim (t , N (0)) * , t then * is a Nash equilibrium of the specified population game





Note that system (3.1) isn't closed, because its right part also depends on N(t). Conception of stable distribution for such system is formally defined in Bogdanov, Vasin (2002) Theorem 3.2 (on asymptotic stability of ESS): Assume that in theorem 3.1 * is evolutionary stable strategy for population game G . Then * is an asymptotically stable distribution of system (3.1) Theorem 3.3 (on the relation between dominating sets of strategies and behavior dynamics): Assume that S is a strictly dominating set of strategies in the game G' S , ln f ( ), s . Then, for any s S and any N (0) 0 lim s (t , N (0)) 0 on the corresponding path of t system 3.1 22
s

S,


Random imitation Replicator Dynamics Model describes activity of evolutionary mechanism that provides direct inheritance of strategies by children. In what degree are the stated results depend on concrete evolutionary mechanism? Turns out that it plays a very important role. Let's consider the mechanism of random imitation as an alternative example This model differs from replicator dynamics only in one point: new individuals do not inherit strategy, they follow a strategy of randomly chosen adult. Then the population N (t ) (t ) ,sS dynamics are described N s (t 1) N s (t ) s (t ) N r (t ) ferr (t ) s s
r


r

N r (t ) r (t )

Such system dynamic corresponds to payoff function s (t ) in the sense of theorems 3.1-3.3. Thus, viability turns out to be an endogenous payoff function of individuals in the corresponding dynamical process. Proceeding from the previous example it seems that we have exchanged arbitrariness in the choice of payoff functions for arbitrariness in the choice of evolutionary mechanisms. However, actual evolutionary mechanisms are subject to natural selection. Only the most efficient mechanisms survive in the process of competition.
23


Model of evolutionary mechanisms competition Consider the corresponding model of a society that includes several populations that differ only in their evolutionary mechanisms. Individuals of all populations interact and do not distinguish population characters in this process. Thus, the evolutionary mechanism of an individual is an unobservable characteristic. Fertility and viability functions describe the outcome of the interaction for each strategy and depend on the total distribution over strategies and the size of the society. The set of strategies S and the functions are the same for all populations. Denote L ­ the set of populations N - the size of population l l N ­ the total population size l l , s -S distribution over strategies in population l. s Then the total distribution over strategies is N N Assume that operator l corresponds to the evolutionary mechanism of population l and determines the dynamics of distribution l . (For example, in one population it is replication dynamics, in an other - random 24 imitation, and so on. Particularly, dynamics may relate to maximization of some payoff function)
l l l


Then the dynamics of the society are governed by equations

N l (t 1) N l (t )


s

sl (t ) f s ( (t ), N (t ))

(3.2)

l (t 1) l ( k (t ), N k (t ), k L), l L
Theorem 3.4. Let there exists a population of replicators in the society. Then the total distribution (t) over strategies meets the following analogs of theorems 3.1 and 3.2: 1) any stable distribution of system (3.2) is a Nash equilibrium of the population game G S , f s ( ), s S 2) if for path N (t ) initial distribution of the population game G for path 3) if is a strict equilibrium of the game G then is an asymptotically stable distribution for system (3.2).



N (0) 0 and

lim ( N (0), t )
t

*

then * is a Nash equilibrium

Thus, the evolutionary mechanisms selection model confirms that individual fitness is an endogenous utility function for self-reproducing populations.

25




The idea of the proof for propositions 1 and 2 is quite easy: if stationary distribution over strategies is not Nash equilibrium of fitness function then nothing can prevent expansion of replicators that use the best reply strategy. That is a contradiction to its stability Generalization of theorem 3 for elimination of dominated strategies is possible under more strict assumptions on the variety of evolutionary mechanisms. For l any evolutionary mechanism and a pair of strategies s,r, let us call an s,rl substitute of mechanism a mechanism ls , r such that for strategies other than s and r the shares of individuals who apply these strategies change as under l mechanism , except that instead of strategy s they always play r. According to (Vasin, 1995), if for any s,r,l the set of mechanisms includes all possible substitutes of ls , r then s (t ) 0 as t 0 for any strictly dominated strategy , s. The results of this section are formulated for a homogenuous population, taking sex or age in consideration. It is easily generalized for populations such structures. Fitness analogue in this case is a rate of balanced growth population, it is determined by the Frobenius number of the Leslie matrix Semevskiy, Semenov, 1982)
26





without with of the (see


Conclusion Models and results of evolutionary game theory show that behavior evolution in a selfreproducing population corresponds to well-known optimality principles -- Nash equilibrium and elimination of dominated strategies Endogenously formed payoff function corresponds to Darwinian concept of individual fitness Problems However in biological and social populations cooperative and altruistic behavior are wellknown. It seems that they don't correspond to optimization of individual fitness Problem of stability of mixed equilibrium, where more than one pure strategy is used with positive probability. This problem appears in case of interpopulation interaction where payoff for one population depends on distribution over strategies in another population. It also appears in case of random contests with role asymmetry between participants. For such games mixed Nash equilibria are never evolutionary stable and strict equilibria may not exist. So, sufficient conditions of stability don't work Relevance of mentioned evolutionary models to social populations. Superindividual is a selfreproducing structure that uses human population as a source for its reproduction. It can influence behavior dynamics in this population

27


II. Stability of equilibrium. Pecularities of social behavior evolution.

28


Stability problem for mixed equilibrium Consider game of two populations with sets of strategies R R1 ,..., Rm and S S1 ,..., S n and payoff functions Ai (q ), i 1,..., m and B j ( p), j 1,..., n that show interaction result for all strategies Assume that individuals of the 1st population interact only with individuals of secon population and vice versa: individuals of the 2nd population interact only with individuals of the 1st . At any moment of time t each individual uses a chosen strategy Assume that m m p (t ) ( p1 (t ),.., pm (t )) p R pi 1 i n n q (t ) ( q1 (t ),.., qn (t )) q R q j 1 j are population distributions over strategies. Point ( p, q) m n is Nash equilibrium of game if for any i, j

( pi 0) i Arg max Bu ( q), (q j 0) j Arg max Bu ( p ) u u pi (t ) 1, q j (t ) 1 Equilibrium is called mixed if for any i, j
29


It is easy to see that For any game with indescrete payoff function there exist a Nash equilibrium In nonsingular case amount of positive coordinates p and q is the same p(t) and q(t) change according to the system


p i c(t , p(0), q(0))Gi ( p, A(q )), i 1,..., m


This system is called - if 1) Functions Giand H j are such as
Gi ( p, A) 0, i 1,..., m H j (q, B) 0, j 1,..., n

q j d (t , p (0), q (0)) H j (q, B ( p )), j 1,..., n

m n for any distributions p , q and payoff vectors A ( A1 ,..., Am ), B ( B1 ,..., Bn ) so i( pi 0) i Arg max Au ; j (q j 0) j Arg max A u That means that any Nash equilibrium is a stable point of system (4.1) 2) Functions c and d are measurable functions over t and continiously differentiable over p(0) and q(0). Derivatives are equibounded over t. m n 3) Set is invariant of system (4.1). Vector-functions A, B, G and H are continiously differentiable. Note that MAIB, and a system with positive functions ..generally correspond to these assumptions

30


Note that system (4.1) can be converted to autonomous system


p i Gi ( p, A(q )), i 1,..., m


(4.2)

if for any t, p(0), q(0)

q j H j (q, B( q)), j 1,..., n

c( p(0), q(0), t ) ( p(0), q(0)) d ( p(0), q(0), t )
This occasion takes place in interaction between populations with fixed sizes or between individuals with different roles in one population, for example, between 'owners' and 'invaders' (Maynard Smith, 1982) Consider game , system (4.1) and corresponding autonomous system (4.2) Stable point of system (4.2) is called singular if there is some eigenvalue of the Jacobian matrix that is equal to 0 Point (p,a) is called a centre if for every eigenvalue Re=0, Im0 Point (p,a) is called a saddle if there is an eigenvalue such that Re>0

31


Theorem 4.1: Any mixed equilibrium (p*, q*) is either a singular point, or a centre point, or a saddle point of (4.2) in the latter case (p*, q*) is an unstable point of the system (4.1) for any functions c, d. This theorem doesn't solve the question of stability for 'centre' points where all eigenvalues of the linearized matrix are purely imaginary. Let's use a method developed by Ritzberger and Vogelsberger (1990) and based on Liouville theorem. Consider system (t ) ( (t )) According to this theorem, the field that is free of divergence (div ( ) 0 ) keeps any volume constant and has noasymptotically stable points.


32


Let's describe MAIB class such that the given method permits to prove the abscence of asymptotically stable equilibria. General MAIB equations for an interpopulation game are:


j rj

j

i: f i f



q ji ji
j

i: f i f



ri i q ji ji , j J
j

Theorem 4.