- 1.EGT and economics: I.Optimality principles and models of
behaviour dynamics Alexander A. Vasin
2.
- Game theory is widely used for description and analysis of
economic players' behavior in microeconomics, public sector
economics, political economics and otherfieldsof economic
theory.
- General ideas of classical game-theoretic analysis:
- Game in normal form(as a model of players interaction)
- - Severalparticipants (players)
- -Several possible strategies for each player
- Principle of Nash equilibrium(as a method to define agents
strategies during their interaction)
- -Nash equilibrium isa basicconcept of game theory . T he
optimal outcome of a game isone whereno player has an incentive to
deviate fromhis or herchosen strategy afterconsidering thechoice s
of other players .
- Principle of elimination of dominated strategies
- -Strategy is called dominated if there existsanalternative
strategy which provides agreater gain nomatter what strategies are
chosen by other players
- -Domination principle means that rational players will not use
dominated strategies
- -Dominance elimination can be made iterative
3. Evolution arygame theory: considered problems
- 1. Correspondenceof real behavior of economic agentstoNash
equilibrium and domina nceelimination principles
-
- In typicalconflicts thesearch of Nash equilibrium and sets of
nondominated strategies deliveressophisticatedmath tasks. It is
necessary to know all sets of strategies and payoff functions for
their solution (seemodels of Cournout and Bertrand economical
competition) .
-
- A u sual 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
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- Whyshould weexpect that his behavior will be relevantto the
principles ?
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- Justification by meansof models of adaptive and imitative
behavior (MAIB)
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-
- These models show that convergence to Nash equilibrium and
domina nceeliminationproceedfrom generalpropertiesof evolution ary
, adaptative and imitative mechanisms of behavior formation
-
-
- In this casecompleteinformation awarness and rationality in
choosing strategies are not required. Itsufficestocompare thepayoff
sforcurrentbehavior strategy and chosen alternative.
-
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- Definition of evolutionary stable strategy and itsrelation
toNash equilibrium .
4.
- 2. How to de termineplayers utility functions
forparticularinteraction s ?Standard approach: use concepts of
particular sciences.
-
- In economics : 'homo economicus concept(P.Samuelson)
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- In the role ofaproduce r:profit maximisation
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- In the role ofaconsumer :maximisation ofthedemandutility
-
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- Discrepancy to real behavior: russian labour market .
-
-
- Alternative: consider evolution of preferences (L.
Samuelson)
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- Model of evolution arymechanisms natural selection
-
-
- In this modela societyof interacting populations with
differentevolutionarymechanisms is considered
-
-
- Analysis of this model shows that if replication is intheset of
competingmechanismsthen behavior dynamics is coordinated
withmaximization ofindividualfitness (Ch. Darwin). What about human
populations?
- 3. Do individualsadjust to other interestsor extensively
influencepayoff function sof otherplayer s? (agressive advertising,
drug distribution)
5.
- Population gameis a static model of interaction ina
largehomogenuous group of individuals.
- Thisconceptis an analog of a game in normal form in classic
game theory, so general noncooperative optimality principles are
summarized:
- Idea ofevolutionary stable strategyis added
- Formally apopulation gameG isa setof parametres
- Jis theset of players strategies
- i s the payoff function for players that use strategyjunder
strategydistribution and other parametres of the model(e.g.t otal
population size and environmental conditions). For social
populations, the payoff function usually corresponds to consumer
utility, income or profit. In this paragraphthefunction is
exogenous.
6.
- Example of population game (M.Smith)
- C onsider pair competitivecontes ts for one resource.
Individuals ofapopulation 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 theowner and another one
isinvader
- -theset sof strategies in each role
- - profits of individuals if choses and-
- -p robability of collision.Itdoesn't depend on startegies
andisde termin ed bythe sizeof population N
- - profit of individuals avoided collision
- Strategy of individual is a pair, whereand. Itis a rule of
behavior choice according to the role .
- Functionshows the averagepayoffof individuals with
strategyj
7. Letanddenote distribution s over alternatives for the
bothrolesandwhich correspond to strategy distribution. Then ,And
for strategy C onsideralsothe situation whenplayers
don'tdistinguish their roles . Then the set of strategiesisthe set
of alternatives:In this case game G is equivalent to the game T he
set of behavioralternativesand individualpayoffs maynot depend
ontheroleintheprevious model too. However ,behavior models for
those similar situations are completely different. 8. Main
staticoptimality principlesNash equilibrium for the populational
game Gis a distribution *such that any startegy which is used with
positive frequence is an optimalreply to thegiven
distributionunderany parameter , i.e. (2.1) Let payoff functions in
gameGbe expansible, i.e.as inthemodelof pair contests .Notethat a
part ofthepayoff function that depends onthestrategy chosen by a
player is independent of the parameter . Then (2.1) is equivalent
to the following condition, which doesn't include the parameter :
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, thatare notrealized in practice. For
this reason we alsoconsiderstrong eroptimality criteria. 9.
Evolutionary stable strategy (ESS)for the populational game Gis a
distribution * such thatistheaveragepayofffor mixed strategy or
distribution when individuals inthepopulation 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 stable population *
. If distribution *is evolutionary stable then the implemented
group can't fix inthepopulation, because its average fitness is
less thenthefitness of initial strategy *. 10. Any ES Sis Nash
equilibri um . Ifis notanequilibrium then mutants with pure
strategy of bestreply to havegrater
payoffthantheaveragepayoffinthemain population. Due to persistance
offo ver it is correct for any group of such mutants. Thisstatement
is correct if theshareofoneindividual inthepopulation is negligible
, i.e. change of his strategy doesn't influnce payoff s order
.Otherwise it is necessary torevisetheESS concept(Schaffer,1989).
Forasymmetric game in normal form withnplayers, set of
strategiesSand payoff function, ESS is defined as a symmetric
situation suchthatany strategy changebyany player do es n't make
his profitmorethantheprofit of other players withtheformer
strategy.I.e.amutant doesn't have any profit benefits.S uchESS can
be not a Nash equilibrium . In particular, in the game which
correspond to symmetric Cournout oligopoly, players use 'market
power' in Nash equilibrium andrealiselowerproduction volumesin
comparison withthecompetitive equilibrium, whiletheESS corresponds
tothecompetitive equilibrium. 11. Strict equilibriumof population
gameGis distribution * such thatall players use one str a tegy,
which is the bestreply to itself : Note, thatany strict equilibrium
is ESS , even for groups withsufficiently large finite size.
Selten(1988) showed that there are no ESS exceptforstrict equilibri
a forrandom contest s with rolea symmetry of players ('owner'
'invader'). For general payoff functionsNash equilibriummay not
exist . In other classesof gamesthere are a lot of equilibri a ,
some of them are unstable. In this case another optimality
principle is appealing dominationthat isalsointernally bound to the
Darwinian principle of natural selection. 12. Strategyjdominates
strategyi on the set of distributionsif for any distributionover
strategiesthe strategyjprovides a greater gainthan strategyiThe set
is called adominating setif it can be obtained by
iterativeexclusionof dominated strategies , i.e. there exists
integerT >1such as 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 >0thenstrategyjstrictly dominatesstrategyi (
),J'is a strictly dominating set . Concepts of domination by mixed
strategy and a set domination in mixed strategies are analogue 13.
Search of Nash equilibrium and dominating sets ofapopulation game
is generally asophisticated optimization problem .For random pair
contests i t is possible toreducethem to the known computational
tasksforappropriatebimatrix games. Proposition 2.1Distribution *is
a Nash equilibrium of gamesuch thatand competitors don't differ
conditions if and only if( * , *) isNash equilibrium in mixed
strategies of symmetrical bimatrix game, i.e Proposition
2.2Distribution *such that is strict equilibrium of gameif and only
if for any ,i.e.(s,s)is strict symmetric Nash equilibrium of
gameinpurestrategies. Proposition 2.3Str a tegysdominates
strategyr()in gameif and only ifin game , i.efor anyProposition
2.4Distributionis Nas hequilibrium of game G for assymetric
paircontestsif and only ifis Nash equilibrium in mixed strategies
of game = = 14.
- So,for any random pair collisions Nash equilibrium of
population game correspond to Na shequilibrium of bimatrix gamethat
describes pair interaction
- Analoguerela tion exists for random collisions withlarger
numberof participants whenaseparate local interaction is
characterizedbythe game of n participants.Theresults are
easilygeneral ized in case of interpopulation collisions when
individuals fromdifferentpopulations or social groups
('predator-prey', 'employer-employees' etc) play different roles
.
- The main condition of correspondence isindependenceof
distributionin interacting groups from players' strategies
15. Model of adaptive- imit ative behavior (MA I B)
- In which cases adeptive- imit ative mechanisms form population
behavior that corresponds to Nash principle and principle
ofdominatedstrategy elimination?
- Let population gamedescribe interaction of population
individuals that happens continiously, at every moment of
time.
- The numberof individuals and external factors arefixed ( do not
depend on )
- With intensity which depends on current player
distributionoverst ra tegies and current payoff vectora player with
strategyjturns into 'adaptive' status whererec onsiders his
behavior
- In adaptive status player with startegyjchoosesias alternative
with probabilit y
- Current and alternative strategies are compared. If chosen for
comparison strategyiis better then initial strategyj(i.e. gives
individual more profitunderthisdistribution over strategies)then
the player changes his strategy toiwith probability
16. Thenis an averageshareof players who change their
strategyjto strategy from setin a moment of time,-the shareof
players who change their strategy from settoj . So, equations of
distribution dynamics look like(2.2) Functionsare such asMentioned
conditions guarantee that pathdoesn't come out set at any moment of
timetand with any initial distribution 17. MA I B. Examples.
Example 1.Let the intensity of status change to adaptive be
constant. Alternative strategy is chosen bymeansof random
imitation. And probability of changing current strategy to
alternative isproportional to the difference between
thecorresponding payoff functions. So,And (2.2) looks like This
system is an analogue of autonomus continuous model of replicator
dynamics (see part 3) Example 2.Alternative strategy is chosen with
equal probabilities fromtheset of possible strategies, i.e.This
example illustrates the mechanism of individual adaptation when
each player knowsthe wholeset of possible strategies, and
adaptation happens according to current payoffvalues . Adaptation
doesn't depend on behavior of other population individuals. It is
evident that there are many different MA I Bs.The f ollowing
theoremsrelateMA I B stablestate s and solution softhecorresponding
population game . Note thatany Nash equilibrium of population game
G isa steady stateof MA I B. 18.
- Relation of MA I B stablestatesand population game
solutions
- Denoteas a set ofbest replies todistribution.
- Theorem 2.1:Let MA I Bmeet the followingconditions 1), 2) and
3) or 3') :
- (intensity of changing status to adaptive is positive for all
strategies)
- 2) For anyfunctionslook likewhere for any
- (probability of strategy choice as alternative is function of
corresponding payoffdifferenceandispositivefor apositive argument
)
- (probability of strategy choice as alternative is positive for
any strategy that gives maximum payoffunder thecurrent
distributionoverstrategies)
- (for any pure strategy with maximum payoff, probability of this
strategy choice as alternative is not less thanits
shareinthepopulation multipliedbysome constant)
- any stable(Lyapunov)point *of system (2.2) is Nash equilibrium
of population game
- b) if initial distributionand for paththere existsthen *is Nas
hequilibriu m ,f or gameG
- c) if *is a point of strict equilibrium for population
gameGthen *is an assymptoticaly stablepointof system(2.2)
19.
- Theorem 2.2: Let MA I B (2.2)meetconditions 1 and 2 of
theorem2.1and moreover
- (alternative strategy ischosen byrandom imitation)
- (intensity of changing status to adaptivedecreaseswith the
raise of pa y off function)
- 3. increase smonotonicallyin x
- (probability of strategy choice as alternative rises
monotonically forpayoff remainder)
- Ifisastrictly dominating set of strategies in populational
gamethen ,for anyand initialdistribution
- Other variants of such consistency conditions of MA I B
dynamics with Nash and domination decisions (see Samuelson L.,
Zhang J.(1992) and Weibull (1995))relate to theconcept of
monotonous dynamicss.t.
- At the same time thereexistadaptation models that
don'tmeettheorem s2.1 and 2.2. Models of natural selection of
evolutional mechanisms considered in the next part explain whywe
nevertheless should expectcoherence of real behavior dynamics
withthenoted optimality principles. Moreover,payofffunctions of
players are endogeneously defined in the frames of these
models.
20.
- Replicator Dynamics Model (RDM)
- Population is characterized with a setSof possible
strategies
- Strategy distribution of individuals at the current moment is
set by vector
- Individuals differ only inbehavior strateg ies , 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 asaseparate population
- Genetical mechanism of inheritance : thestrategy isdeterminedby
genes, connected withsex . Mechanism of imitation : thestrategy is
defined by imitating behavior oftheparent ofthe samesex.
- The r esult oftheinteraction in population during a period of
time is characterized for players with strategys byfertility
functionthatdetermines the average number ofoffsprings. It is also
characterizedbysurvival functionthatdetermines the probability
tosurviv e underdistributionand populationsize N .
21. Denote- the numberof those who use strategys. Then popultion
dynamics, meetsthe following system: (3.1) Where is astrategy
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,thepicture changesif welook at strategy distribution
dynamics. The following theorem shows thatbehavior assymptotics in
such population correspond sto fitness asapayoff functionfor this
population .Particularly, if forstrategy 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 oftheworst strategy
in distribution (t)approaches to 0 while. In this case fitness is
endogenuous objective function of this model. 22. Relation of Nash
equilibr ia withstable MRD points Asymptotic stability of ES S
Relation of dominating sets of strateg ieswith behavior dynamics
Theorem 3.1 (on relation of Nash equilibriumwithstable points of
RDM):Assume that the fitness functionis reperesentable in additive
formThen1) any stable (Lyapunov) distribution * of system (3.1) is
a Nash equilibrium in population game2) if for a certain path, the
initial di s tributionandthen *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 strategyfor population game.
Then *is an asymptotically stable distribution of system (3.1)
Theorem 3.3 (on the relation between dominating sets of strategies
and behaviordynamics ):Assume thatSis a strictly dominating set of
strategies in the game.Then, for anyand anyon the corresponding
path of system 3.1 23. Random imitation Replicator Dynamics
Modeldescribesactivity of evolution arymechanism that provides
direct inheritance of strategies by children. In what degree are
thestatedresults depend onconcrete evolution arymechanism ?Turns
out thatit plays a very important role . Let's considermechanism of
random imitationasanalternative exampleThis 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 dynamics are describedSuch system dynamic
corresponds topayrollfunctionin 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. 24.
- Model of evolutionary mechanism scompetition
- Consider the correspondingmodel of a society that includes
several populationsthat 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 strategiesSand the functions are the same for all
populations.
- - the size of populationl
- N the total population size
- - distribution over strategies in populationl.
- Then the total distribution over strategies is
- Assume that operatorcorresponds to the evolutionary mechanism
of populationland determines the dynamics of distribution.
- (For example, in one population it is replication dynamics, in
an other - random imitation, and so on. Particularly, dynamics may
relate to maximization of some payoff function)
25. Then the dynamics of the society are governed by equations
(3.2) 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
distributionof system (3.2)is a Nash equilibrium of the population
game2) if for pathinitial distributionandthen *is a Nash
equilibrium of the population game G for path3) ifis a strict
equilibrium of the gameGthenis an asymptotically stable
distribution for system (3.2). Thus, the evolutionary mechanisms
selection model confirms that individual fitness is an endogenous
utility function for self-reproducing populations. 26.
- Idea of proof of 1 stand 2d statements is quite easy: if
stationary distribution over strategies is not Nash equilibrium of
fitness function then nothing can prevent expansion of replicators
that usethe best replystrategy. That is a contradiction to its
stability
- G eneralization of theorem 3 for elimination of dominated
strategies is possible under more strict assumptions on the variety
of evolutionary mechanisms. For any evolutionary mechanismand a
pair of strategiess,r , let us call ans,r -substitute of mechanisma
mechanism such that for strategies other thans and rthe shares of
individuals who apply these strategies change as under mechanism,
except that instead of strategysthey always playr . According to
(Vasin, 1995), if for anys,r,lthe set of mechanisms includes all
possible substitutes of, thena s for any strictly dominated
strategys.
- The r esult is stated for homogenuous population, without
taking sex or age in consideration. It is easily generalized for
populations with such structures. Fitness analogue in this case isa
rateof balanced growth ofthepopulation, it isdetermined by
theFrobenius number oftheLeslie matrix (see Semevskiy, Semenov,
1982)
27.
- Models and results of evolutionary game theory show that
behavior evolution ina self-reproducingpopulation corresponds to
well-known optimality principles Nash equilibrium and elimination
of dominated strategies
- Endogenously formed payoff function corresponds to Darwinian
concept of individual fitness
- However in biological and social populations cooperative and
altruistic behavior are well-know n . 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 randomcontestswith role
asymmetry between participants. For such games mixed Nash equilibr
iaare never evolutionary stable and strict equilibr ia may
notexist. So, sufficient conditions of stability don't work
- Relevanceof mentioned evolutiona rymodels to social
populations. Superindividual is a self-rep roducingstructure that
uses human population as a source forits reproduction . It can
influence behavior dynamics in this population
28. II . Stability of equilibrium. Pecularities of social
behavior evolution. 29. Stability problem of mixed strategies Let's
consider game of two populations with sets of strategies and and
payoff functionsand that show interaction result for all strategies
Assume thatindividuals of the 1 stpopulation interact only with
individuals of secon population and vice versa: individuals of the
2 ndpopulation interact only with individuals of the 1 st.At any
moment of timeteach individual uses a chosen strategy Assume that
arepopulation distribution sover strategies. PointisNash
equilibrium of game if for anyi, jEquilibrium is calledmixedif for
anyi, j 30.
- For any game with indescrete payoff function there exist a Nash
equilibrium
- In nonsingular case amount of positive coordinatespandqis the
same
- p(t)andq(t)change according to the system
- 1) Functionsandare such as
- f or any distributionsand payoff vectors
- That means that any Nash equilibrium is a stable point of
system (4.1)
- 2) Functionscanddare measurable functions overtand continiously
differentiable overp(0)andq(0).Derivatives are equibounded overt
.
- 3) Setis invariant of system (4.1). Vector-functionsA, B, G and
Hare continiously differentiable.
- Note that MAIB,and a system with positive functions ..generally
correspond to these assumptions
31.
- Note that system (4.1) can be converted to autonomous
system
- This occasion takes place during interaction between
populations of different sizes or between individuals of different
roles in one population, for example, between 'owner' and 'invader'
(Maynard Smith, 1982)
- Let's consider game , system (4.1) and corresponding autonomous
system (4.2)
- Stable point of system (4.2) is calledsingularif there is a
latent root of Jacobian matrix thatisequal to 0
- Point is calledcentreif for any latent rootRe =0, Im 0
- Point is calledsaddleif there is a latent root such thatRe
>0
32.
- Theorem 4.1:Any mixed equilibrium(P*, Q*)is either
- in this case(P*, Q*)is unstable point of a system (4.1) for any
available functionsc, d.
- This theorem doesn't solve the question of stability of'centre'
points for which all latent roots of linearized matrix are purely
imaginary. Let's use method that was deveped in work of Ritzberger
and Vogelsberger in 1990 and based on Liouville theorem.
- Consider system.According to this theorem, the field that is
free of divergence ( ) saves any volume constant and can't be in
asymptotically stable condition.
33.
- Let's describe MAIB class such that the shown method allows to
make a conclusion about abscence of asymptotically stable
equilibriums.
- Theorem 4.2:Assume that interpopulation MAIB (2.2) is such
that
- don't dependon(intensity of change to adaptive status and
probability of strategy change don't depend on distribution over
strategies in this population. However, it can depend on
distribution over strategies in another population taking part in
interacrion)
- 2)(alternative strategy is chosen with the way of random
imtation of other individuals from population)
- Any mixed equilibrium is asymptotically stable
- If probability of strategy choice as of alternative for
population individuals doen't depend on distribution over
strategies in this population then in general assumption
divergention of vector field of right parts of equation is negative
an it is possible to consider convergence to mixed equilibrium
34.
- Problem of convergence to the mixed equilibrium has been
considered in different literature also for iteration and
indiscrete processes such as fictious play and, particularly, for
Braun process for the game in normal form.
- Denoteas mixed strategies on stept . Thendiscrete Braun process
is described with such correlations :
- whereis payoff function of playerain mixed strategies
- is a set of singular mixed strategies of playerathat
corresponds to a set of its pure strategies
- Speaking about behavior dynamics in interaction population,
process can be interpreted as adaptive in such way:after every
period t>11/t part of each population changes its strategy to
one of the best answers.
35.
- Braun offered and Robinson proved convergence of discrete
process for antagonistiv bimatrix games
- Danskin showed convergence of this process for antagonistic
games with indescrete payoffs over multiplications of compact
spaces
- Fudenberg and Kreps showed convergence of adaptive play model
for nonantagonistic bimatrix game 2x2 with one pure mixed Nash
equilibrium
- and Hirsch developed result for 2x2 games with several nash
equilibrium (can be both mixed or not)
- In Bogdanov's work that was done on the basis of Belenkiy and
others results general rules for convergence of such processes for
bimatrix games are shown. It is proved that convergence can be
guaranteed for all bimatrix games that are converted to
antagonistic with the following development:
-
- a) adding constant to the column of payoff matrix of 1
stplayer
-
- b)adding constant to the line of payoff matrix of 2
ndplayer
-
- c)multiplication of payoff matrix and positive constant
- It is known that those transformations set a game class with
the same sets of Nash equilibrium
- However thes results can't be widespread for nonantagonistic
games of bif dimension
36. Process of fictious play doesn't convergence for Shepley
example, where payoff matrix: If pure strategiesare taken as
initial point then choice of players at other moments of time will
follow a cycle with six strategies: (1,1) -> (1,3) -> (3,3)
-> (3,2) -> (2,2) -> (2,1) -> (1,1) At that amount of
periods, where process is in each of these conditions will rise
exponentially. Obviously the process of fictious play doesn't
convergence. For some nonantagonistic games, particularly for
Shepley example, paths of fictious play process behave as average
over time RDM. It can be shown that RDM is equivalent to MAIB
particular case that corresponds to the theorem 4.2. At the same
time not only paths of these MAIB but also their average over time
don't convergence to equilibrium in Shepley example. 37. Best
properties (in the meaning of convergence to mixed equilibrium) are
shown by some more complex adaptive dynamics. Let's consider the
following modification of indiscrete process of fictious play for
the game of two players: (4.3) Whereis appraisal of mixed strategy
of playeraat the moment of time. is the best answer of playeraover
partners' strategyThe idea is thatapproximateswhenis quite big,
i.e. The best answer is done for the future strategy. For Shepley
example0,0413< /(1- )