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Markov Chain Models in Economics,Management and Finance
Intensive Lecture Course
in High Economic School, Moscow Russia
Alexander S. Poznyak
Cinvestav, Mexico
April 2017
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 1 / 42
4-th Lecture Day
Bargaining
(Negotiation)
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 2 / 42
BargainingDe�nition of barganing
De�nitionBargaining is a negotiation with the aim to arrive at an agreementbetween parties �xing obligations that each promises to carry out andestablishing the terms of a sale or exchange of goods or services.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 3 / 42
BargainingIllustrative photo
Figure: People bargaining in a traditional Indonesian pasar malam (night market)in Rawasari, Central Jakarta.
So, the prase: " to reach a bargain with the antique dealer over some product"can be translated as "Zaklyuchit�sdelku s prodavtzom antikvariata naschetnekotorogo tovara".
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 4 / 42
BargainingMain features of bargaining
Bargaining or haggling (sporit�) is a type of negotiation ("peregovorov") inwhich the buyer and seller of a good or service debate the price and exactnature of a transaction.
Bargaining is an alternative pricing strategy to �xed prices. Optimally, if itcosts the retailer nothing to engage and allow bargaining, the retailer candivine the buyer�s willingness to spend.
Hanggling has largely disappeared in markets where the cost to haggleexceeds the gain to retailers.
The status quo of the classical bargaining problem (or equilibrium) is anagreement reached between all interested parties. It is clear that studyinghow individual parties make their decisions is insu¢ cient for predicting whatagreement will be reached.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 5 / 42
BargainingBargaining as a game
Bargaining games refer to situations where two or more players must reachagreement regarding how to distribute an object or monetary amount. Eachplayer prefers to reach an agreement (favouring their interests) in thesegames, rather than abstain from doing so.J. Nash (The Bargaining Problem // Econometrica.1950. Vol. 18. P.155-162) de�nes a classical bargaining problem as being a set of jointallocations of utility, some of which correspond to what the players wouldobtain if they reach an agreement, and another that represents what theywould get if they failed to do so. The Nash bargaining solution is thebargaining solution that maximizes the product of an agent�s utilities on thebargaining set.A bargaining game for two players is de�ned as a tuple (F , x , ddis ) where
F =nJ1�x (1), x (2)
�, J2
�x (1), x (2)
�ois the set of possible joint utility
allocations (possible agreements), x 2 Xadm and ddis is the disagreementpoint (where participants return if they did not obtain agreement.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 6 / 42
BargainingNash barganing solution
There exists several possible solutions depending on an additional axiomatics.
ProblemBy the Nash barganing solution (NBS) we will understand the solution of thebargaining game (F , x , dis ) for two players possessing the following 4 properties:
- Invariant to a¢ ne transformations or Invariant to equivalent utilityrepresentations.- Pareto optimality.- Independence of irrelevant alternatives (if from the begginning delete irrelevantalternatives, then the solution of the problem remain unchable).
- Symmetry, that is, the players have the same utility functions J1�x (1), x (2)
�= J2
�x (1), x (2)
�and such that J1
�x (2), x (1)
�= J2
�x (1), x (2)
�implying
that in the disagreement point they obtain the same pro�t (meaning that this
point is exactly in the "bisectrice"): i.e., J1�d (1)dis , d
(2)dis
�= J2
�d (1)dis , d
(2)dis
�.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 7 / 42
BargainingThe Nash�s theorem
Theorem (J.Nash, 1950)
The Nash barganing solution (NBS) is the point x�Nash maximizing the function(the Nash product)
PNash�x (1), x (2)
�=h
J1�x (1), x (2)
��J1
�d (1), d (2)
�i hJ2�x (1), x (2)
�� J2
�d (1), d (2)
�ion the Pareto set XPareto :
XPareto :=nx (1), x (2) :
�x (1) (λ) , x (2) (λ)
�=
arg maxx (1)2X (1)adm , x
(2)2X (1)adm
hλJ1
�x (1), x (2)
�+ (1� λ) J2
�x (1), x (2)
�i)0 � λ � 1
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 8 / 42
BargainingThe NBS as a bargaining process
NBS can be explained as the result of the following bargaining process:
A current agreement, say�x (1), x (2)
�, is on the table.
One of the players, say player 2, can raise an objection. An objection is an
alternative agreement�x 0(1), x 0(2)
�, .Probably, the alternative agreement is
better for player 2 (J2�x 0(1), x 0(2)
�> J2
�x (1), x (2)
�) and worse for
player 1 (J1�x 0(1), x 0(2)
�> J1
�x (1), x (2)
�. Suppose that raising such an
objection has some probability of ending the negotiation. This probability pcan be selected by player 2 (e.g, by the amount of pressure he puts on player
1 to agree). The objection is e¤ective only if pJ2�x 0(1), x 0(2)
�>
J2�x (1), x (2)
�, i.e, player 2 prefers the alternative agreement
�x 0(1), x 0(2)
�with a chance p, over the original agreement
�x (1), x (2)
�for sure.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 9 / 42
BargainingThe NBS as a bargaining process (continuation)
Player 1 can then raise a counter-objection by claiming that for him
pJ1�x (1), x (2)
�> J1
�x 0(1), x 0(2)
�. This means that player 1 prefers to
insist on the original agreement even if this might blow up the negotiation.Player 1 prefers the original agreement with a chance p, over the alternativeagreement for sure.
An agreement is NBS if, for every objection raised by one of the players,there is a counter-objection by the other player, that is,
pJ2�x 0(1), x 0(2)
�> J2
�x (1), x (2)
�J1�x 0(1), x 0(2)
�> pJ1
�x (1), x (2)
�It is an agreement which is robust to objections.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 10 / 42
BargainingThe NBS as a bargaining process: illustration
Figure: Nash barganing solution.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 11 / 42
BargainingThe generalization of the Nash�s theorem for the multi-participants case
TheoremThe Nash barganing solution (NBS) is the point
x�Nash=�x�(1)Nash, x
�(1)Nash, ..., x
�(1)Nash
�maximizing the function ( the Nash product)
x�Nash = maxx2XPareto
PNash (x) = arg maxλ2SN
PNash (x (λ))
PNash (x) =N∏i=1[Ji (x)�J i (d)] , SN :=
�0 � λi ,
N∑i=1
λi= 1�
on the Pareto set XPareto :
XPareto :=
(Ji (x (λ)) , i = 1,N : x (λ) = arg max
x (i )2X (i )adm ,
N∑i=1
λiJi (x)
)
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 12 / 42
BargainingAutomated bargaining
FactA bargaining situation is complex (but this takes place practically always) and�nding the Nash barganning equilibrium is di¢ cult using game theory.Evolutionary computation methods have been designed for automated bargaining,and demonstrated e¢ cient and e¤ective for approximating Nash equilibrium.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 13 / 42
BargainingKalai�Smorodinsky bargaining solution
Independence of Irrelevant Alternatives can be substituted with a Resourcemonotonicity axiom (Kalai, Ehud & Smorodinsky, Meir (1975). "Othersolutions to Nash�s bargaining problem". Econometrica. 43 (3): 513�518).This leads to the socalled Kalai�Smorodinsky bargaining solution: it isthe point which maintains the ratios of maximal gains. In other words, ifplayer 1 could receive a maximum of
J�i := maxx (1)2X (1)adm , x
(2)2X (1)adm
Ji�x (1), x (2)
�(i = 1, 2)
with player 2�s help (and vice versa), this de�nes the Utopia Point(J��1 , J
��2 ), then the Kalai�Smorodinsky bargaining solution would yield the
point on the Pareto frontier such that
x�K�S =�x 2 XPareto :
J1(x (1),x (2))�J1(d (1),d (2))J2(x (1),x (2))�J2(d (1),d (2))
=J ��1 �J1(d (1),d (2))J ��2 �J2(d (1),d (2))
�Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 14 / 42
BargainingKalai�Smorodinsky bargaining solution: illustrative �gure
Figure: The Kalai-Smorodinskii solution.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 15 / 42
BargainingKalai�Smorodinsky solution as an Optimization Problem
Generalization for N-participants
The Kalai-Smorodinsky rule, say x�K�S , chooses the maximum individuallyrational payo¤ pro�le at which each agent�s payo¤ gain from disagreement J�i hasthe same proportion to his aspiration ("utopia") payo¤�s gain J��i fromdisagreement J�i . Formally,
x�K�S = arg maxx (i )2X (i )adm ,i=1,N
�mini=1,N
Ji (x (1),...,x (N ))�J �iJ ��i �J �i
�
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 16 / 42
BargainingKalai�Smorodinsky solution as an Optimization Problem
Simplex-format of the KS-solution representation
Lemma
fα := mini=1,N
fi = minλ2SN
N∑i=1
λi fi , λi =
�1 if i = α0 if i 6= α
In view of that
x�K�S = arg maxx (i )2X (i )adm ,i=1,N
minλ2SN
N∑i=1
λiJi (x (1),...,x (N ))�J �i
J ��i �J �i
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 17 / 42
BargainingKalai�Smorodinsky solution as an Optimization Problem: numerical algorithm
λ (n+ 1) = πS
8>>><>>>:λ (n)� γλ
0BBB@J1(x (1)(n),...,x (N )(n))�J �1
J ��1 �J �1...
JN (x (1)(n),...,x (N )(n))�J �NJ ��N �J �N
1CCCA9>>>=>>>;
x (n+ 1) = πXadm
8>>><>>>:x (n) + γx
0BBB@rJ1(x (1)(n),...,x (N )(n))
J ��1 �J �1...
rJN (x (1)(n),...,x (N )(n))J ��N �J �N
1CCCA9>>>=>>>;
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 18 / 42
BargainingEgalitarian bargaining solution of Kalai: the photo of E.Kalai
Figure: Ehud Kalai (2007): born Dec. 7, 1942; nationality: United States andIsrael, �eld: Game theory and economics.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 19 / 42
BargainingEgalitarian Kalai bargaining solution
The egalitarian bargaining solution (Kalai, Ehud (1977), "Proportional solutionsto bargaining situations: Intertemporal utility comparisons". Econometrica. 45(7): 1623�16307) is a third solution which drops the condition of scale invariancewhile including both the axiom of Independence of irrelevant alternatives, and theaxiom of resource monotonicity. It is the solution which attempts to grant equalgain to both parties.
Figure: Kalai egalitarian solution.Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 20 / 42
BargainingComparison table
NamePareto
OptimalitySymmetry
Scale
invariance
Irrelevant
independence
Resource
Monotonicity
Nash (1950)Maximizing the product
of surplus utilities
+ + + + NO
Kalai-Smor.(1975)
Equalizing the ratios
of maximal gains
+ + + NO +
Kalai (1977)Equal payoofs
on Pareto set
+ + NO + +
Table: Table of Comparison
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 21 / 42
BargainingExample of the Nash and Kalai�Smorodinsky bargaining solutions (1)
Example
Alice and George are businessmen and they have to choose between threeoptions a, b and c , that give them the following monetary revenues:24 a b c
Alice $60 $50 $30George $80 $110 $150
35They can also mix these options in arbitrary fractions("predpochteniya"=how often), e.g, they can choose option a for a fractionp1 of the time, option b for fraction p2, and option c for fraction p3, suchthat p1+p2+p3= 1. Hence, the set F of feasible agreements is the convexhull of a(60, 80) and b(50, 110) and c(30, 150):
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 22 / 42
BargainingExample of the Nash and Kalai�Smorodinsky bargaining solutions (2)
The disagreement point is de�ned as the point of minimal utility: this is$30 for Alice and $80 for George, so d = (30, 80).For both Nash and KS solutions, we have to normalize the agents�utilitiesby subtracting the disagreement values, since we are only interested in thegains that the players can receive above this disagreement point. Hence, thenormalized values are:24 a b c
Alice $30 $20 $0George $0 $30 $70
35
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 23 / 42
BargainingExample of the Nash and Kalai�Smorodinsky bargaining solutions (3)
Figure: Possible values of payo¤s.Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 24 / 42
BargainingExample of the Nash and Kalai�Smorodinsky bargaining solutions (4)
The Nash bargaining solution maximizes the product of normalized utilities onthe Pareto set:
x�Nash = arg maxx2XParetoh
J1�x (1), x (2)
��J1
�d (1), d (2)
�i hJ2�x (1), x (2)
��J2
�d (1), d (2)
�i= arg max
p1,p2,p3[30p1+20p2+0p3] [0p1+30p2+70p3]
The maximum on the Pareto frontier is attained when p1= 0, p2 = 7/8 andp3 = 1/8 (i.e, option b is used 87.5% of the time and option c is used in theremaining time). The utility-gain of Alice is $17.5 and of George - $45.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 25 / 42
BargainingExample: the Nash barganing solution (5)
Figure: Nash bargaining solution.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 26 / 42
BargainingExample: the Kalai-Smorodinskii barganing solution (6)
The KS bargaining solution equalizes the relative gains - the gain of eachplayer relative to its maximum possible gain on the Pareto set satisfying therelation
30p1+20p2+0p30p1+30p2+70p3
=3070
Here, the maximum on the Pareto frontier attained when p1 = 0,p2= 21/26 and p3= 5/26. The utility-gain of Alice is $20 and of George- $42.Note that both solutions are Pareto-superior to the "random-dictatorial"solution - the solution that selects a dictator at random and lets him/herselects his/her best option. This solution is equivalent to letting x = 1/2,y = 0 and z = 1/2, which gives a utility of only $15 to Alice and $35 toGeorge.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 27 / 42
BargainingExample: the Kalai-Smorodinskii barganing solution (7)
Figure: KS solution.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 28 / 42
BargainingExample: the Kalai egalitarian barganing solution (8)
It is reached at the point (30, 30).
Figure: Kalai egalitarian solution.Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 29 / 42
Barganing using Markov chain modelsPayo¤ representation in Markov model bargaining problem
For l = 1,N the payo¤ Jl (x1, ..., xN ) in multi-participants bargainingcorresponds to
U l�c1, ..., cN
�:=
N1,M1
∑i1,k1
...NN ,MN
∑iN ,kN
W l(i1,k1,...,iN ,kN )
N∏l=1c l(il ,kl )
W l(i1,k1,...,iN ,kN )
=N1∑j1...NN∑jNU l(i1,j1,k1,...,iN ,jN ,kN )
N∏l=1
πl(il ,jl jkl )
c l :=hc l(il ,kl )
iil=1,Nl ;kl=1,Ml
is a matrix with elements
c l(il ,kl ) = dl(kl jil )P
l�s l=s(il )
�
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 30 / 42
Barganing using Markov chain modelsAdmissible sets representation in Markov model bargaining problem
The admissible sets�x (l)2 X (l)adm
�in Markov model bargaining problem
corresponds to
c l2 C ladm :=
8>>>>>>><>>>>>>>:
kc (l)il ,kl k: c (l)il ,kl� 0,Nl∑il=1
Ml
∑kl=1
c (l)il ,kl= 1
Ml
∑kl=1
c (l)il ,kl=Nl∑il=1
Ml
∑kl=1
π(l)il ,klc ln(i l , k l )
Nl ,Ml
∑il ,kl
ql(jl jil ,kl )cl(il ,kl )
= 0 (for continuos-time MCh.)
9>>>>>>>=>>>>>>>;P l�s l=s(il )
�=
Ml
∑klc l(il ,kl ), d l(kl jil ) =
c l(il ,kl )Ml∑kl
c l(il ,kl )
since in the ergodic caseMl
∑klc l(il ,kl ) > 0 for all l = 1,N . The individual aim of
each player is U l (c l ) ! maxc l2C ladm
.Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 31 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: example (1)
N1 = N2 = 5, and the number of actions M1 = M2 = 2
U l = U l�c1, c2
�:=
N1,M1
∑i1,k1
N2,M2
∑i2,k2
W l(i1,k1,i2,k2)
c1(i1,k1)c2(i2,k2)
, l = 1, 2
We are interested in the point (U1,U2) which is better than the disagreementpoint
�U1�,U2�
�, that is,
U l > U l�
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 32 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: example (2)
U1(i ,j j1)=
26666410 8 13 7 611 19 6 8 109 7 13 19 514 9 15 12 1612 14 9 8 10
377775 , U1(i ,j j2)=
26666412 9 7 10 1516 0 9 14 618 10 16 9 412 16 9 8 1311 9 13 17 10
377775
U2(i ,j j1)=
2666647 9 12 6 1018 12 7 9 105 14 8 11 1610 13 8 14 1016 9 12 10 8
377775 , U2(i ,j j2)=
2666645 17 6 9 118 9 14 12 810 13 9 1 1812 18 15 9 417 13 9 5 6
377775
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 33 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: transition rate matrices (1)
The transition rate matrices for each player are de�ned as follows
q1(i ,j j1)=
266664�0.5366 0.0888 0.0611 0.1893 0.14090.0416 �0.5689 0.0588 0.1331 0.09420.2358 0.1929 �0.3784 0.1878 0.20840.0942 0.1929 0.1244 �0.5963 0.05700.1649 0.0942 0.1342 0.0861 �0.5005
377775
q1(i ,j j2)=
266664�0.8049 0.1333 0.0916 0.2839 0.21140.0624 �0.8534 0.0881 0.1996 0.14120.3538 0.2894 �0.5677 0.2817 0.31270.1414 0.2894 0.1867 �0.8944 0.08550.2474 0.1414 0.2012 0.1292 �0.7508
377775
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 34 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: transition rate matrices (2)
q2(i ,j j1)=
266664�0.5434 0.1635 0.1649 0.2965 0.08710.1090 �0.9561 0.2965 0.0878 0.22640.1005 0.1996 �1.0011 0.1648 0.08770.1641 0.3679 0.3194 �0.8518 0.15710.1697 0.2251 0.2202 0.3027 �0.5585
377775
q2(i ,j j2)=
266664�0.2329 0.0701 0.0707 0.1271 0.03740.0467 �0.4097 0.1271 0.0376 0.09700.0431 0.0856 �0.4290 0.0706 0.03760.0703 0.1577 0.1369 �0.3651 0.06730.0727 0.0964 0.0944 0.1298 �0.2393
377775Π(t j d) = Π(0)eQ (d )t= eQt :=
∞
∑t=0
tnQ n(d )n! !
t!∞Π(d)
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 35 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: disagreement point as Nash equilibrium (1)
c1�=
2666640.1214 0.07880.1214 0.07880.1159 0.08250.1265 0.07540.1185 0.0807
377775 c2�=
2666640.1141 0.08170.1091 0.09350.1091 0.09360.1095 0.09250.1134 0.0836
377775
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 36 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: disagreement point as Nash equilibrium (2)
The mixed strategies obtained for the players are as follows
d1� =
2666640.6065 0.39350.6062 0.39380.5842 0.41580.6264 0.37360.5948 0.4052
377775 d2� =
2666640.5828 0.41720.5384 0.46160.5382 0.46180.5423 0.45770.5757 0.4243
377775With the strategies calculated the resulting utilities in the disagreement point foreach player ψl�(c1, c2) are as follows:
J�1 := J�1 (c1�, c2�) = 126.8675, J�2 := J�2 (c
1�, c2�) = 121.5986
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 37 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: KS-solution (1)
Given γx = γλ = 0.7 we obtain the convergence
to the point
c1K�S =
2666640.1046 0.09560.1045 0.09570.0987 0.09960.1100 0.09210.1015 0.0977
377775 , c2K�S =
2666640.1002 0.09600.0956 0.10680.0955 0.10690.0960 0.10580.0995 0.0977
377775Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 38 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: KS-solution (2)
The players strategies are
d1K�S =
2666640.5224 0.47760.5220 0.47800.4977 0.50230.5443 0.45570.5094 0.4906
377775 , d2K�S =
2666640.5106 0.48940.4722 0.52780.4720 0.52800.4756 0.52440.5045 0.4955
377775and the resulting utilities in the bargaining KS-solution for each player are asfollows:
J1,K�S := J1(c1, c2) = 128.1883, J2,K�S := J2(c1, c2) = 122.1602
The utilities on the utopia point for the bargaining problem are for each player asfollows:
J��1 = 135.7439, J��2 = 126.0856
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 39 / 42
Barganing using Markov chain modelsTwo-person KS-bargaining problem: illustrating �gure
Figure: KS-solution for Markov Chain Model.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 40 / 42
ConclusionWhich topics we have discussed today?
4-rd Lecture Day: Bargaining (Negotiation)� De�nition of Bargaining and the main properties of bargaining.� Bargaining as a game.� Nash�s bargaining solution.� Generalization for N-participants.� Kalai-Smorodinsky bargaining solution.� Expression of KS solution as a result of Optimization Problem.� Egalitarian Kalai-solution.� Comparison table.� Bargaining for Markov Chain models.� Illustrative example.
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 41 / 42
ConclusionNext lecture
Next Lecture Day:
Partially Observable Markov Chain Models and Tra¢ c OptimizationProblem
Thank you for your attention! See you soon!
Alexander S. Poznyak (Cinvestav, Mexico) Markov Chain Models April 2017 42 / 42