Download 30. TYPE OF THE RETAILER PROBLEM WITH COMPLETE INFORMATION WITH NASH EQUALIBRIA REPEATEDLY

Int. J. of Mathematical Sciences and Applications,
Vol. 2, No. 2, May 2012
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TYPE OF THE RETAILER PROBLEM WITH COMPLETE
INFORMATION WITH NASH EQUALIBRIA REPEATEDLY
Prakash Chandra a*
a
K. C. Sharmab
Deptt. of Applied Science & Humanities, Dronacharya College of Engineering, Gurgaon, India.
E-mail:[email protected], [email protected]
b
Deptt. of Mathematics and Computer, MSJ College, Bharatpur, Rajasthan, India.
Email:[email protected]
Abstract
We are presenting in this paper, game theoretic model in between two type of retailers (one is
leader and other is follower) who compete the market demand with substitutable products. In
this game, our part of study, consider the market demand bounded (normalized) and
unbounded and correlation between demands of the two retailers at random demand.
Geometric analysis shows the convexity of the region in which many Nash Equilibrium exits.
We are applying KT condition to find out the solution of the game in each period solution
with Nash Equilibrium. Under the conditions, the leader’s profit does not improve in
Stackelberg game where leader retailer deciding first and other play on opponents’ reaction
function repeatedly.
Keywords: Stackelberg game; Repeated game; Nash equilibrium
INDRODUCTION:
In game theoretic model of this problem, we are considering two type of the retailers(profile), player 1
( )(leader) and player 2 ( ) (follower) which have complete information of the market demand and what
their opponent is going to play after choosing its strategy for the game. They play as per their reaction functions.
In the competitive retailer problem, retailers compete to satisfy the random total demand with substitutable
products. Initially, a customer move to purchase the product from one of them. If he can get the product there,
he buys it. If he cannot get the product in his initial choice then may decide to buy it or not to buy any. For
buying it, move to other retailer and buy the product, if it is available. So the retailer ( )orders according to the
fact the he has an initial allocation of demand to satisfy but he also has an extra demand occurring from the
shortage of ( ) which is called the reallocation demand. If the retailer ( ) order too much, reallocated demand
coming to retailer ( ) decreases so player ( ) may end up with a high inventory holding cost. If the retailer
( ) order too little, reallocation demand coming to retailer ( ) increases so player ( ) may end up with a
high shortage cost. Through this way, we see that the decision of one player affect the other in each period of the
game.
Here ordering decision of the retailers is assumed to be given simultaneously with complete information set;
there is no priority or time sequence among the players’ making decisions. But what we see in reality, decisions
of the one player are observed then the other player make the decision to maximize his payoff. This may be due
to lack of information of a player about the other. If the marginal demands are mobile enough so that unsatisfied
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Prakash Chandra & K. C. Sharma
demand of one can be reallocated to another, then the small firms may want to see the order/production of the
leader before they place their orders. In the Stackelberg version of the competitive retailer problem, there is a
market leader retailer and other is the follower. The leader makes his decision (order quantity) first knowing that
others respond to his decision optimally, who expect for the more profit than the follower.
Profile of the retailers shows their type, natural section of their choice generate different payoff, retailers
maximize rationally their payoffs. The payoff maximization aspect generates the natural type of the retailer to
fulfill the different demand criteria rationally in each period of the game.
In the wide section of the literature, Parler; taken two players where the demands for the competitive firms are
independent random variables that can take any nonnegative value and are substitutable with known rates. He
proves the existence and the uniqueness of the Nash solution under certain conditions.
In this paper, we organized the profit function with some conditions under which the optimal solution of the
competitive retailer problem with two players and the leader-follower type ordering problem have the same
solution and we present several conditions for the study in repeated game. The first condition; if the leader
forces the follower to order in a region for which the follower’s response is flat, then the follower knows that
they actually order according to Nash equilibrium. Then it not intent to collect information about the leader’s
cost parameters. The second condition; if the leader’s profit is not sensitive to the follower’s orders then
ordering first does not bring any advantage repeatedly.
Type of Retailers Matrix:
, where = 1, 2 . we can write in the expanded form
,
,
,….
Let the profile of the retailers =
=
, , ,….
= ′ , ′ , ′ , … . ′ Type of profile matrix of the retailers
of the profile
and )=
be
=∏
=
×
=
, & = 1,2, … ,probability of the first retailer of his type
),
)=
=
, , ,….
), … .
) and Probability of the Second retailer of his type
′ ),
′ ), … .
′ ) . Conditional Probability of second retailer when we
′ , ′ , ′ ,…. ′ =
know
the
probability
distribution
on
the
first
retailer’s
state
of
nature
⁄ )=
⁄ ),
⁄ ),
⁄ ),… ⁄ ) and Conditional Probability of first retailer
when we know the probability distribution on the second retailer’s state of nature
⁄ ′)=
⁄ ′ ),
⁄ ′ ),
⁄ ′ ),… ⁄ ′ ) State of nature matrix of the retailers
!
" ′ # ⋯ "′ % "# "## ⋯ "%#
⋱
⋮
)
# ⋮' ⋮
"% "#% ⋯ "%%
Joint probability of the state of their nature of the retailers
⋮
)
)
′ ) ⋯ * ⋮
)
)
⋯
⋱
⋯
′ )
⋮
)
)
+
Where the joint probabilities of the state of nature can be calculated by the following way)=
)=
)=
,
,
)=
′
,
′
′
)=
)=
⁄
⁄
⁄
).
′
′
′
).
).
)=
′
)=
′
′
744
)=
′ ,
′ ,
′ ,
)=
)=
)=
′
⁄
′ ⁄
′ ⁄
).
).
)
).
)
)
TYPE OF THE RETAILER PROBLEM WITH COMPLETE…
∗-
.= -
,
′
⁄
.= -
′
.. -
′
.= - ′,
.= - ′⁄
..
)
… … …
)=
,
)=
′
⁄
′
).
…
)=
,
′
)=
⁄
′
).
′
)=
′ ,
)=
′ ⁄
).
)
… …
′
)=
′ ,
)=
′ ⁄
).
)
As per the main theme of cooperative game theory, Players plays their best responses to maximize the payoff
which is known as Nash Equilibrium of the game. Rationality works to get Pareto’s Optimality of the game in
each period of the game. Here one mechanism work vice versa from the game, Nash Equilibrium of the game
shows some state of the nature of the retailer at which they are able to have it. This imposes them to have this
state of nature to play Nash. If we look into the picture of the probability distribution of their state of nature,
joint probability of the different combination of their state has maximum probability. We can define this
mathematically- . 9) = ∗ - . = - , ′ . = - ⁄ ′ .. - ′ .
/01ℎ3456 78 5 9) ⇔;<=
,
,..
Assumptions for the game : There is a restriction of the entry to the new player in the game.
RETAILERS’ GAME MODEL
Let number of players - two retailers ( and ) & > , > ) (t): Repeated random demand and continuous
and
respectively.
random vector having a strictly increasing cumulative distribution function of
Realizations of > , > ) are called the initial allocation of demand for
and
respectively. The marginal
density and marginal distribution functions of > 9) are ?@A B ) 9)0CDE@A B ) 9)for retailer and similarly
for retailer . Let F 9),
9), 8 9), 1 9), be the unit cost, unit penalty cost, unit sales revenue and salvage
value, respectively. For 0CD8 > F > 1 ≥ 0, = 1,2. Let J be the order quantity of , = 1, 2. If there is
excess demand for , it is reallocated to with a known substitution rate 0 CK0,1L. Now Effective demand
for
Q
= Initial demand + Reallocated demand = > 9) + 0 9) N> 9) − J 9)P . Let expected profit
R -J , J . 9) = 3 S -J , J . 9) when
R -J , J . 9) = 3 S -J , J . 9) =
orders J 9) and
orders J 9), = 1, 2, ≠ .
∬ S -J , J . 9) + S -J , J . 9) + S -J , J . 9) + S V -J , J . 9) ? -B , B .DB DB ,
With respect to different demand realization B , B ) for a possible order quantity NJ 9), J 9)P. In Figures 1 to
4, all these conditions are represented with the all possible cases with order quantity demanded by the retailers
and market demand by the customers. These four cases are taken with marginal distribution function which is
given below with the following explanation:
Where:
S -J , J . 9) = Revenue at Initial demand + Salvage value at Remaining Quantity – Cost of Order quantity by
the retailer.
S -J , J . 9) = 8 9)> 9) + 1 9)-J 9) − > 9). − F 9)J 9) ?> 9) ≤ J 9), > 9) ≤ J 9)
S -J , J . 9) = Revenue at Initial Demand + Revenue at min of reallocated demand and left quantity to the
retailer + Salvage value at the remaining quantity − Cost of Order quantity by the retailer.
745
Prakash Chandra & K. C. Sharma
S -J , J . 9) = 8 9)> 9) + 8 9)
C J 9) − > 9), 0 9) N> 9) − J 9)P
Q
+ 1 9) YJ 9) − > 9) − 0 9) N> 9) − J 9)P Z − F 9)J 9)
If > 9) ≤ J 9), > 9) ≥ J 9)
S -J , J . 9) = Revenue at Initial Demand (Ordered quantity) − Cost of Order quantity
S -J , J . 9) = 8 9)J 9) − F 9)J 9) ?> 9) ≥ J 9), > 9) ≤ J 9)
S V -J , J . 9) = Revenue at Initial Demand (Ordered quantity) − Cost of Order quantity
S V -J , J . 9) = 8 9)J 9) − F 9)J 9) ?> 9) ≥ J 9), > 9) ≥ J 9)
Nash equilibrium;
If
R -J ′ , J ′ . 9) ≥ R -J , J ′ . 9)?[8066J 9)
orders J ′ 9) , then
orders J ′ 9). the reaction curve of
is the collection of all points
-J , J . 9)\R -J , J . 9) ≥ R -J , J . 9), J 9) ≥ 0 and that is ( NJ J 9)P , J 9) ). Any point of
′
′
intersection of the reaction curves is a Nash Solution. As we know each expected profit function is strictly
concave, each player has a unique response for every decision of the competitor. The first order necessary
condition that is also sufficient for the unique response
]R -J , J .⁄]J = 0
If
orders as much as the total possible demand then
orders the noncompetitive optimal J 9) =
E@^A -_ 9).for i = 1, 2, where _ 9) = 8 +
− F ) 9)⁄ 8 +
− 1 ) 9).
For Stackelberg case:
One retailer is the leader and other is follower, leader player decide the order quantity first which impose to the
follower retailer to decide the rest quantity over her decision. As per the Stackelberg game, Follower responds
according to the leader player response function. We can put the problem in respect to the leader as Max
R -J , J . 9), subject to constraints ]R -J , J .`]J = 0, Competitive Equilibrium is the worst case for the
leader.
Kuhn-Tucker Conditions: as we know that expected profit function is continuous and strictly concave, there is
a unique J on the reaction curve of a for everyJ . ]R -J , J .⁄]J + b] R -J , J .`]J ]J = 0
]R -J , J .`]J + b] R -J , J .`]J
= 0, ]R -J , J .`]J = 0
cde: g, πh -Q h , Q j . t)forXh t) ≤ Q h t), Xj t) ≤ Q j t):
Q j t)
746
Fig:2,πh -Q h , Q j . t)forXh t) ≤ Q h t), Xj t) ≥
TYPE OF THE RETAILER PROBLEM WITH COMPLETE…
cde: p, πh -Q h , Q j . t)forXh t) ≥ Q h t), Xj t) ≤ Q j t) :
Fig: 4,πh V -Q h , Q j . t)forXh t) ≥ Q h t), Xj t) ≥ Q j t)
Geometrical Approach for Bounded market demand:
If we consider the demand distribution on Xh t) + Xj t) ≤ D t), Xh t) ≥ 0, Xj t) ≥ 0, whereD t) > 0 is the
maximum possible demand of the market. And let the correlation between Xh t)andXj t)beρ. Shaded region
of the following fig: 5 has convexity under bounded market demand which has the infinite solution. Point Z of
this region [WhenQ h t) + Q j t) < D t)L is projection of vertex generated by lower bound of the ordered
quantity of the retailers on the bounded the market demand. Open ball at the point Z in ℝ intersection with
convex region S has at least one Nash Equilibrium. In ℝ - open ball of radius ε is the collection of the points of
distance less than the radius from the fixed point Z in ℝ . Bε z) = x: |x − z| < • .
Bε z) ∩ S = NE
And Q h t) + Q j t) ≥ D t) is the case shown in fig: 6, intersection of the convex region uvu′ v ′ (S ′ ) and open
ball at u has at least one Nash Equilibrium
Bε u) ∩ S ′ = NE
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Prakash Chandra & K. C. Sharma
Fig: 5.
) J + J ) 9) < † 9){Bounded Demand} Convexity has Efficiency at Point Z.
Theorem:1 Suppose the demand Xh t) + Xj t) ≤ D t), Xh t) ≥ 0, Xj t) ≥ 0, whereD t) > 0 . if Q h t) +
Q j t) ≥ D t) then Q h ‡ t), Q j ‡ t)) = NQ h ˆ t), Q j ˆ t)P = Q h t), Q j t)).
Proof: If Q h t) + Q j t) ≥ D t) then the reaction function of Pi is independent of Q j t) and constantly equal to
lower bound of Q h t). So if Lower bound of both the order quantity is higher minQ h t) + Q j t) ≥ D t) then
the reaction functions intersect at point which is known a Nash Equilibria in each period of the game ,
where] R -J , J .`]J ]J = 0.
U’
V’
H
L
U
V
M
D
K
Fig: 6.
) J + J ) 9) ≥ † 9) {Bounded Demand} Efficiency at Point uin the region‹Œ‹′ Œ ′ .
Bounded Normalized Demand:
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TYPE OF THE RETAILER PROBLEM WITH COMPLETE…
Suppose the normalization the bounded demand‖†‖ = 1, 9ℎ_C > + > ) 9) ≤ 1, 0CD> 9) ≥ 0, > 9) ≥ 0.
Then the expected profit R -J , J . 9)[?a 7_
•A ••
•A ^=A ^=•
‘
‘
Ž Ž S -J , J . 9) ? -B , B .DB DB + Ž
‘
•• ^=A ^=•
Ž
‘
Ž
•A
Ž
••
^•• ^=A ^=•
S -J , J . 9) ? -B , B .DB DB + Ž
And
•A
< 1
S -J , J . 9) ? -B , B .DB DB +
Ž
••
S V -J , J . 9) ? -B , B .DB DB , ? J + J ) 9)
^•• ••
Ž Ž S -J , J . 9) ? -B , B .DB DB
‘
‘
•A ^=A ^=•
+ Ž
Ž
^••
‘
^=A ^=•
+Ž
Ž
‘
•A
•A ^=A ^=•
S -J , J . 9) ? -B , B .DB DB + Ž
‘
Ž
••
S -J , J . 9) ? -B , B .DB DB S -J , J . 9) ? -B , B .DB DB , ?J + J ≥ 1
Now computing the reaction functions by first order condition, ]R -J , J .⁄]J = 0
•A •A Q•• ^=A
Ž
‘
Ž
‘
•A
1 − F ) 9) ? -B , B .DB DB + Ž
^=A
+Ž Ž
•A
•• ^=A
Ž Ž
‘
•A
‘
‘
^=A
Ž
•A Q•• ^=A
8 − F ) 9) ? -B , B .DB DB +
8 − F ) 9) ? -B , B .DB DB , ? J + J ) 9) < 1
1 − 8 ) 9) ? -B , B .DB DB + 8 − F ) 9) ? J + J ) 9) ≥ 1
For the ordered quantity J , J ) 9) , response functions are continuous. As per the study of Parlar, there is
the convexity of the reaction curves and bounded order quantity
J 9) ≤ J 9) ≤ J 9) is related to the distributed function that is J 9) = E@^A _ )0CDJ 9) = E@^AQ@• _ ).
Theorem 2. Let R J , J ) 9)0CDR J , J ) 9) be the expected profit functions of two retailers, assume that
the probability distribution function of the demands is strictly increasing. If a Stackelberg solution
satisfies] R J , J )⁄]J ]J = 0, ]R J , J )⁄]J = 0, then it be a Nash solution. If a Nash Solution
satisfies] R J , J )⁄]J ]J = 0, ]R J , J )⁄]J = 0, then it is a Stackelberg solution.
Proof: Let -J ’ , J ’ . 9) be a Stackelberg solution then it sure satisfies KT conditions that is
]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0
]R J , J )⁄]J + b] R J , J )⁄]J = 0, ]R J , J )⁄]J = 0,
Then it satisfies ]R J , J )⁄]J = 0, if this holds ] R J , J )⁄]J ]J = 0 then
]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0 reduced to the following equation
]R J , J )⁄]J =0,If ]R J , J )⁄]J = 0 then we can put b = 0 and
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Prakash Chandra & K. C. Sharma
]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0 , Reduces to the following equation ]R J , J )⁄]J =0,
Hence,-J ’ , J ’ . 9) satisfies ]R J , J )⁄]J =0 and ]R J , J )⁄]J =0 and is a Nash solution.
Let -J “ , J “ . 9) be a Nash Solution then it sure satisfies ]R J , J )⁄]J =0 and ]R J , J )⁄]J =0 then
it
satisfies
]R J , J )⁄]J =
0.
If
] R J , J )⁄]J ]J = 0
holds
then
]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0
and
a
b109 1? C” ]R J , J )⁄]J + b] R J , J )⁄]J = 0 can be selected. If ]R J , J )⁄]J = 0 holds
then
b=0
satisfies
]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0,
]R J , J )⁄]J + b] R J , J )⁄]J = 0.
Hence-J “ , J “ . 9) satisfies]R J , J )⁄]J + b] R J , J )⁄]J ]J = 0
]R J , J )⁄]J + b] R J , J )⁄]J = 0, ]R J , J )⁄]J = 0 and Stackelberg solution.
Conclusion:
Study of this paper shows game theoretic aspect of the type of retailer’s problem in which Nash Solution is a
Stackelberg solution of the problem under the condition of normalized bounded market demand in each period
of the game. Solution equilibrium imposes retailers to opt the type according to Equilibrium of the most
efficient. There are more possibilities of the future research considering more than two retailers.
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