Literature Review: Price Equilibrium
1
Introduction
First we present the general problem and mention some important special examples. In
order to solve the problem we propose two theories, as follows. In Section 2 we present
the literature that treats the price equilibrium problem as a vector equilibrium problem.
In Section 3 we present the literature that treats the price equilibrium problem as a
correlated equilibrium problem.
Suppose that there are J products in a market denoted by 1; :::; J. For j 2 f1; :::; Jg,
let pj , sj and cj denote the price, market share and marginal cost of product j, respectively. Suppose that for each j, sj : (0; 1)J ! (0; 1) is a continuously di¤erentiable
function of p = (p1 ; :::; pJ )0 , where x0 denotes the transposed of vector x. We assume
P
that Jj=1 sj (p) < 1 for any p 2 (0; 1)J , that is, there is an outside product with
strictly positive market share.
Suppose further that the J products are produced by F …rms and each …rm f 2
f1; :::; F g sells a subset Gf of the J products. Let pf be the column-vector of own
prices, that is, of those components pj of p for which j 2 Gf ;1 let p f be the columnvector of the rest of the components of p. Each …rm f determines its own vector of
prices pf such that its pro…t
f
(pf ; p
f)
=
X
(pj
cj )sj (p)
j2Gf
is maximized, given p f . The resulting price vector is a Nash equilibrium.
If a Nash equilibrium exists, then it satis…es that
@
f
(pf ; p
@pf
f)
= 0;
which is equivalent to
sj (p) +
X
(ph
mch )
h2Gf
@sh
(p) = 0:
@pj
(1)
In matrix form this is the same as saying that p solves the system of equations
p=c+
(p) 1 s(p);
1
(2)
In general, we use the notation that boldface characters are column-vectors and if they have subscript f then they refer to the column vector corresponding to the products of …rm f .
1
where the matrix
(p) is block-diagonal and can be written as
2
3
0
1 (p)
6
7
...
(p) = 4
5;
0
F (p)
@sf (p)
for f = 1; :::; F . Equation (2) determines a …xed-point equation,
@p0f
which may be used to show that a unique Nash equilibrium exists.
and
1.1
f (p)
=
Special case: The random coe¢ cient logit
The most commonly used version of the model has the market share
Z
exp ( pj + xj + j )
sj (p) =
dF ( ; ) ;
PJ
exp
(
p
+
x
+
)
RK+1 1 +
h
h
h
h=1
(3)
where xj 2 RK , j 2 R, for j = 1; :::; J, and F is a cumulative distribution function.
The interpretation of ; is that they are random coe¢ cients of pj and xj .
Conjecture 1 If the marginal distribution of
there is a unique Nash equilibrium.
has support included in ( 1; 0), then
Proposition 2 If the support of the marginal distribution of
of positive values, then there may be multiple Nash equilibria.
includes some interval
Allon et al. (2010) show uniqueness of price equilibrium, if the coe¢ cient of price
is non-random (in fact is a product-dependent parameter) and the distribution of is
discrete. The key of their proof is a condition by which they exogenously bound the
prices. Their results are challenging because they also prove as a side result that the
pricing game is supermodular, which is not true for the version of the model for which
both , are non-random (Sándor 2001).
1.2
Special case: The simple logit
In this case in (3) both ,
are non-random and
sj (p) =
exp ( pj + xj + j )
PJ
1 + h=1 exp ( ph + xh +
h)
:
Then < 0 is a su¢ cient condition for the existence of a unique equilibrium (Konovalov
and Sándor 2010). The proof can be based on either Kellogg’s (1976) …xed point result
or the Gale and Nikaido (1965) univalence theorem.
2
1.3
Special case: Single-product …rms
In this case each Gf is a singleton. The most important contribution is Caplin and
Nalebu¤ (1991), who prove existence of equilibrium for a class of models that include
(3) in the case where is non-random and is random with a log-concave density.
Their proof is based on Prékopa-Borel results on the quasi-concavity of probabilities.2
For a logit model with non-random and random with a discrete distribution,
Allon et al. (2011) and Aksoy-Pierson et al. (2013) prove that, if the market share is less
than 1=2 and the prices are bounded by some value, then there is a Nash equilibrium.
Further, if the market share is less than 1=3, then there is a unique Nash equilibrium
of prices. The proof of existence is based on reducing the interval of candidate price
equilibria and establishing the concavity of the pro…t function on the reduced price
interval.
Another line of research had been by establishing supermodularity conditions for
the pricing game and applying Tarski’s …xed point theorem (e.g., Milgrom and Roberts
1990). This has turned out to be too restrictive for the model in (3) because according to
Sándor (2001) in the simple logit model the pricing game cannot be either supermodular
or log-supermodular.
Mizuno (2003) extended the Caplin and Nalebu¤ (1991) results by adding some
weak supermodularity conditions for uniqueness. The key condition used is that the
best response function of any …rm is increasing.
References
[1] G. Allon, A.Federgruen, M. Pierson, Price competition under multinomial logit
demand functions with random coe¢ cients, Working paper, Dartmouth College,
Hanover, (2010)
[2] G. Allon, A.Federgruen, M. Pierson, Price competition under multinomial logit
demand functions with random coe¢ cients, Working paper, Dartmouth College,
Hanover, (2011)
[3] M. Aksoy-Pierson, G. Allon, A.Federgruen, Price competition under mixed multinomial logit demand functions, Management Science, 59 (2013) 1817-1835.
[4] A. Caplin and B. Nalebu¤, Aggregation and imperfect competition: on the existence
of equilibrium, Econometrica 59, 25–59 (1991)
2
Quasi-concavity fails when …rms produce multiple products. For example, Hanson and Martin
(1996) provide an example of a model with simple logit market shares in which the pro…t function of a
3-product …rm is not quasi-concave.
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[5] D. Gale, H. Nikaido, The Jacobian matrix and global univalence of mapping. Math
Ann 159, 81–93 (1965)
[6] W. Hanson, K. Martin, Optimizing multinomial logit pro…t functions. Manage Sci
42, 992–1003 (1996)
[7] R.B. Kellogg Uniqueness in the Schauder …xed point theorem. Proc Am Math Soc
60, 207–210 (1976)
[8] A. Konovalov and Z. Sándor, On price equilibrium with multi-product …rms, Econ
Theory 44, 271–292 (2010)
[9] P. Milgrom and J. Roberts, Rationalizability, Learning, and Equilibrium in Games
with Strategic Complementarities, Econometrica, 58, 1255–77.(1990)
[10] T. Mizuno, On the existence of a unique price equilibrium for models of product
di¤erentiation. Int J Ind. Organ 21, 761–793 (2003)
[11] J. Nash. Noncooperative games. Ann Math 54, 286-295, 1951.
[12] Z. Sándor, Computation, e¢ ciency and endogeneity in discrete choice models,
Labyrint Publication, The Netherlands, (2001)
2
Price equilibrium as vector equilibrium
The price equilibrium problem is a Nash equilibrium problem, which is a special case of
a vector equilibrium problem. Let (D; f ) be a game with n players, where Dj are the
strategy sets of the players, D = D1
Dn and Fj : D ! R are the pro…t functions.
The strategy (x1 ; x2 ; :::; xn ) 2 D Nash equilibrium point if
Fj (xj ; x j )
Fj (yj ; x j ) for any yj 2 Dj :
If f (x; y) = (f1 (x; y) ; :::; fn (x; y)) where
fj (x; y) = Fj (xj ; x j )
Fj (yj ; x j )
for any x = (x1 ; x2 ; :::; xn ) ; y = (y1 ; :::; yn ) 2 D and C = Rn+ , then the Nash equilibrium
problem coincides with the next vector equilibrium problem:
Find x 2 D such that f (x; y) 2 C; 8y 2 D:
In order to establish necessary conditions for the existence of the solution, Ansari and
Flores Fazan (2005) use the so-called recession method; in the proof Gong (2006) employs
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the separation theorem for convex sets; Long, Huang and Teo (2008) the Kakutani–
Fan–Glicksberg …xed point theorem. Lin (2005) introduces several notions of maximal
pseudomonotonicity, which he used for obtaining existence results. Fu and Wang (2013)
examine the strong vector equilibrium problems with domination structures.
Gong (2007) studies symmetric strong vector quasi-equilibrium problems, and Yang
and Pu (2013) analyze the system of strong vector equilibrium problems.
References
[1] Ansari Q.H., Flores-Bazán F. , Recession methods for generalized vector equilibrium problems, J. Math. Anal. Appl. 321, 132-146 (2006).
[2] Fu, J., Wang, S.: Generalized strong vector quasi-equilibrium problem with domination structure. J. Glob. Optim. 55, 839-847 (2013)
[3] Gong, X.H.: Strong vector equilibrium problems. J. Glob. Optim. 36, 339-349
(2006)
[4] Gong, X. H.: Symmetric strong vector quasi-equilibrium problems. Math. Meth.
Oper. Res. 65, 305–314 (2007)
[5] Lin, L.J.: Existence Results for primal and dual generalized vector equilibrium
problems with applications to generalized semi-in…nite programming. J. Glob.
Optim. 33, 579-595 (2005)
[6] Long, X.J., Huang, N.J., Teo, K.L., Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Math. Comp. Modelling 47,
445–451 (2008)
[7] Yang, Z., Pu, Y.J, On existence and essential components for solution set for
system of strong vector quasi-equilibrium problems. J Glob Optim 55, 253–259
(2013)
3
Price equilibrium as correlated equilibrium
Let G denote a …nite noncooperative game, let n be the number of players. Each player
k n has a …nite set of strategies, Sk , with jSk j 2. The utility or payo¤ function of
player k is a function uk . The set
n
Y
S=
Sk
k=1
5
is called the set of all joint strategy pro…les, and N = jSj denote the number of outcomes.
Further, let
n
Y
S k=
Sq :
q=1;q6=k
k
Let u (s) be the payo¤ of player k when the joint strategy s is played, and let uk (dk ; s k )
denote the payo¤ to player k when he chooses strategy dk 2 Sk and the others adhere
to s:
The game G is non-trivial if there exist a player k with s 2 S and dk 2 Sk such that
uk (s) 6= uk (dk ; s k ) :
Correlated equilibrium is a generalization of Nash equilibrium that allows the probabilities to be arbitrarily correlated in the strategy spaces. For the …rst time it was
formulated by Aumann (1974), as follows:
A correlated equilibrium of G is a distribution p on S such that for all players
k and all strategies i; j 2 Sk the following is true:
Conditional on the k-th component of strategy pro…le drawn from p being
i, the expected utility for player k of playing i is not smaller than that of
playing j, i.e.
X
X
ukis pis
ukjs pjs :
(4)
s2S
s2S
k
k
The condition that p is a joint distribution on S implies an N
1-dimensional
simplex, henceforth denoted by , consisting of all probability distributions on joint
strategies. The set of all correlated equilibrium distributions determined by (4) is a
convex polytope, henceforth denoted by C, which is a proper subset of , if the game
is non-trivial. The polytope C is of full dimension if it has dimension N 1; the same
as .
The set I of all joint probability distributions that are independent between players is
de…ned by a system of nonlinear constraints, that is, I is the set of all joint distributions
p on S for which there exists a marginal probability distribution xk on Sk such that
Q
p = nk=1 xk .
The geometric relationship between the Nash equilibrium and the correlated equilibrium is formulated in Nau et al (2003). In any …nite, non-trivial game, the Nash
equilibria are on the boundary of the correlated equilibrium. If the polytope is of full
dimension, the Nash equilibria are on its relative boundary. Speci…cally:
(i) In any …nite, non-trivial bimatrix game, if (u; v) is a Nash equilibrium
strategy, then the joint distribution p, where pij = ui vj (i = 1; m; j = 1; n)
is a correlated equilibrium.
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(ii) In any …nite, non-trivial bimatrix game, if the joint distribution p is a
correlated equilibrium and there exists a marginal distribution (u; v) such
that pij = ui vj (i = 1; m; j = 1; n), then (u; v) is Nash equilibrium strategy.
Example 3 The chicken games has 2 players (think of them as very competitive drivers
speeding from di¤erent streets to an intersection), each with two strategies: S1 = S2 =
fStop; Gog. The utilities, tabulated below, re‡ect the situation (in the format u1 (s); u2 (s),
where the strategies of player 1 are the rows and of player 2 the columns):
Stop Go
Stop 6,6
2,7
7,2
0,0
Go
In a 2 2 game, is a 3-dimensional tetrahedron, I is a 2-dimensional saddle and
C is a 3-dimensional convex polytope (see Figure 1).
In the chicken game the correlated equilibrium is a distribution
p11 p12
p21 p22
p=
that satis…es the following inequalities:
8
>
> 6p11 + 2p12 7p11 + 0p12
>
>
7p21 + 0p22 6p21 + 2p22
>
>
<
6p11 + 2p21 7p11 + 0p21
7p12 + 0p22 6p12 + 2p22
>
>
>
>
p + p12 + p21 + p22 = 1
>
>
: 11
pij 0; i; j = 1; 2
(5)
The following …ve distributions are correlated equilibria in the chicken game
SG =
0 1
0 0
p1 =
; GS =
0 0
1 0
2=4 1=4
1=4 0
; M N ash =
; p2 =
4=9 2=9
2=9 1=9
0 2=5
2=5 1=5
The …rst two are pure Nash equilibria, the third is a mixed strategy Nash equilibrium.
The fourth and …fth are only correlated equilibria. Figure 1 shows the geometry of this
game.
The problem of determining a Nash equilibrium is a combinatorial problem; it implies the identi…cation of a support for each player. None of the known algorithms for
determining the supports of the strategies are known to be e¢ cient (in the sense of
determining the supports in a polynomial number of steps). This problem is studied by
Jiang et al. (2013); Papadimitriu et al. (2008).
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Figure 1: The geometry of Example 3: The polytope is the symplex of correlated
equilibria (the set C); the sadle is the set of distributions independent between players
(the set I); and the three intersection points (SG, GS, MNash) are Nash equilibria.
Recently, in the literature there have appeared a number of generalizations of the
notion of correlated equilibrium (Forgo, 2011). Stein et al. (2011) generalized this
concept for games in which each correlated equilibrium has a continuous utility function.
It is interesting to note that the logic for determining the distribution of correlated
equilibria for these games is similar to the estimation algorithm for the BLP model.
The operator equilibrium problems can help the study of existence of correlated equilibrium points. Domokos and Kolumbán (2002) introduced and studied a class of operator variational inequalities. The importance of these operator variational inequalities
stems not only from the fact that they include scalar and vector variational inequalities
as special cases, but they are also interesting as a problem on its own. Inspired by this
work, Kum and Kim (2005, 2007) extend the problem of operator variational inequalities from the single-valued to the multi-valued case. The operator equilibrium problems
were studied by Kazmi and Raouf (2005), Kum and Kim (2008).
References
[1] R. J. Aumann. Subjectivity and correlation in randomized strategies. Journal
of Mathematical Economics, 1: 67-96, 1974.
[2] A. Domokos, J. Kolumbán, Variational inequalities with operator solutions, J.
Global Optim. 23 (2002) 99-110.
[3] F. Forgo. Generalized correlated equilibrium for two-person games in extensive
form with perfect information. CEJOR 19, 201-213, 2011.
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[4] A. X. Jiang, K. Leyton-Brown. Polynomial-time computation of exact correlated equilibrium in compact games. Games and Economic Behavior, 2013.
[5] K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math.
Anal. Appl. 308 (2005) 554-564.
[6] S. Kum, W.K. Kim, Generalized vector variational and quasi-variational inequalities with operator solutions, J. Global Optim. 32 (2005) 581-595.
[7] S. Kum, W.K. Kim, Applications of generalized variational and quasivariational inequalities with operator solutions in a TVS, J. Optim. Theory
Appl. 133 (2007) 65-75.
[8] S. Kum, W.K. Kim, On generalized operator quasi-equilibrium problems, J.
Math. Anal. Appl. 345 (2008) 559-565.
[9] R. Nau, S. G. Canovas, P. Hansen. On the geometry of Nash equilibria and
correlated equilibria. Int. J. Game Theory 32, 443-453, 2003.
[10] C. H. Papadimitriou, T. Roughgarden. Computing correlated equilibria in
multi-player games. J. ACM 55(3), 14, 2008.
[11] N. D. Stein, P. A. Parrilo, A. Ozdaglar. Correlated equilibria in continuous
games: Characterization and computation. Games and Economic Behavior 71,
436-455, 2011.
[12] N. D. Stein, A. Ozdaglar, P. A. Parrilo. Structure of extreme correlated equilibria: a zero-sum example and its implications. LIDS Technical Report 2929,
January 27, 2011.
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