Market entry situations are modelled, where an entrepreneur has to decide for a collection of markets which market to enter and which not. The entrepreneur can improve his prior information by making use of a group of informants, each of them knowing the situation in one or more markets. For such a market entry situation a related cooperative game is introduced, which can be helpful in dealing with the question of how to share the reward of cooperation. The games arising turn out to be elements of the cone of information market games which were introduced for another economic context. This implies that the cooperative solutions of these games have interesting properties. Extra attention is paid to the subcone of information market games arising from market entry situations where for each market only one informant knows the state of the market.cooperative games;information market entry;information market games

Brânzei, R.

Tijs, S.H.

Timmer, J.B.

English

Research Papers in Economics

Center
for
Economic Research
No. 2000-44
CONES OF GAMES ARISING FROM MARKET
ENTRY PROBLEMS
By Rodica Brânzei, Stef Tijs and Judith Timmer
April 2000
ISSN 0924-7815Cones of Games Arising
from Market Entry Problems
Rodica Brˆ anzei1, Stef Tijs2 and Judith Timmer2;3
Abstract
Market entry situations are modelled, where an entrepreneur has to decide for a collection of
markets whichmarkettoenterandwhichnot. The entrepreneurcan improvehispriorinformation
bymakinguseofagroupofinformants,eachofthemknowingthesituationinoneormoremarkets.
For such a market entry situation a related cooperative game is introduced, which can be helpful
in dealing with the question of how to share the reward of cooperation. The games arising turn
out to be elements of the cone of information market games which were introduced for another
economic context. This implies that the cooperative solutions of these games have interesting
properties. Extraattentionispaidtothesubconeofinformationmarketgames arisingfrommarket
entry situationswhere for each market only one informant knows the state of the market.
Journal of Economic Literature ClassiﬁcationNumber: C71, D89.
Keywords: cooperative games, information, market entry, information market games.
1 Faculty of Computer Science, "Al.I. Cuza" University, 11, Carol I Bd., 6600 Ias ¸i, Romania. This author
acknowledges ﬁnancial support from CentER, Tilburg University.
2 CentER and Department of Econometrics, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The
Netherlands.
3 Correspondingauthor. E-mail address: j.b.timmer@kub.nl. This authoracknowledges ﬁnancial support
from the Netherlands Organization for scientiﬁc research (NWO) through project 613–304–059.
11 Introduction
Cooperation in all kinds of economic situations leads not only to an optimization problem for the
agents involved but also to a problem of sharing the beneﬁts or costs. Often the second problem is
tackled by constructinga suitablecooperative game and usingthe solutionsdeveloped in cooperative
game theory, or byﬁnding newsolutionsappealingfor the class of problems at hand. There are many
economic situationsleading to interesting cones of games. Let us mention some.
(i) In Shapley and Shubik(1969) market situationsare considered leading to market games which
turn out to exhaust the cone of totallybalanced games.
(ii) Flowsituations(cf. KalaiandZemel(1982))andlinearproductionsituations(cf. Owen(1975))
lead to the cone of non–negative totallybalanced games.
(iii) Problems of cooperation to make phoning in planes possible (cf. Nouweland (1996)) lead to
the cone generated by unanimity games with two veto–players.
In this paper we consider in Section 2 simple market entry problems of an entrepreneur who has
to decide for each member of a given collection of markets either to enter that market or not. He can
cooperate with a network of informants who know the quality of certain markets, or in our simple
case if the market is good or bad, meaning whether it is worthwhileto enter a market or not.
In Section 3 thesemarket entry situationslead tothe cone of informationmarket games (cf. Muto
et al. (1989)) for which many interesting properties are known. An interesting subcone, where the
Shapley value, the -value and the nucleolus coincide, is obtained by considering markets where for
each market only one agent is informed.
2 Market entry situations
In this paper a market entry situationis a tuple
<f0g;M;N;f( m;r m;1− m;−l m)jm2Mg;K:M−! N>
Here agent 0 is the entrepreneur interested in a non–empty ﬁnite set M of markets, and N =
f1;2;:::;ngis the set of possible informants, which the entrepreneur can consult before making a
decision to enter a market m or not. If the entrepreneur considers market m 2 M withoutconsulting
informants,then he expects withprobabilitym 2 (0;1)a goodmarket withreward rm 2 [0;1)and
with probability1− m a bad market with loss lm. The knowledge of the informants in N about the
markets is described by the correspondence (multi–function) K,w h e r eK ( m )is the non–empty set
of agents in N who know the true state of the market m.
To exclude trivialitiesandto make our mathematical life easier, we supposethroughoutthispaper
the extra conditionsME.1 and ME.2 to be introduced now.
2ME.1 (Each informant knows something) For each i 2 N; there is a market m such that i 2 K(m),
i.e., K : M −! N is surjective.
ME.2 (Information is valuable for the entrepreneur) For each m 2 M the prior expectation mrm −
(1 − m)lm is non–positive.
Problems for the entrepreneur are: which informants to consult, which markets to enter and how
to share the extra earnings with the informants?
Because of ME.1 and ME.2 it is reasonable to assume that player 0 decides to cooperate with all
informants and to tackle his reward sharing problem by looking at the following cooperative game
<N [f 0 g ;v>:
For each coalitionS  N of players the worthv(S)=0and the worth v(S [f0g)is equal to the
expected reward of the markets, given the fact that the knowledge of the informants in S can avoid
agent 0 to enter a bad market m 2 M(S) where M(S)=f m2Mj K( m )\S6 =;g.I nf o r m u l a
v ( S[f 0 g )=
X
m2M(S)
mrm:
This worth can be reached by collecting from the informants in S information about the states of the
markets M(S) and entering m 2 M(S) if the market is good, by not entering if the market’s state is
bad, and also not entering markets outside of M(S).
In Section 3 we look more closely to games arising from market entry situations. But ﬁrst let us
give an example.
Example 1. Consider the market entry situation
<f0g;M;N;f( m;r m;1− m;−l m)jm2Mg;K:M−! N>
withM = fm1;m 2g;N=f1;2;3g; m 1 = m 2 =
1
2
,r m 1 =2 0 ;l m 1 =4 0 ;rm2 = 100;l m 2 = 110;
and K(m1)=f 1 ; 2 g ;K ( m 2 )=f 2 ; 3 g :Then ME.1 and ME.2 hold. The corresponding market
entry game <f0;1;2;3g;v>is given by v(f0g)=0 ;v ( S )=0if S f 1 ;2 ;3 g ;v ( f 0 ;1 g )=1 0 ;
v ( f 0 ;3 g )=50,v(f0;1;3g)=60and v(S)=6 0for all S with f0;2gS:
Now we want to describe a relation between market entry problems and information collecting
(IC) situations, introduced in Brˆ anzei, Tijs and Timmer (2000). Recall that an IC-situation is given
by a tuple
<f0g;N;(Ω;);fIi j i 2 Ng;A;r:ΩA−! IR >:
Here 0 is the action taker with a ﬁnite action set A. The reward corresponding to action a 2 A is
r(!;a), so this reward depends on the state ! 2 Ω which appears with probability (!) 2 (0;1). Ω
is a ﬁnite state space and
P
!2Ω (!)=1 :The action taker can, before choosing an action, collect
information from the informants in N = f1;2;:::;ng. The informationthat i 2 N has about the state
is described by the information partition Ii; which is a partition of Ω.
3ThecorrespondingIC-game<N [f 0 g ;w>isdescribedbyw(S)=0foreachS  N,w(f0g)=
0 ;and
w(S [f 0 g )=
X
I2IS
max
a2A
X
!2I
r(!;a)(!)−max
a2A
X
!2Ω
r(!;a)(!)
where IS consists of non–empty intersections of the form
T
i2S Ii with Ii 2I ifor each i 2 S: Such
a game is monotonic and player 0 is a veto–player (cf. Arin and Feltkamp (1997)).
A market entry situation can be related as follows to an IC-situation as above. In both situations
f0g and N have the same meaning. Let Ω=f 0 ; 1 g M ,w h e r e!=( ! m ) m 2 Mcorresponds to the
markets’ state. We denote by G(!)=f m2Mj! m=1 gthe set of markets which are good when
! is the true state; then the markets in M nG(!) are bad. The probabilitythat the true state is ! 2 Ω,
(!); is given by
(!)=
Y
m 2 G ( ! )
 m
Y
m 2 Mn G ( ! )
(1 − m):
For each i 2 N the correspondinginformation partition Ii has parts (atoms) of the form
I(x)=f !2Ωj! m=x mfor all m 2 M(fig)g
where x 2f 0 ;1 g M( f i g ) . Such a part I(x) corresponds to the situation where the informant i knows
that the state of market m 2 M(fig) is xm: The action set A is f0;1gM,w h e r ea=( a m ) m 2 M
correspondsto the strategy: enter the market m if am =1and don’t enter if am =0 :Finally, r(!;a)
is equal to
(2:1)
X
m2M;am=1
(rm!m − lm(1− !m)):
Note that for S  N, IS is a partition of Ω with parts of the form
I(x)=f !2Ωj! m=x mfor all m 2 M(S)g
where x 2f 0 ;1 g M( S ). Because of conditionME.2, it is for player 0 optimal to choose the following
strategy when workingtogether with S  N:
(i) do not enter markets outsideof M(S);
(ii) do not enter markets m 2 M(S) if ! 2 I(x) 2I Sand if xm =0 ;
(iii) enter a market m 2 M(S) if ! 2 I(x) 2I Sand if xm =1 :
The expected reward of such a strategy is
P
m2M(S)mrm, and this is also the worth v(S [f 0 g )in
the market entry game. Thus we have the followingproposition.
Proposition 1. For each market entry situation and corresponding IC–situation the related games
are the same.
4Example 2. The market entry situation of Example 1, where M = fm1;m 2g; m 1 = m 2 =1 = 2
corresponds to the followingIC-situation
<f0g;N;(Ω;);fIi j i 2 Ng;A;r:ΩA−! IR >:
Here N =f1;2;3g; Ω=f(0;0);(1;0);(0;1);(1;1)g;!=(!1;! 2)2Ωdenotes the market state with
!12f0;1gthestateofm1, !22f0;1gthestateofm2;and where0 standsforbad market and 1 stands
for good market, (!)=1=4for all !2Ω, I1 =ff(0;0);(0;1)g;f(1;0);(1;1)gg; I2 is the discrete
partition of Ω in singletons, I3 = ff(0;0);(1;0)g;f(0;1);(1;1)gg, A = f(a1;a 2) j a i 2f 0 ;1 g ;
i=1 ;2 g ,a n dr( !;a)is as in formula (2.1).
3 Cones of games and market entry situations
Let N = f1;2;:::;ng and T  N: Let <N [f 0 g ;v>be the simple game such that for S  N
the worth v(S)=0a n ds u c ht h a tv ( S[f 0 g )=1if T \ S 6= ;; and v(S [f 0 g )=0otherwise.
Following Muto et al. (1989), we denote this game by u
T;0: Furthermore we denote by IG the
convex cone generated by fu
T;0 j T  Ng. A subcone of this cone is IGC, the cone generated by
fu
T;0 j T N;jTj =1 g ;which is equal to the cone generated by fuf0;ig j i 2 Ng; where uf0;ig is
the unanimitygame with veto player set f0;ig;i.e., uf0;ig(S) 2f 0 ;1 gand uf0;ig(S)=1if and only
if f0;igS . Propositions 2 and 3 below show that these cones IG and IGC are interesting for our
market entry situations.
Proposition 2.
(i) Each market entry game is an element of IG.
(ii) Each v 2 IG is a game correspondingto a market entry situation.
Proof. (i) For a market entry situation as in Section 2, the corresponding market entry game
<N [f 0 g ;v>is given by v(S)=0if 0 = 2 S, v(f0g)=0and
v(S [f 0 g )=
X
m 2 M( S )
 mr m;where M(S)=f m2MjK( m )\S6 =;g:
Obviously,this game can be rewritten as v =
P
m2M mrmu
K(m);0: So v 2 IG:
(ii)Supposethatv 2 IG: Thenthereexistsanumbertinf0;1;2;:::gsuchthatv =
Pt
k=1 cku
Tk;0;
with ck > 0 and Tk  N for all k 2f 1 ; 2 ;:::;tg: (We allow for an empty sum.) We consider the
market entry problem
<f0g;M;N;f( m;r m;1− m;−l m)jm2Mg;K:M −! N>
with M = fm0;m 1;:::;mtg; m=1 = 2for all m 2 M; rm0 = `m0 =0 ;r m k =2 c k ;l m k =3 c k,
K ( m 0 )=Nand K(mk)=T kfor all k 2f 1 ; 2 ;:::;tg: Then the conditions ME.1 and ME.2 are
5satisﬁedand for the correspondingmarket entry game v0 we have v0 =0u 
N;0+
Pt
k=1 cku
Tk;0 = v:
Example 3. For the game v in Example 1 we have
v =1 0u 
f 1 ; 2 g ; 0+5 0u 
f 2 ; 3 g ; 0:
KnowingnowthatthesetofmarketentrygamescoincideswiththeconeIGwecanuseinformation
intheliterature,e.g., Mutoetal. (1989)andMutoetal. (1988). Afewoftheseresultsaresummarized
in the theorem below.
Recall the following properties. A game <N [f 0 g ;v>
(M) is monotonic if v(S)  v(T) for all S  T  N [f 0 g :
(V0) has the 0–veto property if v(S)=0if 0 = 2 S and v(f0g)=0 .
(U) has the union property if v(N [f 0 g )−v ( S) 
P
i 2 N[f0gnSMi(v) for all S  N [f 0 gwith
02S,w h e r eM i( v ): =v ( N[f0g)−v(N[f0gnfig):
Theorem 1. Let <N [f 0 g ;v>be a market entry game. Then
(i) v has the properties(M), (V0) and (U).
(ii) The core C(v) of <N [f 0 g ;v>is given by
C(v)=
(
x2IR N [f0g


 

n X
i=0
xi=v(N [f 0 g ) ;0  x i M i( v)for all i2N
)
:
(iii) The -value (Tijs (1981)) is equal to the nucleolus (Schmeidler (1969)) and is given by
 
v(N [f 0 g )−
1
2
n X
i =1
Mi(v);
1
2
M1(v);:::;
1
2
Mn(v)
!
:
Let us now look at the subclass of games in IG with the property
(CN) v(S [f i g )−v( S)v( N[f 0 g )−v( N[f 0 gnfig)for all S  N [f 0 gnfig.
Then we have
Theorem 2. Let <N [f 0 g ;v>be an element of IG. Then the following assertionsare equivalent
(i) v satisﬁes property(CN).
(ii) v 2 IGC and v =
Pn
i=1 Mi(v)u
fig;0:
(iii) v is convex i.e. v(S [f i g )−v( S)v( T[f i g )−v( T)for all S  T  N [f 0 gnfig:
6Proof. ((i)=)(ii)). Suppose that v has the property (CN).T a k eSN[f 0 gwith 0 2 S: Suppose
that N [f 0 gnS=fi 1;i 2;:::;i kg:Then (CN) implies:
v(N [f 0 g )−v( S)=
k X
r=1
(v(S [f i 1;:::;i rg)−v(S[f i 1;:::;i r−1g))

k X
r=1
(v(N [f 0 g )−v( N[f 0 gnfi rg))=
X
i2N[f0gnS
Mi(v):
Using also the property (U) for v we obtain:
v(N [f 0 g )−v( S)=
X
i 2 N[f0gnS
Mi(v) for 0 2 S  N [f 0 g :
Then
v(S)=( v ( N[f 0 g )−v( f 0 g ))− (v(N [f 0 g )−v( S))
=
X
i2N
Mi(v)−
X
i2N[f0gnS
Mi(v)=
X
i 2 S nf0g
Mi(v) for S 3 0:
Since v(T)=0for all T with 0 = 2 T, we can conclude that
v =
n X
i=1
Mi(v)u
fig;0:
So, v 2 IGC.
((ii)=)(iii)). It is well–known that uf0;ig = u
fig;0 is a convex game for each i 2 N: Then v is
also convex.
((iii)=)(i)). Obviously,a convex game satisﬁes (CN).
From this theorem and the proof of Proposition2, (ii), we can conclude that the next proposition
holds.
Proposition3. Games inIGC correspondtomarketentry situations,where each marketonly hasone
informant.
Furthermore, it follows from Theorem 4.3 in Nouweland et al. (1996) that for games in IGC
the Shapley value (cf. Shapley (1953)), the -value and the nucleolus coincide and are equal to
(
Pn
i=1 ci=2;c 1=2;:::;c n=2) if v =
Pn
i=1 ciuf0;ig 2 IGC:
References
Arin, J., and V. Feltkamp (1997) The nucleolus and kernel of veto–rich transferable utility games.
International Journal of Game Theory, 26, 61–73.
Brˆ anzei, R., S. Tijs, and J. Timmer (2000) Collecting information to improve decision–making.
CentER DP no. 2000–26, Tilburg University, Tilburg, The Netherlands.
7Kalai, E. and E. Zemel (1982)Totallybalancedgames andgamesof ﬂow. Mathematics of Operations
Research, 7, 476–478.
Muto, S., M. Nakayama, J. Potters, and S. Tijs (1988), On big boss games. The Economic Studies
Quarterly 39, 303–321.
Muto, S., J. Potters, and S. Tijs (1989) Information market games. International Journal of Game
Theory, 18, 209–226.
Nouweland, A., van den, P. Borm, W. van Golstein Brouwers, R. Groot Bruinderink, and S. Tijs
(1996) A game theoretic approach to problems in telecommunication. Management Science,
42, 294–303.
Owen, G. (1975) On the core of linear productiongames. Mathematical Programming, 9, 358–370.
Schmeidler, D. (1969), The nucleolus of a characteristic function game. SIAM Journal on Applied
Mathematics, 17, 1163–1170.
Shapley, L.S. (1953), A value for n-person games. Annals of Mathematical Studies, 28, 307–317.
Shapley, L.S., and M. Shubik (1969) On market games. Journal of Economic Theory, 1, 9–25.
Tijs, S.H. (1981), Bounds for the core and the -value. In: O. Moeschlin and D. Pallaschke eds.,
Game Theory and Mathematical Economics, Amsterdam, North Holland, 123–132.
8

Cones of Games arising from Market Entry Problems

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics