This paper considers a two agent model of trade with multiple priors. First, we characterize the existence of an agreeable bet on some event in terms of the set of priors. It is then shown that the existence of an agreeable bet on some event is a strictly stronger condition than the existence of an agreeable trade, whereas the two conditions are equivalent in the standard Bayesian framework. Secondly, we show that the two conditions are equivalent when the set of priors is the core of a convex capacity.multiple priors, convex capacity, agreeing and disagreeing, Choquet integralm

Atsushi Kajii

Takashi Ui

English

Research Papers in Economics

 
 
 
 
 
 
 
Discussion Paper No.581 
 
Agreeable Bets with Multiple Priors 
 
 
by 
Atsushi Kajii 
and 
Takashi Ui 
 
February 2004 
KYOTO UNIVERSITY 
KYOTO, JAPAN 
 
http://www.kier.kyoto-u.ac.jp/index.html 
KYOTO INSTITUTE 
OF 
ECONOMIC RESEARCH 
KIER DISCUSSION PAPER SERIES Agreeable Bets with Multiple Priors∗
Atsushi Kajii
Institute of Economic Research
Kyoto University
kajii@kier.kyoto-u.ac.jp
Takashi Ui
Faculty of Economics
Yokohama National University
oui@ynu.ac.jp
February 2004
Abstract
This paper considers a two agent model of trade with multiple priors, and char-
acterizes the existence of an agreeable bet on some event in terms of the set of priors.
In the model, the existence of an agreeable bet on some event is a strictly stronger
condition than the existence of an agreeable trade, whereas the two conditions are
equivalent in the standard Bayesian framework. The main result shows that the two
conditions are equivalent when the set of priors is the core of a convex capacity.
JEL classiﬁcation numbers: C70, D81.
Key Words: multiple priors; convex capacity; agreeing and disagreeing; Choquet
integral.
∗Kajii acknowledges ﬁnancial support by MEXT, Grant-in-Aid for 21st Century COE Program. Ui
acknowledges ﬁnancial support by MEXT, Grant-in-Aid for Scientiﬁc Research.
11 Introduction and Summary
Imagine two agents, each of whom has a prior distribution over a ﬁnite state space Ω.I f
the two agents do not share priors, there exists an agreeable trade between them: by a
t r a d ew em e a naf u n c t i o nf : Ω → R, and it is said to be agreeable if the expected value
of f for agent 1 is positive and that of −f for agent 2 is positive as well. The converse
is also true: if there exists an agreeable trade, the agents do not share priors. A trade
f is called a bet on event E ⊆ Ω if f is constant over E and Ω\E;i ff is larger on E
than on Ω\E, it can be interpreted that agent 1 wins when E occurs, and agent 2 wins
when E does not occur. It is straightforward to show that if there is no agreeable bet
on any event, the priors of the two agents must coincide. So disagreement of priors, the
existence of an agreeable trade, and the existence of an agreeable bet are all equivalent
conditions in this framework.
Now suppose that two agents have multiple priors over the state space, and they use
the maximin rule a la Gilboa and Schmeidler (1989) to evaluate a trade. In this context,
a trade is agreeable if the minimum expected value of f, where the minimum is taken
over the set of priors, is positive for agent 1, and that of −f is positive for agent 2 as
well. Billot et al. (2000) has shown that there exists an agreeable trade if and only if
there exists a prior which belongs to the set of priors for each agent, i.e., the agents do
not share any priors.
In this paper we show that when the set of priors of each agent is given as the core of a
convex (supermodular) capacity, there exists an agreeable bet on some event if and only
if there exists a prior which belongs to the set of priors for each agent (Proposition 4 in
Section 3). So for the case of convex capacity, disagreement of priors, the existence of an
agreeable trade, and the existence of an agreeable bet are equivalent, which is the same
for the case of single prior. To show the above result, we give a necessary and suﬃcient
condition for the existence of an agreeable bet for a more general class of multiple priors
model, where the set of priors is not necessarily the core of a convex capacity (Lemma 2
in Section 2).
We also show, by an example, that the equivalence of the three conditions fails in
general for multiple priors models. Therefore, the existence of an agreeable bet is a
strictly stronger condition than the existence of an agreeable trade for general multiple
priors models. Unlike in the standard Bayesian framework of single prior, these two
concepts must be distinguished in the multiple priors framework, unless a convex capacity
is assumed.
22 Characterization of Agreeable Bets on Events
Let P1,P 2 ⊆ ∆(Ω) be non empty closed sets,1 which will be referred to as the sets of
priors for agent 1 and agent 2, respectively. We call a function f : Ω → R a trade.A
bet on event E ⊆ Ω is a function f which is constant on both E and Ω\E,s oab e ti sa
special type of trade. Agents evaluate a trade f by the minimum of the expected gain,
as axiomatized by Gilboa and Schmeidler (1989). We interpret that f (ω)i sat r a n s f e r
agent 1 receives from agent 2 when the state is ω. Thus, we say that a trade f is an
agreeable trade if minp∈P1
P
ω∈Ω p(ω)f(ω) > 0a n dm i n p∈P2
P
ω∈Ω p(ω)(−f(ω)) > 0. We
say that there is an agreeable bet on event E if there is an agreeable trade which is a bet
on E.
Let us begin with characterizing the existence of an agreeable trade:2
Lemma 1 Suppose that P1 and P2 are closed and convex. Then, there exists an agreeable
trade if and only if P1 ∩ P2 = ∅.
Proof.S i n c eP1 and P2 are compact and convex, by the separation theorem, P1 ∩P2 = ∅
if and only if there exists f : Ω → R and c ∈ R such that minp∈P1
P
ω∈Ω p(ω)f(ω) >c>
maxp∈P2
P
ω∈Ω p(ω)f(ω). By letting g = f − c, it is clear that the latter statement is
equivalent to the condition that there exists g : Ω → R such that minp∈P1
P
ω∈Ω p(ω)g(ω) >
0a n dm i n p∈P2
P
ω∈Ω p(ω)(−g(ω)) > 0.
Let Pi(E)=m i n p∈Pi p(E)a n dPi(E)=m a x p∈Pi p(E) be the minimum and maximum
probabilities of E ⊆ Ω for agent i =1 ,2. The necessary and suﬃcient condition for the
existence of an agreeable bet on E ⊂ Ω is the following.
Lemma 2 There exists an agreeable bet on E ⊂ Ω if and only if
P1(E) > P2(E) or P2(E) > P1(E).
Proof. Suppose that f is an agreeable bet on E such that f(ω)=a if ω ∈ E and f(ω)=b
otherwise where a,b ∈ R. Note if a = b then f cannot be an agreeable bet. Thus, either
a − b>0o ra − b<0i st r u e .I fa − b>0, we have
min
p∈P1
X
ω∈Ω
p(ω)f(ω)=P1(E)(a − b)+b>0,
min
p∈P2
X
ω∈Ω
p(ω)(−f(ω)) = P2(E)(b − a) − b>0,
1The set ∆(Ω) denotes the collection of all probability distributions over Ω.
2The result is the special case of those of Billot et al. (2000), who considered multiple risk averse agents,
and Kajii and Ui (2004), who considered multiple risk neutral agents with asymmetric information.
3and thus
P1(E) > −b/(a − b) > P2(E).
Similarly, if a − b<0, we have P2(E) > P1(E).
Conversely, suppose that P1(E) > P2(E)o rP2(E) > P1(E). If P2(E) > P1(E),
let f : Ω → R be such that f(ω)=c − 1i fω ∈ E and f(ω)=c otherwise where
P2(E) >c>P1(E). Then,
min
p∈P1
X
ω∈Ω
p(ω)f(ω)=c − P1(E) > 0,
min
p∈P2
X
ω∈Ω
p(ω)(−f(ω)) = P2(E) − c>0,
implying that f is an agreeable bet on E.I fP1(E) > P2(E), let f : Ω → R be such that
f(ω)=1− c if ω ∈ E and f(ω)=−c otherwise where P1(E) >c>P2(E). A similar
calculation shows that f is an agreeable bet.
The conditions in the results above are not equivalent: we provide an example where
there is no agreeable bet on any event whereas the agents do not share any priors, i.e.,
P1∩P2 = ∅.L e tΩ = {1,2,3}, P1 = {p ∈ ∆(Ω):( p(1),p(2),p(3)) = t(0.4,0.4,0.2)+(1−
t)(0.2,0.2,0.6), 0 ≤ t ≤ 1},a n dP2 = {p ∈ ∆(Ω):( p(1),p(2),p(3)) = (0.3,0.25,0.45)}.
Since P1 and P2 are closed convex sets and P1 ∩ P2 = ∅, there exists an agreeable trade
by Lemma 1. However, for any event E ⊂ Ω, P1(E) <P 2(E)=P2(E) < P1(E). Thus,
by Lemma 2, there is no agreeable bet on any event.
3 Case of Convex Capacity
A set function v :2 Ω → R with v(∅)=0i sc a l l e dacapacity if it is monotone and
normalized: that is, v(E) ≥ v(F)i fE ⊇ F and v(Ω) = 1. A capacity v is said to be
supermodular (or convex)i fv(E)+v(F) ≤ v(E ∩ F)+v(E ∪ F)f o rE,F ⊆ Ω,a n d
it is said to be submodular if −v is supermodular. For supermodular and submodular
capacities, the following separation theorem due to Frank (1982) is known.3
3This is a natural extension of the separation theorem for convex functions, taking into account that
the Choquet integral of x ∈ R
Ω with respect to a submodular capacity is a convex function of x ∈ R
Ω.
For a proof, see for instance Murota (2003).
4Theorem 3 Let µ :2 Ω → R be supermodular and ρ :2 Ω → R be submodular. If
ρ(E) ≥ µ(E) for all E ⊆ Ω, then there exists q : Ω → R such that
ρ(E) ≥ q(E) ≥ µ(E) for all E ⊆ Ω
where q(E)=
P
ω∈E q(ω).
T h ec o r eo fac a p a c i t yv is deﬁned as:
Core(v)={q ∈ ∆(Ω):q(E) ≥ v(E) for all E ⊆ Ω}
where q(E)=
P
ω∈E q(ω). The core is a closed convex set. It is known that if v
is supermodular (convex), then Core(v) 6= ∅. The following is the main result of this
paper.
Proposition 4 Suppose that there exists a supermodular (convex) capacity vi such that
Pi =C o r e ( vi) for i =1 ,2. Then, there exists an agreeable bet on some event if and only
if P1 ∩ P2 = ∅.
The “only if” part is an immediate consequence of Lemma 1. The “if” part is an
immediate consequence of Lemma 2 and the following lemma.
Lemma 5 Suppose that there exists a supermodular (convex) capacity vi such that Pi =
Core(vi) for i =1 ,2.I fP1 ∩ P2 = ∅, then there exists E ⊂ Ω such that P1(E) > P2(E).
Proof.D e ﬁne v0
2 :2 Ω → R by the rule v0
2(E)=1−v2(Ω\E)f o rE ⊆ Ω.B yc o n s t r u c t i o n ,
v0
2 is submodular. We shall show that there exists E ⊂ Ω such that v0
2(E) <v 1(E).
Seeking a contradiction, suppose that v0
2(E) ≥ v1(E) for all E ⊆ Ω. By Theorem 3,
there exists q : Ω → R such that
v0
2(E) ≥ q(E) ≥ v1(E) for all E ⊆ Ω.
Since v0
2(Ω)=v1(Ω)=1a n dv0
2(∅)=v1(∅)=0 ,w eh a v eq ∈ ∆(Ω). Thus, q ∈ Core(v1).
In addition,
q(E)=1− q(Ω\E) ≥ 1 − v0
2(Ω\E)=v2(E) for all E ⊆ Ω.
Thus, q ∈ Core(v2), which contradicts to the assumption Core(v1)∩Core(v2)=∅.T h e r e -
fore, there exists E ⊂ Ω such that v0
2(E) <v 1(E). Note that v1(E)=m i n p∈Core(v1) p(E)
and v0
2(E)=1− v2(Ω\E)=1− minp∈Core(v2) p(Ω\E)=m a x p∈Core(v2) p(E). Thus,
P1(E)= m i n
p∈Core(v1)
p(E)=v1(E) >v 0
2(E)= m a x
p∈Core(v2)
p(E)=P2(E),
which completes the proof.
5Remark 1 If a supermodular capacity v is additive in the sense that v(E)+v(F)=
v(E ∩ F)f o rE ∩ F = ∅,t h e nv naturally deﬁnes a probability distribution over Ω,a n d
the core of v is a singleton {v}. In this case, Proposition 4 covers the known result for
the standard Bayesian framework.
Remark 2 If v is a supermodular capacity, then
Z
Ω
fdv =m i n
p∈Core(v)
X
ω∈Ω
p(ω)f(ω)
where the left hand side is the Choquet integral of f with respect to v. The decision
rule based upon the Choquet integral is axiomatized by Schmeidler (1989). Thus, the
characterization in Proposition 4 also applies to the Choquet integral model of trade
with convex capacity.
Remark 3 Lemma 5 can be restated in the following way: if vi :2 Ω → R is supermod-
ular for i =1 ,2a n dC o r e ( v1) ∩ Core(v2)=∅, then there exists f : Ω → {0,1} such that
minp∈Core(v1) p·f>maxp∈Core(v2) p·f. Thus, Lemma 5 is the separation theorem for the
core of supermodular capacities. Murota (1998, Theorem 3.6) obtained the separation
theorem for the core of discrete supermodular set functions: if vi :2 Ω → Z is supermod-
ular for i =1 ,2a n dC o r e ( v1) ∩ Core(v2) ∩ Z = ∅, then there exists f : Ω → {0,1} such
that minp∈Core(v1)∩Z p · f>maxp∈Core(v2)∩Z p · f + 1. The proof of Lemma 5 adopts the
key elements of Murota’s proof.
References
Billot, A., Chateauneuf, A., Gilboa, I., Tallon, J.- M., 2000. Sharing beliefs: between
agreeing and disagreeing. Econometrica 68, 685—694.
Frank, A., 1982. An algorithm for submodular functions on graphs. Ann. Discrete Math-
ematics 16, 97—120.
Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with nonunique prior. J. Math.
Econ. 18, 141—153.
Kajii, A., Ui, T., 2004. Trade with heterogeneous multiple priors. KIER Working Paper
582, Kyoto University.
Murota, K., 1998. Discrete convex analysis. Math. Programming. 83, 313—371.
Murota, K., 2003. Discrete Convex Analysis. SIAM.
Schmeidler, D., 1989. Subjective probability and expected utility without additivity.
Econometrica 57, 571—587.
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