Appendix A: Bilateral Trading in Networks (Condorelli, Galeotti and Renou)
(FOR ONLINE PUBLICATION ONLY)
A.1
Existence of a Regular Equilibrium.
Notations. Throughout, for a given profile of Markov regular strategies, we write Vi,tL (µt )
(resp., Vi,tH (µt )) for the continuation payo↵ of player i at period t, when his valuation is
vL (resp., vH ), the profile of beliefs is µt 2 [0, 1]n , and player i does not own the good at
L
H
period t. We denote by Vi,t
(µt ) (resp., Vi,t
(µt )) a set of continuation payo↵s. Similarly, we
L
H
write Wi,t
(µt ) (resp., Wi,t
(µt )) for the continuation payo↵ of player i at period t, when his
valuation is vL (resp., vH ), the profile of beliefs is µt 2 [0, 1]n , and player i owns the good
L
H
at period t. We denote by Wi,t
(µt ) (resp., Wi,t
(µt )) a set of continuation payo↵s. We write
µ
i,t
for the vector (µj,t )j6=i . For any set A, we denote co A the closed convex hull of A. We
write [a, b] for an interval; if a
b, this is the empty set.
As explained in the text, the proof consists in first showing that every trading network
game with a finite horizon has a regular equilibrium. Then, using the fact bargaining ends in
finite time in all regular equilibria (see Proposition 5), we complete the proof by an argument
analogous to that of Chatterjee and Samuelson (1988). For completeness, we formally state
that every trading network game with a finite horizon has a regular equilibrium.
Proposition A.1. For each T < 1, there exists a regular equilibrium.
A.2
Proof of Proposition A.1
The proof is long and divided into a number of simpler claims to ease its reading. The proof is
by induction on the number of periods remaining. We first characterize the set of equilibrium
L
H
payo↵s when there is only one period remaining, i.e., we characterize Vi,T
(µT ), Vi,T
(µT ),
1
L
H
Wi,T
(µT ), and Wi,T
(µT ) for each belief profile µT , for each i. These sets of equilibrium payo↵s
L
are sustained with Markov regular strategies. Moreover, we show that the sets Vi,T
(µT ) and
H
Vi,T
(µT ) are non-empty, convex, compact valued, and upper hemi-continuous in µT , while
L
H
the sets Wi,T
(µT ) and Wi,T
(µT ) are non-empty, single-valued and upper semi-continuous in
µT (when viewed as functions).
For t < T , we next hypothesize that there are optimal Markov regular strategies and corL
H
L
H
responding equilibrium payo↵ sets Vi,t+1
(µt+1 ), Vi,t+1
(µt+1 ), Wi,t+1
(µt+1 ), and Wi,t+1
(µt+1 ),
which satisfy the properties described above.
We then construct period t-optimal Markov strategies and corresponding subsets of equiL
H
L
H
L
librium payo↵s Vi,t
(µt ), Vi,t
(µt ), Wi,t
(µt ), and Wi,t
(µt ), and prove that the sets Vi,t
(µt ) and
H
Vi,t
(µt ) are non-empty, convex, compact valued, and upper hemi-continuous in µt , while
L
H
the set Wi,T
(µt ) and Wi,t
(µt ) are non-empty, single-valued and upper semi-continuous in µt
(when viewed as a function).
We start with the following claim.
H
Claim 1. Wi,t
(µt ) = {vH } for all µt , for all t.
Proof. This follows directly from Proposition 1.
⌅
Induction Initialization.
Claim 2. Let iT be the owner of the good at period T and µT the belief profile. There
exist optimal Markov regular strategies corresponding to the following sets of payo↵s: For
H
L
all µT , Vi,T
(µT ) = Vi,T
(µT ) = {0} for all i 6= iT , and WiLT ,T (µT ) = {vL }.
Proof. From the definition of the game, no player i 6= iT has an incentive to accept a
strictly positive price, since consumption takes place in period T + 1; no such a period exists
since the game terminates at T . Consequently, player iT is strictly better o↵ consuming the
⌅
good, resulting in a payo↵ of vL or vH .
We have the following properties: the map µT 7! WiLT ,T (µT ) is single-valued and upper
2
H
semi-continuous (when viewed as a function). The correspondence µT 7! Vi,T
(µT ) is non-
empty, convex, compact valued, and upper hemi-continuous.
Induction Hypothesis. There exist optimal Markov regular strategies and associated
sets of payo↵s such that the following is true:
1. The correspondence µt+1 7! WiLt+1 ,t+1 (µt+1 ) is single-valued and upper semi-continuous
(when viewed as a function).
H
2. The correspondence µt+1 7! Vi,t+1
(µt+1 ) is non-empty, convex, compact valued, and
upper hemi-continuous.
Induction Step.
We construct regular Markov strategies, prove that they are optimal and show that the
corresponding sets of equilibrium payo↵s satisfy the induction hypothesis. The proof is then
completed by induction on t.
L
As a first step, assume that for each i, for each µt with µi,t > 0, for each p 2 [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t )),
vH
H
inf Vi,t+1
((µi,t , µ
i,t ))],
there exist µi,t (p, µt ) with the following
properties:
(i) vH
H
p 2 Vi,t
((µi,t (p, µt ), µ
i,t )),
(ii) The map (p, µt ) 7! µi,t (p, µt ) is independent of µi,t , increasing in p, and lower semicontinuous in all its arguments.
We prove the existence of µi,t (p, µt ) in claims 8 and 9. For each i, for each µt with µi,t > 0,
L
for each p 2 [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
claim 8 proves the existence of µ⇤ such that vH
i,t )),
vH
H
p 2 Vi,t
((µ⇤ , µ
H
inf Vi,t+1
((µi,t , µ
i,t ))
i,t ))],
as a standard ap-
plication of Kakutani fixed point theorem. Claim 9 proves the existence of a selection that
satisfies (ii), as an application of corollary 2 of Milgrom and Roberts (1994). We postpone
3
the details of the construction to later claims, as it does not shed light on the logic of the
proof.
The rationale behind our construction is as follows. If player i with valuation vL rejects
all o↵ers p
L
max( Wi,t+1
(µt ), vH
H
inf Vi,t+1
((0, µ
i,t ))),
while player i with valuation vH
accepts with probability
1 µi,t µi,t (p, µt )
,
µi,t 1 µi,t (p, µt )
then µi,t (p, µt ) is the next period belief about player i’s valuation being vH , conditional
on player i having rejected the o↵er p and belief profile µt . By construction, player i is
indi↵erent between accepting and rejecting the o↵er and, thus, can mix between the two. A
similar construction already appears in the classical bargaining problem.
cL
Denote by W
it ,i,t+1 (µt ) the highest payo↵ player it , the owner of the good at period t, can
obtain if he makes an o↵er to player i, who accepts with probability
1 µi,t µi,t (p, µt )
µi,t 1 µi,t (p, µt )
when his valuation is vH and accepts with probability zero when his valuation is vL , when
ciL,i,t+1
the belief profile is µt . Let Pbi,t (µt ) be the set of optimal o↵ers. We will show that W
t
is upper semi-continuous in all its arguments and that Pbi,t (µt ) is non-empty and compact.
(See Claims 10 and 11.)
ciL,i,t+1 to define the acceptance strategies
We now make use of the function µi,t and W
t
of player i, so that he is indi↵erent between accepting and rejecting prices in the range
L
[max( Wi,t+1
(µt ), vH
H
inf Vi,t+1
((0, µ
i,t )),
is vH . Formally, the acceptance strategies
vH
L
i,t
H
inf Vi,t+1
((µi,t , µ
i,t ))]
when his valuation
: [0, m] ⇥ [0, 1]n ! [0, 1] and
H
i,t
: [0, m] ⇥
[0, 1]n ! [0, 1] of player i at period t are defined as follows (with the interpretation that
H
i,t (p, µt )
is the probability with which player i with valuation vH accepts the price p when
the profile of beliefs is µt , a similar interpretation applies to
L
- If µt is such that Wi,t+1
(µt ) < vH
H
sup Vi,t+1
((0, µ
4
H
i,t (p, µt )):
i,t ))
and µi,t > 0,
H
i,t (p, µt )
8
>
>
>
1
>
>
<
µi,t µi,t (p,µt )
> µi,t (1 µi,t (p,µt ))
>
>
>
>
:0
if p < vH
h
if p 2 vH
if p > vH
=
H
sup Vi,t+1
((0, µ
i,t )),
H
sup Vi,t+1
((0, µ i,t )),
H
inf Vi,t+1
((µi,t , µ
vH
inf
H
Vi,t+1
((µi,t , µ i,t ))
i,t )),
i
,
and
8
>
<1
>
:0
L
i,t (p, µt )
=
L
if p  Wi,t+1
(µt ),
L
if p > Wi,t+1
(µt ).
L
cL
- If µt is such that W
it ,i,t+1 (µt ) > Wi,t+1 (µt )
vH
H
sup Vi,t+1
((0, µ
i,t ))
and µi,t > 0,
we have
8
>
>
>
1
>
>
<
µi,t µi,t (p,µt )
> µi,t (1 µi,t (p,µt ))
and
>
>
>
>
:0
H
i,t (p, µt )
L
if p < Wi,t+1
(µt ),
h
L
if p 2 Wi,t+1
(µt ), vH
L
- If µt is such that Wi,t+1
(µt )
>
:0
H
inf Vi,t+1
((µi,t , µ
H
inf Vi,t+1
((µi,t , µ
if p > vH
8
>
<1
µi,t > 0,
=
L
i,t (p, µt )
i,t )),
i,t ))
i
,
=
L
if p < Wi,t+1
(µt ),
if p
L
Wi,t+1
(µt ).
ciL,i,t+1 (µt ), vH
max(W
t
5
H
sup Vi,t+1
((0, µ
i,t )))
and
8
>
>
>1
>
>
<
H
i,t (p, µt )
µi,t µi,t (p,µt )
> µi,t (1 µi,t (p,µt ))
and
>
>
>
>
:0
=
L
if p  Wi,t+1
(µt ),
⇣
L
if p 2 Wi,t+1
(µt ), vH
H
inf Vi,t+1
((µi,t , µ
if p > vH
L
i,t (p, µt )
8
>
<1
H
inf Vi,t+1
((µi,t , µ
i,t )),
i
))
,
i,t
=
L
if p  Wi,t+1
(µt ),
>
:0
L
if p > Wi,t+1
(µt ).
L
- If µi,t = 0, player i accepts all o↵ers up to max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t )))
L
when his valuation is vH and up to Wi,t+1
(µt ) when his valuation is vL , and rejects
all other o↵ers.
Claim 3. The acceptance strategies
L
i,t
: [0, m]⇥[0, 1]n ! [0, 1] and
H
i,t
: [0, m]⇥[0, 1]n !
[0, 1] defined above are Markov, regular and sequentially optimal.
Before proving Claim 3, two remarks are in order. First, the optimality of these strategies
requires to carefully select continuation payo↵s (and therefore continuation strategies). This
is made possible by our induction hypothesis along with the one-shot principle. Second, we
L
carefully define the strategies, with a particular attention at price Wi,t+1
(µt ) (the resale
value). As will be clear later, this is to guarantee the existence of optimal o↵ers by player
it , the owner of the good at period t in state µt . In other words, we need to break ties in
favor of player it if we want to guarantee the existence of an optimal o↵er (mathematically,
we need upper semi-continuity of the payo↵ function).
L
Proof. We first consider a player with valuation vL . If he accepts a price p  Wi,t+1
(µt ),
his payo↵ is max( vL
L
p, p + Wi,t+1
(µt ))
L
0. Since Vi,t+1
(µt ) = 0, it is optimal for the
6
L
player to accept with probability one any p < Wi,t+1
(µt ). The symmetric argument applies
L
L
when p > Wi,t+1
(µt ). Finally, if p = Wi,t+1
(µt ), max( vL
L
Wi,t+1
(µt )
L
p, p + Wi,t+1
(µt )) = 0 since
vL and the player is indi↵erent between accepting or rejecting.
L
We now consider a player with valuation vH . If the o↵er is p  Wi,t+1
(µt ), the payo↵
to player i is vH
p
vH
L
Wi,t+1
(µt )
L
0 if he accepts (since Wi,t+1
(µt )  vH ).
Alternatively, if he rejects the o↵er, the next period beliefs are arbitrary and we can assume
H
that the continuation payo↵ is inf Vi,t+1
((1, µ
L
Wi,t+1
(µt ) < p  vH
H
sup Vi,t+1
((0, µ
i,t )),
i,t )),
which is equal to zero. If the o↵er is
the payo↵ to player i is vH p if he accepts the
H
sup Vi,t+1
((0, µ
o↵er. Alternatively, if player i rejects the o↵er, his payo↵ is at most
i,t )).
Thus, it is optimal for player i to accept the o↵er with probability one. If the o↵er p 2
L
(max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t ))),
H
inf Vi,t+1
((µi,t , µ
vH
i,t ))],
the acceptance
strategies are such that the probability of player i being of valuation vH conditional on
rejecting the o↵er is µi,t (p, µt ). We assume that upon rejection by player i of such an o↵er,
players coordinate on the weak Markov strategies that give player i the payo↵ vH
H
we select the continuation payo↵ Vi,t+1
((µi,t (p, µt ), µ
i,t ))
= vH
p, i.e.,
p. From our induction
⌅
hypothesis, this is possible.
We now turn our attention to the payo↵ of player it , the owner of the good at period t.
L
Claim 4. If µt is such that Wi,t+1
(µt ) < vH
H
sup Vi,t+1
((0, µ
i,t )),
the expected payo↵
to player it if he o↵ers p to player i 2 Nit is
8
>
>
>p
>
>
>
>
>
<µi,t p + (1
L
if p  Wi,t+1
(µt ),
µi,t ) WiL ,t+1 ((0, µ i,t ))
t
µi,t µi,t (p,µt )
1 µi,t
>
>
p 1 µ (p,µ ) + 1 µ (p,µ
WiL ,t+1 (µi,t (p, µt ), µ
>
>
t
t
t)
i,t
i,t
>
>
>
>
: WL
it ,t+1 (µt )
i,t )
L
H
if p 2 ( Wi,t+1
(µt ), vH
sup Vi,t+1
((0, µ i,t ))),
h
i
H
H
if p 2
vH
sup Vi,t+1 ((0, µ i,t )), vH
inf Vi,t+1
((µi,t , µ i,t )) ,
if p > vH
L
cL
If µt is such that W
it ,i,t+1 (µt ) > Wi,t+1 (µt )
H
inf Vi,t+1
((µi,t , µ i,t )).
vH
H
sup Vi,t+1
((0, µ
i,t )),
the expected
payo↵ to player it if he o↵ers p to player i 2 Nit is
8
>
>
>p
>
< µ
µi,t (p,µt )
p i,t
+
1 µi,t (p,µt )
>
>
>
>
:
L
Wi ,t+1 (µt )
1
1 µi,t
WiL ,t+1 (µi,t (p, µt ), µ
µi,t (p,µt )
t
i,t )
L
if p < Wi,t+1
(µt ),
h
L
if p 2
Wi,t+1
(µt ), vH
if p > vH
t
7
i
H
inf Vi,t+1
((µi,t , µ i,t )) ,
H
inf Vi,t+1
((µi,t , µ i,t )).
L
Finally, if µt is such that Wi,t+1
(µt )
ciL,i,t (µt ), vH
max(W
t
H
sup Vi,t+1
((0, µ
i,t ))),
the
expected payo↵ to player it if he o↵ers p to player i 2 Nit is
8
>
>
>p
>
< µ
µi,t (p,µt )
p i,t
+
1 µi,t (p,µt )
>
>
>
>
:
L
Wi ,t+1 (µt )
1
1 µi,t
WiL ,t+1 (µi,t (p, µt ), µ
µi,t (p,µt )
t
i,t )
L
if p  Wi,t+1
(µt ),
⇣
L
if p 2
Wi,t+1
(µt ), vH
if p > vH
t
i
H
inf Vi,t+1
((µi,t , µ i,t )) ,
H
inf Vi,t+1
((µi,t , µ i,t )).
Proof. This follows directly from claim 9 (and the fact that upon selling the good, the
⌅
continuation value of player it is zero).
Claim 5. The maximal expected payo↵ to player it is
L
ciL,i,t (µt )).
WiLt ,t (µt ) = max max(vL , Wi,t+1
(µt ), W
t
i2Nit
Proof. This follows directly from claim 4.
⌅
It follows from claim 5 that player it , the owner of the good at period t, either o↵ers
L
c L (µt )) to player i or
Wi,t+1
(µt ) (the resale value) or pi,t 2 Pbi,t (µt ) (thus obtaining W
it ,i,t
consumes the good. Note that player it makes an o↵er to player i if and only if
L
c L (µt )).
i 2 arg max max(vL , Wj,t+1
(µt ), W
it ,j,t
j2Nit
Claim 6. The maximal expected payo↵ to player it is upper semi-continuous in µt .
Proof. This follows since WiLt ,t (µt ) is the pointwise maximum of a finite family of upper
⌅
semi-continuous functions.
H
We now characterize the set of payo↵s Vi,t
(µt ) induced by the acceptance strategies defined
in Claim 3 and the optimal o↵ers of player it . We introduce the following notation:
L
c L (µt )),
WiLt ,j,t (µt ) := max( Wj,t+1
(µt ), W
it ,j,t
8
i.e., it is the maximal payo↵ player it can obtain from o↵ering to player j (if it is possible to
do so).
There are three cases.
(1) Assume maxj2NiT WiLt ,j,t (µt ) < vL , i.e., player it , the owner of the good, is better o↵
H
consuming the good. We have that Vi,t
(µt ) = {0}.
(2) Assume maxj2NiT WiLt ,j,t (µt ) > vL , i.e, player it is strictly better o↵ by making an o↵er
to one of the players in Nit . There are several sub-cases to consider.
(i) If {i} = arg maxj2Nit WiLt ,j,t (µt ), player i’ set of payo↵ is
8
>
L
>
>
{ vH
Wi,t+1
(µt )}
>
>
<
co { vH Pbi,t (µt )}
>
>
>
>
>
L
:co { vH
Wi,t+1
(µt ), vH
L
ciL,i,t (µt ),
if Wi,t+1
(µt ) > W
t
Pbi,t (µt )}
c L (µt ) > W L (µt ),
if W
it ,i,t
i,t+1
L
c L (µt ).
if Wi,t+1
(µt ) = W
it ,i,t
(ii) If i 2 arg maxj2Nit WiLt ,j,t (µt ), but i is not the only maximizer, player i’s set of
payo↵
8
>
>
>
co {
>
>
<
co {
>
>
>
>
>
:co {
is
vH
vH
vH
L
ciL,i,t (µt ),
if Wi,t+1
(µt ) > W
t
L
Wi,t+1
(µt ), V H
i,t (µt )}
Pbi,t (µt ), V H
i,t (µt )}
L
Wi,t+1
(µt ), vH
Pbi,t (µt ), V H
i,t (µt )}
c L (µt ) > W L (µt ),
if W
it ,i,t
i,t+1
L
c L (µt ),
if Wi,t+1
(µt ) = W
it ,i,t
where V H
i,t (µt ) is the set of continuation payo↵s to player i conditional on any
player in arg maxj2Nit WiLt ,j,t (µt )\{i} having received an o↵er. (A formal definition
is available upon request.)
(iii) If i 2
/ arg maxj2Nit WiLt ,j,t (µt ), his set of payo↵s is co V H
i,t (µt ).
(3) Assume that maxj2NiT WiLt ,j,t (µt ) = vL , i.e, player it is indi↵erent between consuming
the good immediately or selling it. There are three sub-cases.
9
(i) If {i} = arg maxj2Nit WiLt ,j,t (µt ), player i’ set of payo↵ is
8
>
>
>
co { vH
>
>
<
co { vH
>
>
>
>
>
:co { vH
L
c L (µt ),
if Wi,t+1
(µt ) > W
it ,i,t
L
Wi,t+1
(µt ), 0}
Pbi,t (µt ), 0}
L
Wi,t+1
(µt ), vH
Pbi,t (µt ), 0}
L
ciL,i,t (µt ) > Wi,t+1
if W
(µt ),
t
L
c L (µt ).
if Wi,t+1
(µt ) = W
it ,i,t
(ii) If i 2 arg maxj2Nit WiLt ,j,t (µt ) but is not the only maximizer, player i’s set of payo↵
is
8
>
>
>co { vH
>
>
<
co { vH
>
>
>
>
>
:co { vH
L
c L (µt ),
if Wi,t+1
(µt ) > W
it ,i,t
L
Wi,t+1
(µt ), V H
i,t (µt ), 0}
Pbi,t (µt ), V H
i,t (µt ), 0}
L
Wi,t+1
(µt ), vH
Pbi,t (µt ), V H
i,t (µt ), 0}
L
ciL,i,t (µt ) > Wi,t+1
if W
(µt ),
t
L
c L (µt ),
if Wi,t+1
(µt ) = W
it ,i,t
(iii) If i 2
/ arg maxj2Nit WiLt ,j,t (µt ), his set of payo↵s is co {V H
i,t (µt ), 0}.
Note that the operator co appears in the above expressions because player it can randomize when indi↵erent.
H
Claim 7. For each µt , Vi,t
(µt ) is non-empty, convex and compact
⌅
H
Proof. This follows directly from the definition of Vi,t
(µt ).
H,⇤
H
It remains to show that we can find a sub-correspondence Vi,t
of Vi,t
, that is non-empty,
convex, compact valued and upper hemi-continuous. It is immediate to verify that if we
⇤
⇤
restrict attention to prices o↵ered in Pbi,t
(µt ) ✓ Pbi,t (µt ) for all µt with Pbi,t
(µt ) non-empty and
H,⇤
H
compact, then the induced set of payo↵s Vi,t
(µt ) ✓ Vi,t
(µt ) is also non-empty, convex and
compact, for each µt . The end of the proof is devoted to this task.
ciL,i,t . As an initial step, we prove that
To do, we first need to construct µi,t and W
t
p 2
it is possible to find beliefs such that vH
H
sup Vi,t+1
((0, µ
i,t )),
vH
H
inf Vi,t+1
((µi,t , µ
i,t ))].
10
H
Vi,t+1
((µ⇤ , µ
i,t ))
for all p 2 [ vH
H
sup Vi,t+1
((0, µ
Claim 8. For each µt with µi,t > 0, for each p 2 [ vH
H
inf Vi,t+1
((µi,t , µ
i,t ))],
there exists µ⇤ 2 [0, µi,t ] such that vH
b H ((µ, µ
Proof. Define the map µ 7! V
i,t+1
H
bi,t+1
there exists a fixed point µ⇤ 2 V
((µ⇤ , µ
H
Vi,t+1
((µ⇤ , µ
i,t )).
i,t )),
H
p 2 Vi,t+1
((µ⇤ , µ
H
:= Vi,t+1
((µ, µ
i,t ))
i,t ))
vH
i,t )).
vH + p + µ. If
then there exists µ⇤ such that vH
p2
We now prove the existence of such a fixed point.
b H ((0, µ
Note that sup V
i,t+1
H
bi,t+1
0 or sup V
((µi,t , µ
i,t ))
i,t ))
b H ((µi,t , µ
0 and that inf V
i,t+1
i,t ))
b H ((0, µ
 µi,t . If either inf V
i,t+1
µi,t , either 0 or µi,t is a fixed point since the correspondence
H
bi,t+1
is non-empty, convex and compact valued. So, assume that inf V
((0, µ
b H ((µi,t , µ
sup V
i,t+1
i,t )),
i,t ))
> 0 and
< µi,t . For any µ 2 [0, µi,t ], define
8
>
H
H
>
bi,t+1
bi,t+1
>
V
((µ, µ i,t )) \ [0, µi,t ] if V
((µ, µ i,t )) \ [0, µi,t ] 6= ;,
>
>
<
H
ei,t+1
b H ((µ, µ i,t )) > µi,t ,
V
((µ, µ i,t )) = {µi,t }
if inf V
i,t+1
>
>
>
>
>
H
bi,t+1
:{0}
if sup V
((µ, µ i,t )) < 0.
i,t ))
e H ((µ⇤ , µ
From Kakutani fixed point theorem, there exists µ⇤ 2 V
i,t+1
b H ((µ⇤ , µ
{0, µi,t }. It follows that µ⇤ 2 V
i,t+1
i,t )),
i,t )).
From above, µ⇤ 2
/
⌅
which completes the proof.
Claim 8 states the existence of µ⇤ such that vH
H
p 2 Vi,t+1
((µ⇤ , µ
i,t )),
but does not
state uniqueness. We now move to the precise definition of µi,t .
L
For each p 2 [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
define the set Ci,t (p, µt ) := {µ 2 [0, µi,t ] : vH
i,t ))),
vH
H
p 2 Vi,t+1
((µ, µ
H
inf Vi,t+1
((µi,t , µ
i,t ))}.
i,t ))],
The set Ci,t (p, µt )
is thus the set of all beliefs that makes player i indi↵erent between accepting and rejectL
ing p. In what follows, it is understood that p is in the interval [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t ))),
vH
H
inf Vi,t+1
((µi,t , µ
i,t ))].
Claim 9. For each (p, µt ) with µi,t > 0, the set Ci,t (p, µt ) is non-empty and compact. The
map (p, µt ) 7! Ci,t (p, µt ) is upper hemi-continuous.
Proof. Fix (p, µt ). From claim 3, Ci,t (p, µt ) is non-empty. By construction, Ci,t (p, µt ) is a
subset of [0, µi,t ], hence bounded. We now show that it is also closed, hence it is compact. To
11
i,t ))

do so, we prove the stronger property that the correspondence (p, µt ) 7! Ci,t (p, µt ) is upper
hemi-continuous (hence has a closed graph and thus is close-valued).
Take a sequence ((pk , µkt ), µk )k ! ((p, µt ), µ) such that µk 2 Ci,t (pk , µkt ) for all k. For each
H
H
k, there exists Vi,t+1
((µk , µk i,t )) 2 Vi,t+1
((µk , µk i,t )) such that vH
H
pk = Vi,t+1
((µk , µ
i,t )).
H
H
From upper hemi-continuity of Vi,t+1
, the sequence (Vi,t+1
((µk , µk i,t )))k converges to an eleH
ment of Vi,t+1
((µ, µ
i,t )),
and vH
⌅
H
p 2 Vi,t+1
((µ, µ
i,t )),
i.e., µ 2 Ci,t (p, µt ), as required.
For each (p, µt ) with µi,t > 0, define µi,t (p, µt ) := min Ci,t (p, µt ). Note that µi,t (p, (µi,t , µ
µi,t (p, (µ0i,t , µ
i,t ))
whenever µi,t
H
Vi,t+1
((µ, µ
for all µi,t , µ0i,t , µ
i,t
(since Ci,t (p, (µ0i,t , µ
i,t ))
= Ci,t (p, (µi,t , µ
i,t ))
0
i,t ))\[0, µi,t ],
µ0i,t ). It is increasing in p as the lowest zero of the correspondence µ 7!
i,t ))
vH + p, which is continuous but for upward jumps (corollary 2 of Mil-
grom and Roberts (1994)). It is also lower semi-continuous in (p, µt ) (since it is defined as
the infimum of an upper hemi-continuous correspondence). Hence, it is left-continuous in p.
H
sup Vi,t+1
((0, µ
Finally, note that for all prices in [ vH
i,t ))),
vH
H
inf Vi,t+1
((0, µ
i,t ))],
µi,t (p, µt ) = 0.
cL .
We now turn our attention to W
it ,i,t
Define the map (µ, µ
H
inf Vi,t+1
((µi,t , µ
i,t ))],
i,t )
7! pi,t (µ, µ
i,t )
L
2 [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t ))),
as
pi,t (µ, µ
i,t )
:= sup{p : µi,t (p, µt )  µ}.
(If {p : µi,t (p, µt )  µ} = ;, we let pi,t (µ, µ
i,t )
L
= Wi,t+1
(µt ).) The price pi,t (µ, µ
i,t )
is the
highest price that the seller can charge if the next period belief about player i is at most µ.
By construction, the map pi,t is upper semi-continuous in all its arguments and is increasing
in µ (hence it is right continuous in µ, for each µ
i,t ).
Claim 10. For each µt with µi,t > 0, the maximization problem
12
vH
=
ciL,i,t (µt ) := max pi,t (µ, µ
W
t
µ2[0,µi,t ]
i,t )
µi,t µ
1 µi,t L
+
W
(µ, µ
1 µ
1 µ it ,t+1
i,t )
(P)
ciL,i,t (µt ) is upper semi-continuous.
has a solution. The map µt 7! W
t
Proof. By construction, the map µ 7! pi,t (µ, µ
the induction hypothesis, µ 7! WiLt ,t+1 (µ, µ
i,t )
i,t )
is upper semi-continuous. From
is upper semi-continuous. It follows that a
solution exists. From the theorem of the maximum (Ausubel and Deneckere, (1993)), the
c L (µt ) is upper semi-continuous continuous.
function µt 7! W
it ,i,t
⌅
The astute reader will notice that the above maximization problem is nearly identical to
the seller’s maximization problem in Ausubel and Deneckere (1989); a reader is referred to
ci,t (µt ) be the non-empty set of maximizers of P.
this paper for more details. Let M
L
Claim 11. For each µt with µi,t > 0, for each p 2 [max( Wi,t+1
(µt ), vH
H
inf Vi,t+1
((µi,t , µ
p
i,t ))],
H
sup Vi,t+1
((0, µ
there exists µ 2 [0, µi,t ] such that
µi,t µi,t (p, µt )
1 µi,t
+
WiLt ,t+1 (µi,t (p, µt ), µ i,t ) 
1 µi,t (p, µt )
1 µi,t (p, µt )
µi,t µ
1 µi,t L
pi,t (µ, µ i,t )
+
W
(µ, µ i,t ).
1 µ
1 µ it ,i,t+1
Proof. Let µ = µi,t (p, µt ) and observe that pi,t (µ, µt )
⌅
p.
Claim 11 implies that the maximization problem
max p
µi,t µi,t (p, µt )
1 µi,t
+
W L (µi,t (p, µt ), µ
1 µi,t (p, µt )
1 µi,t (p, µt ) it ,t+1
L
subject to p 2 [max( Wi,t+1
(µt ), vH
H
sup Vi,t+1
((0, µ
i,t ))),
has a solution, which is obtained by o↵ering a price pi,t (µ, µ
vH
i,t )
i,t ),
H
inf Vi,t+1
((µi,t , µ
i,t ))]
ci,t (µt ). The
for some µ 2 M
above maximization problem is the one player it faces when he o↵ers a price to player i,
who follows the acceptance strategy defined in Claim 3. For all µt , let pb(µt ) 2 Pbi,t (µt ) :=
{pi,t (µ, µ
i,t ); µ
ci,t (µt )}.
2M
13
i,t ))),
vH
We are ready to state and prove the final claim, Claim 12, which states that we can indeed
find a sub-correspondence with the required properties.
H,⇤
H
Claim 12. There exists a sub-correspondence Vi,t
of Vi,t
, that is non-empty, convex,
compact valued and upper hemi-continuous.
Proof. We prove claim 12 as a limiting (approximation) argument. More precisely, we
consider decreasing sequences (pni,t , WiL,n
)
of continuous functions that converges pointt ,i,t+1 n2N
wise to (pi,t , WiLt ,i,t+1 ) for each (i, it ). Since the functions pi,t and WiLt ,i,t+1 are upper semicontinuous, such sequences exist by Baire theorem. (Here, we slightly abuse notations to
index with n the sequences. This should not create confusion with the cardinality of the set
of players.)
From the maximum theorem, the maximization problem P n :
c L,n (µt ) := max pni,t (µ, µ
W
it ,i,t
µ2[0,µi,t ]
i,t )
µi,t µ
1 µi,t L,n
+
W
(µ, µ
1 µ
1 µ it ,t+1
(P n )
i,t )
c L,n is continuous and the arg max correspondence M
cn is
admits a solution. Moreover, W
i,t
it ,i,t
n
upper hemi-continuous. It follows that the correspondence Pbi,t
is also upper hemi-continuous.
c L,n )n is decreasing and converges
By construction, the sequence of continuous functions (W
it ,i,t
c L , hence W
c L is upper semi-continuous (this is another proof of Claim 6).
pointwise to W
it ,i,t
it ,i,t
cn )n and (Pbn )n converge to sub-correspondences M
c⇤ and Pb⇤ of M
ci,t and Pbi,t ,
Similarly, (M
i,t
i,t
i,t
i,t
respectively, where convergence is defined as outer convergence of their graphs in the sense
of Painlevé-Kuratowski. To see this, write H n for the hypograph of
(µ, µ
i,t )
7! F n (µ, µ
i,t )
:= pni,t (µ, µ
i,t )
µi,t µ
1 µi,t L,n
+
W
(µ, µ
1 µ
1 µ it ,t+1
i,t ),
and observe that H n ◆ H n+1 ◆ . . . , i.e., it is a monotone decreasing sequence. Moreover,
each H n is closed. From Rockafellar and Wets (2009) (page 111, exercise 4.3), the PainlevéKuratowski limit is \n H n , which is the hypograph of
(µ, µ
i,t )
7! F (µ, µ
i,t )
:= pi,t (µ, µ
i,t )
14
µi,t µ
1 µi,t L
+
W
(µ, µ
1 µ
1 µ it ,t+1
i,t ).
Therefore, the sequence of functions (F n )n -converges to F . It follows that every limit points
of sequences of maximizers is a maximizer. (See http://www.tjsullivan.org.uk/pdf/2011-0128-caltech-gamma.pdf - link active in July 2015).
H,n
Let µt 7! Vi,t
(µt ) be the payo↵ correspondence corresponding to (pni,t , WiL,n
). It is
t ,i,t+1
H,n
H,⇤
H
routine to verify that (Vi,t
)n converges to the sub-correspondence Vi,t
of Vi,t
, obtained
⇤
from the prices Pbi,t
(µt ) ✓ Pbi,t (µt ) for all µt . (Note that this is equivalent to the outer limit
H,n
H
of (Vi,t
(µnt ))n converging to an element of Vi,t
(µt ) for all sequences (µnt ) ! µt .)
H,⇤
Since the domain of Vi,t
is [0, 1], a closed set, its range is compact, and has closed values,
it is enough to prove that it has a closed graph to prove upper hemi-continuity. Thus, if
H,n
H,⇤
we prove that each Vi,t
has a closed graph, Vi,t
has a closed graph from Proposition 4.4
(Rockafellar and Wetts, (2009), page 111).
H,n
We now prove that each Vi,t
has a closed graph. Consider a sequence (µk , v k ) ! (µ, v)
H,n
H,n
such that v k 2 Vi,t
(µk ) for all k. We need to prove that v 2 Vi,t
(µ).
1 Assume {i} = arg maxj2Nit WiL,n
(µ). From the continuity of WiL,n
, there exists K1
t ,j,t
t ,j,t
such that for all k > K1 , {i} = arg maxj2Nit WiL,n
(µk ).
t ,j,t
H,n
(i) Assume WiL,n
(µ) < vL , so that Vi,t
(µ) = {0}. From the continuity of WiL,n
,
t ,i,t
t ,j,t
there exists K2 such that for all k > K2 , WiL.n
(µk ) < vL . Therefore, for all
t ,i,t
H,n
k > max(K1 , K2 ), Vi,t
(µk ) = {0} and thus we have the desired result.
(ii) Assume WiL,n
(µ) > vL . From the continuity of WiL,n
, there exists K2 such that
t ,i,t
t ,j,t
for all k > K2 , WiL,n
(µk ) > vL . There are further sub-cases to consider.
t ,i,t
L,n
c n (µ), so that V H,n (µ) = {vH
(a) Assume Wi,t+1
(µ) > W
it ,t
i,t
L,n
Wi,t+1
(µ)}. From
cin,t , there exists K3 such that for all k > K3 ,
the continuity of Wint ,i,t and W
t
cin,t (µk ). It follows that for all k > max(K1 , K2 , K3 ), we have
Wint ,i,t (µk ) > W
t
H,n
that Vi,t
(µk ) = { vH
L,n
L,n
Wi,t+1
(µk )}. From the continuity of Wi,t+1
, we have
the desired result.
L,n
c n (µ), so that V H,n (µ) = co { vH
(b) Assume Wi,t+1
(µ) < W
it ,t
i,t
15
n
Pbi,t
(µ)}. From
c n , there exists K3 such that for all k > K3 ,
the continuity of Wint ,i,t and W
it ,t
c n (µk ). It follows that for all k > max(K1 , K2 , K3 , K4 ), we
Wint ,i,t (µk ) < W
it ,t
H,n
n
have that Vi,t
(µk ) = { vH Pbi,t
(µk )}. Therefore, for each k > max(K1 , K2 , K3 ),
n
there exists p(µk ) 2 Pbi,t
(µk ) such that v k =
vH
n
p(µk ). Since Pbi,t
is
n
upper hemi-continuous, we have that (p(µk ))k ! p(µ) 2 Pbi,t
(µ), so that
(v k )k ! v 2 co { vH
n
P̂i,t
(µ)}.
L,n
c n (µ), so that V H,n (µ) = co { vH
(c) Assume Wi,t+1
(µ) = W
it ,t
i,t
n
Pbi,t
(µ)}. For each k, v k is either vH
Wint ,i,t (µk ) or vH
L,n
Wi,t+1
(µ), vH
pi,t (µk ) for some
n
pi,t (µk ) 2 Pbi,t
(µk ). From (a) and (b) above, the limit of v k must therefore be
H,n
in Vi,t
(µ), as required.
(iii) Assume WiL,n
(µ) = vL . This follows directly from above.
t ,i,t
(2) Assume i 2
/ arg maxj2Nit WiL,n
(µ). From the continuity of WiL,n
, there exists K1 such
t ,j,t
t ,j,t
that for all k > K1 , i 2
/ arg maxj2Nit WiL,n
(µk ).
t ,j,t
H,n
(i) Assume maxj2Nit WiL,n
(µ) < vL , so that Vi,t
(µ) = {0}. There exists K2 such
t ,j,t
H,n
that for all k > K2 , maxj2Nit WiL,n
(µk ) < vL . It follows that Vi,t
(µk ) = {0} for
t ,j,t
all k > max(K1 , K2 ) and thus we have the desired result.
H,n
(ii) Assume maxj2Nit WiL,n
(µ) > vL , so that Vi,t
(µ) = V H,n
i,t (µ). There exists K2
t ,j,t
H,n
k
such that for all k > K2 , maxj2Nit WiL,n
(µk ) > vL , so that Vi,t
(µk ) = V H,n
i,t (µ )
t ,j,t
for all k > max(K1 , K2 ). The result follows from the upper hemi-continuity of
V H,n
i,t (µ); a formal proof is available upon request.
(iii) Assume maxj2Nit WiL,n
(µ) = vL . It follows from similar arguments as above and
t ,j,t
is left to the reader.
(3) Assume i 2 arg maxj2Nit WiL,n
(µ), but is not the unique maximizer. It follows from
t ,j,t
similar arguments as above and is left to the reader.
⌅
16
To summarize, Claims 7 and 12 state the existence of equilibrium payo↵ correspondences
Vi,t that are non-empty, convex, compact valued and upper hemi-continuous. Claim 6 states
the existence of an equilibrium payo↵ correspondence Wit ,t that is single-valued and upper
semi-continuous (when viewed as a function). This completes the proof of Proposition A.1.
A.3
Proof of Proposition 1
To prove existence of a weak Markov equilibrium in infinite horizon games, it suffices to
replicate the arguments in Chatterjee and Samuelson (1988), which exploit the feature that
any finite horizon game has an equilibrium and the game terminates in finite time with
probability one (see part 1 of Proposition 5).
More precisely, from Proposition 5, there exists T ⇤ such that the trading game ends with
probability one before T ⇤ . From Proposition A.1, any trading game with a finite horizon
admits an equilibrium. Consider one such game with a horizon T > T ⇤ . By construction,
a collection of sequential best-reply strategies in the finite horizon game will also be an
equilibrium in the infinite horizon game. (See the appendix of Chatterjee and Samuelson
(1988).)
References
Ausubel, Lawrence M. and Raymond J. Deneckere, “Reputation in Bargaining and
Durable Goods Monopoly,” Econometrica, 1989, 57 (3), pp. 511–531.
and
, “A Generalized Theorem of the Maximum,” Economic Theory, 1993, 3 (1),
pp. 99–107.
Chatterjee, Kalyan and Larry Samuelson, “Bargaining under Two-Sided Incomplete
Information: The Unrestricted O↵ers Case,” Operation Research, 1988, 36 (4), 605–618.
17
Milgrom, P. and J. Roberts, “Comparing Equilibria,” American Economic Review, 1994,
84, 441–459.
Rockafellar, Tyrrel and R. Wets, Variational Analysis, Springer, 2009.
18
Appendix B: Bilateral Trading in Networks (Condorelli, Galeotti and Renou)
(FOR ONLINE PUBLICATION ONLY)
B.1
Trading cycle: an example (detailed version)
0
1
2
3
4
Figure B.1: An example of trading cycle
We illustrate with the help of a simple example the possibility of the object being sold
from one trader to another trader at an earlier date and purchased back from the same
trader at a later date. Consider the trading network in Figure B.1. There 5 traders; we
simplify the notation and write 0 for the initial owner. Assume that 1 > µ3 ≥ µ4 > 0 and
µ2 = µ1 = µ0 = 0, i.e., traders 1 and 2 have valuation vL with probability one. (We later
argue that the same equilibrium outcome obtains if µ1 and µ2 are arbitrarily small, but not
zero.)
The intuition as to why cycling is possible is straightforward. Trader 0, the initial owner
of the object, can extract nearly all the expected surplus by offering the good to trader
1 first, with the promise to buy it back from trader 1 later, if trader 1 does not sale the
object to trader 3. In that contingency, trader 0 can then sale the object to trader 2 at
1
a price corresponding to the expected surplus from trade between traders 2 and 4, i.e.,
δ 2 (µ4 vH + (1 − µ4 )vL ).
More formally, the strategies are as follows:
1. At period t, regardless of the profile of beliefs (µt3 , µt4 ), trader 3 with valuation vL (resp.,
vH ) buys the object from trader 1 at price pt if and only if pt ≤ δvL (resp., pt ≤ δvH ).
Upon purchasing the object at period t, trader 3 consumes the good at period t + 1, if
made an offer.
2. At period t, regardless of the profile of beliefs (µt3 , µt4 ), trader 4 with valuation vL (resp.,
vH ) buys the object from trader 2 at price pt if and only if pt ≤ δvL (resp., pt ≤ δvH ).
Upon purchasing the object at period t, trader 2 consumes the good at period t + 1, if
made an offer.
3. At period t, if µt4 = µ4 , trader 2 buys the object from trader 0 at price pt if and only if
pt ≤ δ 2 (µ4 vH + (1 − µ4 )vL ), if made an offer.
4. At period t, if µt4 = µ4 and trader 2 owns the object, he offers the object to trader 2 at
price δvH .
5. At period t, if µt4 = 0 and trader 2 owns the object, he consumes the object. Note: By
construction of trader 4 strategy and the restriction to passive beliefs upon observing
unexpected actions (e.g., trader 4 rejecting an offer at pt < δvL ), µt4 ∈ {0, µ4} for all t.
6. At period t, if µt3 = µ3 , trader 1 buys the object from trader 0 at price pt if and only if
pt ≤ δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH + (1 − µ4 )vL ),
if made an offer.
7. At period t, if µt3 = µ3 and trader 1 owns the object, he offers the object to trader 3 at
price δvH .
8. At period t, if µt3 = 0 and trader 1 owns the object, he offers the object to trader 0
at price δ 3 (µ4 vH + (1 − µ4 )vL ). Note: By construction of trader 3 strategy and the
2
restriction to passive beliefs upon observing unexpected actions (e.g., trader 3 rejecting
an offer at pt < δvL ), µt3 ∈ {0, µ3 } for all t.
9. At period t, if (µt3 , µt4 ) = (µ3 , µ4 ) and trader 0 owns the object, he offers the object to
trader 1 at price
pt = δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH + (1 − µ4 )vL ).
10. At period t, if (µt3 , µt4 ) = (0, µ4 ) and trader 0 owns the object, he offers the object to
trader 2 at price
pt = δ 2 (µ4 vH + (1 − µ4 )vL ).
11. Note: By construction, whenever µt4 ̸= µ4 , trader 0 is inactive.
12. At period t, if (µt3 , µt4 ) = (0, µ4 ), trader 0 buys the object from trader 1 at price pt if
and only if pt ≤ δ 3 (µ4 vH + (1 − µ4 )vL ), if made an offer
13. At period t, if (µt3 , µt4) = (µ3 , µ4 ), trader 0 buys the object from trader 1 at price pt if
and only if pt ≤ δ 3 (µ4 vH + (1 − µ4 )vL ), if made an offer.
Along the equilibrium path, at period t = 1, trader 0 offers the object to trader 1 at price
δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH + (1 − µ4 )vL ).
Trader 1 accepts the offer. He then offers the object to trader 3 at price δvH at period t = 2.
If the offer is rejected, trader 1 then resale the object to trader 0 at price δ 3 (µ4 vH +(1−µ4 )vL )
at period t = 3. Trader 0 accepts the offer and then offers the object to trader 2 at price
δ 2 (µ4 vH + (1 − µ4 )vL ) at period t = 4. Trader 2 accepts the offer and in turn offers the good
to trader 4 at price δvH at period t = 5. Finally, if the offer is rejected, trader 2 consumes
the object at period t = 6.
It is routine to verify, albeit tedious, that the above strategies form an equilibrium if
3
δ ≥ max
!
vL
vL
,
µ4 vH + (1 − µ4 )vL µ3 vH + (1 − µ3 )vL
3
"
,
which implies that
δ 2 (µ4 vH + (1 − µ4 )vL ) ≥ vL ,
and
δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH + (1 − µ4 )vL ) ≥ δ 2 (µ3 vH + (1 − µ3 )vL ) ≥ vL .
This gives an incentive to traders 0, 1 and 2 to indeed sell the object rather than to consume
it.
It is also important to note that whenever traders 0 and 1 are active, we must have
µt4 = µ4 . Trader 0’s expected payoff is
δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH + (1 − µ4 )vL ),
which is the highest expected surplus from trade accounting for the delays in transaction.
Therefore, trader 0 cannot obtain a higher expected payoff from any period t, where (µt3 , µt4 ) =
(µ3 , µ4). If (µt3 , µt4 ) = (0, µ4 ) and trader 1 owns the object, his expected continuation’s payoff
is δ 2 (µ4 vH + (1 − µ4 )vL ) ≥ vL , which is the highest payoff that trader 0 can obtain by selling
the good to trader 2. Alternatively, if trader 0 consumes the object, his payoff is vL , while if
he offers the good to trader 1, the highest payoff he can obtain is δvL . Therefore, trader 0
has no incentive to deviate in any sub-forms.
A similar reasoning applies to the other traders. The key observation to make is that the
only relevant beliefs are in {0, µ3 } × {0, µ4 }.
So far, we have considered the special case where µ1 = µ2 = 0. Yet, we have the same
equilibrium outcome if we assume that
µ2 <
δ(µ4 vH + (1 − µ4 )vL )
,
vH
and
µ1 <
δµ3 vH + δ 4 (1 − µ3 )(µ4 vH + (1 − µ4 )vL )
,
vH
and that traders 1 and 2 accept all offers up to δvH , when their valuation is vH (and consume
upon purchasing the object). Intuitively, the conditions on µ1 and µ2 guarantee that trader
0 finds it unprofitable to offer the object to trader 2 at a price higher than the resale price
4
δ 2 (µ4 vH + (1 − µ4 )vL ) (resp., to trader 1 at a price higher than δ 2 µ3 vH + δ 5 (1 − µ3 )(µ4 vH +
(1 − µ4 )vL )), since trader 0 would obtain at most µ2 δvH (resp., µ1 δvH ).
5