Appendix C: Technical Results for N = S + 2
In this appendix, we present the technical results for the case of N = S + 2, including the proofs
of the propositions in Section 6 and some additional results to characterize the optimal mechanism.
Note that for the most part, we use the Kuhn-Tucker technique to obtain our results. For example, results characterizing types that obtain rent or types that have slack incentive compatibility
constraints in the optimal solution follow from the respective values of the Lagrangian multipliers. Similarly, the number of types who either obtain rent or have slack incentive compatibility
constraints arises from the number of constraints on the values of the Lagrangian multipliers.
C.1
N2 is a singleton
We first show that the choice of the S basis vectors does not affect the search for the optimal
mechanism; in particular, choosing different basis vectors does not affect the feasible set of the
choice variables {zi ; i ∈ N1 } in (P3).
Proposition C1 Suppose there are two sets of basis vectors in {q i , i ∈ N1 }, the convex hull of
each contains q ñ . That is, there exist Nb ⊂ N1 and Nb′ ⊂ N1 such that both {q i , i ∈ Nb } and
{q j , j ∈ Nb′ } form bases for RS , and there exist λ = (λi , i ∈ Nb ) ≥ 0 and λ′ = (λ′j , j ∈ Nb′ ) ≥ 0
such that q ñ = λQNb and q ñ = λ′ QNb′ . Then using the two sets of basis vectors in constructing
the optimization problem (P3) results in the same optimal mechanism.
Proof. The objective function in problem (P3) is independent of the choice of the basis vectors.
We only need to show that using different sets of basis vectors results in the same restrictions on
{zi , i ∈ N1 } where zi = q i y ñ , i ∈ N1 .
As before, we continue to let m = N1 \ Nb and let {γi , i ∈ Nb } be the unique scalars such that
X
γi q i = q m .
i∈Nb
1
(20)
Similarly, let m′ = N1 \ Nb′ and let {γi′ , i ∈ Nb′ } be the unique scalars such that
X
γi′ q i = q m′ .
(21)
i∈Nb′
Since N1 has exactly S + 1 elements, we know m′ ∈ Nb , m ∈ Nb′ , and N1 = m ∪ Nb = m′ ∪ Nb′ .
We showed in formulating problem (P3) that (20), together with
X
i∈Nb
λi q i = q ne ,
(22)
leads to the following constraints on {zi , i ∈ N1 } in (P3):
X
λi zi = 0,
(23)
γi zi = zm .
(24)
λ′i q i = q ne ,
(25)
λ′i zi = 0,
(26)
γi′ zi = zm′ .
(27)
i∈Nb
X
i∈Nb
Similarly, (21), together with
X
i∈Nb′
implies the following constraints:
X
i∈Nb′
X
i∈Nb′
To prove the Proposition, we need to show that (26) and (27) imply (23) and (24).
Step 1: (27) implies (24). Substituting the value of q m from (20) into (21) and adjusting, we
get
X
i∈Nb′ \m
′
′
γi′ + γm
γi q i + γm
γm′ − 1 q m′ = 0.
(28)
Since (Nb′ \m) ∪ m′ = Nb , the vectors {qj , j ∈ (Nb′ \m) ∪ m′ } are linearly independent. Then (28)
implies
′
γm
=
1
,
γm′
′
γi′ = −γm
γi = −
2
γi
,
γm′
i ∈ Nb′ \ m.
Substituting these into (27), we get
X
−
i∈Nb′ \m
zm
γi
zi +
= zm′ .
γm′
γm′
Multiplying both sides by γm′ and rearranging, we get (24).
Step 2: (26) implies (23). Substituting q m from (20) into (25), we get
X
i∈Nb′ \m
Since (Nb′ \m) ∪
m′
λ′i + λ′m γi q i + λ′m γm′ q m′ = q ne .
= Nb , this equation and (22) imply that
λi = λ′i + λ′m γi ,
i ∈ Nb′ \m
= λ′m γm′
λm′
Hence, we have,
X
λ′i zi =
X
i∈Nb′ \m
i∈Nb′
=
X
λi − λ′m γi zi + λ′m zm
λi zi − λm′ zm′ −
i∈Nb
=
X
X
λ′m γi zi + λ′m zm
i∈Nb′ \m
λi zi −
λ′m γm′ zm′
i∈Nb
−
X
λ′m γi zi + λ′m zm
i∈Nb′ \m


γi zi + γm′ zm′ 
λi zi − λ′m zm − 
=


i∈Nb
i∈Nb′ \m




X
X
γi zi
λi zi − λ′m zm −
=


i∈Nb
i∈Nb
X
=
λi zi
X



X
i∈Nb
where we use the fact that (Nb′ \m) ∪ m′ = Nb in the second and second to last equalities.
Before proving Proposition 7, we provide a lemma characterizing the type in Nb who obtains
rent. Let the Lagrangian for problem (P3) be
L=
X
µi [W (xi ) + u(xi , θi ) − max{0, fi (xñ ) − zi }]
i∈N1
+ µñ [W (xñ ) + u(xñ , θñ )] − α
X
i∈N1
λi zi − β(zm −
X
γi zi ),
i∈N1
where α and −β are the Lagrangian multipliers of constraints λz = 0 and γz = zm respectively.
3
Lemma C1 Suppose allocation xne is given and in the optimal mechanism, θi , i ∈ Nb , obtains rent.
Then we must have
µi
γi
+ β = min
λi
λi
µj
γj
+ β , j ∈ Nb .
λj
λj
(29)
Proof. The Lagrangian is not differentiable everywhere; however it is piecewise linear in z and zm
with kinks appearing only where zi = fi (xñ ) and zm = fm (xñ ). Hence the necessary conditions for
an optimal solution are given by:





αλi − βγi
= µi





αλi − βγi
=0








0 ≤ αλ − βγ ≤ µ
i
i
and
if zi < fi (xñ )
if zi > fi (xñ )
(30)
if zi = fi (xñ )
i





β





β








β
i ∈ Nb
= µm
if zm < fm (xñ )
=0
if zm > fm (xñ )
∈ [0, µm ]
if zm = fm (xñ )
(31)
If θi obtains rent, we know zi < fi (xñ ), which from (30) implies
µi
γi
+ β = α.
λi
λi
(32)
For any other type θj who does not obtain rent, (30) shows that there are two possibilities: αλj −
βγj = 0, implying
µj
γj
µj
+β
=α+
> α,
λj
λj
λj
(33)
µj
γj
+β
≥ α.
λj
λj
(34)
or αλj − βγj ≤ µj , implying
(32) - (34) imply (29).
Proposition 7 Suppose allocation xne is given and full rent cannot be extracted. In the optimal
4
mechanism, generically there is at most one type in Nb who can obtain rent. In particular,
(a) If θm has a slack incentive compatibility constraint of reporting θñ , i.e., if zm > fm (xñ ), then
type θi , i ∈ Nb , can obtain rent (i.e. zi < fi (xne )) only if
µk
µi
= min
, k ∈ Nb ;
λi
λk
(b) If θm obtains rent, i.e., zm < fm (xñ ), then θi , i ∈ Nb , can obtain rent only if
µi
µk
γi
γk
+ µm = min
+ µm , k ∈ N b
λi
λi
λk
λk
Proof. The Proposition follows almost immediately from Lemma C1.
Case (a): zm > fm (xñ ). From (31), we know β = 0. This case then immediately follows from
Lemma C1. Further, if two types, say θi and θj , both obtain rents, we have µi /λi = µj /λj , a
non-generic case. In other words, generically θi is the only type in Nb who obtains rent.
Case (b): zm < fm (xñ ). From (31), we know β = µm , and again this case then follows
immediately from Lemma C1. Further, if two types, say θi and θj , both obtain rents, it must be
that µi /λi + µm γi /λi = µj /λj + µm γj /λj , a non-generic case.
Note that there is a third case, of zm = fm (xñ ), that Proposition 7 does not cover. The following
proposition gives the value of β and shows which type obtains rent in this case.
Proposition C2 Suppose allocation xñ is given and full rent cannot be extracted. Suppose also
that in the optimal mechanism, zm = fm (xñ ). Then generically there are two possible situations.
(a) There are two types, θi and θj , i, j ∈ Nb , such that θi obtains rent and θj has a slack incentive
compatibility constraint of reporting θñ : zi < fi (xñ ) and zj > fj (xñ ), while zk = fk (xñ ) for all
k 6= i, j, k ∈ Nb . In this case,
β=
µi /λi
.
γj /λj − γi /λi
(35)
(b) There are two types, θi and θj , i, j ∈ Nb , who obtain rent, i.e., zi < fi (xñ ) and zj < fj (xñ ),
5
while zk = fk (xñ ) for all k 6= i, j, k ∈ Nb . In this case,
β=
µj /λj − µi /λi
.
γi /λi − γj /λj
(36)
Proof. From (30) and (31), the value of β is entirely determined by conditions in (30) when
zm = fm (xñ ). For each type in Nb that obtains rent or has a slack incentive compatibility constraint
of reporting θñ , there is one equation in (30) (the first or the second equation) that imposes a
condition on β and α, the two unknowns. Since there are only two unknowns, generically for there
to be a solution, there can at most be two such equations. That is, generically only cases (a) and
(b) of the Proposition are possible.
Case (a): From (30), the two conditions determining α and β are αλi − µi − βγi = 0 and
αλj − βγj = 0. Solving for α and β, we get (35).
Case (b): From (30), the two conditions determining α and β are αλi − µi − βγi = 0 and
αλj − µj − βγj = 0. Solving for α and β, we get (36).
Further, in both cases we can verify that β ∈ [0, µm ]. We sketch the argument for β ≤ µm in
case (a); the other case can be shown similarly. So, suppose in case (a), the value of β in (35) is
strictly greater than µm . We will show a contradiction.
From (35), β > µm implies
µi
λi
γ
+ µm λγii > µm λjj . Suppose the principal raises zi by a small
amount dzi , reduces zj by a corresponding small amount dzj , and keeps all the other zk , k ∈ Nb \i, j
fixed, so as to continue to satisfy λz = 0. Since zj > fj (xne ), a small reduction in zj does not
affect θj′ s (zero) rent, and the impact of this change on the principal’s expected payoff is equal
to µi dzi + µm dzm . From the two constraints, λi dzi + λj dzj = 0 and γi dzi + γj dzj = dzm , we can
solve for dzm as a function of dzi , and substituting it into µi dzi + µm dzm , we get the change in
γ
dzi
> 0. Thus if β > µm the mechanism
the principal’s expected payoff as µλii + µm λγii − µm λjj
λi
cannot be optimal.
We next prove Proposition 8.
6
Proposition 8 Generically, in the optimal mechanism,
(a) There cannot be more than two types who obtain rent.
(b) If two types obtain rent, then there cannot be a type whose incentive compatibility constraint of
reporting θñ is slack.
(c) If full rent cannot be extracted, there can be at most one type who has a slack incentive compatibility constraint of reporting θñ .
Proof. From (30) and (31), there is one equation or restriction on the values of α and β corresponding to each type in N1 = Nb
S
m who either obtains rent or has a slack incentive compatibility
constraint of reporting θñ . Since there are two unknowns, α and β, generically there cannot be
three or more such equations. The Proposition follows since whenever any of the three conditions
(a),(b) or (c) are violated we have more than three equations determining the value of the two
variables α and β.
The type in Proposition 8(c), i.e., the one who has a slack incentive compatibility constraint
of reporting θñ , can either be θm or any other type from Nb . In the latter case, this type can be
identified as follows.
Proposition C3 Suppose zm ≤ fm (xñ ) and β > 0. Then type θi , i ∈ Nb , satisfies zi > fi (xñ )
only if
γi
= max
λi
γj
, j ∈ Nb .
λj
(37)
γ
Proof. Since β > 0 and zi > fi (xñ ), (30) implies that α = β λγii while α ≥ β λjj for j ∈ N \ i. Then
(37) follows.
7
N2 contains two elements
C.2
We first show that both θñ1 and θñ2 cannot obtain rents. In particular, if θñ1 obtains rent, θñ2
must have a slack incentive compatibility constraint.
Proposition C4 If θñ1 obtains rent, i.e., if zñ1 < fñ1 (xñ2 ), θñ2 cannot obtain any rent. Further,
θñ2 ’s incentive compatibility constraint of reporting θñ1 must be slack. That is, zñ2 > fñ2 (xñ1 ).
Proof. The incentive compatibility constraints in (P1) between θñ1 and θñ2 are, in terms of the z’s,
u(xñ1 , θñ1 )−tñ1 ≥ u(xñ2 , θñ1 )−tñ2 −zñ1 , and u(xñ2 , θñ2 )−tñ2 ≥ u(xñ1 , θñ2 )−tñ1 −zñ2 . Adding these
two inequalities and adjusting, we get [u(xñ2 , θñ1 ) − u(xñ2 , θñ2 ) − zñ1 ]+[u(xñ1 , θñ2 ) − u(xñ1 , θñ1 ) − zñ2 ] ≤
0, which is [fñ1 (xñ2 ) − zñ1 ] + [fñ2 (xñ1 ) − zñ2 ] ≤ 0. Therefore, if zñ1 < fñ1 (xñ2 ), it must be that
zñ2 > fñ2 (xñ1 ), and thus θñ2 does not obtain rent.
Similar to Section C.1, we form the Lagrangian of the optimization problem in (P4):
L=
X
µi [W (xi ) + u(xi , θi ) − max{0, fi (xñ1 ) − zi (ñ1 ) + (fñ1 (xñ2 ) − zñ1 ), fi (xñ2 ) − zi (ñ2 )}]
i∈N1
+ µñ1 [W (xñ1 ) + u(xñ1 , θñ1 ) − (fñ1 (xñ2 ) − zñ1 )] + µñ2 [W (xñ2 ) + u(xñ2 , θñ2 )]
− α(ñ1 )
X
λi (ñ1 )zi (ñ1 ) − β(ñ1 )(zñ2 −
X
λi (ñ2 )zi (ñ2 ) − β(ñ2 )(zñ1 −
λi (ñ2 )zi (ñ1 ))
X
λi (ñ1 )zi (ñ2 ))
i∈N1
i∈N1
− α(ñ2 )
X
i∈N1
i∈N1
where α(ñi ) and −β(ñi ) are the Lagrangian multipliers of constraints λ(ñi )z(ñi ) = 0 and λ(ñj )z(ñi ) =
zñj respectively, i, j = 1, 2, j 6= i.
Proposition 9 Suppose allocations xñ1 and xñ2 are given. In the optimal mechanism,
(a) If θi , i ∈ N1 , obtains rent from having incentive to report θñ1 , then it must be that θi has the
lowest µ/λ(ñ1 ):
i = arg min
j∈N1
8
µj
λj (ñ1 )
.
(b) If θi , i ∈ N1 , obtains rent from having incentive to report θñ2 , then it must be that




 µ
X
λj (ñ1 ) 
j
,
µk 
+ µñ1 +
i = arg min
j∈N1  λj (ñ2 )
λj (ñ2 ) 
k∈N1 (ñ1 )
where N1 (ñ1 ) ⊆ N1 is the index set of the types who obtain rent from having incentive to report
θñ1 .
Proof. Again, the Lagrangian is piecewise linear in all the zi ’s. Similar to the proof of Lemma C1,
a set of conditions must be satisfied for there to be an optimal solution. The difference is that here
we have two sets of zi ’s, i.e., zi (ñ1 )’s and zi (ñ2 )’s, for the types in N1 , and we are only exploring the
case when θñ1 obtains rent while θñ2 has a slack incentive compatibility constraint. The necessary
conditions are




α(ñ1 )λi (ñ1 ) − β(ñ1 )λi (ñ2 )




if





= µi ,







 α(ñ1 )λi (ñ1 ) − β(ñ1 )λi (ñ2 )
if



= 0,








α(ñ1 )λi (ñ1 ) − β(ñ1 )λi (ñ2 )




if





∈ [0, µi ],









































α(ñ2 )λi (ñ2 ) − β(ñ2 )λi (ñ1 )
if
max {0, fi (xñ2 ) − zi (ñ2 )} ,
(fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 ) <
i ∈ N1 ,
(38)
max {0, fi (xñ2 ) − zi (ñ2 )} ,
(fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 ) =
max {0, fi (xñ2 ) − zi (ñ2 )} ,
fi (xñ2 ) − zi (ñ2 ) >
max {0, (fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 )} ,
= µi ,
α(ñ2 )λi (ñ2 ) − β(ñ2 )λi (ñ1 )
if
fi (xñ2 ) − zi (ñ2 ) <
i ∈ N1 ,
max {0, (fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 )} ,
= 0,
α(ñ2 )λi (ñ2 ) − β(ñ2 )λi (ñ1 )
if
∈ [0, µi ],
(fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 ) >
fi (xñ2 ) − zi (ñ2 ) =
max {0, (fi (xñ1 ) − zi (ñ1 )) + (fñ1 (xñ2 ) − zñ1 )} ,
(39)
9




β(ñ2 )



β(ñ1 )
= µñ1 +
P
i∈N1 (ñ1 ) µi
(40)
= 0,
where (38) and (39) are derived from ∂L/∂zi (ñ1 ) and ∂L/∂zi (ñ2 ), and (40) is derived from ∂L/∂zñ1
and ∂L/∂zñ2 . Although (38) and (39) are rather messy, they are parallel to (30) of the last section.
Equation (40) is parallel to (31), but is simpler since here θñ1 obtains rent and θñ2 has a slack
incentive compatibility constraint of reporting θñ1 .
Repeating the same procedure as in proving Lemma C1, part (a) of the Proposition is obtained
by substituting β(ñ1 ) = 0 into (38), and part (b) is obtained by substituting the value of β(ñ2 ) in
(40) into (39).
Proposition 10 Generically in the optimal mechanism, for k = 1, 2, (a) there is at most one type
in N1 who can obtain rent from having incentive to report θne k ;
(b) if a type obtains rent from having incentive to report θne k , there cannot be a type (with a nonzero
λ(ñk )) whose incentive compatibility constraint of reporting θñk is slack.
Proof. When k = 1, substituting β(ñ1 ) = 0 into (38), we know that if two types, say θi and θj ,
i, j ∈ N1 , obtain rent from having incentive to report θñ1 , it must be that α(ñ1 ) = µi /λi (ñ1 ) =
µj /λj (ñ1 ), a non-generic case. Similarly, if θi obtains rent, we have α(ñ1 ) = µi /λi (ñ1 ) 6= 0. If
θj has a slack incentive compatibility constraint and λj (ñ1 ) 6= 0, we know α(ñ1 ) = 0, which is a
contradiction.
The proof is exactly the same when k = 2.
Essentially, the Proposition is based on the number of constraints that will be imposed on the
values of the α’s and β’s, similar to Section C.1. Since we have assumed that θñ1 obtains rent and
θñ2 has a slack incentive compatibility constraint, the values of the β’s are fixed. That leaves only
the values of the α’s to be determined, limiting the number of restrictions that can be imposed.
10
The following proposition extends Proposition 10.
Proposition C5 Generically, in the optimal mechanism,
(a) there cannot be two or more types in N1 that have slack incentive compatibility constraints of
reporting θñ2 .
(b) If a type θi , i ∈ N1 , has a slack incentive compatibility constraints of reporting θñ2 , it must be
that
λi (ñ1 )
= max
λi (ñ2 )
λj (ñ1 )
, j ∈ N1 .
λj (ñ2 )
Proof. Part (a): From (39) and the value of β(ñ2 ) in (40), if there are two types, say θi and θj ,
i, j ∈ N1 , who have slack incentive compatibility constraints of reporting θñ2 , it must be that
α(ñ2 )/β(ñ2 ) = λi (ñ1 )/λi (ñ2 ) = λj (ñ1 )/λj (ñ2 ),
(41)
which is non-generic.
Part (b): From (39), if a type θj , j 6= i, j ∈ N1 , does not have a slack incentive compatibility
constraint of reporting θñ2 , it must be that α(ñ2 )λj (ñ2 )−β(ñ2 )λj (ñ1 ) ≥ 0, or
λj (ñ1 )
λj (ñ2 )
≤ α(ñ2 )/β(ñ2 ).
This and (41) imply part (b) of the Proposition.
If a type θi , i ∈ N1 , obtains rent from having incentive to report θñ1 , it must have a slack
incentive compatibility constraint of reporting θñ2 . Proposition C5 then indicates that there cannot
be another type in N1 who has a slack incentive compatibility constraint of reporting θñ2 . This
observation, together with Proposition10(b) when k = 2, implies that
Corollary C1 If there is a type in N1 who obtains rent in the optimal mechanism, generically
there is no other type in N1 who has a slack incentive compatibility constraint of reporting θñ2 .
11