An Axiomatic Characterization of Strategyproof Ordinal Mechanisms with Indifferences Timo Mennle;
Sven Seuken;
July 14, 2014
EXTENSION TO (Mennle and Seuken, 2014) Abstract An ordinal mechanism is a mechanism that takes as input ordinal preference orderings over outcomes and selects a lottery over the outcomes, e.g., most matching mechanisms. In (Mennle and Seuken, 2014) we have given an axiomatic characterization of strategyproof ordinal mechanisms when the agent is not indifferent between outcomes: a mechanism is strategyproof if and only if it is swap monotonic, upper invariant, and lower invariant. In this note, we extend the axioms the the case of indifferences. We then show that the extended axioms characterize strategyproof ordinal mechanisms in this larger domain. Our axioms separation monotonicity, separation upper invariance, and separation lower invariance coincide with the original axioms in the domain without indifferences.
1 Model We consider a set of outcomes M . An agent’s weak ordinal preference order over these outcomes is denoted by ¡: for some partition pMk qkPt1,...,K u of M the preference order M1
¡
. . . ¡ Mk
¡
. . . ¡ MK
represents the preferences, where the agent • is indifferent between the outcomes j, j 1 •
P Mk from the same subset, denoted j j 1, prefers outcome j P Mk to outcome j 1 P Mk1 where k k 1 , denoted j j 1 . ¡
We would like to thank Baharak Rastegari for insightful discussions. ; Department of Informatics, University of Zurich, 8050 Zurich, Switzerland, {mennle, seuken}@ifi.uzh.ch.
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When deciding between lotteries over the different outcomes, we assume that agents maximize expected utility with respect to some von Neumann-Morgenstern utility function that is consistent with their preference ordering. For some preference order ¡, the corresponding type is the set of consistent utilities t¡
tu : upaq ¡ upbq ô a
¡
bu,
(1)
and the set of all types is the type space T . A mechanism is a function f : T Ñ X , i.e., a mapping that takes as input an agent’s type report and returns a lottery over the outcomes. Thus, we consider the reporting game from the perspective of a single agent. It is straightforward to extend our analysis to mechanisms involving multiple agents, e.g., one-sided matching mechanisms. In that case, each set of reports from the other agents generates another mechanism from the individual agent’s perspective, i.e., for reports ti pt1 , . . . , ti1 , ti 1 , . . . , tn q, agent i is facing the decision of what to report to the function f pq fi p, ti q. (2) f is strategyproof if it is a best response (or dominant strategy, respectively, in the multiagent case) to report the type truthfully, independent of the underlying vNM utility function, i.e., for all types t, all utilities u P t and all misreports t1 t we have @
u, f ptq f pt1 q
D
¥ 0.
(3)
This is equivalent to strategyproofness in the sense of first order-stochastic dominance (taking indifferences into account): in the domain with indifferences, an allocation x ordinally dominates another allocation y at a type t if for all k¯ P t1, . . . , K u ¯ k ¸
¸
k 1 1 j P M k 1
xj
¥
¯ k ¸
¸
k1 1 j PMk1
yj .
A mechanism is strategyproof if for any type t and any misreport t1 ordinally dominates the allocation f pt1 q at t in the sense of (4).
(4)
P T , the allocation f ptq
2 Axioms The axioms restrict the way in which a mechanism can change the allocation when the agent changes its report, e.g., from t to t1 . Holding t and t1 fixed, for any subset A M we denote by ∆pAq ∆ft,t1 pAq
¸
P
pf pt1q f ptqqpj q
(5)
j A
the change in the probability of getting an object from the set A when reporting t1 instead of t. In general, the types t and t1 may be very different. However, our axioms will only impose restrictions for a particular (simple) report change: let type t correspond to the weak preference
2
p1q
p2q
order t : M1 ¡ . . . ¡ Mκ ¡ . . . ¡ MK . Let Mκ Mκ Y 9 Mκ (i.e., Mκ is the union of the two p 1q p 2q 1 disjoint sets Mκ and Mκ ) and let type t correspond to the weak preference order t 1 : M1
¡
. . . ¡ Mκp1q
¡
Mκp2q
¡
. . . ¡ MK .
We call the pair pt, t1 q a separation. Intuitively, an agent of type t is indifferent between p1q p2q p1q outcomes from Mκ and Mκ , whereas an agent of type t1 prefers outcomes from Mκ over p2q outcomes from Mκ , but for all other outcomes, the preference ordering under the two types is the same. Axiom 1 (Separation Responsive). A mechanism f is separation responsive if for any sepap1q p2q ration pt, t1 q the following holds: ∆pMκ q ¥ 0 and ∆pMκ q ¤ 0. Axiom 2 (Separation Direct). A mechanism f is separation direct if for any separation pt, t1 q p1q p2q the following holds: if ∆pMk q 0 for some k P t1, . . . , K u, then ∆pMκ q 0 and ∆pMκ q 0. Axiom 3 (Separation Monotonic). A mechanism f is separation monotonic if it is separation responsive and separation direct. Axiom 4 (Separation Upper Invariance). A mechanism f is separation upper invariant if for any separation pt, t1 q the following holds: ∆pMk q 0 for all k
κ.
(6)
Axiom 5 (Separation Lower Invariance). A mechanism f is separation lower invariant if for any separation pt, t1 q the following holds: ∆pMk q 0 for all k
¡ κ.
(7)
3 Characterization Result 3.1 Statement of Result Theorem 1. A mechanism is strategyproof if and only if it is separation monotonic, separation upper invariant, and separation lower invariant. Remark 1. If f is separation upper and lower invariant, then separation responsiveness implies separation directness (Lemma 4). Thus, Theorem 1 can be formulated using only separation responsiveness instead of separation monotonicity. The motivation for using the slightly stronger constraint is the interesting relaxation of strategyproofness one obtains by dropping only the lower invariance axiom in the domain without indifferences (Mennle and Seuken, 2014).
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3.2 Proof of Result Lemma 1. f strategyproof
ñ f separation upper invariant.
Proof. Towards contradiction, assume that f is not separation upper invariant. Then for some separation pt, t1 q there exists a (smallest) index k 1 κ such that the change in allocation ∆ p Mk 1 q Suppose, ∆pMk1 q ¡ 0. Since for any k¯ ¯ k ¸ ¸
¸
P
j Mk 1
f pt 1 q f pt q
¯ k ¸
f pt1 q f ptq pj q
P and ∆pMk q 0 for all k k 1 , we obtain 1
P
pj q 0.
(8)
P t1, . . . , K u, ∆ pM k q
(9)
∆pMk q ¡ 0.
(10)
k 1 j Mk
k ¸ ¸
k 1
f pt1 q f ptq pj q
k 1 j Mk
1
k ¸
k 1
Thus, f ptq does not ordinally dominate f pt1 q at t, because the condition (4) is violated for k¯ k 1 . Similarly, if ∆pMk1 q 0, we can show that f pt1 q does not ordinally dominate f ptq at t1 . This contradicts strategyproofness of f . Lemma 2. f strategyproof
ñ f separation responsive.
Proof. Towards contradiction, assume that f is not separation responsive. Then for some p1q p2q p1q separation pt, t1 q either ∆pMκ q 0 or ∆pMκ q ¡ 0. Suppose, ∆pMκ q 0. Lemma 1 implies that ∆pMk q 0 for all k κ. Thus,
κ¸1
¸
P κ¸ 1
f pt1 q f ptq pj q
k 1 j Mk
∆ p Mk q
¸
p 1q j PMκ
f pt1 q f ptq
pj q
(11)
∆pMκp1q q
(12)
k 1
0,
(13)
p1q
p2q
i.e., f pt1 q does not ordinally dominate f ptq at t1 . Similarly, is ∆pMκ q ¥ 0, but ∆pMκ q ¡ 0, we get that f ptq does not ordinally dominate f pt1 q at t. This contradicts strategyproofness. Lemma 3. f strategyproof
ñ f separation lower invariant.
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Proof. Assume towards contradiction that f is not separation lower invariant. Then for some separation pt, t1 q there exists a (smallest) index k 1 such that the change in allocation is ∆ p Mk 1 q
¸
P
j Mk 1
f pt 1 q f pt q
pj q 0.
(14)
It follows from the argument in the proof of Lemma 2 that in fact ∆pMκ q 0 for any separation pt, t1q. In combination with separation upper invariance (Lemma 1), this implies that k1 ¡ κ. Analogous to the proof of Lemma 1, we find that this contradicts strategyproofness. Lemma 4. f strategyproof
ñ f separation direct.
Proof. The fact that f is separation upper and lower invariant implies that f must be separation direct: the allocation never changes under any separation for any Mk with k κ. Hence, if a p1q p2q change occurs, it must occur for Mκ and Mκ . Definition 1 (Separation Strategyproof). A mechanism f is separation strategyproof if for any separation pt, t1 q we have that (1) f ptq ordinally dominates f pt1 q at t, and (2) f pt1 q ordinally dominates f ptq at t1 .
Intuitively, truthful reporting is always weakly better than any separation (or inverse). Definition 2 (L-separation and Multi-separation Strategyproof). Consider a pair of types pt, t1q. • pt, t1 q is called an L-separation if
¡ . . . ¡ M k 1 ¡ Mk ¡ Mk 1t : M1 ¡ . . . ¡ Mk1 ¡ M p1q ¡ . . . ¡ M pLq ¡ Mk k k
t : M1
1
¡
. . . ¡ MK ,
1
¡
. . . ¡ MK ,
plq pl1 q H. A mechanism 1 1 plq where Mk L l1 Mk and for all l, l P t1, . . . , Lu, l l , Mk X Mk is L-separation strategyproof if for any L-separation pt, t1 q we have that
(1) f ptq ordinally dominates f pt1 q at t, and (2) f pt1 q ordinally dominates f ptq at t1 .
• pt, t1 q is called a multi-separation if t : M1
¡
. . . ¡ MK ,
p1q pL q p1q pL q t 1 : M1 ¡ . . . ¡ M 1 1 ¡ . . . ¡ MK ¡ . . . ¡ M K K ,
pl q
1 1 k where for all k P t1, . . . , K u, Mk L l1 Mk and for all l, l P t1, . . . , Lk u, l l , 1 plq pl q Mk X Mk H. A mechanism is multi-separation strategyproof if for any multiseparation pt, t1 q we have that
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(1) f ptq ordinally dominates f pt1 q at t, and (2) f pt1 q ordinally dominates f ptq at t1 .
In Lemmas 5, 6, and 7 let f be a mechanism that is separation monotonic, separation upper invariant, and separation lower invariant. Lemma 5. f is separation strategyproof. Proof. Consider any separation pt, t1 q. By separation upper and lower invariance, the allocation ° does not change for Mk with k κ, and since K k1 ∆pMk q 0, the allocation for Mκ must also remain constant, i.e., ∆pMκ q 0. Thus, f ptq ordinally dominates f pt1 q at t (weakly). On the other hand, by separation responsiveness,
κ¸1
¸
f pt1 q f ptq
P κ¸ 1
k 1 j Mk
¥
∆ p Mk q
pj q
¸
p 1q j PMκ
f pt1 q f ptq
pj q
(15)
∆pMκp1q q
(16)
k 1
0,
(17)
p1q q ¡ 0.
and thus, f pt1 q ordinally dominates f ptq at t1 , and the inequality is strict if ∆pMκ Lemma 6. f is L-separation strategyproof for any L P t2, . . . , mu.
Proof. Consider any L-separation pt, t1 q. As in the proof of Lemma 5, f ptq ordinally dominates f pt1 q at t. We must show that f pt1 q ordinally dominates f ptq at t1 as well. Note that a transition from t to t1 can be achieved by subsequently applying separations. Thus, separation upper and lower invariance already establish that ¯ k ¸ ¸
f pt 1 q f pt q
P
pj q 0
(18)
k 1 j Mk
for any k¯
P t1, . . . , K u. To conclude the proof, we need to verify that ¯ l ¸
∆pMκplq q ¥ 0
(19)
l 1
holds for all ¯l P t1, . . . , Lu. Consider a sequence of separations that leads to the following transition from t to t1 :
t tL : . . . ¡ Mκp1q Y Mκp2q Y . . . Y MκpLq tL1 : . . . ¡ Mκp1q
¡
Mκp2q Y . . . Y MκpLq
.. .
¡
...
(20)
¡
...
(21)
t1 t1 : . . . ¡ Mκp1q ¡ Mκp2q ¡ . . . ¡ MκpLq ¡ . . . .
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(22) (23)
p1q
In the first separation ptL , tL1 q the allocation for Mκ weakly increases by separation rep1q sponsiveness, and in all subsequent separations, the allocation for Mκ does not change by p1q separation upper invariance. Thus, ∆pMκ q ¥ 0, which shows (19) for ¯l 1. Next, consider the transition
t tL : . . . ¡ Mκp1q Y Mκp2q Y Mκp3q Y . . . Y MκpLq
tL1 : . . . ¡ Mκp1q Y Mκp2q .. . t1
t2 : . . . ¡ Mκp1q Y Mκp2q
t1
: . . . ¡ Mκp1q
¡
Mκp2q
¡
¡
Mκp3q Y . . . Y MκpLq
... ¡
(24) ...
(25) (26)
¡
¡
Mκp3q
¡
. . . ¡ MκpLq
. . . ¡ MκpLq
¡
¡
...
....
p1q
(27) (28)
p2q weakly increases in the
Again, by separation responsiveness, the allocation for Mκ Y Mκ first step, but does not change in any subsequent step. Therefore, ∆pMκp1q Y Mκp2q q ∆pMκp1q q
∆pMκp2q q ¥ 0,
(29)
which establishes (19) for ¯l 2. For ¯l P t3, . . . , Lu, (19) follows in the same fashion. Lemma 7. f is multi-separation strategyproof. Proof. As for L-separation strategyproofness, f ptq ordinally dominates f pt1 q at t by separation upper and lower invariance. For ordinal dominance of f pt1 q over f ptq at t1 , we need to show that
¯ 1 Lk k ¸ ¸
¯ l ¸
pl q ∆p M q k
k 1l 1
for all k¯
plq q
∆pMk¯
¥0
(30)
l 1
¯ consider the L¯ -separation pt, tk¯ with P t1, . . . , K u and ¯l P t1, . . . , Lk¯ u. For a given k, k ¯
tk : M1
¡
. . . ¡ Mk¯1
¡
p1q ¡ . . . ¡ M pLk¯ q ¡ M¯
Mk¯
¯ k
k 1
¡
. . . ¡ MK ,
(31)
which is reached via a sequence of separations that split the set Mk¯ , but leave the other sets unchanged. From Lemma 6 we know that ¯ l ¸
plq q ¥ 0
∆ft,tk¯ pMk¯
(32)
l 1
for all ¯l P t1, . . . , Lk¯ u. But by separation upper and lower invariance, the allocation for any of the pl q plq plq Mk¯ does not change when other sets Mk , k k¯ are separated. Thus, ∆ft,tk¯ pMk¯ q ∆ft,t1 pMk¯ q and ¯ l ¸
¯ l
plq q ¸ ∆f pM plq q ¥ 0, ¯ t,t1 k l 1 l 1 i.e., we get ordinal dominance of f pt1 q over f ptq at t1 . ∆pMk¯
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(33)
Lemma 8. For any type t, utility u P t, misreport t1
t, and utility u1 P t1, the line uα p1 αqu αu1 , α P r0, 1s (34) passes through a sequence of types t t0 , t1 , . . . , tS 1 , tS t1 , such that for all s P t1, . . . , S 1u either the pair pts , ts 1 q or the pair pts 1 , ts q is a multi-separation. Proof. For each type ts through which the line tuα : α P r0, 1su passes, select some utility uα P ts . Assume towards contradiction that for some s, neither the pair pts , ts 1 q nor the pair pts 1, tsq is a multi-separation. Then there must exist objects a, b, c, d (not necessarily different) such that under ts , a b and c d, but under ts 1 , a b and c d. This means that uα paq ¡ uα pbq, uα pcq uα pdq, uα paq uα pbq, uα pcq ¡ uα pdq. s
¡
¡
s
s
s 1
s
s 1
s
s 1
s 1
Taking the ‘‘average’’ of uαs and uαs 1 , we get a new utility function u ˜ u 1 pαs 2
q 2 uαs 1
αs
1
uαs
1
,
(35)
which lies on the line between u and u1 , but where
u ˜paq ¡ u ˜pbq and u ˜pcq ¡ u ˜pdq.
(36)
Thus, the line passes through a different type t˜ with u ˜ P t˜ between ts and ts the assumption that the line passes directly from ts to ts 1 .
1,
which contradicts
Proof of Theorem 1. First, we verify that a strategyproof mechanism satisfies all three axioms. Lemmas 1 through 4 establish this. The more involved direction of the proof is to show that any mechanism satisfying the axioms must be strategyproof.1 For type t, utility u P t, and misreport t1 select u1 P t1 according to Lemma 8. Let αs P r0, 1s be such that uαs P ts is a utility function on the line between u and u1 and in type ts of the sequence of types. By construction, αs
1 uαs
αsuα pαs 1 αsqu
(37)
s 1
and from multi-separation strategyproofness of f (Lemma 7), @
D
xuα , f ptsq f pts 1qy ¥ 0 and uα , f pts 1q f ptsq ¥ 0. Therefore, for all s P t0, . . . , S 1u, xu, f ptsq f pts 1qy ¥ 0. s
s 1
(38)
(39)
Summing over all s, we get @
u, f ptq f pt1 q
D
xu, f pt0q f ptS qy ¥ 0,
which concludes the proof. 1
The argument is similar to the proof of the local sufficiency result in (Carroll, 2012).
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(40)
References Carroll, Gabriel. 2012. ‘‘When Are Local Incentive Constraints Sufficient?’’ Econometrica, 80(2): 661--686. Mennle, Timo, and Sven Seuken. 2014. ‘‘An Axiomatic Approach to Characterizing and Relaxing Strategyproofness of One-sided Matching Mechanisms.’’ Working Paper.
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