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Fairness and group-strategyproofness clash
Sophie Bade∗
May 5, 2015
Abstract
No group-strategyproof and ex post Pareto optimal random matching mechanism treats equals equally. Every mechanism that arises out
of the randomization over a set of non-bossy and strategy proof mechanisms is non-bossy. So Random serial dictatorship, which arises out
of the randomization over deterministic serial dictatorships is on the
one hand non-bossy. On the other hand random serial dictatorship
cannot be group strategy proof as it treats equals equally and is ex
post Pareto optimal. In sum, the result that a deterministic matching
mechanism is group strategy proof if and only if it is non-bossy does
not extend to random matching mechanisms.
Keywords: random allocation mechanism, non-bossy, group strategyproof, random serial dictatorship. JEL Classification Numbers:
C78.
∗
Royal Holloway College, University of London and Max Planck Institute for Research
on Collective Goods, Bonn. I would like to thank Anna Bogomolnaia, Annika Johnson,
Debasis Mishra and Herve Moulin.
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1
Introduction
An ideal matching mechanism would be fair, efficient and impossible to manipulate. I show that no mechanism satisfies two rather weak criteria of
fairness, equal treatment of equals and ex post Pareto optimality, together
group-strategyproofness, which is a strong non-manipulability condition. In
a house matching problem a finite set of agents with linear preferences over
a finite set of houses needs to be matched to this set of houses. A random
matching mechanism is group strategyproof if no group can improve their
outcomes by lying about their true preferences. It is ex post Pareto optimal if it maps every profile of preferences to a lottery over Pareto optima.
A mechanism treats equals equally if any two agents who submit the same
preferences face the same lottery over houses.
The result complements Bogomolnaia and Moulin’s [1] result on the incompatibility between fairness, efficiency and non-manipulability. Bogomolnaia and Moulin [1] showed that no strategy proof, ordinally efficient mechanism satisfies equal treatment of equals. So Bogomolnaia and Moulin use
the same weak criterion of fairness: equal treatment of equals. The fairness
and non-manipulability assumptions of the two impossibility results differ:
where I only impose the weak efficiency requirement of ex post Pareto optimality Bogomolnaia and Moulin require ordinal efficiency. Conversely my
non-manipulability requirement is stricter than theirs. While I require that
no group may find it in their interest to submit false preferences, they only
impose this requirement for individuals.
A mechanism is non-bossy if no agent can change the outcome of another
agent without also changing his own. I show that any randomization over a
set of non-bossy and strategy proof mechanisms yields a non-bossy random
matching mechanism, implying that random serial dictatorship, which arises
out of a uniform randomization over the order of agents as dictators in serial
dictatorship, is non-bossy. On the other hand random serial dictatorship is
not group-strategyproof, given that it is ex post Parto optimal and treats
equals equally. This contrasts with Papai’s [2] result that a deterministic
matching mechanism is group strategyproof if and only if it is non-bossy and
strategyproof.
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2
Definitions
There is a set N of n agents and a finite set of houses H. The option of
homelessness ∅ ∈ H is always available. An n-vector x is a matching if
xi = xj and i 6= j imply xi = ∅. The interpretation is that agent i occupies
house xi under x. The set of all matchings is M. Agents are selfish in
the sense that they only consider their own houses when ranking different
matchings. Agent i’s preference on H is a linear order %i . So % is complete
and transitive and xi ∼ yi implies xi = yi . A profile of all agents’ preferences
(%i )i∈N is denoted %. The profile of preferences of all agents in some group
G is %G , the profile of all remaining agents’ preference is %−G . The set of
all profiles of preferences is Ω.
A mechanism φ maps the set of all preference profiles Ω to the set of
matchings M, where agent i obtains φi (%) under φ at the profile %. A
random mechanism ρ maps the set of all profiles of preferences Ω to ∆M,
where I adopt the convention that ∆S denotes the set of all lotteries on S,
for any finite set S. Under ρ agent i faces the lottery ρi (%) ∈ ∆H at % and
ρi (%)(xi ) is the probability that i is matched to xi under ρ(%).
A mechanism ρ : Ω → ∆X is (ordinally) group strategyproof if there
exists no strategy profile %, group of agents G and deviation %0G together
with expected utilities that are consistent with % such that all agents in G
weakly prefer ρ(%0G , %−G ) to ρ(%) while at least one member of G strictly
prefers ρ(%0G , %−G ) to ρ(%). When restricting attention only to groups G
consisting of single agents then ordinal group strategyproofness reduces to
(ordinal) strategyproofness. The mechanism ρ : Ω → ∆X satisfies equal
treatment of equals if any two agents who announce the same preferences
face the same distribution, so ρi (%) = ρj (%) if %i =%j . A mechanism ρ is ex
post Pareto optimal if ρ(%)(x) > 0 implies that x is Pareto optimal at %.
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Group Strategyproofness
Group strategyproofness clashes with even the mildest criteria of fairness and
efficiency.
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Theorem 1 Let there be n ≥ 3 agents and | H |≥ 3. No ex post Pareto
optimal, ordinally group strategyproof mechanism satisfies equal treatment of
equals.
Proof Suppose the ex-post Pareto optimal and group strategyproof mechanism ρ : Ω → ∆X did treat equals equally. Let {a, b, c} ⊂ H. Fix %∗ such
that a ∗i b ∗i c ∗i h for all i and all h ∈ H \ {a, b, c}. Let %01 and %◦2
be such that b 01 a and c ◦2 b, while they are otherwise identical to %∗i ,
so h %01 h0 ⇔ h %∗1 h0 if {h, h0 } =
6 {a, b} as well as h %◦2 h0 ⇔ h %∗2 h0 if
{h, h0 } =
6 {b, c}.
Since all n ≥ 3 agents prefer a, b and c to all other houses at %∗ , (%01 , %∗−1 )
and (%◦2 , %∗−2 ), the ex-post Pareto optimality of ρ implies that a, b and c
must be matched with probability 1 under ρ(%∗ ), ρ(%01 , %∗−1 ) and ρ(%◦2 , %∗−2 ).
Equal treatment of equals implies that under ρ(%∗ ) each agent obtains a, b, c
with probability n1 . Since ρ is ex post Pareto optimal, a may not be assigned
to 1 under ρ(%01 , %∗−1 ). Equal treatment of equals then implies that each agent
1
i 6= 1 receives a with probability n−1
under ρ(%01 , %∗−1 ). Since ρ is ordinally
strategyproof, agent 2 must obtain a with probability n1 under ρ(%◦2 , %∗−2 ).
Equal treatment of equals implies that all other agents equally share the
, implying ρi (%◦2 , %∗−2 )(a) = n1 for all i ∈ N .
remaining probability mass n−1
n
Since ρ is ex-post Pareto optimal b may not be assigned to 2 under ρ(%◦2 , %∗−2 ).
Equal treatment of equals implies that under ρ(%◦2 , %∗−2 ) each agent i 6= 2
1
obtains b with the same probability n−1
.
0
◦
∗
Finally consider ρ(%1 , %2 , %−{1,2} ). By (ordinal) strategyproofness agent
1
1’s probability to be assigned a or b must be n1 + n−1
, which is his probability
◦
∗
to be assigned a or b under ρ(%2 , %−2 ). In a similar vein, ordinal strategyproofness implies that agent 2’s probability to be matched with a under
1
ρ(%01 , %◦2 , %∗−{1,2} ) must be n−1
, which is his probability to be matched with
∗
0
a under ρ(%1 , %−1 ). When agents 1 and 2 deviate to announce %01 and %◦2
at %∗ agent 1’s probability to receive one of his two most preferred objects
and agent 2’s probability to receive his most preferred object both increase.
Consequently, ρ is not group strategyproof.
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4
Non-Bossiness under Randomization
Papai [2] showed that a deterministic mechanism is group strategyproof if
and only if it is strategyproof and non-bossy. For random mechanisms this
equivalence breaks down. Following Satterthwaite and Sonnenschein [3] ρ :
Ω → ∆X is non-bossy if ρi (%) = ρi (%0i , %−i ) ⇒ ρ(%) = ρ(%0i , %−i ) holds
for all triples (i, %, %0i ). Let there be a set of mechanisms M = {ρ1 , ρ2 , · · · ρK }
with ρk : Ω → ∆H for all 1 ≤ k ≤ K. Then the mechanism ρ∗ arises out
of a randomization over the set M if there exists a lottery π on {1, · · · , k}
such that
K
X
ρ∗ (%)(x) =
π(k)ρk (%)(x) for all x ∈ M.
1=k
If the lottery π is a uniform distribution over all mechanism in the set M , then
ρ∗ arises out of uniform randomization over M . An expected utility Ui
on some ∆H is consistent with preference %i over deterministic outcomes H,
if Ui (h) ≥ Ui (h0 ) holds if and only if h % h0 for all h, h0 ∈ H. A mechanism ρ
is ordinally strategy proof if and only if Ui (ρ(%)) ≥ Ui (ρ(%0i , %−i )) holds for
all profiles of preferences %, agents i, deviations %0i and expected utilities Ui
that are consistent with %i .
Theorem 2 Let ρ∗ : Ω → ∆X arise out of the randomization over a set
of ordinally strategyproof and non-bossy mechanisms {ρ1 , · · · , ρK } with ρk :
Ω → ∆X for all k. Then ρ∗ is non-bossy.
Proof Fix (i, %, %0i ) such that ρ∗i (%) = ρ∗i (%0i , %−i ). Suppose we had
∗
∗
ρki (%) 6= ρki (%0i , %−i ) for some 1 ≤ k ∗ ≤ K. Given that each ρk is ordinally
strategyproof, ρki (%) first order stochastically dominates ρki (%0i , %−i ) accord∗
∗
ing to %i . Since ρki (%) 6= ρki (%0i , %−i ) and since %i strictly ranks any two
∗
∗
different assignments ρki (%) must strictly dominate ρki (%0i , %−i ) at %i . So
for any Ui∗ that is consistent with %i we obtain Ui∗ (ρki (%)) ≥ Ui∗ (ρki (%0i , %−i ))
∗
∗
for all 1 ≤ k ≤ K and Ui∗ (ρki (%)) > Ui∗ (ρki (%0i , %−i )). Letting π be the lottery from which the mechanisms {ρ1 , · · · , ρK } are drawn to construct ρ∗ we
PK
PK
∗ k
∗ k
0
obtain Ui∗ (ρ∗ (%)) =
k=1 π(k)Ui (ρ (%)) >
k=1 π(k)Ui (ρ (%i , %−i )) =
Ui∗ (ρ∗ (%0i , %−i )) contradicting ρ∗i (%) = ρ∗i (%0i , %−i ). So ρki (%) = ρki (%0i , %−i )
must hold for all 1 ≤ k ≤ K. Since every ρk is non-bossy the preced5
ing k equalities imply that ρk (%) = ρk (%0i , %−i ) holds for all 1 ≤ k ≤ K.
P
P
So ρ∗ (%)(x) = k π(k)ρk (%)(x) = k π(k)ρk (%0i , %−i )(x) = ρ∗ (%0i , %−i )(x)
holds for any x. This is none other than ρ∗ (%) = ρ∗ (%0i , %−i ), and ρ∗ is
non-bossy.
Theorem 2 directly extends to any space of discrete allocations. To see
that strategyproofness cannot be dropped from Theorem 2 consider a matching problem with n = 3 and and H = {a, b, c}. Let ρ◦ : Ω → ∆X arise out
of the uniform randomization over α, β and γ, defined by the following table
(where %∗1 is such that a ∗1 b ∗1 c):
%1 =%∗1
%1 6=%∗1
α(%)
β(%)
γ(%)
(a, c, b) (b, c, a) (c, b, a)
(b, a, c) (c, a, b) (a, c, b).
Agent 1 alone determines the matching in the mechanisms α, β and γ.
Moreover, in each of these three mechanisms agent 1’s match changes if and
only if all other agents’ matches do so too. Therefore α, β and γ are all nonbossy. For any % agent 1 obtains each house with probability 31 under ρ◦ (%).
However, the other agents’ probabilities over houses do depend on agent
1’s announcement. Agent 2, for example, never obtains house a if agent 1
announces %∗1 and if agent 1 announces a different preference agent 2 obtains
house a with a probability of 23 . So ρ◦ is bossy. In a similar vein, non-bossy
mechanisms might arise out of the randomization over bossy mechanisms. It
does not extend to a larger domain of preferences that allows for indifferences.
Examples to proves these two claims are available on request.
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Random Serial Dictatorship
The serial dictatorship sd is a deterministic mechanism that lets agents sequentially choose from the set of all remaining houses, such that sd1 (%)
is agent 1’s is most preferred house according to %1 , sd2 (%) is the %2 preferred house among all remaining ones and so forth. For any permutation
p : N → N on the set of agents we can generate another serial dictatorship sdp by using p to reorder the agents as dictators. Any such serial dictatorship sdp is strategyproof and non-bossy. Random serial dictatorship
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rsd : Ω → ∆X arises out of a uniform randomization over all deterministic
serial dictatorships {sdp | p : N → N a permutation }. Since serial dictatorship is strategyproof and non-bossy for any order of agents as dictators,
Theorem 2 immediately yields the following corollary.
Corollary 1 Random serial dictatorship is non-bossy.
Since random serial dictatorship is ex post Pareto optimal and treats
equals equally, random serial dictatorship is by Theorem 1 not group strategy
proof. In sum random serial dictatorship shows that the equivalence of groupsrategyproofness and non-bossieness that was established for deterministic
mechanisms by Papai [2] does not extend to the case of random matching
mechanisms.
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This Section is not for publication, it contains the examples that are available on request.
Example that randomizing over bossy mechanisms may yield a nonbossy one.
Let n = 3 and H = {a, b, c} and let ρ arise out of the uniform randomization over the two “bossy serial dictatorships” bsd1 and bsd2 that are
identical to sd with the exception that agent 3 is the second dictator under bsd1 when agent 1 announces the preference a 1 c 1 b and under
bsd2 when agent 1 announces a 1 b 1 c. To see that bsd1 and bsd2 are
indeed bossy fix % such that all three agents prefer a to b to c, and note
that bsd1 (%) = (a, b, c) whereas bsd2 (%) = (a, c, b). Now let agent 1 deviate
to announce a 01 c 01 b . Given that agent 1 ranks a at the top under
%01 , a = bsd11 (%01 , %−1 ) = bsd21 (%01 , %−1 ) holds. However, the second dictator
under % differs from the second dictator under (%01 , %−1 ) under both mechanisms. We therefore obtain bsd1 (%01 , %−1 ) = (a, c, b) 6= (a, b, c) = bsd1 (%)
as well as bsd2 (%01 , %−1 ) = (a, b, c) 6= (a, c, b) = bsd2 (%), so bsd1 and bsd2 are
both bossy.
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To see that ρ is non-bossy, note that ρ also arises out of the uniform
randomization over the classic serial dictatorship sd and another non-bossy
variant of serial dictatorship vsd that differs from the original only insofar
as that agent 3 becomes the second dictator if and only if agent 1 chooses
a. Since sd and vsd are both strategyproof and non-bossy, ρ is non-bossy by
Theorem 1.
Example that randomizing over non-bossy strategy proof mechanisms when the domain of preferences includes preferences according to which agents may be indifferent among houses may yield
a bossy mechanism.
Let n = 4, H = {a, b, c, d} and consider the domain of preference profiles
{%, (%∗1 , %−1 ), (%◦1 , %−1 ) |%∈ Ω} where a ∼∗1 b ∗1 c ∗1 d and a ∼◦1 b ◦1 d ◦1
c. So the only difference between this domain and Ω is that agent 1 might
be indifferent between a and b when he ranks these two houses above the
other two. Letting sd0 be the serial dictatorship in which agent 1 is the first,
3 the second and 2 the third dictator define two mechanisms α, β as follows:
Let α(%∗1 , %−1 ) = sd(%a1 , %−1 ), β(%∗1 , %−1 ) = sd(%b1 , %−1 ), α(%◦1 , %−1 ) =
sd0 (%b1 , %−1 ) and β(%◦1 , %−1 ) = sd0 (%a1 , %−1 ) where a a1 b a1 c a1 d and
b b1 a b1 c b1 d. If according to % agent 1 strictly prefers a to all other
houses then let α(%) = sd(%) and β(%) = sd0 (%). In all remaining cases let
α(%) = sd0 (%) and β(%) = sd(%).
The mechanisms α, β are both strategyproof: agent 1 is always assigned
one of his most favorite houses, so truthfully revealing his preferences is a
dominant strategy for him. For the remaining agents α and β are identical
to serial dictatorships, which also means that for these agents truthfully
revealing their preferences is a dominant strategy. The fact that for agents
2,3 and 4 α and β are identical to serial dictatorships implies that none of
these agents can change anyone else’s assignment without changing their own
assignment. Finally suppose that α1 (%) = α1 (%01 , %−1 ). If α1 (%) = a then
the assignments of all other agents follow sd, so we have α(%) = α(%01 , %−1 ).
If not, that is if α1 (%) 6= a, then the assignments of all other agents follow
sd0 , so also in this case we have α(%) = α(%01 , %−1 ). In sum α is non-bossy.
By the same logic β is also non-bossy.
However the random mechanism ρ that uniformly randomizes over α and
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β is bossy. Let % be such that %1 =%∗1 and a i b i c i d for i = 1, 2, 3.
Under ρ(%) agent 1 obtains houses a and b with equal probability. Agent 2
is the second dictator under α(%) as well as under β(%). Given that agent
2 prefers houses a and b to all other houses, he chooses a if 1 was matched
with b and b if 1 was matched with a. So agent 2 also obtains houses a
and b with equal probability under ρ(%). Now consider ρ(%◦1 , %−1 ). Agent 1
obtains house a under β(%◦1 , %−1 ) and house b under α(%◦1 , %−1 ). So agent
1 faces the same lottery under ρ(%) as under ρ(%◦1 , %−1 ). Agent 3 is the
second dictator under β(%◦1 , %−1 ) as well as under α(%◦1 , %−1 ). For the given
preferences, agent 3 is matched with house a under α(%◦1 , %−1 ) and with
house b under β(%◦1 , %−1 ). Consequently, agent 2 is never matched with
either a or b under ρ(%◦1 , %−1 ). So we have found a profile of preferences
% and a deviation %◦1 for agent 1 such that ρ1 (%) = ρ1 (%◦1 , %−1 ) while at
the same time ρ2 (%) 6= ρ2 (%◦1 , %−1 ). In sum, ρ is bossy; we cannot drop the
requirement that agents strictly rank all possible assignments from Theorem
1.
References
[1] Bogomolnaia, A. and H. Moulin: “A New Solution to the Random Assignment Problem,” Journal of Economic Theory, 100, (2001), 295-328.
[2] Papai, S.: “Strategyproof Assignment by Hierarchical Exchange”,
Econometrica, 68, (2000), pp. 1403 - 1433.
[3] Satterthwaite M. and H. Sonnenschein: “Strategy-Proof Allocation
Mechanisms at Differentiable Points,” Review of Economic Studies, 48,
(1981), pp. 587 - 597.
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