Pareto-Optimal Matching Allocation Mechanisms for Boundedly
Rational Agents
Sophie Bade∗†
December 11, 2014
Abstract
Is the Pareto optimality of matching mechanisms robust to the introduction of boundedly
rational behavior? To address this question I define a restrictive and a permissive notion
of Pareto optimality and consider the large set of hierarchical exchange mechanisms which
contains serial dictatorship as well as Gales top trading cycles. Fix a housing problems
with boundedly rational agents and a hierarchical exchange mechanism. Consider the set of
matchings that arise under all possible assignments of agents to initial endowments in this
mechanism. I show that this set is nested between the sets of Pareto optima according to
the restrictive and the permissive notion. These containment relations are generally strict,
even when deviations from rationality are minimal. In a similar vein, minimal deviations
from rationality suffice for the set of outcomes of Gale’s top trading cycles for all possible
initial endowments to differ from the set of outcomes of serial dictatorship for all possible
orders of agents as dictators.
Keywords: Fundamental Theorems of Welfare, House Allocation Problems, Bounded Rationality, Multiple Rationales. JEL Classification Numbers: C78, D03, D60.
1
Introduction
Boundedly rational behavior should be expected in some of the non-market environments for
which economists have designed matching mechanisms. Take kidney allocation problems as an
∗
Royal Holloway College, University of Londona and Max Planck Institute for Research on Collective Goods,
Bonn [email protected]
†
I would like to thank Christoph Engel, Olga Gorelkina, Martin Hellwig, Aniol Llorente-Saguer, Michael
Mandler, Paola Manzini, and two anonymous referents. I would like to thank David Bielen for thorough research
assistance.
1
example. One difficulty with mechanisms that match donors to recipients is that doctors are
reluctant to state complete and transitive preferences over kidneys. However, the same doctors
do not seem to have any problem choosing the “best” kidney for a particular patient from a
given set. This apparent contradiction could arise from doctors having only limited resources for
tests.1 Alternatively, consider the allocation of elementary school slots. The choices of a family
in which the mother strategically whittles down the options before the father picks a school
are only rationalizable if the parents’ preferences are aligned.2 As a third example consider
the choice of a medical residency program. To reduce the enormous complexity of the choiceproblem med-school graduates might first narrow the set of programmes using a sequence of
incomplete rankings, to then consider a few remaining alternatives in detail.3
Is the Pareto optimality of matching mechanisms robust to the introduction of boundedly
rational behavior? To answer this question, I consider Papai’s [28] hierarchical exchange mechanisms, which comprise many theoretically and practically relevant matching mechanisms.4 I
derive two notions of preference from the agents’ choice functions: Saying that an agent lightly
prefers x to y if the agent chooses x from some set containing both x and y, I construct a
restrictive notion of Pareto optimality; saying that an agent solidly prefers x to y if he never
chooses y when x is also available leads to a permissive notion of Pareto optimality.
In line with standard matching theory I find that any Pareto optimum that satisfies the
restrictive notion can be obtained as the outcome of any fixed hierarchical exchange mechanism
for some initial endowment and any outcome of a hierarchical exchange mechanism is Pareto
optimal according to the permissive notion. However, different from the case of rationalizable
behavior, the set of outcomes of hierarchical exchange is strictly nested between the two Pareto
sets. Different hierarchical exchange mechanisms may, moreover, cover different sets of matchings in housing problems with boundedly rational agents. All findings hold for any degree of
1
These statements reflect a private conversation with Utku Unver, who was involved in the design and practical
implementation of several kidney exchange mechanisms. Consider the task to find the “best” kidney for a patient
out of a set S = {a, b, · · · } of ten kidneys. Financial constraints might force doctors to use some preliminary tests,
to limit the set of kidneys which they test in detail. This procedure yields the choices a = c(S) and b = c({a, b})
if b is eliminated be the preliminary tests, while b turns out to be better than a according to the detailed test.
2
I use the decision theoretic notion of rationalizability: a choice correspondence is rationalizable if there exists
some transitive and complete preference relation %, such that the choice correspondence maps any consumption
set S to the set of %-maximal elements in that set S. Choice functions that can be derived from such interactive
procedures have been characterized by Xu and Zhou [36] and by Apesteguia and Ballester [4].
3
Manzini and Mariotti [26] and by Mandler [25] characterize choices functions that arise out of such procedures.
4
Some subsets of the class of hierarchical exchange mechanisms have been described by Abdulkadiroglu and
Sonmez [2], Svensson [34], Ergin [16], Ehlers, Klaus, and Papai [15], Ehlers and Klaus [13], Kesten [21], Ehlers
and Klaus [14], and Velez [35].
2
irrationality.
2
Matchings and Hierarchical Exchange
Fix N = {1, · · · n}, a set of agents and H, a set of equally many objects, called houses. A
submatching σ : Nσ → Hσ is a bijection with Nσ ⊂ N and Hσ ⊂ H; σ(i) is agent i’s
match under σ. If Nµ = N , µ is a matching. Matchings are also denoted as vectors with
the understanding that the ith component of µ represents µ(i). The sets of all matchings
and respectively of all submatchings, that are not themselves matchings, are M and M. The
submatching under which no one is matched, ∅ is an element of M. The sets of unmatched
agents and houses at some σ ∈ M are N σ and H σ . If σ(i) = σ 0 (i) for all i ∈ Nσ ⊂ Nσ0 holds
for two submatchings σ, σ 0 , then σ is a submatching of σ 0 (σ ⊂ σ 0 ).
I use Pycia and Unver’s [30] ingenious terminology to define Papai’s [28] hierarchical exchange mechanisms. For any fixed σ ∈ M define an onwership function oσ : H σ → N σ ,
with the understanding that agent oσ (x) owns house x at the submatching σ. A hierarchical exchange mechanism is defined through a set of ownership functions o = (oσ )σ∈M where
oσ (x) = oσ0 (x) holds for any two submatchings σ ⊂ σ 0 with oσ (x) ∈
/ Nσ0 and x ∈
/ Hσ0 . So
ownership must persist in the sense that an agent i ∈
/ Nσ0 must own a house x ∈
/ Hσ0 at σ 0 if
i owns x at a submatching σ of σ 0 . The outcome of any hierarchical exchange mechanism is
determined through the following trading process.5
At the start let σ1 = ∅ and k = 1. Round k: each house in H σk points to its owner according
to oσk , each agent i ∈ N σk points to a house in H σk . At least one cycle forms. Calculate σk+1
as σk+1 (i) = σk (i) for i ∈ Nσk and σk+1 (i) = x if i points to x in one of the cycles. Terminate
the mechanism if if σk+1 is a matching. If not, go on to round k + 1.
At the start of a hierarchical exchange mechanism, agents are asked to point to houses.
Houses in turn point to their owners. At least one cycle of agents and houses forms. Any agent
in such a cycle is matched with the house he is pointing to and leaves the mechanism. When an
owner of multiple houses leaves, his unmatched houses are passed on to the remaining agents
according to some fixed inheritance rule, implied by the ownership functions. Agents are once
again asked to point to the remaining houses and the same procedure is repeated until each
5
The restriction to hierarchical exchange mechanisms is not costless. Pycia and Unver [30] define a class of
problems in which hierarchical exchange mechanisms are strictly Lorenz-dominated by some other strategy proof,
Pareto optimal and non-bossy mechanisms. Abdulkadiroglu, Che and Yasuda [3] show that the use of ordinal
mechanisms when agents have cardinal utilities may lead to welfare losses, Pycia [31] shows that these losses can
be arbitrarily large.
3
agent is matched. Serial dictatorships ô and Gale’s top trading cycles mechanism o
(as defined by Shapley and Scarf [33]) are hierarchical exchange mechanisms. The former arises
when ôσ (h) only depends on | Nσ | the number of agents already matched under σ, the latter
arises if o∅ (h) 6= o∅ (h0 ) holds for all h 6= h0 .
For any fixed hierarchical exchange mechanism and any permutation p : N → N define a
permuted hierarchical exchange mechanism po through (po)σ (h) = p(oσ◦p (h)) for all
σ ∈ M.6 Under po agent p(i) takes on the role of agent i under o. If agent i is the ithe dictator
according to the serial dictatorship ô then agent p(i) is the ith dictator according to pô. If in
some hierarchical exchange mechanism o agent 1 is endowed with houses {e, g, h} at the start
of the mechanism (o∅ (e) = o∅ (g) = o∅ (h) = 1), then agent p(1) is endowed with theses houses
at the start of po.
3
Boundedly Rational Behavior
Fixing N and H, a housing problem is a profile c := (ci )i∈N , where ci : P(H) \ ∅ → H is agent
i’s choice function and ci (S) ∈ S is agent i’s choice from the set S.7 Agent i lightly prefers
house x over house y, formally xPi∃ y, if there exists a set of houses S ⊂ H, such that x, y ∈ S
and x = ci (S). Agent i solidly prefers house x over house y, formally, xPi∀ y, if y 6= ci (S) for
all S ⊂ H with x, y ∈ S. So xPi∀ y holds if there exists no set, such that y is chosen when x is
available; conversely, xPi∃ y holds if x is chosen from at least one set that contains y. A matching
µ0 P ∀ -Pareto-dominates (P ∃ -Pareto-dominates) another matching µ0 6= µ if µ0 (i) 6= µ(i)
implies µ0 (i)Pi∀ µ(i) (µ0 (i)Pi∃ µ(i)) for all i. A matching µ is P ∀ -Pareto-optimal (P ∃ -Paretooptimal) if there exists no matching µ0 that P ∀ -Pareto-dominates (P ∃ -Pareto-dominates) it.8
The mechanism o implements o(c) in a housing problem c, if the matching o(c) results
when any agent at any round of the mechanism points to his choice out of all remaining houses.9
6
7
Abusing notation let p be the restriction of the original permutation p under which σ ◦ p is well-defined.
Standard housing problems (profiles of linear orders (%i )i∈N on H) are embedded in the set of housing
problems. Given that agents are represented via choice functions not choice correspondences, the presence of
boundedly rational behavior is the only difference between the present and the standard definition of housing
problems.
8
The notion of solid preference P ∀ is identical with (or very similar to) the notions of preference that Bernheim
and Rangel [10], Mandler [24], and Green and Hojman [19] use to compare outcomes in terms of individual
and collective welfare. Rubinstein and Salant [32] show that this notion may not generate the relevant welfare
preference.
9
In a working paper version I show that Theorem 1 extends to more general assumptions on behavior. Differently from my assumptions on behavior de Clippel [12] assumes that agents base their decision only on one choice
set.
4
If c is rationalizable, agent i points to his most preferred house among the remaining houses
at any round. A mechanism o is said to p-implement a matching µ in housing problem c if
µ = (po)(c) holds for some permutation p.
4
The Result
With boundedly rational behavior hierarchical exchange mechanisms are Pareto optimal in the
following sense:
Theorem 1 Fix a housing problem c and a hierarchical exchange mechanism o. Any P ∃ Pareto optimum in c can be p-implemented by the given mechanism. Any matching that is
p-implementable in c by o is P ∀ -Pareto optimal in c.
Proof
To prove the first part fix a P ∃ -Pareto-optimal µ. To see that there exists an ordering of
agents such that µ(i)Pi∀ µ(k) holds whenever k is later in the ordering than i, suppose µ(i)Pi∀ H
did not hold for any agent i. So for each agent i there exists a set Si ⊂ H, such that µ(i) ∈ Si
and µ(i) 6= ci (Si ). Now let all agents point to the agent who is matched with ci (Si ) under µ.
The matching µ0 , with µ0 (i) = ci (Si ) for any agent i in some pointing cycle and µ(i) = µ0 (i)
otherwise, P ∃ -Pareto dominates µ. Given this contradiction to the P ∃ -Pareto optimality of µ,
some agent, say 1, must solidly prefer his match under µ to all other houses (µ(1)Pi∀ µ(k) for all
k > 1). Since the restriction of the matching µ to the subsets of agents {2, · · · , |N |} and houses
H \ {µ(1)} is also P ∃ -Pareto-optimal, the inductive application of the above arguments implies
that the agents can be ordered such that µ(i)Pi∀ µ(k) holds whenever k is later in the ordering
than i.
For each i let µi ⊂ µ be such that Nµi : = {1, · · · , i−1}. Define p such that (po)µi (µ(i)) = i
for all i ∈ N .10 To see that (po)(c) = µ holds I show that any round of the trading cycles
process must start and end with submatchings of µ. The first round starts with ∅ ⊂ µ. Now
consider any round which starts with some σ ⊂ µ. Fix an agent j ∈
/ Nσ . Since µ(j)Pj∀ µ(i) holds
for all i > j, agent j points to a house µ(k) with k ≤ j.
To see that at σ house µ(i) points to an agent j with j ≤ i suppose that that j > i owned
a house µ(i) at σ under po. Let σ ∗ ⊂ µ be such that Nσ ∪ {1, · · · , i − 1} = Nσ∗ . Since σ ⊂ µ
and µ(i) ∈
/ Hσ agent i is not matched at σ. The definition of p requires (po)µi (µ(i)) = i. Since
po is a hierarchical exchange mechanism, since µi ⊂ σ ∗ , i ∈
/ Nσ∗ and µ(i) ∈
/ Hσ∗ , we have
(po)σ∗ (µ(i)) = i. By the same token σ ⊂ σ ∗ , (po)σ (µ(i)) = j, j ∈
/ Nσ∗ (since j > i) and
10
See Bade [7] for a discussion that p is well-defined.
5
µ(i) ∈
/ Hσ∗ implies (po)σ∗ (µ(i)) = j a contradiction. We can conclude that (po)σ (µ(i)) = j
implies that i ≥ j.
The two preceding paragraphs imply that at σ each unmatched house µ(i) points to an agent
j ≤ i and each unmatched agent j points to a house µ(k) with k ≤ j. Consequently any cycle at
σ involves just one agent j and his match µ(j). So σ 0 ⊂ µ holds for the submatching σ 0 which
is reached through matching cycles at the round starting with σ. In sum we have (po)(c) = µ,
proving the first part of Theorem 1.
To see the second part of Theorem 1 fix any µ = (po)(c) Assume w.l.o.g. that agents
{1, · · · , j} are matched in the first stage of the mechanism. So for each i ≤ j µ(i) is Pi∀ -optimal
in H. By the same argument, house µ(i) is Pi∀ -optimal in H \ {µ(1), · · · , µ(j)} if agent i is
matched in the second stage. Proceeding inductively, we see that µ is P ∀ -Pareto-optimal.
To see that the set matchings implementable through hierarchical exchange is generally
strictly nested between the sets of P ∃ - and P ∀ -Pareto optima consider the following two examples. Example 1 shows that some P ∃ -Pareto-inferior matchings are p-implementable by any
hierarchical exchange mechanism. Example 2 shows that not every P ∀ -Pareto-optimal matching
is p-implementable. For both examples let H = {x, y, z, w}, N = {1, 2, 3, 4} and let c∗i be the
choice function that is rationalizable by x ∗i y ∗i z ∗i w. Arbitrarily fix all choices that are
not explicitly mentioned in the upcoming examples.
Example 1 Let c∗1 = cα1 , c∗2 = cα2 , cα3 (S) = y if y ∈ S, cα3 ({x, z, w}) = w, cα3 ({z, w}) = z,
cα4 (S) = x if x ∈ S, cα4 ({y, z, w}) = z, and cα4 ({z, w}) = w. The matching µα = (x, y, z, w) is not
P ∃ -Pareto-optimal in cα since cα3 ({x, z, w}) = w and cα4 ({y, z, w}) = z imply wP3∃ z and zP4∃ w.
Fix any hierarchical exchange mechanism o and let σ ∗ be the submatching that only matches 1
and x. Define p such that (po)∅ (x) = 1 and (po)σ∗ (y) = 2. If there are exactly two owners
at (po)σ∗ , 3 is the other owner, if there are three owners, let (po)σ∗ (w) = 4. In the first
round of the mechanism agents 1, 2 and 4 point to x and 3 points to y = c3 (H). Each house
h points to its owner (po)∅ (h), so house x points to 1. Moreover, y cannot point to 3, since
since (po)σ∗ (y) = 2 implies (po)∅ (y) 6= 3. Exactly one cycle forms, and the submatching σ ∗
is reached. At σ ∗ agents 2 and 3 both point to y = cαi ({y, z, w}) for i = 2, 3. Given that agent
2 owns house y at σ ∗ , agent 2 and y form a cycle. This is the only cycle: if 4 owns a house at
σ ∗ he points to cα4 ({y, z, w}) = z which is owned by 3 at σ ∗ . Only 3,4, z and w are left in the
next round. Since cα3 ({z, w}) = z and cα4 ({z, w}) = w the desired matching obtains.
Example 2 Let cβi = c∗i for i 6= 3, cβ3 (S) = y if y ∈ S, cβ3 ({x, z, w}) = z, and cβ3 ({z, w}) = w.
The matching µβ = (x, y, z, w) is P ∀ -Pareto optimal in cβ . Suppose o(cβ ) = µβ held for some
6
hierarchical exchange mechanism o. In the first round of o(cβ ) agents 1,2 and 4 point to x and
3 points to y = c3 (H). Since the first round must lead to a submatching σ ⊂ µβ , only agent 1
and house x are matched in that first round. By the same logic, only agent 2 and house y are
matched in the second round. In the third round, agents 3 and 4 point to cβ3 ({z, w}) = w and
cβ4 ({z, w}) = z, contradicting o(cβ ) = µβ .
The next example shows that serial dictatorships and Gale’s top trading cycles may pimplement different sets of matchings with boundedly rational agents. The example sheds some
light on possible extensions of the growing literature on the equivalence between random serial
dictatorship and other random matching mechanisms. According to random serial dictatorship
the order of all agents as dictators is drawn from a uniform distribution over all such orders.
Abdulkadiroglu and Sonmez [1] and Knuth [22] independently found that random serial dictatorship is identical to the “core from random endowments” which starts Gale’s top trading cycles
from an endowment that has been randomly drawn from a uniform distribution over all possible
endowments.11 Example 3 shows that the supports of the two random matching mechanisms
differ with boundedly rational behavior.
Example 3 Let H = {x, y, z}, N = {1, 2, 3} cγ1 ({x, y, z}) = x, cγ1 ({x, z}) = z, cγ2 ({x, y, z}) = y,
and cγ2 ({y, z}) = z. Let agent 3’s behavior be rationalizable by x γ3 y γ3 z. Gale’s top trading
cycles with the initial endowment µγ = (x, y, z) implements µγ in cγ given that agents 1 and 2
choose µγ (1) and µγ (2) out of the grand set. For µγ to be p-implementable via serial dictatorship
either agent 1 or agent 2 has to be the first dictator. If agent 1 is the first dictator, neither one
of the two remaining agents would pick µγ (i) as the second dictator. The same holds for the
case in which agent 2 is the first dictator.
The choice behavior assumed in Examples 1, 2, and 3 is not wildly irrational. Quite to the
contrary the behavior in each of these three examples only minimally deviates from rationalizability according to any theory on the degree of irrationality. Different theories of boundedly
rational behavior entail different measures of the degree of irrationality present in a choice
function. Behavior that is sequentially rationalizable following Manzini and Mariotti [26] is
minimally irrational if two rationales suffice to explain it. Behavior that can be explained as
choices via checklist following Mandler [25] is minimally irrational if the checklist has length two.
The minimal game tree that may explain boundedly rational behavior following Xu and Zhou
[36] has two agents and two nodes. Kalai, Rubinstein and Spiegler [20], Ambrus and Rozen [6],
11
This result has been extended to larger sets of mechanisms by Carroll [11], Pathak and Sethuraman [29], Liu
and Pycia [23] and Bade [9].
7
Apesteguia and Ballester [5] or Manzini and Mariotti [27] define yet further measures for the
degree of irrationality present in a choice function.
All these theories agree that ci has to violate WARP at least once to qualify as boundedly
rational. To judge whether ci is minimally irrational according to any one of theories mentioned
above we need to know ci (S) for all possible choice sets. Consider the example of a choice function
ci on P(X) \ ∅ with X = {x, y, z}, ci ({x, y, z}) = x and ci ({x, y}) = y. For ci to be minimally
irrational some theories require the choice ci ({y, z}) = z others do not.12 What stands out about
Examples 1, 2, and 3 is that all choice functions in these examples are either rationalizable or
they violate WARP exactly once. The omission of some (arbitrarily fixed) choices turns out to be
more than a notational convenience. Given that the omitted choices were not used to establish
any of the points made in Examples 1, 2, and 3 we may fix them to fit any desired notion of
minimal irrationality. In sum we obtain that, no matter how little irrationality we permit in
housing problems and no matter which theory we use to measure the degree of irrationality, some
P ∀ −Pareto-optimal matchings are not implementable by any hierarchical exchange mechanism
and some P ∃ -Pareto dominated matchings are p-implementable by any hierarchical exchange
mechanism. Finally, serial dictatorship and Gale’s top trading cycles p-implement different sets
of matchings - even if we allow only minimal deviations from rationalizability.
5
Conclusion
Hierarchical exchange mechanisms can be viewed as a version of free trade in matching environments, where indivisible goods have to be matched to agents without recourse to prices. At any
moment in the mechanism, each house is owned by someone, in the sense that the owner can
freely appropriate or exchange the house. Hierarchical exchange mechanisms allow for a broad
and fine spectrum of initial endowments, ranging from maximal to minimal inequality (from
serial dictatorship to Gale’s top trading cycles).13 Since the results presented here apply to
all hierarchical exchange mechanisms, they automatically cover any subset thereof. The results
apply in particular if we adopt a more restrictive notion of free trade for matching environments
such as top trading cycles mechanisms following Abdulkadiroglu and Sönmez [2] or Gale’s top
trading cycles.
12
For ci to be rationalizable by the applications of two sequential rationales following Manzini and Mariotti
[26] ci ({y, z}) = z must hold. However, in the framework of Kalai, Rubinstein and Spiegler [20], two rationales
suffice to rationalize ci , whether we let ci ({y, z}) = z or ci ({y, z}) = y.
13
The analogy has its limits. Owners are, for example, neither allowed to destroy their houses nor are they
allowed to determine the heirs of their houses as they leave the mechanism.
8
Identifying hierarchical exchange mechanisms with free trade the main results of the paper
can be interpreted as versions of the first and second fundamental theorem of welfare economics
for the case of boundedly rational behavior.14 The second part of Theorem 1 corresponds to a
First Welfare Theorem for solid preferences: any matching that arises out of free trade is P ∀ Pareto optimal. The first part corresponds to a Second Welfare Theorem for light preferences:15
any P ∃ -Pareto optimum can be p-implemented by any hierarchical exchange mechanism. Examples 1 and 2 show that the respective stronger versions of the two fundamental theorems
do not hold: Some matchings that arise out of free trade are not P ∃ -Pareto optimal and some
P ∀ -Pareto optima cannot be achieved through free trade.
As a further step one could explicitly model the reasons for particular forms of bounded
rationality and/or decision procedures. One could, for example, assume that patients do have
(linear) preferences over kidneys, but that it is costly to learn theses preferences. In this case the
observed bounded rationality can be derived from a fully rational (but unobserved) preference.
An allocation mechanism would then interact with some form of strategic information acquisition. Bade [8] shows that serial dictatorship is the only ex ante Pareto optimal, non-bossy and
strategy proof mechanism in a matching environment with endogenous information acquisition.
Similarly, one could explicitly model the interaction between family members when selecting a
mechanism for school choice.
14
All fundamental theorems of welfare with boundedly rational agents that I am aware of concern market envi-
ronments with divisible goods. Bernheim and Rangel [10] prove a first fundamental theorem of welfare economics
for markets that are standard except for the assumption that the agents’ behavior need not be rationalizable.
Their notion of Pareto optimality relies on a notion of preferences that is very similar to the solid preferences
defined here. This result aligns with the first inclusion relation of Theorem 1. Interestingly, Mandler [24] proves
a version of the second theorem of welfare economics that also defines Pareto optimality with respect to P ∀ preferences. This discrepancy is caused by Mandler’s [24] assumption that agents act fully rational according
their solid preferences. So in his work the choice functions only serve to construct these preferences, individuals
are always willing to select any preference-maximal element of a choice set. Very differently, I not only use the
choice functions of agents to construct the P ∀ -preferences; I also impose that for any agent’s choice in a mechanism there needs to be some set that is consistent with the underlying facts, such that the agent’s choice can
be construed as a choice from this set. The same arguments apply to the difference between my results and the
ones by Fon and Otani [17]. However there is a second difference between their work and mine, they assume
intransitive and/or incomplete preferences. The assumption of such preferences rules out many irregularities that
are permissible under the present framework.
15
Due to the finiteness of matching problems this Second Welfare Theorem does without local nonsatiation or
convex uppercontour sets.
9
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10
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