Technische Universität München
Tayfun Sönmez: Strategy-Proofness and
Essentially Single-Valued Cores
Econometrica, Vol. 67, No. 3 (May, 1999), pp. 677-689.
Seminar Economics and Computation
19.6.2013
Dominik Jurek
Technische Universität München
Two Sides of Game Theory
Non-cooperative
games
Cooperative games
players make decisions
independently
groups of players
("coalitions") may
enforce cooperative
behavior
Today’s Talk:
Providing a link
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What is it about? Repetition
Consider the Housing Market Problem: Each agent owns one
indivisible good, e.g. a house and has strict preferences over all
houses.
Katharina told us: The Top Trading Cycle Mechanism is a Pareto
efficient, individually rational and strategy-proof mechanism.
Roth and Postlewaite (1977) prove: There is one favorable
allocation in the “core”, which is assigned by the Top Trading
Cycle Mechanism.
Question: Are these traits of the Top Trading Cycle Mechanism
related to the outcome?
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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Basic Concepts
Pareto Efficiency: A state of economic allocation of resources in
which it is impossible to make any one individual better off
without making at least one individual worse off.
Individual Rationality: No agent is worse off than he would be on
his own, i.e. every agent has a motivation to participate in the
economy.
Non-cooperative Game Theory:
Cooperative Game Theory:
Strategy-Proofness:
Core:
No agent can ever benefit by
unilaterally misrepresenting
his preferences.
The set of feasible allocations
that cannot be improved upon by
a subset (a coalition) of the
economy's consumers.
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Basic Concepts
A short example for a core:
Treasure Hunt: An expedition of n people has found a treasure in
the mount; each pair of them can carry out one gold bar, but not
more.
The core depends on whether there is an even or odd number of
players.
With 4 Players: The core is { ⁄ , ⁄ , ⁄ , ⁄ }, i.e. everyone gets a half
bar.
With 3 Players: The “fair” allocation { ⁄ , ⁄ , ⁄ }is not in the core
because the coalition of 1 and 2 unanimously prefers the feasible
allocation { ⁄ , ⁄ , 0}. But the coalition of 1 and 3 unanimously prefers
the feasible allocation { ⁄ , 0, ⁄ }.
The core is empty.
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Conceptual Extensions
Weak Coalitional Strategy-Proofness: No group of agents can
benefit by jointly misrepresenting their preferences. At least one
agent does not profit.
Core Correspondence : Correspondence (~Function) that
assigns the set of allocations in the core for a problem.
Essentially Single-Valued Core: The core is never empty and the
allocations in the core are all Pareto indifferent (no allocation is
better than the other for any agent).
Externally Stable Allocations: Every allocation outside the set of
externally stable allocations is dominated by an allocation within
the set.
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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The General Model
A generalized indivisible goods allocation problem or a problem is
a 4-tuple ( , , , ) where
= {1, … , } is a finite set of agents,
= { (1), … , ( )} is an initial endowment of goods,
is the set of feasible allocations,
= { , … , } is a list of preference relations.
x
reads that to individual i, x is at least as good as y. and
respectively denote the symmetric (indifferent) and asymmetric
(strictly prefers) parts of .
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Basic Concepts more formally
Non-cooperative Game Theory:
Cooperative Game Theory:
Strategy-Proofness:
Core:
An allocation rule is strategyproof if for all , and ∈
An allocation # ∈
is in the
core of a problem if no
coalition ⊆ can block #, i.e.
there is no $ ∈
which satisfies
( )
(
,
).
Weak Coalitional StrategyProofness:
For all
⊆
, there exists ∈
such that
( )
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(
",
" ).
1. ∀ ∈ , $( ) ⊆
( ),
2. ∀ ∈ , $ #,
3. ∃' ∈ , $ ( #.
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Remember: is an allocation rule and
$, # are allocations, so:
Does a link exist?
Answer: YES!
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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First Result
Let , , , be such that the core correspondence is essentially
single-valued and the core of the problem is externally stable.
Then any selection from the core correspondence is weakly
coalitionally strategy-proof.
What does this mean?
If there is a core correspondence which is essentially single-valued
and externally stable, there is no group and no individual who can
benefit by lying about their preferences.
*++* , #-- + .-* − 0#-1*2345* ⇒ ∃7#045#$-*#--43#, 4 51-*
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Proof of First Result
Let be a selection from the core correspondence. That is ( ) ∈
8( ) for all . Suppose is not weakly coalitionally strategy-proof.
Then, by definition there exist , ⊆ , " such that
( ).
∀ ∈ , ( " , " )
Since ( ) ∈ 8( ) and the core is essentially single-valued,
( " , " ) ∉ 8( ). Moreover 8( ) is externally stable and therefore
there exists # ∈ 8( ) which dominates ( " , " ) under . That is,
there exists a coalition : ⊆ such that
∀ ∈ :, #( ) ∈ (:),
∀ ∈ :, #
( " , " ),
∃' ∈ :, # ( ( " , " ).
We have # ( ) for all ∈ and this together with the relation above
implies that ∩ : = ∅. But then # dominates ( " , " )under
" , " , contradicting
", " ∈ 8
" , " . Q.E.D.
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What is the problem with this proof?
Answer: The core correspondence being essentially
single-valued is a very strong assumption.
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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Next Step
We need to ensure an essentially single-valued core.
THEOREM 1: If there exists an allocation rule = > that is Pareto
efficient, individually rational and strategy-proof, then
, , , are such that
1) ∀ , ∀ ∈ , ∀#, $ ∈ 8
, # $,
≠ ∅, ∈8
.
2) ∀ ? ,@8
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Proof of an Essentially Single-Valued
Core Correspondence
… is too exhausting to go through it in detail, although it is
mathematically interesting.
Step one: We will prove that the allocation ( ) is Pareto
indifferent to # ∈ 8( ) under for all ∈ . (Remember
Assumption B)
Step two: We will prove that the allocation ( ) is Pareto indifferent
to # under for all ∈ . (By induction!)
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Finishing the Theoretical Part
As a second result: Whenever there exists an allocation rule =
that is Pareto efficient, individually rational and strategy-proof,
the core correspondence is essentially single-valued and the
allocation rule is a selection from the core correspondence, as long
as it is non-empty valued.
What does this mean?
If there is a correspondence with favorable traits, then its allocation
is a subset of the essentially single-valued core.
∃7#045#$-*#--43#, 4 51-* ⇒ *++* , #-- + .-* − 0#-1*2345*
As a further consequence: An essentially single-valued core is a
condition for a favorable allocation rule.
¬*++* , #-- + .-* − 0#-1*2345* ⇒ ∄7#045#$-*#--43#, 4 51-*
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Summary: A Two-Sided Link
Non-cooperative
games
Cooperative games
CSP
PE, IR and SP
allocation rule
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Externally
Stable
ESV Core
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Agenda
1
Important Concepts
2
General Model
3
Coalitional Strategy-Proofness Result
4
Essentially Single-Valued Core Results
5
Applications
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Application: Housing Market
Each agent owns one indivisible good, e.g. a house and has strict
preferences over all houses.
Without indifference, the core is a singleton, i.e. not empty and
externally stable (Roth and Postlewaite (1977)).
The core correspondence is weakly coalitionally
strategy-proof.
On the other hand, the core correspondence is the only allocation
rule that is Pareto efficient, individually rational and strategy-proof
(Ma (1994)).
The core is essentially single-valued (or a singleton).
Sönmez (1999) delivers a theoretical link between the results!
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Application: Housing Market
Example: Consider a market with
= {1324},
= {1423} and
First round of TTC:
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= {1,2,3,4} and
G = {1342}.
Strategy-proofness and essentially single-valued cores
= {2413},
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Application: Housing Market
Example: Consider a market with
= {1324},
= {1423} and
Second round of TTC:
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= {1,2,3,4} and
G = {1342}.
Strategy-proofness and essentially single-valued cores
= {2413},
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Application: Housing Market
Example: Consider a market with
= {1324},
= {1423} and
TTC Result:
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= {1,2,3,4} and
G = {1342}.
Strategy-proofness and essentially single-valued cores
= {2413},
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Application: Housing Market
Example: Consider a market with
= {1324},
= {1423} and
= {1,2,3,4} and
G = {1342}.
= {2413},
Applying the Top Trading Cycle Algorithm: We arrive at the
allocation = (2143). This solution is Pareto efficient,
individually rational and strategy-proof.
The core must be essentially single-valued.
Proof (with Top Trading Cycle Algorithm): Theorem 2 in Roth and
Postlewaite (1977) : If no trader is indifferent between any goods,
then the core is always non-empty, and contains exactly one
allocation.
Note: This in turn implies coalitional strategy-proofness.
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Application: Marriage Problem
There are two sets of agents interpreted as a set of men and a set
of women. Each man has strict preferences over the set of
women and staying single. Similarly each woman has strict
preferences over the set of men and staying single. An allocation
is a matching of men and women.
COROLLARY 4: Consider any subclass of marriage problems with
at least two men and two women. There is no allocation rule in
this context that is Pareto efficient, individually rational, and
strategy-proof and hence there is no strategy-proof selection from
the core correspondence.
Example?
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Application: Marriage Problem Solution
(H ) = {? , ? }; (H ) = {? , ? } and
{H , H }.
:
:
?
>
:
>
>
:
>
= H , H ; (? ) =
One set of individually rational, efficient matchings:
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Application: Marriage Problem Solution
(H ) = {? , ? }; (H ) = {? , ? } and ? = H , H ; (? ) =
{H , H }. Now, ? misrepresents her preferences:
:
:
>
:
>
>
:
>
Then the individually rational matching set changes to:
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Application: Marriage Problem Solution
(H ) = {? , ? }; (H ) = {? , ? } and
{H , H }.
?
= H , H ; (? ) =
One set of individually rational, efficient matchings is {H , ? } and
{H , ? }. So there is an incentive for JK to misrepresent her
preferences as ? = H , i.e. she pretends to prefer being
single to H . Then the individually rational matching set changes to
{H , ? } and {H , ? } .
It is also clear here that these two allocations, both core allocations,
are not Pareto indifferent for all agents. Thus, the core is not
essentially single-valued. (holds with result 2).
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Application: Roommate Problem
A generalization of the marriage problem: The roommate problem.
There is a group of agents, each of whom has strict preferences
over all agents. Here we are assigning two persons to a room.
COROLLARY 5: Consider any subclass of roommate problems
with at least four agents. There is no allocation rule in this context
that is Pareto efficient, individually rational, and strategy-proof.
Example?
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Application: Roommate Problem Solution
L = M, N, O ;
N = L, M, O ;
:
:
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M = N, L, O ;
O = {#5$ ,5#5 }.
>
>
:
>
>
:
>
>
whatever
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Application: Roommate Problem Solution
L = M, N, O ;
N = L, M, O ;
Room 1:
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M = N, L, O ;
O = {#5$ ,5#5 }.
Room 2:
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Application: Roommate Problem Solution
L = M, N, O ;
N = L, M, O ;
Room 1:
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M = N, L, O ;
O = {#5$ ,5#5 }.
Room 2:
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Application: Roommate Problem Solution
L = M, N, O ;
N = L, M, O ;
M = N, L, O ;
O = {#5$ ,5#5 }.
Gale/ Sharpley (1962): “Whoever has to room with O will want to
move out, and one of the other two will be willing to take him in.
There is no stable solution.”
How does this correspond to our results?
A coalition can block any allocation in this context. This means
the core is empty. Thus, there is no favorable allocation rule.
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Key Take Away
There is a link between strategy-proofness and
essentially single-valued cores.
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References
1.: http://en.wikipedia.org/wiki/Cooperative_game.
2. : http://en.wikipedia.org/wiki/Non-cooperative_game.
3. : http://en.wikipedia.org/wiki/Pareto_efficiency.
4. : http://en.wikipedia.org/wiki/Core_%28game_theory%29.
5.: Takamiya, K.: On strategy-proofness and essentially single-valued cores: A
converse result, Soc. Choice Welfare (2003), Vol. 20, pp. 77–83.
6.: Ma, J.:Strategy-Proofness and the Strict Core in a Market with Indivisibilities,
International Journal of Game Theory (1994), Vol. 23, pp. 75-83; Roth, A.\
Postlewaite, A.: Weak versus strong domination in a market with indivisible goods,
Journal of Mathematical Economics (1977), Vol. 4, pp. 131-137, Sharpley, L.\Scarf,
H.: On cores and indivisibility, Journal of Mathematical Economics (1974), Vol. 1,
pp. 23-37.
7.: Alcalde, J.\ Barberà: Top dominance and the possibility of strategy-proof stable
solutions to matching problems, Econ. Theory (1994), Vol. 4, pp. 417-435.
8.: Gale, D.\ Sharpley, L.S.: College Admissions and the Stability of Marriage, The
American Mathematical Monthly (1962), Vol. 69, pp. 9-15.
9.: Sönmez, T: Strategy-proofness and essentially single-valued cores,
Econometrica, Vol. 67, pp. 677-689.
10.: Oliver Schulte: Computational Game Theory
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Appendix
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Basic Concepts
A short example for a core:
Consider a housing market with three agents {1,2,3} and their
preferences
= {321},
= {123},
= {231}.
The allocation {312} is clearly in the core. Why?
Each agent is assigned his most preferred good, so no coalition
can improve upon its own.
If e.g. agent 1 and 2 formed a coalition, agent 1 would only get his
second choice.
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The General Model (II)
ASSUMPTION A: For all and for all #P
#
⇔ # = ( ).
,
ASSUMPTION B: For all and for all #P with #
, there exists
such that
1. ∀$ ∈ \{#}, $ # ⟺ $ #,
2. ∀$ ∈ \{#}, # $ ⟺ # $,
3. ∀$ ∈ \{#}, # $ ⟺ # $, # 2#
$.
Discussion: What could this mean?
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Proof of an essentially single valued core
correspondence (I)
Step one: We will prove that the allocation ( ) is Pareto
indifferent to # ∈ 8( ) under for all ∈ .
By definition 8( ) is Pareto efficient and individually rational.
Suppose now we have a Pareto efficient and individually rational
allocation $ under and ∃ ∈ , $ #.
Consider the coalition = { ∈ |$( ) ≠ ( )}.
Case 1: = ∅. Since 8( ) is by definition individually rational,
∀ ∈ ,#
and thus ∀ ∈ , # $, contradicting ∃ ∈ , $ #.
Case 2: ≠ ∅. We have ∀ ∈ , $( ) ⊆ ( ). By Assumption A we
must have ∀ ∈ , $
. With Assumption B we have ∀ ∈ , $ #.
On the other hand, we have ∀ ∈ \ , # $ as ∀ ∈ \ , $ =
. As we have ∃' ∈ , $ ( #, blocks #, contradicting the core.
Hence, ∀ ∈ , # $. As $ is arbitrary, it can be regarded as the
result of ( ), which proves that ∀ ∈ , ( ) U #.
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Proof of an essentially single valued core
correspondence (II)
Step two: We will prove that the allocation ( ) is Pareto indifferent
to # under for all ∈ .
For
= 1: By step one, we have ( ) ( #. By strategy-proofness,
we have
. This together with the result of step one
(, ( (
proves ∀ ∈ \{'}, ( ( , ( ) U # and ( ( , ( ) ( #.
U #and
For
= - < : Assume we have ∈ \ ,
", "
∈ ,
",
"
#.
Now add to agent ', i.e.
W\" ∪{(} , "\{(} =
" , " : By
strategy-proofness, we have
", " (
W\" ∪{(} , "\{(} . With
our assumption, we get
", " (
W\" ∪{(} , "\{(} ( #. With
Assumption B, we get ∀ ∈ ,
# and with step one
", "
U # . By induction, we arrive at ∀ ∈ , ( ) #.
∀ ∈ \ ,
", "
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Application: Multiple Endowment
COROLLARY 2: Consider any subclass of indivisible goods
exchange economies where at least one agent owns more than
one good. There is no allocation rule in this context that is Pareto
efficient, individually rational, and strategy-proof.
Example?
Hint by Sönmez: One can easily find an example where the core
has at least two allocations that are not Pareto indifferent.
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Application: Multiple Endowment Solution
One possible solution: Apples!
Consider a case with 4 apples and 3 agents: Who gets a second
apple? All three possible solutions are not Pareto indifferent for
all three agents.
Formally: Consider # = 211 , $ = 121 , 3 = 112 as the
feasible core allocations (for individual rationality, every agent must
receive at least one good). # $, 3; $ #, 3 and 3 #, 3 for the set
= 1,2,3 , so the core is not essentially single valued for all
agents.
As a consequence from result 2: There is no favorable allocation
rule.
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