Download Mechanism Design and Social Choice

Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design and Social Choice
Algorithmic Game Theory
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design and Social Choice
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Can selfish agents achieve an agreement on a common outcome?
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How can we set rules that determine an outcome and make sure agents
find it in their interest to follow them?
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How can we motivate rational agents to cooperate?
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design
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Mechanism Design?
Design a (publicly known) set of rules that interact with selfish agents and
implement a common outcome or choice. Mechanism design is sometimes
called “Implementation Theory”.
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Mechanism?
A mechanism is an institution (a function, a set of rules) that collects
private information from selfish agents and determines an outcome.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Applications
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Elections - each voter has preferences, an outcome is the result of the
election
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Markets, E-Commerce - each participant in a market has preferences and
desires, the outcome is an allocation of goods and money.
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Auctions - a small market, single seller, each bidder has an amount she is
willing to pay, the outcome is the identity of the winner
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Goverment Policy - each citizen has his or her preferences, government
must make a single decision
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Internet Protocols - each user has load, protocol must make routing
assignments
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Strategic Voting and the Majority Rule with Two Candidates
Presidential Election: (O)bama, (R)omney
Voter
Preference Order
Voter
Reported Preference Order
1
O
R
1
O
R
2
R
O
2
O
R
3
O
R
3
O
R
Result of Majority Rule: O, R
The Majority rule for two candidates implements many desirable properties:
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Represents the majority of preferences
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Each candidate is in the position he/she appears most often
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Strategic voting is not profitable:
If a winning voter changes his vote, it only becomes worse off.
A losing voter cannot change the outcome by changing his vote.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Three Candidates
Presidential Election: (O)bama, (R)omney, (S)tein
Voter
Preference Order
1
O
S
R
2
R
O
S
3
S
R
O
Majority vote yields a cycle:
2 voters prefer O over S, 2 prefer S over R, 2 prefer R over O ...
This constellation shows that the collective preference can be conflicting (cyclic,
not transitive) although each individual preference is well-defined. It is called
Condorcet’s Paradox and was discovered by Marquis de Condorcet around 1785.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Plurality Voting
Let us examine the Plurality rule, in which each candidate ranks in position
with the highest number of occurences. We break ties w.r.t. alphabet.
Voter
Preference Order
Voter
Reported Preference Order
1
O
S
R
1
O
S
R
2
R
O
S
2
R
O
S
3
S
R
O
3
R
S
O
Plurality: O, R, S
Plurality: R, S, O
Strategic Voting is profitable for the third voter!
How can we avoid strategic voting? A trivial way is to choose one voter as a
dictator who dictates the outcome through his vote...
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Definitions
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Set of candidates (or outcomes, alternatives) A
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Set of n voters (or players) N
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Set of possible preferences (total orders of A) is L
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Each voter i has a preference (or preference order) i ∈ L on the
candidates A
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A social welfare function is a function F : Ln → L.
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A social choice function is a function f : Ln → A.
A social choice function outputs only a single winner, a social welfare function
outputs a complete ranking of all candidates.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Properties of Social Welfare Functions
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Unanimity: For every ∈ L we have F (, . . . , ) =.
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Voter i is a dictator in a social welfare function if for all 1 , . . . , n ∈ L we
have F (1 , . . . , n ) =i . Then F is called a dictatorship.
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Independence of Irrelevant Alternatives (IIA): The social preference
between any two candidates a and b depends only on the voters’
preferences between a and b.
Formally, for every a, b ∈ A and every 1 , . . . , n , 01 , . . . , 0n ∈ L, let
= F (1 , . . . , n ) and 0 = F (01 , . . . , 0n ) then a i b ⇔ a 0i b for all i
implies a b ⇔ a 0 b.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Independence of Irrelevant Alternatives
Plurality violates IIA!
Voter
Preference Order
Voter
Reported Preference Order
1
O
S
R
1
O
S
R
2
R
O
S
2
R
O
S
3
S
R
O
3
R
S
O
Plurality: O, R, S
Plurality: R, S, O
Ordering of pair (O,R) changes although each player ranks O and R pairwise
similarly in both orderings.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Arrow’s Theorem
Theorem (Arrow, 1950)
Every social welfare function over a set of |A| ≥ 3 candidates that satisfies
unanimity and IIA is a dictatorship.
For the proof fix F to be a social welfare function that satisfies the conditions.
Lemma (Pairwise Neutrality)
Let 1 , . . . , n and 01 , . . . , 0n two preference profiles, and = F (1 , . . . , n )
and 0 = F (01 , . . . , 0n ). If for every player i we have a i b ⇔ c 0i d, then
a b ⇔ c 0 d.
Proof:
We first rename our elements to let a b and c 6= b (but possibly a = c,
and/or b = d).
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Pairwise Neutrality
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Now we adjust i and 0i to become identical w.r.t. a, b, c, d by moving c
and d in i and a and b in 0i :
1 :
01 :
..., a, ..., b, ...
c, ..., d, ...
→
→
..., c, a, ..., b, d, ...
c, a, ..., b, d, ...
2 :
02 :
..., b, ..., a, ...
..., d, c, ...
→
→
..., b, d, ..., c, a, ...
..., b, d, c, a, ...
and so on
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IIA guarantees that a and b remain in the same order in ; c and d
remain in the same order in 0 . Similarly, by IIA we can now move all
other elements and assume 0i =i .
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By unanimity now c a and b d, so c d. With i =0i for all i we
(Lemma)
also get c 0 d.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Who is the Dictator?
Pairwise neutrality implies that a social welfare function that satisfies unanimity
and IIA has a general underlying approach of determining a global preference.
This approach is similar for all preference orders and all pairwise comparisons of
elements. This can be used to show that, in fact, the approach boils down to
having one dictator determine the output.
Fix a 6= b and c 6= d.
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If there are no players with a i b, then b a.
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If there are n players with a i b, then a b.
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Breakpoint: i ∗ players
1
. . . i∗ − 1
a i b
a i b
i∗
... n
b i a
b i a
Result
ba
ab
Claim: i ∗ is the dictator!
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
i ∗ is the Dictator
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i ∗ is a dictator if c i ∗ d ⇒ c d for all c 6= d ∈ A.
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Consider an arbitrary set of preferences with c i ∗ d and e ∈ A with
e 6= c and e 6= d.
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Switch third element e s.t. it appears as below in i :
1
...
i∗
...
n
e
e
...
c
...
...
...
...
...
e ...
d
...
e
e
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Because of IIA this does not change order of c and d in .
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(c, e) appears exactly as (a, b) previously, by the lemma on pairwise
neutrality we know c e. Similarly e d.
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Thus c d, and this proves Arrow’s Theorem.
Alexander Skopalik
Mechanism Design and Social Choice
(Theorem)
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Properties of Social Choice Functions
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f can be strategically manipulated by voter i if for some 1 , . . . , n and
some 0i we have that a i b where b = f (1 , . . . , n ) and
a = f (1 , . . . , 0i , . . . , n ). f is called incentive compatible (IC) or
strategyproof if it cannot be manipulated.
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f is monotone if f (1 , . . . , n ) = a 6= b = f (1 , . . . , 0i , . . . , n ) implies
that a i b and b 0i a.
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Voter i is a dictator in f if for all 1 , . . . , n ∈ L we have that if a i b for
all b 6= a, then f (1 , . . . , n ) = a. Then f is called a dictatorship.
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f is onto A if for every candidate a ∈ A there is a set of preferences such
that a is the winner.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Gibbard-Satterthwaite Theorem
Proposition
A social choice function is IC if and only if it is monotone.
Proof: Direct implication of definitions.
Theorem (Gibbard 1973; Satterthwaite 1975)
A social choice function f onto A with |A| ≥ 3 is IC if and only if it is a
dictatorship.
Proof:
We prove the non-trivial direction of the theorem by using a social choice
function f onto A to define a social welfare function F that satisfies IIA and
unanimity.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Extending Social Choice Functions
For a preference order and a set S ⊂ A we denote by S the adjustment of
moving all elements of S in order to the front of .
S = {a, b, c},
A = S ∪ {d, e, f }
a
b
e
f
d
e
c
d
b
a
f
c
→
→
→
a
b
c
a
S
b e
c f
d
e
f
d
We define F as the social welfare function extending f by F (1 , . . . , n ) =,
{a,b}
{a,b}
where a b if and only if f (1
, . . . , n
) = a.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Reaching a Contradiction
Lemma
If f is an incentive compatible social choice function onto A then the extension
F is a social welfare function.
Show antisymmetry and transitivity.
Lemma
If f is an incentive compatible social choice function onto A and not a
dictatorship, then the extension F satisfies unanimity, independence of
irrelevant alternatives, and is not a dictatorship.
A contradiction follows with Arrow’s Theorem
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Proof of Theorem
We prove the theorem by verifying the properties of F :
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Antisymmetry: If a b and b a, then a = b.
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Transitivity: If a b and b c, then a c.
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Unanimity: F (, . . . , ) =.
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IIA
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Non-Dictatorship
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Properties
Claim
For any 1 , . . . , n and any S the winner f (S1 , . . . , Sn ) ∈ S.
Proof:
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f is onto, so there is 001 , . . . , 00n that gives some a ∈ S as winner.
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Iteratively move elements of S to the front, re-sort elements in the back,
re-sort elements of S in the front
⇒ Transformation into S1 , . . . , Sn .
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Monotonicity ensures that no b 6∈ S will ever be a winner in the course of
(Claim)
the transformation.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Properties
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Antisymmetry: If a b and b a, then a = b.
{a,b}
As f (1
I
{a,b}
, ..., n
) ∈ {a, b}.
Transitivity: If a b and b c, then a c.
Suppose for contradiction that a b c a. Take S = {a, b, c} and
w.l.o.g. let f (S1 , . . . , Sn ) = a. Sequential changes to S for S = {a, c}
imply f (S1 , . . . , Sn ) = a, and hence a c. A contradiction follows with
antisymmetry.
Hence, if f is IC onto A, then F is a valid social welfare function.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Properties
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Unanimity: F (, . . . , ) =.
If a i b for all i, then by the claim and monotonicity we have
{a,b}
{a,b}
f (1
, . . . , n
) = a.
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IIA:
Assume a i b ⇔ a 0i b. Note that
0
0
{a,b}
{a,b}
{a,b}
{a,b}
f (1
, . . . , n
) = f (1
, . . . , n
) because by sequential
0
{a,b}
{a,b}
change of i
into i
outcome does not change due to
monotonicity and claim.
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Non-Dictatorship: Obvious.
Alexander Skopalik
Mechanism Design and Social Choice
(Theorem)
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Single-Peaked Preferences
While the Gibbard-Satterthwaite Theorem is devastating, it requires the
generality of preferences. If preferences are restricted, a richer class of IC social
choice rules exist.
Let us consider as set of outcomes the interval A = [0, 1].
Definition
A preference relation i over A is single-peaked if there is a peak pi ∈ A such
that for all x ∈ A\{pi } and λ ∈ [0, 1) we have
(λx + (1 − λ)pi ) i x .
As application consider, e.g., the problem of location of a facility like a grocery
store. Each voter has a residence along a street and would like to have the
facility as close as possible to his residence. Alternatively, consider deciding on
a tempature value for a shared office. For single-peaked preferences the
Gibbard-Satterthwaite theorem does not apply.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Order Mechanisms
k-th Order Mechanism for Single-Peaked Preferences:
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Collect only the peaks p1 , . . . , pn of the agents.
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Order the peaks from 1 to 0 and output the k-th largest peak as location.
Proposition
For any fixed k ∈ {1, . . . , n}, the k-th order mechanism is IC. If n ≥ 2, it is not
a dictatorship.
Proof:
Let p be the outcome if all voters report their order truthfully. If pi > p, voter i
cannot change the outcome with pi0 > pi . If he lies a peak pi0 ≤ p, it results in
a worse outcome p 0 ≤ p. The argument for pi < p is similar. Non-dictatorship
is obvious.
The most prominent rule is the median
Pmechanism with k = b(n + 1)/2c. Note
that taking the average of the peaks ni=1 pi /n is not IC.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Order Mechanisms
By the same argument as above, every k-th order mechanism remains IC if, in
addition to the peaks, we consider any number of apriori fixed locations
yj ∈ [0, 1] and take the k-th largest location of {p1 , . . . , pn , y1 , . . . , ym }.
All k-th order mechanisms are anonymous, i.e., they yield
f (1 , . . . , n ) = f (01 , . . . , 0n ) if (1 , . . . , n ) is a permutation of
(01 , . . . , 0n ).
Theorem (Moulin 1980; Ching 1997)
A social choice rule f for is IC, onto and anonymous for single-peaked
preferences if and only if it is a k-th order mechanism over a set
{p1 , . . . , pn , y1 , . . . , ym }, where pi are the reported peaks and yj ∈ [0, 1] are
fixed locations.
The result is a complete characterization for anonymous IC mechanisms.
Anonymity is required, because every dictatorship is not a k-th order
mechanism but onto and IC (and non-anonymous).
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory
Mechanism Design
Strategic Voting and Social Choice
Impossibility Results
Special Cases
Recommended Literature
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Chapters 9 and 10 in the AGT book.
Alexander Skopalik
Mechanism Design and Social Choice
Algorithmic Game Theory