CS294 Topics in Algorithmic Game Theory
October 18, 2011
Lecture 8
Lecturer: Christos Papadimitriou
Scribe: Sudeep Kamath, Evan Davidson
Individuals have preferences over outcomes.
• When we know their preferences, we obtain the theory of Social Choice.
• When we don’t know their preferences, we obtain the theory of Mechanism Design.
1
Social Choice Theory
Condorcet’s Paradox : Suppose there is a election with three candidates A,B,C competing against
each other. Suppose a third of the voters prefer A > B > C, a third prefer B > C > A and a third
prefer C > A > B. Suppose A gets elected. Then it can be argued that C should win instead because
two-thirds of the voters prefer C to A.
1.1
Social Welfare Function and Arrow’s Impossibility Theorem
Let A denote a set of outcomes and L denote the set of linear orderings over A, i.e. the set of all possible
preferences over A, so that |L| = (|A|)! Let [n] denote the set of all voters.
We wish to construct a social welfare function f : Ln → L, i.e. a function that takes as input the
preferences of all the voters and outputs one preference ordering in some reasonable way. Economists
describe this “reasonableness” by the following two properties that such a function must satisfy.
Notation
: We will use > to denote an element of L. We will often write f (>1 , >2 , . . . , >n ) => .
• Unanimity (U): f (>, >, . . . , >) =>, i.e. if everyone has the same preference order, the social
welfare function must output the same order.
• Independence of Irrelevant Alternatives (IIA): Suppose >1 , >2 , . . . , >n are a set of preferences with
f (>1 , >2 , . . . , >n ) => and >�1 , >�2 , . . . , >�n are another set of preferences with f (>�1 , >�2 , . . . , >�n
) =>� . Suppose that for a, b ∈ A, we have that a >i b ⇔ a >�i b ∀ i ∈ [n]. Then, a > b ⇔ a >� b.
This means that if voters have some set of preferences and they change their preferences somehow
but each voter continues to maintain his previous relative preference of a and b, then the relative
preference of a and b in the output of f must not change.
These two axioms turn out to constrain the function f too strongly.
Theorem 1. (Arrow’s Impossibility Theorem, 1950): If |A| ≥ 3, and if f satisfies U and IIA, then f is
a dictatorship, i.e. f (>1 , >2 , . . . , >n ) =>D for some D ∈ [n].
Remarks
:
• A dictatorship is a particularly bad function that arise out of these “reasonable” axioms. If
the dictator chooses a particular preference ordering and all the other voters choose the reverse
ordering, the social welfare function will output according to the dictator’s wishes.
• There are exactly n such functions that satisfy U and IIA - the dictatorial functions corresponding
to each of the voters.
• The majority function satisfies U but does not satisfy IIA since it is not a dictatorship.
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• The theorem does not require that everyone has a strict preference. There is a version of the
theorem that allows for ties, that is, voters are allowed to be indifferent between two different
outcomes.
• The Impossibility stands for the following: A function satisfying Unanimity, Independence of
Irrelevant Alternatives and non-Dictatorship cannot exist. Such a function is impossible.
Proof:
Suppose f respects U and IIA.
Lemma 1. (Pairwise Unanimity (PU) Lemma): Suppose f (>1 , >2 , . . . , >n ) => . Suppose for some
a, b ∈ A, we have a >i b ∀i. Then a > b.
Proof: (Proof of PU) Suppose a >i b ∀i. Compare the inputs (>1 , >1 , . . . , >1 ) and (>1 , >2 , . . . , >n ).
For the elements a, b, the IIA hypothesis holds. So, if f (>1 , >1 , . . . , >1 ) =>� and f (>1 , >2 , . . . , >n ) =>,
then a >� b if and only if a > b. But by Unanimity, >� =>1 , so it follows that a > b.
�
Lemma 2. (Neutrality lemma - “all pairs are born equal”) Suppose f (>1 , >2 , . . . , >n ) => and f (>�1
, >�2 , . . . , >�n ) =>� . If a >i b ⇔ c >�i d ∀i, then a > b ⇔ c >� d.
Proof: (Proof of Neutrality lemma) Suppose |{a, b, c, d}| ≥ 3, else the lemma is easily seen to be true.
To satisfy the hypotheses of the theorem, we must have a �= b, c �= d.
Case I: a �= d. Without loss of generality, also assume that b �= c, for if b = c, then rename (a, b) to
(c, d) and vice versa and use Case II, namely a = d.
Now, define a new collection of orders >∗1 , >∗2 , . . . , >∗n that are precisely specified only upto the
relative orderings of the elements a, b, c, d. Suppose f (>∗1 , >∗2 , . . . , >∗n ) =>∗ .
Whenever a >i b, we define c ≥∗i a >∗i b ≥∗i d, else we define b ≥∗i d >∗i c ≥∗i a.
Now, compare >i ’s and >∗i ’s. Note that a >i b ⇔ a >∗i b ∀i. By IIA, we have a > b ⇔ a >∗ b. (1)
Similarly, c >�i d ⇔ c >∗i d. So, c >� d ⇔ c >∗ d. (2)
But by PU, c ≥∗ a, b ≥∗ d. (3)
Suppose a > b. By (1), (2) and (3), it follows that c� d.
By symmetry, c >� d implies a > b.
Case II: a = d.
Again, we construct >∗1 , >∗2 , . . . , >∗n , >∗ as follows.
Whenever a >i b, we define c >∗i b >∗i a = d else we define b >∗i a = d >∗i c.
Apply Case I to (a, b) with >i and (c, b) with >∗i . We get a > b ⇔ c >∗ b. (4)
As b >∗i a ∀i, we have by PU lemma, that b >∗ a. (5)
Finally, c >∗i a ⇔ c >�i a. So, by IIA, c >∗ a ⇔ c >� a = d. (6)
Assuming a > b and using (4), (5), (6) yields c >� d.
The converse can be proved similarly.
�
Now, we show how to find the dictator D. Fix a, b.
(k)
(k)
(k)
(k)
For 0 ≤ k ≤ n, define a relative ordering of a, b by a sequence >1 , >2 , . . . , >n as follows: a >i b
(k)
(k)
(k)
(k)
(k)
(k)
if and only if i ≤ n−k. Define f (>1 , >2 , . . . , >n ) =>(k) . >1 , >2 , . . . , >n looks like the following:
(k)
(k)
(k)
(k)
(k)
a >1 b, a >2 b, . . . , a >n−k b, b >n−k+1 a, . . . , b >n a.
(0)
(n)
By pairwise unanimity, a > b and b > a. Let D denote the smallest index such that b >(n+1−D) a.
Then, 1 ≤ D ≤ n. We claim that D is the dictator.
Pick any three distinct elements c, d, z. For f (>1 , >2 , . . . , >n ) =>, we will show that c >D d =⇒
c > d. Suppose S = {i : c >i d}, where D ∈ S. By IIA, we can change the relative ordering of z with
respect to c and d without changing the relative ordering of c and d in > . So, choose the following:
• z >i c >i d for i ∈ S, 1 ≤ i ≤ D − 1
• z >i d >i c for i ∈
/ S, 1 ≤ i ≤ D − 1
• c >i z >i d for i = D
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• c >i d >i z for i ∈ S, D + 1 ≤ i ≤ n
• d >i c >i z for i ∈
/ S, D + 1 ≤ i ≤ n
Now, we know that for a >∗1 b, a >∗2 b, . . . , a >∗D b, b >∗D+1 a, b >∗D+2 a, . . . , b >∗n a, we have a >∗ b
by definition of D.
Use Neutrality Lemma with z, d to get z > d.
Similarly, we know that a >∗1 b, a >∗2 b, . . . , a >∗D−1 b, b >∗D a, b >∗D+1 a, . . . , b >∗n a, we have b >∗ a
by definition of dictator.
Use Neutrality Lemma with z, c to get c > z.
These imply c > d. Thus, the relative ordering in the output of f always matches the relative ordering
in that of D. Thus, D is a dictator.
�
1.2
Social Choice Function and Gibbard-Satterthwaite Theorem
f is a social choice function that takes all voters’ orderings and returns a single outcome, the choice of
the voter collective. That is, f : Ln → A, f (>1 , >2 , . . . , >n ) = a.
Definition 1. We say f is Incentive-Compatible (IC) if it cannot be manipulated. Manipulation means
that there would be some benefit for one or more voters to lie about their preferences in order to improve
the final result from their perspective.
f can be manipulated if there exist preferences >1 , >2 , . . . , >n and outcomes a, b ∈ A with a >i b such
that f (>1 , >2 , . . . , >n ) = b and there exist ordering >�i such that f (>1 , >2 , . . . , >i−1 , >�i , >i+1 , . . . , >n
) = a.
Definition 2. f is called monotone if when there is only one changed preference argument, and that
change causes the function’s output to change from a to b, then the changed preference must have moved b
above a. That is, f (>1 , >2 , . . . , >n ) = a �= b = f (>1 , >2 , . . . , >i−1 , >�i , >i+1 , . . . , >n ) =⇒ a >i b, b >�i a.
Incentive-Compatibility ⇔ Non-manipulatibility ⇔ Monotonicity
Theorem 2. (Gibbard-Satterthwaite Theorem (1973/1975) )
For |A| ≥ 3 and if f is an Incentive Compatible function that is onto, then that function will be a
dictatorship, that is, there exists a voter D whose top preference is always chosen.
Proof:
Definition 3. >S is the ordering > except with all elements of S moved to the top of the ordering with
the relative ordering within S preserved.
Given a social choice function f, we construct a social welfare function fˆ as follows: a > b in
{a,b}
ˆ
f (>i , 1 ≤ i ≤ n) if and only if f (>i
, 1 ≤ i ≤ n) = a. For this definition to make sense and for fˆ to
be a social welfare function, we need the following lemma.
Lemma 3. If f is onto and Incentive-Compatible, then f (>Si , 1 ≤ i ≤ n) ∈ S.
Proof: As f is onto, there exist >�1 , >�2 , . . . , >�n such that f (>�i , 1 ≤ i ≤ n) ∈ S.
Now, for each i = 1, . . . , n step by step, change >�i to >Si . We claim that f will stay in S at each
step. At step i, if the function maps to something outside S, then this violates monotonicity, because
the change only improves the relative ranking of S.
�
This Lemma implies that fˆ is well-defined, i.e. that either a > b or b > a in fˆ. It also shows
transitivity. Assume that fˆ is not transitive. Suppose it results in a > b, b > c and c > a. Without loss
{a,b,c}
{a,b}
of generality, let f (>i
, 1 ≤ i ≤ n) = a. Now, by monotonicity, f (>i
, 1 ≤ i ≤ n) = a. And also
{a,c}
f (>i
, 1 ≤ i ≤ n) = a. This contradicts c > a. Thus, fˆ produces a total order.
It is also easy to show that fˆ satisfies U and IIA.
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{a,b}
{a,b}
{a,b}
Proof of PU (which is stronger than U): Suppose a >i b ∀i. Then (>i
){a} =>i
so f (>i
, 1 ≤ i ≤ n) = a implying that a > b in fˆ.
{a,b}
Proof of IIA: Suppose a >i b ⇔ a >�i b ∀i. Then, by sequential flipping we can see that f (>i
,1 ≤
�{a,b}
i ≤ n) = f (>i
, 1 ≤ i ≤ n). Thus, a > b ⇔ a >� b.
By Arrow’s Impossibility Theorem, fˆ must be a dictatorship. Now, one can easily argue that f (>i
, 1 ≤ i ≤ n) equals the top preference in fˆ. So f is a dictatorship as well.
�
2
Mechanism Design
We have a set of outcomes A, a set of agents [n] with preferences >i that are assumed to be unknown.
We desire to create a function f : Ln → A such that it satisfies
• non-manipulatable, incentive-compatible, truthful,
• f (>i , 1 ≤ i ≤ n) = arg max C(a, {>i , 1 ≤ i ≤ n}),
where C(·, ·) is some objective, say social welfare.
Incentive-Compatible here means that for each agent, the dominant strategy is to tell the truth, i.e.
reveal the true preference.
We know
• it’s impossible
• but it’s possible with Nash Equilibrium
• but that may be too complicated
• but it is possible with money
2.1
Mechanism Design with Money
Money: A = A0 × Rn
where A0 is the set of basic outcomes and the real-valued entries denote payments made to the agents
which could be positive or negative.
The preferences are given by the following form:
vi (a, p1 , p2 , . . . , pn ) = vi0 (a) + pi
The utility to agent i is the value of the basic outcome to the agent plus the payment received by
her.
�
The social welfare agent wishes to maximize C or i vi (a).
Note: Payments are not taken into account.
Examples:
• Single-item auction
• Reverse auction (Procurement)
• Multi-item auction (combinatorial framework)
• Building a bridge
• Buying a path in a network (complex procurement)
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2.1.1
Single-item auction
�
Optimize i vi (a)
Agent i bids price pi . Utility of agent i is Ui (a) = 0 if i loses and vi − pi if i wins the auction. There
are n possible outcomes corresponding to who wins the item.
We want to make this incentive-compatible.
• Bad way to conduct the auction: Keep bidding with bids audible to the other bidders. When no
higher bid is offered, item is given to the highest bidder at the price of his bid.
• Better way: Write your bids in an envelope. Here again, if there are n agents, it may be optimum
to bid slightly lower value than your true bid.
• Vickri auction: Bids are written in an envelope. Highest bidder gets the item at the price offered
by the second-highest bidder.
Theorem 3. Vickrey auction is incentive-compatible.
References
[1] Arrow, K.J., ”A Difficulty in the Concept of Social Welfare”, Journal of Political Economy 58(4)
(August, 1950), pp. 328–346.
[2] Allan Gibbard, ”Manipulation of voting schemes: a general result”, Econometrica, Vol. 41, No.
4 (1973), pp. 587–601.
[3] Mark A. Satterthwaite, ”Strategy-proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions”, Journal of Economic
Theory 10 (April 1975), 187–217.
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