Exercise 12
Algorithmic Game Theory
12.1
Social Choice Functions
Consider the following basic definitions for social choice functions.
Definition 1. For a social choice function f we say
• 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.
• 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.
• f is onto A if for every candidate a ∈ A there is a set of preferences such that a is the winner.
Instead of working with incentive compatibility it is sometimes easier to use monotonicity instead.
Definition 2. f is monotone if f (1 , . . . , n ) = a 6= b = f (1 , . . . , 0i , . . . , n ) implies that a i b
and b 0i a.
One can show that incentive compatibility is equivalent to monotonicity which is formulated in
the following lemma.
Lemma 1. A social choice function is IC if and only if it is monotone.
Your Task: Prove Lemma 1
12.2
Social Choice Functions - Part 2
Definition 3. 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 .
Example 1. S = {a, b, c}, A = {a, b, c, 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 now can use the notion of S to define a method to construct a social welfare function F for
a given social choice function f :
Definition 4 (Extending Social Choice Functions). We define F as the social welfare function ex{a,b}
{a,b}
tending f by F (1 , . . . , n ) =, where a b if and only if f (1 , . . . , n ) = a.
1
We now have to show that the method of extension described above indeed yields a social welfare
function.
Lemma 2. The extension of a social cost function is a social welfare function.
Sketch of Proof. We need to show asymmetry and transitivity:
• Antisymmetry: If a b and b a, then a = b.
• Transitivity: If a b and b c, then a c.
Your Task: Prove Lemma 2
12.3
Auctions
Consider the following setting of an auction. There is one item to be sold. There is a set N of n
bidders (or players). Each bidder i ∈ N values the item with vi (measured in Euro). The value vi is
private information of each player i and, thus, unknown to other bidders or the auctioneer.
The action is run as a Vickrey Second price auction: Each bidder i ∈ N communicates a bid bi
the the auctioneer. No player can observe the bid of another player. The auctioneer gives the item to
the player with the highest bid. This player has to pay the amount of the second highest bid.
a) Show that the Vickrey auction is incentive compatible. That is, show that no player
can improve by submitting a bid that is different from his true value vi .
b) Show that the auction is not incentive compatible if the winning player has to pay
the highest bid.
2