MECHANISM DESIGNINTRODUCTIONEitan Yanovsky

Outline Election Mechanisms with money Incentive compatible mechanism Incomplete information Characterizations of incentive

compatible mechanisms

Election Two candidates election

Intuitive solution fits, take majority vote. Three or more candidates election

Condorcet’s paradox:

The notation means that voter i prefers candidate a to candidate b.

2 2b c a 1 1a b c

3 3c a b

ia b

a b c a The joint majority choice is not consistent

Election Voting methods

Plurality Borda countThese methods encourage

strategic voting

Election Formally:

A - the set of candidates (alternatives). I - the set of n voters. L - the set of linear orders on A ( : is a

total order on A). Preferences of voter i are given by

Definition 1.1: is called social welfare function

(s.w.f). is called a social choice function

(s.c.f).

L

i L

: nF L L

: nf L A

Election Definition 1.2:

s.w.f F satisfies unanimity if Voter i is a dictator in F if

F is not a dictatorship if no i is a dictator in it. s.w.f F satisfies independence of irrelevant

alternatives if the social preferences between any two alternatives a and b only depends on the voters preferences between a and b, formallyif we denote then

, ( ,..., )L F

1 1... , ( ,..., )n n iL F

1 1, , ... , ...n na b A L

1 1( ,..., ) , ( ,..., )n nF F

: i ii a b a b a b a b

Election – Arrow’s theorem Theorem1.13

Every s.w.f over a set A s.t |A|>2 that satisfies unanimity and independence of irrelevant alternatives is a dictatorship.

Arrow’s theorem has devastating strategic implications.

Escape route, mechanism with money.

Mechanisms with money Replace order with valuation A player wishes to maximize its utility which

is derived from its valuation and some price paid or gained at each chosen alternative

Single item auction: . where is the

price player i is “willing to pay” for a. social welfare achieved with second price

auction.

i

, ( )i i aa A u v a p

{ | }A i wins i I

: ( ) , , ( ) 0i i ii I v i wins w j i v j wins iw

: , Ai i iv A v V

is commonly known set of possible valuation functions for player iiV

Incentive compatible mechanisms Notations:

Similarly etc’

Definition 1.14 A mechanism consist of:

a social choice function Payment functions

1 1 1 1( ... ), ( ... , ... )n i i i nv v v v v v v v

1( ... ) ( , )n i iv v v v

1 1 1 1... , ... ...n i i i nV V V V V V V V

,i it x

1: ... nf V V A

1 1... , : ...n i np p p V V

Incentive compatible mechanisms Definition 1.15

A mechanism is called incentive compatible if if we denote

then Intuitively this means that player i whose

valuation is would prefer “telling the truth” rather than any possible “lie” since this gives him higher utility.

1( , ... )nf p p

1 1: ,..., ,n n i ii I v V v V v V

( , ) , ( , )i i i if v v a f v v a ( ) ( , ) ( ) ( , )i i i i i i i iv a p v v v a p v v

iviv

iv

Incentive compatible mechanisms Definition 1.16

A mechanism is VCG mechanism if (f maximizes social welfare)

For some functions ( does not depend on )

The main idea is that each player is paid an amount equal to the sum of the values of all other players, thus this mechanism aligns all players incentives with the social goal of maximizing social welfare.

1( , ... )nf p p

1( , ... ) argmax ( )n a A ii If p p v a

1... , :n i ih h h V ih

iv 1 1 1 1,..., : ( ... ) ( ) ( ( ... ))n n i n i i j nj iv V v V p v v h v v f v v

Incentive compatible mechanisms Theorem 1.17

Every VCG mechanism is incentive compatible. Proof: Let be player i truthful valuation and some different valuation. We need to show that the utility for declaring is higher than declaring . the utility when declaring is

respectively and since f maximizes social welfare over all alternatives then and the same holds when subtracting

iv iv

( , ) , ( , )i i i if v v a f v v a iv iv

( ) ( ) ( ), ( ) ( ) ( )i j i i i j i ij i j iv a v a h v v a v a h v

,i iv v

( ) ( ) ( ) ( )i j i jj i j iv a v a v a v a

( )i ih v

Incentive compatible mechanisms Definition 1.18

A mechanism is individual rational if

A mechanism has no positive transfer if Definition 1.19 (Clark pivot rule)

Each player pays his damage to the others.

1( , ... )nf p p

1 1: ( ( ... )) ( ... ) 0i n i ni I v f v v p v v

1 1... , ( ... ) 0n i nv v p v v

1( ) max ( ), ( ... ) max ( ) ( )i i b A j i n b A j jj i j i j ih v v b p v v v b v a

Incentive compatible mechanisms Lemma 1.20

A VCG mechanism with Clarke Pivot payments makes no positive transfers and if then it is also individually rational. Proof: Let be the alternative maximizing and b the alternative maximizing . Player’s i utility

. No positive transfers since

This down side can be addressed by a modified rule that chooses b as to maximize the social welfare “when i does not participate”.

: ( ) 0i i iv V a A v a

1( ... )na f v v ( )jjv a

( )jj iv b

( ) ( ) ( ) ( ) ( ) ( ) 0i i j j j jj i j i j j

u a v a v a v b v a v b

1( ... ) ( ) ( ) 0i n j jj i j i

p v v v b v a

Incentive compatible mechanisms Examples

A single item auction Bilateral trade

Seller Buyer outcome Public project

Government does a project at price C, each citizen i has a value for the project (may be negative)

Reverse Auction Tender

0 1sv 0 1bv { , }A trade no trade

iv

{ | ( ) 0 : ( ) 0}i i i iV v v i wins j i v j wins

Incomplete information Definition 1.21

A game with strict incomplete information: For every player i, a set of actions For every player i, a set of types , a value

is the private information that player i has.

For every player i, a utility function each player choose action knowing only its own information, i.e.

iX

iT i it T

1: ...i i nu T X X

i ix X

it

Incomplete information Definition 1.22

A strategy of player i is a function A profile is an ex-post-Nash equilibrium

if we have that

A strategy is a dominant strategy ifwe have that

A profile is a dominant strategy equilibrium if each is a dominant strategy

:i i is T X

1... ns s

1, ... ,n ii I t t x ( , ( ), ( )) ( , , ( ))i i i i i i i i i i iu t s t s t u t x s t

is , ,i i it x x

( , ( ), ) ( , , )i i i i i i i i iu t s t x u t x x

1... ns s

is

Incomplete information Definition 1.24

A mechanism for n players is given by: Private information spaces Action spaces Alternative set A Valuations Outcome function Payments

The game induced by the mechanism is by using the private information spaces ,action spaces and the utilities

1... nT T

1... nX X

:i iv T A

1: ... na X X A

1: ...i np X X

1... nT T 1... nX X

1 1 1( , ... ) ( , ( ... )) ( ... )i i n i i n i nu t x x v t a x x p x x

Incomplete information Definition 1.24 cont’

A mechanism implements a s.w.f in dominant strategies if for some dominant strategy equilibrium of the induced game we have that for

Similarly a mechanism implements a s.w.f in ex-post-equilibrium.

1: ... nf T T A

1... ns s1 1 1 1... , ( ... ) ( ( )... ( ))n n n nt t f t t a s t s t

Incomplete information Proposition 1.25

(Revelation principle) If there exists an arbitrary mechanism that implements f in dominant strategies, then there exists an incentive compatible mechanism that implements f. The payment of the players in the incentive compatible mechanism is identical to those, obtained at equilibrium, of the original mechanism.Proof: The new mechanism will simulate the equilibrium strategies. Let be a dominant strategy equilibrium, we define a new direct revelation mechanism:and . Since each is dominant for player i, then for every we have that

this in particular this is true for all and any which gives the definition of incentive compatibility of the mechanism

1... ns s1 1 1( ... ) ( ( )... ( ))n n nf t t a s t s t

1 1 1( ... ) ( ( )... ( ))i n i n np t t p s t s t is, ,i i it x x

( , ( ( ), )) ( ( ), ) ( , ( , ) ( , )i i i i i i i i i i i i i i i iv t a s t x p s t x v t a x x p x x

( )i i ix s t ( )i i ix s t

1( , ... )nf p p

Characterizations of incentive compatible mechanisms

Proposition 1.27 A mechanism is incentive compatible if and

only if it satisfies the following condition for every i and every :

The payment does not depend on , but only on the alternative chosen

The mechanism optimizes for each player. I.e. for every we have that

iv

ip iv( , )i if v v

( , ) argmax ( ( ) )i i a i af v v v a p iv

Characterizations of incentive compatible mechanisms

Proof: <= Denote

The utility of i when telling the truth is which is not less than the utility when declaring ( ) since the mechanism optimize for i ( )

=> (i) If for some but then player with this valuation will increase his gain by declaring

=> (ii) If , fix in the range of and thus for some . Now a player with this valuation will increase his gain by declaring

( , ), ( , ), ( , ), ( , )i i i i a i i a i ia f v v a f v v p p v v p p v v

( )i av a p

iv ( )i av a p ( , ) argmax ( ( ) )i i a i af v v v a p

, , ( , ) ( , )i i i i i iv v f v v f v v ( , ) ( , )i i i i i ip v v p v v

iv( , ) argmax ( ( ) )i i a i af v v v a p

a

argmax ( ( ) )a i aa v a p ( , )if v , ( , )i i iv a f v v

iv

Characterizations – Weak Monoticity

Definition 1.28 A s.c.f f satisfies weak monoticity if we have that

in simply words, if the social choice changes when a single player changes his valuation, then it must be because the player increased his value of the new choice relative to his value of the old choice.

Theorem 1.29 If a mechanism is incentive compatible then f satisfies

WMON. If all preferences are convex sets, then for every s.c.f f that satisfies WMON, there exists such that is incentive compatible.

, ii v( , ) ( , ) ( ) ( ) ( ) ( )i i i i i i i if v v a b f v v v a v b v a v b

iV1... np p

1( , ... )nf p p

Characterizations – Weak Monoticity

Proof (one way) Assume is incentive compatible, fix i

and . Proposition 1.27 implies the existence of prices for all

(that do not depend on ) s.t whenever the outcome is a the bidder i pays it. Assume since a player with valuation does not prefer declaring we have that similarly a since player with does not prefer declaring we have that . By subtracting the second from the first we get as required

1( , ... )nf p p ivap

iv

a A( , ) ( , )i i i if v v a b f v v

iv ( ) ( )i a i bv a p v b p

iv iv( ) ( )i a i bv a p v b p

( ) ( ) ( ) ( )i i i iv a v b v a v b

iv

Characterizations – Weighted VCG

Definition 1.30 A s.c.f f is called an affine maximizer if for

some , for some player weights and for some outcome weights for every we have that

A A

1... nw w

ac a A

1( ... ) argmax ( ( ))n a A a i iif v v c w v a

Characterizations – Single parameter domain Restrict to be one dimensional, a player has

a scalar value for “winning” and 0 value for “losing”.

Definition 1.33 a single parameter domain is defined by a

publicly known and a range of values . is the set of

such that for some for all and otherwise.

Definition 1.34 A s.c.f f is called monotone in if and every

we have that

iV

iW AiV

0 1[ , ]t t iV

iv 0 1, ( )it t t v a t ia W

( ) 0iv a

iv iv i iv v ( , ) ( , )i i i i i if v v W f v v W

Characterizations – Single parameter domain

Definition 1.35 The critical value of a monotone s.c.f f is

the critical value is undefined if i wins for every

Theorem 1.36 A normalized mechanism is incentive

compatible if and only if the following conditions hold: f is monotone in every Every winning bid pays the critical value. Formally,

such that we have that (If is undefined the winner pays some constant value)

: ( , )( ) supi i i ii i v f v v W ic v v

iv

1( , ... )nf p p

iv, ,i ii v v

( , )i i if v v W ( , ) ( )i i i i ip v v c v ( )i ic v

Characterizations – Uniqueness of prices

The payment function is essentially uniquely determined by the social choice function, means that we can modify a payment by adding some function that do not depend on the player valuation.

Conclusion: The only incentive compatible mechanisms that

maximizes social welfare are those with VCG payments.

In the example (bilateral) the only i.c.m that maximizes social welfare, must subsidize the trade.

Characterizations – Randomized mechanisms Definition 1.38

A randomized mechanism is a distribution over deterministic mechanism(all with the same players, types spaces and outcome)

A randomized mechanism is incentive compatible in the universal sense if every deterministic mechanism in the support is incentive compatible

A randomized mechanism is incentive compatible in expectation if truth is dominant strategy in the game induced by expectation. I.e where are random variables denoting the outcome and payment of i when he bids respectively

, , , : [ ( ) ] [ ( ) ]i i i i i i ii v v v E v a p E v a p ( , ), ( , )i ia p a p

,i iv v