Sequential Groves mechanisms for public project problemshomepages.cwi.nl/~apt/slides/tilburg07-sli.pdf · Sequential Groves mechanisms for public project problems Krzysztof R. Apt

Sequential Groves mechanisms forpublic project problems

Krzysztof R. AptCWI, Amsterdam, the Netherlands,

University of Amsterdam

joint work with

Arantza Estevez-FernandezFree University, Amsterdam

Sequential Groves mechanisms for public project problems – p.1/24

Executive Summary

Groves mechanisms allow us to implement desireddecisions by imposing on the players taxes.

In Groves mechanisms truth-telling is a dominantstrategy.

In Groves mechanisms for the public project problemstaxes 6= 0 are unavoidable.

So we study the sequential Groves mechanisms.

We show that other dominant strategies thantruth-telling may then exist.

They can be used to maximize social welfare in publicprojects problems.


Reminder


Decision problems

Assume

players 1, . . ., n,

set of decisions D,

for each player a set of types Θi and a utility function

vi : D × Θi → Rthat he wants to maximize.

Decision rule: a function f : Θ1 × · · · × Θn → D.

We call(D, Θ1, . . ., Θn, v1, . . ., vn, f)

a decision problem.


Decisions, Decisions, . . .

One studies the following sequence of events:

1. each player i receives type θi,

2. each player i announces to the central planner a type θ′i,

3. the central planner takes the decision d := f(θ′1, . . ., θ′n),

and communicates it to each player,

4. the resulting utility for player i is then vi(d, θi).

Problem to solve: Each player i wants to manipulate thechoice of d ∈ D so that his utility vi(d, θi) is maximized.


Decision rules

A decision rule f is called

strategy-proof if for all θ ∈ Θ, i ∈ {1, . . ., n} and θ′i ∈ Θ

vi(f(θi, θ−i), θi) ≥ vi(f(θ′i, θ−i), θi).

efficient if for all θ ∈ Θ and d′ ∈ D

n∑

i=1

vi(f(θ), θi) ≥n

∑

i=1

vi(d′, θi).


Groves Mechanisms

Central authority takes the decision d := f(θ′) andcomputes the sequence of taxes t1(θ

′), . . ., tn(θ′), where

ti(θ′) :=

∑

j 6=i

vj(f(θ′), θ′j) + hi(θ′−i),

hi : Θ−i → R is an arbitrary function.

player’s i ex post utility is

ui((f, t)(θ′i, θ−i), θi) := vi(f(θ′i, θ−i), θi) + ti(θ′i, θ−i).

Intuition:∑

j 6=i vj(f(θ′), θ′j)

is the social welfare with i excluded from decision f(θ′).


Groves Mechanisms, ctd

Groves TheoremSuppose f is efficient. Then in each Grovesmechanism (f, t) is strategy-proof.

Groves mechanism with the tax function t := (t1, . . ., tn)is

feasible if∑n

i=1ti(θ) ≤ 0 for all θ,

pay only if ti(θ) ≤ 0 for all θ and i.


Special Case: VCG Mechanism

One uses hi(θ−i) := −maxd∈D

∑

j 6=i vj(d, θ′j).

So ti(θ′) :=

∑

j 6=i vj(f(θ′), θ′j) − maxd∈D

∑

j 6=i vj(d, θ′j).

Intuition:∑

j 6=i vj(f(θ′), θ′j)

is the social welfare from f(θ′) with i excluded,maxd∈D

∑

j 6=i vj(d, θ′j)

is the maximal social welfare from f(θ′) with i

excluded.

Note If with player i excluded better decision can betaken, i is pivotal. His tax is then 6= 0.

Note VCG mechanism is pay only (so feasible).


Example

Public project problem:

D = {0, 1},

[0, c] ⊆ Θi ⊆ R +,

cost of building the bridge: c,

vi(d, θi) := d(θi −cn),

f(θ) :=

{

1 if∑n

i=1θi ≥ c

0 otherwise

Note f is efficient.


Example, ctd

Suppose c = 300 and n = 3.

player value set of types (Θi) tax ui

A 60 R + -20 -20B 70 R + -10 -10C 150 R + 0 0

The project does not get through (d = 0) since60 + 70 + 150 < 300.

Yet both A and B have to pay a tax.


Optimality of VCG Mechanism

Groves mechanism h′ is superior to h if

for all θ and i

h′i is better for player i than hi:

hi(θ−i) ≤ h′i(θ−i),

for some θ and i

h′i is strictly better for player i than hi:

hi(θ−i) < h′i(θ−i).

TheoremIn the public project problem for no c > 0 and n ≥ 2 a payonly Groves mechanism exists that is superior to VCGmechanism.


Sequential Mechanism Design

The players are randomly ordered.The following revised sequence of events then takes place:

1. each player i receives type θi,

2. each player i in turn announces to the other players atype θ′i,

3. each player takes the decision d := f(θ′1, . . ., θ′n).

Crucial difference: Now each player announces his typeafter he saw the types of earlier players.


Sequential Mechanisms: Dominant Strategies

Set-up: sequential version of Example.

Theorem

si(θ1, . . ., θi) :=

θi if∑i

j=1θj < c and i < n,

0 if∑i

j=1θj < c and i = n,

c if∑i

j=1θj ≥ c

is a dominant strategy for player i.

Strategy si(·) of player i minimizes the tax of every otherplayer(subject to some conditions).


Back to our Example

A: 60, B: 70, C: 150; c = 300.

ordering tA tB tC

A B C 0 0 0

A C B 0 −10 0

B A C 0 0 0

B C A −20 0 0

C A B 0 −10 0

C B A −20 0 0

In each ordering at least one player pays smaller taxes.

In 2 orderings taxes are 0.


Can we Reduce Taxes to 0?

Theorem

Assume each player follows si(·).For all c > 0, n ≥ 2 and θ1, . . ., θn for some permutation ofplayers all taxes equal 0.

Proof idea

Not all players can be pivotal.

Put a non-pivotal player at the end.

However: If a pivotal player is the last one, then he has topay a tax 6= 0.


Theorem: Precise Version

Theorem

si(θ1, . . ., θi) :=

θi if∑i


0 if∑i


c if∑i

j=1θj ≥ c

is a dominant strategy for player i.

If si(·) deviates from truth, then it simultaneouslyminimizes all taxes provided the decision is notchanged.Formally: If si(θ1, . . ., θi) 6= θi, then

tj(si(θ1, . . ., θi), θ−i) ≥ tj(θ′i, θ−i)

for all j 6= i, θ′i, θi+1, . . ., θn such that f(θ′i, θ−i) = f(θi, θ−i).


Maximizing Social Welfare (1)

Fix a Groves mechanism(D × R n, Θ1, . . ., Θn, u1, . . ., un, (f, t)).

Type θ′i is compatible with θ1, . . ., θi if for all θi+1, . . ., θn

ui((f, t)(θ′i, θ−i), θi) ≥ ui((f, t)(θi, θ−i), θi).

Intuition Given the announced types θ1, . . ., θi−1 and thereceived type θi, player i may report θ′i without taking any‘risk’.


Example

Suppose

f is efficient,

for all θ ∈ Θ, f(si(θ1, . . ., θi), θ−i) = f(θi, θ−i).

Then for all θ1, . . ., θi

si(θ1, . . ., θi) is compatible with θ1, . . ., θi.



Strategy si(·) of player i is socially dominant if

it is dominant,

for all θ and all θ′i compatible with θ1, . . ., θi

n∑

j=1

uj((f, t)(si(θ1, . . ., θi), θ−i), θj) ≥n

∑

j=1

uj((f, t)(θ′i, θ−i), θj).

Intuition si(·) is socially dominant if it is dominant and amongall types that player i might report without incurring any risk,si(·) selects the one that yields the ex post maximal social

welfare.



Set-up: sequential version of Example.

Theorem

si(θ1, . . ., θi) :=

θi if∑i


0 if∑i


0 if∑i

j=1θj = c,

∑

j 6=i θj < n−1

nc and i = n,

c otherwise.

is socially dominant strategy for player i.


Example

player value submitted value tax utility (ui)A 110 110 −10 0

B 80 80 0 −20

C 110 110 −10 0

player value submitted value tax utility (ui)A 110 110 0 0

B 80 80 0 0

C 110 0 0 0

By submitting 0 instead of the true value, 110, player Cincreased social welfare from −20 to 0.


Conclusions


Advantages of sequential mechanism design

In sequential mechanism design other dominantstrategies may exist than truth-telling.

Such strategies can be used to maximize social welfare.

Cooperative aspects can be incorporated.

Maxim: First take care of your benefit and then of thebenefit of others.

Dalai Lama: The intelligent way to be selfish is to workfor the welfare of others.(Bowles ’04)

Applicable to various forms of financing of publicprojects.


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Sequential Groves mechanisms for public project problemshomepages.cwi.nl/~apt/slides/tilburg07-sli.pdf · Sequential Groves mechanisms for public project problems Krzysztof R. Apt