Sequential Groves mechanisms for public project problems Krzysztof - - PowerPoint PPT Presentation

Sequential Groves mechanisms for public project problems

Krzysztof R. Apt

CWI, Amsterdam, the Netherlands, University of Amsterdam joint work with

Arantza Est´ evez-Fern´ andez

Free University, Amsterdam

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Executive Summary

Groves mechanisms allow us to implement desired decisions by imposing on the players taxes. In Groves mechanisms truth-telling is a dominant strategy. In Groves mechanisms for the public project problems taxes = 0 are unavoidable. So we study the sequential Groves mechanisms. We show that other dominant strategies than truth-telling may then exist. They can be used to maximize social welfare in public projects problems.

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Reminder

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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 →

that he wants to maximize. Decision rule: a function f : Θ1 × · · · × Θn → D. We call

(D, Θ1, . . ., Θn, v1, . . ., vn, f)

a decision problem.

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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 the choice of d ∈ D so that his utility vi(d, θi) is maximized.

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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).

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Groves Mechanisms

Central authority takes the decision d := f(θ′) and computes the sequence of taxes t1(θ′), . . ., tn(θ′), where

ti(θ′) :=

• j=i

vj(f(θ′), θ′

j) + hi(θ′ −i),

hi : Θ−i →

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=i vj(f(θ′), θ′

j)

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

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Groves Mechanisms, ctd

Groves Theorem Suppose f is efficient. Then in each Groves mechanism (f, t) is strategy-proof. Groves mechanism with the tax function t := (t1, . . ., tn) is feasible if n

i=1 ti(θ) ≤ 0 for all θ,

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

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Special Case: VCG Mechanism

One uses hi(θ−i) := − maxd∈D

• j=i vj(d, θ′

j).

So ti(θ′) :=

j=i vj(f(θ′), θ′ j) − maxd∈D

• j=i vj(d, θ′

j).

Intuition:

• j=i vj(f(θ′), θ′

j)

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

maxd∈D

• j=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 be taken, i is pivotal. His tax is then = 0. Note VCG mechanism is pay only (so feasible).

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Example

Public project problem:

D = {0, 1}, [0, c] ⊆ Θi ⊆

cost of building the bridge: c,

vi(d, θi) := d(θi − c

n),

f(θ) :=

• 1 if n

i=1 θi ≥ c

0 otherwise

Note f is efficient.

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Example, ctd

Suppose c = 300 and n = 3. player value set of types (Θi) tax

ui

A 60

• 20
• 20

B 70

• 10
• 10

C 150

The project does not get through (d = 0) since

60 + 70 + 150 < 300.

Yet both A and B have to pay a tax.

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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).

Theorem In the public project problem for no c > 0 and n ≥ 2 a pay

• nly Groves mechanism exists that is superior to VCG

mechanism.

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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 a

type θ′

i,

• 3. each player takes the decision d := f(θ′

1, . . ., θ′ n).

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

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Sequential Mechanisms: Dominant Strategies

Set-up: sequential version of Example. Theorem

si(θ1, . . ., θi) :=      θi if i

j=1 θj < c and i < n,

if i

j=1 θj < c and i = n,

c if i

j=1 θj ≥ c

is a dominant strategy for player i. Strategy si(·) of player i minimizes the tax of every other player (subject to some conditions).

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Back to our Example

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

• rdering

tA tB tC

A B C A C B

−10

B A C B C A

−20

C A B

−10

C B A

−20

In each ordering at least one player pays smaller taxes. In 2 orderings taxes are 0.

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Can we Reduce Taxes to 0?

Theorem Assume each player follows si(·). For all c > 0, n ≥ 2 and θ1, . . ., θn for some permutation of players 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 to pay a tax = 0.

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Theorem: Precise Version

Theorem

si(θ1, . . ., θi) :=      θi if i

j=1 θj < c and i < n,

if i

j=1 θj < c and i = n,

c if i

j=1 θj ≥ c

is a dominant strategy for player i. If si(·) deviates from truth, then it simultaneously minimizes all taxes provided the decision is not changed. Formally: If si(θ1, . . ., θi) = θi, then

tj(si(θ1, . . ., θi), θ−i) ≥ tj(θ′

i, θ−i)

for all j = i, θ′

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

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Maximizing Social Welfare (1)

Fix a Groves mechanism

(D ×

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 the received type θi, player i may report θ′

i without taking any

‘risk’.

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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.

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Maximizing Social Welfare (2)

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 among all types that player i might report without incurring any risk,

si(·) selects the one that yields the ex post maximal social

welfare.

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Maximizing Social Welfare (3)

Set-up: sequential version of Example. Theorem

si(θ1, . . ., θi) :=          θi if i

j=1 θj < c and i < n,

if i

j=1 θj < c and i = n,

if i

j=1 θj = c, j=i θj < n−1 n c and i = n,

c

• therwise.

is socially dominant strategy for player i.

Sequential Groves mechanisms for public project problems – p.21/24

Example

player value submitted value tax utility (ui) A

110 110 −10

B

80 80 −20

C

110 110 −10

player value submitted value tax utility (ui) A

110 110

B

80 80

C

110

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

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Conclusions

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Advantages of sequential mechanism design

In sequential mechanism design other dominant strategies 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 the benefit of others. Dalai Lama: The intelligent way to be selfish is to work for the welfare of others. (Bowles ’04) Applicable to various forms of financing of public projects.

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