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

Sequential pivotal mechanisms for public project problems

Krzysztof R. Apt

(so not Krzystof and definitely not Krystof)

CWI, Amsterdam, the Netherlands, University of Amsterdam

joint work with

• A. Est´

evez-Fern´ andez

Vrije Universiteit, Amsterdam

Sequential pivotal mechanisms for public project problems – p. 1/2

Executive Summary

We study the public project problem. Our objective: to maximize social welfare. We study strategies in sequential setting. They can yield a higher social welfare.

Sequential pivotal mechanisms for public project problems – p. 2/2

Recap: Direct Mechanisms (1)

Given: set of decisions D, for each player i a set of types Θi, initial utility function vi : D × Θi → R.

Sequential pivotal mechanisms for public project problems – p. 3/2

Recap: Direct Mechanisms (2)

We consider the following sequence of events: each player i has an initial utility vi(d, θi), and a type (e.g., valuation of an item) θi, each player i announces to the central authority a type (e.g., a bid) θ′

i,

the central authority computes decision and taxes

d := f(θ′

1, . . ., θ′ n) and (t1, . . ., tn) := t(θ′ 1, . . ., θ′ n),

and communicates to each player i the pair (d, ti). Player’s i final utility: ui((f, t)(θ), θi) := vi(f(θ), θi) + ti(θ). Social welfare: n

i=1 ui((f, t)(θ), θi).

Sequential pivotal mechanisms for public project problems – p. 4/2

Recap: Direct Mechanisms (3)

A direct mechanism (f, t) is feasible if always n

i=1 ti(θ) ≤ 0.

(External funding not needed.) incentive compatible if no player is better off when submitting a false type (θ′

i = θi).

(Manipulations do not pay off or truth-telling is a dominant strategy.)

Sequential pivotal mechanisms for public project problems – p. 5/2

Public Project Problem

Each person is asked to report his or her willingness to pay for the project, and the project is undertaken if and only if the aggregate reported willingness to pay exceeds the cost of the project. (15 October 2007, The Royal Swedish Academy of Sciences, Press Release, Scientific Background)

Sequential pivotal mechanisms for public project problems – p. 6/2

Public Project Problem Formally

D = {0, 1}

for each player i

Θi = [0, c], where c > 0, vi(d, θi) := d(θi − c

n),

f(θ) :=

• 1 if n

i=1 θi ≥ c

0 otherwise

Sequential pivotal mechanisms for public project problems – p. 7/2

Incentive Compatibility

Theorem (Clarke ’71):

ti(θ′

i, θ−i) :=

• min(0, n−1

n c − k=i θk) if k=i θk + θ′ i < c

min(0,

k=i θk − n−1 n c) otherwise

yields an incentive compatible mechanism. Example

c = 300.

player type submitted type tax

ui

A

110 110 −10

B

80 80 −20

C

110 110 −10

Sequential pivotal mechanisms for public project problems – p. 8/2

Background: an Optimality Result

Theorem [Apt, Conitzer, Guo, Markakis, ’08] Consider the public project problem. No direct mechanism exists that is feasible, incentive compatible, ‘better’ than Clarke’s tax.

Sequential pivotal mechanisms for public project problems – p. 9/2

Sequential Mechanisms

Players move sequentially. Player i submits his/her type after he has seen the types of players 1, . . ., i − 1. The decisions and taxes are computed using a given direct based mechanism.

Sequential pivotal mechanisms for public project problems – p. 10/2

Strategies

Assume a sequential mechanism Seq. A strategy of player i in Seq:

si : Θ1 × . . . × Θi → Θi.

Strategy si(·) of player i is optimal in Seq if for all θ ∈ Θ and θ′

i ∈ Θi

ui((f, t)(si(θ1, . . ., θi), θ−i), θi) ≥ ui((f, t)(θ′

i, θ−i), θi).

Sequential pivotal mechanisms for public project problems – p. 11/2

Intuitions

Strategy of player j is memoryless if it does not depend

• n the types of players 1, . . ., j − 1.

Then si(·) is optimal iff for all θ ∈ Θ it yields a best response to all joint strategies of players j = i assuming players i + 1, . . ., n use memoryless strategies (or move jointly with player i). In particular, an optimal strategy is a best response to truth-telling by players j = i.

Sequential pivotal mechanisms for public project problems – p. 12/2

Optimality Result (1)

Theorem 1 Consider public project problem and Clarke’s tax. Strategy

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

j=1 θj < c and i < n,

0 (!) if i

j=1 θj < c and i = n,

c (!) if i

j=1 θj ≥ c

is optimal for player i in the sequential pivotal mechanism. Under certain natural circumstances si simultaneously maximizes the final utility of the other players.

Sequential pivotal mechanisms for public project problems – p. 13/2

Example 1

c = 300.

Pivotal mechanism: player type submitted type tax

ui

A

110 110 −10

B

80 80 −20

C

110 110 −10

Now: player type submitted type tax

ui

A

110 110 10

B

80 80 −20

C

110 300 −10

Sequential pivotal mechanisms for public project problems – p. 14/2

Example 2

c = 300.

Pivotal mechanism: player type submitted type tax

ui

A

110 110

B

80 80 −10 −10

C

100 100

Now: player type submitted type tax

ui

A

110 110

B

80 80

C

100

Sequential pivotal mechanisms for public project problems – p. 15/2

Optimality Result (2)

Theorem 2 Consider public project problem and Clarke’s tax. Strategy

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

if i

j=1 θj < c and i < n,

0 (!)

if i

j=1 θj < c and i = n,

0 (!!) if i

j=1 θj = c, θi > c n and i = n,

c (!)

• therwise

is optimal for player i in the sequential pivotal mechanism, When all players follow si(·), maximal social welfare is generated in the universe of optimal strategies.

Sequential pivotal mechanisms for public project problems – p. 16/2

Example 3

c = 300.

Before: player type submitted type tax

ui

A

110 110 10

B

80 80 −20

C

110 300 −10

Now: player type submitted type tax

ui

A

110 110

B

80 80

C

110

Sequential pivotal mechanisms for public project problems – p. 17/2

Nash Implementation

Suppose players submit their strategies simultaneously, for each vector of initial types their final utilities are determined using the pivotal mechanism. Game-theoretic interpretation: sequential pre-Bayesian games. Theorem Vectors of strategies from Theorems 1 and 2 form a Nash equilibrium in the universe of optimal strategies. The result does not hold if deviations to non-optimal strategies are allowed.

Sequential pivotal mechanisms for public project problems – p. 18/2

Conclusions

Social welfare can be increased if the players move sequentially. Dalai Lama: The intelligent way to be selfish is to work for the welfare of others. Microeconomics: Behavior, Institutions, and Evolution,

• S. Bowles ’04.

Recent work: similar analysis for sequential Bailey-Cavallo mechanism for single item auctions ([Apt, Markakis, WINE’09]).

Sequential pivotal mechanisms for public project problems – p. 19/2

THANK YOU

Sequential pivotal mechanisms for public project problems – p. 20/2

More on Optimal Strategies

Consider the sequential pivotal mechanism. Lemma si(·) is an optimal strategy for player i iff the following holds: Suppose i

j=1 θj < c and i < n. Then si(θ1, . . ., θi) = θi.

Suppose i

j=1 θj < c and i = n. Then

n−1

j=1 θj + si(θ1, . . ., θn) < c.

Suppose i

j=1 θj = c and i < n. Then si(θ1, . . ., θi) ≥ θi.

Suppose i

j=1 θj > c. Then i−1 j=1 θj + si(θ1, . . ., θi) ≥ c.

Sequential pivotal mechanisms for public project problems – p. 21/2