Sequential mechanism design Krzysztof R. Apt (so not Krzystof and - - PowerPoint PPT Presentation

Sequential mechanism design

Krzysztof R. Apt

(so not Krzystof and definitely not Krystof)

CWI, Amsterdam, the Netherlands, University of Amsterdam

based on joint works with

• A. Est´

evez-Fern´ andez

• E. Markakis

Sequential mechanism design – p. 1/3

Executive Summary

Mechanism design: how to arrange our economic interactions so that, when everyone behaves in a self-interested manner, the result is something we all like. Important question: how to avoid manipulations? This can be done, but is costly. Our objective: minimize these costs. We study the problem in sequential setting for public project problem, single unit auctions.

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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.

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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).

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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 mechanism design – p. 5/3

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)

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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 mechanism design – p. 7/3

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 mechanism design – p. 8/3

Optimality Result (1)

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

Sequential mechanism design – p. 9/3

However . . .

Clarke’s tax is not optimal in the public project problem when the payments per player can differ. Note: Pivotal mechanism then ceases to be anonymous.

Sequential mechanism design – p. 10/3

(Single Item) Sealed Bid Auction

argsmax θ := µi(θi = maxj∈{1,...,n} θj).

D = {1, . . ., n},

for each player i

Θi = R+, vi(d, θi) :=

• θi if d = i
• therwise

f(θ) := argsmax θ.

Sequential mechanism design – p. 11/3

Vickrey Auction as a Direct Mechanism

θ∗: the reordering of θ in descending order. tV

i (θ) :=

• −θ∗

2 if i = argsmax θ

• therwise

Example: player bid tax to authority

ui

A

18

B

24 −21 3

C

21

Theorem : Vickrey auction is incentive compatible.

Sequential mechanism design – p. 12/3

Bailey-Cavallo Mechanism

ti(θ) := tV

i (θ) + (θ−i)∗ 2

n

Example: player bid tax to authority

ui

why? A

18 7

(= 1/3 of 21) B

24 −2 9

(= 24 − 2 − 7 − 6) C

21 6

(= 1/3 of 18) Theorem: Bailey-Cavallo mechanism is feasible and incentive compatible. Warning: Bailey-Cavallo mechanism does not satisfy the participation constraint.

Sequential mechanism design – p. 13/3

Optimality Result (2)

Theorem [Apt, Conitzer, Guo, Markakis, WINE’08] Consider the sealed bid auction. No tax-based mechanism exists that is feasible, incentive compatible, ‘better’ than Bailey-Cavallo mechanism.

Sequential mechanism design – p. 14/3

Groves Auctions

A sealed bid auction with redistribution:

ti(θ) := tV

i (θ) + ri(θ−i).

Theorem [Groves ’73] Each Groves auction is incentive compatible.

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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.

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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).

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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.

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Optimality Result (3)

Theorem [Apt, Estévez-Fernández, SAGT’09] 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.

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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 mechanism design – p. 20/3

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

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Optimality Result (4)

Theorem [Apt, Estévez-Fernández, SAGT’09] 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.

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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 mechanism design – p. 23/3

Proof Idea (1)

Lemma 1 Let s′

i(·) be an optimal strategy for player i.

Suppose i

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

Suppose i

j=1 θj < c and i = n. Then

n−1

j=1 θj + s′ i(θ1, . . ., θn) < c.

Suppose i

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

Suppose i

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

Proof In each case by case analysis.

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Proof Idea (2)

Lemma 2 si(·) maximizes social welfare in the universe of

• ptimal strategies, assuming that players who follow i are

truthful. Proof By Lemma 1 and case analysis: Case 1 i

j=1 θj < c and i < n.

Case 2 i

j=1 θj < c and i = n

Case 3 i

j=1 θj = c, θi > c n and i = n.

Case 4 i

j=1 θj = c, θi ≤ c n and i = n.

Case 5 (i

j=1 θj = c and i < n) or i j=1 θj > c.

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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 mechanism design – p. 26/3

Optimal Strategies in Seq. Groves Auctions

¯ θi := maxj∈{1,...,i−1} θj.

Lemma

si(·) is an optimal strategy for player i iff the following holds:

Suppose θi > ¯

θi and i < n. Then si(θ1, . . ., θi) = θi.

Suppose θi > ¯

θi and i = n. Then si(θ1, . . ., θi) > ¯ θi.

Suppose θi ≤ ¯

θi and i < n. Then si(θ1, . . ., θi) ≤ ¯ θi.

Suppose θi < ¯

θi and i = n. Then si(θ1, . . ., θi) ≤ ¯ θi.

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Optimality Result (5)

Theorem ([Apt, Markakis, AAMAS ’09]). Strategy

si(θ1, . . ., θi) :=

• θi

if θi > maxj∈{1,...,i−1} θj, 0(!) otherwise

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

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Example

Before: player bid tax to authority

ui

A

18

B

24 −21 3

C

21

Now: player bid tax to authority

ui

A

18

B

24 −18 6

C Social welfare: 3 vs 6.

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Optimality Result (6)

Theorem ([Apt, Markakis, WINE’09]).

si(θ1, . . ., θi) :=      θi

if θi > maxj∈{1,...,i−1} θj

(θ1, . . . , θi−1)∗

1(!) if θi ≤ maxj∈{1,...,i−1} θj and i ≤ n − 1

(θ1, . . . , θi−1)∗

2(!) otherwise

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

Sequential mechanism design – p. 30/3

Example: Bailey-Cavallo mechanism

Before: player type tax to authority

ui

why? A

18 7

(= 21/3) B

24 −2 9

(= 24 − 21 + 18/3) C

21 6

(= 18/3) Now: player type tax to authority

ui

why? A

18 6

(= 18/3) B

24 12

(= 24 − 18 + 18/3) C

18 6

(= 18/3) Social welfare: 22 vs 24.

Sequential mechanism design – p. 31/3

Safety-level Equilibrium

Introduced in [Ashlagi, Monderer, Tennenholtz ’06] for pre-Bayesian games. Given θ≤i ∈ Θ≤i and s(·)

min

θ>i∈Θ>i ui((f, t)([s(·), θ]), θi)

is the guaranteed final utility for player i.

s(·) i s′(·) iff for all θ≤i ∈ Θ≤i min

θ>i∈Θ>i ui((f, t)([s(·), θ]), θi) ≥

min

θ>i∈Θ>i ui((f, t)([s′(·), θ]), θi).

s(·) is safety-level equilibrium if for all i and s′

i(·)

(si(·), s−i(·)) i (s′

i(·), s−i(·)).

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Implementation in Safety-level Equilibrium

Theorem ([Apt, Markakis, WINE’09]). Introduced vector s(·) of strategies in sequential Vickrey auction forms a safety-level equilibrium. Introduced vector s(·) of strategies in sequential Bailey-Cavallo mechanism forms a safety-level equilibrium.

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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.

Sequential mechanism design – p. 34/3

THANK YOU

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