Optimal Strategies in Sequential Bidding Krzysztof R. Apt (so not - - PowerPoint PPT Presentation

Optimal Strategies in Sequential Bidding

Krzysztof R. Apt

(so not Krzystof and definitely not Krystof)

CWI, Amsterdam, the Netherlands, University of Amsterdam

based on joint works with

• V. Conitzer, M. Guo and E. Markakis

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Intelligent Design

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Intelligent Design

A theory of an intelligently guided invisible hand wins the Nobel prize WHAT on earth is mechanism design? was the typical reaction to this year’s Nobel prize in economics, announced on October 15th. [...] In fact, despite its dreary name, mechanism design is a hugely important area of economics, and underpins much of what dismal scientists do today. It goes to the heart

• f one of the biggest challenges in

economics: how to arrange our economic interactions so that, when everyone behaves in a self-interested manner, the result is something we all like. (The Economist, Oct. 18th, 2007)

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Summary for Nissim

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 (= maximize social welfare). We study this problem in simultaneous and sequential setting for single unit auctions.

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Recap: Tax-based Mechanisms (1)

The following sequence of events: each player i has a valuation θi of an item, each player i announces to the central authority a bid θ′

i,

the central authority computes decision and taxes

d := f(θ′

1, . . ., θ′ n) ∈ {1, . . ., n},

(t1, . . ., tn) := g(θ′

1, . . ., θ′ n) ∈ Rn,

and communicates to each player i the pair (d, ti). Player’s i final utility: (d = i) · θi + ti. An aside: this setting applies to many other decision problems.

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Recap (2)

A tax-based 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 bid (θ′

i = θi).

(Manipulations do not pay off.) Formally:

(f(θ1, . . ., θi, . . ., θn) = i) · θi + ti ≥ (f(θ1, . . ., θ′

i, . . ., θn) = i) · θi + ti.

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Example: Vickrey Auction

Central authority (seller) organizes a sealed bid auction, so each player submits simultaneously his/her bid in a sealed envelope, the object is allocated to the highest bidder. Vickrey auction: the winner pays the 2nd highest bid. Example: player bid tax to authority util. A

18

B

24 −21 3

C

21

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Formally

θ∗: the reordering of θ is descending order, θ−i: the bids of opponents of player i,

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

f(θ) := argsmax θ, tV

i (θ) :=

• −θ∗

2 if i = argsmax θ

• therwise

Theorem (Vickrey ’61): Vickrey auction is incentive compatible. Social welfare in Vickrey auction: θ∗

1 − θ∗ 2.

Can we do better, while maintaining incentive compatibility and feasibility?

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Bailey-Cavallo Mechanism

Bailey-Cavallo mechanism:

ti(θ) := tV

i (θ) + (θ−i)∗ 2

n

Example: player bid tax to authority util. why? A

18 7

(= 1/3 of 21) B

24 −2 9

(= 24 − 2 − 7 − 6) C

21 6

(= 1/3 of 18) Can we do better than Bailey-Cavallo?

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Optimality Result (1)

Theorem (ACGM ’08): Consider the sealed bid auction. No tax-based mechanism exists that is feasible, incentive compatible, ‘better’ than Bailey-Cavallo mechanism.

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Sequential Mechanisms

Players move sequentially. Player i submits his bid after he has seen the bids of players 1, . . ., i − 1. The decisions and taxes are computed using a given tax-based mechanism.

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Optimality Result (2)

Theorem (AM ’09): Consider Vickrey auction. For i ∈ {1, . . ., n − 1} no dominant strategy exists for player i. Strategy

si(θ1, . . ., θi) :=

• θi if θi > maxj∈{1,...,i−1} θj,
• therwise

is optimal for player i, given other players are myopic. When all players follow si(·), maximal social welfare is generated.

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Example

Before: player bid tax to authority util. A

18

B

24 −21 3

C

21

Now: player bid tax to authority util. A

18

B

24 −18 6

C Social welfare: 3 vs 6.

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Optimality Result (3)

Theorem (AM ’09): Consider Bailey-Cavallo mechanism. For i ∈ {1, . . ., n − 1} no dominant strategy exists for player i.

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, given other players are truthful, When all players follow si(·), maximal social welfare is generated.

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Example: Bailey-Cavallo mechanism

Before: player type tax to authority util. 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 util. why? A

18 6

(= 18/3) B

24 12

(= 24 − 18 + 18/3) C

18 6

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

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Conclusions

Bailey-Cavallo mechanism is optimal for single unit auctions. Social welfare in Vickrey auction and in Bailey-Cavallo mechanism can be increased if the players move sequentially. We also studied this problem for public project problem.

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Useful Quote

Dalai Lama: The intelligent way to be selfish is to work for the welfare of others. Microeconomics: Behavior, Institutions, and Evolution, S. Bowles ’04.

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THANK YOU

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Nissim

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for infecting others with your passion for learning.

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