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 Optimal Strategies in Sequential Bidding – p. 1/2
Intelligent Design Optimal Strategies in Sequential Bidding – p. 2/2
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 of 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) Optimal Strategies in Sequential Bidding – p. 3/2
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. Optimal Strategies in Sequential Bidding – p. 4/2
Recap: Tax-based Mechanisms (1) The following sequence of events: each player i has a valuation θ i 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 } , ( t 1 , . . ., t n ) := g ( θ ′ 1 , . . ., θ ′ n ) ∈ R n , and communicates to each player i the pair ( d, t i ) . Player’s i final utility: ( d = i ) · θ i + t i . An aside: this setting applies to many other decision problems. Optimal Strategies in Sequential Bidding – p. 5/2
Recap (2) A tax-based mechanism ( f, t ) is feasible if always � n i =1 t i ( θ ) ≤ 0 . (External funding not needed.) incentive compatible if no player is better off when submitting a false bid ( θ ′ i � = θ i ). (Manipulations do not pay off.) Formally: ( f ( θ 1 , . . ., θ i , . . ., θ n ) = i ) · θ i + t i ≥ ( f ( θ 1 , . . ., θ ′ i , . . ., θ n ) = i ) · θ i + t i . Optimal Strategies in Sequential Bidding – p. 6/2
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 0 0 B 24 − 21 3 C 21 0 0 Optimal Strategies in Sequential Bidding – p. 7/2
Formally θ ∗ : the reordering of θ is descending order, θ − i : the bids of opponents of player i , argsmax θ := µi ( θ i = max j ∈{ 1 ,...,n } θ j ) , f ( θ ) := argsmax θ, � − θ ∗ 2 if i = argsmax θ t V i ( θ ) := 0 otherwise 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? Optimal Strategies in Sequential Bidding – p. 8/2
Bailey-Cavallo Mechanism Bailey-Cavallo mechanism: i ( θ ) + ( θ − i ) ∗ t i ( θ ) := t V 2 n Example: player bid tax to authority util. why? A (= 1 / 3 of 21 ) 18 0 7 B (= 24 − 2 − 7 − 6 ) 24 − 2 9 C (= 1 / 3 of 18 ) 21 0 6 Can we do better than Bailey-Cavallo? Optimal Strategies in Sequential Bidding – p. 9/2
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. Optimal Strategies in Sequential Bidding – p. 10/2
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. Optimal Strategies in Sequential Bidding – p. 11/2
Optimality Result (2) Theorem (AM ’09): Consider Vickrey auction. For i ∈ { 1 , . . ., n − 1 } no dominant strategy exists for player i . Strategy � θ i if θ i > max j ∈{ 1 ,...,i − 1 } θ j , s i ( θ 1 , . . ., θ i ) := 0 otherwise is optimal for player i , given other players are myopic. When all players follow s i ( · ) , maximal social welfare is generated. Optimal Strategies in Sequential Bidding – p. 12/2
Example Before: player bid tax to authority util. A 18 0 0 B 24 − 21 3 C 21 0 0 Now: player bid tax to authority util. A 18 0 0 B 24 − 18 6 C 0 0 0 Social welfare: 3 vs 6. Optimal Strategies in Sequential Bidding – p. 13/2
Optimality Result (3) Theorem (AM ’09): Consider Bailey-Cavallo mechanism. For i ∈ { 1 , . . ., n − 1 } no dominant strategy exists for player i . s i ( θ 1 , . . ., θ i ) := θ i if θ i > max j ∈{ 1 ,...,i − 1 } θ j ( θ 1 , . . . , θ i − 1 ) ∗ 1 if θ i ≤ max j ∈{ 1 ,...,i − 1 } θ j and i ≤ n − 1 ( θ 1 , . . . , θ i − 1 ) ∗ 2 otherwise is optimal for player i , given other players are truthful, When all players follow s i ( · ) , maximal social welfare is generated. Optimal Strategies in Sequential Bidding – p. 14/2
Example: Bailey-Cavallo mechanism Before: player type tax to authority util. why? A 18 0 7 (= 21 / 3 ) B 24 − 2 9 (= 24 − 21 + 18 / 3 ) C 21 0 6 (= 18 / 3 ) Now: player type tax to authority util. why? A (= 18 / 3 ) 18 0 6 B (= 24 − 18 + 18 / 3 ) 24 0 12 C 18 0 6 (= 18 / 3 ) Social welfare: 22 vs 24. Optimal Strategies in Sequential Bidding – p. 15/2
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. Optimal Strategies in Sequential Bidding – p. 16/2
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. Optimal Strategies in Sequential Bidding – p. 17/2
THANK YOU Optimal Strategies in Sequential Bidding – p. 18/2
Nissim Optimal Strategies in Sequential Bidding – p. 19/2
for infecting others with your passion for learning. Optimal Strategies in Sequential Bidding – p. 20/2
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