Me Merge gers a and Collus Collusion ion in in Al All - Pay Au - PowerPoint PPT Presentation

Me Merge gers a and Collus Collusion ion in in Al All - Pay Au Auctions and Crowdsourcin Crowdsourcing Con g Conte tests sts Omer Lev, Maria Polukarov, Yoram Bachrach & Jeffrey S. Rosenschein AAMAS 2013 St. Paul, Minnesota

Perliminaries Al All - pay auctions Bidders bid and pay their bid to the auctioneer Auction winner is one which submitted the highest bid

Perliminaries Wh Why all all - pay pay auction auctions? Explicit all-pay auctions are rare, but implicit ones are extremely common: Competition for patents between firms Crowdsourcing competitions (e.g., Netflix challenge, TopCoder, etc.) Hiring employees Employee competition (“employee of the month”)

Perliminaries Auctioneer types Au “sum “max profit” profit” Gets the bids Gets only the from all bidders winner’s bid. – regardless of Other bids are, their winning effectively, status “burned” E.g., “emloyee E.g., hiring an of the month” emplyee

Regular all-pay All - pay auction equilibrium Al All bidders give the object in question a value of 1 A single symmetric equilibrium – for n bidders: 1 F n ( x ) = x n − 1 2 − n f n ( x ) = x n − 1 n − 1 1.5 1.25 1 0.75 0.5 0.25 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 Baye, Kovenock, de Vries

Regular all-pay All - pay auction equilibrium Al bidder propert bi dder properties es Expected utility: 0 3 n 2 − 5 n + 2 Utility variance: n (2 n − 1)(3 n − 2) 1 Expected bid: n 2 n − 1 − 1 1 Bid variance: n 2 Baye, Kovenock, de Vries

Regular all-pay All - pay auction equilibrium Al auctioneer propert auct oneer properties es Sum profit 1 expected profit: Sum profit 2 n − 1 − 1 n profit variance: n Max profit n expected profit: 2 n − 1 Max profit n ( n − 1) 2 profit variance: (3 n − 2)(2 n − 1) 2 Baye, Kovenock, de Vries

Regular all-pay Example Exampl e no no co collusi sion n case case 3 bidders Bidders’ c.d.f is and the expected bid is ⅓ , with √ x 2 4 variance of . Expected profit is 0 with variance of . 15 45 Sum profit auctioneer has expected profit of 1 with variance of . 4 15 Max profit auctioneer has expected profit of ⅗ with variance of . 12 175 Baye, Kovenock, de Vries

Me Merge gers k bidders (out of the total n ) collaborate, having a joint (collaboration public knowledge) strategy. All other bidders are aware of this. Mergers

Merge Me ger p prop opertie ties Equilibrium remains the same – but with smaller n Sum Profit Bidder Expected profit: 1 Expected Utility: 0 Profit variance: (collaboration public knowledge) Utility variance: Mergers Max Profit Expected bid: Expected profit: Bid variance: Profit variance:

Exampl Example e no no co collusi sion n case case 3 bidders Bidders’ c.d.f is and the expected bid is ⅓ , with √ x 2 4 variance of . Expected profit is 0 with variance of . 15 45 (collaboration public knowledge) Sum profit auctioneer has expected profit of 1 with variance of . 4 Mergers 15 Max profit auctioneer has expected profit of ⅗ with variance of . 12 175

Example Exampl e merg merger er case case 3 bidders, 2 of them merged Bidders’ c.d.f is uniform, and the expected bid is ½ , with variance of . Expected profit is 0 with variance 1 12 of ⅙ . (collaboration public knowledge) Sum profit auctioneer has expected profit of 1 with Mergers variance of ⅙ . Max profit auctioneer has expected profit of ⅔ with 1 variance of . 18

Collusion Collusions s k bidders (out of the total n ) collaborate, having a joint strategy. Other bidders are not (collaboration private knowledge) aware of this and continue to pursue their previous Collusion strategies.

Collusion Collusion col colluders uders Colluders have a pure, optimal strategy ◆ n − 1 ✓ n − k k − 1 e − 1 b ∗ = k: n: n − 1 Producing an expected profit of: (collaboration private knowledge) ◆ n − 1 ✓ n − k k − 1 ✓ k − 1 ◆ k: n: Collusion n − 1 n − 1 Colluders’ profit per colluder increases as number of colluders grows ◆ 2( n − k ) ◆ n − k ✓ n − k ✓ n − k k − 1 k − 1 Profit variance: − n − 1 n − 1

Collusion Collusion auct auctioneers oneers ◆ n − 1 ✓ n − k n − k k − 1 Sum profit: + n n − 1 k: n: (collaboration private knowledge) 0 1 ◆ 2( n − k ) ✓ n − k n − k k − 1 Max profit: Collusion @ 1 + A 2 n − k − 1 n − 1 k: n: For large enough n exceed non-colluding profits

Collusion Collusion no non - co colludi ding ng bi bidders dders Utility for non-colluding bidders is: n − k ( n − k n − 1 ) k k − 1 n ( n − k ) − n − k (collaboration private knowledge) Collusion For large enough k (e.g., ) this expression is positive. n 2 I.e., non-colluders profit from collusion If a non-colluder discovers the collusion, best to bid a bit above colluders

Exampl Example e no no co collusi sion n case case 3 bidders Bidders’ c.d.f is and the expected bid is ⅓ , with √ x 2 4 variance of . Expected profit is 0 with variance of . 15 45 Sum profit auctioneer has expected profit of 1 with (collaboration private knowledge) variance of . 4 15 Collusion Max profit auctioneer has expected profit of ⅗ with variance of . 12 175

Example Exampl e merg merger er case case 3 bidders, 2 of them merged Bidders’ c.d.f is uniform, and the expected bid is ½ , with variance of . Expected profit is 0 with variance 1 12 of ⅙ . (collaboration private knowledge) Sum profit auctioneer has expected profit of 1 with Collusion variance of ⅙ . Max profit auctioneer has expected profit of ⅔ with 1 variance of . 18

Exampl Example e col collusi usion case on case 3 bidders, 2 of them collude One bidder has c.d.f of (expected bid of ⅓ ), √ x colluders bid ¼ . Colluders’ expected profit is ¼ , while the non-colluder expected profit is ⅙ . (collaboration private knowledge) 7 Sum profit auctioneer expected profit only . Collusion 12 10 Max profit auctioneer has expected profit of . 24

Fu Future re di direct rections ns Adding bidders’ skills to model Detecting collusions by other bidders Designing crowdsourcing mechanisms less susceptible to collusion Adding probability to win based on effort

The he End Thanks for listening! !