Computationally Feasible VCG Mechanisms. by Alpha Chau (feat. MC Bryce Wiedenbeck)

Transcription

1 Computationally Feasible VCG Mechanisms by Alpha Chau (feat. MC Bryce Wiedenbeck)

2 Recall: Mechanism Design Definition: Set up the rules of the game s.t. the outcome that you want happens. Often, it is the socially-best outcome, e.g. highest total welfare (sum of payoffs).

3 Recall: Mechanism Design Applications: Economic applications and e-commerce (auctions) AI (multi-agent systems) Decentralized resource and task allocations Communication Networks

4 Recall: Mechanism Design Incentive-Compatibility: Property of a mechanism in which participants will bid their actual values (i.e. types) for an item during an auction, i.e. truthfulness.

5 Recall: Mechanism Design Revelation Principle: If there exists any mechanism that achieves what you want, there exists a truthful mechanism that achieves want you want. So, we can focus on coming up with a truthful mechanism straightup (if you prove there doesn t exist a truthful mechanism that does what you want, then you prove there doesn't exist a mechanism at all that does what you want).

6 VCG Mechanism: Overview Named after Vickrey, Clarke, & Groves A General Method for Constructing Truthful Mechanisms

7 VCG Mechanism: The Vickrey Auction Vickrey Auction (i.e. Second-Price Auction) Guarantees incentive-compatibility

8 VCG Mechanism: The Vickrey Auction In first-price auctions, increasing your bid increases your chances of winning, but there exists incentive to lower your bid from your actual value to make off with positive payoff

9 VCG Mechanism: The Vickrey Auction Vickrey auctions remove incentive to underbid since someone else s bid determines your payment Increasing your bid still increases your chance of winning, but lowering your bid does not lower your payment

10 VCG Mechanism: The Vickrey Auction Since, rational agents never bid higher than they actually value, in the end, we get... Truthfulness! The VCG Mechanism is a generalization of Vickrey Auction

11 The VCG Mechanism 1) Determine Allocation 2) Determine Prices a) for each winner: i) Remove them ii) Determine allocation iii) They pay imposed cost

12 The VCG Mechanism Examples: VCG mechanism for 1 good VCG mechanism for 3-goods

13 The VCG Algorithm: Step I Allocate auctions to highest bidders 1) Determine Allocation 2) Determine Prices a) for each winner: i) Remove them ii) Determine allocation iii) They pay imposed cost

14 The VCG Algorithm: Step 2 For each auction s winner, remove them from the auction and rerun the allocation 1) Determine Allocation 2) Determine Prices a) for each winner: i) Remove them ii) Determine allocation iii) They pay imposed cost

15 The VCG Algorithm: Step 2 - Imposing Costs Price to be paid by auction winners Everyone else s payoff while winner not in auction = - in Everyone else s payoff while winner auction

16 The Problem with VCG Allocation needs to determine assignment of items to bidders that provides maximum total value across all bidders This is NP-Complete....and we have to solve this NP-Complete problem n+1 times. Like, really?

17 The VCG Mechanism 2.0??? 1) Determine Approximate Allocation 2) Determine Prices a) for each winner: i) Remove them ii) Determine approximate allocation iii) They pay imposed cost

18 So will Heuristics save us once again? Mechanisms still truthful? This paper says

19 So will Heuristics save us once again? Mechanisms still truthful? This paper says No.

20 So will Heuristics save us once again? Mechanisms still truthful? This paper says No. (World, why doth ye be so cruel?)

21 Do not fear, Science is here! But! These folks have found a fair work-around! A heuristic VCG-based method that produces (almost) truthful mechanisms with feasible complexity...

22 For Reference: Notation The Mechanism Design Problem (the Bidders) Auctioned Items Private Values Private Utilities Goal: Max Welfare

23 For Reference: Notation The Mechanism (the Auctioneer) Solution Allocation Function Payment Function

24 For Reference: Notation Incentive-Compatibility / Truthfulness **Note: Requiring a dominant strategy that is truthful is stronger than requiring an equilibrium that is truthful