VCG Mechanism Examples

Game Theory · Axiom Academy

Multi-Item Allocation with Truthful Payments Three bidders (Alice, Bob, Carol) compete for two items (X and Y). Each bidder has private valuations for each item. We will use the Vickrey-Clarke-Groves (VCG) mechanism to: 1. Find the efficient allocation We need to find the allocation that maximizes total social welfare. Since we have 2 items and 3 bidders, we evaluate all possible allocations: Efficient allocation: Alice gets Item X, Bob gets Item Y (total welfare = 180) The VCG payment for each bidder equals the harm they cause to others by participating. Formally: The VCG payment compensates others for the opportunity cost of your participation. This makes truthful bidding a dominant strategy! Without Alice: We re-optimize among Bob and Carol only. • Bob gets X, Carol gets Y: 70 + 50 = 120 • Bob gets Y, Carol gets X: 80 + 60 = 140 ← Optimal With Alice: Alice gets X, Bob gets Y. Others' welfare = Bob's value = 80 Alice pays 60 because her presence prevented Bob and Carol from achieving 140 in total welfare (they only get 80 with Alice). Without Bob: We re-optimize among Alice and Carol only. • Alice gets X, Carol gets Y: 100 + 50 = 150 ← Optimal • Alice gets Y, Carol gets X: 40 + 60 = 100 With Bob: Alice gets X, Bob gets Y. Others' welfare = Alice's value = 100 Bob pays 50 because his winning Item Y prevented Carol from getting it (Carol would have contributed 50 to total welfare). Without Carol: We re-optimize among Alice and Bob only.

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