Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley

Mechanism Design Introduction to Auctions First Price Second Price Revenue Equivalence Vickrey-Clarke-Grove Bidding for QoS

Introduction to Auctions Example: Alice and Bob bid for a House. Alice values the house at v1, Bob at v2. Alice bids b1, Bob bids b2. We examine two auctions: Sealed Bid, First Price Sealed Bid, Second Price

Introduction to Auctions Sealed Bid – First Price: Item goes to highest bidder who pays the highest bid. Alice’s net reward is (v 1 – b 1 )1{b 1 > b 2 }. There is no dominant strategy: Alice’s best bid is min{v 1, b 2 + , which depends on b 2. The unique Nash Equilibrium is b 1 = min{v 1, v 2 +  } b 2 = min{v 2, v 1 +  }

Introduction to Auctions Sealed Bid – First Price: Incomplete information, symmetric.

Envelope Theorem

Introduction to Auctions Sealed Bid – First Price: Incomplete information, symmetric.

Introduction to Auctions Sealed Bid – Second Price: Item goes to highest bidder who pays the second highest bid. [Equivalent to ascending public auction.] Fact (Vickrey): Dominant strategy: Alice bids truthfully: b 1 = v 1. [Incentive compatible] Proof: Alice’s reward is A = (v 1 – z)1{b 1 > z} where z is the second highest bid. Note that (v 1 – z)1{v 1 > z} ≥ (v 1 – z)1{b 1 > z}, as you see by looking at v 1 > z and v 1 < z. Sealed Bid – Second Price: Item goes to highest bidder who pays the second highest bid. [Equivalent to ascending public auction.] Fact (Vickrey): Dominant strategy: Alice bids truthfully: b 1 = v 1. [Incentive compatible] Proof: Alice’s reward is A = (v 1 – z)1{b 1 > z} where z is the second highest bid. Note that (v 1 – z)1{v 1 > z} ≥ (v 1 – z)1{b 1 > z}, as you see by looking at v 1 > z and v 1 < z.

Introduction to Auctions Incomplete information, symmetric – Revenue Equivalence.

Introduction to Auctions Incomplete information, symmetric – Revenue Equivalence.

Vickrey-Clarke-Grove In second price, the winning agent pays the reduction in declared welfare of the other agents. Indeed, if the bids are b 1 > b 2 > …>b n, the first agent gets the item instead of the second. Consequently, the declared welfare of agents {2, 3, …, n} is now zero whereas it would be equal to b 2 if agent 1 were not there. The reduction in welfare of agents {2, 3, …, n} caused by agent 1 is b 2, which is the payment of agent 1 when she wins the bid. This payment internalizes the externality caused by agent 1 and forces her to bid truthfully. The VCG mechanisms generalize this idea. In second price, the winning agent pays the reduction in declared welfare of the other agents. Indeed, if the bids are b 1 > b 2 > …>b n, the first agent gets the item instead of the second. Consequently, the declared welfare of agents {2, 3, …, n} is now zero whereas it would be equal to b 2 if agent 1 were not there. The reduction in welfare of agents {2, 3, …, n} caused by agent 1 is b 2, which is the payment of agent 1 when she wins the bid. This payment internalizes the externality caused by agent 1 and forces her to bid truthfully. The VCG mechanisms generalize this idea.

Vickrey-Clarke-Grove

Bidding for QoS Assume there are K different service classes Class k can accept N(k) connections and offers them a loss rate 1 – p(k). Assume N(K) is infinite. User i bids x i for his connection, i = 1, …, m. Here, x i is the declared value per unit rate of the connection for user i. Mechanism: Assume b(1) > b(2) > …. Place the first N(1) users in class 1, the next N(2) in class 2, and so on. Price: C(i) for user i where C(i) is the reduction in declared value of all the customers i + 1, …, m caused by the presence of i.

Bidding for QoS

Shu and Varaiya, to appear in JSAC 06.