Algorithmic Game Theory
Vickrey Auction Vickrey-Clarke-Groves Mechanisms
Mechanisms with Money Player preferences are quantifiable. Common currency enables utility transfer between players. Preference of player i is given by a valuation function v i : A R from a commonly known set V i R A Mechanism determines an outcome a A and payments, charges each player some money value m i Utility of player i is v i (a) m i, quasi-linear utilities.
Example: Sealed Bid Auction A single item is sold to one customer. Customer 1 2 3 4 5 Value 9 1 20 11 14 Agents report their values as sealed bid. Social Choice: Winner is agent with highest bid. Payments: find payments to ensure incentive compatibility No payments: Players strive to bid unboundedly high values. Payments = Bids: Players try to guess the second highest bid and bid a slightly higher value.
Vickrey Second Price Auction Payment of the winner is the second largest bid. Value 9 1 20 11 14 Payment 0 0 14 0 0 Utility 0 0 6 0 0 Proposition The Vickrey auction is incentive compatible.
Example Value?? 20?? Bid 5 11 x 2 14 Payment 14 Utility 6 Case 1: i wins with true value x = 20, then for all x 14 utility 6, for x < 14 utility 0. Value?? 20?? Bid 5 11 x 2 24 Payment 0 Utility 0 Case 2: i loses with true value x = 20, then for all x < 24 utility 0, for x 24 utility 4.
Vickrey Auction Vickrey-Clarke-Groves Mechanisms
Definitions Direct Revelation Mechanism Denote V = V 1... V n and v V Social choice function f : V A A vector of payment functions p 1,..., p n Each function p i : V R specifies the amount player i pays. Incentive Compatibility (IC) For every player i, profile v V, alternative v i V i, Denote outcomes by a = f (v i, v i ) and b = f (v i, v i ) Mechanism (f, p 1,..., p n) is incentive compatible if the utility. v i (a) p i (v i, v i ) v i (b) p i (v i, v i )
Sealed-Bid Auction Bidder 1 2 3 4 5 Value 9 1 20 11 14 Outcomes A = {1, 2, 3, 4, 5}, where i means i wins Outcome 1 2 3 4 5 v 1 9 0 0 0 0 v 2 0 1 0 0 0 etc. Social Choice: f (v) = arg max i {v i (i)} Payments: p i (v) = 0 if f (v) i, otherwise p i (v) = max j i v j (j).
Mechanism Definition A Vickrey-Clarke-Groves () mechanism is given by f (v) arg max a A i v i(a), and for some functions h 1,..., h n with h i : V i R and all v V we have p i (v) = h i (v i ) j i v j (f (v)) Observations: mechanism picks outcome a that maximizes social welfare j v j(a) h i does not depend on the own bid v i Utility of player i when f (v) = a: v i (a) p i (v) = j v j (a) h i (v i )
is IC Theorem Every mechanism is incentive compatible. Proof: Given types v, for player i a lie v i, and let a = f (v) and b = f (v i, v i ). Utility of i declaring v i is v i (a) + j i v j(a) h i (v i ) Utility of i declaring v i is v i (b) + j i v j(b) h i (v i ) Utility is maximized when outcome maximizes social welfare j v j(x). mechanism maximizes social welfare, j v j(a) j v j(b). By declaring v i player i, picks b which is optimized for his lie, but possibly suboptimal for the real utility. aligns each player incentive with the social incentives.
Desirable Properties of Payments Definition A mechanism is (ex-post) individually rational if players always get non-negative utility, i.e. for all v V we have v i (f (v)) p i (v) 0. A mechanism has no positive transfers if no player is ever paid money, i.e. for all v V and all i we have p i (v) 0. Definition (Clarke Rule) The choice h i (v i ) = max b A j i v j(b) is called Clarke pivot payment. Then the payment of player i becomes p i (v) = max b A v j (b) v j (f (v)) j i j i Payment is the total damage to the other players caused by the presence of i. Each player internalizes externalities.
Clarke Rule Lemma A mechanism with Clarke pivot payments makes no positive transfers. If v i (a) 0 for all v i V i and a A, then it is individually rational. Proof: Let a = f (v) and b = arg max a A j i v j(a ) No positive transfers (by definition) v j (b) v j (a) 0 j i j i Individually rational v i (a) + j i v j (a) j i v j (b) j v j (a) j v j (b) 0
Example: Buying a Path in a Network Reverse auction: Players are edges in a network. Mechanism needs to buy an s-t-path.
Example: Buying a Path in a Network Outcomes are s-t-paths in graph G picks the shortest path P for reported costs c e Payments for e P are h e(c e) + e e c e (P ) Clarke pivot payment: h e(c e) = min P G e e P ce Total payment c(p e) c(p e), where P e is shortest path in G when it would not contain e. An edge e P gets no payment.
Truth-telling is a dominant strategy!
Frugality Incentive compatibility might be EXPENSIVE!