Vickrey Auction VCG Characterization. Mechanism Design. Algorithmic Game Theory. Alexander Skopalik Algorithmic Game Theory 2013 Mechanism Design

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1 Algorithmic Game Theory

2 Vickrey Auction Vickrey-Clarke-Groves Mechanisms Characterization of IC Mechanisms

3 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.

4 Example: Sealed Bid Auction A single item is sold to one customer. Customer Value 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.

5 Vickrey Second Price Auction Payment of the winner is the second largest bid. Value Payment Utility Proposition The Vickrey auction is incentive compatible.

6 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.

7 Vickrey Auction Vickrey-Clarke-Groves Mechanisms Characterization of IC Mechanisms

8 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 )

9 Sealed-Bid Auction Bidder Value Outcomes A = {1, 2, 3, 4, 5}, where i means i wins Outcome v v 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).

10 VCG Mechanism Definition A Vickrey-Clarke-Groves (VCG) 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: VCG 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 )

11 VCG is IC Theorem Every VCG 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). VCG mechanism maximizes social welfare, j v j(a) j v j(b). By declaring v i player i, VCG picks b which is optimized for his lie, but possibly suboptimal for the real utility. VCG aligns each player incentive with the social incentives.

12 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.

13 Clarke Rule Lemma A VCG 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

14 Example: Buying a Path in a Network Reverse auction: Players are edges in a network. Mechanism needs to buy an s-t-path.

15 Example: Buying a Path in a Network Outcomes are s-t-paths in graph G VCG 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.

16 Truth-telling is a dominant strategy!

17 Frugality Incentive compatibility might be EXPENSIVE!

18 Vickrey Auction Vickrey-Clarke-Groves Mechanisms Characterization of IC Mechanisms

19 What IC Mechanisms are there? VCG are IC mechanisms that do social welfare maximization. What other social choice functions can be implemented? Are there any other types of mechanisms besides VCG that are IC?

20 Direct Characterization Proposition A mechanism is incentive compatible if and only if it satisfies the following conditions for every i, v i : 1. The payment p i does not depend on v i, but only on the outcome, i.e., for every v i there exist prices p a R such that for all v i with f (v i, v i ) = a we have p i (v i, v i ) = p a. 2. The mechanism optimizes for each player, i.e., for every v i it holds that f (v i, v i ) arg max a A {v i (a) p a}, where A is the set of alternatives in the range of f (, v i ). Proof: Conditions hold IC: obvious.

21 Proof 1. The payment p i = p a does not depend on v i, but only on the outcome a = f (v i, v i ). 2. The mechanism optimizes for each player. IC Conditions hold: Condition 1: v i v i yield same outcome for fixed v i. Payment p i (v i, v i ) > p i (v i, v i ) then a player with v i is motivated to lie v Condition 2: If not, then there is a better outcome a arg max a(v i (a) p a) and some v i that gives a = f (v i, v i ), so player i is motivated to lie v i. i.

22 Affine Maximizer Definition A social choice function f is an affine maximizer if for some range A A, some player weights w 1,..., w n R and some outcome weights c a R for each a A, we have { } f (v 1,..., v n) arg max a A c a + i w i v i (a). Proposition Suppose f is an affine maximizer. Define for every i the payments p i (v) = h i (v i ) 1 w j v j (a) + c a, w i j i where h i is an arbitrary function independent of v i. Then (f, p 1,..., p n) is incentive compatible.

23 (Only) Affine Maximizers can be implemented Proof: Assume wlog h i = 0. Utility of i if a is chosen v i (a) + 1 w j v j (a) + c a. w i j i Multiply by w i > 0, expression is maximized when c a + j w jv j (a) is maximized. f affine maximizer, i reports truthfully. Theorem (Roberts 1979) If A 3, f is onto A, V i = R A for every i, and (f, p 1,..., p n) is incentive compatible, then f is an affine maximizer.

24 Recommended Literature Chapter 9 and 11 in the AGT book. D. Lehmann, L.I. O Callaghan, Y. Shoham. Truth revelation in approximately efficient combinatorial auctions. Journal of the ACM, 49(5): , 2002.