Vickrey–Clarke–Groves auction

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In auction theory, a Vickrey–Clarke–Groves (VCG) auction is a type of sealed-bid auction of multiple items. Bidders submit bids that report their valuations for the items, without knowing the bids of the other people in the auction. The auction system assigns the items in a socially optimal manner: it charges each individual the harm they cause to other bidders.[1] It also gives bidders an incentive to bid their true valuations, by ensuring that the optimal strategy for each bidder is to bid their true valuations of the items. It is a generalization of a Vickrey auction for multiple items.

The auction is named after William Vickrey,[2] Edward H. Clarke,[3] and Theodore Groves[4] for their papers that successively generalized the idea.

Formal description[edit]

Notation

For any set of auctioned items M = \{t_1,\ldots,t_m\} and any set of bidders N = \{b_1,\ldots,b_n\}, let V^M_N be the social value[clarification needed] of the VCG auction for a given bid-combination. For a bidder b_i and item t_j, let the bidder's bid for the item be v_{i}(t_{j}).

Assignment

A bidder b_i whose bid for an item t_j, namely v_{i}(t_{j}), is an "overbid" wins the item, but pays V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}, which is the social cost of his winning that is incurred by the rest of the agents.

Explanation

Indeed, the set of bidders other than b_i is N \setminus \{b_i\}. When item t_j is available, they could attain welfare V^{M}_{N \setminus \{b_i\}}. The winning of the item by b_i reduces the set of available items to M \setminus \{t_j\}, however, so that the attainable welfare is now V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}. The difference between the two levels of welfare is therefore the loss in attainable welfare suffered by the rest bidders, as predicted, given the winner b_i got the item t_j. This quantity depends on the offers of the rest agents and is unknown to agent b_i.

Winner's utility

The winning bidder whose bid is his true value A for the item t_j, v_{i}(t_{j})=A, derives maximum utility A - \left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right).

Examples[edit]

Example #1[edit]

See the example with apples in the Generalization section of Vickrey Auction.

Example #2[edit]

Assume that there are two bidders, b_1 and b_2, two items, t_1 and t_2, and each bidder is allowed to obtain one item. We let v_{i,j} be bidder b_i's valuation for item t_j. Assume v_{1,1} = 10, v_{1,2} = 5, v_{2,1} = 5, and v_{2,2} = 3. We see that both b_1 and b_2 would prefer to receive item t_1; however, the socially optimal assignment gives item t_1 to bidder b_1 (so his achieved value is 10) and item t_2 to bidder b_2 (so his achieved value is 3). Hence, the total achieved value is 13, which is optimal.

If person b_2 were not in the auction, person b_1 would still be assigned to t_1, and hence person b_1 can gain nothing. The current outcome is 10 hence b_2 is charged 10-10=0.

If person b_1 were not in the auction, t_1 would be assigned to b_2, and would have valuation 5. The current outcome is 3 hence b_1 is charged 5-3=2.

Example #3[edit]

Here we will look at a multiple item auction. Consider the situation when there are n bidders, n houses, and values \tilde v_{ij}, representing the value player i has for house j. Possible outcomes in this auction are characterized by bipartite matchings, pairing houses with people. If we know the values, then maximizing social welfare reduces to computing a maximum weight bipartite matching.

If we do not know the values, then we instead solicit bids \tilde b_{ij}, asking each player i how much he would wish to bid for house j. Define b_i(a) = \tilde b_{ik} if bidder i receives house k in the matching a. Now compute a^*, a maximum weight bipartite matching with respect to the bids, and compute

p_i = \left[ \max_{a \in A} \sum_{j \neq i} b_j(a) \right] - \sum_{j \neq i} b_j(a^*) .

The first term is another max weight bipartite matching, and the second term can be computed easily from a^*.

Optimality of Truthful Bidding[edit]

The following is a proof that bidding one's true valuations for the auctioned items is optimal.[5]

For each bidder b_i, let v_i be his true valuation of an item t_i, and suppose (without loss of generality) that b_i wins t_i upon submitting his true valuations. Then the net utility U_i attained by b_i is given by U_i = v_i-\left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right)=\sum_{i}v_i-\sum_{j\neq i}v_j+V^{M\setminus \{t_i\}}_{N\setminus \{b_i\}}-V^M_{N\setminus\{b_i\}}=\sum_i v_i-V^{M\setminus \{t_i\}}_{N\setminus \{b_i\}}+V^{M\setminus \{t_i\}}_{N\setminus \{b_i\}}-V^M_{N\setminus\{b_i\}}=\sum_i v_i-V^M_{N\setminus\{b_i\}}. As V^M_{N\setminus\{b_i\}} is independent of v_i, the maximization of net utility is pursued by the mechanism along with the maximization of corporate gross utility \sum_i v_i for the declared bid v_i.

Let us form the difference U_i - U_j = \left[v_i + V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right] - \left[v_j + V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right] between net utility U_i of b_i under truthful bidding v_i gotten item t_i, and net utility U_j of bidder b_i under non-truthful bidding v'_i for item t_i gotten item t_j on utility v_j.

\left[v_j + V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right] is the maximum corporate gross utility obtained with the non-truthful bidding. But the allocation assigning t_j to b_i is different from the allocation of t_i to b_i which gets maximum true gross corporate utility. Hence \left[v_i + V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right] - \left[v_j + V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right]\geq 0 and U_i\geq U_j q.e.d.

More general setting[edit]

We can consider a more general setting[6] of the VCG mechanism. Consider a set A of possible outcomes and n bidders which have different valuations for each outcome. This can be expressed as, function

v_i : A \longrightarrow R_+

for each bidder i which expresses the value it has for each alternative. In this auction, each bidder will submit his valuation and the VCG mechanism will choose the alternative a \in A that maximizes  \sum_i v_i(a) and charge prices p_i given by:

p_i = h_i(v_{-i}) - \sum_{j \neq i} v_j(a)

where v_{-i} = (v_1, \dots, v_{i-1}, v_{i+1}, \dots, v_n), that is, h_i is a function that only depends on the valuation of the other players. This alone gives a truthful mechanism, that is, a mechanism where bidding the true valuation is a dominant strategy.

We could take, for example, h_i(v_{-i}) = 0, but we would have all prices negative, which might not be desirable. We would rather prefer that players give money to the mechanism than the other way around. The function:

h_i(v_{-i}) = \max_{b \in A} \sum_{j \neq i} v_j(b)

is called Clark pivot rule.

On the other hand, if we do not know the values v_i, we can solicit bids b_i : A \longrightarrow R_+. The mechanism then chooses a^* maximizing the revenue \sum_i b_i(a^*), treating the bids like the values. We then set

p_i = \left[ \max_{a \in A} \sum_{j \neq i} b_j(a) \right] - \sum_{j \neq i} b_j(a^*).

Intuitively, the mechanism charges player i his externality, or the decrease in optimal social welfare when he is included in the auction.

The Clark pivot rule has some very good properties as:

  • it is individually rational, i.e., v_i (a) - p_i \geq 0. It means that all the players are getting positive utility by participating in the auction. No one is forced to bid.
  • it has no positive transfers, i.e., p_i \geq 0. The mechanism does not need to pay anything to the bidders.

See also[edit]

References[edit]

  1. ^ von Ahn, Luis (2008-10-07). "Sponsored Search" (PDF). 15–396: Science of the Web Course Notes. Carnegie Mellon University. Retrieved 2008-11-05. 
  2. ^ Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders". The Journal of Finance 16 (1): 8–37. doi:10.1111/j.1540-6261.1961.tb02789.x. 
  3. ^ Clarke, E. (1971). "Multipart Pricing of Public Goods". Public Choice 11 (1): 17–33. doi:10.1007/bf01726210. 
  4. ^ Groves, T. (1973). "Incentives in Teams". Econometrica 41 (4): 617–631. doi:10.2307/1914085. 
  5. ^ http://www.cs.cmu.edu/~arielpro/15896/docs/notes14.pdf
  6. ^ Nisan, Roughgarden, Tardos and Vazirani, Algorithmic Game Theory, Cambridge, 2007