# Vickrey–Clarke–Groves auction

"VCG" redirects here. For the heartbeat recording method see Vectorcardiography. In auction theory, a Vickrey–Clarke–Groves (VCG) auction of multiple goods is a sealed-bid auction wherein bidders report their valuations for the items. The auction system assigns the items in a socially optimal manner. This system charges each individual the harm they cause to other bidders,[1] and ensures that the optimal strategy for a bidder is to bid the true valuations of the objects. 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

For any set of auctioned items $M = \{t_1,\ldots,t_m\}$ and a set of bidders $N = \{b_1,\ldots,b_n\}$, let $V^M_N$ be the social value of the VCG auction for a given bid-combination. A bidder $b_i$ who wins an item $t_j$ 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.

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 given the winner $b_i$ got the item $t_j$. In other words it is the social cost incurred by the rest as predicted. This quantity depends on the offers of the rest agents and is unknown to agent $b_i$. According Vickrey-Clarke-Groves recipe which is an improvement of Vickrey' s auction recipe it is the payment for $t_j$ paid by the winning bidder $b_i$.

The highest bidder is the winning, but he pays for the auctioned object the social cost incurred by the rest of the agents.

The winning bidder who has value $A$ for the item $t_j$ derives therefore utility $A - \left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right).$

## Examples

### Example #1

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

### Example #2

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

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

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.

Note that the size bid of $b_i$ has no effect on his utility as long as he wins the item (see the utility function above). Hence, we assume that $b_i$ does not bid truthfully, and receives item $t_j$ because of his non-truthful bidding. In the truthful bidding case, $b_i$ has total utility $U_i = v_i-\left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right)$. In the untruthful bidding case, $b_i$ has total utility $U_j = v_j-\left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right)$. Hence, we must prove that $U_i - U_j \geq 0$, which shows that the utility received from truthful bidding is always at least that received from untruthful bidding.

$U_i - U_j = v_i - \left(V^{M}_{N \setminus \{b_i\}} - V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right) - v_j + \left(V^{M}_{N \setminus \{b_i\}}-V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right) = \left[v_i + V^{M \setminus \{t_i\}}_{N \setminus \{b_i\}}\right] - \left[v_j + V^{M \setminus \{t_j\}}_{N \setminus \{b_i\}}\right]$

However, the first term there is the maximum total social value achieved when $b_1$ received $t_1$, and the second term there is the maximum total social value achieved when $b_1$ received $t_j$. However, we assumed that the VCG auction gave $b_1$ item $t_1$; hence, the first term must be greater, and $U_i - U_j \geq 0$.

The following is an alternative and simpler proof of Vickrey-Clark-Groves auction theorem.[6]

If our bidder offers anything, he may sometimes win, sometimes don' t get the item, that is zero balance, and sometimes lose. It depends on the incurred social loss of the rest agents. We shall prove that if he offers his true valuation he has the greatest probability to gain something and at the same time zero probability to lose. Let $v_i$ be the actual bid of our bidder for item $t_i$ and $A$ his honest value for the same item and let $u_i$ be the social cost suffered by the rest agents. If $v_i < A$ he can hope to get the object at a gain $A-u_i$ if $v_i > u_i$. The relative probability of this to happen is $v_i \div A$. This is maximized to $1$ for $v_i = A$. If $v_i > A$ he runs the risk to lose $u_i - A$ if $v_i > u_i > A$ and the relative probability of this to happen is $(v_i-A)\div v_i$. This is zeroed again for $v_i = A$.

The theorem thus proven has as conclusion that the utility value obtained by honest bidding is maximized. In other words the private interest of the highest bidder is optimally served. But as we see from the foregoing mathematical transformations also the total social value achieved is maximum. This is the magic of Vickrey-Clarke-Groves recipe: It coordinates the private interest with the public!

## More general setting

We can consider a more general setting[7] 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 Clarke 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.