Web Science/Part2: Emerging Web Properties/Web Search Ecosystem/Finding the true value of an advertisement/script

How should a Web Search engine or any other publisher come up with a CPC? Of course one could look for demand and try to play around with it. But especially in a Web Search context many keywords that a person searches for will yield very different CPCs. One way of figuring out the CPC is by creating an auction system. The auction system within Google is quite complex but the main Idea can be brought down to a Second Price Auction or Vickery auction.

A Vickrey auction is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. The auction was first described academically by Columbia University professor William Vickrey in 1961.

The second price auction has a very nice property: From a game theoretic perspective the dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item. This property is so easy to prove that we will do it right now:

Let $$v_i$$ be bidder i's value for the item. Let $$b_i$$ be bidder i's bid for the item.

The payoff for bidder i is $$ \begin{cases} v_i-\max_{j\neq i} b_j & \text{if } b_i > \max_{j\neq i} b_j \\ 0 & \text{otherwise} \end{cases} $$

The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids $$ b_i > v_i $$.

If $$\max_{j\neq i} b_j < v_i $$ then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.

If $$\max_{j\neq i} b_j > b_i $$ then the bidder would lose the item either way so the strategies have equal payoffs in this case.

If $$v_i < \max_{j\neq i} b_j < b_i $$ then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding.

The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids $$ b_i < v_i $$.

If $$\max_{j\neq i} b_j > v_i $$ then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case.

If $$\max_{j\neq i} b_j < b_i $$ then the bidder would win the item either way so the strategies have equal payoffs in this case.

If $$b_i < \max_{j\neq i} b_j < v_i $$ then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding.

Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy.