Let’s think about a simple case first. For example, there are four advertisers 1, 2, 3, 4
- Advertiser 1,2,3,4 has a profit of v1, v2, v3, v4 per click
- Advertiser 1,2,3,4 bid b1,b2,b3,b4 amount
- Highest bidder gets position 1, second highest position 2, and so on
- Payoff to bidder at position i is then (vi-bi+1)xi, where xi is the CTR. (CTR is the number of clicks that your ad receives divided by the number of times your ad is shown: clicks ÷ impressions = CTR. For example, if you had 5 clicks and 100 impressions, then your CTR would be 5%.)
After all the advertisers know their positions and bids, they may want to change their positions according to the known information. If the person who gets position 2 want to move up to position 1, he will need to pay a price of b1+e(e is an extremely small value). But if he wants to move down to position 3. He just needs to pay b4+e. Because people at position 4 bid b4.
In conclusion, if a bidder wants to move up to position i, he will need to pay bi+e. If a bidder wants to maintain or move down to position i, he will bid bi+1+e. ————————(*). This rule (*) can be expanded to n advertisers.
But how will those advertisers really pay? All of them want to maximize their payoff. Is there any Nash equilibrium points for them. For bidder at position i, it will be a dominant strategy for him to stay at position i if he makes at least as much profit by being in the position i than in any other positions. That is:
(Vi-bi+1)xi >= (Vi-bj)xj for j<i ——–(1)
(Vi-bi+1)xi >= (Vi-bi+1)xj for j>i ———(2)
Therefore, a set of bids(bi) that satisfies both (1) and (2) of them will be a Nash equilibrium.
I think this Nash equilibrium in auction is very interesting because it expands the Nash equilibrium to n players(bidders). I think this topic here is closely related to the game theory topic but also expands our thinking.
Links:
https://www.inmobi.com/blog/2018/10/24/what-is-a-second-price-auction-and-how-does-it-work-video