Vickery Auction

Motivation. A first-price auction (=give the highest bidder the item, then charge them their bid) is a bad idea because, assuming the highest bidder was truthful ($v_i=b_i$) then their utility is $v_i-b_i=0$. Thus this is not DISC and not a good auction. It’s also very hard to reason about. Therefore we have…

def. Second Price Auction. (=Vickery Auction) Give to the highest bidder; charge them the second highest price.

• An ascending bid auction (=an e-bay auction) is actually a second price auction if you think about it. The last two bidders $(v_1,b_1)\ge (v_2,b_2)$ will increase their bids continually until they reach $b_2$, and then bidder $1$ bids just slightly above ($b_1=v_2+\delta$) to get the item.
• ! Not to be confused with Ascending Price Auction; they’re ascending two different things.

thm. Second Price Auction is DSIC.

proof. There are only two cases where a bidder may lie.

1. Case one: a bidder lies to reduce the price of the item. But $p_i(\vec{b})$ does not depend on $\vec{v}$. Thus this is impossible
2. Case two: a bidder lies to get the item. But lying to get the item results in negative utility. ◆

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