1
want to maximize surplus (sum of benefit for both seller and buyer)
first-price auction: difficult to analyze, will do so in ch 2/6
ascending-price auction strategy:
- increase price from 0 until all but one person drops out, then give that last person the current price
- assume that agents act to maximize their own benefit, then clearly there is no benefit to dropping out when price < WTP
- therefore agent will drop out at price , so agent will gain and seller will gain , thus achieving the maximum possible benefit of
second-highest bid strategy:
- have everyone bid, offer second-highest price to highest bidder
- importantly, this encourages everyone to bid truthfully
- prove sketch: fix x to be the max of everyone elseโs bids. if x < v_i, any bid >= x is optimal. and if v_i < x, any bid โ x is optimal. therefore, v_i is always an optimal bid (and the only bid that is optimal in all scenarios)
maximizing consumer surplus rather than surplus is difficult, as if is constant, lottery is best: but if and all other , second-highest bid is better. In general, no single mechanism works, whereas when maximizing surplus, second-highest bid is clearly optimal.
2
let and denote whether an agent receives a good and their payment, respectively
in order to be in equilibrium:
must be monotone as bid increases
intuitively, this makes sense: why would you bid higher only to get less chance of getting the item?
RSOP
b/c price given to each person doesnโt depend on what they bid, itโs clearly optimal for them to always tell the truth
1/4 lower bound: consider 2 bidders with . With probability they are in different sets, and therefore seller gains revenue of with probability , so expected revenue is , which is of maximum revenue .
4.68 bound paper: = how many of the first bidders are in the sample = fraction of first bidders in market to sample =
- 1/15 proof considers scenarios where , and shows that this occurs with high probability
โ probability that the maximum ratio of does not exceed
and are positively correlated for any ,