Single-item auctions

Scenario

a seller wants to sell an item

n potential buyers (interested in the item, ready to act strategically to get it)

Question: how do we design how the item should be sold?

Intermediate problem: what do the buyers want?

buyer i has no-negative val $v_i$ of the item
valuate is private, unknown to other potential buyers and the seller
utility
- 0 if buyer doesn’t get the item
- $v_i-p$ if buyer gets the item and pays $p$

Sealed bid Auction

every buyer i submits bid $b_i$ to the seller
seller decides who should get the item
the seller decides the price of items

First-price auction

highest bidder gets/wins the item and pays equal to bid

for bidder: difficult to decide how to behave

for seller: difficult to predict what will happen

Second-price auction

highest bidder wins the item and pays the second highest bid

dominant strategy: if it maximizes utility for agent, no matter what other buyers decide to do

dominant strategy is the subset of best response

Theorem: in a second-price auction, every buyer i has as dominant strategy to submit private valuation as bid for the item $b_i=v_i$

Proof?

Let B be the highest bid among other buyers
2 cases
$v_i<B$ utility: 0. no incentive to bid $v_i>B$ since it will lead utility to negative.
$v_i\ge B$ utility: $v_i-B\ge0$

for bidder: easy to decide how to behave

for seller: easy to predict what will happen

Theorem: in second-price auction, truthful bidder is guaranteed non-negative utility

DSIC

DSIC aka dominant strategy incentive compatible: if truthful bidding is a dominant strategy for every potential buyer i and gives non-negative utility.

efficiency issues