Week 11 - Chap 15A

What is an auction + what does auction states

• Auction: Any transaction where the final price of the object(s) for sale is arrived at by way of competitive bidding

Auction states: 

1. Allocation rule (who gets the prize) 

2. Payment rule (who pays)

What is a procurement auction

• An auction in which multiple bidders compete to supply an item

• Bids are prices that bidders are willing to receive to supply the good

• The lowest bidder wins and is paid her bid

What is multi-unit and combinatorial auction

1. Multi-unit auction 

• Multiple identical objects are sold

2. Combinatorial auction 

• Multiple dissimilar objects in which bidders are able to bid on and win combinations of objects

What is a single-object and all-pay auction

1. Single-object auction 

• Single indivisible object is sold

2. All-pay Auction

• Every bidder pays their submitted bid, but only the highest bidder receives the object

Describe mixed strategy equilibria for all-pay auction

• For any bid x considered, expected payoff = 0 (same as not bidding) 

• Expected payoff = (chance you win) × 1 − x = 0 -> (chance you win) = x

• Need all (n−1) opponents below x

• [Prob(bid < x)]^(n−1) = x -> Prob(bid < x) = x^(1/(n−1))

• Eg. n = 10 → Prob(bid < x) = x^(1/9) → most bids cluster near 0 because winning against 9 opponents is unlikely 

What is war of attrition and reserve price

1. War of attrition 

• Contestants choose when to retreat, victor is whoever stays longest, staying is costly

2. Reserve price 

• Minimum price set by the seller of an item up for auction, if no bids exceed the reserve, the item is not sold

Compare war of attrition VS all-pay AND how it is applied in example: Firm A bids $200, Firm B bids $150

• The winner in a war of attrition never needs to pay their full bid (only need to outlast the last person who quits) -> less costly than an all-pay auction

• Example: Firm A bids $200, Firm B bids $150:

  • All-pay: both envelopes handed over simultaneously → official keeps everything → $200 + $150 = $350 total paid

  • War of attrition: firms pay in installments over time → Firm B drops out at $150 → Firm A only needs to match that to win → $150 + $150 = $300 total paid

What is jump bidding and shill bidder

1. Jump bidding 

• Submitting a bid that is significantly higher than the previous bid and beyond whatever minimum bid increment exists

2. Shill bidder 

• A fake bidder created by sellers at an auction to place fictitious bids for an object they are selling

What is sniping and shading

1. Sniping 

• Waiting until the last moment to make a bid

2. Shading 

• Strategy in which bidders bid slightly below their true valuation of an object

Describe ascending-price auction and its optimal strategy

• Price rises continuously, each bidder decides when to drop out, last remaining bidder wins and pays the final price

• Optimal strategy: enter only if r < V, then stay in until price reaches your value V, drop out at p = V (weakly dominant strategy)

Explain why ascending-price = sealed-bid 2nd auction

Highest bidder pays the 2nd-highest bidder's value (price at which the last competitor dropped out)

Describe descending price auction and its optimal strategy

• Auctioneer starts very high and lowers the price until someone pays that price 

• Optimal strategy: wait until price falls below your value V (bidding at V gives 0 profit)

Eg. V = $100, others' bids uniform on [50, 150] → optimal jump-in price P* = 75, find expected payoff

• Differentiate -> P* = 75 

• Format equivalent to the sealed-bid first-price auction

Describe 1st price sealed-bid auction

• Each bidder privately submits one bid

• The highest bidder wins and pays their own bid

• Because you pay what you bid, rational bidders shade their bids below their true value

Describe 2nd price sealed-bid auction

• Each bidder privately submits one bid

• The highest bidder wins but pays only the second-highest bid

• Bidders have a weakly dominant strategy to submit b = V

1st price auction is equivalent to… AND 2nd price auction is equivalent to…

1. First-price sealed bid ↔ Dutch auction (same dominant strategy) 

2. Second-price sealed bid ↔ English auction

• Identical outcomes only when each bidder's value is their own private information

• Breaks down under common values

State payoff equations for:

1. 1st, 2nd and all pay auctions

2. P(winning)

1. Payoffs 

• 1st price payoff = P(winning)(V-b)

• 2nd price = P(winning)(V − E[b₂ | b₂ ≤ b])

• All pay = P(winning)(V) − b

2. P(winning) = b^(m-1) for uniform distribution

State equations for: 1st, 2nd equilibrium bid

1. First price equilibrium bid:

• b(v)=(m-1)/m⋅v

• m = number of bidders in the auction

2. Second price equilibrium bid: b(v) = v

• Take FOC of Eπ to find optimal bid