1. Protocols for Multiagent Resource Allocation: Auctions
MS-E2142 Optimointiopin seminaari: Peliteoria ja tekoäly
Ari Orre
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2. Single-good auctions
Auction is a problem of allocating (discrete) resources among selfish agents in a multiagent system
Auctions are an important application of mechanism design
Various different flavors of auctions, including single-good, multiunit, and combinatorial auctions
Auctions are widely used in real life, in consumer, corporate, as well as government settings
Millions of people use auctions daily on Internet consumer Web sites to trade goods.
More complex types of auctions have been used by governments around the world to sell important public resources such as access to electromagnetic spectrum. .

3. English auctions The best known family of auctions
The auctioneer sets a starting price for the good
Agents then have the option to announce successive bids, each of which must be higher than the previous bid
May have some minimum increment set by the auctioneer
The rules for the auction closes can vary:
In some instances the auction ends at a fixed time, or it ends after a fixed period during which no new bids are made
In others at the latest of the two, and in other instances at the earliest of the two.
The final bidder with the highest bid must purchase the good for the amount of his final bid
Dominant strategy for each buyer whose price is above the displayed price to always bid the minimal allowed above the displayed price.
Most competitive auction. Much information about demand is revealed.

4. Japanese auctions Ascending auction like the english auction
The auctioneer sets a starting price for the good
Each agent must choose whether or not to be in with that price
The auctioneer then calls out successively increasing prices in a regular fashion, and after each call each agent must announce whether he is still in
When an agent drops out it is irrevocable: cannot reenter the auction
The auction ends when there is exactly one agent left in
The agent must purchase the good for the current price
Jump-bidding is not possible in a Japanese auction -> easier to analyze
A Japanese auction is an extensive form game with chance imperfect information
Japanese auctions are dominant-strategy truthful when agents have independent private values.
Buyers can see the exit prices of the other buyers

5. Dutch auctions Descending auction
Auctioneer begins by announcing a high price
Proceeds to announce successively lower prices in a regular fashion
In practice, the descending prices are indicated by a clock that all agents can see
The auction ends when the first agent signals the auctioneer and stops the clock
The signaling agent must then purchase the good for the displayed price.
Most often used in settings where goods must be sold quickly
Strategically equivalent to a first-price auction
Little information about demand is revealed

6. Sealed-bid auctions Simultaneous bids
Each agent submits to the auctioneer a sealed bid for the good that is not accessible to any of the other agents.
The agent with the highest bid must purchase the good
The price at which he does so depends on the type of sealed-bid auction:
In a first-price auction the winning agent pays an amount equal to his own bid.
In second-price auction he pays an amount equal to the next highest bid

7. Sealed-bid auctions
The second-price auction is also called the Vickrey auction
In general, in a kth-price auction the winning agent purchases the good for a price equal to the kth highest bid
In the first price auction bidders strategy is shade bidding: a bid that is below what they believe a good is worth for positive profit and to avoid winner’s curse.
In the second price auction bidders strategy is (weakly) dominant strategy incentive compatibility: the true value. Less fear for winner’s curse.
Little information about demand is revealed

8. Proxy bidding in Ebay
a bidder tells the system the maximum amount he is willing to pay
The user can then leave the site, and the system bids as the bidder’s proxy: every time the bidder is outbid, the system will respond with a bid one increment higher until the bidder’s maximum is reached
It is equivalent to second-price auction if all bidders use the proxy service and update it only once

9. Optimal strategies English Sealed-fp Sealed-sp Dutch Japanese With IPV
remain active until price exceeds own valuation
N bidders:
For agent with own valuation v and lowest possible valuation L
b = v-(v-L)/n.
Less than valuation
Bid agent’s own valuation
Same as sealed-fp
Same as english

10. Auctions as structured negotiations
There are lots of different types of auctions
Two examples:
1. elimination auction that is different from earlier
In the first round the lowest bidder drops out
his bid is announced and becomes the minimum bid in the next round for the remaining bidders.
This process continues until only one bidder remains
This last bidder wins and pays the minimum bid in the final round.

11. Auctions as structured negotiations
2. Procurement reverse auction
an initial sealed-bid auction is conducted among the interested suppliers, and then a reverse English auction is conducted among the three cheapest suppliers to determine the ultimate supplier.
This two-phase auction is common in industry

12. What is an auction?
A structured framework for negotiation with certain rules which can be broken down into three categories
Bidding rules: How are offers made?
by whom, and when, what can their content be
Clearing rules: When do trades occur, or what are those trades as a function of the bidding
who gets which goods, and what money changes hands
Information rules: Who knows what and when about the state of negotiation
Choosing an auction that has desired properties is a mechanism design problem

13. Auctions as Bayesian mechanisms
Auction as a quasilinear mechanism:
set of agents N
set of outcomes O = X × Rn all possible ways to allocate the good
set of actions Ai available to each agent i ∈ N
will vary in different auction types.
choice function X that selects one of the outcomes given the agents’ actions
all possible ways of charging the agents
payment function ℘ that determines what each agent must pay given all agents’ actions
As a mechanism design problem: the choice and payment functions x and ℘ depend on the objective of the auction like achieving an efficient allocation or maximizing revenue.

14. First-price and Dutch auctions
Dutch auction and the first-price auction are actually the same auction
They are strategically equivalent
In both auctions each agent must select an amount without knowing about the other agents’ selections
The agent with the highest amount wins the auction and must purchase the good for that amount
Strategic equivalence is a very strong property: it says the auctions are exactly the same no matter what risk attitudes the agents have and no matter what valuation model describes their utility functions.
They have a trade-off between time complexity and communication complexity.
First-price auctions require each bidder to send a message to the auctioneer, which could be unwieldy with a large number of bidders
First-price auctions can be held asyncronously
Dutch auctions require only a single bit of information to be communicated to the auctioneer, but requires the auctioneer to broadcast prices
Dutch auctions are fast and require minimal communication
Bidders don't have a dominant strategy
Trade of between probability of winning and amount paid upon winning

15. Risk Bidders are usually assumed risk neutral
A risk averse i, will value sure payment more highly than risky payment
-> will bid more aggressively in first-price auction, causing it to yield the auctioneer a higher revenue than second-price auction.
Change in first-price auction behaviour when risk attitude changes
Risk attitude change does not affect second-price auction behaviour
Risk seeking i would bid less in first-price auction -> less profit to auctioneer
Second-price auction better for auctioneer
The variations in bidders’ payments are greater in second-price auctions than they are in first-price auctions, because the former depends on the two highest draws from the valuation distribution, while the latter depends on only the highest draw.
A similar dynamic holds if the bidders are all risk neutral, but the seller is either risk averse or risk seeking.
Relationships between revenues

16. Revenue equivalence theorem
How to verify an equilibrium strategy
Theorem: In a first-price auction with two risk-neutral bidders whose valuations are drawn independently and uniformly at random from the interval [0, 1], ( ½ v1, ½ v2) is a Bayes–Nash equilibrium strategy profile
>first-price auctions are not incentive compatible and are not equivalent to second-price auctions
Theorem: In a first-price sealed-bid auction with n risk-neutral agents whose valuations are independently drawn from a uniform distribution on the same bounded interval of the real numbers, the unique symmetric equilibrium is given by the strategy profile : 𝐵= 𝑛−1 𝑛 𝑣
the unique equilibrium of the auction occurs when each player bids (n-1)/n * v of his valuation.

17. Revenue equivalence theorem
Assume that each of n risk-neutral agents has an independent private valuation for a single good at auction, drawn from a common cumulative distribution F(v) that is strictly increasing and atomless on [v_, v]. Then any auction mechanism in which
the good will be allocated to the agent with the highest valuation
any agent with valuation v_ has an expected utility of zero
yields the same expected revenue, and hence results in any bidder with valuation vi making the same expected payment.
If bidders are risk neutral and have independent private valuations English, Japanese,Dutch, and all sealed bid auction protocols are revenue equivalent

18. Revenue equivalence theorem
By revenue equivalence theorem it does not much matter which auction to hold
It is used to identify equilibrium bidding strategies for auctions that meet the theorem’s conditions
The revenue equivalence theorem doesn't say that every revenue-equivalent strategy profile is an equilibrium
First and second-price auctions satisfy the requirements of the revenue equivalence theorem
every symmetric game has a symmetric equilibrium
in a symmetric equilibrium of this auction game higher bid == higher valuation
Vickrey proved a version of the revenue-equivalence theorem, namely that
in the independent private value case, the English, Dutch, first-price, and second price auctions all produce the same expected revenue for the seller.

19. Collusion Bidders make agreements to get the object at a lower price
They can select designated winner and other follow specified strategy like abstaining from bidding -> kartel ring

20. Terminologia
Single-sided auction=multiple agents on only one side of the market
Open-outcry=bidding is done by calling out the bids in public
Sealed-bid auctions= bidding is done by submitting to the auctioneer a secret bid
IPV=independent private values; bidder knows his valuation but not other bidders’. No dependence on other bidders.
Social efficiency= max total value to byuers
English auction=The auctioneer sets a starting price for the good, and agents have the option to announce successive bids,
each of which must be higher than the previous bid. May close after fixed time or after no new bids are made. Purchase of good for the amount of final bid.
Japanese auction=ascending bid auction. Auctioneer sets a starting price for the good, agent must choose whether or not to be “in”. Auctioneer calls out successively increasing prices. Drop out agent cannot reenter the auction. The auction ends when there is exactly one agent left in. The agent must then purchase the good for the current price.
Dutch auction=the auctioneer begins by announcing a high price and then proceeds to successively lowering prices in a regular fashion. The auction ends when the first agent signals the auctioneer. The signaling agent must then purchase the good for the displayed price.
Sealed-bid auctions=each agent submits to the auctioneer a “sealed” bid for the good not known to any of the other agents. The agent with the highest bid must purchase the good. The price at which he does so depends on the type of sealed-bid auction.
first-price sealed-bid auction=the winning agent pays an amount equal to his own bid.
second-price sealed-bid auction=the winning agent pays an amount equal to the next highest bid. Also called “Vickrey auction”.
kth-price sealed-bid auction=the winning agent purchases the good for a price equal to the kth highest bid.

21. Kotitehtävät
Consider a second-price, sealed-bid auction with one seller who has one unit of the object which he values at s and two buyers 1,2 who have values of v1 and v2 for the object. The values s, v1, v2 are all independent, private values. Suppose that both buyers know that the seller will submit his own sealed bid of s, but they do not know the value of s. Is it optimal for the buyers to bid truthfully? Should they each bid their true value?
The seller has one unit of the good which will be sold using a second-price, sealed-bid auction. Assume that there are three bidders who have independent, private values for the good v1, v2, v3 which are uniformly distributed on the interval [0,1]. Bidders behave rationally and submit optimal bids. Which bidder in terms of values wins the auction and how much does this bidder pay?