Chapter 7:
Reaching Agreements
In chapter six, we had a “one shot”
decision with no way of binding the
negotiation. In this chapter, we
rethink those decisions.
7-1
Reaching Agreements
How do agents reach agreements when they are self interested?
In an extreme case (zero sum encounter) no agreement is possible in
which both win — but in most scenarios, there is potential for
mutually beneficial agreement on matters of common interest
The capabilities of negotiation and argumentation are central to the
ability of an agent to reach such agreements
2
Mechanisms, Protocols, and Strategies
Negotiation is governed by a particular mechanism, or
protocol
The mechanism defines the “rules of encounter” between
agents
Mechanism design is designing mechanisms so that they
have certain desirable properties
Overview – auctions we are familiar with
At seats, what would desirable properties of the mechanism
be?
Given a particular protocol, how can a particular strategy
be designed that individual agents can use?
3
Example
Suppose we want to design a mechanism for dividing up items in an estate.
What is the protocol?
Could protocol have some negative ramifications?
2007 Nobel Prize in Economics – for mechanism design
the market, under ideal conditions, ensures an efficient allocation of scarce
resources. But in practice conditions are usually not ideal; for example,
competition is not completely free, consumers are not perfectly
informed and privately desirable production and consumption may
generate social costs and benefits
4
Auctions – a specific form of negotiation
An auction takes place between an agent known as the auctioneer and a
collection of agents known as the bidders
The goal of the auction is for the auctioneer to allocate the goods to the
bidders
In most settings the auctioneer desires to maximize the price; bidders desire to
minimize price
5
History of Auctions
Some point to Babylon in 500 B.C. as the origin of the auction. In
these early auctions, women, sought after as brides, were the
commodities offered for sale. Attractive brides would command a
great price; unfortunately, less desirable women would have to be
accompanied by a dowry, making the winning bid price negative.
In other words, the winning bidder would actually have to be paid
to marry the women!
Some early auctions had incredibly high stakes. In 193 A.D., the
entire Roman Empire was actually auctioned off! The highest
bidder, Didius Julianus, won the bidding at a price of 6,250
drachmas for each Roman guard. He didn’t get to enjoy his
purchase for long though, since he was beheaded two months
later by Septimius Severus during his conquest of Rome.
6
Mechanism Design
As an example, suppose a person has an estate auction.
Desirable properties of mechanisms:
– Convergence/guaranteed success
– Maximizing social welfare: auctioneers sell all items if anyone wants
them. Maximize total happy people.
– Pareto efficiency: the item must sell to the buyer with the highest
evaluation as profit to auctioneer is best and bidder is happiest.
– Individual rationality (encourages bidders to behave rationally)
– Stability (won’t desire to change mind once outcome is known)
– Simple, quick
– Distribution (no central control)
– Ability to set Reservation Price (a seller specified bid level below which
no sale is made)
7
Pareto Efficient Solutions: f represents
possible solutions for two players.
Assuming utility is amount more than minimal sale price for seller and less that maximum
purchase price for buyer.
U2
f1
f2
f3
f4
U1
8
Assumption is –
If you buy the item for exactly what it is worth, there is no utility.
If you sell the item for exactly what is it worth, there is no utility.
We are trying to do better than value – a good deal for both.
9
Pareto Efficient Solutions
U2
f1
f 2 Pareto
dominates f 3
f2
f3
f4
U1
10
Pareto Efficient Solutions
U2
f1
f2
The Pareto frontier
f3
f4
U1
11
Auction Parameters
Goods can have
– private value (Aunt Bessie’s Broach)
– public/common value (oil field to oil companies)
– correlated value (partially private, partially values of
others): consider the resale value
Winner pays
– first price (highest bidder wins, pays highest price)
– second price (to person who bids highest, but pay value
of second price)
Bids may be
– open cry
– sealed bid
Bidding may be
– one shot
– ascending
– descending
12
A reserve price auction operates in the same manner as a straight auction,
with bidders trying to outbid each other for the item. The major difference
is that the seller establishes a secret “reserve price,” the lowest prices that
he will accept for the item offered. If the bidding does not reach the
reserve price, the seller is not obligated to sell.
Example: Cy knows that he could sell his grandfather’s antique dresser for
$2,000 to a neighbor; however, he hopes to make more than that at an
auction. So, he tells the auctioneer that the lowest he will accept for the
dresser is $2,000. Cy asks the auctioneer not to announce this price to the
bidders because he wants them to bid as high as they think the dresser is
worth, potentially much more than $2,000. If the bidding comes in lower
than $2,000, Cy will not sell the dresser at the auction, but instead to his
neighbor. If the bidding ends up being higher than $2,000, Cy will sell the
dresser at the auction.
13
What is an Auction?
Auctions are mediated negotiation mechanisms in which one negotiable
parameter is price
Note:
– Mediated implies messages are sent to mediator, not directly between
participants
– Is the mediator is important? Why?
– Mediator follows a strict policy for determining outcome based on
messages
– Single seller auctions are a special case
14
Auction settings
Private value : value of the good depends only on the agent’s own
preferences
– E.g. cake which is not resold or showed off
Common value : agent’s value of an item determined entirely by
others’ values
– E.g. treasury bills
Correlated value (Affiliated value): agent’s value of an item
depends partly on its own preferences & partly on others’
values for it
– E.g. painting when bidders can keep it and like the colors
or reauction it to others
15
English Auctions
Most commonly known type of auction:
– first price
– open cry
– ascending
– Open exit (openly declare exit, cannot re-enter)
– Real time
Dominant strategy is for agent to successively bid a small amount more than the
current highest bid until it reaches their valuation, then withdraw
Efficient (pareto sense) as person who values item most gets it
Susceptible to:
– winner’s curse – get excited and bid too much
– shills (no intention of buying. Bid up the price. Work for auctioneer on
commission. Illegal in most cases.)
– The earliest use of the word 'shill' actually dates back to
Elizabethan England when theatre owners would pay a 'shilling' to
a theatre goer who would applaud and cheer loudly at the end of a
performance. Since applause is contagious, this would help ensure
the success of a production.
16
The key point is that the winner pays no more than the highest price that
the second-last bidder remaining would pay (note, there is an
assumption that the bid increases in quite small intervals so that the last
bidder recognizes when the second-last bidder drops out of the auction.
)
From a pareto efficiency standpoint, the bidder that values the item the
most ends up with the item. This makes economists happy. Not all
auctions are efficient in that sense.
17
Dutch (Aalsmeer)
flower auction
18
Dutch Clock
The hand of the clock starts always on the top.
The hand of the clock runs anti clockwise, from left to right.
The price drops from high to low.
The first man, out of the 300 buyers, who pushes the button first is the
buyer of the goods
19
Auction protocols:
Dutch (open-cry descending)
20
Protocol: Auctioneer continuously lowers the price until a bidder
takes the item at the current price
Strategically equivalent to first-price sealed-bid protocol in all
auction settings (we will discuss this option shortly)
Time efficient (real-time)
Strategy: Bid as a function of agent’s private value and his prior
estimates of others’ valuations
Best strategy: No dominant strategy in general (without more info)
– Lying (down-biasing bids) & counterspeculation
– Possible to determine Nash equilibrium strategies via common
knowledge assumptions regarding the probability distributions
of others’ values
– Requires multiple rounds of posting current price
Dutch flower market, Ontario tobacco auction, Filene’s basement,
Waldenbooks
21
Questions
What are advantages and disadvantages of such an auction?
Why would you choose such a mechanism?
Private: Learn only who values it the most, not the values that others have
Quick
22
How do you recommend the auction is handled?
Erin opens her birthday presents and realizes that she has
received six Yoda figurines. She only wants one, so decides to
auction off the other five in one auction.
23
Descending price Multiple item
A descending price auction is a type
of auction used to sell multiple
units of the same item at the
same time. In a descending price
auction, multiple items are
offered at an opening bid price. A
potential buyer can bid on one or
more units of the item. The price
is then lowered in successive
decrements until the entire lot
has been purchased.
Erin opens her birthday presents and realizes
that she has received six Yoda figurines. She
only wants one, so decides to auction off the
other five in one auction.
$80, but lower price by $10 each hour.
• 3 bidders at $60
• 1 bidder at $50
• Many bidders at $40, first wins
24
Ebay Multiple Item Dutch Auction - Ascending
Multiple identical items are for sale
Winning bids are selected in order of bid price per item.
You cannot lower your “total bid value” (your bid price per item
times the number of items on which you’re bidding) if you raise
your bid in this type of Multiple Item Auction.
Example:
For a listing with 10 available items and 4 bidders:
Bidder A bid for 2 items at $4 each
Bidder B bid for 8 items at $5 each
Bidder C bid for 3 items at $6 each
Bidder D bid for 2 items at $7 each
25
Therefore, the outcome of this listing is:
Bidder D wins 2 items at $5 each.
Bidder C wins 3 items at $5 each.
Bidder B wins 5 items at $5 each.
Bidder A wins no items.
The ranking of the bids affects the allocation of the items. Winning bidders have the
right to refuse partial quantities. This means that if you win some, but not all, of the
quantity you bid for, you don't have to buy any of them. In the above example,
Bidder B bid on 8 items, but won only 5 of them. Bidder B can refuse to complete
the purchase, because Bidder B did not win the quantity he or she bid on.
26
The Yankee auction
This is another type of auction that is used to sell multiple units of the same item at the same time.
It operates identically to the ascending multiple-item auction except that the winning bidders
each pay their winning bid price instead of the lowest winning bid price. In other words, in a
Yankee auction it is possible to have buyers that pay different prices; whereas in a Dutch
auction, everybody pays the same price.
So for the previous example:
Bidder D wins 2 items at $7 each.
Bidder C wins 3 items at $6 each.
Bidder B wins 5 items at $5 each.
Bidder A wins no items.
27
Auction protocols:
First-price sealed-bid
Example – Outdoor rec - kayac
Protocol: Each bidder submits one bid without knowing others’ bids. The highest bidder wins the
item at the price of his bid
– Single round of bidding
– (once you bid, there are no counter offers)
Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations
Best strategy: No dominant strategy in general
– Strategic underbidding & counter speculation
– Can determine Nash equilibrium (not do anything different, knowing what others
would do) strategies via common knowledge assumptions about the probability
distributions from which valuations are drawn
– Goal is to try to maximize the expected profit.
No relevant information is revealed – not even price or buyer (if you aren’t the winner)
Bidder uncertainty of valuation is a factor
No dominant strategy – as may not be pareto optimal with the “best
strategy”.
Efficient in real time as each person takes minimal time (as bidding happens in
parallel).
28
Vickrey Auctions
Vickrey auctions are:
– one shot
– second-price
– sealed-bid
Good is awarded to the agent that made the highest bid; at the price of
the second highest bid
Bidding to your true valuation is dominant strategy in Vickrey auctions.
Why?But in practice, you bid less due to the winner’s curse.
Vickrey auctions susceptible to antisocial behavior (bid really high to
guarantee win, someone else bids somewhat high to stick you with it)
Effort not wasted in counter-speculation as just bid true value.
Widely advocated for computational multiagent systems
Old method [Vickrey 1961], but not widely used among humans
29
What is learned?
Dutch auctions are tense events, but are not very informative. You do
learn that the winner values the item at no less than the price bid. But
you learn nothing about how others value the item.
As with the FPSB auction, you learn only that you are (are not) the high
bidder.
Even the SPSB (Vickrey) auction yields more information as it reveals to
the winner the second highest bid
Why do we care that the bidder’s don’t learn anything? If the bidders are
unhappy (always lose the bid but don’t know why) or win (but suffer
winner’s remorse), they may not choose to frequent your auction.
That could be bad for you.
30
How do you counterspeculate?
Consider a dutch auction
While you don’t know what the other’s valuation is, you know a range and
guess at a distribution (uniform, normal, etc.)
For example, suppose there is a single other bidder whose valuation lies in
the range [a,b] with a uniform distribution. If your valuation of the item
is v, what price should you bid?
Thinking about this logically, if you bid above your valuation, you lose. If
you bid lower than your valuation, you increase profit.
If you bid very low, you lower the probability that you will ever get it.
31
What is your expected profit (dutch auction)?
It seems natural to try to maximize your expected profit.
Expected profit (as a function of a specific bid) is the
probability that you will win the bid times the amount of
your profit at that price.
Let p be the price you bid for an item. v be your valuation.
[a,b] be the uniform range of others bid.
The probability that you win the bid at this price is the
fraction of the time that the other person bids lower
than p. (p-a)/(b-a)
The profit you make at p is v-p
Expected profit as a function of p is the function
= (v-p)*(p-a)/(b-a) + 0*(1- (p-a)/(b-a))
32
Finding maximum profit is a simple calculus
problem
Expected profit as a function of p is the function (v-p)*(pa)/(b-a)
Take the derivative with respect to p and set that value to
zero. Where the slope is zero, is the maximum value. (as
second derivative is negative)
f(p) = 1/(b-a) * (vp -va -p2+pa)
f’(p) = 1/(b-a) (v-2p+a) = 0
p=(a+v)/2 (half the distance between your bid and the min
range value)
33
Are you surprized?
The results make sense. You never bid higher than your valuation. You can’t
win these cases, so we’ll ignore them.
Of the remaining cases, if you bid halfway between the low evaluation and
your valuation, you expect to win half the time and lose half the time [in
the cases where you have a chance].
When you do win, you pay considerably less than your valuation, and hence
make a handsome profit.
You have to bid more often as you won’t get everything you bid on – but this
is a good plan.
As there are more bidders, how would that affect your decision?
34
In general, with uniform distribution on range
[0,Max]
If there are n total agents, what should each agent bid?
For simplicity, lets assume a=0. In order to win the bid at price p,every
other person would need to bid below p. What is the chance of that?
Since we want bidder 1 to bid below p and bidder 2 (and so forth), we
multiply the probabilities for each of the other n-1 bidders:
Expected valued = (p/b)n-1 * (p-v) = (1/bn-1)*(p(n) – vpn-1)
derivative wrt p = (1/bn-1*((n)p(n-1) – (n-1)vp(n-2)) = 0
(n)p(n-1) – nvp(n-2) +vp(n-2) = p(n-2)(pn –nv + v)
so either p = 0 (obviously a minimum) or p=((n-1)/n) * v
When n=2, we get the results on the previous slide. Idea is that with more
bidders (randomly bidding within the range), you have to bid closer to
your valuation to win; the idea of competition.
35
Revenue for the Auctioneer
Which protocol is best for the auctioneer?
• Revenue-equivalence Theorem (Vickrey, 1961):
All four protocols give the same expected revenue for private value auctions amongst riskneutral bidders with valuations independently drawn from a uniform distribution.
• Intuition: revenue second highest valuation:
– Vickrey: clear 
– English: bidding stops just after second highest valuation 
– Dutch/FPSB: because of the uniform value distribution and
counter speculation, top bid second highest valuation 
• But: this applies only to an artificial and rather idealized
situation; in reality there are many exceptions.
36
How does this happen
Consider buying something on Craig’s list.
The Craigslist auction strategy is not as easy as EBay because you can’t
view other bids. A starting point is realizing that you don’t want to bid higher
than your value. If your value is $100, it does not make sense to bid $101.
You might not win, but if you do, you are sure to overpay. The next step is
realizing you don’t want to bid exactly your value. If you do that, then you’ll
only break even when you win. So the idea is to bid some amount lower than
your value. How much lower?
In this auction, you know that you only win if you’re the highest value, and
you ideally would want to pay the least amount—one bid above the
maximum for the second highest person. It’s impossible to know where your
relative value is, so what you do is the following. You guess that you have the
highest value, and then estimate the second highest value and bid just
higher.
37
What is a Rational Decision?
We introduce notation to talk about preferences over choices.
We assume that agents have preferences over states of the world
– A > B A is strictly preferred to B
– A ~ B agent is indifferent between A & B
– A ≥ B A is weakly preferred to B (could be equal)
38
Lotteries
A lottery is a combination of a probability and an outcome (like considering a
mixed choice)
– L = [p, A; 1 – p, B] (p% chance of picking A)
– L = [1, A]
– L = [p, A; q, B, 1 – p – q, C] (percent chance of picking each of A, B or C)
Lotteries can be used to represent a human’s preference structure
39
Millionaire Scenario
You have just achieved $500,000
You have have no idea on the last question
If you guess
– [3/4, $100,000; 1/4, $1,000,000]
If you quit
– [1, $500,000]
What do you do?
40
Maximizing the Expected Payoff
Maximize expected value (EV):
– EV(guess) = pcorrect * U(guess correct) + pwrong * U(guess wrong)
–
=1/4(1,000,000) + 3/4 (100,000)
–
= 325,000
– EV(quit) = 500,000
What if you had narrowed the choice to two alternatives?
41
Properties of Preferences
Orderability
– For any two states, either A > B, B > A, or A~B (equivalent)
Transitivity
– If A > B and B > C, then A > C
Continuity
– If A > B > C, then there is some p, s.t.
[p, A; (1-p) C] ~ B
42
More Properties
Substitutability
– If A~B, then [p, A; (1-p) C] ~ [p, B; (1-p) C]
for any value of p
Monotonicity
– If A > B and p ≥ q then [p, A; (1-p) B] ≥ [q, A; (1-q) B]
Decomposibility
– Compound lotteries can be reduced to simpler ones using laws of
probability
43
Revelation Principle
You can transform any auction into an “equivalent” one
which is direct and incentive compatible (i.e., bidder will
bid the true valuation)
Do you believe that? That is quite a statement.
Rather than lie (bid less than your true valuation), the
mechanism will “lie” for you
Example: assume two bidders (with valuations drawn from a
uniform distribution on a fixed interval [0,max]). The
optimal strategy is to bid ½ your true value. But if the rule
is changed so that the winner only pays half his bid, it is
optimal to bid your true value.
44
As an example of the revelation principle
Proxy Bidding on ebay
Once you have found an item you want to buy on eBay and decided you are willing to pay
£25 for it but the current price is £2.20, what should you do?
Well, you could bid just £2.40 and probably be the high bidder... but what happens when
you leave your computer and someone else comes in and bids £2.60? Do you have to
sit in front of your computer day and night until the auction ends to make sure you win?
Thanks to eBay's automatic proxy bidding tool, the answer is no. You allow the system
to increase you bid up to a max.
How does Proxy bidding work?
Here's a step by step guide:
An Auction is listed that starts at £1.
I come along and submit a maximum bid of £100. The proxy server executes this bid, and as
there are no other bids yet (mine is the first), the bid is on me for £1.
Now you come along and see that the current bid is £1 and you decide to bid £5. You enter
the bid and then you get an immediate outbid notice. Why? Because the proxy system
has my £100 maximum bid to execute while keeping the bid at the lowest possible
amount.
So you see an outbid notice, and the bid goes up to £5.50, with me as the high bidder.
45
You rebid £10 and the same thing happens. I am still the high bidder at
£10.50 and will remain so until you or someone else surpasses my
initial maximum bid.
Now, say you bid £150. The highest bid then falls to you at £101, as you
have passed my maximum bid (I'm now out of the running until I place
a new maximum bid).
If someone should come by and decide to bid £103, then they would get
an immediate outbid notice because the proxy system automatically
outbid the new bidder…and so on until the new bidder gives up or
places a maximum bid which outstrips your £150.
Once your maximum bid is reached, you receive an outbid notification by
email. You can decide then whether or not you want to increase your
maximum bid.
Thus, you can tell the truth (your real valuation) and the mechanism lies
for you (bids a competitive bid without overpaying)
Revelation principle – by changing the mechanism, we can convince
bidders to reveal their true valuation
46
Twenty-Dollar Auction
Object for sale: a $20 bill
Rules
– Highest bidder gets it
– Highest bidder and the second highest bidder pay their bids
– New bids must beat old bids by 50¢.
– Bidding starts at $1
– What would your strategy be?
47
This is sometimes called the entrapment game--that is
particularly nasty. Suppose that anyone who bids at the
auction of our $20 bill must pay the amount of the bid
whether he wins or not. Someone will open the bidding
low at $.50 in hopes of getting a real bargain. Someone
else will top the bid with a $1 bid. Bidding will usually
proceed up to about $10 and then pause. The second
bidder must now decide whether to lose his $8 or $9
bid, or continue. If he continues, the bidding will usually
advance up to $20 and then pause again. The second
highest bidder now realizes that he is not going to gain
anything on this auction, but has the potential for a
substantial loss, so he has a strong temptation to up his
bid beyond $20.
48
Here is how Frank and Cook describe this game:
"One might be tempted to think that any intelligent, well-informed
person would know better than to become involved in an auction
whose incentives so strongly favor costly escalation. But many of the
subjects in these auctions have been experienced business
professionals; many others have had formal training in the theory of
games and strategic interaction. For example, psychologist Max
Bazerman reports that during the past ten years he has earned more
than $17,000 by auctioning $20 bills to his MBA students at
Northwestern University.... In the course of almost two hundred of his
actions, the top two bids never totaled less than $39, and in one
instance totaled $407."
49
Other types of Auctions
Continuous Double Auction (CDA)
– Multiple buyers and sellers
– Clears continuously (determines who trades and at what price)
– A double auction market can be carried out by open outcry: buyers and sellers
call out prices they are willing to buy and sell at, and a match is made if a buyer
and seller call out the same price. NYSE, Pit (the game).
– Under double auction rules, any buyer who makes a bid must raise his/her hand
and be recognized. All bids and offers are written on the blackboard as they are
made.
– Only most attractive bid/offer has "standing" or can be accepted. Any buyer is
free at any time to accept a standing offer, and any seller can accept a standing
bid.
– It is common practice to add an "improvement rule"; that is, that a new bid be
greater than the standing bid and that a new offer be lower than the standing
offer.
– This is a double auction in a sense that bids rise and offers fall at the same time.
– Like Haggling: each suggest price
– Trading does not stop as each auction is concluded
Try it!
50
Other types of Auctions
Reverse Auction
– Single buyer
– Lowest seller gets to sell the object
– Used in many procurement situations
51
Japanese auction
Similar to an English auction in which an auctioneer regularly raises the current
price.
Participants must signal at every price level their willingness to stay in the
auction and pay the current price.
The auction concludes when only one bidder indicates his willingness to stay in.
also known as the button auction.
52
Call auction
Investors can place orders with their financial intermediary
throughout the day until 4.30 pm.
orders dispatched to the central order book for a 30-minute
accumulation phase.
this phase ends at 5.00 pm; match buy and sell orders
anonymously and centrally in order to establish a price.
The price determined by the auction procedure is the
reference price and serves as opening price for the
following day’s auction.
53
Taxonomy
http://gsb.haifa.ac.il/~sheizaf/ecommerce/auction.types.html
54
Core Auction Activities
Receive bids
– Enforce any bidding rules
Release intermediate information (optional)
– Produce quotes
– List of winning bidders
Clear
– Determine who trades with who and at what price
55
PROTOCOL EQUIVALENCE
Strategic equivalence
– Same expected revenue for the auctioneer
– Same bidding strategy for the bidder
English and Vickrey auctions have same strategic equivalence if we have
independent values (meaning my valuation doesn’t increase by hearing
your bid) – though they are constructed differently.
With a common value, English and Vickrey are no longer equivalent as
information is gained through the open cry bidding that is not obtained
from Vickrey.
First-price sealed-bid and Dutch auctions are strategically equivalent. The
differences are superficial. The essential features (you pay the price you
bid, and you have no information about others when you bid) are the
same. Therefore, you are gaining no real advantage from observing the
auctioneer’s price fall in a Dutch auction. The optimum bid strategies for
bidders are the same in each.
56
Perfect Information
Perfect information occurs when each bidder knows the value of an asset to
himself and to the other bidders.
With perfect information, all auctions have the same results. The second
highest valuation (or an infinitesimal bit above it)
57
To see that all auctions are the same with
perfect information…
Order the valuations (high to low): v1, v2, v3, …
In English, the winning bid is v2+
In FPSB, all know the values, so v2+  wins here also.
In Dutch auction, the bidder knows v2+  is the price that should be held out for
In Vickrey, just bid your true evaluation, and you’ll get it for the second price.
58
Results for private value auctions
English and Vickrey auctions - Most efficient (as those
that value item the most, get item)
All four protocols allocate item efficiently
– (assuming no reservation price for the auctioneer)
English & Vickrey have dominant strategies =>
no effort wasted in counterspeculation
59
Bidders may desire private valuation to remain
private
Reminder: the Vickrey auction’s dominant strategy in private-value auctions is
bidding truthfully.
May reveal sensitive information as identity of first and second bidders and
price of second bidder may be known (a main reason why the Vickrey
auction protocol is not widely used).
Doesn’t occur in first-price sealed-bid auctions, as no one even needs to know
who won the bid.
60
RISK: For each of the three deals which choice
would you pick?
Deal 1
– I give you $1 OR
– We pick a random integer 1..100 and if it comes out 1 you get
$100 otherwise you get nothing.
Deal 2
– I give you $10 OR
– We flip a coin and if it is heads, you get $20, otherwise you get
nothing.
Deal 3
– I give you $1
– We pick a random integer 1..100 and if it comes out 1 you get
$1000 otherwise you get nothing.
61
Risk
Let
S = [1, x] get x with probability of 1
Where x =ExpectedValue(L) = py + (1 – p)z
L = [p, y; 1 – p, z]
Suppose the expected values are the same!
Risk averse: Utility(S) > Utility(L) (a sure thing is valued more
than choice with same expected value)
Risk neutral: Utility(S) = Utility(L) (a sure thing is valued the
same as a choice with same exp value)
Risk seeking: Utility(S) < Utility(L) (a sure thing is valued less
than a chance at more)
62
Risk Averse: Which would you choose?
Gamble 1:
win $400 with a probability ½ and nothing with probabiltiy
½
Gamble 2: win $225 with a probability of ½ and win $36
with probability ½
Gamble 3: win $200 guaranteed
If our u(w) = w
The expected utility of gamble 1 is
½ sqrt(400) + 1/2(0) = ½ 20 = 10
The expected utility of gamble 2 =
½*sqrt(36) + ½ sqrt(225) = ½(6+15) =10.5
The expected utility of gamble 3=sqrt(200) = 14.14
63
Risk SeekingWhich would you choose?
Gamble 1:
win $400 with a probability ½ and nothing with probabiltiy
½
Gamble 2: win $225 with a probability of ½ and win $36 with
probability ½
Game 3: win $200 guaranteed
If our u(w) = x2
The expected utility of gamble 1 is
½(400)2 + 1/2(0) = ½ 20 = 80,000
The expected utility of gamble 2 =
½*(36)2 + ½(225)2 = ½(51921) =25,960
The expected utiltiy of gamble 3=(200)2 = 40,000
64
Individuals have different tolerance for risk.
An individual who ranks lotteries according to their
expected value (rather than expected utility) is said to be
risk neutral. In other words, an risk neutral individual
who is offered $100 outright or a 50% chance of winning
$200 will value the choices EQUALLY!
65
If the utility function over wealth is linear
u(w) = aw + b
the person is risk neutral
If the utility function is concave(line between points is under
curve), the individual is risk averse.
If the utility function is convex(line between points is above
curve), the individual is risk seeking. Note, gambling is like
staying on the line as the two endpoints are picked with
probability p or (1-p).
66
So u(w) = w is risk neutral
u(w) =
is risk averse
u(w) = w2 is risk seeking (as large amount of money is worth much more than
small amounts)
w
67
Expected Utility Theory
describes behavior under uncertainty
If people are risk neutral or risk averse, they would never play the lottery or
gamble (as return there is usually negative)
The expected value of Powerball lottery (if tickets cost $1 and jackpot is 7
million) is
7000000 * 1/85000000 -1(84999999/85000000) = -.917647
68
But people do play powerball - Why?
Loss is so small, people often ignore it.
If losses were larger, people may behave very differently.
People who buy lottery tickets may behave in very risk averse manner in other
situation
69
Risk Averse
With English and SPSB, risk aversion makes no difference – as you bid your true valuation,
but may not pay that much. Thus, you automatically have potential for profit.
With Dutch and FPSB, to generate profit, you must take the risk of letting the price go below
your valuation. If you are risk averse, you would rather win a little money than run the
risk of making nothing or making a lot. So you bid higher and revenues for auctioneer
increase.
Expected revenues for risk averse bidders:
Dutch = FPSB > English=SPSB
– Since the risk averse bidder values a little money more than the potential of
more, he will be happier with a sure thing at less profit.
For risk seeking, Dutch = FPSB < English=SPSB
– Since the risk seeking bidder values a big profit more than the expected vaue of
less, he will be happier with the gamble.
70
Winner’s Curse
In auctions where bidders have common values, the winner
tends to have overestimated asset value. He/she may
come to regret the bid (curse).
You know you’ve bid too much as others bid less.
Recognizing this, all bidders may adjust their bids
downwards.
The winner’s curse works against the seller, especially if the
bidders are risk seeking. It is then best to try to release as
much information about true worth to bidders.
71
On cheating in sealed-bid auctions
7-72
Bidder Collusion
– Collusion: bidders agreeing together to control bids to their
advantage. (Let’s keep price low and split profit.)
– None of the four is collusion-proof
– First-price sealed-bid and Dutch auctions make it harder to
conspire against auctioneer (as hard to know who is going to
bid)
– Bidders need not identify each other ahead of time to collude in
English auctions, unlike in the others. Could have a way of
signalling or contacting the active bidders to arrange something.
– Also, can enforce collusion in English auction as can respond to
defector (who bids higher than agreed upon)
73
With Vickrey, if collude
Say one bidder values item at 20, the rest value it at 18. They agree to have
one bid 20 and the rest bid 5. The high bid gets it for 5.
No reason for anyone to bid any higher as they wouldn’t get it anyway (and
they wouldn’t want it for over 20).
74
Lying Auctioneer
– Problem in Vickrey auction – auctioneer overstates second bid –
electronic signatures (or have trusted third party handle bids)
– Non-private value auctions – English auction – auctioneer’s shills
(someone who bids up the price to increase perceived value,
but never wants to take it home. Works for auctioneer - illegal)
– Overstated reservation price (minimum price that the
auctioneer will accept) – sometimes risky to the auctioneer as
he may not sell it
– No risk in first price sealed bid, as know how much you offered
and are not affected by other bidders.
75
First-price sealed bid auction, cheating bidder
Consider the case the seller is honest, but there is a chance the other
agents will look at the bids before submitting their own.
Notice that this kind of cheating is pointless in second-price auctions.
76
Complements and Substitutes
The value an agent assigns to a bundle of goods may relate to the
value it assigns to the individual goods in a variety of ways . . .
• Complements: The value assigned to a set is greater than the
sum of the values assigns to its elements.
A standard example for complements would be a pair of shoes
(a left shoe and a right shoe).
• Substitutes: The value assigned to a set is lower than the sum
of the values assigned to its elements.
A standard example for substitutes would be a ticket to the
theatre and another one to a football match for the same night.
In such cases an auction mechanism allocating one item at a time is
problematic as the best bidding strategy in one auction may depend
on whether the agent can expect to win certain future auctions.
77
Combinatorial Auction Protocol
• Setting: one seller (auctioneer) and several potential buyers
(bidders); many goods to be sold
• Bidding: the bidders bid by submitting their valuations to the
auctioneer (not necessarily truthfully)
• Clearing: the auctioneer announces a number of winning bids
The winning bids determine which bidder obtains which good,
and how much each bidder has to pay. No good may be
allocated to more than one bidder.
78
Interrelated auctions
Strategies might be different when interrelated items are auctioned at a
time instead of each item separately.
Say – bid for two tasks, but second is cheaper if already doing the first.
Lookahead is a key feature in auctions of multiple interrelated items.
Auctioneers often allow
bidders to pool all of
the interrelated items
under one bid.
79
Interrelated auctions (cont.)
Sometimes auctioneers allow bidders to backtrack from commitments by
paying penalties. This is helpful if you win bid on one item hoping to get
interrelated item (but don’t get it).
Different kind of speculations: trying to guess what items will be auctioned in
the future, and which agents are going to win in those auctions.
Trade-off: (partial) lookahead vs. cost.
80
Bidding Languages
• As there are 2n − 1 non-empty bundles for n goods, submitting
complete valuations may not be feasible.
• We assume that each bidder submits a number of atomic bids
(Bi, pi) specifying the price pi the bidder is prepared to pay for a
particular bundle Bi.
• The bidding language determines what combinations of
individual bids may be accepted. Today, we (mostly) assume
that at most one bid of each bidder can be accepted.
• In general, we may think of the bidding language as
determining a conflict graph: bids are vertices and edges
connect bids that cannot be accepted together.
• The bidding language also determines how to compute the
overall price
81
The Winner Determination Problem
The winner determination problem (WDP) is the problem of
finding a set of winning bids
(1) that is feasible and
(2) that will maximize the sum of the prices offered.
The sum of prices can be given different interpretations:
(1) If the simple pricing rule is used where bidders pay what they
offered, then it is the revenue of the auctioneer.
(2) If the prices offered are interpreted as individual utilities, then it
is the utilitarian social welfare of the selected allocation.
82
Example
• Each bidder submits a number of
bids describing their valuation.
• Each bid (Bi, pi) specifies which
price pi the bidder is prepared to
• pay for a particular bundle Bi. The
auctioneer may accept at most one
atomic bid per bidder (other
bidding languages are possible).
• In this example, we can
remove clearly inferior bids
• ({a, b}, 5) is inferior to ({b}, 5),
({a, b, c, d}, 12) is inferior to
({a, d}, 7) plus ({b, c}, 7)
Agent 1: ({a, b}, 5), ({b, c}, 7),
({c, d}, 6)
Agent 2: ({a, d}, 7), ({a, c, d}, 8)
Agent 3: ({b}, 5), ({a, b, c, d},
12)
What would be the optimal
solution?
83
Complexity of Winner Determination
The decision problem underlying the WDP is NP-complete:
Theorem 1 Let K  Z. The problem of checking whether there
exists a solution to a given combinatorial auction instance
generating a revenue exceeding K is NP-complete.
(Note, they have changed the optimization problem into a simpler boolean form
of the problem.)
M.H. Rothkopf, A. Peke˘c, and R.M. Harstad. Computationally Manageable
Combinational Auctions. Management Science, 44(8):1131–1147, 1998.
84
Solving the Winner Determination Problem
We have seen that the WDP is intractable (NP-complete) in its
general form. Nevertheless, sophisticated search algorithms often
manage to solve even large CA instances in practice.
There are two types of approaches to optimal winner determination in the
general case:
• Use powerful general-purpose mathematical programming software (next
slide)
• Develop search algorithms specifically for winner
determination, combining general AI search techniques and
domain-specific heuristics (rest of this lecture)
Other options include developing special-purpose algorithms for
tractable subclasses and approximation algorithms for the general case
85
Integer Programming Approach
Suppose bidders submit n bids as bundle/price pairs (Bi, pi) with
the implicit understanding that the auctioneer may accept any
combination of non-conflicting bids and charge the sum of the
associated prices (this is the so-called OR bidding language).
Introduce a decision variable xi  {0, 1} for each bid (Bi, pi).
The WDP becomes the following Integer Programming problem:
Maximize  pi · xi subject to each item only being in one bid that is accepted.
Highly optimized software packages for mathematical programming
(such as CPLEX/ILOG) can often solve such problems efficiently.
86
Search for an Optimal Solution
Next we are going to see how to customise well-known search
techniques developed in AI so as to solve the WDP.
This part of the lecture will largely follow the survey article by
Sandholm (2006).
T. Sandholm. Optimal Winner Determination Algorithms. In P. Cramton et
al. (eds.), Combinatorial Auctions, MIT Press, 2006.
87
Search Techniques in AI
A generic approach to search uses the state-space representation:
• Represent the problem as a set of states and define moves
between states. Given an initial state, this defines a search tree.
• The goal states are states that correspond to valid solutions.
• Each move is associated with a cost (or a payoff ).
• A solution is a sequence of moves from the initial state to a goal state with
minimum cost (maximum payoff ).
• Example: route finding (states are cities and moves are directly connecting
roads), but it also applies to CAs . . .
A search algorithm defines the order in which to traverse the tree:
• Uninformed search: breadth-first, depth-first, iterative deepening
• Heuristic-guided search: branch-and-bound, A*
88
State Space and Moves
There are (at least) two natural ways of representing the state
space and moves between states:
• Either: Define a state as a set of goods for which an allocation
decision has already been made. Then making a move in the
state space amounts to making a decision for a further good.
• Or: Define a state as a set of bids for which an acceptance
decision has already been made. In this case, a move amounts
to making a decision for a further bid.
What is the initial state? What are the goal states?
According to Sandholm (2006), the bid-oriented approach tends to
give better performance in practice.
89
Moves in Bid-oriented Search
We represent bids as triples (ai,Bi, pi): agent ai is offering to buy
the bundle Bi for a price of pi.
The initial state is when no decisions on bids have been made.
A move amounts to making a decision (accept/reject) for a new bid.
The bidding language specifies which bids (if any) must be accepted/rejected
given earlier decisions.
We are in a goal state once a decision for every bid has been made
(some of which will be consequences of the explicit choices).
Observe that that the search tree will be binary (accept or reject?).
90
Problem
Suppose we have the following independent bids in a combinatorial auction:
{1,2} 10
{3} 4
{2,3} 12
{1,3} 12
{4,5,6} 12
{4,5} 9
{5,6} 10
{3,5} 6
91
Example from Sandholm (2006) shows a PARTIAL
conflict graph for our example
Conflict graph
Use it (in) or Don’t use it (out)
Show the conflict
graph at this point.
92
Applying g(n) +h(n) as the heuristic for A* search
93
Applying g(n) +h(n) as the heuristic for A* search
94
Uninformed Search
Uninformed search algorithms (in particular depth-first search) can be used to
find a solution with a given minimum revenue: traverse the tree and keep
the best solution encountered so far in memory. Optimality can only be
guaranteed if we traverse the entire search tree (not feasible for interesting
problem instances).
95
Uniform Cost
Strategy: expand lowest
path cost
The good: UCS is
complete and optimal!
The bad:
Explores options in every
“direction”
No information about goal
location
96
Best First
•
Strategy: expand nodes
which appear closest to goal
Heuristic: function which
maps states to distance
•
A common case: Best-first
takes you straight to the
(wrong) goal
• Worst-case: like a badlyguided DFS
• In our case, explore the bids
with the most items.
97
Heuristic-guided Search
In the worst case, any algorithm may have to search the entire
search tree. But good heuristics, that tell us which part of the tree
to explore next, often allow us to avoid this in practice.
For any node N in the search tree, let g(N) be the revenue generated by
accepting (only) the bids accepted according to N. g(N) is the revenue
earned from previous decisions.
We are going to need a heuristic that allows us to estimate for every node N
how much revenue over and above g(N) can be expected if we pursue the
path through N. This will be denoted as h(N). h(N) is future possible
revenue.
The more accurate the estimate, the better — but the only strict requirement
is that h never underestimates the true revenue.
We are going to describe two algorithms using such heuristics:
• depth-first branch-and-bound
• the A* algorithm
98
Slides from: Dan Klein – UC Berkeley
Many slides over the course adapted from either Stuart
Russell or Andrew Moore
99
Heuristic Upper Bounds on Revenue
Sandholm (2006) discusses several ways of defining a heuristic
function h such that g(N) + h(N) is guaranteed to be an upper
bound on revenue for any path through node N.
Here is one such heuristic function:
• For each good g, compute its maximum contribution as:
c(g) = max{p/|B| | (B, p) Bids and g  B}
• Then define h(N) as the sum of all c(g) for those goods g that have not yet
been allocated in N.
This is indeed an upper bound (why?). This assumes the revenue from a
bundle is credited equally. The only way this wouldn’t be true is if a partner
could get more, and then the one in question gets less.
The quality of this heuristic can be improved by recomputing c(g)
after every step (need to balance accuracy and computing time). Bids is the set
of Bids not yet ruled out.
100
Depth-first Branch-and-Bound
This algorithm works like basic (uninformed) depth-first search, except that
branches that have no chance of developing into an optimal solution get
pruned on the fly:
• Traverse the search tree in depth-first order.
• Keep track of which of the nodes encountered so far would generate
maximum guaranteed revenue . Call that node N*.
• If a node N with g(N) + h(N) <=g(N*) is encountered, remove that node and
all its offspring from the search tree.
This is correct (guarantees that the optimal solution does not get removed)
whenever the heuristic function h is guaranteed
101
The A* Algorithm
The A* algorithm (Hart et al., 1968) is probably the most famous
search algorithm in AI. It works as follows:
• The fringe is the set of leaf nodes of the subtree visited so far
(initially just the root node).
• Compute f(N) = g(N) + h(N) for every node N in the fringe.
• Expand the node N with the largest f(N); that is, remove it from the fringe and
add its (two) immediate children instead to the visited subtree.
By a standard result in AI, A* with an admissible heuristic function
(here: h never underestimates marginal revenue) is optimal: the first solution
found (when no bids are left) will generate maximum revenue.
P. Hart, N. Nilsson, and B. Raphael. A Formal Basis for the Heuristic Determination
of Minimum Cost Paths. IEEE Transactions on Systems Science
and Cybernetics, 4(2):100–107, 1968.
102
Branching Heuristics
So far, we have not specified which bid to select for branching in
each round (for any of our algorithms). This choice does not affect
correctness, but it may affect speed.
There are two basic heuristics for bid selection:
• Select a bid with a high price and a low number of items.
• Select a bid that would decompose the conflict graph of the remaining bids (if
available).
103
Tractable Subproblems
As a final example for possible fine-tuning of the algorithm, we can
try to identify tractable subproblems at nodes and solve them using
special-purpose algorithms.
Here are two very simple examples:
• If the bid conflict graph is complete, i.e. any pair of remaining bids is in
conflict, then only one of them can be accepted. Simply pick the one with
the highest price.
• If the bid conflict graph has no edges, then there is no conflict between any
of the remaining bids. Accept all remaining bids (assuming positive prices).
104
Mechanism Design
Mechanism design is concerned with the design of mechanisms for
collective decision making that favor particular outcomes despite
of agents pursuing their individual interests.
Mechanism design is sometimes referred to as reverse game theory.
While game theory analyses the strategic behavior of rational
agents in a given game, mechanism design uses these insights to
design games inducing certain strategies (and hence outcomes).
We are going to concentrate on mechanism design questions in the
context of (private value) combinatorial auctions.
105
Revelation Principle
This is somewhat simplified and informal:
Theorem 1 Any outcome that can be implemented through some
indirect mechanism with dominant strategies can also be implemented by
means of a direct mechanism (where agents simply reveal their
preferences) that makes truth-telling a dominant strategy.
Intuition: Whatever the agents are doing in the indirect mechanism to transform
their true preferences into a strategy, we can use as a “filter” in the
corresponding direct mechanism. So, first apply this filter to whatever the
agents are reporting and then simulate the indirect mechanism with the filtered
input. The outcome will be the same as the outcome we’d get with the indirect
mechanism iff the agents report their true preferences.
Discussion: we can concentrate on searching for a one-step mechanism
Example: the (direct) Vickrey auction may be regarded as a direct
implementation of the (indirect) English auction
106
Quasi-linear Utilities
• Each agent i has a valuation function vi mapping agreements x
(e.g. allocations) to the reals. This could be any such function.
• The actual utility ui of agent i is a function of its valuation
vi(x) for agreement x and a possible price p the agent may
have to pay in case x is chosen. In principle, this could be any
such function.
• However, we make the (common) assumption that utility functions are quasilinear (linear in one of the parameters):
ui(x, p) = vi(x) − p
That is, utility is linear in both valuation and price paid.
107
Reminder: Vickrey Auction
Dominant strategy: bid your true valuation
– if you bid more, you risk paying too much
– if you bid less, you lower your chances of winning while still having to pay the
same price in case you do win
• How can we generalize this idea to combinatorial auctions?
108
Reinterpreting the Vickrey Pricing Rule
for Combinatorial Auction
• Distinguish allocation rule and pricing rule
• Allocation rule: highest bid wins
• Pricing rule: winner pays price offered, but gets a discount
• The amount of the discount granted reflects the contribution to
overall value made by the winner. How can we compute this?
– Without the winner’s bid, the second highest bid would
have won. So the contribution of the winner is equal to the
difference between the winning and the second highest bid.
– Subtracting this contribution from the winning bid yields
the second highest bid (the Vickrey price).
109
Vickrey-Clarke-Groves Mechanism
This idea is used in the so-called Vickrey-Clarke-Groves mechanism, which we are
going to introduce next.
We are going to concentrate on the variant introduced by Edward H. Clarke (for combinatorial auctions), but
also mention the more general form of the mechanism as put forward by Theodore Groves.
For example, suppose that we want to auction two apples, and we have three
bidders. Bidder A wants one apple and bids $5 for that apple. Bidder B wants
one apple and is willing to pay $2 for it. Bidder C wants two apples and is
willing to pay $6 to have both of them, but is uninterested in buying only one
without the other. First, we decide the outcome of the auction by maximizing
bids: the apples go to bidder A and bidder B. Next, to decide payments, we
consider the opportunity cost that each bidder imposed on the rest of the
bidders. Currently, B has a utility of $2. If bidder A had not been present, C
would have won, and had a utility of $6, so A pays $6-$2 = $4. For the payment
of bidder B: currently A has a utility of $5 and C has a utility of 0. If bidder B
had been absent, C would have won and had a utility of $6, so B pays $6-$5 =
$1. C does not need to pay anything because he doesn’t get anything.
W. Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders.
Journal of Finance, 16(1):8–37, 1961. E.H. Clarke. Multipart Pricing of Public Goods. Public Choice, 11(1):17–33,
1971. T. Groves. Incentives in Teams. Econometrica, 41(4):617–631, 1973.
110
Notation
Set of bidders: A = {1, . . . , n}
• Set of possible agreements (allocations): X
• (True) valuation function of bidder i A: vi : X  R
• Valuation function reported by bidder i  A: ˆvi : X  R
• Top allocation as chosen by the auctioneer:
x*  arg maxxX
n
 vj^ ( x)
j 1
Allocation that would be chosen if agent i were not to bid:
x *i  arg maxxX
n
 vj^ ( x)
j 1
111
VCG Mechanism for Combinatorial Auctions
Allocation rule: solve the WDP and allocate goods accordingly
• Pricing rule: Again, the idea is to give each winner a discount
reflecting its contribution to overall value. In short, bidder i
should pay the following amount:
bidi − (max-value − max-value−i)
The same more formally:
The value of the best allocation possible without me minus
the value to everyone else of the best allocation with me
112
Strategy-Proofness
Theorem 2 In the VCG mechanism, reporting their true valuation
is a dominant strategy for each bidder.
Proof: Consider the situation of bidder i.
Remark: Contrast this with the Gibbard-Satterthwaite Theorem,
which (roughly) says that in the context of voting there is no such
strategy-proof mechanism. The crucial difference is that here we
can use money to affect people’s incentives.
113
114
Efficiency
By construction, if all bidders submit true valuations (dominant
strategy), then the outcome maximizes utilitarian social welfare (the benefit to
society in general in terms of their utility):
• payments (including the auctioneer’s) sum up to 0; and
• the sum of valuations is being maximized.
But note that this does not mean that revenue gets maximized as well (unlike
for the basic combinatorial auction without special pricing rules).
115
So, does it ever work well?
Consider the following example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0)
Agent 4: ({a}, 0), ({b}, 2), ({a, b}, 2)
Best allocation is agent 1 ({a, b}, 4)
Agent 1 pays: 3 – 0 = 3 So it works just like we thought it would
Lots of the problems come from very few bidders – which would always be a
problem for Vickrey auctions.
116
Another Example
Consider the following example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0)
Agent 4: ({a}, 0), ({b}, 2), ({a, b}, 2)
Best allocation is agent 1 ({a, b}, 4)
Agent 1 pays: 4 – 0 = 4 So it works just like we thought it would
Second price bid is really the same as the first price bid
Lots of the problems come from very few bidders – which would always be a
problem for Vickrey auctions.
117
Collusion
The VCG mechanism is not collusion-proof: if bidders work
together they can manipulate the mechanism. Consider the
following example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 1), ({a, b}, 0)
Who wins? What do they pay?
But if the two losing bidders collude and increase their two bids to
({a}, 4) and ({b}, 4), respectively, they can obtain the items for free.
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 4), ({a, b}, 0)
Who wins? What do they pay?
118
Collusion ANSWERS
The VCG mechanism is not collusion-proof: if bidders work
together they can manipulate the mechanism. Consider the
following example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 1), ({a, b}, 0)
Agent 1 wins and pays 2-0= 2.
But if the two losing bidders collude and increase their two bids to
({a}, 4) and ({b}, 4), respectively, they can obtain the items for free.
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 4), ({a, b}, 0)
Agents 2 and 3 win and pay 4-4= 0. Can you explain the logic?
119
Example
The following example shows problems if we drop the free disposal
assumption:
Agent 1: accept one of ({a}, 90), ({b}, 10), ({a, b}, 10)
Agent 2: accept one of ({a}, 20), ({b}, 30), ({a, b}, 50)
We end up with the following payments in all items must be disposed of:
Agent 1: 50 -30 = +20(the best without my bid is 50, the value to others of
the best allocation computed with my bid is 30)
Agent 2: 10-90 = −80 (the best without my bid is 10, the value to others of
the best allocation computed with my bid is 90)
That is, agent 2 should receive money from the auctioneer!
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Problems with the VCG Mechanism
Despite their nice game-theoretical properties, CAs using the
Clarke tax to determine payments have several problems:
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Low (and possibly even zero) revenue for the auctioneer
Non-monotonicity: “better” bids don’t entail higher revenue
Collusion amongst (losing) bidders
False-name bidding: bidders may benefit from submitting bids using
multiple identities
The following examples illustrating these problems are adapted
from Asubel and Milgrom (2006).
L.M. Asubel and P. Milgrom. The Lovely but Lonely Vickrey Auction. In
P. Cramton et al. (eds.), Combinatorial Auction, MIT Press, 2006.
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Zero Revenue
There are cases where the VCG mechanism gives zero revenue:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2)
Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0)
Payments are computed as follows:
Agent 1: 0
Agent 2: 2 − (4 − 2) = 0 (the best without my bid is 2, the value to others of the
best allocation computed with my bid is 2)
Agent 3: 2 − (4 − 2) = 0 (the best without my bid is 2, the value to others of the
best allocation computed with my bid is 2)
Note that this problem is independent from whether or not we
admit free disposal.
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Non-monotonicity
Revenue is not necessarily monotonic in the set of bids or the amounts that are being bid.
Consider again the following example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2)
Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0)
As seen before, revenue for this example is 0.
If we either remove agent 3 or decrease the amount agent 3 is
offering for item b, then revenue will increase.
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2)
Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0)
Agent 3: ({a}, 0), ({b},1), ({a, b}, 0)
Revenue for this example
Agent 2 pays = 2-1 = 1 (the best without my bid is 2, the value to others of the best
allocation computed with my bid is 1)
Agent 3 pays = 2-2 = 0 ( the best without my bid is 2, the value to others of the best
allocation computed with my bid is 2)
New Example
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 1 pays 0
Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 2 pays 2 -0 = 2
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False-name Bidding
False-name bidding (aka. shill or pseudonymous bidding) is yet another
form of manipulation the VCG mechanism is exposed to. Example:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 1), ({b}, 1), ({a, b}, 2)
Agent 1 wins. But agent 2 could instead submit bids under two names:
Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)
Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0)
Agent 2’: ({a}, 0), ({b}, 4), ({a, b}, 0)
Agent(s) 2 (and 2’) will win and not pay anything! This form of
manipulation is particularly critical for electronic auctions, as it is easier
to create multiple identities online than it is in real life.
M. Yokoo. Pseudonymous Bidding in Combinatorial Auctions. In P. Cramton
et al. (eds.), Combinatorial Auction, MIT Press, 2006.
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Computational Issues
Observe that computing the Clarke tax requires solving an
additional n winner determination problems.
• That means, the auctioneer has to solve n + 1 NP-hard optimization
problems.
• If allocations and prices are not being computed according to the optimal
solutions to these problems, then we cannot guarantee strategy-proof-ness
anymore.
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Summary
We have introduced the Vickrey-Clarke-Groves mechanism, a
mechanism for collective decision making that makes truth-telling
the dominant strategy.
• Distinguish most general form of the VCG mechanism and the
variant where the Clarke tax is used to determine payments
• Additional properties: efficiency and weak budget balance
(the latter under suitable conditions)
• Drawbacks: high complexity, potential for low revenue,
manipulation through collusion or use of false-name bids, . . .
• Restriction: applies to agents with quasi-linear utilities only
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Conclusions
Pros
VCG processes have great theoretical appeal. They are a dominant strategy mechanism. This
means that, in theory, a bidder’s decision to use the strategy they call for does not
depend on what the bidder thinks her competitors’ strategies are, and she need spend no
effort in trying to find them out or to keep her competitors from learning her strategy.
In some circumstances, they produce, in theory, expected revenue equivalent to other
common auction forms.
Cons
However, VCG processes are just not practical. They do not work the way the (simple) theory
says they should.
So Why do we study VCG processes ?
Because finding equilibrium strategies in combinatorial auctions is extraordinarily difficult
except in VCG processes, there may well be useful insights to be had from such
knowledge. For example, Mishra and Parkes (2007) analyze an iterative version of the
VCG process.
In addition, computerized bidding agents may be able to be programmed to avoid some of
the 13 problems discussed here.
Rothkopf: Thirteen Reasons Why the Vickrey-Clarke-Groves Process Is Not Practical
Operations Research 55(2), pp. 191–197, ©2007 INFORMS
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Lecture 7: Reaching Agreements