Thousands of advertisers Handful of publishers B1 B2 B3 Bn q1 q2 A bstract view Publishers Determine allocation of slots to advertisers based on budgets given to them Cannot adjust size of supply ID: 526757
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Slide1
Competition Among Asymmetric Sellers with Fixed Supply
Uriel
Feige
(
Weizmann Institute of Science
)
Ron Lavi (
Technion and Yahoo! Labs
)
Moshe
Tennenholtz
(
Technion and MSR
)
*
Work conducted at MSR
HerzliyaSlide2
Motivation: markets for ads
. . .
Thousands of advertisers
Handful of publishers
B1
B2
B3
Bn
q1
q2
A
bstract view:
Publishers
: Determine allocation of slots to advertisers, based on budgets given to them.
Cannot adjust size of supply!
Advertisers
: Based on publishers’ policies, assign budgets.
Motivation
:
How should publishers
choose allocation policies
to maximize revenue?Slide3
Motivation: markets for ads
. . .
n advertisers
m publishers
B
B
B
B
q
1
q
m
T
his work: Symmetric advertisers, non-symmetric publishers:
. . .Slide4
The Model
. . .
n buyers
m sellers
B
B
B
B
q
1
q
m
T
his work: Symmetric buyers, non-symmetric sellers:
. . .Slide5
This model was studied previously:Friedman (1958):Assumes sellers’ policies are “proportional”:seller j gives buyer i* (bi*j /
i
b
ij
)qj
(bij is the budget that seller j receives from buyer i)
This is a “Blotto game” (Generals send soldiers to battlefields)Studied many times over the years…
Also applied to political campaigns, e.g., Snyder (1989)“Seller” = voting district (supply = number of district voters); “Buyer” = political party that partitions its budget among the various districts.
And much more…Slide6
Key DifferencePrevious work: Sellers with fixed exogenous allocation policiesMakes sense for battlefields, voting districts, etc.
But publishers are strategic and are free to determine their own allocation policies
T
his work
:
Sellers endogenously determine allocation policies to maximize revenue(In the language of Blotto games: a battlefields decides on its winning function, to maximize the number of soldiers it gets)Slide7
Summary of the gameSellers determine allocation policies: given budget assignments, how much items each buyer receivesGiven allocation policies, buyers partition budgets among sellersResulting utility of sellers: number of dollars obtained
Resulting utility of buyers: number of items obtained
Same assumption as in previous work, this not quasi-linear
Assumptions
(following previous
work):Complete informationBuyers play a Nash eq. (pure or mixed) of the resulting ”buyers game” (
after observing the allocation policies).Slide8
Convenient Change of Units
. . .
n buyers
m sellers
B=1
q
1
q
m
. . .
B=1
B=1
i
q
i
= nSlide9
“Fair Share”
. . .
n buyers
m sellers
B=1
q
1
q
m
. . .
B=1
B=1
i
q
i
= n
Overall: n dollars, n items
A reasonable outcome:
each item costs one dollar
This is the unique Walrasian outcome
In this outcome, the revenue of seller j is q
j
We call
q
j
the “fair share” revenue of seller j.Slide10
“Fair Share”Overall: n dollars, n items
A reasonable outcome:
each item costs one dollar
This is the unique Walrasian outcome
In this outcome, the revenue of seller j is q
j
We call q
j the “fair share” revenue of seller j.
THM (Friedman 1958) If all sellers use proportional allocation, there is a unique Nash equilibrium of the buyers game whose outcome is the fair share outcome.
Proportional allocation:seller j gives buyer i* (bi*j
/ ibij)q
j (bij
: budget to seller j
from buyer i)Slide11
“Fair Share”Overall: n dollars, n items
A reasonable outcome:
each item costs one dollar
This is the unique Walrasian outcome
In this outcome, the revenue of seller j is q
j
We call q
j the “fair share” revenue of seller j.
THM (Friedman 1958) If all sellers use proportional allocation, there is a unique Nash equilibrium of the buyers game whose outcome is the fair share outcome.
Thus, collectively, sellers can implement the fair share outcomeBut, acting strategically, some sellers may use other policies. Can a seller
guarantee her fair share?Slide12
Safety LevelTHM 1: If seller j uses proportional allocation, her revenue in any Nash equilibrium of the resulting buyers game will be strictly larger than qj – 1, regardless of the allocation policies used by the other sellers.
Remarks
:
Proportional allocation guarantees
q
j – 1 but does not guarantee qj (we give an example for that).No strategy can guarantee more than q
j+1We do not know what is the maximal possible guarantee in the interval [ q
j – 1 , qj +
1 ], and what strategy achieves it.Slide13
Extremely Asymmetric SupplyConclusion: a seller with large supply can guarantee almost her fair share, and that’s almost the best she can do (up to an additive +/-1 term).But for a seller with extremely small supply this additive term is very significant.Extreme example
: q
H
= n – ( 1 - 1/n), q
L
= 1 - 1/n The THM is meaningless for LSuppose H uses proportional allocation. By placing all budget with H,
every buyer can obtain at least qH /n > q
LTherefore all buyers will put all their budget with HSlide14
Are Extreme Asymmetries Common?Consider an Example: 800 buyers, 2 sellers, qH = 720, q
L
= 80
On the face of it, supply is not extremely asymmetric (e.g. L can guarantee $79 using proportional allocation).
But, suppose this market is actually divided equally to 100 separate sub-markets
The Internet advertising market is divided to many small markets (based on properties like geo, gender, age, smarter targeting…)Each sub-market
has 8 buyers , qH =72
, qL = 0.8H can offer at least 7.2/8 = 0.9 > 0.8 to each buyer.Thus, H obtains all budgets, leaving L with 0 revenue!Slide15
A special case with two sellersHow extreme should the asymmetries be, for this to happen?The “first” case where the answer is non-trivial is the case of two sellers,
q
H
= n-1, q
L
= 1THM 1 is still meaninglessBut how can H 1 obtain more than her fair share?The answer is not straight-forward, and includes two key ingredients: (i) Budget rigidity
(ii) Elimination of Pure NashSlide16
(i) Budget RigidityA rigid buyer will not split his budget. A rigid seller will accept only rigid buyers.Rigidity can be enforced on the seller’s side by technological and marketing means. It can also originate from the buyer’s side, especially in more traditional markets.
This is also called “single-homing” in the literature
I
t is common even in Internet advertising [Ashlagi et al.]
Remarks
:Proportional allocation with budget rigidity gives the same guarantee as proportional allocationTypically,
rigidity “favors” large sellersExample: q1 = 13; q2
, …, q9 = 1.5. n=25. All fractional surplus goes to seller 1. (Her revenue is 17,
the revenue of every other seller is 1.)Slide17
Back to the special case(Reminder: two sellers, qH = n-1, qL
=
1)
If the H enforces rigidity, multiple equilibria emerge.
In a pure Nash: One buyer goes to the L, all others go to HThis is the fair share outcome. Rigidity does not help…However there are also mixed Nash equilibria, including the natural symmetric equilibrium.
THM 2: If the large seller enforces rigidity, in every mixed Nash equilibrium the revenue of the large seller is O(n – 1/n) while the revenue of the small seller is O(1/n).
Surprising and significant separation of resulting revenueSlide18
(ii) Elimination of pure NashProportional allocation among rigid buyers, with a gamble:If a single buyer is rigid and declares he “gambles”,
he
gets
1
+ O(1/n
) items. If multiple bidder gamble, they get nothing.All other rigid buyers equally split the remaining items.
THM 3:
If H uses this, and regardless of the policy of L, H’s revenue in every Nash equilibrium
is O(n - 1/n).Technically, an allocation policy is a function of n numbers, without extra bits. The “gamble”
can be a bid of 1-.Such allocation is non-monotone (a buyer can get more for less budget); we show that such non-monotonicity is required.Slide19
Additional RemarksRestricting H to be monotone:L cannot guarantee a revenue of 1, even if H is restricted to be monotone.
H cannot
guarantee a revenue of
n-1
if it is restricted to be monotone
.H can guarantee a revenue of n-1-O(1/n2) with a monotone policy.
One way to generalize the case of two extremely asymmetric sellers is to have one large seller and
multiple small sellers. In this case the small sellers’ aggregate supply must very small: THM: If
all sellers in a set S use proportional allocation, their total revenue in any Nash equilibrium of the resulting buyers game will be strictly larger than
(jSqj
)– 1, regardless of the allocation policies used by the other sellers.Slide20
SummaryStudy competition among sellers that cannot adjust supply sizeDetermine allocation based on budgets given to themLike a Blotto game with endogenous winning functions for battlefieldsA
seller that uses proportional allocation can guarantee almost the fair share revenue (the revenue of the Walrasian outcome).
Holds for arbitrary supply sizes and any number of sellers
When differences in supply are extreme, the large seller can drive the revenue of the small seller to be almost zero
Two sellers large and small
Optimal policy uses bid rigidity and elimination of pure Nash
Main open question: how large supply differences must be, for the large seller to exercise such a extra power?Slide21
A bit of political scienceThis model is relevant (among so many other things) tothe question of assigning parliament seats as a function
of the votes
that a party
receives in the elections?
The analogy:
Parties are sellers, votes that a party receives are its supply.Seats are rigid
buyers (each seat has a budget of 1, to be given to the party who is assigned this seat).Pure Nash
of proportional allocation with rigid budgets exactly implements a method invented by D’Hondt (
1878)Well known to favor large partiesUsed until this day in many
European countriesAlready used by Thomas Jefferson (1792)Other methods (e.g. the Hare-Neimeyer
method) can be represented as other natural allocation policies.