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 Herzliya
Motivation: markets for ads
Thousands of
advertisers
B1
B2
B3
Bn
. . .
Abstract 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.
Handful of
publishers
q1
q2
• Motivation: How should
publishers choose
allocation policies to
maximize revenue?
Motivation: markets for ads
This work: Symmetric advertisers, non-symmetric publishers:
n advertisers
B
B
B
B
. . .
m publishers
. . .
q1
qm
The Model
This work: Symmetric buyers, non-symmetric sellers:
n buyers
B
B
B
B
. . .
m sellers
. . .
q1
qm
This model was studied previously:
• Friedman (1958):
– Assumes sellers’ policies are “proportional”:
seller j gives buyer i* (bi*j / ibij)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…
Key Difference
• Previous work: Sellers with fixed exogenous allocation policies
– Makes sense for battlefields, voting districts, etc.
– But publishers are strategic and are free to determine their
own allocation policies
• This 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)
Summary of the game
• Sellers determine allocation policies: given budget assignments,
how much items each buyer receives
• Given allocation policies, buyers partition budgets among sellers
• Resulting 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 information
– Buyers play a Nash eq. (pure or mixed) of the resulting
”buyers game” (after observing the allocation policies).
Convenient Change of Units
B=1
B=1
B=1
. . .
n buyers
m sellers
. . .
q1
qm
i qi = n
“Fair Share”
B=1
B=1
B=1
. . .
n buyers
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 qj
m sellers
. . .
q1
qm
i qi = n
We call qj the “fair share”
revenue of seller j.
“Fair Share”
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)qj
(bij : budget to seller j from buyer i)
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 qj
We call qj the “fair share”
revenue of seller j.
“Fair Share”
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.
Overall: n dollars, n items
• Thus, collectively, sellers can
implement the fair share outcome
In this outcome, the
revenue of seller j is qj
• But, acting strategically, some
sellers may use other policies. Can
a seller guarantee her fair share?
We call qj the “fair share”
revenue of seller j.
A reasonable outcome:
each item costs one dollar
This is the unique
Walrasian outcome
Safety Level
THM 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 qj – 1 but does not
guarantee qj (we give an example for that).
• No strategy can guarantee more than qj+1
• We do not know what is the maximal possible guarantee in
the interval [ qj – 1 , qj + 1 ], and what strategy achieves it.
Extremely Asymmetric Supply
• Conclusion: 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: qH = n – ( 1 - 1/n), qL = 1 - 1/n
– The THM is meaningless for L
– Suppose H uses proportional allocation. By placing all
budget with H, every buyer can obtain at least qH /n > qL
– Therefore all buyers will put all their budget with H
Are Extreme Asymmetries Common?
• Consider an Example: 800 buyers, 2 sellers, qH = 720, qL = 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 = 0.8
– H can offer at least 7.2/8 = 0.9 > 0.8 to each buyer.
• Thus, H obtains all budgets, leaving L with 0 revenue!
A special case with two sellers
• How extreme should the asymmetries be, for this to happen?
• The “first” case where the answer is non-trivial is the case of
two sellers, qH = n-1, qL = 1
– THM 1 is still meaningless
– But 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 Nash
(i) Budget Rigidity
• A 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
– It is common even in Internet advertising [Ashlagi et al.]
Remarks:
• Proportional allocation with budget rigidity gives the same
guarantee as proportional allocation
• Typically, rigidity “favors” large sellers
– Example: 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.)
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 H
– This 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 revenue
(ii) Elimination of pure Nash
Proportional 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.
Additional Remarks
1. Restricting 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.
2. 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 (jSqj)– 1, regardless of the
allocation policies used by the other sellers.
Summary
• Study competition among sellers that cannot adjust supply size
– Determine allocation based on budgets given to them
– Like a Blotto game with endogenous winning functions for battlefields
• A 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?
A bit of political science
• This model is relevant (among so many other things) to
the 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 parties
– Used until this day in many European countries
– Already used by Thomas Jefferson (1792)
• Other methods (e.g. the Hare-Neimeyer method) can be
represented as other natural allocation policies.