Seminar in Auctions and Mechanism Design
Presented by: Miki Dimenshtein & Noga Levy
Based on J. Hartline’s book:
Approximation in Economic Design
Lecture topics
1. Introduction to multi-dimensional environments.
*Item pricing against m-agents auction.
2. Reduction to single dimensional preferences.
3. Lottery pricing.
Background
In previous chapters, we’ve consider agents' types as
single dimensional - agent’ private preference is given by a
single value for receiving a service
We now turn to multi-dimensional environments where the
agents’ preferences are given by a multi-dimensional type
Example – multi-dimensional environment
• a home buyer may have distinct values for different houses on the market.
generally multi-dimensional
• Consider selling m items and n agents
• Each agent has a valuation function that is defined across all
bundles.
I.e.,
if agent i receives bundle S ⊂ {1, ...,m} then he has value
vi(S).
• A generally multi-dimensional auction assigns to agent i , bundle Si
and payment pi.
• Agent i’s utility is given by vi(Si) − pi. This called quasi-linear.
• The mechanism chooses the outcome that maximizes social
surplus and it charges each agent the externality imposed on the
remaining agents
Theorem
For agents with (generally multi-dimensional) quasi-linear
preferences, the surplus maximization mechanism is
dominant strategy incentive compatible and maximizes
the social surplus.
Even though the surplus maximization mechanism is
optimal (for social surplus), it is
sometimes infeasible to run.
We will show that posted-pricing mechanisms
can approximate the optimal social surplus in some
relevant environments (not all of them).
Matching market
•
n agents
and
m items
• Each agent i has a value vij for item j
V13 = 30,000$
V11 = 10,000$
V12 = 20,000$
Matching market (cont.)
• The agents are unit-demand - each wants at most one item
• The items are unit-supply - each can be sold to at most one
agent
• Agent values are drawn independently at random,
vij ∼ Fij .
Item Pricing
We start with only one agent, n = 1.
We need to identify revenue optimal pricings
I.e., identify a pricing p = (p1, . . . , pm) such that:
when the agent buys the item that
maximizesvj − pj → revenue is maximized.
How we do that?
Finding an upper bound on the revenue of an
optimal pricings.
if a pricing approximates this upper bound, it
also
approximates the optimal pricing.
Finding an upper bound
First, notice that instead of:
Single agent, unit-demand preferences,
with m items
We create an environment of m (singledimensional) agents, who each want their specific
the constraint that at most one item, but with
can be served.
single-agent, m-item pricing
Vi ~ F i
V1
V5
V2
V4
V3
Vi ~ F i
single-item, m-agent auction
Only one can be served!
V1
V2
V3
V4
V5
Theorem
For any product distribution F = F1 × · · ·× Fm, the
expected revenue of the optimal single-agent, m-item
pricing when the agent’s values for the items are
drawn from F, is at most that of the optimal singleitem,
m-agent auction when the agents’ values for the item
are drawn from F
Expected revenue of the optimal
single-agent, m-item pricing
≥
V1
V2
V3
V4
V5
Expected revenue of the optimal
single-item, m-agent auction
Pricing problem : seller can only post a
price on each item.
Auction problem: the competition between agents can
drive the price up.
Revenue in the (single-dimensional) auction
environment is an upper bound on the revenue in the
single buyer (multi-dimensional) pricing
environment.
Proof
Any item pricing p can be converted into a single-item auction Ap such
that the expected revenue from the item pricing is at most that of the
auction.
• The auction Ap assigns the item to the agent j that maximizes vj − pj .
• For any fixed values of the other agents, v−j , this allocation rule is
monotone in agent j’s value and therefore ex post incentive compatible. It
is also deterministic.
• Our auction is deterministic, DSIC Mechanism
•
•
So according to 2.18: there is a critical value j for agent j which is the
infimum of values for which the agent wins the auction; the agent pays
exactly this critical value on winning. Of course j ≥ pj
Proof (cont.)
Also, notice that the allocation rule of the auction Ap is identical to the
allocation rule of the pricing p:
* For the pricing the agent chooses the item that maximizes vj − pj
* For the auction the winner is selected to maximize vj −pj .
* The revenue for the pricing is exactly the pj that corresponds to this j
whereas in the auction it is τj which, is at least pj .
Therefore,
the auction Ap obtains at least
revenue of the pricing p.
Now we can say.
Revenue in the (single-dimensional) auction
environment is an upper bound on the revenue in the
(multi-dimensional) pricing environment.
Reminder:
Theorem 4.7.
There exists a threshold strategy such that the
expected prize of the gambler is at least half the
expected value of the maximum prize.
Moreover, one such threshold strategy is the one
where the probability that the gambler receives no
prize is exactly 1/2.
Moreover, this bound is invariant to the tie-breaking
rule.
Theorem 4.9. For any independent, single-item
environment the second-price auction with
a uniform ironed virtual reserve price is a
2-approximation to the optimal auction revenue.
Theorem 4.10. For any independent, single-item
environment a sequential posted pricing
of uniform ironed virtual prices is a 2-approximation to
the optimal auction revenue.
7.2 Reduction: Unit-demand to Single-dimensional
Preferences
Goal: Show a reduction from multi-dimensional unit-demand
preferences to single-dimensional preferences from the perspective
of approximation.
We assume independence of the agents values.
Define a general unit-demand environment:
 n agents, m services.
 Agent i has value 𝑣𝑖𝑗 for service j.
 Indicator x with
1 𝑖𝑓 𝑖 𝑟𝑒𝑐𝑖𝑣𝑒𝑠 𝑠𝑒𝑟𝑣𝑖𝑐𝑒 𝑗
𝑥𝑖𝑗 =
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 C() – cost function for the implicit constraint that each agent
can only receive one service:
𝑥𝑖𝑗 = 𝑥𝑖 ′ 𝑗 = 1
𝑓𝑜𝑟 𝑖 ′ ≠ 𝑖 ℎ𝑎𝑣𝑒 𝑐 𝑥 = ∞
7.2.1 Single-dimensional Analogy
• Definition 7.4 the representative environment for the n agent, m
services unit demand environment given by F and c() is the
single-dimensional environment given by F and c() with nm
single-dimensional agents indexed by coordinates ij.
7.2.2 Upper bound
The restriction that only one representative of each unit demand
agent can be served at once induces competition between
representatives.
Intuitively this competition should result
in an increased revenue in the optimal mechanism for the
representative environment over
the original unit-demand environment.
Theorem 7.5. For any independent, unit-demand environment, the
optimal deterministic mechanism’s revenue is at most that of the
optimal mechanism for the single-dimensional representative
environment.
Vi j ~ Fij
Vi1
V1j
V2j
V3j
V4j
V5j
single-item, nm-agent auction
Vi2
Vi3
Vi4
Vi5
Reminder
Theorem 4.10. For any independent, single-item environment a
sequential posted pricing
of uniform ironed virtual prices is a 2-approximation to the
optimal auction revenue.
Sequential posted pricings, i.e., where the agents arrive in any
order, approximate the optimal multi-agent single-item auction.
7.2.3 Reduction
We will now show a lower bound:
In unit-demand pricing, item is allocated
that 𝑀𝑎𝑥(𝑣𝑗 − 𝑃𝑗 )
In sequential posted pricing ties are broken in worst-case order to
𝑀𝑎𝑥(−𝑃𝑗 )
I.e. worst case is that the first agent arrives and pays the lowest price.
Definition 7.6. A sequential posted pricing is an pricing of
services (specialized) for each agent with the semantics that
agents arrive in any order and take their favorite service that
remains feasible. The revenue of such a pricing is given by the
worst-case ordering.
A sequential posted pricing is given by prices p with𝑝𝑖𝑗 the price
offered to agent i for item j.
After the valuations are realized, the agents arrive in sequence
and take their utility maximizing item that is still feasible.
Example
We assume worst case
Theorem 7.7. The expected revenue of a sequential posted pricing
for unit-demand environments is at least the expected revenue
of the same pricing in the representative single-dimensional
environment.
Proof
When comparing the two environments, in the representative environment
the nm agents can arrive in any order.
In the original environment, an agent arrives an considers the prices on
services ordered by utility.
The set of orders in the representative environment contains the set of
orders in the original.
For worst-case the representative is worse.
Corollary 7.8. If a sequential posted pricing is approximately
optimal in the representative
(single-dimensional) environment it is approximately optimal in
the original (unit-demand) environment.
7.2.4 Instantiation
we need to show that there are good sequential posted pricing
mechanisms for single-dimensional environments.
Here we will give such an instantiation for independent, regular,
matching markets, i.e., where
the services are items, and each item has only one unit of supply.
The representative environment for
matching markets:
• nm agents.
• Agent ij with value 𝑣𝑖𝑗
~𝐹desires
item j.
𝑖𝑗
• For item j and original agent i, at most one representative ij can
win.
• VSM is the optimal mechanism.
𝑉𝑆𝑀
• 𝑞𝑖𝑗 - the probability that VSM serves ij.
𝑉𝑆𝑀
= 𝐹𝑖𝑗−1 (1 −. 𝑞𝑖𝑗𝑉𝑆𝑀 ) 𝑉𝑆𝑀
• 𝑝𝑖𝑗
• 𝑝𝑖𝑗
=
𝐹𝑖𝑗−1
1 − 𝑞𝑖𝑗 𝑓𝑜𝑟 𝑞𝑖𝑗 =
𝑞𝑖𝑗
2
Definition 7.9. For the representative matching market
environments, the simulation prices, p, satisfy
1
𝑝𝑖𝑗 = 𝐹𝑖𝑗−1 1 − Pr 𝑡ℎ𝑒 𝑜𝑝𝑡𝑖𝑚𝑎𝑙 𝑚𝑎𝑐ℎ𝑎𝑛𝑖𝑠𝑚 𝑠𝑒𝑟𝑣𝑒𝑠 𝑖𝑗
2
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑗
Theorem 7.10. For regular distributions in the representative matching market
environment, the sequential posted pricing with the simulation prices p is an 8approximation to the revenue of the optimal mechanism.
Lemma 7.11. For regular distributions in the representative matching market
environment, the expected revenue of the optimal mechanism, VSM, is at most
. 𝑖𝑗 𝑝𝑖𝑗𝑉𝑆𝑀 𝑞𝑖𝑗𝑉𝑆𝑀
Lemma 7.12. For regular distributions in the representative matching market
the expected revenue from the sequential posted pricing of the
1environment,
𝑉𝑆𝑀 𝑉𝑆𝑀
𝑞𝑖𝑗 is at least
𝑖𝑗 𝑝𝑖𝑗 prices
8simulation
Upper Bound
Consider an un-constrained mechanism that allocates to representative ij with probability
at most𝑞𝑖𝑗𝑉𝑆𝑀
𝑉𝑆𝑀
𝑝
Because if regularity the un-constrained mechanism posts price𝑖𝑗
to each representative ij and the expected revenue is𝑖𝑗 𝑝𝑖𝑗𝑉𝑆𝑀 𝑞𝑖𝑗𝑉𝑆𝑀
VSM is the optimal mechanism for the constrained environment and a valid solution to the
un-constrained environment.
Lower Bound
• 𝑞𝑖𝑗 =
• 𝑝𝑖𝑗 ≥
•
𝑞 𝑖𝑗𝑉𝑆𝑀
.
2
𝑝𝑖𝑗𝑉𝑆𝑀since prices increase with a lower selling probability.
The expected revenue from agent ij is𝑞𝑖𝑗 𝑝𝑖𝑗 ≥
𝑉𝑆𝑀 𝑉𝑆𝑀
𝑞𝑖𝑗
𝑝𝑖𝑗
2
Lower Bound
We need to show that the probability that it is feasible to offer service to
representative ij is at least ¼.
Representative ij is last and can be served if none of the other
representatives has be served yet, and no one been served item j.
𝑣𝑖 ′ 𝑗 < 𝑝𝑖𝑗 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖′ ≠ 𝑖
Lower Bound
Show that each of the events happens with probability at least 1/2, since
the events are independent, the probability for both is ¼ .
𝑣𝑖 ′ 𝑗 < 𝑝𝑖𝑗 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖′ ≠ 𝑖
Consider the event
.
𝑞𝑖 ′ 𝑗
The possibility that 𝑣𝑖 ′ 𝑗 ≥ 𝑝is𝑖𝑗a bad event happens with probability
1
𝑉𝑆𝑀
The of all the bad events is bounded by 𝑖 ′ 𝑞𝑖 ′ 𝑗 ≤ 1 → 𝑖 ′ 𝑞𝑖 ′ 𝑗 ≤ 2
Therefore, probability that none of them happen is at least ½.
Lottery Pricing and
Randomized
Optimal
mechanism may notMechanisms
be a deterministic pricing
of items. We now show another mechanism using
pricing of items including pricing randomized outcome.
Lotteries – Giving a price to randomized outcomes.
A lottery is a probability distribution over outcomes.
For instance, for the m = 2 item case, a lottery could
assign
either item 1 or item 2 with probability 1/2 each.
A lottery pricing is then a set of lotteries and prices for
each.
For such a lottery pricing, the agent then chooses the lottery and price that give
the highest utility for his given valuations for the items.
Lottery pricings can give
higher revenue than item pricings
Example
There are 2 items (m=2), 1 agent (n=1)
• The agent’s value for each item is distributed independently and uniformly
from the interval [5,6].
• Set a uniform price of 5.097 for each item. Agent is offered to buy an item at
the price of 5.097
• The agent buys the item he values the most, and also at least 5.097
•
V1
V2
Min price: 5.097
Allocation rule without lottery
Adding the lottery option
•
We now provide another option to agent - buying at price 5.057 a lottery with
probability ½ to get an item.
Without the lottery option if the agent had average value bigger than 5.057 but
no individual value over 5.097, the agent would buy nothing.
V1
V2
Lottery – 5.057
Min price: 5.097 or lottery (price: 5.057)
Allocation rule with lottery with price p’=5.057
Optimal lottery pricing
There are examples where the optimal lottery pricing obtains more
revenue than the optimal single-item auction for the representative
environment.
Unlike representative environment that increased competition to obtain more
revenue than that of the original unit-demand environment. It is not entirely
correct when randomized mechanisms are allowed.
Homework - Exercise 7.2
Prove Theorem 7.5:
For any independent, unit-demand environment, the optimal
deterministic mechanism’s revenue is at most that of the optimal
mechanism for the single-dimensional representative
environment.