On Revenue Maximization for Agents with
Costly Information Acquisition
Extended Abstract
L. Elisa Celis1 , Dimitrios C. Gklezakos2, and Anna R. Karlin2
1
2
[email protected], Xerox Research Centre India
{gklezd, karlin}@cs.washington.edu, University of Washington
Abstract. A prevalent assumption in traditional mechanism design is
that buyers know their precise value for an item; however, this assumption is rarely true in practice. In most settings, buyers can “deliberate”,
i.e., spend money or time, in order improve their estimate of an item’s
value. It is known that the deliberative setting is fundamentally different
than the classical one, and desirable properties of a mechanism such as
equilibria, revenue maximization, or truthfulness, may no longer hold.
In this paper we introduce a new general deliberative model in which
users have independent private values that are a-priori unknown, but can
be learned. We consider the design of dominant-strategy revenue-optimal
auctions in this setting. Surprisingly, for a wide class of environments,
we show the optimal revenue is attained with a sequential posted price
mechanism (SPP). While this result is not constructive, we show how
to construct approximately optimal SPPs in polynomial time. We also
consider the design of Bayes-Nash incentive compatible auctions for a
simple deliberative model.
1
Introduction
In many real-world scenarios, people rarely know precisely how they value an
item, but can pay some cost (e.g., money, time or effort) to attain more certainty.
This not only occurs in online ad markets (where advertisers can buy information
about users), but also in everyday life. Suppose you want to buy a house. You
would research the area, school district, commute, and possibly pay experts such
as a real estate agent or an inspection company. Each such action has some cost,
but also helps better evaluate the worth of the house. In some cases, e.g., if you
find out your commute would be more than an hour, you may simply walk away.
However, if the commute is reasonable, you may choose to proceed further and
take more actions (at more cost) in order to gain even more information. This
continues until you take a final decision. A deliberative agent as defined in this
paper has this kind of multiple-round information-buying capability.
Previous work shows that mechanism design for deliberative agents is fundamentally different than classical mechanism design due to the greater flexibility
in the agents’ strategies. A classical agent has to decide how much information
to reveal; a deliberative agent has to additionally decide how much information
to acquire. This affects equilibrium behavior. For example, in second-price auctions, deliberative agents do not have dominant strategies [20], and standard
mechanisms and techniques do not apply. Revenue-optimal mechanisms have remained elusive, and the majority of positive results are for simple models where
agents can determine their value exactly in one round, often restricting further
to binary values or single-item auctions [2, 5, 3, 24, 7]. Positive results for more
general deliberative settings restrict agents in other ways, such as forcing agents
to acquire information before the mechanism begins [22, 8], to commit to participate before deliberating [13], or to deliberate in order to be served [4]. (Also see
[1, 19, 14].) Furthermore, impossibility results exist for certain types of dominant
strategy3 deliberative mechanisms [20, 21]. This result, however, relies crucially
on the fact that agents are assumed to have the ability to deliberate about other
agent’s values. In an independent value model, this is not a natural assumption.
In this paper we continue a line of research begun by Thompson and LeytonBrown [24, 7] related to dominant strategy mechanism design in the independent
value model. Specifically, we extend results to a general deliberative model in
which agents can repeatedly refine their information. Our first main result is that
the profit maximizing mechanism (a.k.a. optimal mechanism) is, without loss of
generality, a sequential posted price mechanism (SPP).4 Our second main result
is that, via a suitable reduction, we can leverage classical results (see [10, 11,
18]) that show revenue-optimal mechanisms can be approximated with SPPs, in
order to construct construct approximately optimal SPPs in our setting. These
are first results in an interesting model that raises many more questions than it
answers. In the final section, we take first steps towards understanding BayesNash incentive-compatible mechanisms in a simpler deliberative setting.
2
Deliberative Model
In our model, each agent has a set of “deliberative possibilities” that describe
the ways in which they can acquire information about their value.
Definition 1 (Deliberative Possibilities). An agent’s deliberative possibilities are represented by tuples (F , D, c) where
– F is a probability distribution over the possible values the agent can have;
– D is a set of deliberations the agent can perform; and
– c : D → R+ , where c(d) > 0 represents the cost to perform deliberation d.
3
4
Dominant strategy equilibria occur when each agent has a strategy that is optimal
against any (potentially suboptimal) strategies the other agents play.
In a sequential posted price mechanism (SPP), agents are offered take-it-or-leave-it
prices in sequence; the mechanism is committed to sell to an agent at the offered
price if she accepts, and will not serve the agent at any point point if she rejects.
In any given state (F , D, c), the agent may choose one of the deliberations d ∈ D
to perform. A deliberation is a random function that maps the agent to a new
state (F ′ , D′ , c′ ), where F ′ is the new prior the agent has over his value, D′ is
the new set of deliberations the agent can perform, and c′ (·) the corresponding
costs. The distribution over new priors is such that the marginals agree with F ;
i.e., while v ∼ F ′ is drawn from a new distribution (the updated prior), v ∼ d(F )
is identically distributed as v ∼ F . 5
We focus on the design of mechanisms in single-parameter environments [17,
16], where each agent has a single private (in our case, unknown) value for
“service”, and there is a combinatorial feasibility constraint on the set of agents
that can be served simultaneously.6
A mechanism in the deliberative setting is a (potentially) multi-stage process in which the mechanism designer interacts with the agents. It concludes
with an allocation and payment rule (x, p) where xi = 1 if agent i is served
and is 0 otherwise7 and agent i is charged pi .8 The mechanism designer knows
the agents’ deliberative possibilities and initial priors. At any point during the
execution of the mechanism, an agent is free to perform any of her deliberation
possibilities according to her current state (F , D, c). Indeed, it may be in the
mechanism designer’s best interest to incentivize her to do certain deliberations.
We focus in this paper on a public communication model; i.e., every agent
observes the interaction between any other agent and the mechanism. Versions
of our results also extend to the private communication model. We also make
the standard assumption that agents have full knowledge of the mechanism to
be executed. Crucially however, if and when an agent deliberates, there is no
way for other agents or the mechanism to certify that the deliberation occurred.
Moreover, the outcome of a deliberation is always private. Hence, the mechanism designer must incentivize the agent appropriately in order to extract this
information.
If, over the course of the execution, an agent performs deliberations d1 , . . . , dk ,
then her expected utility is
X
xi · E[F k |result of {d1 , . . . , dk } ] − pi −
c(dj ).
1≤j≤k
We assume that agents choose their actions so as to maximize their expected
utility. However, note that it is possible that an agent’s realized utility turns out
to be negative.
5
6
7
8
For example, the initial prior might be U [0, 1], and the agent might be able to
repeatedly determine which of two quantiles her value is in, at some cost. Note
that in general we do not impose any restriction on the type of distributions or
deliberations except, in some cases, that they be finite (see, e.g., Definition 4).
Examples include single-item auctions, k-item auctions, and matroid environments.
Clearly, this allocation must satisfy the feasibility constraints
P of the given singlex ≤ 1.
parameter environment; e.g. in a single item auction, then,
i i
As is standard, the mechanism does not charge agents that are not served. Hence,
pi = 0 when xi = 0.
In classical settings, the revelation principle [15, 23] is a crucial step in narrowing the search for good mechanisms. In our deliberative setting, we cannot
simply apply the revelation principle to “flatten” mechanisms to a single stage,
since the mechanism cannot simulate information gathering on behalf of the
agent. However, a minor variant of the revelation principle developed by Thompson and Leyton-Brown [24] can be shown to also apply to our general setting.
To state it formally, we consider the following type of mechanism:
Definition 2 (Simple Deliberative Mechanism9 ). A simple deliberative mechanism (SDM) is a multi-stage mechanism where at each stage either
– a single agent (or set of agents) is asked to perform and report the result of
a specified deliberation10 ,
– the mechanism outputs an allocation and payment rule (x, p). (Note that
allocation can be made to agents that were never asked to deliberate.)
Note that for deterministic dominant strategy mechanisms, SDMs can interact
with a single agent at a time and restrict interactions to soliciting and receiving
the results of their deliberations without loss of generality.
A strategy of a deliberative agent against an SDM consists of either
1. not deliberating and reporting a result rb,
2. performing each requested deliberation and reporting a result rb (which may
depend on the real result r of the deliberation), or
3. performing other or additional deliberation(s) and reporting a result rb (which
may depend on the real result(s) r of the deliberation(s)).
Definition 3. A truthful strategy always takes option (2) and reports rb = r,
the true result of the deliberation. A truthful SDM is one in which truthtelling is a dominant strategy for every agent. That is, no matter how other
agents behave, it is in her best interest to execute the truthful strategy.
A crucial step in narrowing the search for good mechanisms is the Revelation
Principle. A version for simple deliberative agents generalizes to our setting
without complication.
Lemma 1 (Revelation Principle [24]). For any deliberative mechanism M
and equilibrium σ of M, there exists a truthful SDM N which implements the
same outcome as M in equilibrium σ.
3
Revenue Domination by SPPs
Theorem 1 (SPPs are Optimal) Any deterministic truthful mechanism M
in a single-parameter deliberative environment is revenue-dominated by a sequen9
10
This definition was given in [24] under the name Dynamically Direct Mechanism.
The deliberation the agent is asked to perform must be one of her current deliberative
possibilities, assuming she did as she was told up to that point.
tial posted price mechanism M′ under the assumption that M does not exploit
indifference points.11
In other words, any optimal mechanism can be transformed into a revenueequivalent SPP.
We use the following lemma which extends results from [7, 24] to our more
general setting. We omit the proof, which is a straightforward extension.
Lemma 2 (Generalized Influence Lemma).
Consider a truthful SDM M, and a bounded agent who has performed t deliberations resulting in a prior distribution F (t) . Let S = support(F (t) ). If the
agent is asked to perform deliberation d, then there exist thresholds Ld , Hd ∈ S
such that
– If an agent deliberates and reports value ≤ Ld she will not be served.
– If an agent deliberates and reports value ≥ Hd she will be served.
– It is possible that the deliberation results in a value ≤ Ld or ≥ Hd .
Note that “value” above refers to an agent’s effective value E[F (t+1) ] where
F (t+1) ∼ d(F (t) ).
We now show that at the time the agent is first approached, the price she is
charged does not depend on the future potential actions of any agent.
Lemma 3. Let M be a dominant-strategy truthful SDM for deliberative agents.
If M asks an agent to deliberate, we can modify M so that there is a single
price p determined by the history before M’s first interaction with this agent.
Moreover, the agent will win if her effective value is above p, and lose if it is
below p. The modification preserves truthfulness and can only improve revenue.
Proof. Let M be an SDM as above. Note that M can be thought of as a tree
where at each node the mechanism asks an agent to perform a deliberation (say
d). Each child corresponds to a potential result reported by the agent, at which
point, the mechanism either probes an (potentially the same) agent or terminates
at a leaf with an allocation and payment rule. With some abuse of notation, we
say an agent reports a value in H if her value is above Hd , reports a value in L
if her value is below Ld , and reports a value in M otherwise.
Consider an execution of M in which agent i is asked to deliberate. Consider
the path of execution.We show that we can modify M without any loss to revenue
so that after i is first asked to deliberate, the price at which she gets the item
(assuming she is served) is effectively fixed and does not depend on the rest of
the path.
Firstly, note that if an agent i reports a value in H, from Lemma 2, then
she is served. Assume that i determines her value is in H after deliberation.
11
I.e., if the agent is indifferent between receiving and not receiving the item, the
mechanism commits to either serving or not serving this agent without probing
other agents.
Consider the case where there is only one possible value i can report in H and
she is not asked to deliberate again after this point. If this can result in multiple
possible prices, the difference must depend only on the behavior of other agents,
and is not intrinsic to i. In fact, all such prices must be acceptable to her due to
truthfulness, since, for a fixed set of strategies the other agents take, it should not
be the case that she prefer to lie and say her value is in L and hence avoid service
at a price that is too high. Therefore, any such mechanism can be modified by
replacing all these prices with the maximum such price.
Otherwise, if there are two different prices, p and p′ , that are be reached
depending on i’s report(s), then this again contradicts dominant-strategy truthfulness. This is due to the fact that an agent’s behavior must be truthful against
any set of fixed strategies for the other agents. Thus, whenever i reports a value
in H, she must charged the same price p.
If agent i is never served when i reports v ∈ M , the proof is complete. Assume
otherwise. Let pm be some price that she is served at along a path in M and let
ph be the price she is charged if she reports a value in Hd . Clearly, if pm > ph
then if i’s value is in M she would have incentive to lie. Additionally, if pm < ph ,
then there is a set of strategies we can fix for the other agents for which i would
again have incentive to lie. Hence, by dominant strategy truthfulness, pm = ph .
We conclude by observing that, by truthfulness, it is straightforward to see
that an agent must be served at the above price p whenever her effective value
is above p (and not served otherwise).
⊓
⊔
Lemma 4. Let M be a truthful SDM such that the price it charges i, assuming
i is served, only depends on the history before M’s first interaction with i. Then,
M is revenue-equivalent to an SPP N .
Proof. Given M, we construct N . Consider any node in the SDM M where
some agent, say i, is asked to deliberate. Let hb
i be the history leading up to this
node. Lemma 3 implies that from this point forward, whenever agent i is served,
she is charged pi . Let N , under the same hb
i , offer price pi to agent i.
Since M is truthful, each deliberation she asks i to perform is in her best
interest. Additionally, there is no deliberation i would like to perform which the
mechanism did not ask her to perform. Let this final effective value be b
vi . Clearly,
by truthfulness, i gets the item if and only if pi ≤ vbi . Similarly, when N offers
her price pi it will be in her interest to take the same sequence of deliberations,
otherwise M was not truthful to begin with. She accepts if and only if pi ≤ vbi ,
matching the scenario under which she accepts M’s offer.
The above holds for any deliberative node. Hence, M and N have the same
expected revenue, completing the proof.
⊓
⊔
Combining Lemmas 4 and 1 concludes the proof of Theorem 1.
4
Approximating Optimal Revenue
While our result above is nonconstructive, it turns out to be easy to construct
approximately-optimal SPPs. The basic approach is simple:
1. For any price p ∈ [0, ∞), determine the utility-maximizing set of deliberations the agent would perform, and the probability α(p) that the agent
accepts this price when she deliberates optimally. We denote the agent’s
optimal expected utility when offered a price of p by u(p).
2. Note that f (p) = 1 − α(p) defines a cumulative distribution function on
[0, ∞].
3. Observe that this implies any SPP has the same expected revenue in the
deliberative setting as it does in the classical setting when agents’ values are
drawn from the distribution v ∼ 1 − α(·).12
4. Use known approximation results [9–11, 18] that show how to derive approximately optimal SPPs in the classical setting to derive an approximately
optimal SPP in the deliberative setting.
We apply this recipe to bounded deliberative agents for which we can efficiently compute the distribution f (p) = 1 − α(p).
Definition 4 (Bounded Deliberative Agent). An deliberative agent is bounded
if the following holds:
1.
2.
3.
4.
Every prior has bounded expectation, i.e, E[F(t) ] < ∞ for all i, t.
Every set of deliberative actions is finite, i.e, D(t) < ∞ for all i, t.
Every deliberative action results in one of finitely-many potential priors F (t) .
There is some finite T such that D(t) = ∅ for all t ≥ T , i.e., no further
deliberation is possible.
We now define a lemma that contains the key insight for this result.
Lemma 5. The probability and utility functions α(p) and u(p) have the following properties:
1. u is piecewise linear and convex.
2. α is a step function and decreasing.
3. If the agents are bounded, α and u can be constructed in polynomial time in
the size of the deliberation tree.
Proof. Consider the deliberative decision-making an agent faces when offered
service at a particular price p. We think of her as alternating between decisions
(deliberate, accept the price or reject it), and receiving the random results of a
deliberation. This process defines a deliberation tree (which is finite, under the
boundedness assumption above).
We call the decision nodes choice nodes, and each has a corresponding deliberation possibilities (F , D, c). From each choice node, the agent can proceed to
an accept or reject leaf. When such a node is selected, the deliberation process
terminates with the agent having effective value equal to the expected value of
his updated prior. Alternately, the agent can proceed to a deliberation node. One
12
Clearly, the expected revenue of this SPP in the classical setting is generally less than
the expected revenue of the optimal mechanism in the classical setting for agents
with values are drawn from these distributions.
such node exists for each potential deliberation d ∈ D, and proceeding to such
a node comes at cost c(d). The strategy for the agent at a choice node consists
of deciding which child to select. Obviously, under optimal play, the child that
yields the maximum expected utility is chosen.
At a deliberation node, the chosen deliberation d is performed. The children
of a deliberation node d are the set of possible (F ′ , D′ , c′ ) that can be returned
when d is applied to F . Note that the agent’s expected utility at a deliberation
node is simply a convex combination of the expected utilities of its children.
Consider a specific deliberation subtree T . Let uT (p) be the agent’s optimal
expected utility conditioned on reaching the root of this subtree. Also let αT (p)
be the probability the agent accepts an offer of p conditioned on reaching the
root of this subtree when she uses an optimal deliberation strategy (from this
point onwards). The lemma states that for any T , uT (p) is a piecewise linear
and convex function of p, αT (p) is a decreasing step function in p, and that both
functions can be constructed in polynomial time in the size of the deliberation
tree. The proof is by induction on the height of the deliberation subtree.
Base Case: The base case is a single node, which is either an accept or reject
node. For an accept node A reached via a sequence of deliberations d1 , . . . , dt
withPfinal prior F (t) , the probability of acceptance is 1 and uA = E[F (t) ] −
t
p − i=1 c(di ). For a reject node, the probability of acceptance is 0 and uR =
Pt
− i=1 c(di ). In this case, all three propositions of the theorem hold trivially.
Inductive step: Let us first consider the claims about u(p). By the inductive
hypothesis, for any tree of height h, the utility function is convex and piecewise
linear. A tree of height h + 1 is constructed by conjoining a set of m trees
{T1 , ..., Tm }, the maximum height of which is h, via a single root. The convexity
of the utility function is immediate from the fact that it is is either a convex
combination of convex functions or the maximum of a set of convex functions.
When taking the convex combination of the children, the utility function of the
root has a breakpoint (a price where the utility function changes slope) for each
breakpoint a child node has, and thus if b(Ti ) is the number of breakpoints in
the utility curvePfor the child Ti , then the total number of breakpoints at the
root is |b(T )| ≤ m
i=1 |b(Ti )|.
When the utility function at the root is the pointwise maximum of the utility
functions
associated with the children, order the set of all breakpoints |b(T )| ≤
Pm
i=1 |b(Ti )| associated with any of the children of the root by their p value.
Within any of these intervals, the utility function is the maximum of m lines,
which can generate at most m − 1Pnew breakpoints. Thus, the total number of
m
breakpoints is at most |b(T )| ≤ m i=1 |b(Ti )|. Inductively, this implies that the
total time to compute the utility function is O(|T |2 ).
Now consider α(p). Let αTi (p) be the probability that the agent will accept
an offer at price p conditioned on reaching the root of Ti . The inductive hypoth′
esis is that αTi (p) = −uTi (p). where we extend the derivative (−u′Ti (p)) to all
breakpoints by right continuity.
If the root of T is a deliberation node, then in each interval in the partition
defined by the breakpoints the slope of the linear function representing uT (p) is
the convex
of the slopes of thePindividual utilities uTi (p). Therefore
Pcombination
′
′
m
m
uT (p) = i=1 q(Ti )uTi (p), and αT (p) = i=1 q(Ti )αTi (p). where q(Ti ) is the
probability of the deliberation
outcome associated with Ti . It then follows from
Pm
′
′
the IH that αT (p) = − i=1 q(Ti )uTi (p) = −uT (p).
When T is a choice node, the acceptance probability for each p is precisely
that associated with the Ti that has maximum utility for that p, and we obtain
′
′
αT (p) = αTi (p) = −uTi (p) = −uT (p)
′
′
′
Finally, by convexity, for p < p′ , we have uT (p ) ≥ uT (p) which, by the
′
′
previous equalities implies that −αT (p ) ≥ −αT (p) or equivalently αT (p ) ≤
αT (p), concluding the proof.
⊓
⊔
The following lemma is then immediate.
Lemma 6. Let M be any SPP in a single-parameter deliberative setting. Then
the expected revenue of M in this deliberative setting is equal to the expected
revenue of M in the classical setting with agents whose values v are drawn from
the distribution F (v) = 1 − α(v).
This gives us a direct connection between the deliberative and classical settings, and, crucially, allows us to apply results from the classical setting. Specifically, using the fact that optimal mechanisms in the classical settings are wellapproximated by SPPs [9–11, 18] and that SPPs are optimal for deliberative
settings, we obtain the following theorem.
Theorem 2 (Constructive SPP 2-Approximation) Consider a collection
of bounded deliberative agents. An SPP that 2-approximates the revenue of optimal deterministic deliberative mechanisms for public communication matroid
environments can be constructed in polynomial time.
5
Bayes-Nash Incentive Compatible Mechanisms
In this section, we consider the design of Bayes-Nash incentive compatible (BIC)
mechanisms in the following simplified setting.
Definition 5. A public communication, single deliberation environment is a setting where an agent’s value is drawn from a distribution with continuous and
atomless density function f (·), and each agent has a single deliberative possibility after which they learn their exact value v.13
Suppose that agent is asked to deliberate at some point during the execution
of the mechanism. Denote by a(v) the probability that an agent receives allocation when her value is v.14 Recall we assume a public communication model,
hence the agent knows the reports of all agents that deliberated before her. Her
13
14
Note that a mechanism will ask an agent to deliberate at most once.
This probability is taken over the values of all agents asked to deliberate at the same
time or later than this agent.
expected payment p(v) can be characterized as in the classical setting [23] and
her utility is u(v) = a(v)v − p(v).
Note that an SDM is Bayes-Nash incentive compatible (BIC) if and only if
each agent’s expected utility is maximized by complying with the requests of the
mechanism whenever other agents are also compliant. We provide a characterization for BIC mechanisms.
Proposition 1. Consider a set of single-parameter agents in a single deliberation enviornment. A simple deliberative mechanism is BIC if and only if for
each agent i:
1. If i is asked to deliberate then
′
(a) Rai (v)R≥ 0, with payment ruleRas in the classical setting.
∞
v
µi
(b) 0
0 ai (x)dx fi (vi )dvi ≥ 0 ai (x)dx + ci .
2. If
R e i is offered the item at price e without being asked to deliberate, then
F (v)dv ≤ ci .
0 i
Proof. For notational simplicity, we present the proof when Fi = F and ci = c
for all i.
Condition 1a: Follows as in the classical setting [23] after deliberation.
Condition 1b: In the last case, we have to consider what the agent could
gain from this action. The utility of an agentR in BNE with allocation probability
v
a(x) that performs a deliberation is u(v) = 0 a(x)dx − c. Given that v ∼ F , the
expected utility ud (v) of an agent that performs a deliberation with cost c is:
Z ∞ Z v
a(x)dx f (v)dv − c.
ud =
0
0
The utility of a player that does not deliberate and reports valuation w is:
Z w
a(x)dx
uE (w) = µa(w) − p(w) = µa(w) − wa(w) +
0
where µ = E[F ]. Observing that uE is maximized at w = µ, and simplifying the
constraint that ud ≥ uE (w), we obtain
Z µ
Z ∞ Z v
a(x)dx f (v)dv ≥
a(x)dx + c
0
0
0
Condition 2: If the mechanism offers the item to the agent and expects her
to take it without deliberation at price e, it must be that her utility uE = µ − e
is greater than the utility she could obtain from deliberating, that is:
Z ∞
Z e
µ−e≥
(v − e)f (v)dv − c which after simplification is
F (v)dv ≤ c.
e
0
⊓
⊔
To give an example, consider a single item auction in the classical setting,
with two agents whose values are drawn uniformly on [0, 1]. Hence, the revenueoptimal mechanism is a Vickrey auction with reserve price 1/2, which achieves an
expected revenue of 5/12. Note that this acution is dominant strategy truthful.
In the deliberative setting this is no longer the case, since that would require
that, ex-post, an agent has “no regrets”. However, a deliberative agent will regret
having paid a deliberation cost c if she ends up losing. It follows though from
1
Proposition 1 that VCG is BIC for for c < 12
. It also follows from condition 2
√
that an agent will take the item without deliberating at a price up to 2c. Thus,
when c = 1/12 − ǫ the following mechanism raises more revenue than VCG:
– Approach√agent 1, ask her to deliberate and report her value v1 . If it is above
p∗ = 1 − 2c, sell her the item at price p.
√
– Otherwise, approach agent 2, and offer her the item at price 2c without
deliberation.
Surprisingly, the problem of designing an optimal mechanisms, even for 2 agents
and a single item, seems to be difficult.
6
Future Work
We view this as very preliminary work in the setting of deliberative environments; numerous open problems remain. In the specific model studied, directions
for future research include understanding other communication models, and the
power of randomization. Beyond dominant strategies, revenue maximization using other solution concepts is wide open. It would also be interesting to study
objectives other than revenue maximization. Finally, it would be interesting to
derive “price of anarchy” style results that compare optimal revenue in deliberative and non-deliberative settings.
Acknowledgements: We would like to thank Kevin Leyton-Brown and Dave
Thompson for introducing us to the fascinating setting of deliberative agents.
References
1. M. Babaioff, R. Kleinberg, and R. P. Leme. Optimal mechanisms for selling information. In 12th International World Wide Web Conference, 2012.
2. D. Bergemann and J. Valimaki. Information acquisition and efficient mechanism
design. Econometrica, 70(3), 2002.
3. D. Bergemann and J. Valimaki. Information acquisition and efficient mechanism
design. Econometrica, 70(3), 2002.
4. S. Bikhchandani. Information acquisition and full surplus extraction. Journal of
Economic Theory, 2009.
5. R. Cavallo and D. C. Parkes. Efficient metadeliberation auctions. In AAAI, pages
50–56, 2008.
6. L. E. Celis, D. C. Gklezakos, and A. R. Karlin. On revenue maximization for
agents with costly information acquisition. homes.cs.washington.edu/~ gklezd/
publications/deliberative.pdf, 2013. Full version.
7. L. E. Celis, A. Karlin, K. Leyton-Brown, T. Nguyen, and D. Thompson. Approximately revenue-maximizing mechanisms for deliberative agents. In Association for
the Advancement of Artificial Intelligence, 2011.
8. I. Chakraborty and G. Kosmopoulou. Auctions with edogenous entry. Economic
Letters, 72(2), 2001.
9. S. Chawla, J. Hartline, and R. Kleinberg. Algorithmic pricing via virtual valuations. In Proc. 9th ACM Conf. on Electronic Commerce, 2007.
10. S. Chawla, J. Hartline, D. Malec, and B. Sivan. Sequential posted pricing and multiparameter mechanism design. In Proc. 41st ACM Symp. on Theory of Computing,
2010.
11. S. Chawla, D. Malec, and B. Sivan. The power of randomness in bayesian optimal
mechanism design. In ACM Conference on Electronic Commerce, pages 149–158,
2010.
12. O. Compte and P. Jehiel. Auctions and information acquisition: Sealed-bid or
dynamic formats? Levine’s Bibliography 784828000000000495, UCLA Department
of Economics, Oct. 2005.
13. J. Cramer, Y. Spiegel, and C. Zheng. Optimal selling mechanisms wth costly
information acquisition. Technical report, 2003.
14. J. Cremer and R. P. McLean. Full extraction of surplus in bayesian and dominant
strategy auctions. Econometrica, 56(6), 1988.
15. A. Gibbard. Manipulation of voting schemes: a general result. Econometrica,
41:211–215, 1973.
16. J. Hartline. Lectures on approximation and mechanism design. Lecture notes,
2012.
17. J. Hartline and A. Karlin. Profit maximization in mechanism design. In N. Nisan,
T. Roughgarden, É. Tardos, and V. Vazirani, editors, Algorithmic Game Theory,
chapter 13, pages 331–362. Cambridge University Press, 2007.
18. R. Kleinberg and S. M. Weinberg. Matroid prophet inequalities. In Symposium on
Theoretical Computer Science, 2012.
19. K. Larson. Reducing costly information acquisition in auctions. In AAMAS, pages
1167–1174, 2006.
20. K. Larson and T. Sandholm. Strategic deliberation and truthful revelation: an
impossibility result. In ACM Conference on Electronic Commerce, pages 264–265,
2004.
21. R. Lavi and C. Swamy. Truthful and near-optimal mechanism design via linear
programming. In Proc. 46th IEEE Symp. on Foundations of Computer Science,
2005.
22. D. Levin and J. L. Smith. Equilibrium in auctions with entry. American Economic
Review, 84:585–599, 1994.
23. R. Myerson. Optimal auction design. Mathematics of Operations Research, 6:58–
73, 1981.
24. D. R. Thompson and K. Leyton-Brown. Dominant-strategy auction design for
agents with uncertain, private values. In Twenty-Fifth Conference of the Association for the Advancement of Artificial Intelligence (AAAI-11), 2011.