Robust Virtual Implementation under
Common Strong Belief in Rationality∗
Christoph Müller†
Carnegie Mellon University
October 18, 2013
Abstract
Bergemann and Morris (2009b) show that static mechanisms cannot robustly virtually
implement any non-constant social choice function if preferences are sufficiently interdependent. Without any knowledge of how agents revise their beliefs this impossibility
result extends to dynamic mechanisms. In contrast, we show that if the agents revise
their beliefs according to the forward induction logic embedded in strong rationalizability,
admitting dynamic mechanisms leads to considerable gains. We show that all ex-post incentive compatible social choice functions are robustly virtually implementable in private
consumption environments satisfying a weak sufficient condition, including in all auction
environments with generic valuation functions, regardless of the level of preference interdependence. This result derives from the key insight that in such environments, in
any belief-complete type space under common strong belief in rationality (Battigalli and
Siniscalchi, 2002), dynamic mechanisms can distinguish all payoff type profiles by their
strategic choices. Notably, dynamic mechanisms can robustly virtually implement the
efficient allocation of an object even if static mechanisms cannot.
Keywords: Strategic distinguishability, common strong belief in rationality, extensive form
rationalizability, strong rationalizability, robust implementation, virtual implementation.
∗
I thank my advisor Kim-Sau Chung for his dedication and encouragement and for his invaluable guidance
throughout this project, and David Rahman, Itai Sher and Jan Werner for their continuous support. I also
thank an anonymous associate editor and two referees for their excellent comments and suggestions, Pierpaolo Battigalli, Dirk Bergemann, Narayana Kocherlakota, Eric Maskin, Konrad Mierendorff, Stephen Morris,
Guillermo Ordoñez, Alessandro Pavan, Andrés Perea and Christopher Phelan for helpful discussions and suggestions, and the participants of the Workshop on Information and Dynamic Mechanism Design (HIM, Bonn),
the SED Annual Meetings (Istanbul), the ESEM (Barcelona), the 11th SAET Conference (Ancao) and seminar
audiences at ANU, Bocconi, Bonn, CMU, Maastricht, Minnesota, Oxford, Penn State, Pitt, Royal Holloway,
Sabanci and Western Ontario. Financial support through a Doctoral Dissertation Fellowship of the Graduate
School of the University of Minnesota is gratefully acknowledged. All errors are my own.
†
Contact: cmueller@cmu.edu
1
1
Introduction
Through their stringent necessary conditions for implementation, many results from a growing
literature on robust implementation reveal an unwelcome trade-off for mechanism designers:
Either choose one of the comparatively few social choice functions that are robustly implementable, or give up robustness. The former choice is severely limiting in many environments
and can even confine the designer to choose among constant social choice functions. The
latter choice entails reverting to non-robust mechanisms which depend on fine details about
the agents’ assumed beliefs and higher order beliefs about private information. Results by
Bergemann and Morris (2009b) (henceforth BM) establish this trade-off for robust virtual implementation by static mechanisms (normal game forms). In this paper, we show that using
dynamic mechanisms (roughly, extensive game forms) and adopting a natural generalization
of BM’s notion of robustness allows designers to escape this trade-off in important cases by
enlarging the set of robustly virtually implementable social choice functions.
Notion of Robustness. Assumptions about the agents’ beliefs and higher order beliefs
about payoff types are usually described by a type space, often implicitly via the assumption
of a common prior over a payoff type space. Motivated by the desire to avoid such assumptions,
BM define robust virtual implementation (rv-implementation) as (full) rationalizable virtual
implementation. This is justified by the well-known equality between the union of all Bayesian
Nash equilibrium strategies across all type spaces on the one hand and the set of rationalizable strategies on the other hand (Brandenburger and Dekel, 1987; Battigalli and Siniscalchi,
2003). If a social choice function is rationalizably implementable, it is (fully) implementable
in Bayesian Nash equilibrium for all type spaces, freeing the designer from the task of guessing
the “correct” type space.1
Instead of anchoring the concept of robust implementation in an equilibrium concept, a
second justification for BM’s definition formulates the notion of implementation directly in
terms of explicit assumptions about the agents’ strategic reasoning. This justification derives
from the characterization of rationalizability by the epistemic condition of rationality and
common belief in rationality (RCBR). Just as the equilibrium based approach, implementation
with respect to RCBR frees the designer from guessing the correct type space, as for any type
space, any strategy that is consistent with “RCBR in the type space” is rationalizable.2 And
1
We are interested in robustness with respect to type spaces describing beliefs about payoff types. Epistemic
results are often stated in terms of “epistemic” type spaces describing interactive beliefs about payoff types and
strategies (see e.g. Battigalli and Siniscalchi, 1999). In what follows, we do not distinguish between these two
notions; rather, it is understood that we are only interested in “epistemic” type spaces that correspond to type
spaces about (initial) beliefs about payoff types and that we tacitly rule out all other “epistemic” type spaces.
2
BM adopt a special case of ∆-rationalizability (Battigalli, 1999, 2003; Battigalli and Siniscalchi, 2003)
called belief-free rationalizability. A strategy is consistent with RCBR precisely if it is belief-free rationalizable
2
since RCBR itself does not impose any restrictions on belief hierarchies about payoff types,
implementation with respect to RCBR can justly be called robust.
The approach of examining implementation with respect to an epistemic condition lends
itself to generalization to dynamic mechanism.3 Generalizing from above, it is natural to
assume that agents are (sequentially) rational and that there is common belief in (sequential)
rationality at the beginning of the mechanism. But with dynamic mechanisms, one also
needs to take a stand on how agents revise their initial beliefs while playing mechanisms,
a decision that did not arise with static mechanisms. We assume that agents revise their
beliefs via Bayesian updating whenever possible. This still leaves open how an agent revises
his beliefs at information sets at which Bayesian updating cannot be applied because they
surprise him, that is, at information sets that should have been reached with probability zero
according to the agent’s earlier beliefs. One can make no assumption about this belief revision,
which corresponds to adopting the epistemic assumption of rationality and common initial
belief in rationality (RCIBR). Alternatively, in the spirit of the logic of backward induction,
one can assume rationality and common belief in future rationality (RCBFR) (Penta, 2013;
Perea, 2013). However, in Müller (2012) we prove that under RCIBR, dynamic mechanisms
cannot rv-implement any more social choice functions than static mechanisms. A similar
result can be proved for RCBFR. Therefore, under these assumptions, designers interested in
rv-implementation gain nothing by admitting dynamic mechanisms.
In this paper, we assume that the agents revise their beliefs according to the epistemic
condition of rationality and common strong belief in rationality (RCSBR) (Battigalli and
Siniscalchi, 2002), where we restrict attention to belief-complete type spaces. An agent strongly
believes in an event if he initially believes in the event, and continues to believe in the event “as
long as possible.” For example, if i strongly believes in j’s rationality then i believes that j is
rational at all information sets, including those that he did not expect to occur, save for those
that can have resulted only from irrational play of j. By revising their beliefs according to
RCSBR, agents engage in forward induction reasoning. We show that in contrast to the results
for RCIBR and RCBFR, this leads to considerable gains from admitting dynamic mechanisms.
(see Battigalli, Di Tillio, Grillo, and Penta, 2011, and references therein). Moreover, a strategy is consistent
with the restrictions on belief hierarchies about payoff types expressed by a given type space and RCBR if
and only if it is interim correlated rationalizable for that type space (Dekel, Fudenberg, and Morris, 2007;
Battigalli, Di Tillio, Grillo, and Penta, 2011). Any strategy that is interim correlated rationalizable for some
type space is also belief-free rationalizable.
3
Another approach to robust implementation in dynamic mechanisms is to generalize the equilibrium based
justification of BM’s definition. Viable approaches emerge from the equivalence of the set of weak perfect
Bayesian equilibrium strategies to the set of weakly rationalizable strategies (Battigalli, 1999, 2003), and the
equivalence of the set of interim perfect equilibrium strategies to the set of backwards rationalizable strategies
(Penta, 2013). The resulting implementation concepts correspond to the notions of implementation under
RCIBR and RCBFR described below.
3
Belief-Completeness. A type space is belief-complete if it admits all belief systems. Restricting attention to belief-complete type spaces allows us to use a result by Battigalli and
Siniscalchi (2002) which shows that in such type spaces, RCSBR is characterized by strong
rationalizability 4 (Battigalli, 1999, 2003). Using their result, we circumvent the formalism of
type spaces. While we implicitly analyze virtual implementation under RCSBR that is robust
with respect to all belief-complete type spaces, we define rv-implementation directly as virtual
implementation under strong rationalizability.
Although without loss of generality in static mechanisms,5 assuming belief-completeness
may be with loss of generality in dynamic mechanisms under RCSBR. In general games,
the set of strategies consistent with RCSBR can be larger in a “small” type space than in a
belief-complete type space. This is a consequence of the non-monotonicity of the strong belief
operator (Battigalli and Siniscalchi, 2002; Battigalli and Friedenberg, 2012). Suppose player
1 moves first and chooses action a, and that a is rational only for his payoff type θ1 . Upon
observing a, in a belief-complete type space, a player 2 that strongly believes in 1’s rationality
infers that 1’s payoff type is θ1 . In a “small” type space, player 2 might simply never be
“allowed” to believe in θ1 . In this case, upon observing a, 2 concludes that 1 is irrational and
forms any new belief about 1’s payoff type, potentially leading to a larger set of strategies
consistent with RCSBR than in the belief-complete type space. Under RCSBR, a “small” type
space can thus affect how agents revise their beliefs.
Focusing on belief-complete type spaces can be interpreted as focusing on cases in which
the mechanism designer and the agents have symmetric information about which payoff type
profiles are conceivable. The notion of robustness corresponding to strong rationalizability
presupposes this symmetry. It is important to note that this notion does not, however, rule
out any beliefs that the agents might initially hold about payoff types (including those that
place probability zero on some of the opponents’ payoff types).
Robust Virtual Implementation and Strategic Distinguishability. BM show how to
rv-implement social choice functions in environments with little preference interdependence.
But BM also point out that if an environment exhibits sufficient preference interdependence,
it is impossible to find a static mechanism that rv-implements any non-constant social choice
function. In such environments, mechanism designers using static mechanisms face the stark
trade-off between robustness and implementability described in the introductory paragraph.
4
Strong rationalizability is a version of Pearce’s (1984) extensive-form rationalizability for incomplete information environments with correlated beliefs such as ours.
5
The “larger” a type space, the larger the number of possible belief hierarchies that it permits, the larger the
set of strategies that are consistent with RCBR and the type space, and the more difficult is implementation
with respect to the type space. Implementation with respect to the “large” belief-complete type spaces (such
as the universal type space) thus implies implementation with respect to all “small” type spaces.
4
BM show that this trade-off in fact transcends the implementation question. A mechanism
implements a social choice function if any play of the mechanism results in the outcome that
the social choice function prescribes for the “true” (but unknown) payoff type profile. Suppose
for a moment that a mechanism implements a social choice function that assigns a different
outcome to each payoff type profile. Then an observer of any play of the mechanism can
infer the “true” payoff type profile from the realized outcome. In other words, it is possible to
tell apart any two payoff type profiles by their strategic choices, or to strategically distinguish
(BM) all payoff type profiles. More generally, if the social choice function can assign the same
outcome to several payoff type profiles, it is necessary (but not sufficient) for implementation
that there exists a mechanism that strategically distinguishes enough payoff type profiles.
BM show that if preferences are sufficiently interdependent it is impossible to find a static
mechanism that strategically distinguishes at least some payoff types of some agent.
Strategic distinguishability captures how much of the payoff-relevant private information
of a group of agents can be robustly revealed, and can be viewed as a multi-person analogue
to the single-person revealed preference theory. We generalize strategic distinguishability to
dynamic mechanisms and show that under our epistemic assumptions there are substantial
gains from considering dynamic mechanisms. We first illustrate this in the context of auction environments or, more precisely, environments with quasilinear preferences and private
consumption that describe the assignment of an object to one of finitely many agents (QPCA
environments). In contrast to the result for static mechanisms, we show that all payoff types of
all agents can be strategically distinguished by dynamic mechanisms in QPCA environments
with generic valuation functions, and therefore (essentially) regardless of the degree of preference interdependence (Proposition 4). We thus show how to successfully harness the forward
induction logic embedded in strong rationalizability to induce agents to robustly reveal their
private payoff-relevant information.
We then use this result to characterize rv-implementation. BM prove that ex-post incentive compatibility (epIC) and robust measurability are necessary and, under an economic
assumption, sufficient for rv-implementation in static mechanisms. A social choice function is
robustly measurable if it assigns the same outcome to any payoff types that are strategically
indistinguishable by static mechanisms. We show that if we admit dynamic mechanisms and
appropriately generalize robust measurability, analogous necessary (Proposition 1) and, with
a minor qualification, analogous sufficient conditions (Proposition 3) emerge. Consequently,
the fact that dynamic mechanisms can strategically distinguish more payoff types immediately implies that they can also rv-implement more social choice functions. In particular, in
QPCA environments, non-constant epIC social choice functions can be rv-implemented by
static mechanisms only if the agents’ preferences are sufficiently little interdependent, but
rv-implemented by dynamic mechanisms (essentially) regardless of the degree of preference
5
interdependence.
Obtaining general results when using strong rationalizability can be challenging due to
the non-monotonicity of the strong belief operator. Nonetheless, we show that strategic distinguishability is well-behaved in an important sense in general environments (Proposition
2). Moreover, we generalize the results described in the previous two paragraphs to private
consumption environments (Corollary 2). In all of our analysis, we rule out badly behaved
mechanisms by restricting attention to finite dynamic mechanisms, and focus on finite payoff
type spaces.6
QPCA Environments. Example 1.1 previews some results in the context of a QPCA
environment with specific valuation functions. The example highlights that under RCSBR,
dynamic mechanisms can robustly virtually allocate a single object in an efficient manner in
many cases in which the degree of preference interdependence prevents static mechanisms to
achieve the same. We tacitly assume belief-completeness.
Example 1.1 (compare BM, Section 3) An object is to be allocated among I < ∞ agents.
Each agent i has a finite payoff type space Θi , {0, 1} ⊆ Θi ⊆ [0, 1], and receives utility
vi (θi , θ−i )qi + ti if the payoff type profile is (θi , θ−i ), where qi is the probability that i will
P
receive the good, ti a monetary transfer and vi (θi , θ−i ) = θi + γ j6=i θj the value of the object
to i. The parameter γ ≥ 0 measures the degree of preference interdependence in the environ1
ment. If γ < I−1
static mechanisms can strategically distinguish all payoff type profiles (see
1
BM, Section 3). But if γ ≥ I−1
neither static mechanisms nor dynamic mechanisms under
RCIBR can strategically distinguish any payoff type profiles. In contrast, Proposition 4 im1
1
plies that for all γ < I−1
and for almost all γ ≥ I−1
, all payoff type profiles are strategically
distinguishable by dynamic mechanisms if there is RCSBR.
1
If γ ≥ I−1
, only constant social choice functions are rv-implementable by static mechanisms or by dynamic mechanisms under RCIBR. This is because only constant social choice
functions are robustly measurable (BM): only constant social choice functions assign the same
outcome to payoff type profiles that are strategically indistinguishable by static mechanisms,
or, equivalently, strategically indistinguishable by dynamic mechanisms under RCIBR. In contrast, under RCSBR, all epIC social choice functions can be rv-implemented in dynamic mechanisms for almost all γ by Proposition 3. This includes in particular the efficient allocation of
the object (a non-constant social choice function), which is epIC if the single-crossing condition
γ < 1 holds.7
6
Finiteness of the payoff type spaces is a standard assumption in the virtual implementation literature,
compare BM, Abreu and Matsushima (1992a,b), Artemov, Kunimoto, and Serrano (2013). BM provide an
example of strategic distinguishability with continuous payoff type spaces.
Q
P
7
More precisely, an allocation rule is a profile (qi ) of functions qi : Θ = Θi → [0, 1] such that Ii=1 qi (θ) =
6
1.1
1.1.1
Detailed Preview of Results
Strategic Distinguishability
Two payoff types of an agent are strategically indistinguishable by a static mechanism if they
have a rationalizable strategy in common. If the common rationalizable strategy is played, the
mechanism designer is unable to draw an inference about the agent’s payoff type. Since the
designer can observe individual strategies, strategic distinguishability can be defined agent by
agent. In a dynamic mechanism the designer observes terminal histories instead of strategies,
and whether or not the designer can infer agent i’s payoff type may depend on which strategies
′
the agents −i follow, and thus indirectly on −i’s payoff types. In particular, if θ−i and θ−i
are two profiles of −i’s payoff types then it is possible that two of i’s payoff types lead to the
′ , but to different
same terminal history for some strongly rationalizable strategy profile of θ−i
terminal histories for all strongly rationalizable strategy profiles of θ−i (see Example 4.1).
This is why we will sometimes need to talk about the strategic distinguishability of payoff
type profiles (θi , θ−i ) rather than payoff types θi .
Generalizing BM’s notion, we call two payoff type profiles strategically indistinguishable if
in any mechanism some terminal history is reached by strongly rationalizable strategy profiles
of both payoff type profiles (and strategically distinguishable otherwise). If such a terminal
history occurs, it is impossible to tell which (if any) of the two payoff type profiles has played the
mechanism. Analogously, we call two payoff types of an agent strategically indistinguishable
if in any mechanism some terminal history is reached by strongly rationalizable strategies of
both payoff types and if in addition, this terminal history is admitted by strongly rationalizable
strategies of some payoff types of the other agents.8
We show in Proposition 2 that strategic distinguishability is well-behaved in the following
sense. We never need to worry about being able to strategically distinguish some payoff type
profiles by a first, but some other payoff type profiles only by a second, different mechanism.
There always exists a maximally revealing mechanism that strategically distinguishes all payoff
type profiles that can be strategically distinguished.
Next, we address in Section 5 which payoff type profiles maximally revealing mechanisms
actually do strategically distinguish. To begin, we focus on QPCA environments. Lotteries
over allocations are available. Preferences are interdependent, meaning that an agent’s valuation for the object can depend on his own and on the other agents’ payoff types. Proposition
4 shows that under a weak sufficient condition, there is a fully revealing mechanism, that is,
1 for all θ ∈ Θ. It is efficient if for each payoff type profile θ and each agent i, qi (θ) > 0 implies vi (θ) =
maxj=1,...,I vj (θ). Any efficient allocation rule is epIC for γ < 1 if combined with generalized VCG transfers,
P
that is, if the transfer to i is ti (θ) = −qi (θ)(γ j6=i θj + maxj6=i θj ) (Dasgupta and Maskin, 2000).
8
If in these definitions we replace “strongly rationalizable” with “weakly rationalizable,” we obtain the notion
of strategic indistinguishability we use in Müller (2012).
7
a mechanism which strategically distinguishes all payoff type profiles. This mechanism has
a simple structure: One after another, agents announce a possible ex-post valuation. Each
agent only moves once.
The sufficient condition of Proposition 4 is that for every agent, the sets of ex-post valuations are disjoint for any two distinct payoff types. In QPCA environments, an ex-post
valuation of a payoff type is any valuation the payoff type can have if he holds a degenerate
belief about the other agents’ payoff types. The sufficient condition holds for almost all valuation function profiles, and merely requires that the set of ex-post valuations be disjoint across
an agent’s payoff types — distinct payoff types can have the same valuation for non-degenerate
beliefs. On top of that, the sufficient condition is not necessary. While there is a measure zero
set of valuation function profiles for which no fully revealing mechanism exists — if a valuation
is an ex-post valuation of two payoff types for the same degenerate belief about the others’
payoff types, then the two payoff types cannot be strategically distinguished (Proposition 5)
— a fully revealing mechanism can exist even if the sets of ex-post valuations of all payoff
types of all agents intersect (Example 5.1).
Finally, we generalize the sufficient condition for the existence of a fully revealing mechanism to private consumption environments that satisfy the economic property (Corollary 2).
It is interesting to note that the mechanism from Corollary 2 would also be fully revealing if
we were to use iterated admissibility instead of strong rationalizability as the solution concept.
1.1.2
Robust Virtual Implementation
We say that a social choice function is robustly implementable if it is fully implementable
under strong rationalizability. A social choice function is robustly virtually implementable
(rv-implementable) roughly if arbitrarily close-by social choice functions are robustly implementable.9
Section 4 examines rv-implementation in general environments in which outcomes are lotteries over a finite set of pure outcomes and agents have expected utility preferences. The necessary (Proposition 1) and sufficient (Proposition 3) conditions reveal a simple relation between
rv-implementability and strategic distinguishability that originates in Abreu and Matsushimas’ (1992b) characterization of (non-robust) virtual implementation in static mechanisms.
Our sufficiency proof builds on the proofs by Abreu and Matsushima (1992a,b).
There are two necessary conditions. First, only social choice functions which assign the
same outcome to strategically indistinguishable payoff type profiles can be rv-implementable.
We call such social choice functions dynamically robustly measurable, or briefly dr-measurable.
9
Replacing “strongly rationalizable”/“strong rationalizability” with “weakly rationalizable”/“weak rationalizability” in the previous sentences yields the implementation concept we study in Müller (2012), and replacing
“dynamic mechanism” with “static mechanism” yields the implementation concept studied by BM.
8
A second necessary condition is ex-post incentive compatibility (epIC), which requires that in
the direct mechanism, truth-telling is a best response for every payoff type of an agent that expects his opponents to tell the truth, regardless of his belief about his opponents’ payoff types.
Ex-post incentive compatibility has been shown to be necessary for robust implementation in
static mechanisms by Bergemann and Morris (2005), and is a strong incentive compatibility
condition.10 Conversely, under an economic assumption, any epIC social choice function that
is strongly dr-measurable is rv-implementable. Strong dr-measurability is a somewhat stronger
version of dr-measurability and requires that for any agent, the social choice function treats
any two strategically indistinguishable payoff types the same.
From our results on strategic distinguishability (Proposition 4 and Corollary 2) we know
that in private consumption environments satisfying the economic property, dr-measurability
and strong dr-measurability are very weak conditions. Indeed, under a sufficient condition
which is for example satisfied by any QPCA environment with generic valuation functions,
all social choice functions are both dr-measurable and strongly dr-measurable, and dynamic
mechanisms can rv-implement a social choice function if and only if it is epIC (Corollaries 1
and 2).
1.2
Related Literature
Robust Virtual Implementation. Carrying out the Wilson doctrine (Wilson, 1987) in
the context of mechanism design, Bergemann and Morris (2005, 2009a,b, 2011), Chung and
Ely (2007) and others examine which social choice functions are implementable if the beliefs
about others’ private information are not common beliefs. Within this literature on robust
mechanism design, Artemov, Kunimoto, and Serrano (2013), BM and the present paper all
require full (and not just partial) implementation but slightly weaken the implementation
concept to virtual, that is, approximate implementation. Our analysis is less general than
that of BM and Artemov, Kunimoto, and Serrano (2013) inasmuch as we restrict attention
to private consumption environments, at least for the purpose of determining which payoff
type profiles are strategically distinguishable. At the same time, our analysis is more general
inasmuch as we admit dynamic mechanisms and not only static ones.
As discussed, BM impose no assumptions on the belief hierarchies over payoff type profiles.
This makes their mechanisms “totally” robust but permits virtual implementation only in
environments with little preference interdependence. Artemov, Kunimoto, and Serrano (2013)
present more permissive sufficient conditions for virtual implementation which do not impose
a limit on the preference interdependence. They achieve this by giving up “total” robustness
and adopting an “intermediately” robust approach. Specifically, they assume that a (finite)
10
On the strength of epIC, see e.g. Jehiel, Meyer-Ter-Vehn, Moldovanu, and Zame (2006), but also Bikhchandani (2006) and Dasgupta and Maskin (2000).
9
subset of first-order beliefs about payoff type profiles is common knowledge. In the present
paper we leave the initial belief hierarchies over payoff type profiles unrestricted and thus
adopt a generalization of BM’s “total” robustness approach to dynamic mechanisms. Despite
maintaining “total” robustness, we still offer sufficient conditions that do not impose a limit
on the preference interdependence.
Robust Dynamic Mechanisms. Bergemann and Morris (2007) point out the potential of
dynamic mechanisms to improve on static mechanisms in complete information environments
(in which bidders commonly know the payoff type profile). They present an ascending price
version of the generalized VCG mechanism and show that it robustly virtually allocates an
object in an efficient manner even if there is so much preference interdependence that a static
mechanism cannot. Bergemann and Morris (2007) focus on robustness solely in terms of
the uncertainty about others’ strategies. Both because they use backward induction as their
solution concept and because they assume complete information, inferences about others’
payoff types, while central to our analysis, are absent from theirs.
Penta (2013) explores robust implementation in incomplete information environments with
multiple stages, but under different epistemic assumptions than those used in the present
paper. He introduces backwards rationalizability, a solution concept that incorporates a belief
revision assumption as expressed by common belief in future rationality at each decision node.
In each stage of his model, an agent learns part of his payoff type and participates in a static
mechanism. As a special case that is similar to our set-up, an agent may learn everything
about his payoff type in the first stage. In this case, the agent subsequently participates in
a sequence of static mechanisms and so de facto in a dynamic mechanism with observable
actions. But due to Penta’s (2013) use of backwards rationalizability, in his model, dynamic
mechanisms cannot strategically distinguish more payoff type profiles than static mechanisms.
Implementation under RCSBR. Our paper is not the first to consider implementation
under RCSBR. One mechanism design problem for which a solution under RCSBR has already
been suggested is King Solomon’s dilemma (see e.g. Perry and Reny, 1999). The dilemma is
to give an object to the one of two agents who values it most, at zero cost. Who values
the object most is assumed to be common knowledge among the agents, an informational
assumption that sets the dilemma apart from the paradigm of “total” robustness. Battigalli
and Siniscalchi (2003, Section 3.2.2) show that a first-price auction with an entry fee and a
preceding opt-in stage solves a specific parametrization of the dilemma if the agents engage in
forward induction reasoning.
10
2
Example
Example 2.1 argues in a simple environment that static mechanisms and, equivalently, dynamic
mechanisms under RCIBR can strategically distinguish less payoff type profiles than dynamic
mechanisms under RCSBR. Again, we tacitly assume belief-completeness.
Example 2.1 There are two agents i ∈ {1, 2} with two conceivable payoff types each, θ̂i ∈
{θi , θi′ }, and four outcomes, w, x, y, z. The utility functions are given in Figure 1. We only say
u1 (·, θ1 , θ2 )
u1 (·, θ1 , θ2′ )
u1 (·, θ1′ , θ2 )
u1 (·, θ1′ , θ2′ )
w
x
y
z
5
1
1
5
1
1
0
1
0
0
2
0
2
0
0
2
u2 (·, θ1 , θ2 )
u2 (·, θ1 , θ2′ )
u2 (·, θ1′ , θ2 )
u2 (·, θ1′ , θ2′ )
w
x
y
z
1
0
0
0
0
1
1
1
3
3
3
0
2
2
2
1
Figure 1: Utility functions
that some information is revealed if it is revealed for all strategy profiles that are consistent
with the epistemic assumption under consideration. For this demanding notion of information
revelation, we claim that neither a) static mechanisms nor b) dynamic mechanisms under
RCIBR can reveal anything about the state of the nature (θ̂1 , θ̂2 ), but that c) the dynamic
mechanism shown in Figure 2 fully reveals the state of nature if there is RCSBR.
1
2
ϑ2
w
ϑ1
ϑ′1
ϑ′2
ϑ2
2
x
y
ϑ′2
z
Figure 2: Mechanism that strategically distinguishes all payoff type profiles
a) To see that static mechanisms cannot reveal anything about the state of nature we show
that all payoff type profiles are strategically indistinguishable by static mechanisms (see BM,
Proposition 1). This is the case if in any static mechanism there is a strategy profile which
can be played by every payoff type profile. In accordance with the robust approach we assume
that a strategy profile can be played if it is consistent with RCBR, where a strategy is rational
for θ̂i if it is a best response to some, arbitrary belief about θ̂−i (and −i’s strategy).
Let S1 and S2 be finite strategy sets for agents 1 and 2, respectively, and Γ : S1 × S2 →
{w, x, y, z} be a static mechanism. To show that some strategy profile can be played by all
11
payoff type profiles, we first construct a strategy profile (s11 , s12 ) which is rational for all payoff
type profiles, or, equivalently, which survives one round of iterated elimination of never-best
responses for all payoff type profiles. Let s11 ∈ S1 be a best response for a payoff type θ1 who
is certain that agent 2’s payoff type is θ2 and that agent 2 will play some arbitrarily chosen
s02 ∈ S2 . Since u1 (·, θ1 , θ2 ) = u1 (·, θ1′ , θ2′ ), s11 must then also be a best response for a payoff type
θ1′ who is certain that agent 2’s payoff type is θ2′ and that agent 2 will play s02 ∈ S2 . Therefore,
s11 is rational for both payoff types of agent 1. Similarly, since u2 (·, θ1 , θ2′ ) = u2 (·, θ1′ , θ2 )
there is a s12 ∈ S2 that is rational for both payoff types of agent 2. Next, we construct
a profile (s21 , s22 ) which is consistent with rationality and mutual belief in rationality for all
(θ̂1 , θ̂2 ), or, equivalently, survives two rounds of iterated elimination of never-best responses
for all payoff type profiles. To that end, simply let s21 be a best response for θ1 against the
degenerate belief in (s12 , θ2 ). Then s21 is also a best response for θ1′ , to the degenerate belief
in (s12 , θ2′ ), and thus rational for both payoff types of agent 1. Moreover, if agent 1 is certain
of (s12 , θ2 ) or of (s12 , θ2′ ), then agent 1 believes that agent 2 is rational. We can derive s22 ∈ S2
analogously. Finally, we can iterate this argument. The iterated elimination procedure stops
in some round k < ∞. The corresponding strategy profile (sk1 , sk2 ) is consistent with RCBR
for every payoff type profile, or, equivalently, rationalizable for every payoff type profile. Since
strong rationalizability equals rationalizability in any static mechanism, formally, (sk1 , sk2 ) is a
terminal history of Γ that is strongly rationalizably reached by all payoff type profiles. This
does not change if we admit lotteries over {w, x, y, z} as outcomes of mechanisms.
b) To see that dynamic mechanisms under RCIBR cannot reveal anything about the state
of nature we show that all payoff type profiles are strategically indistinguishable by dynamic
mechanisms if there is only RCIBR (see Müller, 2012). The argument is essentially as in a):
Take any dynamic mechanism, and let s11 be a sequential best response for a payoff type θ1
whose beliefs are as follows: initially, θ1 is certain that agent 2’s payoff type is θ2 and that
agent 2 will play some arbitrarily chosen s02 . If surprised, θ1 continues to be certain that agent
2’s payoff type is θ2 and believes that agent 2 plays some arbitrarily chosen strategy that
admits the current information set. Then, s11 must also be a sequential best response for a
payoff type θ1′ who at each information set is certain that agent 2’s payoff type is θ2′ and that
agent 2 plays the strategy that θ1 believes in. Since there is also a s12 that is a sequential
best response for both θ2 and θ2′ , we can again iterate the argument to find a strategy profile
(sk1 , sk2 ) that is consistent with RCIBR for every payoff type profile.
c) To show that the dynamic mechanism presented in Figure 2 is fully revealing under
RCSBR we prove that it strategically distinguishes all payoff type profiles. To that end, we
iteratively delete the strategies which are not sequentially rational, not consistent with sequential rationality and mutual strong belief in sequential rationality and so on. This corresponds
to determining the strongly 1-rational, strongly 2-rational, ... strategies as formally defined
12
later in Definition 3. For ease of exposition, instead of saying that any strategy that prescribes
ϑ′2 at history (ϑ′1 ) is a never-best sequential response (or conditionally dominated) for θ2 we
say that action ϑ′2 is never rational (or conditionally dominated) for θ2 at (ϑ′1 ), and so on.11
• It is never rational for θ2 to play ϑ′2 if agent 1 announced ϑ′1 and
it is never rational for θ2′ to play ϑ2 if agent 1 announced ϑ1 .
1
Agent 2
2
ϑ2
u2 (θ1 , θ2 ) u2 (θ1 , θ2′ )
u2 (θ1′ , θ2 ) u2 (θ1′ , θ2′ )
1 0
0 0
ϑ1
ϑ′1
ϑ′2
ϑ2
2
3 3
3 0
0 1
1 1
ϑ′2
2 2
2 1
In the figure above, we replaced the outcomes assigned to the terminal histories of Γ
with agent 2’s payoffs, with θ2 ’s payoffs filling the left and θ2′ ’s payoffs the right columns.
Conditional on reaching the history (ϑ′1 ) playing ϑ′2 is dominated for payoff type θ2 , as
independent of θ2 ’s belief about 1’s payoff type the action ϑ′2 leads to a payoff of 2 while
the action ϑ2 leads to a payoff of 3. Similarly, ϑ2 is dominated for θ2′ conditional on
reaching (ϑ1 ).
No strategies are eliminated for agent 1, so we move on to the next step.
• If agent 1 (strongly) believes in agent 2’s rationality, ϑ′1 is never rational for θ1 .
1
ϑ1
Agent 1
2
ϑ2
u1 (θ1 , θ2 ) u1 (θ1′ , θ2 )
u1 (θ1 , θ2′ ) u1 (θ1′ , θ2′ )
5 1
1 5
ϑ′2
1 0
1 1
ϑ′1
2
ϑ2
0 2
0 0
ϑ′2
2 0
0 2
We now display agent 1’s payoffs, graying out payoffs to which agent 1 assign probability
zero because he is certain that agent 2 does not follow any strategy which was deleted
in the previous step (any sequentially irrational strategy). If we ignore the grayed out
payoffs, then strategy ϑ′1 leads to a payoff of 0 for θ1 and is thus dominated by ϑ1 , which
leads to a payoff of at least 1.
11
For a formal definition of conditional dominance see Shimoji and Watson (1998) or the proof of Proposition
2 in Appendix B.
13
• If agent 2 strongly believes in agent 1’s rationality and 1’s (strong) belief in 2’s rationality,
and if agent 1 announced ϑ′1 , then ϑ2 is never rational for θ2′ .
1
Agent 2
2
ϑ2
u2 (θ1 , θ2 ) u2 (θ1 , θ2′ )
u2 (θ1′ , θ2 ) u2 (θ1′ , θ2′ )
ϑ1
ϑ′1
ϑ′2
ϑ2
2
0 1
1 1
1 0
0 0
3 3
3 0
ϑ′2
2 2
2 1
We switch back to agent 2 and gray out payoffs that correspond to strategies of agent
1 that already were deleted. We see that at this point, independent of his initial belief
about θ̂1 , agent 2 concludes that 1’s payoff type is θ1′ if (ϑ′1 ) is reached. In essence,
RCSBR allows us to predict agent 2’s belief about agent 1’s payoff type at this history.
Therefore, ϑ2 is dominated for θ2′ conditional on reaching (ϑ′1 ). This is the first time
we crucially rely on the forward induction logic embedded in RCSBR. If nothing were
known about how agent 2 revises his beliefs, we could not exclude the case that θ2′ , once
surprised by the action ϑ′1 , believes to face payoff type θ1 (rationalizing ϑ2 at (ϑ′1 )).
• If agent 1 (strongly) believes in ..., ϑ1 is never rational for θ1′ .
1
Agent 1
2
ϑ2
u1 (θ1 , θ2 ) u1 (θ1′ , θ2 )
u1 (θ1 , θ2′ ) u1 (θ1′ , θ2′ )
ϑ1
ϑ′1
ϑ′2
ϑ2
2
1 0
1 1
5 1
1 5
0 2
0 0
ϑ′2
2 0
0 2
Given 1’s beliefs at this stage, telling the truth and announcing ϑ′1 leads to a payoff of
2 for θ1′ and thus dominates ϑ1 , which leads to a payoff of at most 1.
• If agent 2 strongly believes in ..., and if agent 1 announced ϑ1 , then ϑ′2 is never rational
for θ2 .
1
Agent 2
2
ϑ2
u2 (θ1 , θ2 ) u2 (θ1 , θ2′ )
u2 (θ1′ , θ2 ) u2 (θ1′ , θ2′ )
1 0
0 0
14
ϑ1
ϑ′1
ϑ′2
ϑ2
0 1
1 1
2
3 3
3 0
ϑ′2
2 2
2 1
Again, RCSBR allows us to predict agent 2’s belief about agent 1’s payoff type, this
time at the history (ϑ1 ). Agent 2 concludes at (ϑ1 ) that 1’s payoff type is θ1 . Given this
restriction on beliefs, ϑ′2 is dominated for θ2 conditional on reaching (ϑ1 ).
We see that for both payoff types of both agents only truth-telling survives the iterated elimination procedure — only truth-telling is strongly rationalizable. Therefore one can infer
(θ̂1 , θ̂2 ) from observing (ϑ̂1 , ϑ̂2 ), and Γ strategically distinguishes all payoff type profiles. Full
revelation is possible because, by engaging in forward induction, the agents “learn” each others’
payoff types during the course of the mechanism, independently of their initial belief hierarchies about each others’ payoff types. In particular, by the time agent 2 moves, his payoff
types must have different (expected) preferences, as by then agent 2 already “learned” agent
1’s payoff type and has a fixed, degenerate belief about agent 1’s payoff type.
The difference in the revelation properties of static and dynamic mechanisms (under
RCSBR) in Example 2.1 translate to a difference in terms of implementation. Consider
the non-constant epIC social choice function f defined by f (θ1 , θ2 ) = w, f (θ1 , θ2′ ) = x,
f (θ1′ , θ2 ) = y and f (θ1′ , θ2′ ) = z. The mechanism of Figure 2 robustly implements f , but
no static mechanism can robustly, or even robustly virtually implement f (all payoff type profiles are strategically indistinguishable by static mechanisms, hence only constant social choice
functions are rv-implementable by static mechanisms). Proposition 3 will provide a dynamic
mechanism that rv-implements any epIC social choice function in Example 2.1 if lotteries over
{w, x, y, z} are admitted as outcomes.
3
Environment and Preliminaries
There is a finite set I = {1, . . . , I} of at least two agents. Each agent i ∈ I has a nonempty and
finite payoff type space Θi . We let Θ denote the set of payoff type profiles (θ1 , . . . , θI ). More
Q
generally, if (Zi )i∈I is a family of sets Zi , we let Z denote the Cartesian product i∈I Zi .12 It
is also understood that z denotes (z1 , . . . , zI ) whenever zi ∈ Zi for all i ∈ I. If Zi = Ai × Bi for
all i ∈ I, we at times ignore the correct order of tuples and write ((a1 , . . . , aI ), (b1 , . . . , bI )) ∈ Z
for (ai , bi )i∈I ∈ Z.
There is a nonempty and finite set X of pure outcomes; the set of outcomes is the set
P
Y = {y ∈ R#X : y ≥ 0, #X
n=1 yn = 1} of lotteries over X. We let ui (x, θ) denote the von
Neumann-Morgenstern utility that i derives from the pure outcome x if the payoff type profile
is θ ∈ Θ, and, in a slight abuse of notation, ui (y, θ) the expected utility that i derives from
lottery y if the payoff type profile is θ. If m, n ∈ N = {0, 1, . . .} and m > n, then {m, . . . , n}
is the empty set.
12
As an exception to this rule, Hi will denote the set of non-terminal histories at which i is active, and
Q
H 6= i∈I Hi the set of all histories (compare Definition 9).
15
3.1
Mechanisms
A (dynamic) mechanism Γ = hH, (Hi )i∈I , P, Ci is a finite extensive game form with perfect
recall that has no trivial decision nodes. The set of dynamic mechanisms includes the set of
static mechanisms or normal game forms as a proper subset. We relegate most definitions to
Appendix A, but summarize some important notation here. A mechanism’s first component,
H, is a finite set of histories h = (a1 , . . . , an ), which are finite sequences of actions. We
let ∅ denote the initial history and T the set of terminal histories. At non-terminal history
h = (a1 , . . . , an ) ∈ H\T , the agent determined by the player function P : H\T → I chooses
an action from the set {a : (h, a) ∈ H}. Here, (h, a) denotes the history (a1 , . . . , an , a). Once
a terminal history h ∈ T is reached the lottery C(h) ∈ Y obtains as the outcome of the
mechanism. The set Hi partitions the set Hi of all histories at which i moves into information
sets H. Whenever i moves he knows the information set, but not the history he is at.
A strategy si for player i specifies an action for each information set H ∈ Hi . The set of
i’s strategies is Si . The terminal history induced by strategy profile s ∈ S is denoted by ζ(s).
We use the symbol to indicate precedence among histories, and also to indicate precedence
among i’s information sets. We let Si (h) = {si ∈ Si : ∃s−i ∈ S−i , h ζ(s)} be the set of i’s
strategies admitting history h, Si (H) be the set of i’s strategies admitting j’s information set
H, j ∈ I, and S−i (H) be the set of −i’s strategy profiles admitting H. Moreover, Σi = Si ×Θi ,
Σ−i = S−i × Θ−i , Σi (h) = Si (h) × Θi and Σ−i (H) = S−i (H) × Θ−i .
For any J ⊆ I, we let both H((sj , θj )j∈J ) and H((sj )j∈J ) denote the set of histories admitted by (sj )j∈J , and Hi ((sj , θj )j∈J ) = Hi ((sj )j∈J ) = {H ∈ Hi : ∃h ∈ H, h ∈ H((sj )j∈J )}
the set of i’s information sets admitted by (sj )j∈J . For A ⊆ Σ, H(A) denotes the union of
sets H(s, θ) and Hi (A) the union of sets Hi (s, θ), where both times (s, θ) ∈ A. Similarly for
A ⊆ Σi . Finally, we let lh denote the length of history h ∈ H and H =t = {h ∈ H : lh = t} the
set of histories with length t. The sets Hi=t , H ≤t etc. are defined analogously. Combinations
of such notation have the obvious meaning. For example, H =1 (s) is the singleton consisting
of the history of length 1 which lies on the path induced by strategy profile s.
3.2
Beliefs and Sequential Rationality
Player i’s beliefs about his opponents’ strategies and payoff types are captured by a family of
probability measures on Σ−i , with each measure representing i’s belief at one of his information
sets. Player i also holds a belief at the initial history, even if it does not comprise one of his
information sets. Formally, i’s beliefs are indexed by the members of H̄i = Hi ∪{{∅}} and form
a conditional probability system. For later, note that we let H̄i (s−i ) consist of the elements of
H̄i admitted by s−i etc.
16
Definition 1 (Rényi, 1955) A conditional probability system (CPS) on Σ−i is a function
µi : 2Σ−i × H̄i → [0, 1] such that
a) for all H ∈ H̄i , µi (·|H) is a probability measure on (Σ−i , 2Σ−i )
b) for all H ∈ H̄i , µi (Σ−i (H)|H) = 1.
c) for all H, H′ ∈ H̄i , if H′ H then µi (A|H)µi (Σ−i (H)|H′ ) = µi (A|H′ ) for all A ⊆
Σ−i (H).
Condition b) requires that at information set H, agent i places zero (marginal) probability
on any strategy of −i which would have prevented that H occurs. Condition c) says that i
uses Bayesian updating “whenever applicable:” Suppose that H′ H, and that at H′ , agent i
believes that A will happen with probability µi (A|H′ ). The play proceeds and i finds himself
at H. If H was no surprise to him — that is, if µi (Σ−i (H)|H′ ) > 0 — he now believes in A
with probability
µi (A|H′ )
µi (A|H) =
.
µi (Σ−i (H)|H′ )
If, on the other hand, H did surprise him — if µi (Σ−i (H)|H′ ) = 0 —, Bayesian updating
“does not apply” and condition c) allows any µi (A|H) ∈ [0, 1], that is, any new estimate of the
likelihood of A. We let ∆(Σ−i ) denote the set of probability measures on Σ−i and ∆H̄i (Σ−i )
denote the set of conditional probability systems on Σ−i .
Given a CPS µi ∈ ∆H̄i (Σ−i ),
Uiµi (si , θi , H) =
X
ui (C(ζ(s)), θ)µi ((s−i , θ−i )|H)
(s−i ,θ−i )∈Σ−i (H)
denotes agent i’s expected utility if he plays strategy si ∈ Si (H), is of payoff type θi and holds
beliefs µi (·|H).
Definition 2 Strategy si ∈ Si is sequentially rational for payoff type θi ∈ Θi of player i with
respect to the beliefs µi ∈ ∆H̄i (Σ−i ) if for all H ∈ Hi (si ) and all s′i ∈ Si (H)
Uiµi (si , θi , H) ≥ Uiµi (s′i , θi , H).
We let ri : Θi × ∆H̄i (Σ−i ) ։ Si denote the correspondence that maps (θi , µi ) to the set of
strategies that are sequentially rational for payoff type θi with beliefs µi , and ρi : ∆H̄i (Σ−i ) ։
Σi denote the correspondence that maps µi to the set of strategy-payoff type pairs that includes
(si , θi ) if and only if si is sequentially rational for payoff type θi with beliefs µi . For each i ∈ I,
ri and ρi are nonempty-valued.
17
3.3
Strong Rationalizability
Battigalli (1999, 2003) defines strong rationalizability for multi-stage games. We extend his
definition to mechanisms in the obvious way.
Definition 3 For i ∈ I let Fi0 = Σi and Φ0i = ∆H̄i (Σ−i ) and recursively define the set Fik+1
of strongly (k + 1)-rationalizable pairs (si , θi ) for agent i by
Fik+1 = ρi (Φki ),
and the set Φk+1
of strongly (k + 1)-rationalizable beliefs for agent i by
i
o
n
k+1
k+1
6= ∅ ⇒ µi (F−i
|H) = 1 ,
Φk+1
= µi ∈ Φki : ∀H ∈ H̄i Σ−i (H) ∩ F−i
i
T
k
k ∈ N. Finally, let Fi∞ = ∞
k=0 Fi be the set of strongly rationalizable strategy-payoff type
T
∞
k
pairs for player i, and Φ∞
i =
k=0 Φi be the set of strongly rationalizable beliefs for player i.
The strongly rationalizable strategies are determined by iteratively deleting never-best
sequential responses, where it is required that at each of his information sets an agent believes
in the highest degree of his opponents’ rationality that is consistent with the information set
(best-rationalization principle). For convenience, we let Rik (θi ) = {si ∈ Si : (si , θi ) ∈ Fik }
Q
denote the set of strongly (k-)rationalizable strategies for θi ∈ Θi , and Rk (θ) = i∈I Rik (θi ),
k ∈ N ∪ {∞}. The sets Ri∞ (θi ) and Φ∞
i are nonempty for all i ∈ I and θi ∈ Θi .
4
Robust Virtual Implementation
A social choice function f : Θ → Y assigns a desired outcome to each payoff type profile. It
is robustly implementable if there exists a mechanism such that for every payoff type profile
θ, every strategy profile that is strongly rationalizable for θ leads to f (θ). A social choice
function is robustly virtually implementable if it can be robustly approximately implemented
in the following sense, where k · k denotes the Euclidean norm on R#X .
Definition 4 Social choice function f is robustly ε-implementable if there is a mechanism
Γ such that kC(ζ(s)) − f (θ)k ≤ ε for all (s, θ) ∈ F ∞ . Social choice function f is robustly
virtually implementable (rv-implementable) if it is robustly ε-implementable for every ε > 0.
4.1
Necessary Conditions for Robust Virtual Implementation
The robust approach to implementation gives rise to the following incentive compatibility
condition.
18
Definition 5 Social choice function f is ex-post incentive compatible (epIC) if for all i ∈ I,
all θ ∈ Θ and all θi′ ∈ Θi
ui (f (θ), θ) ≥ ui (f (θi′ , θ−i ), θ).
Bergemann and Morris (2005) show that epIC is necessary for robust implementation in
static mechanisms. Roughly, if f is robustly implementable then it is implementable regardless
of what agent i believes about −i’s payoff types. Payoff type θi must never have an incentive
to deviate, and in particular not if he holds the degenerate belief in θ−i . A payoff type θi
with this belief must prefer playing one of his rationalizable strategies and obtaining f (θ) over
imitating payoff type θi′ and obtaining f (θi′ , θ−i ). Generalizing this argument, one can show
that epIC is also necessary for rv-implementation in static mechanisms (BM, Theorem 2), and,
as we will see in Proposition 1, for rv-implementation in dynamic mechanisms.
Admitting dynamic mechanisms does not change the incentive compatibility condition. But
it weakens the second condition necessary for rv-implementation. For rv-implementation in
static mechanisms, this second condition is called robust measurability (BM). We now define
strong dynamic robust measurability and dynamic robust measurability. These conditions
are both weaker than robust measurability and will turn out to be sufficient and necessary,
respectively, for rv-implementation in dynamic mechanisms.
First, we generalize BM’s notion of strategic indistinguishability. We write θi ∼Γi θi′ and say
′ ∈
that θi and θi′ are strategically indistinguishable by the mechanism Γ if there exist θ−i , θ−i
Θ−i such that ζ(s) = ζ(s′ ) for some s ∈ R∞ (θ), s′ ∈ R∞ (θ ′ ). We write θi ∼i θi′ and say that
θi and θi′ are strategically indistinguishable if θi ∼Γi θi′ for every mechanism Γ. Otherwise, we
call θi and θi′ strategically distinguishable.
Definition 6 Social choice function f is strongly dynamically robustly measurable (strongly
dr-measurable) if for all i ∈ I, θi , θi′ ∈ Θi and θ−i ∈ Θ−i , θi ∼i θi′ implies f (θi , θ−i ) = f (θi′ , θ−i ).
Strong dr-measurability is a direct generalization of BM’s robust measurability. Robust
measurability requires that f treats two payoff types the same if no static mechanism strategically distinguishes them, and strong dr-measurability requires this only if no mechanism at
all (static or dynamic) strategically distinguishes them. Proposition 3 will establish strong
dr-measurability as a sufficient condition for rv-implementability.
Strong dr-measurability is, however, not necessary for rv-implementation. Example 4.1 will
illustrate why. To obtain a necessary condition, we further weaken robust measurability by
introducing a notion based on the strategic indistinguishability of payoff type profiles instead
of payoff types. We write θ ∼Γ θ ′ if Γ is a mechanism and ζ(s) = ζ(s′ ) for some s ∈ R∞ (θ),
s′ ∈ R∞ (θ ′ ). We write θ ∼ θ ′ and say that θ and θ ′ are strategically indistinguishable if θ ∼Γ θ ′
for every mechanism Γ. Otherwise, we call θ and θ ′ strategically distinguishable. The binary
relations ∼ and ∼Γ are reflexive and symmetric, but not necessarily transitive.
19
Definition 7 Social choice function f is dynamically robustly measurable (dr-measurable) if
for all θ, θ ′ ∈ Θ, θ ∼ θ ′ implies f (θ) = f (θ ′ ).
Example 4.1 There are two agents i ∈ {1, 2} with two payoff types each, Θi = {θi , θi′ }, and
three pure outcomes, X = {x, y, z}. Player 1 prefers “not z” when he is of payoff type θ1 and
z when he is of payoff type θ1′ :
u1 (x, θ1 , ·) = u1 (y, θ1 , ·) > u1 (z, θ1 , ·)
u1 (z, θ1′ , ·) > u1 (x, θ1′ , ·) = u1 (y, θ1′ , ·)
Player 2 is indifferent between all outcomes unless the payoff type profile is (θ1 , θ2 ), in which
case he favors x, or (θ1 , θ2′ ), in which case he favors y:
u2 (x, θ1 , θ2 ) > u2 (y, θ1 , θ2 ) = u2 (z, θ1 , θ2 )
u2 (y, θ1 , θ2′ ) > u2 (x, θ1 , θ2′ ) = u2 (z, θ1 , θ2′ )
u2 (x, θ1′ , ·) = u2 (y, θ1′ , ·) = u2 (z, θ1′ , ·)
Clearly (θ1′ , θ2 ) ∼ (θ1′ , θ2′ ),13 and therefore θ2 ∼2 θ2′ . The social choice function f given in
θ2
θ2′
θ1
x
y
θ1′
z
z
1
ϑ1
2
ϑ2
x
ϑ′2
ϑ′1
z
y
Figure 3: f (left) and mechanism Γ that rv-implements f (right)
Figure 3 is not strongly dr-measurable because f (θ1 , θ2 ) 6= f (θ1 , θ2′ ) even if θ2 ∼2 θ2′ . But f
is dr-measurable, and indeed robustly and therefore robustly virtually implementable via the
mechanism Γ. This is because in Γ, (ϑ1 , θ1′ ) ∈ Σ1 and (ϑ′1 , θ1 ) ∈ Σ1 are eliminated via strict
dominance — for any payoff type of agent 1, truth-telling strictly dominates lying. Hence if
agent 2 gets to move he infers by forward induction that 1’s payoff type is θ1 . Given that he
believes to face θ1 , any payoff type of 2 strictly prefers truth-telling over lying.
Robust measurability implies strong dr-measurability, which itself implies dr-measurability.
In general, the converse of these implications is not true. Example 4.1 presented a social choice
function that is dr- but not strongly dr-measurable, and a main implication of Section 5 will be
that strong dr-measurability can be considerably weaker than robust measurability. Example
13
Formally, this follows from the proof of Proposition 5 which, if we let i = 2, applies without change to the
current example.
20
1.1 already described some environments in which all social choice functions are strongly drmeasurable, but only constant social choice function robustly measurable.
We prove the following proposition in Appendix B.
Proposition 1 Every rv-implementable social choice function is epIC and dr-measurable.
4.2
Sufficient Conditions for Robust Virtual Implementation
In this subsection, we show that epIC and strong dr-measurability are sufficient for rvimplementation. Like BM and Artemov, Kunimoto, and Serrano (2013), we adapt the logic
behind the mechanisms of Abreu and Matsushima (1992a,b) in constructing our implementing
mechanisms. What is new is that our mechanisms are dynamic. This matters in particular
whenever we need to “combine” several mechanisms in such a way that the combined mechanism strategically distinguishes at least as many payoff type profiles as each of the individual
mechanisms (see the proofs of Propositions 2 and 3). Combining dynamic mechanisms can
change the information available at some information sets as well as which information sets
are admitted by strongly rationalizable strategies, and we need to make sure that this does
not affect which payoff type profiles are being strategically distinguished.
1
First, we let ȳ denote the uniform lottery that assigns probability #X
to all x ∈ X, and
recall the economic property used by BM.
Definition 8 (Economic Property) The economic property is satisfied if there exists a profile of lotteries (zi )i∈I such that, for each i ∈ I and θ ∈ Θ, both ui (zi , θ) > ui (ȳ, θ) and
uj (ȳ, θ) ≥ uj (zi , θ) for all j 6= i.
The economic property allows us to single out and reward agent i. If ȳ is the reference
outcome, we can reward agent i without rewarding any other agent at the same time, namely
by substituting zi for ȳ. The economic property is satisfied in quasilinear environments (we can
reward i by transfering some of −i’s money to i) and in Example 2.1, if we admit lotteries.14
Second, we recall some facts about the environment from BM (their Lemma 2) in Lemma
1, and generalize an observation of BM (their Lemma 4) to dynamic mechanisms in Lemma 2.
Lemma 1 (BM) There exist C0 > 0 and, if the economic property holds, c0 > 0 such that
a) |ui (y, θ) − ui (y ′ , θ)| ≤ C0 for all i ∈ I, y, y ′ ∈ Y and θ ∈ Θ.
b) ui (zi , θ) > ui (ȳ, θ) + c0 for all i ∈ I and θ ∈ Θ. Here, (zi )i∈I are the lotteries from the
economic property.
14
In Example 2.1, the degenerate lotteries placing probability one on w for agent 1 and on z for agent 2
satisfy the requirements of the economic property.
21
The existence of the uniform upper bound on utility differences in part a) of the preceding
lemma follows directly from the finiteness of I, X and Θ, and the existence of the minimal
utility difference between zi and ȳ in part b) from the finiteness of I and Θ. Lemma 2, proved
in Appendix B, implies that in any mechanism, the loss in utility from playing a strategy that
is not strongly (k + 1)-rationalizable is uniformly bounded below. It is a consequence of the
finiteness of Σ−i and Hi , the resulting compactness of Φki , k ∈ N, and the continuity of i’s
expected utility with respect to his belief system.
Lemma 2 For any mechanism Γ there exists ηΓ > 0 such that for any i ∈ I, k ∈ N, (si , θi ) ∈
/
k+1
k+1
k
′
Fi
and µi ∈ Φi there are H ∈ Hi (si ) and si ∈ Ri (θi ) ∩ Si (H) such that
Uiµi (s′i , θi , H) > Uiµi (si , θi , H) + ηΓ .
Third, suppose that f is strongly dr-measurable (and epIC), and that f (θ) 6= f (θ ′ ) and
f (θ̃) 6= f (θ̃ ′ ). Since f is strongly dr-measurable it is dr-measurable. Thus there is a mechanism Γ which strategically distinguishes θ and θ ′ , and a mechanism Γ̃ which strategically
distinguishes θ̃ and θ̃ ′ . Potentially, Γ 6= Γ̃. If we want to rv-implement f , a first step is to find
a single mechanism that strategically distinguishes both θ and θ ′ and θ̃ and θ̃ ′ . Does such a
mechanism exist?
If Γ and Γ̃ are static mechanisms, then letting the agents play Γ and Γ̃ simultaneously
strategically distinguishes all payoff type profiles that are strategically distinguishable by either
of the two mechanisms. For this to be true, it is irrelevant how we weight the outcomes of Γ
and Γ̃. In particular, BM (their Lemma 5) show that if C and C̃ are the outcome functions
of Γ and Γ̃, respectively, then it suffices to let the new, combined outcome function equal
εC + (1 − ε)C̃ for some ε ∈ (0, 1).
If Γ or Γ̃ are genuinely dynamic, letting the agents play the mechanisms simultaneously
becomes impossible and some strategic interdependence between the mechanisms unavoidable.
Inevitably, at some information set an agent chooses after having observed some choices made
in the other mechanism. Such an interdependence generally changes the set of strongly rationalizable strategies. However, if we let the agents play Γ first and Γ̃ second and let ε be
close to 1, then the realized terminal histories remain unaffected despite possible changes in
the strongly rationalizable strategies, and the strategic distinguishability of θ and θ ′ and θ̃
and θ̃ ′ is preserved. This result has surprisingly profound roots and relies on recent work by
Chen and Micali (2011). We make it precise in the proof of the upcoming Proposition 2 (see
Appendix B).
We generalize from the case of two mechanisms and ask whether there is a mechanism which
reveals to the mechanism designer at least as much information about the true payoff type
profile as any other mechanism. Formally, we say that mechanism Γ is maximally revealing if
22
it strategically distinguishes any two payoff type profiles that are strategically distinguishable
by some mechanism, that is, if for all θ, θ ′ ∈ Θ, θ ≁ θ ′ implies θ ≁Γ θ ′ .
Proposition 2 There exists a maximally revealing mechanism.
Armed with the knowledge that a maximally revealing mechanism exists, we can now
establish strong dr-measurability and epIC as sufficient conditions for rv-implementation.
Proposition 3 Suppose the environment satisfies the economic property. Then every ex-post
incentive compatible social choice function that is strongly dr-measurable is rv-implementable.
We conclude this section by proving Proposition 3. Compared with the necessary conditions from Proposition 1, we make two additional assumptions. First, we assume the economic
property in order to guarantee the existence of small punishments and rewards which we use
in constructing our implementing mechanisms. This is much like in Abreu and Matsushima
(1992a,b), Artemov, Kunimoto, and Serrano (2013) and BM. Second, we assume strong drmeasurability instead of dr-measurability. Our mechanisms will resemble a direct mechanism
in which i expects any j 6= i to announce his true payoff type θj or a θ̂j such that θ̂j ∼j θj .
By epIC, truth-telling is a best response for i if i believes that all j 6= i tell the truth. Strong
dr-measurability ensures that truth-telling remains a best response even if i expects some j to
lie by announcing a θ̂j which is strategically indistinguishable from j’s true payoff type θj .
In Section 5, we will consider environments that we call QPCA environments. Any such
environment satisfies the economic property, and if its valuation functions are generic, we
will be able to find a mechanism that strategically distinguishes all payoff type profiles. In
this case, the difference between dr- and strong dr-measurability vanishes (both are trivially
satisfied by every social choice function), the additional assumptions of Proposition 3 have
no bite and an exact characterization of rv-implementation obtains: a social choice function
is rv-implementable if and only if it is epIC (Corollary 1). Maintaing the assumption of the
economic property, we will even be able to generalize this simple characterization to a broad
class private consumption environments (Corollary 2).
Proof. Let f be an epIC and strongly dr-measurable social choice function and Γ∗ =
(H ∗ , (H∗i )i∈I , P ∗ , C ∗ ) be a maximally revealing mechanism. Let δ > 0 and L ∈ N\{0} be such
that
δ2 C0 < δηΓ∗
(1)
and
1
1
C0 < δ2 c0 ,
(2)
L
I
where C0 and c0 are the constants from Lemma 1 and ηΓ∗ the constant from Lemma 2. The
√
following mechanism Γ = (H, (Hi )i∈I , P, C) will turn out to robustly 2(δ + δ2 )-implement
23
f . First, each agent is asked to L times submit his payoff type. An agent can lie, and his l-th
submission can differ from is m-th submission. His submissions are not revealed to the other
agents during the entire mechanism. Second, the agents play Γ∗ . Formally, the set of histories
is15
L ∗
∗
H = {h : h (θ1 , . . . , θI , h∗ ) for some (θi )i∈I ∈ ΘL
1 × . . . × ΘI , h ∈ H }.
The player function is defined by P (θ1 , . . . , θi−1 ) = i and P (θ1 , . . . , θI , h∗ ) = P ∗ (h∗ ) for
i ∈ I and (θ1 , . . . , θI , h∗ ) ∈ H. Agent i’s information sets are Hi∅ = {(θ1 , . . . , θi−1 ) : θj ∈
ΘL
j for j < i} and
∗
∗
[θi , H∗ ] = (θ1 , . . . , θI , h∗ ) : θj ∈ ΘL
,
j for j 6= i, h ∈ H
∗
∗
∀θi ∈ ΘL
i , H ∈ Hi .
∗
∗
In summary, Hi = Hi∅ ∪ [θi , H∗ ] : θi ∈ ΘL
i , H ∈ Hi . The agents’ l-th submissions of
their payoff types determine nearly L1 -th of the outcome via a direct mechanism. Hence, if all
agents truthfully announce their payoff type L times, the outcome of the mechanism virtually
equals the outcome stipulated by f . Because f is strongly dr-measurable this is also true if the
agents lie, but their lies are “inconsequential.” We call payoff type θi ’s lie θ̂i inconsequential
if θi «i θ̂i , where «i denotes the transitive closure of ∼i . If on the other hand θi ffi θ̂i , then
there might exist a θ−i for which f (θi , θ−i ) 6= f (θ̂i , θ−i ) and we call i’s lie “serious.” The small
part of the outcome that is not determined by direct mechanisms incentivizes the agents to not
ever seriously lie in their submissions of their payoff types. This part consists of the outcome
of Γ∗ , and to a smaller extent of some reward terms. More precisely, the outcome function
C : T → Y assigns the lottery
L
C(h) = (1 − δ − δ2 )
1X
1X
ri (h),
f (θ l ) + δC ∗ (h∗ ) + δ2
L
I
i∈I
l=1
to terminal history h = ((θi1 , . . . , θiL )i∈I , h∗ ), where agent i’s reward ri (h) is defined by
ri ((θi1 , . . . , θiL )i∈I , h∗ ) =


ȳ













zi
if ∃m ∈ {1, . . . , L} s.t.
a) h∗ ∈
/ H ∗ (Ri∗,∞ (θ̂i )) for all θ̂i s.t. θ̂i «i θim and
.
b) ∀l ∈ {1, . . . , m − 1}∀j 6= i :
∗,∞
∗
∗
l
h ∈ H (Rj (θ̂j )) for some θ̂j s.t. θ̂j «j θj
otherwise
The reward term ri (h) either punishes agent i with ȳ or rewards him with the lottery zi
guaranteed to exist by the economic property. It punishes i exactly if one of his announcements
θim meets the following two conditions. First, the history h∗ observed in the Γ∗ -part of Γ is
For a finite sequence h with codomain A and a1 , . . . , an ∈ A′ , let (a1 , . . . , an , h) denote the finite sequence
g with codomain A ∪ A′ and length n + lh such that g(k) = ak , k = 1, . . . , n, and g(n + k) = h(k), k = 1, . . . , lh .
15
24
not admitted by any strongly rationalizable strategy of θim or of an inconsequential lie of θim
in the stand-alone mechanism Γ∗ . Second, for all l < m, h∗ is admitted in Γ∗ by some strongly
l or of an inconsequential lie of θ l .
rationalizable strategies of θ−i
−i
The upcoming Lemma 3 implies that Γ strategically distinguishes any two payoff type
profiles that Γ∗ strategically distinguishes, and therefore that Γ is maximally revealing. For
si ∈ Si , let ϕi (si ) ∈ Si∗ be the strategy induced by the Γ∗ -part of si that is not prevented by
si itself. Formally, ϕi : Si → Si∗ satisfies
ϕi (si )(H∗ ) = si ([si (Hi∅ ), H∗ ]),
∀H∗ ∈ H∗i .
For convenience, let ϕ(s) = (ϕ1 (s1 ), . . . , ϕI (sI )) for any s ∈ S.
Lemma 3 For all (s, θ) ∈ F ∞ we have ζ ∗ (ϕ(s)) ∈ H ∗ (R∗,∞ (θ)).
We prove Lemma 3 in Appendix B. To gain some intuition for why the lemma holds,
note that at the time an agent reaches his information sets in the Γ∗ -part of Γ he has already
made the L announcements of his payoff type and can only influence two components of Γ’s
P
outcome: δC ∗ (h∗ ) and the reward terms δ2 I1 i∈I ri (h). Suppose θi is certain that θ−i follows
∗,k−1
extensions of the strategies in R−i
(θ−i ) to Γ. By (1), seeking a more favorable reward by
∗
following a strategy in the Γ -part of Γ that is suboptimal in the stand-alone mechanism Γ∗
can never offset the loss the suboptimal strategy causes in δC ∗ (h∗ ). Hence, θi will only follow
extensions of the strategies in Ri∗,k (θi ). In the Γ∗ -part of Γ, we can eliminate at least as many
strategies from Fi∗,k−1 as in Γ∗ . Maybe we can actually eliminate strictly more strategies, but
we choose not to. This choice can change the set of strategies surviving the iterated removal
of never-best sequential responses, but Chen and Micali (2011) proved that the set of induced
terminal histories remains unaffected. As a consequence, any path of play observed in the
Γ∗ -part of Γ might be played in Γ∗ , as well.
The following lemma shows that as a consequence of Lemma 3, the agents will never
seriously lie.16
Lemma 4 (s, θ) ∈ F ∞ implies si (Hi∅ )l «i θi for all l ∈ {1, . . . , L} and all i ∈ I.
Proof. Suppose the lemma is false. Pick i ∈ I, µi ∈ Φ∞
i and (si , θi ) ∈ ρi (µi ) such that
∅ m
si (Hi ) ffi θi , where m is the minimal element in {1, . . . , L} for which there are i ∈ I and
(ŝi , θ̂i ) ∈ Fi∞ such that ŝi (Hi∅ )m ffi θ̂i .
As a first case, suppose that i believes that none of his opponents will seriously lie, that
is, suppose that for all (s−i , θ−i ) ∈ Σ−i , µi ((s−i , θ−i )|Hi∅ ) > 0 implies sj (Hj∅ )l «j θj for
16
If one uses a mechanism with slightly more complicated rewards, one can even show that the agents will
never announce payoff types that are strategically distinguishable from their true payoff types (see Müller,
2010).
25
all l ∈ {1, . . . , L} and all j 6= i. Then, given the beliefs µi , the strategy s′i defined by
s′i (Hi∅ ) = (θi , . . . , θi ) and s′i ([θi , Hi∗ ]) = si ([si (Hi∅ ), Hi∗ ]) for all [θi , Hi∗ ] gives strictly higher
expected utility to payoff type θi at Hi∅ than si , contradicting (si , θi ) ∈ ρi (µi ): Because
f is epIC and strongly dr-measurable, s′i maximizes the expected utility from (1 − δ −
P
∅ l
17 The strategies s and s′ yield the same expected utility
δ2 ) L1 L
i
i
l=1 f ((ŝi (Hi ) )i∈I ) in Si .
from δC ∗ (ζ ∗ (ϕ(ŝ))). For any profile (s−i , θ−i ) that i expects with strictly positive probability
we have s ∈ R∞ (θ), and Lemma 3 implies that the terminal history induced by both ϕ(s) and
(ϕi (s′i ), (ϕj (sj ))j6=i ) is in H ∗ (R∗,∞ (θ)). Hence for any such (s−i , θ−i ), ri (ζ(s′i , s−i )) = zi while
ri (ζ(s)) = ȳ. Finally, rj (ζ(s′i , s−i )) = rj (ζ(si , s−i )) = zj for any j 6= i and any (s−i , θ−i ) that
i expects with strictly positive probability.
As a second case, suppose that the set
o
n
l ∈ {1, . . . , L} : ∃(s−i , θ−i ) ∈ Σ−i , ∃j 6= i µi ((s−i , θ−i )|Hi∅ ) > 0 and sj (Hj∅ )l ffj θj
is nonempty and n its minimum. Then n ≥ m is the smallest l for which i expects some j 6= i
to seriously lie. Define the strategy s′i by s′i (Hi∅ ) = (θi , . . . , θi , si (Hi∅ )n+1 , . . . , si (Hi∅ )L ) and
s′i ([θi , Hi∗ ]) = si ([si (Hi∅ ), Hi∗ ]) for all [θi , Hi∗ ]. Then given µi , s′i gives strictly higher expected
utility to θi at Hi∅ than si , contradicting (si , θi ) ∈ ρi (µi ): Because f is epIC and strongly drPn−1
measurable, s′i maximizes the expected utility from (1 − δ − δ2 ) L1 l=1
f ((ŝi (Hi∅ )l )i∈I ) in Si .
P
∅ l
Strategy s′i yields the same expected utility as si from (1 − δ − δ2 ) L1 L
l=n+1 f ((ŝi (Hi ) )i∈I ) +
δC ∗ (ζ ∗ (ϕ(ŝ))). Since agent i seriously lies for “first” time “later” (if at all) under s′i than under
si , agent i weakly prefers rj (ζ(s′i , s−i )) over rj (ζ(si , s−i )) for any j 6= i and any s−i ∈ S−i .
With probability p > 0 agent i believes in profiles (s−i , θ−i ) such that sj (Hj∅ )n ffj θj for some
j 6= i. For such (s−i , θ−i ), ri (ζ(s′i , s−i )) = zi and ri (ζ(s)) = ȳ. By Lemma 1 b) and (2), the
expected utility difference between zi and ȳ strictly outweighs the possible expected utility
loss from playing s′i instead of si in the n-th direct mechanism (1 − δ − δ2 ) L1 f ((ŝi (Hi∅ )n )i∈I ).
With probability 1 − p, i expects no j to seriously lie in his n-th announcement of his payoff
type. In this case, as si prescribes a serious lie in the m-th submission and hence leads to
ri (ζ(si , s−i ) = ȳ for any s−i under consideration, agent i expects s′i to lead to no worse reward
than si . Moreover, in this case, because f is epIC and strongly dr-measurable, i believes that
s′i will be at least as good as si with respect to the term (1 − δ − δ2 ) L1 f ((ŝi (Hi∅ )n )i∈I ).
By strong dr-measurability of f , θj′ «j θj for all j ∈ I implies f (θ) = f (θ ′ ), for any
θ, θ ′ ∈ Θ. Therefore, by Lemma 4, (s, θ) ∈ F ∞ implies
kC(ζ(s)) − f (θ)k ≤ δkC ∗ (ζ ∗ (ϕ(s))) − f (θ)k + δ2 k
√
1X
ri (ζ(s)) − f (θ)k ≤ 2(δ + δ2 ),
I
i∈I
We say that s′i maximizes the expected utility from g(ŝ) in Si (for payoff type θi with beliefs µi ), where
P
g : S → R#X , if s′i ∈ arg maxŝi ∈Si (ŝ−i ,θ−i )∈Σ−i ui (g(ŝ), θ)µi ((ŝ−i , θ−i )|H∅i ).
17
26
√
and Γ robustly 2(δ + δ2 )-implements f . Since δ can be chosen arbitrarily small, f is rvimplementable. This completes the proof of Proposition 3.
5
Strategic Distinguishability
In this section we examine how much information a mechanism designer can infer about the
agents’ private payoff types by observing their actions in a mechanism of the designer’s choice.
In short, we ask which payoff type profiles are strategically distinguishable.
BM (Theorem 1) show that two payoff types are strategically distinguishable by static
mechanisms if and only if they are “separable.” Roughly, how many payoff types are separable
depends on the degree of preference interdependence in the environment. If the agents’ preferences are too interdependent, static mechanisms cannot strategically distinguish any payoff
types. In such situations, dynamic mechanisms may help to reveal the agents’ payoff types
without having to give up robustness.
We first illustrate this in Subsection 5.1 by means of environments in which a single good
is to be allocated among the agents. The set of outcomes is
X
Y = {(q, t) = (q1 , . . . , qI , t1 , . . . , tI ) ∈ [0, 1]I × [−B, B]I :
qi ≤ 1},
i∈I
where qi is the probability that agent i receives the good, ti a monetary transfer to agent i
and B > 0. Agent i’s utility is given by ui ((q, t), θ) = vi (θ)qi + ti , where vi : Θ → R is his
valuation function. We call such an environment a QPCA environment (an environment with
quasilinear utility and private consumption, describing an assignment problem). By Proposition 2 there always exists a maximally revealing mechanism which per definition uncovers as
much of the agents’ private information as possible. Proposition 4 will show that in a typical
QPCA environment, “as much information as possible” is indeed tantamount to all private
information: typically, and thus regardless of the degree of preference interdependence, there
exists a mechanism that strategically distinguishes all payoff type profiles. We call such a
mechanism fully revealing.
To construct a mechanism that is fully revealing, we harness the forward induction logic
embedded in RCSBR. Something that can only happen in a dynamic mechanism is that agent
i observes agent j’s action before choosing an action himself. Say j’s action was rational only
for a particular payoff type θj , then we may reasonably expect that i infers j’s payoff type
from observing j’s action, and that subsequently, i holds the degenerate belief in θj . Precisely
this kind of reasoning occurs if i strongly believes in j’s rationality, and in the mechanism of
Proposition 4. We note, however, that this mechanism would also be fully revealing if we were
to use iterated admissibility instead of strong rationalizability as the solution concept.
27
The necessary and sufficient conditions for rv-implementation from Section 4 established a
link between how many payoff type profiles dynamic mechanisms can strategically distinguish
and how many social choice functions they can rv-implement. This link implies that their
greater ability to cope with interdependent preferences allows dynamic mechanisms to rvimplement a wider range of social choice functions than static mechanisms. In fact, we will
conclude in Corollary 1 that in a typical QPCA environment, dynamic mechanisms can rvimplement all epIC social choice functions.
Secondly, in Subsection 5.3 we generalize from QPCA to private consumption environments.
In particular, under the assumption of the economic property, Corollary 2 generalizes the
sufficient condition for the existence of a fully revealing mechanism. The generalized condition
is also sufficient for epIC to characterize rv-implementability.
5.1
A Sufficient Condition in QPCA Environments
Proposition 4 is the main result of the paper: Under a weak sufficient condition, all payoff type
profiles can be strategically distinguished. We prove this result for QPCA environments here,
and generalize it to the more abstract case of private consumption environments in Subsection
5.3. Let Vi (θi ) denote the set {vi (θi , θ−i )|θ−i ∈ Θ−i } of θi ’s ex-post valuations, that is, the set
of θi ’s expected valuations that can arise from degenerate beliefs.
Proposition 4 If a QPCA environment satisfies Vi (θi ) ∩ Vi (θi′ ) = ∅ for all i ∈ I, θi , θi′ ∈ Θi ,
θi 6= θi′ , then there exists a fully revealing mechanism.
We can view a valuation function vi : Θ → R as a point in R#Θ , and a profile of valuation
functions v = (v1 , . . . , vI ) as a point in RI·#Θ . Then the set
n
o
V = v ∈ RI·#Θ : ∀i ∈ I, θi , θi′ ∈ Θi , θi 6= θi′ Vi (θi ) ∩ Vi (θi′ ) = ∅
is open and its complement has Lebesgue measure zero.18 Hence we can call V generic, and
Propositions 3 and 4 imply the following corollary.
Corollary 1 In QPCA environments with generic valuation functions, social choice function
f is rv-implementable if and only if f is ex-post incentive compatible.
The necessary and sufficient conditions for rv-implementation were derived under the assumption that outcomes are lotteries over pure outcomes. In order to deduce Corollary 1, we
verify in Appendix C that the outcome space Y of a QPCA environment as defined above is, for
our purposes, indeed equivalent to a space of lotteries over an appropriately defined set of pure
outcomes. The existence of a fully revealing mechanism implies that all social choice functions
18
The complement of V is a subset of the measure-zero set {v ∈ RI·#Θ : (∃i, j ∈ I, θ, θ′ ∈ Θ)(vi (θ) = vj (θ′ ))}.
28
are strongly dr-measurable and, since QPCA environments also satisfy the economic property
which was assumed in Proposition 3, therefore that epIC characterizes rv-implementability.
We now prove Proposition 4.
S
Proof. For i ∈ I, let Vi = θi ∈Θi Vi (θi ) be the set of agent i’s ex-post valuations, let
ni
1
vm
i denote the m-th smallest element of Vi and let ni = #Vi (so that Vi = {vi , . . . , vi }
and v1i < . . . < vni i). Also, let v0i = v1i − 1. With each mi ∈ {1, . . . , ni } associate the
i
unique τi (mi ) ∈ Θi for which there exists θ−i ∈ Θ−i such that vi (τi (mi ), θ−i ) = vm
i . For all
Q
m ∈ i∈I {1, . . . , ni }, i ∈ I and θi ∈ Θi , let m|0 = ∅ and m|i = (m1 , . . . , mi ), and let
Vi (θi km|i−1 ) = {vi (θi , θ−i ) : θj = τj (mj ) for all j < i, θj ∈ Θj for all j > i}
be the set of θi ’s valuations that can arise from degenerate beliefs given i is certain that the
payoff types of the agents j < i are τ1 (m1 ), . . . , τi−1 (mi−1 ). Note that V1 (θ1 km|0 ) = V1 (θ1 ).
Define the following mechanism Γ = (H, (Hi )i∈I , P, C). Agent 1 moves first and announces
a number m1 ∈ {1, . . . , n1 }. Agent 2 follows by announcing a number m2 ∈ {1, . . . , n2 }, but
is only allowed to choose m2 if vm2 is among his possible ex-post valuations if agent 1’s payoff
type is τ1 (m1 ). And so on. The set of histories is
n
Y
o
i
∈
V
(θ
km|
)
H = h ∈ F : ∃m ∈
{1, . . . , ni } h m and ∀i ∈ I, ∃θi ∈ Θi vm
i
i
i−1
i
i∈I
where F is the set of finite sequences with codomain N.19 The player function is P : H\T → I
such that P (m|i−1 ) = i for all i ∈ I and m ∈ H =I . Every action taken is immediately visible
to all other agents, so that Hi = {{m|i−1 } : m|i−1 ∈ Hi }.
In order to define the mechanism’s outcome function C, we introduce some additional
language. An “option” oi = (qi , ti ) ∈ [0, 1] × [−B, B] for agent i consists of a probability qi
that i wins the good and a transfer ti to i. As Figure 4 illustrates, we represent oi by a line in
a diagram with i’s utility ui = vi qi + ti on the vertical and i’s valuation vi on the horizontal
1
m m
axis. For i ∈ I and m ∈ {1, . . . , ni }, we let om
i = (qi , ti ) ∈ [0, I ] × [−B, B] be as shown in
Figure 5 and o0i = (0, 0). We obtain the following lemma. Its proof and a formal definition of
the options can be found in Appendix B.
Lemma 5 For i ∈ I and m ∈ {1, . . . , ni },
a)
m
l 0
0
vliqm
i + ti < viqi + ti = 0
m
l 0
0
for l ∈ {1, . . . , m − 1} and vliqm
i + ti > viqi + ti = 0 for
l ∈ {m, . . . , ni }. That is, option om
i gives i a strictly negative utility if his valuation is
l
vi, l < m, and a strictly positive utility if his valuation is vli, l ≥ m.
19
We assume that #Θi ≥ 2 for all i ∈ I. In this case, the game form we define cannot have trivial decision
nodes and is thus a mechanism (see Definition 10). The case in which Θi is a singleton for some i ∈ I can be
easily accommodated at the cost of additional notation.
29
ui
qi
1
oi = (qi , ti )
vi
ti
Figure 4: Some option for agent i
ui
oni i
b
v0i
oini−2
o1i
b
v1i
b
...
b
v
oini−1
Figure 5: The options
b)
v
ni−2
i
b
ni−1
i
vni i
vi
o1i , . . . , oni i for agent i
m
m
m l
l
vm
i qi + ti > vi qi + ti for all l ∈ {0, . . . , ni }\{m}.
he strictly prefers om
i to any of his other options.
That is, if i’s valuation is
vm
i then
For i ∈ I, let > order Ni lexicographically: m|i > m′ |i if there is j ∈ {1, . . . , i} such that
mk = m′k for all k < j and mj > m′j . For m ∈ H =I , i, k ∈ I and j ∈ {0, . . . , I}, let m̄(m|j )
be the largest and m(m|j ) be the smallest element m′ in H =I such that m′ |j = m|j . Let
m̄i (m|j ) and mi (m|j ) denote the i-th components of those histories. If mk 6= mk (m|k−1 ), let
m↓ (m|k ) denote the largest element m′ in H =I such that m′ |k−1 = m|k−1 and m′k < mk , and
let m↓i (m|k ) denote the i-th component of m↓ (m|k ).
30
At history m|i−1 , agent i can secure himself oi i i−1 by announcing the smallest number
he can possibly announce, mi (m|i−1 ). Announcing a different mi makes the option he receives
contingent on −i’s future actions mi+1 , . . . , mI . He ends up with option o0i if his ex-post value
vi (τ1 (m1 ), . . . , τI (mI )) associated with the profile of announcements m is strictly smaller than
m̃i
m̃i
mi
i
vm
i , and with option oi if vi (τ1 (m1 ), . . . , τI (mI )) = vi ≥ vi . Consequently, the outcome
function is C : T → Y such that C(m) = (o1 , . . . , oI ), where
m (m|
oi =





i (m|i−1 )
om
i
i
om̃
i
o0i
)
if mi = mi (m|i−1 )
i
.
if vi (τ1 (m1 ), . . . , τI (mI )) = vm̃
i , m̃i ∈ {mi , . . . , ni } and mi 6= mi (m|i−1 )
otherwise
Let us call a strategy of agent i “truthful” if, at any history, the strategy asks i to announce
i
mi only if vm
i arises as an ex-post valuation of i’s true payoff type. In order to show that Γ
is fully revealing, we summarize agent i’s truthful strategies by the set
Fi = (si , θi ) ∈ Σi : ∀m|i−1 ∈ H =i
visi{m|i−1 } ∈ Vi (θi km|i−1 )
and show that F ∞ ⊆ F. For m ∈ NI let
n
o
F(m) = (s, θ) ∈ Σ : ∀m ∈ H =I ∀i ∈ I (si {m|i−1 } = mi and m|i ≥ m|i) ⇒ θi = τi (mi )
summarize the strategy profiles at which all agents are truthful along all histories m ≥ m.
Lemma 6 We have
a) F ∞ ⊆ F(m̄(n1 )).
b) If F ∞ ⊆ F(m1, . . . , mI ),
m ∈ H =I , then F ∞ ⊆ F(m1, . . . , mI−1, 1).
c) If F ∞ ⊆ F(m|i, 1, . . . , 1), m|i ∈ H =i and
mi 6= mi (m|i−1), then F ∞ ⊆ F(m↓ (m|i)).
Proof. a) Suppose (s1 , θ1 ) ∈ ρ1 (µ1 ), µ1 ∈ Φ∞
1 , is such that s1 {∅} = n1 . If agent 1 follows s1
the only two options that he can end up with are o01 and on1 1, and there is λ ∈ ∆({0, n1 }×Θ−1 )
P
such that U1µ1 (s1 , θ1 , {∅}) = l∈{0,n1 },θ−1 ∈Θ−1 (v1 (θ)ql1 + tl1)λ{(l, θ−1 )}. Let s′1 ∈ S1 be such
P
that s′1 {∅} = 1. Then U1µ1 (s′1 , θ1 , {∅}) > θ−1 ∈Θ−1 (v1 (θ)q01 + t01)(margΘ−1 µ1 )(θ−1 ) = 0, as
agent 1 can secure himself o11 by playing s′1 and he strictly prefers o11 to o01 by Lemma 5 a).
Hence (s1 , θ1 ) ∈ ρ1 (µ1 ) implies that
X
l∈{0,n1 },θ−1 ∈Θ−1
(v1 (θ)ql1 + tl1)λ{(l, θ−1 )} >
X
θ−1 ∈Θ−1
By Lemma 5 a), this is possible only if τ1 (n1 ) = θ1 .
31
(v1 (θ)q01 + t01)(margΘ−1 µ1 )(θ−1 ) = 0.
Suppose now that for j < i, (sj , θj ) ∈ Fj∞ and sj {m̄(n1 )|j−1 } = m̄j (n1 ) imply τj (m̄j (n1 )) =
θj . Suppose further that (si , θi ) ∈ ρi (µi ), µi ∈ Φ∞
i , is such that si {m̄(n1 )|i−1 } = m̄i (n1 ). Analogously to above, at {m̄(n1 )|i−1 }, the expected utility from si has to be greater or equal than
the expected utility from any s′i ∈ Si ({m̄(n1 )|i−1 }) such that s′i {m̄(n1 )|i−1 } = mi (m̄(n1 )|i−1 ),
implying that
X
(vi (θ)qli + tli)λ{(l, θ−i )} >
(vi (θ)q0i + t0i )(margΘ−i µi )(θ−i ) = 0
X
θ−i ∈Θ−i
l∈{0,m̄i (n1 )},θ−i ∈Θ−i
for some λ ∈ ∆({0, m̄i (n1 )} × Θ−i ), which can hold only if τi (m̄i (n1 )) = θi .
b) It suffices to show that for any mI ∈ {1, . . . , nI } such that (m|I−1, mI ) ∈ H =I , (sI , θI ) ∈
FI∞ and sI {m|I−1} = mI imply τI (mI ) = θI . But this is immediate from Lemma 5 b),
considering that at {m|I−1}, agent I is certain that −I’s announcements were truthful, µI (S−I ×
{(τ1 (m1), . . . , τI−1 (mI−1))}|{m|I−1}) = 1 for all µI ∈ Φ∞
I , and hence I’s expected valuation is
in VI (θI km|I−1).
c) We show that F ∞ ⊆ F(m|i, 1, . . . , 1), m|i ∈ H =i and mi 6= mi (m|i−1) imply F ∞ ⊆
F(m|i−1, m↓i (m|i), ni+1 + 1, . . . , nI + 1). The claim then follows by an inductive argument
analogous to the one made in the proof of a).
↓
Suppose that (si , θi ) ∈ ρi (µi ), µi ∈ Φ∞
i , is such that si {m|i−1} = mi (m|i), but that
m↓(m| )
τi (m↓i (m|i)) 6= θi . If µi (S−i × {θ−i ∈ Θ−i : vi (θ) ≥ vi i i }|{m|i−1}) = 0 then m↓i (m|i) 6=
mi (m|i−1) and playing si leads to a (weakly) worse expected utility for θi at {m|i−1} than
that of o0i by Lemma 5 a). Since at {m|i−1}, any strategy prescribing mi (m|i) yields a strictly
higher expected utility than that of o0i , a contradiction to the sequential rationality of si
i (m|i)
}|{m|i−1}) > 0, let m̃i be the smallest
vm
i
m↓(m| )
mi
mi
mi ∈ {1, . . . , ni } for which vi ∈ Vi (θi km|i−1) and vi > vi i i , and let s′i ∈ Si ({m|i−1}) be
such that s′i {m|i−1} = m̃i . Using the supposition F ∞ ⊆ F(m|i, 1, . . . , 1),
↓
obtains. If µi (S−i × {θ−i ∈ Θ−i : vi (θ) ≥
Uiµi (s′i , θi , {m|i−1}) =
X
X
(vliqli + tli)µi (S−i × {θ−i }|{m|i−1})
l∈{m̃i ,...,ni } θ−i ∈Θ−i s.t. vi (θ)=vli
+
X
θ−i ∈Θ−i s.t. vi (θ)<v
m̃i
i
(vi (θ)q0i + t0i )µi (S−i × {θ−i }|{m|i−1}),
(3)
where the second sum on the right-hand side equals zero in QPCA environments. If m↓i (m|i) =
mi (m|i−1), this is strictly greater than Uiµi (si , θi , {m|i−1}) (note that in this case Vi (θi km|i−1) ⊆
ni
′
i
{vm̃
i , . . . , vi } and thus si yields the “best possible” expected utility, while by Lemma 5 b)
and τi (mi (m|i−1)) 6= θi the expected utility from si has to be strictly smaller). If m↓i (m|i) 6=
mi (m|i−1) there exists a λ ∈ ∆({0, . . . , ni } × Θ−i ) with the same marginal on Θ−i as µi such
32
that
Uiµi (si , θi , {m|i−1}) =
X
(vi (θ)qli + tli)λ{(l, θ−i )}
l ∈ {0} ∪ {m↓i (m|i), . . . , ni }\{l : vli ∈ Vi (θi km|i−1)},
m̃
θ−i ∈ {θ−i ∈ Θ−i : vi (θ) ≥ vi i }
+
X
(vi (θ)qli + tli)λ{(l, θ−i )}.
l ∈ {0} ∪ {m↓i (m|i), . . . , ni }\{l : vli ∈ Vi (θi km|i−1)},
m̃
θ−i ∈ {θ−i ∈ Θ−i : vi i > vi (θ)}
The first sum is strictly smaller than the first sum in (3) by Lemma 5 b) and the second sum
i
is (weakly) smaller than the second sum in (3) by Lemma 5 a) because {θ−i ∈ Θ−i : vm̃
i >
m↓i (m|i)
vi (θ) ≥ vi
} = ∅. Contradiction to si being sequentially rational.
From Lemma 6 we can conclude that F ∞ ⊆ F(1, . . . , 1) = F. Since Vi (θi km|i−1 ) ⊆ Vi (θi )
for all m|i−1 ∈ Hi and by assumption Vi (θi ) ∩ Vi (θi′ ) = ∅ for all θi , θi′ ∈ Θi , θi 6= θi′ , this
means that Γ strategically distinguishes all payoff type profiles. This completes the proof of
Proposition 4.
It is actually true that F = F ∞ (for a proof, see Müller (2010)), which implies that every
history in the mechanism of Proposition 4 is admitted by strongly rationalizable strategies of
some payoff type profile. All information sets are consistent with “common belief in rationality,”
and at no information set is there only a finite-level mutual belief in rationality.
While generic, the sufficient conditions of Proposition 4 can still be weakened. From the
proof of Proposition 4, it is clear that some payoff types of some agents can share an ex-post
valuation but still be strategically distinguishable. Instead of stipulating that for every agent
no two payoff types share an ex-post valuation, it suffices to require that
• ∃i ∈ I such that Vi (θi ) ∩ Vi (θi′ ) = ∅ for all θi , θi′ ∈ Θi , θi 6= θi′ , and
• ∀mi ∈ {1, . . . , ni } ∃j = j(mi ), j 6= i, such that Vj (θj kmi ) ∩ Vj (θj′ kmi ) = ∅ for all
θj , θj′ ∈ Θj , θj 6= θj′ , and
S
mj
• ∀mi ∈ {1, . . . , ni }∀mj ∈ {1, . . . , nj(mi ) } such that vj(m
∈
θj ∈Θj(mi ) Vj(mi ) (θj kmi )
)
i
′
∃k = k(mi , mj ), k 6= i, j(mi ), such that Vk (θk kmi , mj ) ∩ Vk (θk kmi , mj ) = ∅ for all
θk , θk′ ∈ Θk , θk 6= θk′ , and
• etc.
where Vj (θj kmi ) is defined as in the proof of Proposition 4, and analogously, Vk (θk kmi , mj )
is the set of θk ’s valuations that can arise from degenerate beliefs given k is certain that the
payoff type of i and j are τi (mi ) and τj (mj ). More surprisingly, even these relaxed sufficient
33
conditions are not necessary for the existence of a fully revealing mechanism. In the following
example, all payoff types of all agents share an ex-post valuation. Still, all payoff type profiles
are strategically distinguishable.
1
Example 5.1 Let I ≥ 3, Θi = {0, 1} and vi (θ) = θi + I−1
i ∈ I, the set of ex-post valuations is
Vi (0) = {0,
for payoff type 0 and
P
j6=i θj
for all i ∈ I. Then for any
2
1
,
, . . . , 1}
I −1 I −1
I
I +1
,
, . . . , 2}
I −1 I −1
for payoff type 1. Since Vi (0) ∩ Vi (1) = {1}, even the weakened sufficient conditions for
strategic distinguishability of all payoff type profiles are violated. The mechanism Γ defined in
Appendix D and depicted for the case I = 3 in Figure 6 nonetheless strategically distinguishes
all payoff type profiles. First, truth-telling at every node is strongly rationalizable for every
Vi (1) = {1,
1
1
0
2
1
2
0
3
1
o1.5
1
o1.5
2
o1.5
3
1
3
0
o1.5
1
o1.5
2
o13
1
o1.5
1
o12
o13
0
3
0
1
o11
o12
o0.5
3
o01
o12
o13
Figure 6: Fully revealing mechanism Γ. The options
of Proposition 4.
3
0
o01
o12
o0.5
3
1
o01
o0.5
2
o13
0
o01
o0.5
2
o0.5
3
om
i are defined as in the proof
payoff type of every agent. (Independently of a payoff type’s marginals on the others’ payoff
types at his information sets, truth-telling at every node is a best response to the belief that
everybody else tells the truth at all nodes.) Second, truth-telling at every node is the unique
strongly rationalizable strategy for every payoff type of every agent, as we can see as follows:
• if (sI , 1) ∈ FI∞ then sI {(1, 0, . . . , 0)} = 1, therefore
• if (s1 , 0) ∈ F1∞ then s1 {∅} = 0, therefore
• if (si , θi ) ∈ Fi∞ , i ∈ {2, . . . , I − 1} then sI {(1, h2 , . . . , hi−1 )} = θi for all (h2 , . . . , hi−1 ) ∈
{0, 1}i−2 , and
if (sI , θI ) ∈ FI∞ , then sI {(1, h2 , . . . , hI−1 )} = θi for (h2 , . . . , hI−1 ) 6= (0, . . . , 0), therefore
34
• if (sI , 0) ∈ FI∞ , then sI {(1, 0, . . . , 0)} = 0, therefore
• if (s1 , 1) ∈ F1∞ then s1 {∅} = 1, therefore
• if (si , θi ) ∈ Fi∞ , i ∈ {2, . . . , I}, then sI {(0, h2 , . . . , hi−1 )} = θi for all (h2 , . . . , hi−1 ) ∈
{0, 1}i−2 .
5.2
A Necessary Condition in QPCA Environments
While a fully revealing dynamic mechanism exists for almost all valuation functions, there
are some valuation functions for which some payoff types are strategically indistinguishable.
In particular, if vi is such that for a fixed degenerate belief two payoff types have the same
ex-post valuation, then those two payoff types are strategically indistinguishable.
Proposition 5 For any i ∈ I, θi , θi′ ∈ Θi and θ−i ∈ Θ−i , if vi (θi , θ−i ) = vi (θi′ , θ−i ) then
(θi , θ−i ) ∼ (θi′ , θ−i ).
∞ (θ ). Let δ
Proof. Take any mechanism Γ and pick an arbitrary s−i ∈ R−i
−i
(s−i ,θ−i ) ∈
′
∞
Σ
∆(Σ−i ) denote the degenerate belief in (s−i , θ−i ) and let µi ∈ Φi . Define µi : 2 −i ×H̄i → [0, 1]
by µi (·|H) = δ(s−i ,θ−i ) for H ∈ H̄i (s−i ) and µi (·|H) = µ′i (·|H) for H ∈
/ H̄i (s−i ). Note that µi
′
∞
is a CPS, and, since µi ∈ Φi and all the mass of δ(s−i ,θ−i ) concentrates on strongly rational′
′
izable strategy-payoff type pairs, µi ∈ Φ∞
i . Let si ∈ ri (θi , µi ), then there exists si ∈ ri (θi , µi )
∞ , (s , θ ), (s′ , θ ′ ) ∈ F ∞ and
such that si |Hi (s−i ) = s′i |Hi (s−i ) . In summary, (s−i , θ−i ) ∈ F−i
i i
i i
i
′
ζ(si , s−i ) = ζ(si , s−i ).
5.3
A Sufficient Condition in Private Consumption Environments
In this subsection we generalize the sufficient condition for the existence of a fully revealing
mechanism to environments in which (some) outcomes correspond to tuples (y1 , . . . , yI ) of private components yi , but that do not necessarily share all the structure of QPCA environments.
An environment with a set X̃ of pure outcomes and functions ũi : X̃ × Θ → R, i ∈ I, as the
agents’ von Neumann-Morgenstern utility functions is a private consumption environment if
there are 1) for each i ∈ I a subset Yi of a real topological vector space and a utility function
ui : Yi × Θ → R, 2) a subset Ỹ ′ ⊆ Ỹ of lotteries over pure outcomes and 3) a linear surjection
g = (gi )i∈I : Ỹ ′ → Y such that
ũi (ỹ, θ) = ui (gi (ỹ), θ),
∀i ∈ I, θ ∈ Θ, ỹ ∈ Ỹ ′ ,
and such that for each i ∈ I, there is some interior point of Ỹ ′ that gi maps to an interior
Q
point of Yi .20 In particular, any environment such that X̃ = i∈I X̃i for some sets X̃i with
20
We endow Ỹ ′ ⊆ Ỹ = {y ∈ R#X̃ : y ≥ 0,
P
yn = 1} with the relative Euclidean topology.
35
at least two elements each and
ũi (x̃, θ) = ũi (x̃′ , θ)
∀i ∈ I, θ ∈ Θ, x̃, x̃′ ∈ X̃ s.t. x̃i = x̃′i ,
is a private consumption environment. Moreover, any QPCA environment is a private consumption environment.21
We consider private consumption environments that satisfy the economic property and
whose corresponding sets of private components Yi satisfy the following condition.
For all i ∈ I there are wi , xi , yi ∈ Yi such that for all θ, θ ′ ∈ Θ such that θi 6= θi′ ,
ui (yi , θ) − ui (xi , θ) ui (wi , θ ′ ) − ui (xi , θ ′ ) 6= ui (yi , θ ′ ) − ui (xi , θ ′ ) ui (wi , θ) − ui (xi , θ) .
(4)
In QPCA environments (4) is equivalent to the requirement that Vi (θi )∩Vi (θi′ ) for all θi , θi′ ∈ Θi
such that θi 6= θi′ . If vi (θ) = vi (θ ′ ) and θi 6= θi′ then (4) is clearly violated for θ, θ ′ ∈ Θ. And if
Vi (θi )∩Vi (θi′ ) = ∅ for all θi , θi′ ∈ Θi such that θi 6= θi′ then wi = (0, ε) (agent i receives the good
with probability zero and a transfer of ε > 0), xi = (0, 0) and yi = (qi , 0) with qi > 0 satisfy
(4). Condition (4) thus generalizes the sufficient condition for the existence of a maximally
revealing mechanism from Proposition 4 to private consumption environments.
We obtain the following lemma, proved in Appendix B.
Lemma 7 Take any private consumption environment that satisfies the economic property
and (4). Then for every player i ∈ I there exist ziH , ziM , ziL ∈ Yi such that
a) ui (ziH , θ) > ui (ziM , θ) > ui (ziL , θ) for all θ ∈ Θ,
b) ziM is in the interior of Yi , and
c) ξi (θ) 6= ξi (θ ′ ) for all θ, θ ′ ∈ Θ such that θi 6= θi′ , where
ξi (θ) =
ui (ziH , θ) − ui (ziM , θ)
,
ui (ziM , θ) − ui (ziL , θ)
21
∀θ ∈ Θ.
To verify the first claim, let Yi be the space of lotteries over X̃i , endowed with the relative Euclidean
topology, Ỹ ′ = Ỹ , gi (ỹ) = margX̃i ỹ and ui (yi , θ) = ũi (⊗j∈I yj , θ), where ⊗j∈I yj is the product measure of yi
and some arbitrary yj ∈ Yj , j 6= i. As mentioned earlier, formally, a QPCA environment is not an environment
in the sense of sense of Section 3. But every QPCA environment maps to such an environment (see Appendix C).
The precise version of the second claim is that for any QPCA environment, the corresponding environment in
Q
the sense of sense of Section 3 is a private consumption environment. To see this, let Y = i∈I [0, 1I ] × [−B, B],
−1
Ỹ ′ = gC
(Y ), where gC denotes the utility-preserving surjection introduced in Appendix C, and g = gC |Ỹ ′ .
Note that the first and the second claim are not nested, as for any QPCA environment there can be no sets X̃i
Q
such that X̃ = i∈I X̃i is the corresponding set of pure outcomes. If there were, then if x1 ∈ X̃1 represented a
case in which agent 1 obtains the good and x2 ∈ X̃2 a case in which agent 2 obtains the good, (x1 , x2 , . . .) ∈ X̃.
But this cannot be, as assigning the good to both agents is impossible.
36
We can now quite easily generalize the main results from Subsection 5.1. The ratios ξi
assume the role that the ex-post valuations played in QPCA environments. Let Ξi = {ξi (θ) :
θ ∈ Θ}, and number the elements of this set so that Ξ = {ξi1 , . . . , ξini }, where ξi1 < . . . < ξini .
With each mi ∈ {1, . . . , ni } associate the unique τi (mi ) ∈ Θi for which there exists θ−i ∈ Θ−i
such that ξi (τi (mi ), θ−i ) = ξimi . Then the following generalization of Lemma 5 obtains. It is
proved in Appendix B.
Lemma 8 Take any private consumption environment that satisfies the economic property
and (4). For each i ∈ I there exist o0i , o1i , . . . , oni i ∈ Yi such that for m ∈ {1, . . . , ni },
m−1
0
1
a) ui (om
} and ui (om
i , θ) < ui (oi , θ) for all θ ∈ Θ such that ξi (θ) ∈ {ξi , . . . , ξi
i , θ) >
ni
0
m
ui (oi , θ) for all θ ∈ Θ such that ξi (θ) ∈ {ξi , . . . , ξi }.
l
m
b) ui (om
i , θ) > ui (oi, θ) for all l ∈ {0, . . . , ni }\{m} and all θ ∈ Θ such that ξi (θ) = ξi .
Given Lemma 8, the proof of Proposition 4 generalizes without any substantial change,
with the options from Lemma 8 replacing the options from Lemma 5, ξim , ξi (θ), . . . replacing
m
m
m
0
vm
i , vi (θ), . . ., and ui (oi , θ) replacing vi (θ)qi + ti . In the generalized version, ui (oi , θ) does
not necessarily equal zero, but this is of no consequence. Thus the following corollary obtains.
Corollary 2 Take any private consumption environment that satisfies the economic property
and (4). Then there exists a fully revealing mechanism, and social choice function f is rvimplementable if and only if f is ex-post incentive compatible.
The main insights from QPCA environments thus extend to private consumption environments. Generalizing Corollary 2 further, to some class of environments without private
consumption, is however not straightforward, even though it is clear that fully revealing mechanisms exist in at least some such environments. Above, the privacy of consumption allowed
us to change an outcome by just changing its i-th component, thus affecting only agent i. If
consumption is not private this is no longer possible, and whenever one assigns an outcome to
a terminal history of a mechanism, one has to consider the effect on all agents’ incentives.
A
Mechanisms
This appendix formally defines a mechanism. We first recall the definition of an extensive
game form (see e.g. Kuhn (1953); our notation is close to that of Osborne and Rubinstein
(1994)).
Definition 9 An extensive game form is a tuple Γ = hH, (Hi )i∈I , P, Ci such that
37
• H is a nonempty finite set of finite sequences with codomain A (where A is a nonempty
set of actions) such that with h every initial subsequence of h is in H.22 We let A(h) =
{a ∈ A : (h, a) ∈ H} for h ∈ H, T = {h ∈ H : A(h) = ∅} and call ∅ ∈ H the initial
history. We write h′ h if h′ ∈ H is an initial subsequence of h ∈ H.
• P : H\T → I. We let Hi = {h ∈ H\T : P (h) = i} for all i ∈ I.
• for each i ∈ I, Hi is a partition23 of Hi such that
– for all H ∈ Hi and all h, h′ ∈ H, A(h) = A(h′ ).
– for all H ∈ Hi and all h, h′ ∈ H, if h ∈ H ∈ Hi , h′ h and h′ 6= h then h′ ∈
/ H.
• C :T →Y.
Note that we do neither allow infinitely many time periods (histories are finite sequences)
nor infinitely many actions at any history (H is finite), and that partially orders H. We
define a binary relation on Hi by H′ H if there are h′ ∈ H′ and h ∈ H such that h′ h,
and extend this relation to H̄i = Hi ∪ {{∅}} (if necessary) by letting {∅} H for all H ∈ H̄i .
A strategy for player i in an extensive game form Γ is a function si : Hi → A such that for
all H ∈ Hi , there is an h ∈ H such that si (H) ∈ A(h). The set of player i’s strategies admitting
information set H ∈ Hj , j ∈ I, is defined as Si (H) = {si ∈ Si : ∃s−i ∈ S−i ∃h ∈ H, h ζ(s)}.
Q
For J ⊆ I and θ ∈ Θ, the set of histories admitted by (sj )j∈J ∈ j∈J Sj is defined by
Y
H((sj , θj )j∈J ) = H((sj )j∈J ) = h ∈ H : ∃(sj )j∈I\J ∈
Sj , h ζ(s) .
j∈I\J
To ensure that our definition of a Bayesian agent (made in Subsection 3.2) is sensible, we
restrict attention to extensive game forms with perfect recall and no trivial decision nodes.
Definition 10 A mechanism is an extensive game form Γ = hH, (Hi )i∈I , P, Ci such that
• (perfect recall) for all i ∈ I, si ∈ Si and H ∈ Hi , if H ∩ H(si ) 6= ∅ then H ⊆ H(si ).
• (no trivial decisions) for all (h, a) ∈ H there exists an action a′ 6= a such that (h, a′ ) ∈ H.
A mechanism is static if each agent has exactly one information set, if at any two nonterminal histories of equal length the same player is active, and if all terminal histories have
the same length.
22
Let A be a nonempty set. A finite sequence h of length n ∈ N with codomain A is a function h : {1, . . . , n} →
A. A finite sequence g : {1, . . . , k} → A is an initial subsequence of the finite sequence h : {1, . . . , n} → A if
k ≤ n and gl = hl for all l ∈ {1, . . . , k}. Note that ∅ (the unique finite sequence mapping {1, . . . , 0} to A) is an
initial subsequence of every finite sequence with codomain A. For h : {1, . . . , n} → A and a ∈ A, (h, a) denotes
the finite sequence that maps {1, . . . , n + 1} into A, has h as an initial subsequence and maps n + 1 to a.
SN
23
A partition of Hi is a family (Hn )N
n=1 of nonempty, pairwise disjoint sets Hn ⊆ Hi such that
n=1 Hn = Hi .
38
B
B.1
Proofs
Proof of Proposition 1
Proof. Suppose f is rv-implementable. We first show that f has to be dr-measurable. Take
ε > 0, then there is a mechanism Γ that robustly ε-implements f . Suppose θ ∼ θ ′ , then
there are s ∈ R∞ (θ) and s′ ∈ R∞ (θ ′ ) such that ζ(s) = ζ(s′ ). By robust ε-implementation,
kC(ζ(s)) − f (θ)k ≤ ε and kC(ζ(s′ )) − f (θ ′ )k ≤ ε and thus kf (θ) − f (θ ′ )k ≤ 2ε. Since this is
true for all ε > 0, f (θ) = f (θ ′ ).
Second, we will establish that f is epIC. Take any i ∈ I, θi , θi′ ∈ Θi and θ−i ∈ Θ−i . We
need to show that
(5)
ui (f (θ), θ) ≥ ui (f (θi′ , θ−i ), θ).
If f (θi′ , θ−i ) = f (θ) then (5) is trivially satisfied. Thus consider the case where f (θi′ , θ−i ) 6=
f (θ). In this case, intuitively, if i correctly believes that θ−i plays the strongly rationalizable
strategy s−i then i can induce (an outcome arbitrarily close to) f (θ) by following a strategy that
is strongly rationalizable for θi and (an outcome arbitrarily close to) f (θi′ , θ−i ) by mimicking θi′
and following a strategy that is strongly rationalizable for θi′ . And since f is rv-implementable,
he must prefer f (θ) to f (θi′ , θ−i ), implying (5). Formally, let ε satisfy 0 < ε < 21 kf (θi′ , θ−i ) −
f (θ)k. Since f is rv-implementable, there is a mechanism Γ = (H, (Hi )i∈I P, C) that robustly
ε-implements f , that is, such that kC(ζ(s̃)) − f (θ̃)k ≤ ε for all (s̃, θ̃) ∈ F ∞ . For each j 6= i,
pick some sj ∈ Rj∞ (θj ). Let λi ∈ ∆(Σ−i ) denote the degenerate belief in (s−i , θ−i ), and let
Σ−i × H̄ → [0, 1] by µ (·|H) = λ for H ∈ H̄ (s ) and
µ′i be an element of Φ∞
i
i
i
i −i
i . Define µi : 2
′
′
µi (·|H) = µi (·|H) for H ∈
/ H̄i (s−i ). Then µi is a CPS, as H H implies that either µi (·|H) is
the Bayesian update of µi (·|H′ ) or µi (Σ−i (H)|H′ ) = 0: If H, H′ ∈
/ H̄i (si ) then this is the case
′
′
because µi is a CPS, if H, H ∈ H̄i (si ) then µi (·|H) trivially is the Bayesian update of µi (·|H′ ),
and if H ∈
/ H̄i (si ) and H′ ∈ H̄i (si ) then µi (Σ−i (H)|H′ ) = 0. Indeed, since all the mass of λi
concentrates on a profile of strongly rationalizable strategy-payoff type pairs, µi ∈ Φ∞
i . Hence
∞
ρi (µi ) ⊆ Fi — if s̃i is sequentially rational for θ̃i with respect to µi then (s̃i , θ̃i ) is strongly
rationalizable.
Pick some si ∈ ri (θi , µi ) and some s′i ∈ ri (θi′ , µi ). Since Γ robustly ε-implements f
kC(ζ(s′i , s−i )) − C(ζ(s))k ≥ kf (θi′ , θ−i ) − f (θ)k − 2ε > 0,
and thus C(ζ(s′i , s−i )) 6= C(ζ(s)). So there is a (unique) information set H′ ∈ Hi (s) such that
si (H′ ) 6= s′i (H′ ) and si (H) = s′i (H) for all H ∈ Hi that strictly precede H′ . By the definition of
sequential rationality, ∀H ∈ Hi (si )∀s̃i ∈ Si (H) : Uiµi (si , θi , H) ≥ Uiµi (s̃i , θi , H). In particular,
Uiµi (si , θi , H′ ) ≥ Uiµi (s′i , θi , H′ ).
39
Since Γ robustly ε-implements f , kui (C(ζ(s)), θ) − ui (f (θ), θ)k ≤ K · ε, where K denotes the
Lipschitz constant of ui (·, θ).24 Since µi (·|H′ ) = λi , we have Uiµi (si , θi , H′ ) = ui (C(ζ(s)), θ).
Similarly, kUiµi (s′i , θi , H′ ) − ui (f (θi′ , θ−i ), θ)k ≤ K · ε, and so
ui (f (θ), θ) − ui (f (θi′ , θ−i ), θ)
≥ ui (f (θ), θ) − Uiµi (si , θi , H′ ) + Uiµi (s′i , θi , H′ ) − ui (f (θi′ , θ−i ), θ)
≥ −kui (f (θ), θ) − Uiµi (si , θi , H′ )k − kUiµi (s′i , θi , H′ ) − ui (f (θi′ , θ−i ), θ)k
≥ −2K · ε.
Since this holds for every sufficiently small ε > 0, (5) follows.
B.2
Proof of Lemma 2
Proof. Since Σ−i and H̄i are finite we can view ∆H̄i (Σ−i ) as a subspace of the #Σ−i · #H̄i dimensional Euclidean space. For any i ∈ I, k ∈ N, (si , θi ) ∈
/ Fik+1 and µi ∈ Φki there
exist H ∈ Hi (si ) and s′i ∈ Si (H) such that Uiµi (s′i , θi , H) − Uiµi (si , θi , H) > 0. Without loss of
generality, choose H and s′i such that s′i ∈ Rik+1 (θi ). By continuity of f (i,k,si,θi ,µi ) : ∆H̄i (Σ−i ) →
µ′
µ′
R, µ′i 7→ Ui i (s′i , θi , H)−Ui i (si , θi , H) there is ε(i, k, si , θi , µi ) > 0 such that f (i,k,si,θi ,µi ) assumes
a strictly positive minimum ηΓ (i, k, si , θi , µi ) on the closed ball B̄ε(i,k,si,θi ,µi ) (µi ) with radius
ε(i, k, si , θi , µi ) and center µi . Since Φki is compact, the cover (Bε(i,k,si ,θi ,µi ) (µi ))µi ∈Φk of open
n(i,k,si ,θi )
balls Bε(i,k,si ,θi ,µi ) (µi ) has a finite subcover (Bε(i,k,si ,θi ,µm
(µm
i ))m=1
i )
k+1
finitely many distinct F
, it now suffices to let
ηΓ =
B.3
min
min
i∈I,k∈N,(si ,θi )∈F
/ ik+1 m∈{1,...,n(i,k,si,θi )}
i
. As there are only
ηΓ (i, k, si , θi , µm
i ).
Proof of Proposition 2
Proof. Suppose Γ̂ and Γ̃ are two mechanisms. We will construct a mechanism Γ∗ that strategically distinguishes any two payoff type profiles that Γ̂ strategically distinguishes as well as any
two payoff type profiles that Γ̃ strategically distinguishes. Since Θ is finite, this establishes the
existence of a maximally revealing mechanism. Informally, Γ∗ lets the agents play Γ̂ first and
then Γ̃ second and assigns much more weight to the outcome of Γ̂ than to that of Γ̃. Formally,
let25
H ∗ = {h∗ ∈ F ∗ : h∗ (ĥ, h̃) for some ĥ ∈ T̂ , h̃ ∈ T̃ },
24
25
That is, K is such that kui (y, θ) − ui (y ′ , θ)k ≤ Kky − y ′ k for all y, y ′ ∈ Y .
If ĥ = (a1 , . . . , an ) and h̃ = (b1 , . . . , bm ) then (ĥ, h̃) denotes the history (a1 , . . . , an , b1 , . . . , bm ).
40
be the set of Γ∗ ’s histories, where F ∗ is the set of finite sequences that have the union of Γ̂’s
and Γ̃’s action sets as codomain. Let
o
n
o n
H∗i = Ĥ : Ĥ ∈ Ĥi ∪ [ĥ, H̃] : ĥ ∈ T̂ , H̃ ∈ H̃i ,
be the set of i’s information sets, where [ĥ, H̃] = {(ĥ, h̃) ∈ F ∗ : h̃ ∈ H̃}. Let the player
function P ∗ : H ∗ \T ∗ → I be defined by P ∗ (ĥ) = P̂ (ĥ) for ĥ ∈ Ĥ\T̂ and P ∗ (ĥ, h̃) = P̃ (h̃) for
ĥ ∈ T̂ , h̃ ∈ H̃\T̃ . Finally, let κ > 0 be such that κC0 < (1 − κ)ηΓ̂ , where C0 is the constant
from Lemma 1 and ηΓ̂ the constant from Lemma 2, and the outcome function
C ∗ : T ∗ → Y be such that C ∗ (ĥ, h̃) = (1 − κ)Ĉ(ĥ) + κC̃(h̃).
First, recall that the set of strongly rationalizable strategy-payoff type profiles is obtained
by the iterated removal of never-best sequential responses, or equivalently (see Shimoji and
Watson, 1998), by the iterated removal of conditionally dominated strategies. In an important contribution, Chen and Micali (2011, Theorem 1) establish that if, at each iteration,
instead of simultaneously removing all conditionally dominated strategies as in Shimoji and
Watsons’ procedure, one only removes some conditionally dominated strategies, then one ends
up with strategy-payoff type profiles which lead to the same terminal histories as the strongly
rationalizable strategy-payoff type profiles. Shimoji and Watson (1998) and Chen and Micali
(2011) focus on finite, complete information games with perfect recall, but their results easily
extend to our incomplete information set up.26 In our notation, if Γ = hH, (Hi )i∈I , P, Ci is a
mechanism:
• Let D = (Di,θi )i∈I,θi ∈Θi be a profile of strategy sets Di,θi ⊆ Si , and for each i, θi and H,
let Di,θi (H) = Di,θi ∩ Si (H). Let θi ∈ Θi and si ∈ Di,θi . We say that si is conditionally
dominated for θi within D if there exist an information set H ∈ Hi (si ) that is admitted
by D−i,θ−i for some θ−i ∈ Θ−i , and a mixed strategy σi ∈ ∆(Di,θi (H)) such that for
P
each θ−i ∈ Θ−i and each s−i ∈ D−i,θ−i (H), s′ ∈Di,θ (H) ui (C(ζ((s′i , s−i ))), θ)σi (s′i ) >
i
i
ui (C(ζ(s)), θ).
26
A standard argument shows that strategy si is conditionally dominated for θi within D if
and only if there exists a H ∈ Hi (si ) that is admitted by D−i,θ−i for some θ−i ∈ Θ−i such
that for no belief λi ∈ ∆(Σ−i (H)) with λi {D−i,θ−i ×{θ−i } : θ−i ∈ Θ−i } = 1, si maximizes
θi ’s expected utility with respect to λi over Di,θi (H). Therefore, if strategy si is not
S
conditionally dominated for θi within D then for every H ∈ Hi (si )∩ θ−i ∈Θ−i Hi (D−i,θ−i )
H
there is a belief λH
i ∈ ∆(Σ−i (H)) with λi {D−i,θ−i × {θ−i } : θ−i ∈ Θ−i } = 1 such that
si maximizes θi ’s expected utility with respect to λH
i over Di,θi (H). For later, note
′
H′
H
that without loss of generality, these beliefs satisfy λi (A)λH
i (Σ−i (H)) = λi (A) for all
S
A ⊆ Σ−i (H), for all H, H′ ∈ Hi (si ) ∪ θ−i ∈Θ−i Hi (D−i,θ−i ) such that H′ H.
Shimoji and Watson (1998) point this out for their results in their footnote 5.
41
• We say that a profile of strategy sets D = (Di,θi )i,θi survives iterative elimination of
conditionally dominated strategies if there exists a finite sequence D = (D 0 , . . . , D K ) of
profiles of strategy sets such that
0 = S for all i and θ , and D K = D;
1. Di,θ
i
i
i
k \D k+1 6= ∅ and ∀i, θ : (D k+1 ⊆ D k
2. ∀k < K, ∃i, θi : Di,θ
i
i,θi and every strategy in
i,θi
i,θi
i
k+1
k
k \D
Di,θ
i,θi is conditionally dominated for θi within D ); and
i
K contains no strategy that is conditionally dominated for θ within D K .
3. each Di,θ
i
i
We refer to D as an elimination order, and to D as a resilient solution.
• Let D = (Di,θi )i,θi be a profile of strategy sets, θi ∈ Θi and si , s′i ∈ Si . We write si ≃D s′i
≃D denote the
if for each θ−i ∈ Θ−i and each s−i ∈ D−i,θ−i , ζ(s) = ζ(s′i , s−i ). Let Di,θ
i
i
set of equivalence classes of Di,θi under ≃D , and for si ∈ Di,θi , let s≃D,θ
denote the
i
equivalence class to which si belongs.
′ )
are equivalent
• We say that two profiles of strategy sets D = (Di,θi )i,θi and D ′ = (Di,θ
i i,θi
′
′≃D
≃D
if there exists a profile of bijections φi,θi : Di,θi → Di,θi such that for any θ ∈ Θ and
1
I
)).
any strategies si ∈ Di,θi , i ∈ I, we have H(s) = H(φ1,θ1 (s≃D,θ
) × . . . × φI,θI (s≃D,θ
1
I
• Any resilient solution is equivalent to the profile of sets of strongly rationalizable strategies (Ri∞ (θi ))i,θi .
k )
We claim that the finite sequences of profiles of strategy sets (D k ) = ((Di,θ
) and
i i,θi k
∗
k
k
(E ) = ((Ei,θi )i,θi )k defined by the following conditions are elimination orders for Γ .
a) Let k̂ be the smallest k for which F̂ k = F̂ ∞ . For each k ≤ k̂, i ∈ I, θi ∈ Θi and s∗i ∈ Si∗ ,
let
k
s∗i ∈ Di,θ
if and only if s∗i |Ĥi ∈ R̂ik (θi ),
(6)
i
where s∗i |Ĥi is the restriction of s∗i to Ĥi . The first k̂ elements of (D k ) exactly “mimick”
the sequence of sets of strongly k-rationalizable strategy profiles in Γ̂. For k > k̂, obtain
k−1
k
Di,θ
by eliminating from Di,θ
all strategies which are conditionally dominated for θi
i
i
k−1
k−1
contains no strategy which is conditionally
within D . Stop when for all i and θi , Di,θ
i
k−1
dominated for θi within D .
b) Let k̃ be the smallest k for which F̃ k = F̃ ∞ . For each k ≤ k̃, i ∈ I, θi ∈ Θi and s∗i ∈ Si∗ ,
let
k
if and only if ∀ĥ ∈ T̂ (s∗i |Ĥi ) : si∗,ĥ ∈ R̃ik (θi ),
(7)
s∗i ∈ Ei,θ
i
where T̂ (s∗i |Ĥi ) = T̂ ∩ Ĥ(s∗i |Ĥi ) and for each ĥ ∈ T̂ (s∗i |Ĥi ), si∗,ĥ ∈ S̃i denotes the strategy
k by eliminating
satisfying si∗,ĥ (H̃) = s∗i ([ĥ, H̃]) for all H̃ ∈ H̃i . For k > k̃, obtain Ei,θ
i
42
k−1
all strategies which are conditionally dominated for θi within E k−1 . Stop
from Ei,θ
i
k−1
when for all i and θi , Ei,θ
contains no strategy which is conditionally dominated for θi
i
k−1
within E
.
In order to verify that (D k ) is an elimination order, it suffices to show that for all k ≤
k̂ − 1, i ∈ I and θi ∈ Θi , if s∗i ∈ Si∗ is such that s∗i |Ĥi ∈ R̂ik (θi )\R̂ik+1 (θi ), then s∗i is
conditionally dominated for θi within D k . Suppose that this is not the case, and let k̄ be the
smallest k for which for some i, θi and s∗i , s∗i |Ĥi ∈ R̂ik (θi )\R̂ik+1 (θi ) but s∗i is not conditionally
k̄
) for some θ−i ∈ Θ−i }
dominated within D k . Let Gi∗ = {H∗ ∈ H∗i (s∗i ) : H∗ ∈ H∗i (D−i,θ
−i
∗
∗
k̄
∗
∗
ˆ
and Gi = Ĥi ∩ Gi = Ĥi (si |Ĥi ) ∩ Ĥi (F̂−i ). Then, for every H ∈ Gi , there exists a belief
k̄
× {θ−i } : θ−i ∈ Θ−i } = 1 such that s∗i maximizes
λ∗i (H∗ ) ∈ ∆(Σ∗−i (H∗ )) with λ∗i (H∗ ){D−i,θ
−i
k̄ (H∗ ). As remarked above, without loss
θi ’s expected utility with respect to λ∗i (H∗ ) over Di,θ
i
of generality,
λ∗i (H∗ )(A) · λ∗i (H∗′ )(Σ∗−i (H∗ )) = λ∗i (H∗′ )(A) for all A ⊆ Σ∗−i (H∗ ),
for all H∗ , H∗′ ∈ Gi∗ such that H∗′ H∗ . For all Ĥ ∈ Gˆi let λ̂i (Ĥ) denote the “projection” of
λ∗i (Ĥ) to Γ̂, that is, let λ̂i (Ĥ) ∈ ∆(Σ̂−i (Ĥ)) satisfy
λ̂i (Ĥ)(ŝ−i , θ−i ) =
∗
s∗−i ∈S−i
X
s.t.
s∗−i |Ĥ =ŝ−i
−i
λ∗i (Ĥ)(s∗−i , θ−i )
for all (ŝ−i , θ−i ) ∈ Σ̂−i (Ĥ).
Note that if Ĥ, Ĥ′ ∈ Gˆi , Ĥ′ Ĥ and λ̂i (Ĥ′ )(Σ̂−i (Ĥ)) > 0 then λ̂i (Ĥ) is the Bayesian update
of λ̂i (Ĥ′ ).
Let µ̂ik̄−1 ∈ Φ̂ik̄−1 be a CPS against which s∗i |Ĥi is a sequential best response for θi . Since
by (6) for k̄, for each Ĥ ∈ Gˆi the belief λ̂i (Ĥ) assigns probability one to the set of strongly
k̄-rationalizable strategy-payoff type pairs, there exists a CPS µ̂i ∈ Φ̂k̄i such that
– if Ĥ ∈ Gˆi then µ̂i (·|Ĥ) = λ̂i (Ĥ), and
k̄ , then µ̂ (·|Ĥ) = µ̂k̄−1 (·|Ĥ).
– if Ĥ is not admitted by F̂−i
i
i
One can construct such a µ̂i as follows. Let µ̂k̄i ∈ Φ̂k̄i . Start by letting µ̂i (·|{∅}) = λ̂i ({∅}) or,
if {∅} ∈
/ Gˆi , by letting µ̂i (·|{∅}) = λ̂i (Ĥ) for some Ĥ ∈ Gˆi that does not have a predecessor
ˆ . If Ĥ is such a successor
in Ĥi . Then proceed to the immediate successors of {∅} in H̄
i
and µ̂i (Σ̂−i (Ĥ)|{∅}) > 0, then let µ̂i (·|Ĥ) be the Bayesian update of µ̂i (·|{∅}) (implying
that µ̂i (·|Ĥ) = λ̂i (Ĥ) if Ĥ ∈ Gˆi ). If, on the other hand, µ̂i (Σ̂−i (Ĥ)|{∅}) = 0, then let
k̄ ) and µ̂ (·|Ĥ) = µ̂k̄ (·|Ĥ) if
µ̂i (·|Ĥ) = λ̂i (Ĥ) if Ĥ ∈ Gˆi , µ̂i (·|Ĥ) = µ̂ik̄−1 (·|Ĥ) if Ĥ ∈
/ Ĥi (F̂−i
i
i
k̄ )\Ĥ (s∗ | ). Next, define µ̂ in a similar fashion for the immediate successors of
Ĥ ∈ Ĥi (F̂−i
i i Ĥi
i
k̄ ),
the information sets just considered, and so on. To see that µ̂i ∈ Φ̂k̄i , note that if Ĥ ∈
/ Ĥi (F̂−i
k , k ∈ N, that admit Ĥ. That is, µ̂k̄−1 (·|Ĥ)
then µ̂ik̄−1 (·|Ĥ) must place probability one on all F̂−i
i
43
and therefore µ̂i (·|Ĥ) place probability one on the highest degree of rationality of −i consistent
with Ĥ.
Because s∗i |Ĥi is not strongly (k̄ + 1)-rationalizable for θi , s∗i |Ĥi cannot be a sequential
best response against µ̂i for θi . Therefore, by Lemma 2, there exist Ĥ′ ∈ Ĥi (s∗i |Ĥi ) and
ŝ′i ∈ R̂ik̄+1 (θi ) ∩ Ŝi (Ĥ′ ) such that under µ̂i , ŝ′i promises strictly more than ηΓ̂ higher expected
utility at Ĥ′ to θi than s∗i |Ĥi . By construction we must have Ĥ′ ∈ Gˆi and thus µ̂i (·|Ĥ′ ) = λ̂i (Ĥ′ ).
k̄+1
Consider an arbitrary extension of ŝ′i to H∗i . By (6) for k̄, this extension is in Di,θ
(Ĥ′ ) and
i
k̄ (Ĥ′ ). And because κC < (1 − κ)η , the expected utility from this extension
hence in Di,θ
0
Γ̂
i
for θi with respect to λ∗i (Ĥ′ ) strictly exceeds the expected utility from s∗i , contradicting that
k̄ (Ĥ′ ). Thus, (D k ) is an elimination
given λ∗i (Ĥ′ ), s∗i is expected utility maximizing over Di,θ
i
order.
The proof that (E k ) is an elimination order is similar. Suppose that for some k ≤ k̃ − 1
there are i ∈ I, θi ∈ Θi , s∗i ∈ Si∗ and ĥ ∈ T̂ (s∗i |Ĥi ) such that si∗,ĥ ∈ R̃ik (θi )\R̃ik+1 (θi ), but s∗i is
not conditionally dominated within E k . Let k̄ be the smallest such k, and Ei∗ = {H∗ ∈ H∗i (s∗i ) :
k̄
k̄ ). Then, for every H∗ ∈ E ∗
H∗ ∈ H∗i (E−i,θ
) for some θ−i ∈ Θ−i } and E˜i = H̃i (si∗,ĥ )∩ H̃i (F̃−i
i
−i
∗
∗
∗
∗
∗
∗
k̄
there exists a belief λi (H ) ∈ ∆(Σ−i (H )) with λi (H ){E−i,θ−i × {θ−i } : θ−i ∈ Θ−i } = 1 such
k̄ (H∗ ). Without loss
that s∗i maximizes θi ’s expected utility with respect to λ∗i (H∗ ) over Ei,θ
i
of generality,
λ∗i (H∗ )(A) · λ∗i (H∗′ )(Σ∗−i (H∗ )) = λ∗i (H∗′ )(A) for all A ⊆ Σ∗−i (H∗ ),
for all H∗ , H∗′ ∈ Ei∗ such that H∗′ H∗ . For all H̃ ∈ E˜i , let λ̃i (H̃) ∈ ∆(Σ̃−i (H̃)) denote the
“projection” of λ∗i ([ĥ, H̃]) to Γ̃, that is, let
λ̃i (H̃)(s̃−i , θ−i ) =
X
∗ (ĥ) s.t. s∗,ĥ =s̃
s∗−i ∈S−i
−i
−i
λ∗i ([ĥ, H̃])(s∗−i , θ−i )
for all (s̃−i , θ−i ) ∈ Σ̃−i (H̃).
Then λ̃i (H̃) is the Bayesian update of λ̃i (H̃′ ) if H̃, H̃′ ∈ E˜i , H̃′ H̃ and λ̃i (H̃′ )(Σ̃−i (H̃)) > 0.
Let µ̃ik̄−1 ∈ Φ̃ik̄−1 be a CPS against which si∗,ĥ is a sequential best response for θi . Then,
since for each H̃ ∈ E˜i , λ̃i (H̃) assigns probability one to the set of strongly k̄-rationalizable
strategy-payoff type pairs, there exists a CPS µ̃i ∈ Φ̃k̄i such that
– if H̃ ∈ E˜i then µ̃i (·|H̃) = λ̃i (H̃), and
k̄ , then µ̃ (·|H̃) = µ̃k̄−1 (·|H̃).
– if H̃ is not admitted by F̃−i
i
i
Since si∗,ĥ is not strongly (k̄ + 1)-rationalizable for θi , it is not a sequential best response
against µ̃i for θi . So there must exist H̃′ ∈ E˜i and s̃′i ∈ R̃ik̄ (θi ) ∩ S̃i (H̃′ ) such that under µ̃i , s̃′i
promises strictly higher expected utility at H̃′ to θi than si∗,ĥ . Note that µ̃i (·|H̃′ ) = λ̃i (H̃′ ).
′∗,ĥ′
′
′∗
∗
k̄
Define s′∗
= s̃′i for all ĥ′ ∈ T̂ . Then the expected
i ∈ Ei,θi ([ĥ, H̃ ]) by si |Ĥi = si |Ĥi and si
44
∗
′
∗
utility from s′∗
i for θi with respect to λi ([ĥ, H̃ ]) strictly exceeds the expected utility from si ,
k̄ ([ĥ, H̃′ ]). Thus,
contradicting that given λ∗i ([ĥ, H̃′ ]), s∗i is expected utility maximizing over Ei,θ
i
(E k ) is an elimination order.
Suppose now that θ̂ ≁Γ̂ θ̂ ′ and that s̄∗ ∈ R∗,∞ (θ̂) and s̄′∗ ∈ R∗,∞ (θ̂ ′ ). By Chen and Micalis’
k̂
k̂
K
result there exist s∗i ∈ Di,Kθ̂ ⊆ D
for each i ∈ I and s′∗
i ∈ Di,θ̂ ′ ⊆ D ′ for each i ∈ I such
i,θ̂i
i
i
i,θ̂i
that ζ ∗ (s̄∗ ) = ζ ∗ (s∗ ) and ζ ∗ (s̄′∗ ) = ζ ∗ (s′∗ ). Since Γ̂ strategically distinguishes θ̂ and θ̂ ′ , (6)
implies that ĥ ≡ ζ̂(s∗1 |Ĥ1 , . . . , s∗I |ĤI ) 6= ζ̂(s′∗1 |Ĥ1 , . . . , s′∗I |ĤI ) ≡ ĥ′ . As ĥ precedes ζ ∗ (s∗ ) and ĥ′
precedes ζ ∗ (s′∗ ), s∗ and s′∗ and hence s̄∗ and s̄′∗ must lead to different terminal histories in
Γ∗ . Therefore, Γ∗ strategically distinguishes θ̂ and θˆ′ .
Similarly, suppose that θ̃ ≁Γ̃ θ̃ ′ and that s̄∗ ∈ R∗,∞ (θ̃) and s̄′∗ ∈ R∗,∞ (θ̃ ′ ). Then there
k̃
k̃
k̃
k̃
exist s∗ ∈ E1,θ̃ × . . . × EI,θ̃ and s′∗ ∈ E1,θ̃′ × . . . × EI,θ̃′ such that ζ ∗ (s̄∗ ) = ζ ∗ (s∗ ) and
1
ζ ∗ (s̄′∗ )
I
1
I
=
Suppose that ĥ ∈ T̂ precedes both ζ ∗ (s∗ ) and ζ ∗ (s′∗ ), then θ̃ ≁Γ̃ θ̃ ′ implies
via (7) that ζ̃(s1∗,ĥ , . . . , sI∗,ĥ ) 6= ζ̃(s′ 1∗,ĥ , . . . , s′ I∗,ĥ ), and s̄∗ and s̄′∗ lead to different terminal
histories in Γ∗ . Hence Γ∗ strategically distinguishes θ̃ and θ̃ ′ .
B.4
ζ ∗ (s′∗ ).
Proof of Lemma 3
Proof. The structure of the proof is similar to that of Proposition 2. We use the terminology
and results by Shimoji and Watson (1998) and Chen and Micali (2011) outlined there, and
k )
) for Γ. Let k be the smallest k for
start by defining an elimination order (D k ) = ((Di,θ
i i,θi
∗,k
∗,∞
k by
which F = F
. For k ≤ k, i ∈ I, θi ∈ Θi and si ∈ Si , define Di,θ
i
k
if and only if ϕi (si ) ∈ Ri∗,k (θi ).
si ∈ Di,θ
i
(8)
k by eliminating from D k−1 all strategies which are conditionally domiFor k > k, obtain Di,θ
i,θi
i
k−1
k−1
does not contain a strategy which is
nated for θi within D . Stop when for all i and θi , Di,θ
i
k−1
conditionally dominated for θi within D
and denote the last element of the finite sequence
k
K
(D ) by D .
To verify that (D k ) is an elimination order, suppose there is a k ≤ k − 1 for which for some
i ∈ I, θi ∈ Θi and si ∈ Si , ϕi (si ) ∈ Ri∗,k (θi )\Ri∗,k+1 (θi ) but si is not conditionally dominated
for θi within D k . Let k̄ be the smallest such k. Let Gi ⊆ Hi consist of all information sets
k̄
H ∈ Hi (si ) that are admitted by D−i,θ
for some θ−i ∈ Θ−i , and Gi∗ ⊆ H∗i consist of all
−i
∗,k̄
information sets H∗ ∈ H∗i (ϕi (si )) that are admitted by F−i
. Then, for every H ∈ Gi , there
k̄
exists a belief λi (H) ∈ ∆(Σ−i (H)) with λi (H){D−i,θ−i × {θ−i } : θ−i ∈ Θ−i } = 1 such that si
k̄ (H). Without loss of generality,
maximizes θi ’s expected utility with respect to λi (H) over Di,θ
i
λi (H)(A) · λi (H′ )(Σ−i (H)) = λi (H′ )(A) for all A ⊆ Σ−i (H),
45
for all H, H′ ∈ Gi such that H′ H. For any H∗ ∈ Gi∗ let λ∗i (H∗ ) ∈ ∆(Σ∗−i (H∗ )) denote the
“projection” of λi ([si (Hi∅ ), H∗ ]) to Γ∗ , that is, let
∗
λ∗i (H∗ )(s∗−i , θ−i ) = λi ([si (Hi∅ ), H∗ ])(ϕ−1
−i (s−i ) × {θ−i })
for all (s∗−i , θ−i ) ∈ Σ∗−i (H∗ ).
Then λ∗i (H∗ ) is the Bayesian update of λ∗i (H∗′ ) whenever H∗ , H∗′ ∈ Gi∗ , H∗′ H∗ and
λ∗i (H∗′ )(Σ∗−i (H∗ )) > 0.
Let µi∗,k̄−1 ∈ Φi∗,k̄−1 be a CPS against which ϕi (si ) is a sequential best response for θi .
∗,k̄
Since λ∗i (H∗ )(F−i
) = 1 for all H∗ ∈ Gi∗ , there exists a CPS µ∗i ∈ Φi∗,k̄ such that
– if H∗ ∈ Gi∗ then µ∗i (·|H∗ ) = λ∗i (H∗ ), and
k̄ , then µ∗ (·|H∗ ) = µ∗,k̄−1 (·|H∗ ).
– if H∗ is not admitted by F−i
i
i
Since ϕi (si ) is not strongly (k̄ + 1)-rationalizable for θi , it is not a sequential best response
∗,k̄
∗
∗′
against µ∗i for θi . By Lemma 2 there are H∗′ ∈ Gi∗ and s∗′
i ∈ Ri (θi ) ∩ Si (H ) such that
∗
µ∗i ∗′
µ
Ui (si , θi , H∗′ ) > Ui i (ϕi (si ), θi , H∗′ ) + ηΓ∗ . Note that µ∗i (·|H∗′ ) = λ∗i (H∗′ ). But then by (1),
∗
∗
s′i ∈ Si ([si (Hi∅ ), H∗′ ]) defined by s′i (Hi∅ ) = si (Hi∅ ) and s′i ([θi , H∗ ]) = s∗′
i (H ) for all [θi , Hi ]
gives strictly higher expected utility under λi ([si (Hi∅ ), H∗′ ]) than si , as playing s′i instead
of si leads to an expected utility gain of at least δηΓ∗ in δC ∗ (ζ ∗ (ϕ(ŝ))) and a possible exP
pected utility loss of at most δ2 C0 in δ2 1I j∈I rj (ζ(ŝ)). By (8) for k̄ the strategy s′i is in
k̄ ([s (H∅ ), H∗′ ]), contradicting that given λ ([s (H∅ ), H∗′ ]), s is expected utility maximizDi,θ
i
i
i
i
i
i
i
k̄ ([s (H∅ ), H∗′ ]). Thus, (D k ) is an elimination order.
ing over Di,θ
i
i
i
K ⊆ Dk ,
Consequently, for any θ ∈ Θ and s ∈ R∞ (θ) there are strategies s̄i ∈ Di,θ
i,θi
i
k
i ∈ I, that induce the same terminal history as s in Γ. By the definition of D , we have
ϕ(s̄) ∈ R∗,∞ (θ). As ϕ(s) induces the same history in Γ∗ as ϕ(s̄), the lemma follows.
B.5
Proof of Proposition 4: Options and the Proof of Lemma 5
1
Let (qni i, tni i) = ( I1 , − 2I
(vni i + vini−1 )), and
q
m
i
=
min
l=m+1,...,ni
vliqli + tli
m−1
2vli − (vm
)
i + vi
m−1
m m
tm
),
i = −0.5qi (vi + vi
m = ni − 1, . . . , 1. Moreover, let (q0i , t0i ) = (0, 0) and
Then:
om
i
,
m
= (qm
i , ti ) for m ∈ {0, . . . , ni }.
m−1 m
a) 0.5(vm
)qi + tm
i + vi
i = 0 for m ∈ {1, . . . , ni }
m
By definition of ti for m ∈ {1, . . . , ni − 1}, and by definition of
46
m
tm
i and qi for m = ni .
b)
1
qm
i ∈ (0, I ] for m ∈ {1, . . . , ni }
By definition, qni i > 0. For m ∈ {1, . . . , ni − 1}, if qli > 0 for all l ∈ {m + 1, . . . , ni },
m−1
l
l
then both 2vli − (vm
) > 0 and vliqli + tli > 0.5(vli + vl−1
i + vi
i )qi + ti = 0 for all
ni
1
l ∈ {m+1, . . . , ni }, and therefore qm
i > 0. By definition, qi ≤ I . For m ∈ {1, . . . , ni −1},
qm
i ≤
c)
vni iqni i + tni i
m−1
2vni i − (vm
)
i + vi
<
vni iqni i + tni i
ni
ni
ni−1
1
)]
2 [2vi − (vi + vi
=
1
.
I
m
m
vm
i qi + ti > 0 for m ∈ {1, . . . , ni }
mI
1
By b) and the definition of q0i , any profile of options (om
1 , . . . , oI ) with mi ∈ {0, . . . , ni }
for all i ∈ I is an element of Y and thus can be assigned as the outcome of a mechanism (if
1 mi
i
i
/ [−B, B] for some i ∈ I and some mi ∈ {0, . . . , ni }, redefine om
tm
i ∈
i as K oi for all i ∈ I,
mi ∈ {0, . . . , ni } and some sufficiently large K > 0).
We can now prove Lemma 5:
m−1 m
m
m
Proof. a) Let l ∈ {1, . . . , m − 1}, then vliqm
)qi + tm
i + ti < 0.5(vi + vi
i = 0. Let
l
m
m
m
m
m
l ∈ {m, . . . , ni }, then viqi + ti ≥ vi qi + ti > 0.
m
m
m m
m
b) The claim is true for l = 0 because vm
i qi + ti > 0. For l ∈ {m+1, . . . , ni }, vi qi + ti >
l
l
l
0 > vm
i qi + ti by a). For l ∈ {1, . . . , m − 1}, by the definition of qi,
qli ≤
and therefore
B.6
m
m
vm
i qi + ti
l−1
l
2vm
i − (vi + vi )
<
m
m
vm
i qi + ti
l−1
l
vm
i − 0.5(vi + vi )
l−1
l
l l
m l
l
m m
m
vm
i qi − 0.5qi(vi + vi ) = vi qi + ti < vi qi + ti .
Proof of Lemma 7
Proof. Take i ∈ I. By the economic property, there exists z̃ ′ ∈ Ỹ such that ũi (z̃ ′ , θ) > ũi (ȳ, θ)
for all θ ∈ Θ. Let ỹ be an interior point of Ỹ ′ that gi maps to an interior point of Yi , then for
β > 0 small enough z̃i ≡ (1 − β)ỹ + β z̃ ′ and ṽi ≡ (1 − β)ỹ + β ȳ are interior points of Ỹ ′ , and
zi ≡ gi (z̃i ) and vi ≡ gi (ṽi ) interior points of Yi . Moreover, ui (zi , θ) > ui (vi , θ) for all θ ∈ Θ.
Because gi and ũi are linear, ui satisfies the following linearity property. For all αn ∈ R,
P
P
yn ∈ Yi , n = 1, . . . , N , such that
αn yn ∈ Yi and
αn ỹn ∈ Ỹ ′ for some ỹn ∈ gi−1 (yn ),
n = 1, . . . , N ,
ui
X
n
X
X
αn ỹn , θ =
αn gi (ỹn ), θ = ui gi
αn yn , θ = ui
n
n
= ũi
X
n
αn ỹn , θ =
47
X
αn ũi (ỹn , θ) =
X
αn ui (yn , θ)
for all θ ∈ Θ. Let wi , xi , yi ∈ Yi be outcomes that satisfy (4) for i. For some small enough
ε > 0 and any δ, γ > 0 in some neighborhood of zero, (1 − 2ε)ṽi + 2εz̃i + γ(ỹi − x̃i ) + δ(w̃i − x̃i )
and (1 − ε)ṽi + εz̃i are in Ỹ ′ for some x̃i ∈ gi−1 (xi ), ỹi ∈ gi−1 (yi ) and z̃i ∈ gi−1 (zi ), and
ziH ≡ (1 − 2ε)vi + 2εzi + γ(yi − xi ) + δ(wi − xi ),
ziM ≡ (1 − ε)vi + εzi ,
ziL ≡ vi
are in Yi and satisfy 1) and 2). (By the linearity property, 1) is satisfied for γ ≪ ε and δ ≪ ε.)
Now observe that for any θ, θ ′ ∈ Θ, ξi (θ) = ξi (θ ′ ) if and only if
γ(ui (yi , θ) − ui (xi , θ)) + δ(ui (wi , θ) − ui (xi , θ))
ui (zi , θ) − ui (vi , θ)
γ(ui (yi , θ ′ ) − ui (xi , θ ′ )) + δ(ui (wi , θ ′ ) − ui (xi , θ ′ ))
=
ui (zi , θ ′ ) − ui (vi , θ ′ )
Let
(9)
n
ui (yi , θ) − ui (xi , θ)
ui (yi , θ ′ ) − ui (xi , θ ′ ) o
Ni = (θ, θ ′ ) ∈ Θ2 : θi 6= θi′ and
.
=
ui (zi , θ) − ui (vi , θ)
ui (zi , θ ′ ) − ui (vi , θ ′ )
Since ξi (θ) 6= ξi (θ ′ ) is equivalent to (9) failing, ξi (θ) 6= ξi (θ ′ ) for any (θ, θ ′ ) ∈
/ Ni such that θi 6=
′
′
θi , as long as δ > 0 is sufficiently small compared to γ. For (θ, θ ) ∈ Ni , ui (yi , θ) − ui (xi , θ) = 0
if and only if ui (yi , θ ′ ) − ui (xi , θ ′ ) = 0. Since (4) rules out that both differences are zero, both
must be non-zero, in particular, ui (yi , θ ′ ) − ui (xi , θ ′ ) 6= 0. Suppose that ξi (θ) = ξi (θ ′ ), then
ui (zi , θ) − ui (vi , θ)
ui (yi , θ) − ui (xi , θ)
ui (wi , θ) − ui (xi , θ)
=
=
,
′
′
′
′
ui (wi , θ ) − ui (xi , θ )
ui (zi , θ ) − ui (vi , θ )
ui (yi , θ ′ ) − ui (xi , θ ′ )
where the first equality follows from (9) and (θ, θ ′ ) ∈ Ni , and the second from (θ, θ ′ ) ∈ Ni . But
by (4), the first fraction cannot equal the third fraction. Hence if (θ, θ ′ ) ∈ Ni then ξi (θ) 6= ξ(θ ′ ).
In summary, by choosing δ sufficiently small compared to γ, we can guarantee that 3) holds. B.7
Proof of Lemma 8
Proof. Let
o0i ≡ ziM .
Let εni > 0 be sufficiently small so that
oni i ≡ εni ziH +
and εni z̃iH + 1 − εni − εni
H, M, L, and let
1 − εni − εni
n
n −1
ξi i +ξi i
2
πini ≡
ξ ni + ξini −1 L
ξini + ξini −1 M
zi ∈ Yi
zi + εni i
2
2
n
n −1
z̃iM + εni
ξi i +ξi i
2
min
ui (oni i , θ) − ui (o0i , θ).
n
θ∈Θ s.t. ξi (θ)=ξi i
48
z̃iL ∈ Ỹ ′ for some z̃iA ∈ gi−1 (ziA ), A =
Recursively, for k = ni − 1, . . . , 1, let
oki ≡ εk ziH +
1 − εk − εk
πik ≡
min
ξik + ξik−1 M
ξ k + ξik−1 L
zi ,
zi + εk i
2
2
θ∈Θ s.t. ξi (θ)=ξik
ui (oki, θ) − ui (o0i , θ),
where ξ 0 ≡ ξ 1 − 1 and εk > 0 is small enough so that
ξ k +ξ k−1
ξ k +ξ k−1 M
z̃i + εk i 2i z̃iL ∈ Ỹ ′ and
εk i 2i
max
θ∈Θ s.t.
ξi (θ)>ξik
oki
∈ Yi and εk z̃iH + 1 − εk −
ui (oki, θ) − ui (o0i , θ) < min{πik+1 , . . . , πini }.
(10)
Using the linearity property of ui (see the proof of Lemma 7), it is easy to see that
ui (oki, θ) − ui (o0i , θ) R 0
⇔
ξi (θ) R
ξik + ξik−1
,
2
∀θ ∈ Θ, k ∈ {1, . . . , ni }.
′
This implies that εk is well-defined, as πik > 0 for all k′ > k, for all k ∈ {1, . . . , ni − 1}. It
also implies that a) is satisfied. Condition (10) together with a) guarantee b).
C
QPCA Environments
In this appendix we show that any QPCA environment corresponds to an environment as
described in Section 3, and therefore that the results on rv-implementation from Section 4
apply to QPCA environments (introduced in Section 5). Formally, we verify that for any
QPCA environment there exist an environment with an outcome space Ỹ which is a space of
lotteries over an appropriately defined set of pure outcomes, and a utility-preserving surjection
g from Ỹ to the outcome space Y of the QPCA environment. Because of the utility-preserving
surjection, we can view the outcome space Y of a QPCA environment as the “reduced form”
of the outcome space Ỹ .
Let Y be the outcome space of a QPCA environment with B > 0 and valuation functions
vi : Θ → R, i ∈ I. Define a set of pure outcomes by
X
q̃i ≤ 1},
X̃ = {(q̃, t̃) = (q̃1 , . . . , q̃I , t̃1 , . . . , t̃I ) ∈ {0, 1}I × {−B, B}I :
i∈I
and a von Neumann Morgenstern utility function for agent i by
ũi : X̃ × Θ → R, ((q̃, t̃), θ) 7→ vi (θ)q̃i + t̃i .
Agent i’s preferences over the space Ỹ of lotteries over X̃ are the expected utility preferences
with respect to ũi . Let g : Ỹ → Y map ỹ to the (q, t) for which
X
X
qi =
ỹ(q̃, t̃)q̃i , and ti =
ỹ(q̃, t̃)t̃i ,
∀i ∈ I.
(q̃,t̃)∈X̃
(q̃,t̃)∈X̃
49
Then g is utility-preserving and surjective27 : First, for every i ∈ I and every ỹ ∈ Ỹ ,
X
X
ũi (ỹ, θ) = vi (θ)
ỹ(q̃, t̃)q̃i +
ỹ(q̃, t̃)t̃i = ui (g(ỹ), θ).
(q̃,t̃)∈X̃
Second, let (q, t) ∈ Y . Let q0 = 1 −
(−B)ri + B(1 − ri ). Define ỹ ∈ Ỹ by
ỹ(q̃, t̃) =
(q̃,t̃)∈X̃
P
Y
X
qi q̃i
q0 (1 − q̃i ) +
i∈I
i∈I
and for each i ∈ I let ri ∈ [0, 1] solve ti =
i∈I qi ,
!
Y
1{t̃|t̃i =−B} (t̃)ri + 1{t̃|t̃i =B} (t̃)(1 − ri ) ,
i∈I
where 1A , A ⊆ {−B, B}I , is the indicator function of A. Then g(ỹ) = (q ′ , t′ ) for
Y
X
1{t̃|t̃i =−B} (t̃)ri + 1{t̃|t̃i =B} (t̃)(1 − ri ) = qi ,
qi
qi′ =
t̃∈{−B,B}I
t′i =
X
t̃∈{−B,B}I
i∈I
Y
1{t̃|t̃i =−B} (t̃)ri + 1{t̃|t̃i =B} (t̃)(1 − ri )
t̃i
i∈I
·
X
q̃∈{q̂∈{0,1}I :
P
i∈I
q0
q̂i ≤1}
Y
i∈I
(1 − q̃i ) +
X
i∈I
qi q̃i
!
= (−B)ri + B(1 − ri ) = ti .
That is, g(ỹ) = (q, t).
D
Mechanism of Example 5.1
Define Γ = hH, (Hi )i∈I , P, Ci as follows. The agents publicly and in sequence announce their
payoff types:
H = {h ∈ F : h h′ for some h′ ∈ {0, 1}I },
where F is the set of finite sequences with codomain {0, 1}. The player function is P : H\T →
I such that P (h) = i if lh = i − 1, for all i ∈ I and h ∈ H, and Hi = {{h} : h ∈ Hi }. The
outcome function C : T → Y maps h to the lottery C(h) = (o1, . . . , oI ) such that

I

 oI−1 if h1 = 1 and (h2 , . . . , hI ) 6= (0, . . . , 0)
,
o1 = o1
if h = (1, 0, . . . , 0)

 0
if h1 = 0
o

I


 oI−1 if h1 = 1 and hi = 1
oi = o1
if (h1 = 1 and hi = 0) or (h1 = 0 and hi = 1) ,

I−2

 oI−1
if h = 0 and h = 0
1
i
27
The function g is not bijective. For example, if I = 2, then g maps both ỹ such that ỹ(1, 0, B, −B) =
ỹ(0, 1, −B, B) = 0.5 and ỹ ′ such that ỹ ′ (1, 0, −B, B) = ỹ ′ (0, 1, B, −B) = 0.5 to (q, t) = (0.5, 0.5, 0, 0). This
does not matter, however, as for any y ∈ Y , all agents are indifferent between all elements of g −1 (y).
50
and
oI =

















oI−1
o1
I
o
1
oI−1
I−2
I−1
if h1 = 1, hI = 1 and (h2 , . . . , hI−1 ) 6= (0, . . . , 0)
if (h1 = 1, hI = 0 and (h2 , . . . , hI−1 ) 6= (0, . . . , 0)) or
h = (1, 0, . . . , 0, 1) or (h1 = 0 and hI = 1)
,
if h1 = 0 and hI = 0
if h = (1, 0, . . . , 0, 0)
where i ∈ {2, . . . , I − 1} and om is defined as in the proof of Proposition 4 (we omit subscripts
since the options are the same for all agents).
References
Abreu, D., and H. Matsushima (1992a): “Virtual Implementation in Iteratively Undominated Strategies: Complete Information,” Econometrica, 60(5), 993–1008.
(1992b): “Virtual Implementation in Iteratively Undominated Strategies: Incomplete
Information,” Working Paper.
Artemov, G., T. Kunimoto, and R. Serrano (2013): “Robust Virtual Implementation:
Toward a Reinterpretation of the Wilson Doctrine,” Journal of Economic Theory, 148, 424–
447.
Battigalli, P. (1999): “Rationalizability in Incomplete Information Games,” Working Paper.
(2003): “Rationalizability in infinite, dynamic games with incomplete information,”
Research in Economics, 57, 1–38.
Battigalli, P., A. Di Tillio, E. Grillo, and A. Penta (2011): “Interactive Epistemology
and Solution Concepts for Games with Asymmetric Information,” The B.E. Journal of
Theoretical Economics, 11, Article 6.
Battigalli, P., and A. Friedenberg (2012): “Forward induction reasoning revisited,”
Theoretical Economics, 7, 57–98.
Battigalli, P., and M. Siniscalchi (1999): “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games,” Journal of Economic Theory, 88, 188–230.
(2002): “Strong Belief and Forward Induction Reasoning,” Journal of Economic
Theory, 106, 356–391.
(2003): “Rationalization and Incomplete Information,” Advances in Theoretical Economics, 3(1), Article 3.
51
Bergemann, D., and S. Morris (2005): “Robust Mechanism Design,” Econometrica, 73(6),
1771–1813.
(2007): “An Ascending Auction for Interdependent Values: Uniqueness and Robustness to Strategic Uncertainty,” American Economic Review Papers and Proceedings, 97(2),
125–130.
(2009a): “Robust Implementation in Direct Mechanisms,” Review of Economic Studies, 76, 1175–1204.
(2009b): “Robust virtual implementation,” Theoretical Economics, 4(1), 45–88.
(2011): “Robust implementation in general mechanisms,” Games and Economic Behavior, 71(2), 261–281.
Bikhchandani, S. (2006): “Ex post implementation in environments with private goods,”
Theoretical Economics, 1, 369–393.
Brandenburger, A., and E. Dekel (1987): “Rationalizability and Correlated Equilibria,”
Econometrica, 55(6), 1391–1402.
Chen, J., and S. Micali (2011): “The Robustness of Extensive-Form Rationalizability,”
Working Paper.
Chung, K.-S., and J. C. Ely (2007): “Foundations of Dominant-Strategy Mechanisms,”
Review of Economic Studies, 74, 447–476.
Dasgupta, P., and E. S. Maskin (2000): “Efficient Auctions,” Quarterly Journal of Economics, 115(2), 341–388.
Dekel, E., D. Fudenberg, and S. Morris (2007): “Interim correlated rationalizability,”
Theoretical Economics, 2, 15–40.
Jehiel, P., M. Meyer-Ter-Vehn, B. Moldovanu, and W. R. Zame (2006): “The Limits
of Ex Post Implementation,” Econometrica, 74(3), 585–610.
Kuhn, H. W. (1953): “Extensive Games and the Problem of Information,” in Contributions
to the Theory of Games, Vol. II, ed. by H. W. Kuhn, and A. W. Tucker, vol. 28 of Annals
of Mathematics Studies, pp. 193–216. Princeton University Press.
Müller, C. (2010): “Robust Implementation in Dynamic Mechanisms,” Ph.D. thesis, University of Minnesota.
52
(2012): “Robust Implementation in Weakly Rationalizable Strategies,” Working Paper.
Osborne, M. J., and A. Rubinstein (1994): A Course in Game Theory. The MIT Press.
Pearce, D. G. (1984): “Rationalizable Strategic Behavior and the Problem of Perfection,”
Econometrica, 52(4), 1029–1050.
Penta, A. (2013): “Robust Dynamic Mechanism Design,” Working Paper.
Perea, A. (2013): “Belief in the Opponents’ Future Rationality,” Working Paper.
Perry, M., and P. J. Reny (1999): “A General Solution to King Solomon’s Dilemma,”
Games and Economic Behavior, 26, 279–285.
Rényi, A. (1955): “On a new axiomatic theory of probability,” Acta Mathematica Academiae
Scientiarum Hungaricae, 6, 285–335.
Shimoji, M., and J. Watson (1998): “Conditional Dominance, Rationalizability, and Game
Forms,” Journal of Economic Theory, 83, 161–195.
Wilson, R. (1987): “Game-theoretic analyses of trading processes,” in Advances in Economic Theory: Fifth World Congress, ed. by T. F. Bewley, vol. 12 of Econometric Society
Monographs, chap. 2, pp. 33–70. Cambridge University Press.
53