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Mezzetti, Claudio and Renou, Ludovic. (2017) Repeated Nash implementation. Theoretical Economics, 12 (1). pp. 249-285.
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Repeated Nash implementation
ClaudioMezzetti
School of Economics, University of Queensland
LudovicRenou
School of Economics and Finance, Queen Mary University of London
We study the repeated implementation of social choice functions in environments with complete information and changing preferences. We definedynamic mono-tonicity, a natural but nontrivial dynamic extension of Maskin monotonicity, and show that it is necessary and almost sufficient for repeated Nash implementation, regardless of whether the horizon is finite or infinite and whether the discount factor is “large” or “small.”
Keywords. Dynamic monotonicity, Nash implementation, Maskin monotonic-ity, repeated implementation, repeated games.
JELclassification. C72, D71.
1. Introduction
Many economic and social interactions are repeated: the same buyers and sellers of-ten trade with one another multiple times, teams of contractors regularly work for the same procurement agencies, and voters repeatedly elect representatives, to name just a few. The central theme of this paper is the design of institutions, or contractual arrange-ments, that generate “socially desirable” outcomes in settings where agents repeatedly interact and preferences change over time.
To illustrate the type of economic problems this paper addresses, consider for ex-ample the situation in which a buyer and a seller interact more than once. Are there contractual arrangements that (in all equilibria) allow the seller to extract all the surplus from trade? As another example, consider the case in which two (or more) agents may work on a number of tasks that are profitable to a principal. Can we design arrange-ments that (again, in all equilibria) induce the agents to work on the most profitable tasks at each point in time, even if it is costly to them? In all these problems, an es-sential difficulty is the multiplicity of equilibria, including “undesirable” equilibria, that repeated interactions make possible to sustain. The aim of the paper is to character-ize the social outcomes that are implementable; that is, those outcomes for which there exist contractual arrangements that only yield equilibria consistent with them.
Claudio Mezzetti:[email protected]
Ludovic Renou:[email protected]
We would like to thank an associate editor and anonymous referees for their insightful comments and sug-gestions. Mezzetti’s work was funded in part by Australian Research Council Grant DP120102697.
More formally, we study the problem of repeated, full implementation of social choice functions in environments with complete information and a changing state of the world. A social choice function is repeatedly implementable in Nash equilibrium if there exists a sequence of (possibly history-dependent) mechanisms such that for any period, for any profile of preferences at that period, the set of equilibrium outcomes corresponds to the social choice function at that profile of preferences.
Full implementation in a static environment (i.e., with a single period) has been ex-tensively studied.1The seminal contribution isMaskin(1999), which states that Maskin monotonicity is necessary and almost sufficient for full implementation. In this paper, we provide a condition, calleddynamic monotonicity, and show that it is necessary and almost sufficient for repeated Nash implementation, regardless of whether the horizon is finite or infinite and whether the discount factor is “large” or “small.”
Dynamic monotonicity is a natural but nontrivial dynamic extension of Maskin monotonicity. It reduces to Maskin monotonicity in single-period settings, but is weaker in all other finitely repeated implementation problems. Thus, perhaps surprisingly, finitely repeated implementation is “easier” to achieve than single-shot implementa-tion. For example, while full-surplus extraction by a seller cannot be implemented in a static problem, it can be if there are at least two periods in which the buyer and the seller interact (seeExample 1inSection 3).
We also show that in infinitely repeated problems with patient enough agents, dy-namic monotonicity implies that the social choice function is weakly efficient from the agents’ point of view. However, no efficiency condition is necessary in infinitely repeated problems with impatient enough agents and in all finitely repeated problems. For exam-ple, collusion among agents in a team can be deterred in all finite horizon problems and in infinite horizon problems with impatient enough agents (seeExample 2inSection 3). In a repeated implementation problem, the designer’s choice of a mechanism in each period may depend on the agents’ actions and mechanisms in all previous periods; agents need not be playing the same stage game in each period. Intuitively, contractual arrangements may be used to compensate an agent when he deviates before periodt from a collusive strategy profile that would induce socially undesirable outcomes from periodtonward. This possibility of inducing preemptive deviations from future collu-sion facilitates implementation and is the reason why finitely repeated implementation is easier than static implementation. Indeed, it is only when the horizon is infinite and the discount factor is close to1that the gain from a future collusive agreement domi-nates any preemptive punishment and only outcomes that are efficient for the agents can be implemented. This insight is at the heart ofLee and Sabourian’s (2011) work on infinitely repeated implementation problems (to be discussed shortly).
Unlike the literature on dynamic mechanism design, which has recently seen a flurry of papers (e.g., see the survey byBergemann and Said 2011), the literature on full imple-mentation in dynamic environments is in its infancy. Two papers have studied repeated setting where, unlike in this paper, the state of the world does not change over time.
1SeeJackson(2001),Maskin and Sjöström(2002), andSerrano(2004) for recent surveys on
Kalai and Ledyard(1998) study infinitely repeated implementation in dominant strate-gies; they show that every social choice function can be repeatedly implemented start-ing from some (possibly distant) point in the future. Chambers (2004) studies virtual repeated Nash implementation in continuous time.
In an important recent paper,Lee and Sabourian(2011) consider environments in which, like in our paper, the state of the world changes over time.2 Unlike us, they
fo-cus on infinitely repeated settings with patient agents; that is, agents with a discount factor arbitrarily close to1. Their main result is that weak efficiency of the social choice function relative to any other function with an equal or smaller range is necessary for infinitely repeated implementation. Under some mild additional assumptions on the environment, they also show that if the discount factor is larger than1/2, then strict effi-ciency in the range is sufficient for infinitely repeated implementation from period two onward (but the designer may fail to implement the correct outcome in the first period). Maskin monotonicity and weak efficiency in the range are very different conditions, and thus it is perhaps a puzzle that the first is necessary and almost sufficient in the static case and the second is necessary and almost sufficient in the polar case of infi-nite interactions with patient enough agents. In this paper, we solve this puzzle by in-troducing the condition of dynamic monotonicity and showing that it is necessary and almost sufficient in all repeated implementation problems, including the so-far unex-plored, but clearly empirically important, case of a finite number of interactions and the case of infinitely repeated interactions with general discount factors. In the static case, dynamic monotonicity is equivalent to Maskin monotonicity. In infinitely repeated problems with an arbitrarily high enough discount factor, dynamic monotonicity is es-sentially equivalent to weak efficiency in the range. As we illustrate in Examples1and 2, neither Maskin monotonicity nor an efficiency condition are necessary for repeated implementation in general.
The paper is organized as follows. Section 2defines the problem of repeated im-plementation.Section 3presents two examples motivating our investigation.Section 4 introduces the condition of dynamic monotonicity.Section 5presents the main results of the paper.Section 6provides some extensions of our results andSection 7concludes. All proofs are given in theAppendix.
2. Definitions
Single-shot implementation.A static orsingle-shotimplementation problemPis a tuple I X (ui)i∈I, whereI= {1 I}is a set ofI agents,X is the set of alternatives— a compact subset of a finite dimensional Euclidean space,is a finite set of states of the world, and for each agenti∈I,ui:X×→Ris a state-dependent continuous utility
function. LetLi(x θ)= {y∈X:ui(x θ)≥ui(y θ)}be agenti’s lower contour set ofxat
stateθ. A social choice function (henceforth, scf )f :→X associates with each state of the worldθthe alternativef (θ)∈X.
2See alsoRenou and Tomala(2015) andLee and Sabourian(2013) for the problem of approximate
A static mechanismGis a pair(MiG)i∈I gwhereMiGis the set of messages of agent
iandg: ×i∈IMiG→Xis the allocation rule. LetMG= ×j∈IMjGandM−Gi= ×j∈I\{i}MjG,
wheremand m−i are generic elements. The mechanismMG gand the state θ
in-duce the strategic-form gameG(θ)= I (ui(g(·) θ) MiG)i∈I. LetNEG(θ)⊆X be the
set of (pure) Nash equilibrium outcomes of the gameG(θ). The social choice function f is single-shot implementable in Nash equilibrium if there exists a static mechanismG such thatNEG(θ)= {f (θ)}for allθ∈.
A necessary and almost sufficient condition for static Nash implementation is Maskin monotonicity. InDefinition 1, we present two equivalent, slightly unusual, for-mulations of Maskin monotonicity, as they foreshadow and will help understanding our definition of dynamic monotonicity. Call any mapπ:→a (static)deceptionand let 1be the set of static deceptions. The interpretation is that when the state isθ, agents act as if the state wereπ(θ)instead.
Definition 1. A social choice functionf is Maskin monotonic when it satisfies(MA) or, equivalently,(MB).
(MA) For allπ∈1, for allθ∈,
∀i∈I Li
fπ(θ) π(θ)⊆Li
fπ(θ) θ ⇒ fπ(θ)=f (θ)
(MB) For allπ∈1, for allθ∈,
fπ(θ)=f (θ)
⇒ ∃(i∈I x∈X):ui
fπ(θ) π(θ)−ui
x π(θ)
≥0> uifπ(θ) θ−ui(x θ)
The intuition for the necessity of Maskin monotonicity is simple. Suppose thatf is implementable and letπbe a deception. At stateπ(θ), there must exist an equilibrium m∗that implementsf (π(θ)). However, iff (π(θ))=f (θ),m∗should not be an equilib-rium at stateθ, so that at least one agent must have a profitable deviation; that is, he must have a unilateral deviation fromm∗that induces an alternativexstrictly preferred tof (π(θ))at stateθ. And sincem∗is an equilibrium at stateπ(θ), the deviation can-not be profitable atπ(θ); that is,f (π(θ))is preferred toxat stateπ(θ). Condition (MB) precisely captures this intuition.
Repeated implementation.Arepeated implementation problem, denotedPT, repre-sents theT-time repetition of the implementation problemP;Tcan be finite or infinite. At the beginning of each periodt∈T = {1 T}, the state of the world is drawn from with probability mass functionp, withp(θ) >0for allθ∈. In each period, the realized state is commonly observed by all agents, but not the designer.
Let(x(t θ))t∈Tθ∈be a sequence of alternatives, wherex(t θ)is the alternative
payoff of agentifrom(x(t θ))t∈Tθ∈is given by3
Ui
x(t θ)t∈Tθ∈= 1−δ
1−δT
t∈T
θ∈
δt−1ui
x(t θ) θp(θ)
The aim of the designer is to repeatedly implement a social choice function f. A dynamic mechanism regime specifies a mechanism in each periodt, contingent on the profile of mechanisms offered and messages played up to periodt (excluding pe-riodt). A designer historyhtDis a sequence of mechanisms and corresponding messages (G1 m1 Gτ mτ Gt−1 mt−1)such thatGτis the mechanism adopted at periodτ
andmτ∈MGτis the corresponding message profile, for allτ < t. The set of all possible
histories observed by the designer at periodtis denotedHtD. The set of initial histories H1
Dis the singleton{∅}and the set of all possible designer histories isHD=
T t=1HtD.
A dynamic mechanism regime, or regime for short, specifies a lottery over static mechanisms as a function of the designer history. We writer(G;htD)for the probability that mechanismGis chosen after historyhtD.4
We assume perfect monitoring.5 At the beginning of periodt, each agent knows the entire profile of mechanisms chosen up to periodt−1, the entire profile of messages sent up to period t−1, the entire profile of states of the world realized up to period t−1, and the periodt’s mechanism selected as well as the realized state of the world for periodt. Writeθt=(θ1 θt−1)for a profile of realized states of the world up to period
t−1. A history for agentiis thusht=(ht
D θt). LetHtbe the set of all possiblet-period
agent histories and ketH=Tt=1Ht be the set of all such histories. The only possible initial history is the empty set:H1= {∅}.
A pure strategysifor agentispecifies a message in each periodt as a function of
the historyht, the mechanismGt currently selected, and the current stateθt; that is,
si(ht Gt θt)∈MiGt for all(ht Gt θt). Lets=(s1 sI)be a strategy profile. The
strat-egy profiles, the random draw of a state in each period, and the regimer generate a random sequence of historiesht.
Given a regimer, we write q(ht;s)for the probability that history ht occurs when
the strategy profile iss. Throughout, we slightly abuse notation and writer(Gt;ht)for
r(Gt;htD)for anyht=(htD θt). The expected payoff of agentiwhen the profile of
strate-gies issis
Ui(s)=
1−δ
1−δT
t∈T
ht∈Ht
Gt∈G
θt∈ δt−1ui
gs(ht Gt θt)
θt
q(ht;s)r(Gt;ht)p(θt)
3When computing payoffs starting from any periodt, we use the normalizing factor(1−δ)/(1−δT−t+1),
so that the discounted payoff fromtis measured on the same scale as the single-shot payoff.
4We assume that, for eachht
D,r(·;htD)has finite support.
5In other words, we assume that the designer truthfully and publicly reveals all his information (i.e.,
A profile of pure strategiess∗=(s∗i s−∗i)is a pure Nash equilibrium of the dynamic game induced by regimerifUi(s∗)≥Ui(si s∗−i)for all strategiessi, for all agentsi∈I
(wheres∗−idenotes the strategy profile of agenti’s opponents).
Definition 2. A social choice function f is repeatedly implementable if there exists a dynamic mechanism regimersuch that (i) there exists a Nash equilibriums∗ of the dynamic game induced byrand (ii) for each Nash equilibriumsinduced byr, we have g(s(ht Gt θt))=f (θt)for allθt∈, for all(ht Gt)such thatq(ht;s) >0, andr(Gt;ht) >
0, for allt∈T.
Intuitively, a social choice function is repeatedly implementable if we can construct a dynamic mechanism whose unique equilibrium outcome isf (θ)in all periods where the state isθ. As is customary in the literature,Definition 2does not rule out mixed strategy equilibria with outcome realizations different fromf (θ). In Section 6we will show that it is possible to rule out such undesirable mixed strategy equilibria.
We end this section with three notions of efficiency of an scf. The expected payoff of agentiwhenfis repeatedly implemented isvfi =θ∈ui(f (θ) θ)p(θ). LetF(f )= {f:
→X:f()⊆f ()}be the set of social choice functions with a range (weakly) smaller thanf, letV (f )= {(vi)i∈I: vi=vf
i for alli∈I, for somef∈F(f )}be the associated
(expected) payoff profiles, and let co(V (f ))be the convex hull ofV (f ).
The social choice functionf isweakly efficient in the rangeif there does not exist a payoff profile(vi)i∈I∈co(V (f ))such thatvi> vfi for alli∈I;f isefficient in the rangeif
there does not exist a payoff profile(vi)i∈I∈co(V (f ))such thatvi≥vfi for alli∈Iand
vi> vfi for somei∈I;f isstrictly efficient in the range if it is efficient in the range and
there exists nof∈F(f ),f=f, such thatvif=vfi for alli∈I.6
3. Two examples
This section illustrates repeated implementation with the help of two simple examples.
Example1 (Trading a Good). This is a multiperiod variation of the leading example of Aghion et al.(2012).
There are two periods,t=12, a buyerB, and a sellerS. In each period, the seller has a good for sale; the qualityθof the good is independently drawn in each period and equally likely to beθL=10orθH=14. The buyer and the seller have a common discount
factorδand observe the good’s quality at the beginning of each period.
As in Aghion et al., payments to and from a third party are allowed. Hence, the set of outcomesX is the set of triplets(z pB pS)withz∈ {01}representing whether the
good is traded(z=1)or not(z=0),pB∈P representing the price paid by the buyer,
andpS∈Prepresenting the price paid to the seller, wherePis a (arbitrarily large) closed
interval inR. For any outcome(z pB pS), the (per-period) buyer’s utility isu(zθ−pB)
Seller
Buyer
θL NθL NθH θH
θL (11010) (11010) (011) (011) →G2 →(11111) →(000) →(000)
NθL
(11410) (11414) (100) (011) →(0−y0) →(11414) →(100) →(000)
NθH (011) (100) (11414) (11410) →(000) →(100) →(11414) →(000)
θH (011) (011) (11414) (11414) →(000) →(000) →(11111) →G2
Table1. The first-period allocation and the transition →to the second-period allocation or game played inExample 1.
when the good quality isθ, withu(0)=0andua strictly increasing, strictly concave function. The seller’s utility ispS.
We want to implement the efficient allocation prescribing that in each period the good is traded and the buyer pays the seller the true quality,pB=pS=θ; that is, the
scf we want to implement isf (θL)=(11010)andf (θH)=(11414).7 Sincef is not
Maskin monotonic, it cannot be implemented in Nash equilibrium in a static setting.8 We now present a simple dynamic mechanism that repeatedly implements f in Nash equilibrium. In the first period, the buyer and the seller report a message in {θL NθL NθH θH}. We interpret the reportθk as stating that “the quality isθk.” The
reportsNθk are objections that lead to different first-period allocations and
second-period mechanisms than announcing eitherθHorθL. In the second period, the buyer
and the seller have the opportunity to make an additional report in{θL θH}if and only
if they have reported the same quality in the first period. In all other cases, the second-period allocation is chosen without requiring buyer and seller to make reports. Table 1 gives the allocation rule in the first period along with the regime.
Table 1has16cells, one for each possible report profile in the first period; the row (resp., column) report is the buyer (resp., seller) report. Each cell has two elements. The top element gives the first-period allocation, while the bottom element (indicated with the symbol→) gives the transition to the second-period mechanism. For instance, if the buyer reportsθLand the seller reportsNθL, the first-period allocation is(11010),
while the second-period mechanism implements(11111) and requires no second-period reports. When the buyer and the seller report the same quality in the first second-period, the second-period mechanism isG2, given inTable 2:9
7Note that ifu(0)=1, then this allocation also maximizes total surplus. 8Formally, we have that L
B(f (θL) θL)= {(z pB pS):u(0)≥u(zθL−pB)} ⊆ {(z pB pS):u(4)≥ u(zθH−pB)} =LB(f (θL) θH), whileLS(f (θL) θL)=LS(f (θL) θH). Sincef (θL)=f (θH), we have a
vio-lation of Maskin monotonicity.
9On the equilibrium path, mechanismG
2guarantees that trade takes place in the second period and the
θL θH
θL (11010) (000) θH (000) (11414)
Table2. The second-period mechanismG2inExample 1.
We claim that wheneveryis chosen so that−u(−4) > δu(y) > u(4), the unique pure strategy equilibrium implements the efficient allocation in both periods.10 This is ver-ified in theAppendix, which presents the two reduced strategic-form games that are obtained by conditioning on the first-period quality.11
A notable feature of our mechanism is that it provides at least one agent with the incentive to deviate early (att=1) from future (att=2) coordination on undesirable equilibria (coordinating on announcingθLwhen the good’s quality isθH). It is precisely
the ability to provide such incentives in a dynamic setting that allows the repeated Nash implementation of social choice functions, like the one in this example, that are not
implementable in a static setting. ♦
Example2 (Task assignment). In each of a possibly infinite number of periods, a princi-pal needs to assign two agents (experts),1and2, to one of two tasks,AandB. There are two states of the world,θ∈ {θA θB}. The agents know the state of the world, but not the
principal. In stateθA(resp.,θB), taskA(resp.,B) yields the principal a benefitvgreater
than the cost to undertake it, while the other task yields zero benefit and cost. An allo-cation is a quadruplet(a1 a2 w1 w2), withai∈ {A B}the assignment of agenti∈ {12}
andwi≥0his wage. When the state isθ, the assignment is (ai a−i), and the wage is
wi, agenti’s payoff iswi−ci(ai a−i θ), whereci(ai a−i θ)is agenti’s cost of executing
taskaiwhen the other agent is assigned to taska−i, at stateθ. There are
complementar-ities: the more agents work on a task, the less costly it is:ci(ai a−i θ)=1if(ai a−i θ)=
(A A θA)=(B B θB),ci(ai a−i θ)=3if(ai a−i θ)=(A B θA)=(B A θB), and the
cost is zero otherwise; in addition,vis sufficiently large, e.g.,v >4, so that it is profitable for the principal to induce the agents to work on the right task.
The principal wants to maximize his ex post profit in each period, subject to giving the agents at least their per-period outside option payoff, which we normalize to zero. This corresponds to the scfsf (θA)=(A A11)andf (θB)=(B B11). Note thatf
maximizes social surplus in each period and state.
The scff is Maskin monotonic, but it is not efficient relative to social choice func-tions having (weakly) smaller ranges. For instance, the funcfunc-tionsf∗(θA)=(B B11)
andf∗(θB)=(A A11), with agents being paid to work on the unprofitable task, give
a strictly higher expected utility to both agents thanf. Thus, if the agents are sufficiently patient, thenf cannot be repeatedly implemented in infinite horizon problems (Theo-rem1,Lee and Sabourian 2011).
10The existence ofyfollows from observing thatu(4)+u(−4) <0, sinceuis strictly concave andu(0)=0. 11As we argue inSection 6, undesirable mixed strategy equilibria could also be ruled out, at the cost of
Agent2
Agent1
θA θB
θA (A A11) (B A22)
θB (A B22) (B B11)
Table3. The static mechanism inExample 2.
θA θB
Agent1 Agent2 Agent1 Agent2
(A A11) 0 0 1 1
(B B11) 1 1 0 0
(B A22) 2 −1 −1 2
(A B22) −1 2 2 −1
Table4. Agents’ payoffs inExample 2.
At the end ofSection 5we will show thatf is infinitely repeatedly implementable if the discount factor is not too large. We now argue that thefis repeatedly implementable in any finite horizon problem. Consider the static mechanism where each agent has two messages,θAandθB, and the allocation rule is represented as inTable 3;Table 4displays
the payoffs to each agent of each alternative in each state.
At state θA, the mechanism induces a prisoner’s dilemma, with (θA θA) as the
unique Nash equilibrium and equilibrium outcome(A A11). Similarly, at stateθB,
the mechanism induces a prisoner’s dilemma, with(θB θB)as the unique Nash
equi-librium and equiequi-librium outcome(B B11). Sofis implementable whenT =1. More fundamentally, at statesθA andθB, the unique equilibrium payoff coincides with the
min-max payoff. Consequently, repeated play of the stage game equilibrium is the only Nash equilibrium of the finitely repeated game (e.g., seeBenoît and Krishna,1987, and González-Díaz 2006), and by selecting the mechanism regime that uses the static mech-anism in each round, f can be finitely repeatedly implemented in Nash equilibrium, regardless of the number of periods.
This shows that there is an important difference between what can be implemented in finitely repeated problems and what can be implemented in infinitely repeated prob-lems with an arbitrarily large discount factor, as studied byLee and Sabourian(2011).♦
4. Dynamic monotonicity
Consider any periodt and any sequence(uτi)τ≥t of payoffs from periodtonward. We
can write agenti’s discounted payoff at periodtas
1−δ
1−δT−t+1 u t i+δ
T
τ=t+1
δτ−t−1uτi
wherevi(t)is the (normalized) discounted continuation payoff andβtT is the
(normal-ized) discount factor at periodt: that is,
vi(t)= 1−δ
1−δT−t T
τ=t+1
δτ−t−1uτi and βtT =δ−δ T−t+1
1−δT−t+1
When the horizon is infinite, i.e., T = ∞, we haveβt∞=δ. The lowest and highest
expected payoff agentican obtain are
vi=
θ∈
min
x∈Xui(x θ)p(θ) vi=
θ∈
max
x∈Xui(x θ)p(θ)
For each t∈T \ {T}, let Vi(t) be the closed interval [vi vi]with the convention that
Vi(T )= {0}if T <∞. The set Vi(t) corresponds to the set of feasible agenti’s
(nor-malized) continuation payoffs at periodt. Denote byvfi(t)the (normalized) expected discounted payoff of agentiwhenf is implemented from periodt+1onward. Thus, vfi(t)=vfi =θ∈ui(f (θ) θ)p(θ)ift < T andvfi(T )=0ifT <∞.
We now generalize the important concept ofdeceptionto the dynamic setting. At each periodt, a deception specifies a stateθˆt as a function of the realized stateθt and
the history of realized states up to periodt,θt. Formally, a deceptionπis a sequence of maps(πt:t×→)Tt=1. Intuitively, suppose that each agent is asked to directly
re-port a state at each period (as in a direct mechanism). A deception then corresponds to a situation where the agents coordinate their reports toθˆt=πt(θt θt)at periodt, when
the current state isθtand the profile of realized states isθt.12 (If reports are not
coordi-nated, the designer detects a lie and can punish the agents.) Of course, the mechanism does not have to be direct. Nonetheless, the concept of a deception remains important: agents can play at periodtand realized statesθt as if the current state isπt(θt θt)and
notθt. A special deception isπ∗, given byπt∗(θt θt)=θtfor all(θt θt), for allt. This
corresponds to truth-telling. LetT be the set of deceptions.
We define the (normalized) expected discounted continuation payoff of agentifrom following the deceptionπafter state history(θt θt)recursively as
vfπ
i (θt θt)=
θt+1∈
(1−βt+1T)uifπt+1
(θt θt) θt+1
θt+1
+βt+1Tvfiπ(θt θt) θt+1p(θt+1)
This is agenti’s discounted continuation payoff if, in all periodsτ > t, the designer uses the social choice functionf at the reported stateπτ(θτ θτ)to determine the periodτ
alternative. Note that the discounted continuation payoff vfπ∗
i (θt θt)from the
truth-telling deceptionπ∗is equal tovfi(t), regardless of(θt θt).
For any history of realized statesθtand deceptionπ, we define the dynamic lower
contour set ofxatθtas
Lfπ
iθt(x θt)=
y vi(t)
∈X×Vi(t):(1−βtT)ui(y θt)+βtTvi(t)
≤(1−βtT)ui(x θt)+βtTvfiπ(θt θt)
Dynamic lower contour sets are defined in the space of alternatives and continuation payoffs. Intuitively, for any deceptionπand history of statesθt, the dynamic lower con-tour set atθtis composed of all the pairs of alternatives and continuation payoffs that
give agentia smaller expected discounted payoff than whenxis implemented at state θtin periodtand agents continue to follow the deceptionπfrom periodt+1onward.
Note thatLfπ∗
iθt(x θt)does not depend onθt, since the truth-telling deceptionπ∗does
not. With a slight abuse of notation, we therefore writeLfit(x θt)forLfiθπ∗t(x θt).
We are now ready to present two equivalent definitions of dynamic monotonicity, the dynamic generalization of Maskin monotonicity.
Definition3 (Dynamic monotonicity). A social choice functionf is dynamic mono-tonic if it satisfies(DMA)or, equivalently,(DMB).
(DMA) For allπ∈T, for allθT∈T,
∀(i∈I t∈T) Lfitfπt(θt θt) πt(θt θt)⊆Lfiθπt
fπt(θt θt) θt
⇒ ∀(t∈T) fπt(θt θt)=f (θt)
(DMB) For allπ∈T, for allθT∈T,
∃(t∈T):fπ
t(θt
θt)
=f (θt)
⇒ ∃i∈I t∈T x∈X vi∈Vi(t):
(1−βtT)uifπt(θt θt) πt(θt θt)−uix πt(θt θt)
+βtT
vfi(t)−vi
≥0
0> (1−βtT)uifπt(θt θt) θt−ui(x θt)
+βtT
vfπ
i (θt θt)−vi
Intuitively, dynamic monotonicity says that if agents coordinate on a deception that induces an undesirable alternative at some periodt(for some profile of realized states), then at least one agent must have a profitable deviation starting at some timet. Since the problem is dynamic, the profitable deviation does not have to start att; it could start before or after;tneed not equalt. For instance, inExample 1, the seller has a profitable deviation at the first period from the second-period coordination on trading the high quality good at the low price.
It is worth noting that we can restrict attention to deceptions that weakly domi-nate truth-telling in checking for dynamic monotonicity, i.e., to deceptionsπsuch that vfπ
i (θt θt)≥v f
A few additional observations are worth making. First, for T =1, dynamic mono-tonicity reduces to Maskin monomono-tonicity. Second, observe that whenT = ∞,βtT =δ
for allt, and the dynamic lower contour sets do not vary witht. Consequently, when checking for dynamic monotonicity, it is sufficient to considert=1. Third, an easy-to-check sufficient condition for dynamic monotonicity is as follows. For each agenti, definevmaxi =maxπ:→θui(f (π(θ)) θ)p(θ)as the highest payoff that agentican
ob-tain if all agents coordinate on the most favorable static deceptionπ for agenti(vmaxi is also the highest payoff that agent ican obtain by maximizing over all dynamic de-ceptions). Suppose thatf (θ)=f (θ∗). Using (DMB), a sufficient condition for dynamic
monotonicity is that for all deceptions such thatπt(θt
θ∗)=θfor someθt ∈t and
t∈T, there existt∈T,i∈I,x∈X, andvi(t)∈Vi(t)that satisfy
(1−βtT)uif (θ) θ+βtTvif≥(1−βtT)ui(x θ)+βtTvi(t)
and
(1−βtT)ui
f (θ) θ∗+βtTvmaxi < (1−βtT)ui(x θ∗)+βtTvi(t)
The example illustratingRemark 5shows that this condition is easy to check.
We end this section with a series of remarks. The message we want to convey is that dynamic monotonicity is the “general” condition for repeated Nash implementation. It reduces to Maskin monotonicity when there is a single period and essentially corre-sponds toLee and Sabourian’s (2011) efficiency in the range when there are an infinite number of periods and a discount factor close to1.
The first remark gives another easy-to-check sufficient condition for the dynamic monotonicity of a social choice function. The second remark states that, in finite hori-zon problems, dynamic monotonicity is weaker than Maskin monotonicity. The con-verse is false;Example 1demonstrates that dynamic monotonicity is strictly weaker than Maskin monotonicity.
Remark1. If the social choice functionf is strictly efficient in the range and(vfi)i∈Iis
an extreme point of co(V (f )), thenfis dynamic monotonic wheneverT≥2.
Remark2. SupposeT <∞. Iffis Maskin monotonic, then it is dynamic monotonic.
Remark3. SupposeT= ∞. There existsδH∈(01)such that for allδ∈(δH1), iff is
dynamic monotonic, then it is weakly efficient in the range.
Remark4. SupposeT = ∞. Iffis Maskin monotonic and efficient in the range, then it is dynamic monotonic.
Remark 5. There are social choice functions, which are neither efficient nor Maskin monotonic, and yet are dynamically monotonic.
θ θ
a 33 17
b 60 33
c 1010 1010 d −10−10 00 e 00 −10−10
Table5. Agents’ payoffs in the example illustratingRemark 5.
The social choice functions aref (θ)=a andf (θ)=b, and the associated payoff profile is(vf1 v2f)=(33). It is not Maskin monotonic sinceLi(f (θ) θ)⊆Li(f (θ) θ)
for alli, and yetf (θ)=f (θ). It is also not efficient in the range since if players coordinate onθ(resp.,θ) when the state isθ(resp.,θ), then they each obtain a payoff of7/2. Yet,f is dynamic monotonic. To see this, remember thatvimaxis the highest payoff that agenti can obtain if all agents coordinate on the most favorable deception for agenti, and note thatv1max=9/2, whilevmax2 =5. It is immediate to check that the pair(d10)satisfies
u1
f (θ) θ+vf1=3+3≥ −10+10=u1(d θ)+v1
maxu1
f (θ) θ u1
f (θ) θ+v1max=3+9/2<0+10=u1(d θ)+v1
Similarly, the pair(e10)satisfies
u2f (θ) θ+vf2=3+3≥ −10+10=u2(e θ)+v2
maxu2f (θ) θ u2f (θ) θ+v2max=3+5<0+10=u1(e θ)+v2
We have the necessary preference reversals in the first period and, therefore, the social choice function is dynamic monotonic.
The final remark states that in finitely repeated settings the set of social choice func-tions that are dynamic monotonic is weakly increasing inT.
Remark6. SupposeT <∞andf is dynamic monotonic overT periods. Thenfis also dynamic monotonic overT+1periods.13
5. Main results
This section presents our main results, stating that dynamic monotonicity is necessary and almost sufficient for repeated Nash implementation. We begin with necessity.
Theorem1 (Necessity). If the social choice functionfis repeatedly implementable, then it is dynamic monotonic.
The intuition forTheorem 1is simple and analogous to the intuition for the neces-sity of Maskin monotonicity in static implementation problems. If the social choice
function f is implementable, there must exist a mechanism and an equilibrium such thatf (θt)is implemented at periodtand stateθt, and the continuation payoff to any
agent i isvfi(t), for anyt∈T. Moreover, for any realized profile of states θt, all de-viations at period t and state θt must give to agent ian alternative x and a
contin-uation payoffvi inLfiθt(f (θt) θt). Consider a deceptionπ and a “collusive”
equilib-rium in which agents follow the deception (on the equilibequilib-rium path) and revert to the original equilibrium after unilateral deviations. In particular, agents pretend that the state isπt(θt θ∗t)=θtwhen the realized state at period tisθ∗t and the history of
real-ized states up to periodtisθt. As a result,f (θt)=f (πt(θt θ∗t))is implemented attin
stateθ∗t, and the expected payoff of agentiis(1−βtT)ui(f (θt) θ∗t)+βtTvfiπ(θt θ∗t). If
Litf (f (θt) θt)⊆Liθfπt(f (θt) θt∗), then agentihas no profitable deviation from the
collu-sive equilibrium. For otherwise, he would have had a profitable deviation at stateθtfrom
the original equilibrium. Hence, forfto be implemented, it must be thatf (θ∗t)=f (θt);
that is,f must be dynamic monotonic.
We now consider sufficient conditions. As in static implementation problems, we distinguish between the case of two and more than two agents. We need to introduce some additional definitions.
For eachY ⊆X, definemaxθiY = {x∈Y :ui(x θ)≥ui(y θ)for ally∈Y}as agenti’s
maximal set inY at stateθ. A social choice functionf satisfiesno-veto power if, for all θ∈,x∈maxθiXfor alli∈I∗with|I∗| ≥I−1impliesf (θ)=x. Maskin monotonicity and no-veto power are sufficient for static Nash implementation when there are at least three agents. A similar results holds in the repeated setting once we replace Maskin monotonicity with dynamic monotonicity.
Theorem 2 (Sufficiency I ≥3). Let I ≥3. If the social choice function f is dynamic monotonic and satisfies no-veto power, then it is repeatedly implementable.
The proof is constructive. The main building block of our construction is the static mechanism G∗, a close relative toMaskin’s (1999) canonical mechanism. The mech-anism G∗ requires the agents to report a state, an alternative, a continuation payoff, and an integer. At periodt, “unanimous” reports(θt f (θt) vfi(t)0)result in the
realiza-tion off (θt)and in the adoption ofG∗in the next period. A unilateral deviation from
unanimity by agentj att,(θjt xjt vjt njt), results in the realization ofxjt att and
in the continuation payoffvjt thereafter, if(xjt vjt)is in agentj’s dynamic contour set
Lfjt(f (θt) θt)(whereθt is the common state report of all agents but agentj).
Alterna-tively, the deviation results in the realization off (θt)at periodtand in the continuation
payoffvfj(t)thereafter. To guarantee that agentjobtainsvjt(orvjf(t)) in the future, the
regime appropriately randomizes between adopting a mechanism where agentjis dic-tatorial (i.e., chooses the alternative), which would guarantee he receivesvj, and a
pun-ishment mechanism where agentjwould get less thanvjt(orvfj(t)). Any other report
our mechanism regime does not rule out undesirable mixed strategy equilibria. As we discuss inSection 6, under a mild additional assumption we can eliminate them.14
The dynamic mechanism regime we construct only uses stage mechanisms that are deterministic functions of the agents’ messages, but permits random transitions be-tween these mechanisms. Without making further assumptions, it seems impossible to proveTheorem 2without the help of stochastic transitions or, alternatively, stochastic stage mechanisms.15 Yet, in environments with transfers and quasi-linear preferences, there is no need for stochastic transitions; we can always adjust the transfers to guaran-tee that the agent obtains the appropriate continuation payoff.
AsMaskin’s (1999) theorem for static Nash implementation,Theorem 2requires no-veto power. We can weaken the no-no-veto power requirement. For instance,Theorem 2 remains valid if we replace no-veto power withAssumption A, stated below, which is closely related to the conditionsμ(ii) andμ(iii) of Moore and Repullo.16 We first need some additional notation. Letϕt: {t+1 T} ×→X be a time-dependent social
choice function and writevϕt
i the continuation payoff of implementingϕt from period
t+1onward, that is,
vϕt
i :=
1−δ
1−δT−t T
τ=t+1
δτ−t−1uiϕt(τ θ) θp(θ)
For anyvi∈Vi(t), defineλ(vi)=(vi−vi)/(vi−vi). We are now ready to state
Assump-tion A.
AssumptionA. A social choice functionfsatisfiesAssumption Aif the following state-ments hold:
(i) For all(x vi(t))∈Lit(f (θ) θ)withi∈I,θ∈andt∈T and for all pairs(ϕt ϕt)
such that
(a1) eitherλ(vi(t))=0orϕt(τ θ)∈jmaxθj Xfor allθ∈, for allτ > t17
(a2) ϕt(τ θ)∈
j=imaxθjXfor allθ∈, for allτ > t
(b) x∈j=imaxθj∗X
(c) βtTui(x θ∗)+(1−βtT)[λ(vi(t))vϕit+(1−λ(vi(t)))viϕt] ≥βtTui(y θ∗)+(1−
βtT)vifor all(y vi)∈Lit(f (θ) θ),
we have thatx=f (θ∗), andϕt(τ·)=ϕt(τ·)=f for allτ > t.
(ii) For allxsuch thatx∈jmaxjθ∗X, we have thatx=f (θ∗).
14SeeMezzetti and Renou(2012) for an alternative definition of static implementation in mixed Nash
equilibrium.
15Azacis and Vida¯ (2015) use random mechanisms and random transitions in their analysis of infinitely
repeated implementation problems.
16We prove this and the following claim in footnotes22and23.
17We thank Helmuts Azacis and Peter Vida for pointing out the need to addλ(v
Condition (i) is similar to conditionμ(ii) of Moore and Repullo. It states that ifx maximizes the payoff of all agents but agentiat stateθ∗, if ϕt maximizes the
contin-uation payoff of all agents whileϕtmaximize the continuation payoff of all agents but
agenti, and if the pair(x λ(vi(t))vϕit+(1−λ(vi(t)))vϕit)is maximal in the dynamic lower
contour setLit(f (θ) θ)at stateθ∗, then not only alternativexmust coincide withf (θ∗)
at state θ∗, but alsoϕt(τ·) andϕt(τ·)must coincide with f for allτ > t. Note that
condition (i) is weaker than no-veto power and is almost identical to conditionμ(ii) at periodT, whenT <∞. Condition (ii) is a unanimity condition.
We now consider the two-agent case. As shown byDutta and Sen(1991) andMoore and Repullo (1988), for the static case with two agents, self-selection is a necessary condition for Nash implementation.18 Our sufficiency result for two agents requires a strengthening of self-selection.19
Assumption B. There exists an alternative w such thatui(w θ) < ui(f (θ) θ) for all
(θ θ)∈×, for alli∈ {12}.
Assumption Brequires that there exists a bad outcome (relative tof) for both agents. For instance, in pure exchange economies with strictly monotone preferences, the zero consumption bundle is a bad outcome relative to any social choice function that gives positive consumption to each consumer in at least one state of the world. Other exam-ples satisfyingAssumption Binclude environments with transferable utilities, like our two examples inSection 3. We have the following theorem.
Theorem3 (SufficiencyI=2). LetI=2. Suppose AssumptionsAandBhold. If a social choice functionfis dynamic monotonic, then it is repeatedly implementable.
We now briefly return to Examples1and2.
Example 1 (revisited). The setV (f )of expected (ex ante) payoff vectors that the two parties would obtain with an scf whose range is a subset of{(11010) (11414)}, the range off, is{(u(4)/210) (012) ((u(−4)+u(4))/212) (u(−4)/214)}. Thusf, which yields expected payoffs(vfB vSf)=(012), is strictly efficient and an extreme point in the convex hull ofV (f ). ByRemark 1,f is dynamic monotonic. Since AssumptionsAand Bhold,fis repeatedly implementable in Nash equilibrium irrespective of the discount
factor, as long as there are at least two periods. ♦
Example 2 (revisited). Consider an infinitely repeated setting. To show under which condition f is dynamic monotonic when T = ∞, we can use the sufficient condition provided afterDefinition 3. Observe thatvfi =0for alli∈I and that the best possible collusive deception isπt(θt θA)=θBandπt(θt θB)=θAfor allθt, for allt∈T. Under
such a deception,vfπ
i (θt θ)=v fπ
i =1for alli∈I. (This corresponds tovmaxi .) Given the
18InProposition 1in theAppendix, we show that a weaker condition,dynamic self-selection, is necessary
for repeated Nash implementation.
19Self-selection: LetI=2. There existsx(θ
symmetry of the setup, we only need to consider the pairs(θA θB)withπt(θt θB)=θA.
Sincef (θB)=f (θA), dynamic monotonicity (DMB) requires that there existi∈I,x∈X,
andvi∈Vi(t)such that
(1−δ)ui
f (θA) θA
−ui(x θA)
+δ[0−vi] ≥0
0> (1−δ)ui
f (θA) θB
−ui(x θB)
+δ[1−vi]
This is equivalent to
−(1−δ)ui(x θA)≥δvi>1−(1−δ)ui(x θB) (1)
By symmetry, we may takeito be any agent, say agent1. The only alternativesxthat may satisfy (1) for agent 1 assign agent 1to task A and agent 2 to taskB. Letting x=(A B w1 w2), (1) becomes(1−δ)(3−w1)≥δvi>1−(1−δ)w1, which holds if
and only ifδ <2/3. This shows thatf satisfies dynamic monotonicity if the discount factor is less than2/3. Thus, dynamic monotonicity does not imply weak efficiency in infinite horizon problems when the discount factor is not too large. Since the setting of the example satisfies AssumptionsAandB, with an infinite time horizon,f can be repeatedly implemented, and collusion among the agents avoided, as long asδ <2/3.♦
6. Discussion
This section discusses some important aspect of our analysis.
Mixed strategies.The proof ofTheorem 2does not rule out undesirable mixed strat-egy equilibria. We now show that the theorem extends to mixed strategies under the mild additional assumption ofno indifference, which states that no agent is totally indif-ferent between all alternatives at all states.
We say that a scf f is repeatedly implementable in mixed Nash equilibrium if it it is repeatedly implementable in Nash equilibrium and, in addition, there are no mixed strategy Nash equilibria that yield in some periodtan outcomey /∈f (θ)with positive probability, when the state isθ.
Theorem 4. LetI ≥3. Assume no indifference holds. If the social choice functionf is dynamic monotonic and satisfies no-veto power, then it is repeatedly implementable in mixed Nash equilibrium.
Maskin and Sjöström(2002), which allows agents to propose alternatives contingent on the state report of their opponents. This guarantees that no undesirable equilibria exist. Subgame perfection. The solution concept adopted in this paper is Nash equilib-rium. All our results extend straightforwardly to subgame perfection. First, it is easy to check that the Nash equilibriumsE constructed in the proof ofTheorem 2(and
The-orem 3) is subgame perfect. Since there are no undesirable Nash equilibria, hence no undesirable subgame-perfect Nash equilibria, this implies that dynamic monotonic-ity together with no-veto power (orAssumption A) are sufficient for subgame-perfect implementation. Dynamic monotonicity is also necessary as long as the mechanism adopted in each period is a static mechanism. To see this, suppose thatf is repeat-edly implementable in subgame-perfect Nash equilibrium, and letsbe an implement-ing equilibrium. Assume that there exists a deceptionπ such that for allt∈T, for all θt∈t, for all pairs(θt θt∗)withπt(θt θ∗t)=θt, we haveLfit(f (θt) θt)⊆Lfiθπt(f (θt) θ∗t)
for alli∈I. As in the proof ofTheorem 1, we can construct a Nash equilibriums that implementsf (πt(θt·))at all periodstand at all profilesθtof realized states up to period
t. Moreover, off the equilibrium path,sagrees withs, so thatsis also a subgame-perfect equilibrium and hencef must be dynamic monotonic.20
Time-dependent social choice functions. We have assumed that the designer wants to implement the same social choice functionf in each period. A more general objec-tive would be to implement a sequence(ft)t∈T of social choice functions. It is
straight-forward to modify the definitions of continuation payoffs, dynamic lower contour sets, and dynamic monotonicity to account for time-dependent social choice functions. With these modifications, dynamic monotonicity remains necessary and almost sufficient for repeated Nash implementation.
Social choice correspondences.The analysis extends to the implementation of social choice correspondences. LetF :→2X \ {∅}be a social choice correspondence; de-note byFthe set ofallpossible social choice functions that are selections ofF. A social choice correspondence is implementable if there exists a dynamic mechanism such that for every selectionf∈F, there exists a Nash equilibrium that repeatedly implementsf, and every Nash equilibrium repeatedly implements a selectionf ∈F. A social choice correspondenceFis dynamic monotonic when it satisfies the following criterion:
(DMAC) For allf∈F, for allπ∈T, for allθT∈T,
∀(i∈I t∈T) Lfitfπt(θt θt)
πt(θt θt)
⊆Lfπ
iθt
fπt(θt θt)
θt
⇒ ∃f∗∈F: ∀t∈T fπt(θt θt)=f∗(θt)
Note that the concept of dynamic monotonicity (for correspondences) is equiva-lent to Maskin monotonicity (for correspondences) in static implementation problems, and clearly equivalent toDefinition 3whenF is single-valued. To see the necessity of
20It is important to stress that the restriction to static mechanisms within a period rules out the
the modified condition of dynamic monotonicity, suppose thatF is repeatedly imple-mentable and assume that there exist a selectionf∈F, a deceptionπ such that for all t∈T, for allθt∈t, for all pairs(θt θ∗t)withπt(θt θ∗t)=θt, we haveLfit(f (θt) θt)⊆
Lfπ
iθt(f (θt) θ∗t)for alli∈I. As in the proof ofTheorem 1, we can construct an
equilib-rium that implementsf (πt(θt·))at all periodstand at all profilesθtof realized states
up to periodt. Consequently, there must existf∗∈Fsuch thatf (πt(θt·))=f∗for all
θt∈t, for allt∈T, i.e.,Fmust be dynamic monotonic. To show sufficiency, we need to augment the dynamic mechanism regime in the proof ofTheorem 2with an initial stage (periodt=0) in which all agents announce a selectionf ∈F. If all agents announce the same selectionf∈Fat periodt=0, then our dynamic mechanism regime takes ef-fect fromt=1withf the social choice function adopted in the canonical mechanism G∗t. If not all agents make the same announcement att=0, then our dynamic mecha-nism regime takes effect fromt=1with an arbitraryf∗∈Fas the social choice function adopted inG∗t.
7. Conclusions
Our main contribution is to introduce the condition of dynamic monotonicity, a natural but nontrivial dynamic extension of Maskin monotonicity, and to show, in Theorems 1–4, that dynamic monotonicity is necessary and almost sufficient for repeated Nash implementation of social choice functions, regardless of whether the horizon is finite or infinite and whether the discount factor is large or small.21
Many economic applications of implementation theory, for example most of the contracting literature (e.g., seeAghion et al.,2012orMaskin and Tirole 1999), focus on static problems. One of the main insights of our paper is that the (finitely) repeated im-plementation of desirable social choice functions is easier than static imim-plementation, as last-period, or late periods, planned deviations from truth-telling can be avoided by rewarding defection in early periods. For instance, we can implement full surplus ex-traction by a seller as long as there are at least two periods, while full surplus exex-traction is not implementable in static problems (seeExample 1).
Appendix
This appendix contains the proofs of all our results and the reduced strategic-form games associated withExample 1.
Example1 (The strategic-form games). Conditional on a realized first-period quality, the buyer and the seller have64strategies each. An agent is active at the initial history as well as at the histories(θH θH)and(θL θL). At the initial history, the agent has four
actions. At histories(θH θH)and(θL θL), an agent has two actions for each realization
of the second-period quality. All strategies where an agent playsNθLin the first period
21Indeed,Theorem 1also remains true if we adopt a different criterion than the discounting criterion to
are payoff equivalent (there are16strategies of that form), and similarly, for all strategies where an agent playsNθHin the first period. We writeNθLandNθHfor those strategies.
If the first-period reports do not match, then the game essentially ends. Thus, all strate-gies where an agent reportsθL at the initial history, reportsθL at the history(θL θL)
conditional on second-period qualityθL, and reportsθLat the history(θL θL)
condi-tional on second-period qualityθH are payoff-equivalent. We writeθLθLθL for those
strategies. Similarly, for all other strategies. For instance,θHθHθL represents all
strate-gies where an agent reportsθH at the initial history, reportsθHat the history(θH θH)
conditional on second-period qualityθL, and reportsθL at the history(θH θH)
condi-tional on second-period qualityθH. Each reduced strategic-form game has therefore10
“strategies.” Tables6and7represent the two reduced strategic-form games associated with each first-period qualityθLandθH. The buyer is the row player, while the seller is
the column player. In each cell, the top payoff is the buyer’s payoff, while the bottom
payoff is the seller’s payoff. ♦
Throughout the proofs, we use the following observation. For any deceptionπ˜ ∈T and state historyθ˜T∈T such that for alli∈Iandt∈T,Lfit(f (π˜t(θ˜tθ˜t))π˜t(θ˜tθ˜t))⊆
Lfπ˜
iθ˜t(f (π˜t(θ˜
tθ˜
t))θ˜t), there exists a deceptionπ∈T such that for alli∈I,t∈T and,
importantly, for all(θt θt)∈t×, Lfit(f (πt(θt θt)) πt(θt θt))⊆Liθfπt(f (πt(θt θt))
θt). The deceptionπagrees withπ˜ atθ˜T and withπ∗at all other state histories. Thus, if
f is dynamic monotonic, thenf (πt(θt θt))=f (θt)for allt∈T and(θt θt)∈t×. As
the converse is also true, we have an equivalent formulation of dynamic monotonicity.
Proof of Remark 1. Note that since f is strictly efficient in the range, for each v∈ co(V (f ))such that v=vf =(vif)i∈I, there existsi∗∈I such thatvi∗ < vfi∗. Moreover,
since v=f∈F(f )αfvf withf∈F(f )αf =1and αf ≥0for allf∈F(f ), it follows from strict efficiency offand the fact thatvf is an extreme point of co(V (f ))thatαf =
1wheneverv=vf, i.e., v corresponds to the implementation off. Consequently, for any deceptionπsuch thatπt(θt θt∗)=θt=θ∗t,vfπ∈co(V (f ))andvfπ=vf. Therefore,
for somei∗we havevfπ
i∗ < v f
i∗ and hence(f (θt) v f i∗)∈L
f
i∗t(f (θt) θt), but(f (θt) v f i∗) /∈
Lfπ
i∗θt(f (θt) θ∗t).
Proof of Remark 2. Suppose that f is Maskin monotonic and assume that there exists a deception π such that for allt ∈T, for allθt∈t, for all pairs (θ
t θ∗t) with
πt(θt θ∗t)=θt, we haveLfit(f (θt) θt)⊆Lfiθπt(f (θt) θ∗t) for alli∈I. We need to show
thatf (θ∗t)=f (θt)for allθt∈t, for allt∈T. The argument is by induction. Consider
the last periodT, anyθT, and pairs(θT θ∗T)withπT(θT θ∗T)=θT. SinceVi(T )= {0}, the
nestedness of the dynamic lower contour sets, i.e.,LfiθT(f (θT) θT)⊆LfiθπT(f (θT) θT∗),
is equivalent to the nestedness of the static lower contour sets, i.e., Li(f (θT) θT)⊆
Li(f (θT) θ∗T). From Maskin monotonicity, it follows thatf (θ∗T)=f (θT), as required. To
e tical E c onomics 1 2 (2017) R e peated N a sh implementation 269
θLθLθH θLθLθL θLθHθL θLθHθH NθL θHθLθH θHθLθL θHθHθL θHθHθH NθH
θLθLθH 0 0 0 0
δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10+12δ 10+5δ 10 10+7δ 10+11δ 1 1 1 1 1
θLθLθL
0 δu(24) δu(24) 0 δu(−1)2+δu(3) u(−1) u(−1) u(−1) u(−1) u(−1)
10+5δ 10+10δ 10+5δ 10 10+11δ 1 1 1 1 1
θLθHθL 0
δu(4)
2
δu(4)+δu(−4)
2
δu(−4)
2
δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10 10+5δ 10+12δ 10+7δ 10+11δ 1 1 1 1 1
θLθHθH 0 0
δu(−4)
2
δu(−4)
2
δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10+7δ 10 10+7δ 10+14δ 10+11δ 1 1 1 1 1
NθL
u(−4)+δu(y) u(−4)+δu(y) u(−4)+δu(y) u(−4)+δu(y) (2+δ)u(2 −4) u(−1) u(−1) u(−1) u(−1) (2+δ)u(102)+δu(14)
10 10 10 10 14+14δ 1 1 1 1 0
θHθLθH
u(−1) u(−1) u(−1) u(−1) u(−1) u(−4) u(−4) u(−4) u(−4) 2u(−4)+δu(2−1)+δu(3)
1 1 1 1 1 14+12δ 14+5δ 14 14+7δ 14+11δ
θHθLθL u(−1) u(−1) u(−1) u(−1) u(−1) u(−4)
2u(−4)+δu(4)
2
2u(−4)+δu(4)
2 u(−4)
2u(−4)+δu(−1)+δu(3)
2
1 1 1 1 1 14+5δ 14+10δ 14+5δ 14 14+11δ
θHθHθL u(−1) u(−1) u(−1) u(−1) u(−1) u(−4)
2u(−4)+δu(4)
2
(2+δ)u(−4)+δu(4)
2
(2+δ)u(−4)
2
2u(−4)+δu(−1)+δu(3)
2
1 1 1 1 1 14 14+5δ 14+12δ 14+7δ 14+11δ
θHθHθH
u(−1) u(−1) u(−1) u(−1) u(−1) u(−4) u(−4) (2+δ)u(2 −4) (2+δ)u(2 −4) 2u(−4)+δu(2−1)+δu(3)
1 1 1 1 1 14+7δ 14 14+7δ 14+14δ 14+11δ
NθH u(−1) u(−1) u(−1) u(−1)
(2+δ)u(10)+δu(14)
2 u(−4) u(−4) u(−4) u(−4)
(2+δ)u(−4)
2
1 1 1 1 0 10 10 10 10 14+14δ
M e zz etti and R enou Theor e tical E c onomics 1 2 (2017)
θLθLθH θLθLθL θLθHθL θLθHθH NθL θHθLθH θHθLθL θHθHθL θHθHθH NθH
θLθLθH u(4) u(4) u(4) u(4)
2u(4)+δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10+12δ 10+5δ 10 10+7δ 10+11δ 1 1 1 1 1
θLθLθL u(4)
(2+δ)u(4)
2
(2+δ)u(4)
2 u(4)
2u(4)+δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10+5δ 10+10δ 10+5δ 10 10+11δ 1 1 1 1 1
θLθHθL
u(4) (2+δ)u(2 4) (2+δ)u(42)+δu(−4) 2u(4)+2δu(−4) 2u(4)+δu(2−1)+δu(3) u(−1) u(−1) u(−1) u(−1) u(−1)
10 10+5δ 10+12δ 10+7δ 10+11δ 1 1 1 1 1
θLθHθH u(4) u(4)
2u(4)+δu(−4)
2
2u(4)+δu(−4)
2
2u(4)+δu(−1)+δu(3)
2 u(−1) u(−1) u(−1) u(−1) u(−1)
10+7δ 10 10+7δ 10+14δ 10+11δ 1 1 1 1 1
NθL δu(y) δu(y) δu(y) δu(y)
δu(−4)
2 u(−1) u(−1) u(−1) u(−1)
(2+δ)u(14)+δu(10)
2
10 10 10 10 14+14δ 1 1 1 1 0
θHθLθH
u(−1) u(−1) u(−1) u(−1) u(−1) 0 0 0 0 δu(−1)2+δu(3)
1 1 1 1 1 14+12δ 14+5δ 14 14+7δ 14+11δ
θHθLθL
u(−1) u(−1) u(−1) u(−1) u(−1) 0 δu(24) δu(24) 0 δu(−1)2+δu(3)
1 1 1 1 1 14+5δ 14+10δ 14+5δ 14 14+11δ
θHθHθL u(−1) u(−1) u(−1) u(−1) u(−1) 0
δu(4)
2
δu(−4)+δu(4)
2
δu(−4)
2
δu(−1)+δu(3)
2
1 1 1 1 1 14 14+5δ 14+12δ 14+7δ 14+11δ
θHθHθH u(−1) u(−1) u(−1) u(−1) u(−1) 0 0
δu(−4)
2
δu(−4)
2
δu(−1)+δu(3)
2
1 1 1 1 1 14+7δ 14 14+7δ 14+14δ 14+11δ
NθH
u(−1) u(−1) u(−1) u(−1) (2+δ)u(142)+δu(10) 0 0 0 0 δu(2−4)
1 1 1 1 0 10 10 10 10 14+14δ
have thatf (θ∗τ)=f (θτ). It follows that in periodtthe continuation payoffvfiπ(θt θt)is
equal tovfi(t)for all agentsi, for all(θt θt)and, thus,Lfiθπt(f (θt) θ∗t)=L f
it(f (θt) θ∗t)for
all(θt θt θ∗t). As a result,L f
it(f (θt) θt)⊆L fπ
iθt(f (θt) θt∗)is equivalent toL f
it(f (θt) θt)⊆
Lfit(f (θt) θ∗t). In turn, this is equivalent to the nestedness of the static lower contour
sets, i.e.,Li(f (θt) θt)⊆Li(f (θt) θ∗t). Maskin monotonicity then impliesf (θt∗)=f (θt).
This concludes the proof.
Proof of Remark 3. Assume to the contrary thatf is dynamic monotonic but not weakly efficient in the range; that is, there exists ε >0and a payoff profile (vi)i∈I ∈
co(V (f ))such thatvi> vif+2εfor alli∈I. Using a standard argument about
convexi-fying the set of payoffs without public randomization (e.g., see Lemma 3.7.2 inMailath and Samuelson 2006), it follows that there existsδH2such that for allδ∈(δH21)there exists an infinite sequence of social choice functions{f1 f2 }withft∈F(f )for all
in-tegerst(i.e., the range offtis a subset of the range off), and(1−δ)∞τ=tδτ−tv fτ
i > vi−ε,
for all i∈I, for all t. Since ft ∈F(f ), there exist mappings πt :→ such that
f◦πt=ft. Consider the deceptionπ such thatπt(θt θt∗)=πt(θ∗t)for allθ∗t, for allθt,
for allt. It follows thatvfπ
i (θt θ∗t)=(1−δ)
∞
τ=t+1δτ−t−1v fτ
i > vi−ε > vfi +εfor alli.
Letρ=maxi∈Iθθ∗∈|ui(f (θ) θ)−ui(f (θ) θ∗)|, and letδH=max(ρ/(ρ+ε) δH2). Then, for δ∈(δH1), for alli, for all pairs (θt θ∗t)withπt(θt θ∗t)=θt, for all θt∈t, for all
t∈T , it is(1−δ)ui(f (θt) θ∗t)+δv fπ
i (θt θ∗t)≥(1−δ)ui(f (θt) θt)+δvfi or, equivalently,
Lfit(f (θt) θt)⊆Lfiθπt(f (θt) θ∗t). Dynamic monotonicity then implies thatf◦πt=ft=f
for allt, contradicting the assumed weak inefficiency off.
Proof of Remark4. Assumef is Maskin monotonic and efficient in the range, and suppose that there exists a deception π such that for all t∈T, for allθt∈t, for all
pairs(θt θ∗t)with πt(θt θ∗t)=θt, we haveLitf (f (θt) θt)⊆Lfiθπt(f (θt) θ∗t)for alli∈I.
Recall that vfπ
i (θt θt)is the (normalized) expected discounted continuation payoff of
agent i from following the deception π from the history induced by π and (θt θ t).
Thus, vfπ
i (θt θt) is an element of the convex hull of V (f ), the set of payoff profiles
of social choice functions with a range contained in the range of f. First, suppose that(vfπ
i (θt θt))i∈I=(v f
i)i∈I. Sincef is efficient in the range, it follows that there
ex-ists an agent i∗ such thatvfπ
i∗(θt θt) < v f
i∗. Consequently, we have that (f (θt) v f i∗)∈
Lfi∗t(f (θt) θt)(by definition) and (f (θt) v f i∗) /∈L
fπ
i∗θt(f (θt) θ∗t), a contradiction. So it
must be thatvfπ
i (θt θt)=v f
i for alli∈I. It then immediately follows that the
nested-ness of the dynamic lower contour sets (i.e.,Lfit(f (θt) θt)⊆Lfiθπt(f (θt) θ∗t)) implies the
nestedness of the static lower contour sets (i.e.,Li(f (θt) θt)⊆Li(f (θt) θ∗t)). Maskin
monotonicity then implies thatf (θ∗t)=f (θt). This shows thatf (πt(θt·))=f for all
θt∈t, for allt∈T, and hencef must be dynamic monotonic.
f (θt)for at least onet∈ {1 T+1}, while the dynamic lower contour sets are nested,
i.e., for alli∈I, for allt∈T,
Lfitfπt(θt θt) πt(θt θt)⊆Lfiθπt
fπt(θt θt) θt (2)
We first argue thatf (πt(θt θt))=f (θt)for allt∈ {2 T +1}. Fix the first-period
stateθ1in the profileθT+1and consider any deceptionπ∗∗∈T such thatπt∗∗(θt θt)=
πt+1((θ1 θt) θt)for allt∈ {1 T}. In words,π∗∗mirrors the lastTperiods ofπ, given
that the first-period state wasθ1.
By (2) andβtT =βt+1T+1, for alli∈Iandt∈ {1 T},
Lfitfπt∗∗(θt θt) πt∗∗(θt θt)⊆Lfiθπ∗∗t
fπt∗∗(θt θt) θt
Sincefis dynamic monotonic overTperiods, this implies thatf (π∗∗t (θt θt))=f (θt)for
allt∈ {1 T}or, equivalently,f (πt(θt θt))=f (θt)for allt∈ {2 T+1}. It follows
thatvfπ
i ((θt θt))=v fπ∗∗
i ((θt θt))=v f
i(t)for allt≥1.
Therefore, we must havef (π1(θ1))=f (θ1). We now argue that this cannot be the
case either. Consider any deceptionπ◦such thatπt◦(θt θ
t)=πt(θt θt)for all(θt θt),
that is,π◦coincides with the firstT periods ofπ.
Sincef is dynamic monotonic overT periods (and the fact thatf (π1◦(θ1))=f (θ1)),
there existi∈I,t∈ {1 T}, and(x vi)such that
(1−βtT)ui
fπt◦(θt θt)
π◦t(θt θt)
+βtTvif(t)≥(1−βtT)ui
x πt◦(θt θt)
+βtTvi
and
(1−βtT)uifπt◦(θt θt) θt+βtTvfi(t) < (1−βtT)ui(x θt)+βtTvi
Using the definition ofπ◦, this is equivalent to (remember thatβtT+1∈(01))
(1−βtT+1)uifπt(θt θt) πt(θt θt)−uix πt(θt θt)
≥1−βtTβ
tT
(1−βtT+1)
vi−vfi(t)
> (1−βtT+1)
ui
fπt(θt θt) θt−ui(x θt)
Letvˆibe given by
βtT
1−βtT
1−βtT+1
βtT+1
vi+
1− βtT 1−βtT
1−βtT+1
βtT+1
vfi(t)
SinceβtT ≤βtT+1, we have thatvˆi∈ [vi vi]. It follows that there exists(xvˆi)∈X×Vi(t)
such that
(1−βtT+1)ui
fπt(θt θt) πt(θt θt)+βtT+1vfi(t)
and
(1−βtT+1)uifπt(θt θt) θt+βtT+1vfi(t) < (1−βtT+1)ui(x θt)+βtT+1vˆi
This is equivalent toLfit(f (πt(θt θt)) πt(θt θt))Lfiθπt(f (πt(θt θt)) θt), a
contradic-tion with (2). Therefore,f (π1(θ1))=f (θ1), as required.
Proof of Theorem 1. Suppose thatf is repeatedly implementable by the dynamic mechanism regimer. Fix an equilibriums. Consider a historyht and a mechanism
Gt= MGt gthaving positive probability of occurring on the equilibrium path at
pe-riodt; that is, such thatq(ht;s) >0andr(Gt;ht) >0. Since the dynamic regimer
im-plementsf, the profile of actionss(ht G
t θt)at periodtmust satisfygt(s(ht Gt θt))=
f (θt) for eachθt∈, and the continuation payoff must be vfi(t). LetQi(ht Gt θt;s)
be the set of current alternative and continuation payoff pairs that agent i is able to generate by any deviation starting at t, given that all other agents follow s. For-mally,(x vi)∈X ×Vi(t)belongs toQi(ht Gt θt;s)if there exists mi∈MiGt such that
x=g(mi s−i(ht Gt θt))and there existsvi∈Vi(t)that corresponds toi’s expected
dis-counted continuation payoff when (starting att, in stateθt, after historyht) agenti
fol-lows some continuation strategy (which prescribes sending messagemiatt), while all
other agents continue to follows−i.
Sincesis an equilibrium, for eachi∈I, for eachθt∈, we must have that
(1−βtT)uif (θt) θt+βtTvif(t)≥(1−βtT)ui(x θt)+βtTvi
for each (x vi) ∈ Qi(ht Gt θt;s). Consequently, it must be that Qi(ht Gt θt;s) ⊆
Lfit(f (θt) θt)for allht θt, andGtsuch thatq(ht;s) >0andr(Gt;ht) >0.
Now consider a deceptionπsuch that for allt∈T, for allθt∈t, for all pairs(θt θ∗t)
withπt(θt θ∗t)=θt, we haveLfit(f (θt) θt)⊆Lfiθπt(f (θt) θ∗t)for alli∈I. In the
remain-der of the proof, we will show that there exists an equilibriums that implements the social choice functionf (πt(θt·))at each periodtfor eachθt. Since the regimer
repeat-edly implementsf, it must be thatf (πt(θt·))=ffor allθt∈t, for allt∈T. Hence, we
may conclude thatf is dynamic monotonic and the theorem holds.
We now construct the strategy profiles. First, consider the equilibrium path. Let h1=h1
π= {∅}and for allθ1, for allG1, for alli, define
si(h1 G1 θ1)=sih1π G1 π1(θ1)
Then assume that the strategy profilesand the historieshτandhτπ have been defined
up to periodτ=t. Letht+1=(ht G
t θt s(ht Gt θt)), withht=(htD θt), be a period
t+1history corresponding to the history of realized statesθt. Associate the historyht+1 withhtπ+1=(hπt Gt πt(θt θt) s(hπt Gt πt(θt θt)), and for allθt+1, for allGt+1, for alli,
define
