Temporary Implementation Takashi Hayashi University of Glasgow E-mail:
[email protected] Michele Lombardi University of Glasgow E-mail:
[email protected] November 10, 2016 Abstract The paper examines problems of implementing social choice objectives in a dynamic environment, in which society can only decide and execute a policy variable at hand period by period. The social objective that society wants to achieve is represented by a social choice function (SCF) that maps each state of the world into a dynamic process mapping every history into a social outcome. This social process is temporarily implementable if there exists a process of one-period game forms (with observed actions and simultaneous moves) each of which generates a social outcome only at one given period after a given history, such that at each state of the world there is a subgame-perfect Nash equilibrium in which the social objective is ful…lled at every period, after every history, as a unique equilibrium outcome process. The paper identi…es necessary conditions for SCFs to be temporarily implemented, the folding condition and temporary Maskin monotonicity, and shows that they are also su¢ cient under auxiliary conditions when there are three or more individuals. Finally, it provides an account of welfare implications of temporary implementability in the contexts of sequential trading and sequential voting.
Introduction This paper takes as an institutional reality the fact that despite intertemporal coordination of policies is fundamentally desirable, social decisions and their execution cannot be done in an all-inclusive manner and society has to decide and execute a policy variable at hand period by period in a temporary manner. Therefore, achieving a desirable social outcome while being mindful of what happens along the way seems to be a compelling argument. Thus, we are interested into a dynamic process of decisions and executions that forms a multi-fold layer. At every period, after a given history, society has to decide a current policy variable without seeing the future ones and execute it before the next period comes. This current decision and the past ones form an augmented history in the next period. In such dynamic environments, preferences over sequences of social outcomes are naturally non-separable. Even when preferences over consumption sequences are time-separable and stationary, the natural situation is that preferences induced over sequences of social outcomes (e.g., trades and taxes) are non-separable and non-stationary. This is because a current decision a¤ects how agents substitute consumptions between periods. Moreover, even if individuals’preferences over sequences of social outcomes are time-separable, social choices in general should not be done in a time-separable manner since they involve intertemporal coordination of policies across periods. Therefore, one cannot decompose a dynamic social choice problem into a time-sequence of mutually separate issues. Even if there is no asymmetric information among agents and there is no reputation e¤ects among them, such an environment creates a problem of potential manipulation by means of hiding the intention of manipulation. In dynamic environments as described above, agents may have di¤erent reasons for manipulating a social outcome at a given period in the same way. For example, consider voting over tax rates period by period, and that an agent is trying to manipulate the current tax rate, let’s say to be higher. It may be because he purely prefers the high tax rate in this period, or it may be because he wants to have the social choice outcome in the future period following as a consequence of the high tax rate in this period. For another example, consider consumption-saving coordination in society, and that an agent is trying to save more in the current period. It may be because he is
1
purely patient, or it may be because he is trying to manipulate equilibrium interest rate in the future period, as saving/borrowing in the current period a¤ects endowments in the next period. In fact, in the section of application we provide an example of sequential trading that an agent’s saving choice reveals no information to the social planner. In the mechanism design literature there are three ways to incorporate dynamics. One is to allow that only decision processes are dynamic, but once a decision is reached society simply commits to it, as is done by the researches on subgame-perfect implementation (Moore and Repullo, 1988; Abreu and Sen, 1990; Herrero and Srivastava, 1992; Vartiainen, 2007). Second is to consider that new pieces of information, such as health status, arrive in every period and the mechanism design is concerned with inducing agents to reveal them correctly in every period, where preferences over sequences over policy variables are assumed to be separable and such new information is relevant only to evaluation of current policy variables (Lee and Sabourian, 2011; Mezzetti and Renou, 2016).1 Third is to allow for the role of intertemporal linkages due to reputation e¤ects in the setting of in…nitely repeated games. To our knowledge, however, the problem of potential manipulability as explained above has not been paid attention. Note also that in the second and third approaches time preferences are assumed to be known to the planner, whereas in our approach it is a part of information that the planner needs to elicit from agents. To motivate further, let us give two notable examples. Consider the Arrow-DebreuMcKenzie model (Arrow and Debreu, 1954; McKenzie, 1954), which is the most prominent framework in economic theory. When it is interpreted literally, it says that all the agents meet on the …rst day of their life and write down a contract on all the deliveries of consumption contingent on every date-event, and simply commit to it. A more realistic description of trading over time is by Radner (1972, 1982), which considers that at each period agents can trade only between current consumption and assets to be carried over to the next period.2 1
Kalai and Ledyard (1998) study the problem of in…nitely repeated implementation with time invariant preferences. They show that every social choice function can be repeatedly implemented in dominant strategies from a point in time onwards provided the planner is su¢ ciently patient. See also Chambers (2004). 2 Grandmont (1977) provides a model of dynamic process of temporary equilibrium, from which we borrowed the word “temporary,” in which the agents trade only between current consumption and assets to be carried over to the next period, while it imposes no rationality restriction on expectation formation about spot prices in future. Arrow (1964) was the …rst to observe that sequential trading and trading at a single
2
However, to our knowledge the Radner-type description of temporary trading has not been given a strategic foundation. The reason is more fundamental. In the Radner model prices are de…ned only for on-path situation and it is left unclear what prices should be formed in o¤-path situations. The competitive model is silent about what prices and allocation should be formed after the society makes mistake. The other example is voting over time. It is institutionally realistic to consider that in each period we can cast a vote against the current tax policy without being able to vote for future ones. In the literature of positive political economy, it is known that such lack of commitment leads to ine¢ cient outcomes. Similar examples can be found in situations where there is no supranational legal framework, such as in the context of international trade. Based on the above motivation, we consider the following implementation problem: (i) to focus on the problem of hiding the intention of manipulation, we consider that there is complete information among agents and time horizon is …nite; (ii) a social choice objective is a social choice function (SCF) that maps each state of the world into a dynamic process mapping every history into a social outcome at each given period, so that we are required to prescribe something reasonable even after society make a mistake; this corresponds to the above question about what prices should be formed in o¤-path situations; (iii) a dynamic mechanism is a process of one-period game forms (with simultaneous moves and observed actions), each of which generates a social outcome only at a given period after a given history; (iv) the de…nition of implementability is that there is a subgame-perfect Nash equilibrium such that the social objective is ful…lled at every period, after every history, as a unique equilibrium outcome process. When it is so we say that the SCF is temporarily implementable. We provide necessary conditions for implementability. The …rst and key necessary condition is what we call folding condition, which says that the SCF at a given period should depend only on agents’preferences over current social outcomes, which are induced by means of backward induction. In other words, the planner must ignore why an agent wants a particular social outcome at a given period, while there are in general many reasons for wanting it as explained above. The planner must do this folding period by period, which transforms a dynamic social choice objective into a process of temporary social choice objectives, which point in time are equivalent when markets are complete.
3
are apparently static. Then we show that such a process of temporary SCFs has to satisfy Maskin monotonicity at each period, after every history. Together with the auxiliary conditions that the process of temporary SCFs satis…es unanimity and weak no veto power at each period, we show that the necessary conditions are also su¢ cient. The construction of the dynamic mechanism is simple: after every history, just run the Maskin mechanism over current social outcomes, in which each agent reports a preference pro…le de…ned over just current social outcomes as well as a tie-breaking information. Here the planner does not question why an agent prefer a particular current social outcome, and it is not the planner’s business. Further, the paper provides an account of welfare implications of its su¢ ciency result in the context of temporary trading and temporary voting. Firstly, we consider a borrowing-lending model with no liquidity constraints, in which individuals trade in spot markets and transfer wealth between any two periods by borrowing and lending. In this set-up, intertemporal pecuniary externalities arise because trades in the current period change the spot price of the next period, which, in turn, a¤ects its associated equilibrium allocation. The quantitative implication of this is that every individual’s induced preference concerns not only her own consumption/saving behavior but also the consumption/saving behavior of all other individuals. We show that, under such a pecuniary externality, the standard dynamic competitive equilibrium solution is not temporarily implementable because it fails to satisfy the folding condition. We have also identi…ed preference domains –which involve no pecuniary externalities –for which the no-commitment version of the dynamic competitive equilibrium solution is de…nable and temporarily implementable. Secondly, we consider a bi-dimensional policy space where an odd number of individuals vote sequentially on each dimension and where an ordering of the dimensions is exogenously given. We assume that each voter’s type space is unidimensional, that a majority vote is organized around each policy dimension and that the outcome of the …rst majority vote is known to the voters at the beginning of the second voting stage. This temporary resolution is common in political economy models (see, e.g., Persson and Tabellini, 2000). In this environment, we show that the simple majority solution, which selects the Condorcet winner in each voting stage, is temporarily implementable. In this process, we explicitly state the 4
conditions on the utility function of each voter that are needed for this SCF to be wellde…ned and show that this is the case. As established by De Donder et al (2012) for the case where there is a continuum of voters, the assumption that both dimensions are strategic complements, as well as the requirement that the induced utility of both dimensions is increasing in the type of the voter, are particularly important for guaranteeing the existence of a Condorcet winner in each voting stage. The remainder of the paper is organized as follows. Section 2 sets out the theoretical framework and outlines the basic implementation model. Necessary and su¢ cient conditions are presented in section 3. Section 4 covers temporarily implementable SCFs in the context of trading and voting problems. Section 5 concludes. Appendix includes proofs not in the main body.
2. Basic framework Let us imagine that a set of individuals indexed by i 2 I what outcome is best in each time period indexed by t 2 T
f1;
; Ig have to decide
f1; 2;
; T g. Let us denote
the universal set of period-t outcomes by X t , with xt as a typical outcome. Thus, the universal set of outcome paths available to individuals is the space: Y
X
X t,
t2T
with x as a typical outcome path. The t-head x omitting the last t components, that is, x omitting the …rst t
t
(x1 ;
1 components, that is, x+t
t
is obtained from the path x 2 X by ; xt 1 ), the t-tail is obtained from x by xt ;
; xT , and we identify (x t ; x+t )
with x. The same notational convention will be followed for any pro…le of outcomes. We will refer to the t-head x
t
as the past outcome history x t .
The feasible set of period-t + 1 outcomes available to individuals depends upon past outcome history x
(t+1)
, that is, X t+1 x
(t+1)
X t+1 for every period t 6= T .
5
We write F t for the collection of functions de…ned as follows: Ft
f t jf t : X
t
! X t such that f t x
We also write F for the product space X 1
F2
t
2 Xt x
t
, for all t 6= 1.
FT .
The information held by the individuals is summarized in the concept of a state, which is a complete description of the variable characterizing the environment. Write domain of possible states, with
as a typical state. For every period t
2, the description
of the variable characterizing the environment after the outcome history x jx t . Moreover, for every t
for the
t
is denoted by
2 we write jx t ; x+(t+1) for a complete description of the
variable characterizing the environment in period t after the outcome history x
t
and the
future sure outcome path x+(t+1) . In the usual fashion, individual i’s preferences in state
are given by a complete and
transitive binary relation, subsequently an ordering, Ri ( ) of elements of X . The corresponding strict and indi¤erence relations are denoted by Pi ( ) and Ii ( ), respectively. The statement xRi ( ) y means that agent i judges x to be at least as good as y. The statement xPi ( ) y means that agent i judges x to be better than y. Finally, the statement xIi ( ) y means that agent i judges x and y as equally good, that is, she is indi¤erent between them.
2.1 Implementation model
Dynamic social objectives The goal of the central designer is to implement a social choice function (SCF) f : that assigns to each state
!F
a dynamic “socially optimal”process f [ ] = f1 [ ] ; f2 [ j ] ;
; fT [ j ] ,
where: f 1 [ ] 2 X 1 is the period-1 socially optimal outcome and 6
f t [ j ] 2 F t is the period-t socially optimal process that selects the socially optimal outcome f t [ jx t ] in period t
2 at the state
after the past outcome history x
t
2
X t. To save writing, for every period t 6= 1 and every past outcome history x t , we write f +t [ jx t ] for the t-tail path of socially optimal outcomes in state
that follows the past
outcome history x t , whose period- element is the value of the composition f at jx t ; that is:
where f [ jx t ]
f +t (f
jx
f
jx
jx
t
ft
jx
t
j 2
ft
t
t. The image or range of
the period-t function f t of the SCF f at the past outcome history x ft
1
t
f t ) [ jx t ] for every period
1
f
t
f
, for every x
t
2X
t
t
is the set:
with t 6= 1.
The image or range of the period-1 function f 1 of the SCF f is the set f 1 [ ]
ff 1 [ ] j 2
g.
Dynamic mechanism The central designer delegates the choice to individuals according to a process of oneperiod mechanisms (or game forms) with observed actions and simultaneous moves and then execute that choice. In other words, we assume that the actions of every individual are perfectly monitored by every other individual as well as that every individual chooses an action in period t without knowing the period t action of any other individual. More formally, in the …rst period all individuals i 2 I choose actions from nonempty choice sets Ai (h1 ), where h1
? denotes the initial history. Thus, the period-1 action space
is the product space: A h1
Y
Ai h1 ,
i2I
with a (h1 )
(a11 (h1 ) ;
; a1I (h1 )) as a typical period-1 action pro…le.
In the second period, individuals know the history h2
a1 , and the actions that every
individual i 2 I has available in period 2 depends on what has happened previously. Then, 7
let Ai (h2 ) denote the period-2 nonempty action space of individual i when the history is h2 and let A (h2 ) denote the corresponding period-2 nonempty action space, which is de…ned by: A h2
Y
Ai h2 ,
i2I
with a (h2 )
(a1 (h2 ) ;
; aI (h2 )) as a typical period-2 action pro…le.
Continuing iteratively, we can de…ne ht , the (nontrivial) history at the beginning of period t > 1, to be the list of t
1 action pro…les, ht
; at
a1 ; a2 ;
1
,
identifying actions played by individuals in periods 1 through t
1. We let Ai (ht ) be
individual i’s nonempty action set in period t when the history is ht and let A (ht ) be the corresponding period-t action space, which is de…ned by: A ht
Y
Ai ht ,
i2I
with a (ht )
(a1 (ht ) ;
; aI (ht )) as a typical pro…le of actions.
We assume that in each period t, every individual knows the history ht , this history is common knowledge at the beginning of period t, and that every individual i 2 I chooses an action from the action set Ai (ht ). We also assume that in each period t, all individuals i 2 I choose actions simultaneously. We let H t be the set of all period-t histories, where we de…ne H 1 to be the null set, and let H
[
Ht
t2T
be the set of all possible histories. For any nontrivial history ht sequence of the form (a1 ;
; at 1 ) 2 H, de…ne a subhistory of ht to be a
(a1 ; a2 ;
; am ) with 1
m
t
1, and the trivial history consisting of
no actions is denoted by ?. The delegation to individuals is made by means of a dynamic mechanism
(I; H; A (H) ; g),
8
where H is the set of all possible histories, A (H) is the set of all pro…les of actions available to individuals, de…ned by
[
A (H)
A (h) ,
h2H
; g T is a sequence of outcome functions, one for each period t 2 T , with
g1;
and g
the property that: a) the outcome function g 1 assigns to period-1 action pro…le a (h1 ) 2 A (h1 ) a unique outcome in X 1 , and b) for every period t 6= 1 and every nontrivial history ht
; at 1 ) 2 H t , the outcome function g t assigns to each period-t action pro…le
(a1 ; a2 ;
a (ht ) 2 A (ht ) a unique outcome in X t (g
t
(ht )).
The submechanism of a dynamic mechanism
that follows the history ht is the dynamic
mechanism ht
I; Hjht ; A Hjht ; g +t ,
where Hjht is the set of histories for which ht is a subhistory for every h 2 Hjht , A Hjht
[
A (h)
h2Hjht
is the set of all pro…les of actions available to individuals from period t to period T , and g +t is t-tail of the sequence g that begins with period t after the history ht such that for every hT g
t
; aT
a1 ;
hT ; a hT
1
2 H T jht and every a hT 2 A hT it holds that g hT ; a hT
; g +t hT ; a hT
=
.
Temporary implementation A dynamic mechanism
and a state
induce a dynamic game ( ; ) (with observed
actions and simultaneous moves). The subgame of the dynamic game ( ; ) that follows the history ht 2 H is the dynamic game ( (ht ) ; ). S Let Ai Ai (h) be the set of all actions for individual i 2 I. A (pure) strategy for h2H
individual i is a map si : H ! Ai with si (h) 2 Ai (h) for every history h 2 H. Individual i’s space of strategies, Si , is simply the space of all such si . A strategy pro…le s
(s1 ;
; sI ) is a list of strategies, one for each individual i 2 I. 9
The strategy pro…le s
i
= (s1 ;
; sI ), and we identify (si ; s i ) with s.
; si 1 ; si+1 ;
is obtained from s by omitting the ith component, that is, s
i
For any strategy si of individual i and any history ht in the dynamic mechanism , the strategy that si induces in the dynamic subgame ( (ht ) ; ) is denoted by si jht . Individual i’s space of strategies that follows history ht is denoted by Si jht . The period-t strategy of individual i is sometimes denoted by sti . For every dynamic game ( ; ), the strategy pro…le s is a Nash equilibrium of ( ; ) if for every individual i 2 I it holds that: g si ; s
i
Ri ( ) g si ; s
i
for every si 2 Si .
Let N E ( ; ) denote the set of Nash equilibrium strategy pro…les of ( ; ). Moreover, for every dynamic game ( ; ) and every nontrivial history ht 2 H, the strategy pro…le s jht is a Nash equilibrium of ( (ht ) ; ) if for every individual i 2 I and a given past outcome history x
t
x t ; g +t si jht ; s i jht
2X
t
it holds that:
Ri ( ) x t ; g +t si jht ; s i jht
for every si jht 2 Si jht .
Let N E ( (ht ) ; ) denote the set of Nash equilibrium strategy pro…les of ( (ht ) ; ). A strategy pro…le s is a subgame perfect equilibrium (SPE) of a dynamic game ( ; ) if it holds that: s jht 2 N E
ht ;
for every history ht 2 H.
,
Let SP E ( ; ) denote the set of SPE strategy pro…les of ( ; ), with s as a typical element. Definition 1 A dynamic mechanism in SPE if for every
ft
2
jg
(I; H; A (H) ; g) implements the SCF f :
!F
, f 1 [ ] = g1 s t
ht
= gt s
ht
h1
, and
, for every ht 2 H t with t 6= 1,
if and only if s 2 SP E ( ; ) 10
If such a mechanism exists, the SCF f is said to be temporarily implementable.
3. Necessary and su¢ cient conditions 3.1 Folding In this section, we …rst propose a property, folding, and show that this is a necessary condition for temporary implementation. While this property is heavy in notation, its idea is simple. This property reduces the dynamic implementation problem into a process of “temporary” implementation problems, one for each period, where the planner takes only the individuals’preferences induced over current social outcomes into account. This necessary condition is derived by using the approach developed by Moore and Repullo (1990) and thus it is stated in terms of the existence of certain sets. These sets are denoted by Y t , Y 1 and Y t (y t ) and represent respectively the set of feasible past outcome histories up to period t 6= 1, the set of period-1 attainable outcomes and the set of period-t attainable outcomes after the past outcome history y t . Moreover, the condition consists of three parts: the …rst part characterises the period-T implementation problem, the second one relates to the implementation problem of period t 6= 1; T and the third one relates to the period-1 implementation problem. Solving backward, for any feasible past outcome history y dering of individual i in state
at y
T
, that is, at jy
T
T
, the period-T induced or-
, denoted by Ri
jy
T
, is equal
to: y T Ri
jy
T
z T ()
We denote by R D
jy
T
jy
y
T
; y T Ri ( ) y
T
; z T , for every y T ; z T 2 Y T y
T
the pro…le of period-T induced orderings at jy
the period-T domain of induced orderings at D
jy
T
R
jy
T
j 2
jy
T
T
. T
(1) and by
, that is:
.
(2)
Therefore, the …rst part of the condition can be formulated as follows:
11
(i) The preference domain D D
jy
T
!YT y
T
jy
T
is not empty, and there is a period-T function 'T :
such that:
'T R
jy
T
= fT
jy
T
, for every
2
.
(3)
To introduce the second part of the condition, let us suppose that in our way back to t
period 1 we have reached period t 6= 1; T and that y
is a feasible past outcome history.
Given that in our framework temporary rationality is common knowledge between the players and given that the objective of the planner is to implement a dynamic social choice process prescribed by the SCF f , every player will "look ahead" and a period-t outcome y t will be evaluated at the past outcome history y
t
as well as at the future sure outcome path f +(t+1)
prescribed by the SCF in response to the outcome history path (y t ; y t ). On this basis, the period-t induced ordering of individual i in state
at the past outcome history y
t
and at the
future sure outcome path prescribed by the social process f +(t+1) , that is, at jy t ; f +(t+1) , denoted by Ri y t Ri
jy t ; f +(t+1) , is equal to: jy t ; f +(t+1) z t () y t ; y t ; f +(t+1) [ j (y t ; y t )] Ri ( ) y t ; z t ; f +(t+1) [ j (y t ; z t )] ,
(4)
for every y t ; z t 2 Y t (y t ). Let us denote by R
jy t ; f +(t+1) the pro…le of period-t induced orderings at jy t ; f +(t+1) jy t ; f +(t+1) the period-t domain of induced orderings at jy t ; f +(t+1) ,
for t 6= 1; T and by D that is: D
jy t ; f +(t+1)
R
jy t ; f +(t+1) j 2
.
(5)
Therefore, as for the …rst part of the condition, the second part can be stated as follows: (ii) The preference domain D 't : D
jy t ; f +(t+1) is not empty, and there is a period-t function
jy t ; f +(t+1) ! Y t (y t ) such that: 't R
jy t ; f +(t+1)
= ft
jy
t
, for every
2
.
(6)
12
Reasoning like that used in the preceding paragraphs, the period-1 induced ordering of individual i in state
at the outcome path prescribed by the social process f +2 , that is, at
jf +2 , denoted by Ri [ jf +2 ], is equal to: y 1 Ri
jf +2 z 1 ()
Ri ( ) z 1 ; f +2
jy 1
y 1 ; f +2
jz 1
, for every y 1 ; z 1 2 Y 1 .
(7)
Denoting the pro…le of period-1 induced orderings at jf +2 by R [ jf +2 ] and de…ning the period-1 domain of induced orderings at
jf +2 by:
jf +2
D
jf +2 j 2
R
,
(8)
the third part of folding can be stated as follows: (iii) The preference domain D [ jf +2 ] is not empty, and there is a period-1 function '1 : D [ jf +2 ] ! Y 1 such that: '1 R
jf +2
= f 1 [ ] , for every
2
.
(9)
In summary, if the SCF f is temporarily implementable, then the following condition must be satis…ed: Definition 2 The SCF f :
! F satis…es the folding condition if there is a collection of
spaces of sequences of past outcomes fY t gt2T nf1g and if there is a period-1 outcome space n o Y1 Y 2 and there is a collection of period-t outcome spaces fY t (y t )gy t 2Y t t2T nf1g
such that:
f1 [ ]
Y 1 and f t [ jy t ]
Y t (y t ) for every t 6= 1;
for every t 6= 1, it holds that y
t
2Y
t
() y 1 2 Y 1 and y 2 Y
part (i) is satis…ed for every y
T
2Y
T
y
for every 2
t;
; 13
part (ii) is satis…ed for every y
t
2Y
t
with t 6= 1; T ;
part (iii) is satis…ed. A SCF satisfying the above condition is said to be a folding SCF. Our …rst main result can thus be stated as follows: Theorem 1 If I
2 and the SCF f :
! F is temporarily implementable, then it satis…es
the folding condition. Proof. See Appendix.
This is not an obvious condition. In intertemporal environments there may be various di¤erent reasons for sending an identical message in a single period. For example, consider that an agent sends a message telling that he prefers high tax rate in the current period. It may be because he purely prefers to have high tax in the current period, or it may be because he desires the tax rate to be chosen in the next period following the high current tax rate, and so on. The folding condition says that the planner does not distinguish between those reasons, and does not need to question why either.
3.2 Temporary Maskin monotonicity A condition that is central to the Nash implementation thanks to Maskin (1999) is an invariance condition, now widely referred to as Maskin monotonicity. This condition says that if an outcome x is socially optimal at the state
and this x does not strictly fall in
preference for anyone when the state is changed to 0 , then x must remain a socially optimal outcome at
0
. An equivalent statement of Maskin monotonicity follows the reasoning that
if x is socially optimal at
but not socially optimal at
fallen strictly in someone’s ordering at the state
0
0
, then the outcome x must have
in order to break the Nash equilibrium via
some deviation. Therefore, there must exist some (outcome-)preference reversal if a Nash equilibrium strategy pro…le at
is to be broken at
0
. Let us formalize that condition as
follows: For any state and any individual i and any outcome x 2 X, the weak lower contour set of Ri ( ) at x is de…ned by L (x; Ri ( ))
fy 2 XjxRi ( ) yg. Therefore: 14
Definition 3 The SCF F : all ; 2
, if L(f
; Ri
! X is Maskin monotonic provided that for all x 2 X and )
; Ri ( )) for every i 2 I, then f
L(f
= f ( ).
We basically require an adaptation of Maskin monotonicity to each temporary implementation problem. In other words, temporary Maskin monotonicity requires that every period-t temporary social choice function 't that results from the folding of the SCF is Maskin monotonic. Therefore, the condition of temporary Maskin monotonicity can be stated as follows: Definition 4 A folding SCF f : (i) the period-T function 'T over D
! F is temporary Maskin monotonic provided that: jy
T
is Maskin monotonic for every y
for every t 6= 1; T , the period-t function 't over D every y
t
T
2Y
T
; (ii)
jy t ; f +(t+1) is Maskin monotonic for
2 Y t ; (iii) the period-1 function '1 over D [ jf +2 ] is Maskin monotonic.
Our second main result is that only temporary Maskin monotonic SCFs are temporary implementable. Theorem 2 If I
2 and the SCF f :
! F is temporarily implementable, then it is
temporary Maskin monotonic. Proof. See Appendix.
3.3 The characterization theorem In the abstract Arrovian domain, the condition of no veto-power says that if an outcome is at the top of the preferences of all agents but possibly one, then it should be chosen irrespective of the preferences of the remaining agent: that agent cannot veto it. The condition of no veto-power implies two well-known conditions: unanimity and weak no vetopower. The property of unanimity can be stated as follows for an abstract outcome space X: Definition 5 The SCF F :
! X satis…es unanimity provided that for all
2
and all
x 2 X if xRi ( ) y for all i 2 I and all y 2 X, then x = F ( ). A SCF that satis…es this 15
property is said to be a unanimous SCF. In other words, it states that if an outcome is at the top of the preferences of all individuals, then that outcome should be selected by the SCF. As a part of su¢ ciency, we require an adaptation of the above de…nition to each period-t temporary implementation problem. In other words, temporary unanimity requires that each of period-t temporary social function 't de…ned over period-t domain of induced orderings is unanimous. Thus, the condition can be stated as follows: Definition 6 A folding SCF f :
! F satis…es temporary unanimity provided that the
following requirements hold: (i) the period-T function 'T over D every y
T
2Y
T
jy
T
; (ii) for every t 6= 1; T , the period-t function 't over D
unanimous for every y
t
is unanimous for jy t ; f +(t+1) is
2 Y t ; (iii) the period-1 function '1 over D [ jf +2 ] is unanimous.
Furthermore, the condition of no veto-power implies the condition of weak no vetopower, which states that if an outcome x is socially optimal at the state changes from x at Ri of Ri
to
and if the state
in a way that under the new state an outcome y that was no better than
for some agent i is weakly preferred to all outcomes in the weak lower contour set at x according to the ordering Ri ( ) and this y is maximal for every other agent j
in the set X according to Rj ( ), then this y should be socially optimal at . Formally, for an abstract outcome space X: Definition 7 A SCF F : ;
2
if y 2 L f
! X satis…es weak no veto-power provided that for every L (y; Ri ( )) for some i 2 I and X
; Ri
L (y; Rj ( )) for
every j 2 In fig, then f ( ) = y. As a part of su¢ ciency, we require the following adaptation of the weak no veto-power condition: Definition 8 A folding SCF f :
! F satis…es temporary weak no veto-power provided
that the following requirements hold: (i) the period-T function 'T over D weak no veto-power for every y over D
T
2Y
T
jy
T
satis…es
; (ii) for every t 6= 1; T , the period-t function 't
jy t ; f +(t+1) satis…es weak no veto-power for every y
t
2 Y t ; (iii) the period-1
function '1 over D [ jf +2 ] satis…es weak no veto-power. 16
Our characterization of temporary implementable SCFs can thus be stated as follows: Theorem 3 If I
3 and the SCF f :
! F satis…es the folding condition and temporary
Maskin monotonicity and if the SCF satis…es temporary weak no veto-power as well as temporary unanimity, then it is temporarily implementable. Proof. See Appendix.
4. Implications 4.1 sequential trading In this section, we investigate whether the trading rule as considered in the dynamic general equilibrium framework is indeed temporarily implementable. When it is literally understood, the concept of Arrow-Debreu-McKenzie (ADM) equilibrium says that all the agents meet on the …rst day of their life and write down a contract on all the deliveries of consumption contingent on every date-event, and simply commit to it. A more realistic description of trading over time is by Radner, which considers that at each period agents can trade only between current consumption and assets to be carried over to the next period. To our knowledge, however, the Radner-type model has not been given a strategic foundation. In the Radner model prices are de…ned only for on-path situation and it is left unclear what prices should be formed in o¤-path situations, while a strategic outcome function in a dynamic environment must specify prices and allocations even at o¤path histories. In fact, as far as the markets are sequentially complete ADM equilibrium and Radner equilibrium are equivalent. This means that from strategic viewpoints the Radner model cannot escape the problem which the ADM model has. The competitive models are silent about what prices and allocations should be formed after the society makes mistake. Strategic implementation of competitive solutions in general involves a strange story: each agent is supposed to behave as a price-taker, despite he is aware that message he sends may a¤ect the market price. In the static setting, these apparently contradicting natures can be made compatible by making the mechanism nicely so that agents face a kind of
17
coordination game in which they are induced to agree on prices in equilibrium. In fact, the (feasibility-constrained version of) ADM solution is Nash-implementable. Being a price-taker is harder in dynamic environments, however, when social decision and execution can be made only in a temporary manner. It requires that every agent perceives that he cannot a¤ect spot price/interest rate at any period, in particular that the amount of asset to carry over to the future does not a¤ect the spot prices/interest rates in the future, despite he is aware that messages he sends may a¤ect the market price in both the current period and the future periods, and that equilibrium prices and allocations in the future periods are a function of whole allocations in the current period including his own. Below we explain the nature of the problem and see whether the Radner-type solution can clear this bar. For the sake of convenience, we assume that there are only three consumption periods (CPs), and so two trading periods (TPs), and that there is one perfectly divisible commodity in each CP. In TP1 agents transfer consumption between CP1 and CP2, and in TP2 they transfer consumption between CP2 and CP3. In TP1, agents sell/buy consumption in CP1 and buy/sell consumption in CP2. In TP2, agents sell/buy consumption in CP2 and buy/sell consumption in CP3. Let q 1 be the TP1 spot price, the relative price of CP2 consumption for CP1 consumption, and q 2 be the TP2 spot price, the relative price of CP3 consumption for CP2 consumption. Each agent i is endowed with an amount ! ti of the commodity in CPt. The total endowment of the commodity in CPt is denoted by ! t . Agent i’s consumption set is R3+ , and her consumption in CPt is denoted by cti . In state , this agent has preference ordering Ri ( ) over consumption sequences in her consumption set. Endowments are given once and for all, and therefore an economy is described by a state . The domain assumption is that at each economy
2
agent i’s preference ordering
is represented by an additively separable utility function Ui ( ; c1i ; c2i ; c3i ) = vi1 ( ; c1i ) + vi2 ( ; c2i ) + vi3 ( ; c3i ). This guarantees that all consumption goods in CP1, CP2 CP3 are gross-substitutes of each
18
other and the ADM and Radner equilibrium is unique. We describe feasible allocations by using net trade vectors. Let
H=
(
z 2 RI j
X
)
zi = 0 ,
i2I
which is the set of closed net trades. Thus, the set of closed net trade vectors for TPt can be de…ned by Zt = Ht
H t+1 ,
for t = 1; 2.
A TPt net trade allocation is thus a vector z t = (z tt ; z tt+1 ) in Z t , where the ith element zitt of z tt denotes agent i’s net trade of consumption in CPt, and where the ith element zitt+1 of z tt+1 denotes agent i’s net trade of consumption in CP(t + 1). The set of feasible net trade allocations over the two trading periods is denoted by Z and de…ned by Z = (z 1 ; z 2 ) 2 Z 1
Z 2 j! 1i + zi11
0; ! 2i + zi12 + zi22
0; ! 3i + zi23
0; 8i 2 I .
The set of feasible TP1 net trade allocations is given by Z 1 = fz 1 2 Z 1 j(z 1 ; z 2 ) 2 Z for some z 2 2 Z 2 g, while the set of TP2 net trade allocation, conditional on z 1 , is given by Z 2 (z 1 ) = fz 2 2 Z 2 j(z 1 ; z 2 ) 2 Zg, for all z 1 2 Z 1 . In economy
2
, agent i’s preference ordering
a preference ordering Ri ( ) over the set of feasible net trade allocations Z in the natural way: for all z; z^ 2 Z,
19
()
zRi ( ) z^
Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + zi22 ; ! 3i + zi23 ) Ui ( ; ! 1i + z^i11 ; ! 2i + z^i12 + z^i22 ; ! 3i + z^i23 ).
Though the preference ordering
2
and every z 1 2 Z 1 , the net trade allocation
f 2 [ jz 1 ] 2 Z 2 (z 1 ) constitutes a TP2 competitive net trade allocation, conditional on z 1 , if there is a TP2 spot price q 2 [ jz 1 ] such that for every agent i this allocation f 2 [ jz 1 ] solves the following problem: Max
z 2 2Z 2 (z 1 )
Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + zi22 ; ! 3i + zi23 ),
subject to zi22 + q 2 [ jz 1 ]zi23
0. (10)
Let Ri1 [ ; f 2 ] denote agent i’s TP1 induced preference ordering over the set of feasible TP1 net trade allocations, Z 1 , and be de…ned by 2 12 22 1 3 23 1 x1 Ri1 [ ; f 2 ]y 1 () Ui ( ; ! 1i + x11 i ; ! i + xi + fi [ jx ]; ! i + fi [ jx ])
(11)
Ui ( ; ! 1i + yi11 ; ! 2i + yi12 + fi22 [ jy 1 ]; ! 3i + fi23 [ jy 1 ]),
for all x1 ; y 1 2 Z 1 .
In contrast to static pure exchange economies where each agent’s preferences are de…ned over her own net trade vectors, in sequential trading, each individual must have preferences over whole TP1 net trade allocations. This is due to the presence of intertemporal pecuniary externalities. Indeed, an outcome of the trading rule in TP2 depends on the net trade 3
Note that this is a feasibility-constrained version. As it is known that the ADM solution fails to satisfy Maskin monotonicity when it results in boundary allocations, and it is necessary to modify the solution by truncating each agent’s consumption set by the set of feasible allocations. Here each individual’s admissible set of trades is truncated by Z 2 z 1 , although it does not matter when we can restrict attention to interior allocations.
20
allocation assigned in TP1, because trading in TP1 a¤ects the values of endowments in the next trading period. Moreover, the induced ordering Ri1 [ ; f 2 ] may be non-convex. In order for it to be a convex preference ordering, it is required that the TP2 function f 2 that maps every economy, conditional on past trades, into a TP2 net trade allocation be a concave function, but this requirement fails for any reasonable trading rule. As is known, although convexity is no more than a su¢ cient technical condition for things to work, it becomes extremely di¢ cult to establish any reasonable solution once it is violated. We may proceed in two ways. First, we can still de…ne a concept of competitive equilibrium following the tradition of dynamic general equilibrium theory. Definition 10 For every economy
2
, a TP1 net trade allocation f 1 [ ] 2 Z 1 constitutes
a TP1 competitive net trade allocation if there is a TP1 spot price q 1 [ ] such that for every agent i the net trade allocation pro…le (f 1 [ ] ; f 2 [ jf 1 [ ]]) solves the following problem: Max Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + zi22 ; ! 3i + zi23 ) z2Z
subject to zi11 + q 1 [ ]zi12
0
zi22 + q 2 [ jf 1 [ ]]zi23
0.
(i) (ii)
This is consistent with the existing dynamic general equilibrium framework, in the sense that individuals take the price path as given. Note that it assumes that each individual perceives that her saving choice does not a¤ect either TP1 spot price q 1 [ ] or TP2 spot price q 2 [ jf 1 [ ]], despite that in the next period the spot price q 2 [ jz 1 ] is a¤ected by whole z 1 which includes his own trade vector zi1 . The path of consumptions given by this solution is equivalent to the ADM solution.4 This solution is not temporarily implementable, however. We prove this by means of an example. 4
Note again that this is the feasibility-constrained version, while it does not matter when we can restrict attention to interior allocations.
21
Claim 1 Let I
2. Then, the Radner solution, de…ned over
, does not satisfy the folding
condition. Proof. Suppose that there are three individuals, i, j and k. Assume that agents’intertemporal endowments are as follows: ! i = (! 1i ; 0; 0), ! j = (0; ! 2j ; 0) and ! k = (0; 0; ! 3k ), where ! 1i ; ! 2j ; ! 3k > 1. Each economy
2
= (0; 1] speci…es a preference pro…le over consumption paths
represented by: Ui ( ; c1i ; c2i ; c3i ) = c1i + ln c3i Uj ( ; c1j ; c2j ; c3j ) = ln c1j + c2j Uk ( ; c1k ; c2k ; c3k ) = ln c2k + c3k . Then, the TP2 spot price equilibrium is given by: q 2 [ jx1 ] = x12 i ,
for all x1 2 Z 1 ,
and the TP2 competitive net trade allocation is given by: fi22 [ jx1 ] =
x12 i
fi23 [ jx1 ] = 1 fj22 [ jx1 ] = 0 fj23 [ jx1 ] = 0 fk22 [ jx1 ] = x12 i fk23 [ jx1 ] =
1,
for all x1 2 Z 1 .
The TP1 orderings over Z 1 induced by TP2 competitive net trade allocations are rep-
22
resented respectively by: Ui1
; x1 jf 2
= ! 1i + x11 i
Uj1
; x1 jf 2
2 12 = ln x11 j + ! j + xj
Uk1
; x1 jf 2
3 = ln x12 i + !k
For every economy
2
1,
for all x1 2 Z 1 , for all
2
.
, the TP1 equilibrium spot price is: q1[ ] = ,
which results in the following TP2 equilibrium spot price: q 2 [ jf 1 [ ]] = 1, and in the following competitive equilibrium net trade allocations: fi11 [ ] = fi12 [ ] = 1 fi22 [ jf 1 [ ]] =
1
fi23 [ jf 1 [ ]] = 1 fj11 [ ] = fj12 [ ] =
1
fj22 [ jf 1 [ ]] = 0 fj23 [ jf 1 [ ]] = 0 fk11 [ ] = 0 fk12 [ ] = 0 fk22 [ jf 1 [ ]] = 1 fk23 [ jf 1 [ ]] =
1.
23
We have found that f 1 [ ] 6= f 1 [ 0 ] for all , utility pro…les are identical across economies in
0
2
with
6=
0
, though TP1 reduced
, in violation of part (iii) of the folding
condition. When an agent reveals an intention to save more, there may be di¤erent reasons to do so. It may be because he is simply patient, or it may be because he wants to manipulate the Radner equilibrium outcome in the next period. The planner does not distinguish between those reasons. In particular, the above example is the case that no information is revealed to the planner after TP1. While we can induce agents to behave as price-takers in the static setting, it is thus in general impossible to do that in the realistic dynamic setting in which social decision and execution can be done only in a temporary manner. The problem will disappear when there is a large number of traders, as each individual tends to be small and unable to manipulate through intertemporal pecuniary externalities. Then we would say that the dynamic general equilibrium model should be understood as such a limit model rather than an exact …niteperson model. The second way is …nd a domain in which an exact …nite-person implementation is possible. It is the domain such that there are no intertemporal pecuniary externalities.5 Condition 1 For all
2
, the TP2 spot price q 2 [ jx1 ] is constant in x1 2 Z 1 .
Note that when the above condition is met, a TP2 competitive net trade vector assigned to individual i depends only on her own past saving/borrowing behavior. For this reason, we write fi22 [ jzi12 ] and fi23 [ jzi12 ] for fi22 [ jz 1 ] and fi23 [ jz 1 ] respectively. Here are examples of domains which satisfy Condition 1. In what follows, let us focus on economies where the quantity ! ti is strictly positive for every individual i and every consumption period t = 1; 2; 3. Assumption 1 (
1
) Assume that aggregate endowment is constant over time; that is, ! 1 =
! 2 = ! 3 . Also, assume that the individuals have identical discount factors, while they may exhibit di¤erent elasticities of intertemporal substitution. That is, for every economy
2
1
5
Such situation emerges also when constant returns to scale in intertemporal production prevails, since interest rate in such economy is constant.
24
1
it holds that ! 1 = ! 2 = ! 3 and that there is (
2
;
) such that every i’s preference over
consumptions is represented in the form:
Ui ( ; c1i ; c2i ; c3i ) = vi ( ; c1i ) +
1
vi ( ; c2i ) +
1 2
vi ( ; c3i ),
where: the sub-utility vi ( ; ) is twice continuously di¤erentiable, strictly increasing and strictly concave over R++ . the limit of the …rst derivative of the sub-utility vi ( ; ) is positive in…nity as cti approaches 0; that is, limcti !0
@vi ( ;cti ) @cti
= 1.
the limit of the …rst derivative of the sub-utility vi ( ; ) is zero as cti approaches positive in…nity; that is, limcti !1
@vi ( ;cti ) @cti
= 0.
the sub-utility vi ( ; ) satis…es the requirement that
@ 2 vi ( ;cti ) t @vi ( ;cti ) ci = @ct @ 2 cti i
< 1 for all
cti 2 R++ . For this domain, we obtain that the TP2 competitive spot price, net trade allocations and consumption allocations prescribed for every q2
jz 1
=
fi22
jzi12
=
fi23
jzi12
ci 2
jz 1
2
1
are:
2 2
1+ 1 = 1+ 2 = ci 3
zi12 + ! 2i
2
zi12 + ! 2i
jz 1 =
! 3i ! 3i
zi12 + ! 2i + 1+ 2
2
! 3i
, 8i 2 I and 8z 1 2 Z 1 .
Note that period-1 reduced utility on Z 1 is represented by: Ui
1
; z jf
2
= vi
; ! 1i
+
zi11
+
1
1+
2
vi
z 12 + ! 2i + ; i 1+ 2
2
! 3i
, 8i 2 I and 8z 1 2 Z 1 .
25
Assumption 2 (
2
) In this domain we drop the assumption of constant aggregate endow-
ment over time, but we assume that individuals have identical CES preferences. That is, for every
2
2
1
there is a triplet (
2
;
; ) such that every i’s preference ordering over
consumptions is represented in the form: Ui ( ; c1i ; c2i ; c3i ) =
(c1i )1 1
2 1 1 (ci )
+
+
1
3 1 1 2 (ci )
1
,
with
> 0.
When agents have identical CES preferences, we obtain that the TP2 competitive equilibrium spot price, net trade allocations and consumption allocations prescribed for every 2
2
are: q2 fi22
fi23
jz 1 jzi12 jzi12
=
=
2
!2 !3
zi12 + ! 2i 1+
1 2
!2 1 !3
! 3i
2 12 ! 3 zi + ! i = !2 1+ 2
ci 2
jz 1
=
ci 3
jz 1
=
zi12 + ! 2i + 1+ !3 c2 !2 i
!2 !3
! 3i
2
2
!2 !3
!3 1 !2
! 3i
!2 !3
!3 1 !2
jz 1 ,
8i 2 I and 8z 1 2 Z 1 .
Next, let us de…ne a TP1 competitive equilibrium when Condition 1 is satis…ed. Definition 11 For every economy
satisfying Condition 1, a TP1 net trade allocation
f^1 [ ] 2 Z 1 constitutes a backward TP1 competitive net trade allocation if there is a TP1
spot price q 1 [ ] such that for every agent i the net trade allocation f^1 [ ] solves the following problem: Max
z 1 2Z 1
Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + fi22
jzi12 ; ! 3i + fi22
jzi12 ), subject to zi11 + q 1 [ ]zi12
0.
26
Using this de…nition, we obtain that the competitive equilibrium spot prices prescribed for every economy
2
1
are: 1
1
q [ ]=
and q
2
h
i 1 ^ jf [ ] =
2
,
and so the competitive net trade allocations and the equilibrium consumption allocations are for every individual i 2 I as follows: f^i11 [ ] = f^i12 [ ] = h
i
1 1
1+ 1+
1
1 +
! 1i 1 +
1 2
2
! 1i 1 +
1 2
+
2
! 2i
2
2
! 2i
! 3i
! 3i
2
f^i12 [ ] + ! 2i ! 3i 1+ 2 h i 1 fi23 jf^i12 [ ] = f^i12 [ ] + ! 2i ! 3i 1+ 2 h i h i !1 + 1!2 + i ci 1 [ ] = ci 2 jf^i12 [ ] = ci 3 jf^i12 [ ] = i 1 1+ + fi22
jf^i12 [ ]
For economies in 2
2
=
2
1 2 1 2
! 3i
.
, we obtain that the equilibrium spot prices prescribed for every
are: q1 [ ] =
1
!1 !2
and q 2
h
i jf^1 [ ] =
2
!2 !3
.
Thus, the competitive net trade allocations are:
f^i11 [ ] =
! f^i12 [ ] = !1
fi23
1
!3 !2
! 1i
2
!3 !2
f^i12 [ ] + ! 2i
h
i jf^i12 [ ] =
h
2 ^12 i ! 3 fi [ ] + ! i 12 ^ jfi [ ] = !2 1+ 2
1 2
!1 1 !2
1 1
! 3i
1
2 !1 !2
+1 !2 1 !1
2
! 3i
! 2i +
2
! 2i +
+
1
1+
1+
!1 !2
+1 1+
2
fi22
2
! 1i
+
!2 !3
!3 1 !2
1 2
! 3i
!2 !3
!3 1 !1
!2 !3
!2 1 !3
! 3i !3 1 !2
!2 !3
,
27
while the corresponding equilibrium consumption allocations are:
ci 1 [ ] = ci 2
jz 1 [ ]
ci 3
jz 1 [ ]
! 1i +
1
1+
1
! 2i
!1 !2
!2 1 !1
+ +
1 2 1 2
! 3i
!1 !3
!3 1 !1
!2 ci 1 [ ] 1 ! !3 = c 1[ ]. !1 i
=
The backward competitive solution of an economy is a SCF f = f 1 [ ] ; f 2 [ j ] associat-
ing the period-1 function f 1 [ ] with the backward TP1 competitive net trade allocation f^1 [ ],
that is, f 1 [ ] = f^1 [ ] 2 Z 1 , and the period-2 function f 2 [ j ] with the TP2 competitive net trade allocation for any TP1 net trade allocation in the set Z 1 , that is, f 2 [ jz 1 ] = f 2 [ jz 1 ] for every z 1 2 Z 1 . Thanks to Condition 1, we can now state and prove the following permissive results. Claim 2 Assume that I
3. Suppose that the quantity ! ti is strictly positive for every
individual i and every consumption period t = 1; 2; 3. Then, the backward competitive solution f is temporarily implementable if it is de…ned either over
1
or over
2
.
Proof. Let the premises hold. To show that f is temporarily implementable when it is de…ned either over
1
or over
2
, we need to show that this solution satis…es the folding
condition and temporary Maskin monotonicity. Moreover, we need also to show this solution satis…es temporary unanimity and temporary weak no veto-power. First, let us show that f satis…es the folding condition. To this end, let Y 1 = Y and let Y 2 (z 1 ) = Z 2 (z 1 ) for every z 1 2 Z 1 . Then, the sets Y 1 = Y empty sets. Note that for k = 1; 2, it holds that f 1
k
Z 1 and f 2
2
2
= Z1
and Y 2 (z 1 ) are not k
jz 1
Z 2 (z 1 ) for
every z 1 2 Z 1 . Let us de…ne the TP2 induced ordering of individual i in state
at z 1 2 Z 1 , denoted
by Ri [ jz 1 ], as follows: x2 Ri
jz 1 y 2
3 23 Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + x22 i ; ! i + xi )
() Ui ( ; ! 1i + zi11 ; ! 2i + zi12 + yi22 ; ! 3i + yi23 ), 28
for every x2 ; y 2 2 Y 2 (z 1 ). We denote by R [ jz 1 ] the pro…le of TP2 induced orderings at jz 1 , by D [
1
jz 1 ] the TP2 domain of induced orderings at
domain of induced orderings at '2 : D
k
2
1
jz 1 and by D [
2
jz 1 ] the TP2
jz 1 . For every k = 1; 2, let us de…ne the TP2 function
jz 1 ! Y 2 (z 1 ) as follows: '2 R
jz 1
= f2
jz 1 ,
k
8 2
.
The TP1 induced ordering of individual i in state , denoted by Ri in (11). Let us denote by R
jf 2 the pro…le of TP1 induced orderings at jf 2 , by D
the TP1 domain of induced orderings at orderings at
2
jf 2 , is de…ned as
1
jf 2 , and by D
2
1
jf 2
jf 2 the TP1 domain of induced
jf 2 . For every k = 1; 2, let us de…ne the TP1 function '1 : D
k
jf 2 ! Y 1
as follows: '1 R
jf 2
= f1 [ ] ,
8 2
k
.
By the above de…nitions and by the fact that competitive equilibrium exists in each TP, one can check that the backward competitive solution f satis…es the folding condition. To see that f also satis…es temporary Maskin monotonicity, it su¢ ces to observe that in each TP the competitive net trade allocation is unique and always an interior allocation, that the TP1 competitive solution '1 on D that the TP2 competitive solution '2 on D
k k
jf 2 is Maskin monotonic for k = 1; 2, and
jz 1 is also Maskin monotonic for k = 1; 2.
Finally, to see that the backward competitive solution f satis…es temporary unanimity and temporary weak no veto-power it su¢ ces to observe that they are vacuously satis…ed since individuals’induced orderings are strictly monotonic in consumption.
4.2 Sequential voting In this section, we consider a bi-dimensional policy space where an odd number of individuals vote sequentially on each dimension and where an ordering of the dimensions is exogenously given. We assume that a majority vote is organized around each policy dimension and that the outcome of the …rst majority vote is known to the voters at the beginning of the second voting stage. This temporary resolution is common in political
29
economy models (see, e.g., Persson and Tabellini, 2000). We are interested in temporarily implementing the simple majority solution, which selects the Condorcet winner in each voting stage. A policy choice is an ordered pair (x1 ; x2 ) 2 X 1 X 2 , where the policy space of dimension d = 1; 2 is an open interval.6 Each voter i is described by a one-dimensional type type space is the open interval
;
i.
The
.
Definition 12 The voter i’s utility function U :
;
X1
X 2 ! R is a twice-
continuously di¤erentiable satisfying: (a) Strict concavity, that is: @ 2 U ( i ; x1 ; x2 ) < 0 and @ 2 x1
@ 2 U ( i ; x1 ; x2 ) < 0, for every x1 ; x2 2 X 1 2 2 @ x
X 2.
(b) induced single-crossing property, that is: @ 2 U ( i ; x1 ; x2 ) > 0 and @ i @x1
@ 2 U ( i ; x1 ; x2 ) > 0, for every x1 ; x2 2 X 1 X 2 and 2 @ i @x
i
2
;
(c) Strategic complementarity, that is: @ 2 U ( i ; x1 ; x2 ) @x1 @x2
0, for every x1 ; x2 2 X 1
X 2.
The induced single-crossing property simply requires that the induced utility of both dimensions is increasing in the type of voter. This property can also be found in De Donder et al. (2012). We now introduce the de…nition of a Condorcet winner for an arbitrary policy space P : Definition 13 Suppose that individuals in I votes over the set of policies P . We say that p 2 P is a majority voting outcome, also known as a Condorcet winner (CW ), if there does not exist any other distinct outcome p0 2 P that is strictly preferred by more than half of voters to the outcome p. 6
The choice of a bi-dimensional policy space is motivated by convenience.
30
.
For any integer k
2, the set of states
takes the structure of the Cartesian product
of allowable independent types for voters, that is,
2k 1
;
, with
as typical element.
It simpli…es the argument, and causes no loss of generality, to assume that 2k 1 .
Therefore, the type
k
is the median type, denoted by
med ,
1
2
at state .
At the state , each voter is assumed to have an ordering preference relation Ri ( ) over the policy space X 1
X 2 which is represented by U ( i ; ; ).
Solving by backward induction when the state
is the prevailing state, if x1 2 X 1 is
the outcome of the …rst majority voting, then the stage-2 induced ordering of voter i on X 2 in state
at x1 is denoted by Ri [ jx1 ] and is represented by U ( i ; x1 ; ).
The pro…le of the stage-2 induced orderings in state
at x1 is denoted by R [ jx1 ]. Let
D [ jx1 ] be the stage-2 domain of induced ordering preferences induced by the set
as well
as by the outcome x1 ; that is: D
jx1
jx1 j 2
R
, for every x1 2 X 1 .
(12)
If x1 2 X 1 is the outcome of the …rst majority voting, then the stage-2 majority voting function f 2 : D [ jx1 ] ! X 2 is de…ned as follows: f2
jx1 = CW R
jx1
,
where CW (R [ jx1 ]) denotes the Condorcet winner under the pro…le R [ jx1 ]. It will be shown below that this outcome is the most-preferred outcome of the median type. Let us suppose that the stage-2 majority voting function is well-de…ned for every outcome x1 2 X 1 . Then, in stage-1, the utility of a voter i at state
for the outcome z 1 2 X 1
is: U
i; z
1
; f2
jz 1
.
Then, the stage-1 induced ordering of voter i on X 1 in state at the majority voting function f 2 [ j ], denoted by Ri [ jf 2 ], is given by: y 1 Ri
jf 2 z 1 ()
y1; f 2
jy 1
Ri ( ) z 1 ; f 2
jz 1
, for every y 1 ; z 1 2 X 1 . 31
As usual, the pro…le of the stage-1 induced orderings in state
at the majority voting
function f 2 [ j ] is denoted by R [ jf 2 ]. Let D [ jf 2 ] be the stage-1 domain of induced ordering preferences induced by the set
as well as by the majority voting function f 2 ; that
is: D
jf 2
R
jf 2 j 2
.
(13)
Thus, the stage-1 majority voting function f 1 : D [ jf 2 ] ! X 1 is de…ned as follows: f 1 [ ] = CW R
jf 2
, for every
2
,
where CW (R [ jf 2 ]) denotes the Condorcet winner under the pro…le R [ jf 2 ]. Definition 14 The SCF f ( ) = (f 1 [ ] ; f 2 [ j ]) on every
2
is the majority voting solution if for
:
f 1 [ ] = CW R
jf 2
and f 2
jx1 = CW R
jx1
for every x1 2 X 1 .
The following lemma shows that the majority voting solution is a single-valued function. The intuition behind it is similar to that of Proposition 4 of De Donder et al. (2012) for the case where there is a continuum of voters. Firstly, the assumption of strict concavity assures the existence and unicity of the Condorcet winner in the second voting stage. This assumption, combined with the assumption of strategic complementarity and with the induced single-crossing property, assures that the stage-1 induced ordering of voter i on X 1 in state
at the majority voting function f 2 [ j ] is single-crossing. This guarantees the
existence and unicity of the Condorcet winner in the …rst voting stage. Lemma 1 Suppose that the cardinality of I is 2k utility function Ui on
X1
1 with k
2. Suppose that voter i 2 I’s
X 2 meets the requirements of De…nition 12 and depends
only on her own type. Then, the majority voting SCF f ( ) = (f 1 [ ] ; f 2 [ j ]) over
is a
single-valued function on each policy dimension.
32
Proof. See Appendix.
Thanks to the above lemma, we can now state and prove the main result of this section. Claim 3 Suppose that the cardinality of I is 2k utility function Ui on
X1
1 with k
2. Suppose that voter i 2 I’s
X 2 meets the requirements of De…nition 12 and depends
only on her own type. Then, the majority voting solution is temporarily implementable. Proof. Let the premises hold. By Theorem 3, it su¢ ces to show that the majority voting solution satis…es the folding condition and temporary Maskin monotonicity and, moreover, it satis…es temporary unanimity and temporary weak no veto-power. Thus, T = f1; 2g. Let Y 1 = X 1 and Y 2 (x1 ) = X 2 for every x1 2 X 1 . De…ne D [ jx1 ] as in (12) and de…ne D [ jf 2 ] as in (13). For every x1 2 X 1 , de…ne the second stage function '2 : D [ jx1 ] ! X 2 by '2 (R [ jx1 ]) = CW (R [ jx1 ]) for every R [ jx1 ] 2 D [ jx1 ]. Moreover, de…ne the …rst stage function '1 : D [ jf 2 ] ! X 1 by '1 (R [ jf 2 ]) = CW (R [ jf 2 ]) for every R [ jf 2 ] 2 D [ jf 2 ]. These functions are single-valued by Lemma 1. This shows that the majority voting solution satis…es the folding condition. By de…nitions of the preceding paragraph and by the fact that in each period individuals have single crossing preferences, one can see that the majority voting solution satis…es temporary Maskin monotonicity. Since unanimity and weak no veto-power are satis…ed, we conclude that the majority voting solution on
is temporarily implementable.
5. Conclusion Summary. The paper has examined problems of implementing social choice objectives in a dynamic environment, in which society can only decide and execute a policy variable at hand period by period. The social objective that society wants to achieve is represented by a social choice function (SCF) that maps each state of the world into a dynamic process mapping every history into a social outcome. This social process is temporarily implementable if there exists a process of one-period game forms (with observed actions and simultaneous moves)
33
each of which generates a social outcome only at one given period after a given history, such that at each state of the world there is a subgame-perfect Nash equilibrium in which the social objective is ful…lled at every period, after every history, as a unique equilibrium outcome process. We have identi…ed two necessary conditions for temporary implementability, the folding condition and temporary Maskin monotonicity. The …rst condition states that a temporarily implementable SCF can be decomposed into a sequence of temporary social choice functions, each of which is de…ned only over induced preferences induced over outcomes at hand. Each induced preference is constructed in the manner of backward-induction. This means that a period-t induced preference over the current component set depends on past decisions as well as on the socially optimal path that the dynamic process will bring about in the future. The second condition states that every such temporary social function needs to satisfy a remarkably strong invariance condition for Nash implementation, now widely referred to as Maskin monotonicity (Maskin, 1999). We have also shown that under two auxiliary conditions the two necessary conditions are su¢ cient, as well. The implementing mechanism we have constructed is simple. After each history, we run the canonical Maskin mechanism just to decide the current outcome, in the "apparently static" manner. Participants report messages consisting of a preference pro…le de…ned only over current outcomes, an outcome as well as a tie-breaking device. The last decades have seen impressive advances in the theory of implementation. One conclusion is that the use of re…nements in implementation leads to permissive results. This is so because implementation in re…nements of Nash equilibrium (Moore and Repullo, 1988; Abreu and Sen, 1990; Palfrey and Srivastava, 1991; Jackson, 1992; Vartiainen, 2007) allow us to circumvent the limitations imposed by Maskin monotonicity. As Sjöström (1994) pointed out, ‘With enough ingenuity the planner can implement “anything”’(p. 503). In contrast to this, we have found that sequential rationality, when used in a context where society has to decide and execute a policy variable at hand period by period in a temporary manner, does not allow the planner to escape the limitations imposed by Maskin monotonicity. Furthermore, we have also shown (in Claim 1) that the folding condition imposes non-trivial restrictions on the class of social dynamic processes that are temporarily implementable. 34
Indeed, we have applied our analysis to two prominent dynamic problems, voting over time and temporary trading. In the voting application, we have shown that on the domain satisfying the single-crossing property the simple majority solution, which selects the Condorcet winner in each voting stage (after every history), is temporarily implementable. In a borrowing-lending model with no liquidity constraints, in which individuals trade in spot markets and transfer wealth between any two periods by borrowing and lending, we have noted that intertemporal pecuniary externalities arise because trades in the current period change the spot price of the next period, which, in turn, a¤ects its associated equilibrium allocation. The quantitative implication of this is that every individual’s induced preference ordering concerns not only her own consumption/saving behavior but also the consumption/saving behavior of all other individuals. In this set-up, we have shown that, under such pecuniary externalities, the standard dynamic competitive equilibrium solution is not temporarily implementable. However, we have also identi…ed preference domains –which involve no pecuniary externalities – for which the no-commitment version of the dynamic competitive equilibrium solution is de…nable and temporarily implementable. It remains an open question how we should deal with intertemporal pecuniary externalities. We hope that this and other topics related to this paper will be investigated in future research. In this paper, we have considered temporary implementation in SPE. One may consider that agents follow an alternative equilibrium concept, which is known to be less restrictive. This does not ease the restrictiveness of temporary implementability, however, since it does not change the restrictiveness of the folding condition that is orthogonal to the degree of restrictiveness of implementability in a temporary social choice problem.
References Abreu D, Sen A, Subgame perfect implementation: a necessary and almost su¢ cient condition, J Econ Theory 50 (1990) 285-299 Arrow KJ, Debreu G, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954) 265–290
35
Arrow KJ, The Role of securities in the optimal allocation of risk-bearing, Rev Econ Studies 31 (1964) 91-96 Chambers CP, Virtual repeated implementation, Econ Letters 83 (2004) 263-268 De Donder P, Le Breton M, Peluso E, Majority voting in multidimensional policy spaces: Kramer-Shepsle versus Staskelberg, J Public Econ Theory 14 (2012) 879-909 Grandmont JM, Temporary general equilibrium theory, Econometrica 45 (1977) 535-572 Herrero MJ, Srivastava S, Implementation via backward induction, J Econ Theory 56 (1992) 70-88 Jackson MO, Implementation in Undominated Strategies: A Look at Bounded Mechanisms, Rev Econ Stud 59 (1992) 757-775 Kalai E, Ledyard JO, Repeated Implementation, J Econ Theory 83 (1998) 308-317 Lee J, Sabourian H, E¢ cient Repeated Implementation, Econometrica 79 (2011) 1967-1994 Maskin E, Nash equilibrium and welfare optimality, Rev. Econ. Stud. 66 (1999) 23-38 McKenzie LW, On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems, Econometrica. 22 (1954) 147–161 Mezzetti C, Renou L, Repeated Nash Implementation, Theoretical Economics, forthcoming Moore J, Repullo R, Subgame perfect implementation, Econometrica 56 (1988) 1191-1220. Moore J, Repullo R, Nash implementation: A full characterization, Econometrica 58 (1990) 1083-1100 Osborne MJ, Rubinstein A, A Course in Game Theory, The MIT Press (1994) Palfrey T, Srivastava S, Nash-implementation using undominated strategies, Econometrica 59 (1991) 479-501 Persson T., Tabellini G, Political Economics, The MIT Press (2002) Radner R, Existence of equilibrium of plans, prices, and price expectations in a sequence of markets, Econometrica 40 (1972) 289-304
36
Radner R, Equilibrium under uncertainty, Ch. 20 in Handbook of Mathemical Economics, eds KJ Arrow and MD Intriligator, Vol II, Amsterdam: North-Holland, (1982) 9231006 Sjöström T, Implementation in undominated Nash equilibria without using integer games, Games Econ Behav 6 (1994) 502–511 Vartiainen H, Subgame perfect implementation: a full characterization, J Econ Theory 133 (2007) 111-126.
Appendix Proof of Theorem 1 Proof of Theorem 1. Let the premises hold. Thus, there exists a dynamic mechanism (I; H; A (H) ; g) that temporarily implements the SCF f . Therefore, for every f1 ft
jg
if and only if s 2 SP E of
(ht ) ;
t
ht
= g1 s
= gt s
ht
h1
,
and
for every ht 2 H t with t 6= 1
. Fix any s 2 SP E
;
2
;
. Then, s jht is a Nash equilibrium
for every history ht 2 H. Moreover, by temporary implementability of f , it
also follows that: f +t
jg
t
(h) = g +t s jh , for every h 2 H t with 2
Fix any period t 6= 1. Let us de…ne the set Y 1 , the set Y
t
t
T.
and the set Y t (g
(14) t
(h)) as
follows: Y1
2 X t jfor some a h1 2 A h1
g 1 a h1 Y
t
g
t
(h) 2 X t jfor some h 2 H t ,
,
(15) (16)
37
and for every g
t
Y t (g
(h) 2 Y t : t
fg t (a (h)) 2 X t (g
(h))
t
(h)) ja (h) 2 A (h) for some h 2 H t g .
By their de…nitions as well as by the assumption that the dynamic mechanism in SPE the SCF f , one can check that f t [ jg Moreover, given that
t
(h)]
Y t (g
t
(17)
implements
(h)) and that f 1 [ ]
Y 1.
is a dynamic mechanism, one can also check that for every period
t 6= 1: t
g for every
ht 2 Y
t
() g 1 a1 2 Y 1 and g (a ) 2 Y
such that 2
For every y
T
2Y
t T
1, for every ht
a1 ;
g
;a
1
; at 1 ) 2 H t .
(a1 ;
, the period-T preference domain D
jy
follows from its de…nition in (2) and from the fact that Y T y
T
T
is nonempty, and this is not empty. Let the
period-T function 'T : D
jg
T
(h) ! Y T g
T
(h)
be de…ned by: 'T R
jg
T
(h)
= g T (s (h)), for every history h 2 H T and state
2
,
(18)
t
for some
where s 2 SP E ( ; ). Fix any period t 6= 1; T and any t-head outcome path y h 2 H t . Since the set Y t (g
t
(h)) is not empty and since
can see that the period-t domain of induced orderings D
t
g
t
(h) 2 Y
temporarily implements f , one jy t ; f +(t+1) as de…ned in (5) is
not empty. Similarly, one can see that period-1 domain of induced orderings D [( jf +2 )] as de…ned in (8) is not empty. For every t 6= 1; T , let the period-t function 't : D
jg
t
(h) ; f +(t+1) ! Y t g
t
(h)
38
be de…ned by: 't R
t
jg
= g t (s (h)) for every h 2 H t and every
(h) ; f +(t+1)
2
.
(19)
Let the period-1 function '1 : D
jf +2 ! Y 1
be de…ned by: h1 ), for every
= g 1 (s
jf +2
'1 R
2
.
(20)
To complete the proof, we need to show that the period-t function 't is a function for every t 2 T . The following claim establishes it for the case where t 6= 1; T . The same arguments, suitably modi…ed, can be used to show that '1 and 'T are functions. Claim 4 If the SCF f over for some y Proof. R
0
t
t
2Y
with t 6= 1 and some ;
Suppose that y
jy t ; f +(t+1) = R
is temporarily implementable and R
t
jy t ; f +(t+1) for some ;
= g 0
2
t
0
2
, then f t [ jy t ] = f t [ 0 jy t ].
(h) for some h 2 H t and that R
jy t ; f +(t+1)
0
jy t ; f +(t+1)
=
.
Since s 2 SP E ( ; ) and since, moreover, R
jy t ; f +(t+1) = R
0
jy t ; f +(t+1) , we
have that: s (h) 2 N E
jy t ; f +(t+1)
(h) ; R
\ NE
(h) ; R
0
jy t ; f +(t+1)
,
and so, for every i 2 I and ai (h) 2 Ai (h), it holds that: s (h) Ri From the de…nition of Ri
0
0
jy t ; f +(t+1)
ai (h) ; s
i
(h) .
jy t ; f +(t+1) and from (14), it follows that for every i 2 I
and ai (h) 2 Ai (h) it holds that: 0
g
t
(h) ; g t s (h) ; g +(t+1) s j h; s (h)
g
t
(h) ; g t ai (h) ; s
i
0
Ri ( 0 )
(h) ; g +(t+1) s j h; ai (h) ; s
(21) i
(h)
.
39
0
Let si denote individual i’s strategy according to which this i plays si (h0 ) = si (h0 ) for every history h0 6= h and according to which this i plays sti = si (h) after the history h. Note 0
that sjh0 is a Nash equilibrium of ( (h0 ) ; 0 ) for every history h0 6= h since s is a strategy
pro…le in SP E ( ; 0 ). Thus, to have that the strategy pro…le s is a SPE strategy pro…le of ( ; 0 ), we need to show that sjh is a Nash equilibrium of ( (h) ; 0 ). Since the action pro…le s (h) is a Nash equilibrium of
0
(h) ; R
jg
t
(h) ; f +(t+1) ,
it follows that (21) holds for every i 2 I and every ai (h) 2 Ai (h). Thus, no individual i can gain by deviating from the action si (h) and thereafter conforming to si . Since the one deviation property (see, e.g., Osborne and Rubinstein, 1994; Lemma 98.2) holds for a …nite-horizon multi-period game with observed actions and simultaneous moves, it follows that the strategy pro…le sjh 2 SP E ( (h) ; 0 ), and so sjh 2 N E ( ; 0 ). Therefore, we have that s 2 SP E ( ; 0 ). Since the dynamic mechanism
g s (h) = g (s (h)) we have that f t [ 0 jg
t
implements the SCF f in SPE and
(h)] = f t [ jg
t
(h)].
The statement follows by the above arguments.
Proof of Theorem 2 Proof of Theorem 2. Let the premises hold. Thus, there exists a dynamic mechanism (I; H; A (H) ; g) that temporarily implements the SCF f . Therefore, for every f1 ft
t
ht
if and only if s 2 SP E
;
jg
s jht is a Nash equilibrium of
= gt s
= g1 s ht
h1
,
and
for every ht 2 H t with t 6= 1
. Consider any state . Fix any s 2 SP E (ht ) ;
2
. Then,
;
for every history ht 2 H. Moreover, by temporary
implementability of f , it also follows that: f +t
jg
t
(h) = g +t s jh , for every h 2 H t with 2
t
T.
Since the SCF f satis…es the folding condition, de…ne the set Y 1 , the set Y
t
and the
40
set Y t (g
t
(ht )) as in (15), (16) and (17) of the proof of Theorem 1, respectively. T
Fix any g
(h) 2 Y
T
with h 2 H T and suppose that for every i 2 I and every
a (h) 2 A (h), it holds that: 'T R
jg
T
(h)
T
jg
Ri
(h) g T (a (h)) =)
'T R for some R
jg
T
0
(h) and R
T
jg
jg
T
T
(h)
(h) in D
Since the dynamic mechanism 'T R
jg
T
jg
g T (s (h))Ri
jg
T
(22)
(h) g T (a (h)) ,
implements the SCF f in SPE, we have that:
(h)
T
jg T
jg
(h) .
= g T (s (h)) = f T
and that action pro…le s (h) is a Nash equilibrium of From the de…nitions of Ri
0
Ri
0
(h) and Ri
jg
T
(h) , jg
(h) ; R
jg
T
T
(h) .
(h) given in (1), we have that:
(h) g T (a (h)) () T
g
T
(h) ; g (s (h)) Ri ( ) g
(23)
T
T
(h) ; g (a (h)) ,
T
(h) ; g T (a (h)) .
and that: g T (s (h))Ri
0
jg
T
(h) g T (a (h)) () T
g
(h) ; g T (s (h)) Ri ( 0 ) g
(24)
If there exist i 2 I and ai (h) 2 Ai (h) such that: g T ai (h) ; s
i
(h) Pi
0
jg
T
(h) g T (s (h)),
it follows from (22)-(24) that: g T ai (h) ; s
i
( ) (h) Pi
jg
T
(h) g T (s (h)), hT ; R
which contradicts the fact that the action pro…le s (h) is a Nash equilibrium of Thus, this action pro…le s (h) is also a Nash equilibrium of
(h) ; R
0
jg
T
jg
(h) . Also,
41
T
(h) .
note that this pro…le s (h) is also a Nash equilibrium of ( (h) ; 0 ). Since the period-T SCF f T is a function and since the action pro…le s (h) is a Nash equilibrium of ( (h) ; 0 ), it needs to be the case that g T s (h) = f T
0
jg
T
(h) . It
follows from the fact that the SCF f satis…es the folding condition that g T s (h) 'T R
0
T
jg
(h) , as was to be proved. t
Fix any t 6= 1; T and consider any g pro…le R
jg
=
t
t
(h) 2 Y
(h) ; f +(t+1) and any pro…le R
0
with h 2 H t . Furthermore, consider any t
jg
(h) ; f +(t+1) in D
jg
t
(h) ; f +(t+1) .
Suppose that for every i 2 I and every a (h) 2 A (h): 't R
jg
t
(h) ; f +(t+1)
Ri
jg
't R
jg
Since the dynamic mechanism 't R
jg
t
t
t
g
t
jg
t
= ft
(h) ; f +(t+1) t
jg
0
Ri
jg
t
(h) ; f +(t+1) g t (a (h)) . (25)
t
jg
(h) = g t s (h) .
(h) ; f +(t+1) and Ri
0
jg
t
(h) ; f +(t+1) given in
temporarily implements the SCF f , one can see that the action
pro…le s (h) is a Nash equilibrium of g t s (h) Ri
(h) ; f +(t+1)
implements the SCF f in SPE, we have that:
Moreover, from the de…nitions of Ri (4) and from the fact that
(h) ; f +(t+1) g t (a (h)) =)
(h) ; R
jg
t
(h) ; f +(t+1) , that:
(h) ; f +(t+1) g t (a (h)) ()
(h) ; g +t s jh
Ri ( ) g
t
(h) ; g t (a (h)) ; g +(t+1) s j (h; a (h))
(26) ,
and that: g t s (h) Ri g
t
0
jg
t
(h) ; f +(t+1) g t (a (h)) () 0
(h) ; g t s (h) ; g +(t+1) s j h; s (h)
Ri ( 0 ) g
t
0
(h) ; g t (a (h)) ; g +(t+1) s j (h; a (h)) (27)
If there exist i 2 I and ai (h) 2 Ai (h) such that: g t ai (h) ; s
i (h) Pi
h
0
jg
t
0
(h) ; g +(t+1) s j (h; )
i
g t (s (h)),
42
.
it follows from (25)-(27) that: g t ai (h) ; s
i
jg
(h) Pi
t
g t (s (h)),
(h) ; g +(t+1) s j (h; )
which contradicts the fact that s (h) is a Nash equilibrium of Thus, the action pro…le s (h) is also a Nash equilibrium of
0
(h) ; R
t
jg
(h) ; R
t
jg
(h) ; f +(t+1) .
(h) ; f +(t+1) . 0
Let si denote individual i’s strategy according to which this i plays si (h0 ) = si (h0 ) for every history h0 6= h and according to which this i plays sti = si (h) after the history h. Note 0
that sjh0 is a Nash equilibrium of ( (h0 ) ; 0 ) for every history h0 6= h since s is a strategy
pro…le in SP E ( ; 0 ). Thus, to have that the strategy pro…le s is a SPE strategy pro…le of ( ; 0 ), we need to show that sjh is a Nash equilibrium of ( (h) ; 0 ). Since the action pro…le s (h) is a Nash equilibrium of
(h) ; R
0
jg
t
(h) ; f +(t+1) , it
follows from (27) that for every i 2 I and every ai (h) 2 Ai (h): g
t
(h) ; g +t (s) Ri ( 0 ) g
t
(h) ; g t (ai (h) ; s
i
(h)) ; g +(t+1) (sj (h; (ai (h) ; s
i
(h)))) .
Thus, no individual i can gain by deviating from the action pro…le s (h) and thereafter conforming to si , and so the strategy pro…le s (h) is a NE of ( (h) ; 0 ). It follows that s 2 SP E ( ; 0 ). Since the dynamic mechanism
implements the SCF f in SPE and since, moreover, the
strategy pro…le s 2 SP E ( ; 0 ), it follows that f t [ 0 jg the folding condition, we have g t (s (h)) = 't R
0
jg
t
t
(h)] = g t (s (h)). Since f satis…es
(h) ; f +(t+1) , as was to be shown.
Consider some R [ jf +2 ] and some R [ 0 jf +2 ] in D [ jf +2 ]. Suppose that for every i 2 I and every a (h1 ) 2 A (h1 ): '1 R
jf +2
Ri
jf +2 g 1 a h1
=) '1 R
jf +2
0
Ri
jf +2 g 1 a h1
.
(28)
Since f satis…es the folding condition, we have that '1 R
jf +2
= f 1 [ ] = g1 s
h1
.
43
Moreover, it also follows from the de…nitions of Ri [ jf +2 ] and Ri [ 0 jf +2 ] given in (7) and temporarily implements the SCF f that the action pro…le s (h1 ) is a
from the fact that
Nash equilibrium of ( (h1 ) ; R [ jf +2 ]), that: '1 (R [ jf +2 ]) Ri [ jf +2 ] g 1 (a (h1 )) () g 1 s (h1 ) ; g +2 s js (h1 )
Ri ( ) g 1 (a (h1 )) ; g +2 s ja (h1 )
, (29)
and that: '1 (R [ jf +2 ]) Ri [ 0 jf +2 ] g 1 (a (h1 )) () 0
g 1 s (h1 ) ; g +2 s js (h1 )
0
Ri ( 0 ) g 1 (a (h1 )) ; g +2 s ja (h1 )
. (30)
Suppose that g 1 ai h1 ; s
i
h1
0
Pi
jf +2 g 1 s
h1
for some i 2 I and some ai (h1 ) 2 Ai (h1 ). Thus, it follows from (28)-(30) that: g 1 ai (h1 ) ; s
i
(h1 ) Pi ( jf +2 ) g 1 s (h1 )
g 1 ai (h1 ) ; s
i
()
(h1 ) ; g +2 s j ai (h1 ) ; s
i
(h1 )
Pi ( ) g 1 s (h1 ) ; g +2 s js (h1 )
,
which contradicts the fact that action pro…le s (h1 ) is a Nash equilibrium of ( (h1 ) ; R [ jf +2 ]). Therefore, the pro…le s (h1 ) is also a Nash equilibrium of ( (h1 ) ; R [ 0 jf +2 ]). As we did previously, let si which this i plays s1i 0
strategy si ; that is, sti
(si )
1
denote the individual i’s strategy according to
si (h1 ) at the start of the game and thereafter she conforms to the si
0
t
for every t
2.
Note that sjh0 is a Nash equilibrium of ( (h0 ) ; 0 ) for every nontrivial history h0 2 H 0
since s is a strategy pro…le in SP E ( ; 0 ). Thus, to have that the strategy pro…le s is a SPE of ( ; 0 ), we need to show that s is also a Nash equilibrium of ( ; 0 ). Since the action pro…le s (h1 ) is a Nash equilibrium of ( (h1 ) ; R [ 0 jf +2 ]), it follows from (30) that for every i 2 I and every ai (h1 ) 2 Ai (h1 ): (g (s)) Ri ( 0 ) g 1 ai h1 ; s
i
h1
; g +2 sj ai h1 ; s
i
h1
.
44
Thus, no individual i can gain by deviating from si (h1 ) and thereafter conforming to si , and so the strategy pro…le s is a SPE of ( (h) ; 0 ). Since the dynamic mechanism
implements the SCF f in SPE, we have that f 1 [ 0 ] =
g 1 (s (h1 )). Since f satis…es the folding condition, we have g 1 (s (h1 )) = '1 (R [ 0 jf +2 ]), as was to be shown.
Proof of Theorem 3 Proof of Theorem 3. The proof is based on the construction of a dynamic mechanism , where each period-t mechanism is a canonical mechanism.
Period-1 mechanism: Individual i’s period-1 action space is de…ned by: Ai H 1
D
jf +2
Y1
Z+ ,
where Z+ is the set of nonnegative integers and H 1 is the null set. Thus, a period-1 action of individual i consists of an element of the set Y 1 , an element of the period-1 domain of induced preferences induced by the set
at the socially optimal 2-tail outcome paths f +2 ,
and a nonnegative integer. A typical period-1 action played by individual i is denoted by ai (h1 )
R
jf +2
i
i
; (x1 ) ; (z)i .
Period-1 action space of individuals is the product space: A H1
Y
Ai H 1 ,
i2I
with a (h1 ) as a typical period-1 action pro…le. The period-1 outcome function g 1 is de…ned by the following three rules: Rule 1: If ai (h1 )
R
jf +2 ; x1 ; 0 for every i 2 I and x1 = '1 R
jf +2 , then
g 1 (a (h)) = x1 . Rule 2: If n
1 individuals play aj (h1 )
R
jf +2 ; x1 ; 0 with x1 = '1 R
jf +2
but 45
individual i plays ai (h1 )
R
jf +2
i
i
; (x1 ) ; (z)i 6= aj (h1 ), then we can have two cases:
i
i
jf +2 (x1 ) , then g 1 (a (h1 )) = (x1 ) .
1. If x1 Ri i
2. If (x1 ) Pi
jf +2 x1 , then g 1 (a (h1 )) = x1 .
Rule 3: Otherwise, an integer game is played: identify the individual who plays the highest integer (if there is a tie at the top, pick the individual with the lowest index among them.) This individual is declared the winner of the game, and the alternative implemented is the one she selects.
Period-t mechanism with t 6= 1; T : Individual i’s period-t action space after history h 2 H t such that g
t
t
(h) 2 Y
is
de…ned by: D
Ai (h)
jg
t
(h) ; f +(t+1)
Yt g
t
(h)
Z+ ,
where Z+ is the set of nonnegative integers. Thus, a period-t action of individual i after history h 2 H t consists of an element of the set Y t (g domain of induced preferences induced by the set
t
(h)), an element of the period-t
at the t-head outcome path g
t
(h)
and at the socially optimal t + 1-tail outcome paths f +(t+1) , and a nonnegative integer. A typical period-t action played by individual i after history h 2 H t is denoted by ai (h) R
jg
t
i
(h) ; f +(t+1)
i
; (xt ) ; (z)i .
Period-t action space of individuals after history h 2 H t is the product space: A (h)
Y
Ai (h) ,
i2I
with a (h) as a typical period-t action pro…le after history h 2 H t . The period-t outcome function g t is de…ned by the following three rules for every h 2 H t such that g
t
(h) 2 Y t :
Rule 1: If ai (h)
R
jg
t
(h) ; f +(t+1) ; xt ; 0 for every i 2 I and xt = 't R
jg
t
(h) ; f +(t+1) ,
then g t (a (h)) = xt . 46
Rule 2: If n
1 individuals play aj (h) xt = 't R
but individual i plays ai (h)
R
jg
R jg t
jg
t
t
(h) ; f +(t+1) ; xt ; 0 with
(h) ; f +(t+1) i
(h) ; f +(t+1)
i
; (xt ) ; (z)i 6= aj (h), then we can
have two cases: 1. If xt Ri
jg
i
2. If (xt ) Pi
t
i
i
(h) ; f +(t+1) (xt ) , then g t (a (h)) = (xt ) .
jg
t
(h) ; f +(t+1) xt , then g t (a (h)) = xt .
Rule 3: Otherwise, an integer game is played: identify the individual who plays the highest integer (if there is a tie at the top, pick the individual with the lowest index among them.) This individual is declared the winner of the game, and the alternative implemented is the one she selects.
Period-T mechanism: Individual i’s period-T action space after history h 2 H T such that g
T
(h) 2 Y
T
is
de…ned by: Ai (h)
D
jg
T
YT g
(h)
T
(h)
Z+ ,
where Z+ is the set of nonnegative integers. Thus, a period-T action of individual i after history h 2 H T consists of an element of the set Y T g domain of induced preferences induced by the set
T
(h) , an element of the period-T
and the T -head outcome path g
T
(h),
and a nonnegative integer. A typical period-T action played by individual i after history h 2 H T is denoted by ai (h)
R
jg
T
(h)
i
; xT
i
; (z)i .
Period-T action space of individuals after history h 2 H T is the product space: A (h)
Y
Ai (h) ,
i2I
with a (h) as a typical period-T action pro…le after history h 2 H T . The period-T outcome function g T is de…ned by the following three rules for every h 2 H T such that g
T
(h) 2 Y
T
: 47
Rule 1: If ai (h)
jg
R
T
(h) ; xT ; 0 for every i 2 I and xT = 'T R
jg
T
(h) , then
g T (a (h)) = xT . Rule 2: If n 1 individuals play aj (h) but individual i plays ai (h)
R
R T
jg
jg (h)
T i
(h) ; xT ; 0 with xT = 'T R ; xT
i
; (z)i
jg
T
(h)
6= aj (h), then we can have
two cases: 1. If xT Ri 2. If xT
i
jg Pi
T
(h)
jg
T
i
i
xT , then g T (a (h)) = xT .
(h) xT , then g T (a (h)) = xT .
Rule 3: Otherwise, an integer game is played: identify the individual who plays the highest integer (if there is a tie at the top, pick the individual with the lowest index among them.) This individual is declared the winner of the game, and the alternative implemented is the one she selects.
Let
[
H
Ht
t2T
S
be the set of all possible histories, let Ai
Ai (h) be the set of all actions for individual
h2H
i 2 I, let A (H) be the set of all pro…les of actions available to individuals, de…ned by [
A (H)
A (h) ,
h2H
and let g
g1;
; g T be the sequence of outcome functions, one for each period t 2 T .
Note that g satis…es the following properties: a) the outcome function g 1 assigns to period-1 action pro…le a (h1 ) 2 A (h1 ) a unique outcome in Y 1 , and b) for every period t 6= 1 and every nontrivial history ht 2 H t , the outcome function g t assigns to each period-t action pro…le a (ht ) 2 A (ht ) a unique outcome in Y t (g
t
(ht )). Thus, by construction,
(I; H; A (H) ; g)
is a dynamic mechanism. We now prove that (a) for every such that g 1 s (h1 ) = f 1 [ ], f t [ jg
2 t
, there exists a SPE strategy s 2 S of ( ; )
(ht )] = g t s (ht ) for every nontrivial ht 2 H t , and 48
(b) for every
and for every s 2 SP E ( ; ), g 1 s (h1 ) = f 1 [ ] and f t [ jg
2
g t s (ht ) for every nontrivial ht 2 H t . Thus, …x any state
2
t
(ht )] =
.
Let us …rst prove (a). Since the SCF satis…es the folding condition, we have that f 1 [ ] = '1 (R [ jf +2 ]), that f t [ jg
t
(h)] = 't R
h 2 H t and every t 6= 1; T and that f T
jg
T
jg
t
(h) ; f +(t+1)
(h) = 'T R
jg
T
for every nontrivial for every h 2 H T .
(h)
Let us de…ne individual i 2 I’s strategy si : H ! Ai by: si h1 = R jg
si (h) = R
t
(h) ; f +(t+1) ; 't R
si (h) = R
jg
T
jf +2 ; '1 R jg
(h) ; 'T R
t
jf +2
(h) ; f +(t+1) jg
t
(h)
;0 ,
; 0 , for every h 2 H t with t 6= 1; T ,
; 0 , for every h 2 H T .
For every period t and history ht 2 H t , to show that s jht
s1 jht ;
; sI jht is a SPE
of ( (ht ) ; ) it su¢ ces to show that no individual i can gain by deviating from si jht in a t and conforming to si jht thereafter. To this end, …rst note that for every
single period
history h 2 H, the strategy pro…le s (h) falls into Rule 1. Thus, by construction and the fact that the SCF satis…es the folding condition, one can check that g 1 s (h1 ) = f 1 [ ], f t [ jg
t
(ht )] = g t s (ht ) for every ht 2 H t and every t 6= 1.
Fix any period t and any history ht 2 H t . Suppose that individual i deviates from si jh with h 2 Hjht by changing only the action si (h ) into ai (h ) 2 Ai (h ). Given that no unilateral deviation from s (h ) can induce Rule 3, the outcome is thus determined by Rule 2. But then, under this rule the outcome would only change to be the periodoutcome announced by this i in her deviation if this outcome is not better than the outcome g
s (h ) according to the period- induced ordering Ri [ jf +2 ] if
induced ordering Ri Ri [ jg
(h )] if
jg
(h ) ; f +(
(h ) ; f +(t+1) if
6= 1; T , and to the period- induced ordering
= T . By noting that Ri [ jf +2 ] is the true period-1 induced ordering of
individual i in state Ri
jg
= 1, to the period-
+1)
at the socially optimal 2-tail outcome paths f +2 [ j ] if
= 1, that
is the true period- induced ordering of individual i in state
head-path g
(h ) and the socially optimal -tail outcome paths f +(
+1)
that Ri [ jg
(h )] is the true period- induced ordering of individual i in state
[ j ] if
at the
6= 1; T and at the
49
head-path g
(h ) if
= T , individual i will not bene…t from such a deviation. Since the
choice of individual i as well as of the history h 2 H jht are arbitrary, we conclude that the strategy pro…le s jht is a SPE of ( (ht ) ; ). Hence, the proposed strategy pro…le s jh is a SPE of ( (h) ; ) for every history h 2 H, whose outcomes are such that g 1 s (h1 ) = f 1 [ ], f t [ jg
t
(ht )] = g t s (ht ) for every ht 2 H t and every t 6= 1. This proves our goal (a) stated
above. The rest of the proof shows that our goal (b) holds, too. To see this, assume that the strategy pro…le s is a SPE of ( ; ). Moreover, …x any history h 2 H. Thus, the strategy pro…le sjh is a SPE of ( (h) ; ). Assume, to the contrary, that there is a period t 2 T as well as a history ht 2 Hjh such that either f t [ jg
t
(ht )] 6= g t (s (ht ))
if t 6= 1 or f 1 [ ] 6= g 1 (s (h1 )) if t = 1. Among all such histories, let h 2 Hjh be one of the longest histories. Thus, it must be the case that f [ jg that f ^
jg
case where g ^ s h^
^
h^
(h )] 6= g (s (h )) and, moreover,
for every h^ 2 Hj (h ; s (h )) if
= g ^ s h^
6= T . Note that for the
6= T the folding condition mplies that: = '^ R
^
jg
g T s hT
h^ ; f +(^+1)
= 'T R
jg
T
for every h^ 2 Hj (h ; s (h )) with ^ 6= T , and that: hT
for every hT 2 Hj (h ; s (h )) .
Also, note that the true pro…le of period- induced orderings at true state R jg
R
R Let us suppose that (h ) ; R
jg
(h ) ; f +(
jf +(
+1)
if
(h ) ; f +(
+1)
jg
(h ) if
= 1, if
R
jg
(h) ; f +(
+1)
6= 1; T ,
= T.
6= 1; T . Then, the action pro…le s (h ) is a Nash equilibrium of +1)
.
Suppose that s (h ) falls into Rule 1 of period'
is:
mechanism. Thus, g (s (h )) =
for some , and this outcome is an element of Y (g
(h )).
Since f satis…es the folding condition, an immediate contradiction is obtained if g (s (h )) = '
R
jg
(h ) ; f +(
+1)
. Therefore, let us suppose that g (s (h )) 6= '
R
jg
(h ) ; f +(
50
+1)
.
Since f satis…es temporary Maskin monotonicity and since '
(h ) ; f +(
jg
R
+1)
6= '
(h ) ; f +(
jg
R
there exists an individual i and a period- outcome y 2 Y (g '
R
(h ) ; f +(
jg
+1)
Ri
+1)
,
(h )) such that
jg
(h ) ; f +(
jg
(h ) ; f +(
+1)
y
and (h ) ; f +(
+1)
By changing si (h ) into ai (h ) = R
jg
y Pi
jg
Rule 2 and obtain g (ai (h ) ; s
i
R
'
(h ) ; f +(
+1)
+1)
.
; y ; 1 , individual i can induce
(ht )) = y , thereby contradicting the fact that the action
pro…le s (h ) is a Nash equilibrium of
(h ) ; f +(
jg
;R
+1)
.
Suppose that s (h ) falls into Rule 2 of period- mechanism. Thus, for every individual j 6= i, the period- outcome determined by this rule is maximal for this j in Y (g according to her period- induced ordering Rj
jg
the action pro…le s (h ) is a Nash equilibrium of
(h ) ; f +( ;R
+1)
. Moreover, given that
(h ) ; f +(
jg
(h ))
+1)
, for individual
i it holds that the outcome g (s (ht )) is such that g (s (ht )) is an element of the weak lower contour set of Ri
jg
(h ) ; f +( g
+1)
s ht
at ' Ri
jg
R jg
for every x in the weak lower contour set of Ri
(h ) ; f +(
(h ) ; f +( jg
+1)
+1)
and that
x
(h ) ; f +(
+1)
at '
R
jg
(h ) ; f +(
+1)
Since the SCF f satis…es the temporary weak no veto-power, this implies that g (s (h )) = ' The folding condition implies that '
R
R jg
jg
(h ) ; f +( (h ) ; f +(
+1)
+1)
. = f [ jg
(h )], which is a
contradiction. Suppose that s (h ) falls into Rule 3 of period- mechanism. Thus, for every individual j, the period- outcome determined by this rule is maximal for this j in Y (g
(h )) according 51
.
to her period-
induced ordering Rj
(h ) ; f +(
jg
temporary unanimity, we have that g (s (h )) = ' condition implies that '
(h ) ; f +(
jg
R
+1)
+1)
. Since the SCF f satis…es the
R
jg
(h ) ; f +(
= f [ jg
+1)
. The folding
(h )], which is a contradiction.
We conclude the proof by mentioning that, suitably modi…ed, the above proof provided for the case where
6= 1; T applies to the case where
= 1 as well as to the case where
= T.
Proof of Lemma 1 Proof of Lemma 1. Let the premises hold. Fix any x1 2 X 1 and any
2
. Let x2 [ jx1 ]
be the solution to: @U ( ; x1 ; x2 ) = 0. @x2 By the implicit function theorem, we have that: @ 2 U ( ;x1 ;x2 [ jx1 ]) @ 2 x2 @ 2 U ( ;x1 ;x2 [ jx1 ]) @ @x2
@x2 [ jx1 ] = @ Therefore, the peak for the median type stage for each x1 2 X 1 . Write x2 [
med jx
= 1
> 0.
is always the peak in the second voting
med
] for the peak of the median type in the second
voting stage conditional on x1 . Since it holds that: 1 2 med ; x ; x @x2
@U (
[
med jx
1
])
= 0,
from the implicit function theorem we obtain that: @x2 [ med jx1 ] = @x1 Let us show that x2 [
med jx
For every allowable type
1
@2U (
med ;x
1 ;x2
[
@x1 @x2 @2U (
1 ;x2 [ @ 2 x2
med ;x
])
med jx
1
med jx
1 ])
0.
] is the Condorcet winner under R [ jx1 ] for every x1 2 X 1 .
2
;
( ; x1 ; x2 ) = U
and policy (x1 ; x2 ), let: ; x1 ; x2
med jx
1
U ( ; x1 ; x2 ).
52
Then, for every x2 < x2 [
(
1 med ; x ],
we have that:
1
Z
2
med ; x ; x ) =
x2 [
med jx
1
] @U (
1 2 med ; x ; z ) dz 2 . @z 2
x2
Furthermore, for every 1
2
( ;x ;x )
(
med ,
>
1
med ; x
it holds that:
2
;x ) =
Z
k
Z
x2 [
med jx
1
x2
] @ 2 U ( ; x1 ; z 2 ) dz 2 d > 0. 2 @ @z
Since 1 2 med ; x ; x )
(
= U(
1 2 med ; x ; x
med jx
1
)
U(
1 2 med ; x ; x )
0,
it follows that: ( ; x1 ; x2 ) > 0, which, in turn, guarantees that: U ( ; x1 ; x2
med jx
Therefore, for every voter j = k + 1;
; 2k
U ( j ; x1 ; x2 Likewise, for every x2 > x2 [
med jx
med jx 1
1
) > U ( ; x1 ; x2 ).
1, it holds that: 1
) > U ( j ; x1 ; x2 ).
], one can show that for every voter j = 1;
;k
1
it holds that: U ( j ; x1 ; x2 Therefore, x2 [ x2 [ 2
med jx
1
med jx
1
med jx
1
) > U ( j ; x1 ; x2 ).
] is a Condorcet winner under R [ jx1 ], that is, CW (R [ jx1 ]) =
], and so the majority voting function f 2 [ j ] is a single-valued function for every
and every x1 2 X 1 . Let x [
med ]
= (x1 [
we show that x1 [
med ]
2 med ] ; x
[
med ])
be the global peak for the median type
med .
Next,
is the Condorcet winner under R [ jf 2 ].
Solving backward, given that the majority voting function f 2 [ jx1 ] = x2 [
med jx
1
] for
53
every x1 2 X 1 , we have that the reduced utility of type V ( ; x1 ) = U ( ; x1 ; x2
is:
med jx
1
).
Then, we have that: @V ( ; x1 ) @U ( ; x1 ; x2 [ = @x1 @x1
med jx
1
])
@U ( ; x1 ; x2 [ @x2
+
med jx
1
]) @x2 [ med jx1 ] , @x1
and so, by De…nition 12, it follows that: @ 2 U ( ; x1 ; x2 [ med jx1 ]) @ 2 U ( ; x1 ; x2 [ med jx1 ]) @x2 [ med jx1 ] @ 2 V ( ; x1 ) = + > 0. @ @x1 @ @x1 @ @x2 @x1 Then, for every x1 2 X 1 , let: ( ; x1 ) = V ( ; x1 [ Next, take any x1 < x1 [
(
med ].
Then, it holds that: 1
med ; x ) =
Z
x1 [
med ]
@V (
x1
Moreover, for every
> 1
( ;x )
med ,
(
V ( ; x1 )
med ])
med ; z @z 1
1
)
dz 1 .
it also holds that: 1
med ; x ) =
Z
med
Z
x1 [
med ]
x1
@ 2V ( ; z1) 1 dz d > 0. @ @z 1
med ])
V(
Since (
1 med ; x )
=V(
1 med ; x
[
1 med ; x )
0,
we have that: ( ; x1 ) > 0, which, in turn, guarantees that: V ( ; x1 [
med ])
> V ( ; x1 ).
54
Therefore, for every voter j = k + 1; V ( j ; x1 [ Likewise, for every x1 > x1 [ V ( j ; x1 [
med ])
We conclude that x1 [ x1 [
med ],
med ]
; 2k med ])
1, we have that:
> V ( j ; x1 ).
one can also show that:
> V ( j ; x1 ), for every voter j = 1;
med ]
;k
1.
is a Condorcet winner under R [ jf 2 ], that is, CW (R [ jf 2 ]) =
and so the majority voting function f 1 [ ] is a single-valued function for every 2
.
55
