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DISCUSSION PAPERS IN ECONOMICS
Working Paper No. 05-12
Does It Take a Tyrant to Carry Out a Good Reform?
Anna Rubinchik-Pessach
Department of Economics, University of Colorado at Boulder
Boulder, Colorado
Ruqu Wang
Department of Economics, Queen’s University
Kingston, Ontario, Canada
December 5, 2005
Center for Economic Analysis
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© 2005 Anna Rubinchik-Pessach and Ruqu Wang
Does it Take a Tyrant to Carry Out a Good
Reform?∗
Anna Rubinchik-Pessach, Ruqu Wang†
December 5, 2005
Abstract
In this model a reform is a switch from one norm of behavior (equilibrium) to
another and agents have to endure private costs of transition in case of a reform. An
authority, which coordinates the transition, can enforce transfers across the agents and
is capable of imposing punishments upon them. A transfer is limited, however, by an
agent’s payoff, and a punishment can not exceed an upper bound. Carrying out a good,
Pareto improving, reform can be hindered by asymmetric information about the costs
of transition, which are privately known to the agents and can not be verified by the
authority. In this case execution of some good reforms is impossible without a credible
threat of a punishment, even if Bayesian mechanisms can be used. Allowing for harsher
punishments in that framework reduces to ‘softening’ individual rationality constraints,
thus widening the range of feasible reforms. The flip-side of increasing the admissible
punishment is making ‘bad’ reforms possible as well as lowering well-being of selected
individuals. We, thus, formulate a trade-off between successful implementation of good
reforms and severity of acceptable punishments.
Key words: reform, mechanism design, incentive compatibility
JEL Classification numbers: D78, E61.
∗
We are grateful to Eckhard Janeba for extensive and insightful discussions. We also wish to thank
Alex Cukierman, Murat Iyigun, Alexander Karaivanov, Sergio O. Parreiras, Jamele Rigolini as well as the
participants of ‘Political Economy Group’ at the University of Colorado at Boulder, seminar participants
at Tel-Aviv University and those at the Second Game Theory World Congress in Marseille, Canadian
Economic Theory Meetings in Vancouver, Public Economic Theory Meetings in Peking, 16th International
Game Theory Conference in Stony Brook. Wang acknowledges financial support from the Social Sciences
and Humanity Research Council of Canada.
†
Anna Rubinchik-Pessach, [email protected], University of Colorado at Boulder, Department of Economics, UCB 256, Boulder, CO, 80309; Ruqu Wang, [email protected], Queen’s
University, Department of Economics, Kingston, Ontario, Canada K7L 3N6.
1
1
Motivation
Why are some potentially good reforms never implemented? What can explain fruitfulness
of recent economic reforms in China, overwhelming success of the newly industrialized
countries in the 1970’s-1980’s along with a turbulent and murky path followed by India and
Russia in the past decade, for example?1 Our goal is not to provide an ultimate answer,
but, rather, to illuminate some possible connections between political and economic changes.
We envision the role of a reformer — be it a single “dictator" or a democratic government
— as a one-time intervention, with the sole purpose of changing the “norm of behavior"
in a country.2 For example, a norm could describe production-consumption choices under
a given market structure, degree of openness to the international trade and a monetary
regime.3 Even if two different norms can be ranked Pareto, a (decentralized) switch to
a dominant one might not occur due to the reluctance of some individuals to cover the
transition costs, which range from the effort of re-structuring one’s investment portfolio to
the ‘loss of cultural identity’ (as in Janeba (2003)). There is no doubt that a tyrant with
an unlimited power can implement any change, however whimsical. We aim at describing
the least severe threat necessary to “convince" all the individuals to change their actions in
accordance with the new norm. This bound can be also viewed as a measure of how hard
it is to implement a reform, as well as an indicator of how tyrannical a reformer has to be
in order to succeed. We formulate the bound on punishments for two types of reformers:
a benevolent one, who prefers to implement only good reforms with overall gains from
transition outweighing the costs; and an eccentric one, who chooses to reform, no matter
what the costs are.
The threat used by a reformer has to be credible, in other words, the potential punishment can not constitute a violation of the laws protecting human dignity. This means
that in the presence of some exogenous constraint on punishments certain reforms will be
impossible to implement. Existence of such constraints can be justified, in particular, by
the concern of the international community that in the absence of human rights monitoring an eccentric authority will get a lee-way to implement undesirable reforms. This same
constraint, however, might prevent some good transitions from going through.
In his overview of economic reforms (1960-1980) around the world, Rodrik (1996) finds it
puzzling that often times sound economic reforms are not popular, moreover, he mentions
that “...the implementation of good economic policy is often viewed as requiring ‘strong’
1
For a general overview see Rodrik (1996), Velasco and Tommasi (1996) among others; Russian economic
reforms are discussed in more detail in Lang and Weber (2000).
2
In the spirit of Binmore (1998), we view a social norm to be self-sustainable in a sense that, once in
place, it prescribes each individual to act in his own best interest.
3
Analyzing a policy that supports a switch from one equilibrium to another has a long-standing tradition
in development economics, see Ray (2000) for an overview and additional examples of the relevant policies.
2
and autonomous’ (not to say authoritarian) leadership."4 Similar observations are offered
by Harberger (1993) in his overview of the Latin American reforms. He stresses personal
charisma and outstanding leadership of “key group of individuals," often times acting — as
in case of Roberto Campos, who is now given credit for the ‘Brazilian Miracle,’ — “...in
spite of adverse circumstances and at high personal cost." Not surprisingly, much of the
recent literature have been devoted to the political determinants of reforms, thus, shifting
traditional focus from normative suggestions to understanding the barriers to implementing
the prescriptions. Fernandez and Rodrik (1991) show that individual specific uncertainty,
i.e., inability of an individual to assess with certainty whether she will be a winner or a
loser, can hinder good reforms, if no transfers are allowed. Jain and Mukand (2003) allow
for the transfers that are sustainable in a citizen-candidate political equilibrium. It is the
uncertainty with respect to the identity of the politician in power, and thus, to whether
the transfers will eventually be realized, that drives the resistance to a reform. While these
models can provide an introspection as to why actual polities that rely heavily on majority
voting are unable to deliver the desired reform, we focus on feasibility of a transition under
any social choice procedure that respects the laws protecting individual well-being. Note
that if the decision of whether to reform is based on a majority voting, for example, it is
implicitly assumed that those who voted against the majority, nevertheless, comply with
the will of the decisive coalition. In other words, even if a majority of the voters supports
the reform, it can still fail, provided the opposition is not convinced (or even coerced) to
follow up. On the other hand, as simple majority voting is not sensitive to individual
intensity of preferences (say, to how high or how low the costs are), beneficial reforms might
not be accepted just because this simple rule is chosen, and not due to the presence of
asymmetric information per se. To avoid these difficulties, we allow for any possible way
of extracting the relevant information from individuals and any procedure of aggregating
this information to formulate a common decision, while making sure that it is in the best
interest of all involved to comply with that decision.
It is natural to choose a mechanism design framework to tackle the question. The
connection to the related literature on the subject is easier to present once the model is
developed, and so we defer this discussion to sections 4.3 and 6.
Another related strand of literature is devoted to potential failures to coordinate actions
by a large group of individuals. Morris and Shin (1998) develop a model of speculative currency attacks, in which the value of investor’s holding crucially depends on the actions of the
other investors. Each gets an imperfect signal from the government about “fundamentals,"
indicating the desire of the government to support local currency. The noise in the signal
destroys common knowledge that the currency is in the stable region, even when it, indeed,
4
p.10.
3
is. This creates a chance that the investors abandon local currency, even if the fundamentals
are good, i.e., it creates a possibility of the switch to a Pareto dominated equilibrium. This
can be remedied by a costly action of the “policy maker," as shown in Angeletos, Hellwig,
and Pavan (2003). Inability of individuals to synchronize their actions can also lead to the
failure of a (de-centralized) switch to the efficient equilibrium, as in Morris (1995). While
strategic manipulation of individual beliefs can be interesting to explore, we leave it for future investigation, resorting, instead, to a common knowledge environment. This choice is
dictated, in part, by our desire to relieve the pressure on necessary punishments by adhering
to a less restrictive solution concept (Bayesian Nash).
The rest of the paper is organized as follows.
After setting up the model in section 2, we proceed with the full information model,
in which individual costs of transition are known to the reformer (local authority). In
section 3 we show that the authority does not need to use punishments to implement good
(Pareto improving) reforms, moreover an eccentric authority may be incapable of forcing
undesirable reforms (i.e., a switch to a Pareto dominated equilibrium) without resorting to
a punishment. Under asymmetric information, introduced in section 4, the authority may
need to credibly threat individuals with punishments. The punishment might be higher
for more divided countries and in case of bad reforms, as illustrated in section 4.2 for the
discrete distribution case. We generalize the main results for the case of continuous costs
distribution thereafter. Extensions and conclusions follow. The proofs are in the appendix.
2
The Setup
A country consists of N individuals (agents). Their everyday interactions are reduced to a
simultaneous move coordination game G with two actions {A, B} and real-valued payoffs,
which depend on i0 s choice of action and which are symmetric with respect to the actions
of others,
(1)
ui (s) = u (si , s−i ) ,
where s−i is the action profile chosen by all the players but i, s ∈ {A, B}N is the strategy
profile. If one thinks about a strategy as representing a sequence of actions over time, the
payoff can be viewed then as a (discounted) sum of future payoffs that individual i receives,
if he follows the chosen strategy, si , for example, driving on the right side of the road, or
accepting local currency. We would like to introduce the simplest possible framework and
analyze reforms as coordinated switches between the two pure strategy — sA = (A, A, .., A),
4
sB = (B, B, .., B) − Nash equilibria.5 These equilibria are ‘Pareto’ ranked as follows:
¡ ¢
¡ ¢
u sA = a > b = u sB ≥ 0,
(4)
and assume both dominate a mixed strategy payoff.
Definition 1 A reform is a switch from one equilibrium (norm) to another.
Agent i has a cost, ci ∈ [c, c̄] ⊂ R+ , associated with switching her action. In this model
a switch from sB to sA is a Pareto improving (good) reform, provided the average cost is
below the gain, a − b. Otherwise a switch is a bad, or an undesirable one.
An authority, however, may have distinct interests from the rest of the society. It has
the ability of coordinating a switch, or announcing the reform, besides, it has an access to
two tools: (1) transfers to the agents, (ti )i ∈ RN ; (2) punishments, (mi )i ∈ RN
+ . There are
no outside sources of financing the reform so that
Σi ti ≤ 0
(BB)
Both the transfer and the punishment schemes, we assume, are anonymous, they are
independent on the “names" of individuals, but rather, on the observed actions and on
verifiable individual characteristics. More precisely, the transfers and the punishment vary
only with the action, s1i , taken by individual i, actions taken by the rest of the players, s1−i
and the cost of transition, ci , if observed:
ti = t(s1i , s1−i , ci , c−i , I (c));
mi = m(s1i , s1−i , ci , c−i , I (c)).
(5)
In particular, costs of transition might influence the decision with respect to the reform,
indicated by I (c) , which is unity in case the reform is announced and zero otherwise,
c ∈ [c, c̄]N .
5
In addition, there could be "knife edge" assymetric equilibria of the form: proportion p (pN is an integer)
of the agents are choosing A and the rest are choosing B :
Ã
!
Ã
!
u A, .., A, B, ..B
pN
=
w > u B, A, .., A, B, ..B
(1−p)N
w
>
Ã
pN −1
u A, .., A, B, ..B
These are assumed away for simplicity.
5
pN +1
;
(2)
(1−p)N
!
(1−p)N −1
.
(3)
While the authority announces its recommendation “switch to strategy A" or “continue
with B, ” it also has to make sure that the agents are sure to follow. This implies that the
prescribed action s∗i should satisfy6
¡
¡
¢
¢
s∗ ∈ arg max u s1i , s1−i − ci ι s0i , s1i + ti − mi , ∀i
(IC)
s1i ∈{A,B}
¡
¢
over the available (new) actions s1i ∈ {A, B}, with ι s0i , s1i is the switching index, it is
unity, if i switched the action, so that pre- and post- reform actions are different, s0i 6= s1i ;
and zero otherwise.
There is no doubt that with the threat of a punishment harsh enough, any request
of the authority will be “convincing enough," in other words, if the punishment (mi ) for
disobeying the prescription is sufficiently large, any prescription will be followed. One of
our goals is to understand just how much punishment is needed to motivate the agents to
follow the suggestions of the authority.
Another way of looking at it is to assume that during the transition “human rights"
constraints should abided, as those are strictly enforced by an “international community,"7
mi ≤ m̄,
(IRH)
where m̄ ∈ R+ denotes the upper bound on a credible punishment. Thus, we will be seeking
to define the smallest such bound m̄ that will allow for good reforms. This could be of
interest to a benevolent international community, viewed as a “meta-mechanism designer"
whose objective is to prevent bad reforms and not to inhibit good reforms with limited
tools, those being just the bound on punishments, m̄. Indeed, it might be impossible for an
outsider to judge whether the “reformer" is benevolent or not and to dictate precisely how
to use the transfers and whether to undertake the reform, i.e., intervening in the internal
affairs of a country.
Clearly, if there are no additional constraints, and if taxes (transfers) can be expropriated by the reformer or simply burnt, the (IRH) constraint is irrelevant (not binding).
However, “financial" punishments are not unlimited. We, therefore, impose another, “positive,” assumption — the individual resource constraint — imposing a lower bound on the
amount of transfers that can be collected.
ti ≥ −u (s) ,
6
(RC)
Here and in what follows we restrict attention to “weak implementation," our objective being to formulate
the smallest necessary punishment, in particular, to determine whether any punishment is needed at all.
Requiring “full implementation," for example, might require more pressure on the punishment stemming
from a more demanding solution concept, although the latter exercise can be interesting to perform on
positive grounds.
7
See the related discussion in the introduction.
6
for any strategies profile s. It amounts to saying that no more than an individual’s “income"
can be extracted from each.
To sum up, a reformer formulates a mechanism, (I, t, m) , where I indicates whether the
reform is declared based on the profile of individual costs,
I : [c, c̄]N → {0, 1} ,
(6)
and t is the transfer profile, and m is a punishment profile, both based on the costs reports,
the decision with respect to the reform and the actions takes by the individuals:
t = (ti )N
i=1 ;
m = (mi )N
i=1 ,
(7)
where ti and mi are functions defined in (5) .
3
The Benchmark
We will start with the full information case, in which the authority observes individual costs
of transition.
We will assume that a benevolent reformer calls for the reform only if the sum of the
individual costs of transition is below the total surplus from the switch, or, if µ ≤ a − b,
with average cost of transition being µ :
½
1, if µ ≤ a − b
.
(8)
I1 (c) =
0, otherwise
An eccentric reformer wants the reform no matter what the costs are, I2 (c) = 1.
Definition 2 A rule I is (weakly) implementable (in Nash strategies) with allowable punishment m̄, if there exist a transfer profile t and a punishment profile m, satisfying BB, IC, IRH, RC.
Thus, we are interested in finding out whether there exist a mechanism that satisfies all
the listed constraints, without an additional restriction pertaining to ‘full’ or ‘exact’ implementation requiring the only equilibrium of the “transition game" to be the one envisioned
by the designer. Reason for that is, once again, that we are after the smallest punishment
making reforms feasible.
In this case an appropriate choice of the level of punishment can allow for a good reform
to be implemented and, sometimes can prevent bad reforms.
7
Proposition 3 The objective of a benevolent planner, I1 , is implementable with allowable
punishment of at least max {0, m̄1 } , m̄1 ≡ −a + µ. The same punishment is sufficient to
implement the reform always, rule I2 .
A way to implement the reform is to redistribute the surplus so that every individual
ends up paying the average cost of transition. In case the reform is good, µ ≤ a − b, there
should be enough surplus to implement the reform with no punishment, m̄1 ≤ 0. Also, if
µ > a, no punishment (imposing m̄ = 0) will prevent undesirable reforms.
The result hinges on several key assumptions: (1) full observability (and verifiability)
of individual costs, (2) no limits on transfers, and (3) common knowledge of the timing
and the decision of the reformer. If they are satisfied, there should be no reason for a
country not to carry out a good reform, moreover, there should be no reason to expect a
tyrant, threatening agents with punishments to head the desirable transition. Interestingly,
the mere presence of such threats signals that reform might be too costly and, thus, not
worthwhile. As we will show in what follows this conclusion might be erroneous if the
first assumption is relaxed, so that the costs are privately observable and are impossible
to verify.8 We, thus, retain the ability of a reformer to transfer utility and show that she
might need to resort to a punishment, even if she has benevolent intentions.
4
Asymmetric Information with respect to the Costs of Transition
Now assume costs of transition are privately known to the citizens. They share a common
belief that the costs are drawn independently from distribution F : [c, c̄] → [0, 1] , c ≥ 0.
The reformer announces a mechanism, (I, t, m) , where I indicates whether the reform is
declared based on the profile θ of reports of individual costs,
I : [c, c̄]N → {0, 1} ,
(9)
and both the punishment and the tax depend on the announced valuations,
ti = t(s1i , s1−i , θi , θ−i , I (θ));
mi =
m(s1i , s1−i , θi , θ−i , I
(θ)).
(10)
(11)
Agents privately observe the realization of the costs of transition, and report them
simultaneously to the reformer. Based on the reports, the reformer might either call for
8
Implications of assumptions (2) and (3) have been studied in the literature, see introduction for a brief
overview.
8
a reform or not. Endowed with the common knowledge of the reformer’s decision, the
agents choose one of two actions {A, B} . They get transfers and are subject to punishment
according to the mechanism thereafter.
A rule is implementable if every agent is choosing his best response given his cost and
his beliefs about the costs of the others, the costs are truly revealed and everybody chose
the action as instructed by the authority, i.e., according to I (θ) . Note that due to linearity
of the agents’ preferences, we can separate the incentive to choose the requested action (at
the last stage) from the incentive to report the costs truthfully, therefore we can apply the
revelation principle to the latter. Therefore, after the announcements are made, and thus,
the true costs revealed, the reformer will have to make sure that the agents are motivated
to act as she requires, i.e., the prescribed action s∗i should maximize
¡
¡
¢
¢
u s1i , s1−i − ci ι s0i , s1i + ti − mi , ∀i
as under full observability. Given true revelation of costs, this condition is equivalent
to (IC) . In addition, the agents should be motivated to tell the truth, so that Bayesian
Incentive compatibility is satisfied,
ci
∈ arg max Ec−i U (I (θi , c−i )) − ci I (θi , c−i ) + ti − mi ;
θi
U (I (θ)) ≡ aI (θ) + b (1 − I (θ)) .
(12)
(13)
Here θi is citizen i0 s announcement about his cost of transition and Ec−i [·] denotes
expectation formed by citizen i over the costs of his fellow citizens conditional on his cost
being ci .
Definition 4 Rule I is implementable in Bayesian strategies with allowable punishment m̄,
if there exist transfers t and punishments m, satisfying BB, IC, 12, IRH, RC.
The following lemma helps to simplify the analysis by reducing the number of constraints.
Lemma 5 The constraints IC, IRH, RC imply
¡
¢
a − ci + t sA , θ, 1 ≥ −m̄ for all i,
(14)
in case the reform is announced and
otherwise.
¢
¡
b + t sB , θ, 0 ≥ 0 for all i
9
(15)
In particular, the latter constraints imply,
Ec−i [U (I (θi , c−i )) − ci I (θi , c−i ) + τ (θi , c−i )] ≥ −m̄Ec−i I (θi , c−i ) for all i,
(IIR)
where
¡
¢
¡
¢
τ (θi , c−i ) ≡ t sA , θi , c−i , 1 I (θi , c−i ) + t sB , θi , c−i , 1 (1 − I (θi , c−i )) .
(16)
Luckily, this is nothing but an interim individual rationality constraint from standard
mechanism design literature, if m̄ = 0. Allowing for m̄ > 0, thus, “softens" this constraint,
undeniably “helping" the reformer.
As we demonstrate below, the minimal punishment might be above zero even for implementing a benevolent rule I1 and it crucially depends on the shape of distribution F.
However, an eccentric ruler has to be the most tyrannical, as she needs to resort to a punishment above the one pertinent to a benevolent rule. First, we calculate the latter “upper"
bound, and then proceed by deriving the minimal threat to be granted to a benevolent
reformer in order to be always successful.
Proposition 6 The eccentric rule I2 is implementable with punishment of at least max {0, m̄2 } ,
m̄2 = c̄ − a,
(17)
where c̄ is the upper support of the cost distribution.
It is worth noting that m̄2 is not necessarily strictly positive, so that even in the asymmetric information case an eccentric ruler might not need to resort to strictly positive
punishments. For example, if the improvement, (a − b) , is quite small relative to the costs,
but the level of the new benefit a is sufficiently high, c̄ < a, no punishment will be necessary. A mechanism supporting such a reform is very simple. Impose no transfers if an agent
complies with the request to switch his action. In case an agent obeys the authority, the
new payoff is then a − ci ≥ a − c̄; in case he pursues B, all his income is transferred away
and, potentially, the harshest punishment is applied. But if a − c̄ > 0, there is no need to
resort to punishment, because even in its absence, the non-compliance payoff is zero.
In most of what follows we focus on the complementary case, c̄ > a.
4.1
Carrying Out a Good Reform
Recall that our objective here is to describe the smallest punishment consistent with implementing the “first best," i.e., rule I1 with the smallest m̄, defined in (IIR) and consistent
10
with truthtelling at the same time. In order to determine this bound, we will first analyze the case of a discrete distribution sections and then proceed to the continuous case in
Section (4.3).
4.1.1
The Two Types Case
Suppose that each agent’s switching cost is either c (with probability ρ) or c̄ (with the
complimentary probability) and is distributed independently and identically, so that the
the costs are driven from distribution D :
⎧
⎨ 0, if x < c
ρ, if c ≤ x < c̄ .
(18)
D (x) =
⎩
1, otherwise
If a − c ≤ b, then switching from sB to sA is never beneficial. If a − c̄ ≥ b, then switching
from sB to sA is always beneficial. Each of these two cases is straightforward.
We will focus on the most interesting case where a − c > b and a − c̄ < b. Then there
exits an integer 0 < n∗ < N , such that the switch from sB to sA is beneficial if and only if
the number of agents having c is at least n∗ . In other words,
(n∗ − 1)c + (N − n∗ + 1)c̄ > N (a − b) ≥ n∗ c + (N − n∗ )c̄.
In most of the analysis, we ignore the integer problem for n∗ and set
N (a − b) = n∗ c + (N − n∗ )c̄.
(19)
Thus, n∗ can be viewed as the smallest number of low cost individuals needed to make the
regime switch welfare improving.
We can now calculate the lowest punishment necessary to implement the benevolent
rule.
Proposition 7 Assume the costs are distributed D independently, with a − c > b and
a − c̄ < b. Then I1 is implementable with allowable punishment of at least max {0, m̄1 } ,
m̄1 =
−a Pr{A} − b Pr{B} + c̄ Pr{nL ≥ n∗ } + c Pr{c1 = c} Pr{nL = n∗ − 1}
,
Pr{nL ≥ n∗ }
(20)
where Pr{A} is ex-ante probability that the reform is worthwhile, Pr{B} is the complementary probability, nL is the number of low cost agents excluding agent 1.
11
Note that if c̄ is high and a or b are sufficiently small, m̄1 is positive. This is because it
is expensive to make the high cost agents to switch, and the tax revenue that is available
for transfers is not enough to cover the expense. In this case, some punishments have to be
imposed to make the switch implementable, so that more “costly" reforms might require
higher punishments. Conversely, higher gross payoffs from the transition, a − b, decrease
the necessary threat. As will be shown for the continuous case, the bound m̄1 is closely
related to the “informational rents" (that a low cost agent can extract), often blamed for the
inefficiency in the standard public good provision problem,9 i.e., the impossibility to find an
efficient incentive compatible, budget balanced and individually rational mechanism, which
corresponds to the case of m̄1 = 0 in this framework.
Interestingly, the boundaries on punishments can be ordered,10
m̄1 ≤ m̄2 .
(23)
If the allowable punishment is in the range [m̄1 , m̄2 ] , the benevolent rule I1 is implementable
(for all realizations of individual costs), while the eccentric ruler might sometimes fail to
reform. However, this boundary hinges on the exact knowledge of the distribution F , from
which the costs are driven. If a uniform bound is to be set for a group of countries with
different underlying primitives (reduced to the cost generating distribution), this boundary
might be set too high for some countries and too low for the others.
Moreover, as we show below, ‘big’ reforms and more dispersed distributions of costs
(describing a “more divided country") require more severe punishment m̄1 in order to be
implemented, thus, potentially calling for m̄1 > 0, thereby setting the “right" bound on the
allowable punishment a non-trivial task.
4.2
Comparative Statics
Recall that lowest boundary m̄1 on allowable punishment that allows good reforms to go
through might be either positive or negative, it’s sign depending on the underlying distribution of costs, number of people involved in the transition, etc. One would want to
understand what affects this boundary, which, — as you might remember from the discussion in the introduction, — can also be thought of as an indicator of how difficult it is to
9
See Ledyard and Palfrey (2003), Mailath and Postlewaite (1990), Myerson and Satterthwaite (1983).
Indeed, it amount to showing that a Pr {A}+b Pr {B}−c Pr {c1 = c} Pr {nL = n∗ − 1} > a Pr {nL ≥ n∗ } ,
which follows from the inequality a > c and
10
Pr {A} − Pr {nL ≥ n∗ }
=
=
Pr {c1 = c} Pr {nL ≥ n∗ − 1} − Pr {c1 = c} Pr {nL ≥ n∗ } =
Pr {c1 = c} Pr {nL = n∗ − 1} .
12
(21)
(22)
carry out a reform. That is why we compare boundary m̄1 across different environments
(countries). These comparisons are reproduced for the continuous case in section (4.3) .
4.2.1
Reforms in Divided Countries are Harder to Implement
We can compare two ‘countries’ that differ by the shape of their costs distribution. One is
more ‘divided’ than the other, if the possible realizations of costs are further apart. This,
for example, corresponds to the variation in attitudes towards the reform: if some people
favor the transition (view its costs as rather small), while others perceive it as undesirable,
or very costly. The bigger is this gap — we show — the harder it is to implement the reform.
It happens as higher difference in costs increases the “informational rents," which, in turn,
call for a higher minimal punishment. As an illustration one could rely on economic success
of (relatively) homogenous Far Eastern countries (Taiwan, Singapore) in the mid-1980’s and
challenges of economic reforms in the vastly diverse India.
As in the previous case, we want to keep the social decision with respect to reform,
i.e., the smallest number of low cost announcements to execute the reform, n∗ , constant.
In order to do so, we can only consider cases in which low cost and high cost realizations
are equally likely and the gain from reform is exactly between the costs, thus making the
“majority rule" an optimal decision.
Lemma 8 Assume the costs are distributed D independently,
with a − c > b and a − c̄ < b.
¡ ¢
Assume, in addition, that ρ = 1/2 and a − b = N c̄+c
.
Then
a mean preserving spread
2
of the costs, i.e., if an individual cost either c̄ + δ or c − δ with equal probabilities for any
δ > 0, leads to an increase in the required punishment, m̄1 , to implement the corresponding
benevolent rule.
4.2.2
Smaller Reforms are Easier to Implement
In this section we show that smaller reforms are easier to implement as opposed to big leaps.
Relatively successful reforms in China and a painful transition in Russia can be seen as an
illustration of this relation.
To make such a comparison we have to introduce “intermediate steps," or to extend the
initial coordination game to generate additional equilibria. Let the initial action set in game
G now include action X, and we assume, that every agent choosing action X constitutes
a new (pure strategy) Nash equilibrium, sX , in that game with the corresponding payoff
x ∈ (b, a) to each. Therefore, switching from sB to sX captures a proportion of the benefit
of the big switch (sB to sA ). Let α denote this proportion. That is, x − b = α(a − b).
13
We think about a transition to X as a “scaled down" reform, so it is natural to think that
the costs for this transition are also proportionally smaller. Let c(x) and c̄(x) denote that
switching cost respectively for a low cost agent and a high cost agent to sX . Then
cj (x) = αcj (a) = αcj , j ∈ {L, H} .
(24)
In this set up the smallest number of the low cost agents needed for a reform to be
worthwhile stays constant from switch to switch. Indeed, let n∗ (x) denote the minimum
number of low-cost agents required for the switch to sX to be beneficial. Then
N (x − b) = n∗ (x)c(x) + (N − n∗ (x))c̄(x).
Since x − b = α(a − b), c(x) = αc, and c̄(x) = αc̄, we can conclude that n∗ (x) = n∗ .
Define the benevolent rule for small reforms, I1α , accordingly, with x replacing a and the
new average cost being αµ.
Proposition 9 Let b > 0. Assume the costs are distributed D independently, with a − c > b
and a − c̄ < b and that an agent’s switching cost is proportional to the gain from a switch.
Then I1α is implementable with allowable punishment of at least max {0, m̄α1 } ,
m̄α1 =
1
Pr (nL ≥ n∗ )
(αm̄1 − (1 − α) b) ,
(25)
where m̄1 is the punishment needed for a big (original) reform. Therefore for any a, b, c, c̄
there exists α∗ > 0, rule I1α is implementable with no punishment (m̄α1 ≤ 0) .
One might not find it surprising that a smaller reform requires less coercion. As the size
of the reform decreases, gross payoff from the switch, a − b, as well as the switching costs
all decrease at the same proportion, thereby decreasing the gain from misrepresenting one’s
costs (informational gains). What is striking, however, is that for a scaled down version of
any reform the implementation of the benevolent rule requires no punishment whatsoever.
Interestingly, an eccentric authority can not avoid threats altogether if the big leap requires
a positive punishment, i.e., if m̄2 = c̄ − a > 0. The best she can do is to decrease the
punishment for a small reform to α (c̄ − a) , which is, clearly, positive for any α > 0.
Along the same lines, if one compares a reform with a big gain and a high cost with
another reform, in which both the gain and the costs are smaller (not necessarily proportional to the first one), then the punishment needed to implement a bigger reform is higher
provided the threshold n∗ of the low cost agents is identical in both cases. See lemma 14 in
the appendix.
14
One may reasonably expect that breaking up a large reform into smaller ones would
require lower total punishment. But showing this proves to be difficult. This is because
earlier reforms reveal information regarding agents’ private costs of reform. Therefore, if
agents expect that the transfers in later reforms depend on the information they revealed
in earlier reforms, they would have more incentive to misrepresent themselves in the earlier
reforms. Therefore, more punishments than those indicated by the above proposition are
required, and it is not clear whether the total punishment is lower.
4.3
Continuous costs case
Let the private costs of transition ci be i.i.d. with compact support. Note that in the
discussion of the bad reform we did not rely on the discreteness of the distribution, so it is
left to derive m̄1 , the the bound on the minimal punishment, for a benevolent reformer.11
First, we follow the standard mechanism design argument (Mirrlees (1971), as presented
in Fudenberg and Tirole (1996)) to develop implications of incentive compatibility and then
incorporate all other constraints to calculate the bound on the minimal punishment, m̄1 .
Proposition 10 Let ci be i.i.d. F on [c, c̄] , F be logconcave with corresponding marginal
distribution f . Then I1 is implementable with allowable punishment of at least max {0, m̄1 } ,
½Z c̄ ∙
¶¸
¾
µ
F (s)
1
¯
(26)
+ s I (s) f (s) ds + (1 − Q (∆)) b ,
m̄1 = −
a−
I (c̄)
f (s)
c
where Q is the cumulative distribution of the sum of the costs, and ∆ = N (a − b) , i.e., the
sum of the gains from transition; I¯ (θi ) = Q (∆|ci = θi ) probability of reform conditional on
individual i’s announcement of his cost, θi .
It is possible to demonstrate that this is, indeed, a counterpart of m̄1 derived for the
two types case.12
11
We use the same notation for the upper bound of the punishment in the continuous case. This is justified
by the next footnote.
12
Note that Q(∆) = Pr{A}. Let c̄ = c̄, and c = c. Furthermore, for the two type case
Z c̄
sf (s) I¯F (s) ds =
c
=
c̄ Pr{A|c1 = c̄} Pr{c1 = c̄} + c Pr{A|c1 = c} Pr{c1 = c}
=
c̄ Pr{nL ≥ n∗ } Pr{c1 = c̄} + c Pr{nL ≥ n∗ − 1} Pr{c1 = c}.
For the two point distribution, D(s) = Pr{c1 = c} for s ∈ (c, c̄). Furthermore, I¯D (s) = Pr{nL ≥ n∗ } for
15
Note that this bound, m̄1 , is the negative of two terms. The first is the expected
‘virtual’ payoff in case of reform, and the second one is its counter-part in case no reform is
undertaken. The first term is familiar from the mechanism design literature. Assume b = 0,
then m̄1 > 0 only when the objective is not implementable in the standard framework, i.e.,
if the standard individual rationality constraint is incompatible with incentive compatibility
and budget balance constraints.13 Softening restrictions on the punishment, is identical (in
this case) to relaxing the ex-post individual rationality constraint, thus, it extends the range
of feasible reforms. Recall that without the individual rationality constraint, benevolent rule
is implementable using d’Aspremont and Gérard-Varet (1979) mechanism.
The next proposition generalizes some of the comparative statics results for this case.
If costs distributions can be ordered according to the first order stochastically dominance
criterion, then the dominating distribution corresponds to a more ‘expensive’ reform, in
particular, with higher average cost of transition. In particular, it asserts that ‘bad’ reforms
require harsh punishments. The second part of the proposition compares punishments under
two distributions that are ordered by “more peaked" order. The following definition is
adopted from Shaked and Shanthikumar (1994), p.77.
Definition 11 Consider two unimodal distributions, F and H, symmetric about µ. F is
more peaked than H, if H (x) ≥ F (x) for all x ≤ µ, i.e., if F (x|x ≤ µ) first order stochastically dominates H (x|x ≤ µ) .
This order indicates which distribution is more spread, thus, corresponding to a more
‘heterogeneous’ society. Therefore, this is a counter-part of proposition (8) .
Proposition 12
1. Assume H first order stochastically dominates (FOSD) F, then the
allowable punishment necessary to implement I1 under H is higher than that under
F;
2. The same conclusion is also true if
(a) f, h are symmetric and unimodal around d =
(b) F is more peaked than H.
s ∈ (c, c̄]. Therefore,
Z
c̄
c
1
2
(c̄ + c) = (a − b);
F (s) I¯F (s)ds = Pr{c1 = c} Pr{nL ≥ n∗ }(c̄ − c).
13
In terms of Myerson and Satterthwaite (1983), all the individuals are “sellers" that obtain the same
“price" (a) and have private information about their costs.
16
To formulate a generalization of proposition (9) , note that a ‘small’ step reform that
generates a fraction α ∈ (0, 1) of the original gain, (a − b) , and requires a fraction α of the
original costs, cαi ≡ αci for all agents i will require a punishment
µ
¶
¶¸ µ
¶
Z αc̄ ∙ µ
1
1
1
1
α
sα + sα f
sα I¯
sα dsα − b − α (a − b) Q (∆) (27)
F
m̄1 ≡
α
α
α
α
αc
(28)
= αm̄1 − (1 − α) b.
Clearly, with α small enough and b > 0, the small step reform will require no punishments,
mα1 ≤ 0, then identical argument to that in proposition (9) establishes the rest of the result.
5
Extensions
5.1
Outcome Uncertainty
Here we demonstrate that it is easy to re-formulate this model to capture some cases of
common uncertainty with respect to the outcome of a reform for the two types case.
Suppose that there are two different levels of payoffs (aH and aL ) in the outcome of sA ,
both of which are higher than b. Each agent’s payoff in sA is independently and identically
distributed. Every agent has the same switching cost c. Let ñH be the number of agents
other than agent 1 that have the high payoffs. Let ñ∗ be the cut-off number of agents with
high payoffs such that the switch is beneficial. Thus,
ñ∗ aH + (N − ñ∗ )aL = N c.
Then we have the following result.
Proposition 13 Assume that there are only two levels of benefits. When aH − c > b and
aL − c < b, the minimum level of punishment to always implement the first best outcome is
given by max {0, m̄1 } ,
m̄1 =
−aH Pr{a1 = aH } Pr{ñH = ñ∗ − 1} − aL Pr{ñH ≥ ñ∗ } − b Pr{B} + c Pr{A}
. (29)
Pr{ñH ≥ ñ∗ }
The case where agents differ in benefits from reform is parallel to the case where agents
differ in costs. The reason is that an agent cares about only the reform’s net gain, which
is benefit minus cost. In both cases above, the agents’ information structure about this net
gain is the same. So the same methodology applies.
17
Consider another situation where the payoff is the same for every agent in sA but
different agent receives different information about it. Suppose that all agents have the
same switching cost. Let a be the common payoff that every of them will receive in sA ,
and agent i receives a signal ai . It is easy to see that it is a dominant strategy for them
to reveal their information in the absence of any transfers. Therefore, the truthtelling
condition is automatically satisfied. Suppose that conditional on the revealed information
(a1 , a2 , ..., aN ), it is beneficial to switch to sA . Then the incentive compatibility constraint
for switching becomes
E(a|a1 , a2 , ..., aN ) − c ≥ −m̄.
Therefore, we require that
m̄ ≥ c − E(a|a1 , a2 , ..., aN )
for all combinations of (a1 , a2 , ..., aN ) such that switching is beneficial in expectation term.
Since switching is beneficial only if
E(a|a1 , a2 , ..., aN ) − c ≥ b,
the above determined m̄ is negative. That is, no positive punishment is required.
Therefore, provided the reforming authority can credibly announce all the individual
signals (example being “free press"), a worthwhile reform that imposes identical costs across
agents should be implementable with no punishments.
This result is quite intuitive. If every agent has the same benefit and cost from reform,
it becomes a common interest game. Therefore, it is either everyone wants the reform to go
ahead, or everyone does not want the reform to go ahead. Even though they have different
information on whether the reform should go ahead, they have no incentive to misrepresent
this information. It is in their own interest to find out truthfully whether the reform should
or should not go ahead. Hence, no punishment is necessary for agents to either reveal their
information honestly or carry out the reform when it is beneficial.
This also demonstrates that there should be some additional restrictions that prevent
desirable transitions in this case, i.e., incomplete information alone is not sufficient to explain
why some good reforms are not carried out in this context. In particular, restrictions on
transfers across individuals, — which might reflect some constraints existing in practice—
might be to blame (Fernandez and Rodrik (1991)).
5.2
Foreign Aid
As an additional extension, one could also easily introduce a source of an outside financing or
aid, T, so as to modify the budget constraint BB to read Σi ti ≤ T. It is easy to check that in
18
this case, the necessary punishments, m̄1 and m̄2 , (if positive) each decrease by T /(NI (c̄)),
thus, making it easier to implement both a benevolent and an eccentric rule.14 Provided the
interim well-being of the highest cost individual is exactly equal to −m̄i , this outside transfer
T, may improve the (expected) utility of the least fortunate.15 Clearly, this improvement is
not guaranteed by the mere presence of transfer, as the reformer has to be encouraged to
use it appropriately. If the latter is assured, our results can be re-interpreted as suggesting
that preserving the level of “human rights" might be costly (i.e., require external financial
assistance) especially at the outset of a reform involving substantial individual adjustments,
besides, more aid might be required for more heterogeneous societies.16 The model then
introduces a way to formulate a trade-off between the standards of human rights and foreign
aid. Besides, it also provides a rationale to condition international aid on human rights
protection in the recipient country.
6
Discussion
We chose a standard mechanism design framework to characterize the lowest bound on
punishments that make the set of implementable ex-post efficient mechanisms feasible, i.e.,
(ex-ante) budget balanced and (ex-post) individually rational (in terms of lemma 5). The
existence of efficient Bayesian mechanisms (d’Aspremont and Gérard-Varet (1979)) coupled
with the non-existence results (Myerson and Satterthwaite (1983), Mailath and Postlewaite
(1990) and Rob (1989)) in the presence of individual rationality constraints suggest, in
particular, that there should be just the right way to ‘soften’ these constraints, which
amounts to increasing the lower bound on punishments in our environment.
We used Bayesian-Nash framework, as it is easier to implement the objective of a designer, when the choice of the agents’ strategies is not restricted (to dominant strategies,
say), thus, imposing less pressure on the threats that the designer might need.17 Even in
this framework a designer has to credibly threaten with a punishment in order to implement
good reforms under some circumstances.
As usual, many of the assumptions were made for simplicity. Introducing additional aspects that vary across individuals will hardly simplify the problem of finding the appropriate
lower bound on punishment. However, in some environments multidimensionality of the rel14
Follows from the appropriately modified proofs of proposition 10 and IIR correspondingly.
With high enough transfers (in this model) an individual will be induced to switch to any action, so
that any reform, however whimsical, can be implemented.
16
under the assumptions of proposition 12.
17
The assumption of common knowledge of distribution of types (costs) shared by all the agents may,
indeed, be problematic (see Myles (1995)), but it “helps" the designer, thus, might decrease the necessary
punishment. Provided lower bound on punishment is our objective, this assumption is reasonable.
15
19
evant individual characteristics might relieve the pressure on this boundary. Bundling, or
linking independent decisions (public goods) can improve efficiency, see Jackson and Sonnenschein (2003), Fang and Norman (2003). However in the context of this model, provided
the reform is interpreted as a single (global) public good the above results are not applicable.
So far we have assumed that a reform entails a coordinated response of all the agents in
a society. No doubt, it might be a close description of some real-life transitions, for example,
altering the alphabet, or exchanging the acceptable currency, a switch from driving on the
left to driving on the right hand side and vice versa. However, some other reforms, say,
privatization, rely on just a subset of individuals to substantially alter their actions for the
reform to be “successful." It could be interesting to extend the framework by allowing some
of the agents to retain their old action, for example, if their costs are high enough, i.e., to
incorporate partial reforms.
In the view of the contribution by Ledyard and Palfrey (2003), we conjecture that our
results can be re-formulated for independent but not identical distributions. Introducing
correlation in individual costs of transition, however, can substantially change the results. It
is well known that in this case it is possible to (approximately) achieve ex-post efficiency, as
in Crémer and McLean (1985) and McAfee and Reny (1992). Although the implementation
abides interim individual rationality constraint, it might violate the ex-post one (requiring
infinitely high taxes), which is crucial to the current framework, requiring the “limited liability" constraint (RC) and restricting available punishments (IRH) . Similar investigation
for correlated environments, therefore, is left for the future research.
7
Conclusions
We found that reforms can be hindered by asymmetric information about costs of transition
if some individuals have to be compensated for the transition, if there is no outside funding
for a reform, and if individual resources available for transfers are limited. In this case a
reformer might need to be able to credibly threaten an agent with a punishment to assure
compliance. The minimal level of credible punishment should be higher for ‘big’ transitions
and in more divided societies, as we have illustrated.
This might explain the puzzling link between economic success of reforms and the ‘authoritarian’ rulers in power mentioned in the introduction. However, even in our simple
set up, the threat of punishment (associated with these rulers) does not necessarily have to
be severe in order for any desirable reform to be implemented. Moreover, unfortunately,
as was mentioned above, allowing for more punishments makes it easier to carry out any
reform, independent of costs and benefits.
20
It is then natural to expect that the international community will come up with some
mechanisms to protect individuals against bad reforms in their countries. With direct
foreign intervention (determining which reforms to undertake, or dictating the identity of
the ruler) being often impossible or undesirable, the outsiders can settle on enforcing human
rights protection instead. As our results suggest, human rights, indeed, may be a sensible
indicator to monitor. If the level of maximal individual punishment is set at a ‘correct’ level,
bad reforms will be impossible to implement, while good ones can still go through. This
punishment level hinges on the knowledge of local costs of transition and their distribution,
which may differ by country. If this level is to be set internationally, i.e., it is to be the same
across all countries, it will prevent good reforms in some countries and enable bad ones in
the others.
This analysis also suggests that there is a trade-off between a successful implementation
of good reforms from the utilitarian perspective and well-being of selected individuals in the
society, who are affected by the harshest punishment available. This trade-off is not driven
by ad-hoc restrictions on transfers, but, rather, by asymmetric information with respect to
private costs of transition. We are far from being able to contribute to the ‘moral calculus’
resolving the trade-off, but the credit is ours for its formulation.
A
Proofs for the Benchmark
Proof of proposition 3. First, assume the benevolent reformer observes µ ≤ a −b. Then,
to make the good reform (from norm b to norm a) implementable, the following conditions
should hold: BB IRH, RC and the incentive constraint becomes (restricting attention to
pure strategies)
¢
¡
¢
¡
(30)
U sA , ci , c−i , 1 ≥ U B, sA
−i , ci , c−i , 0 for all i, where
U (s, ci , c−i , y) = u (s) − ci y + t (s, ci , c−i , y) − m (s, ci , c−i , y) ,
where sA
−i is subprofile of actions corresponding to the case in which all players but i choose
action A, so that
¡
¢
B, sA
(31)
−i = (B, A, .., A) .
Then the reformer will set
¡
¢
m sA , c, 1 = 0;
¢
¡
= m̄;
m B, sA
−i , c, 1
¢
¡
¡
¢
A
t B, s−i , c, 1 = −u B, sA
−i
21
(32)
(33)
(34)
Therefore the incentive constraint becomes
¡
¢
¡
¢
¡
¢
A
a − ci + t sA , ci , c−i , 1 ≥ u B, sA
−i − u B, s−i − m̄ for all i,
which implies
¡
¢
a − ci + t sA , ci , c−i , 1 ≥ −m̄ for all i,
(35)
(36)
Sum up over i and divide by N, and get
a − µ + t̄ ≥ −m̄,
(37)
where t̄ is the average tax and µ is the average switching cost. But in the view of (BB) ,
t̄ = 0, so if all the incentive constraints hold if
m̄ ≥ −a + µ.
(38)
conversely, if (38) holds, the reformer can pick taxes in such a way as to equalize the after
tax switching cost across agents and thus, implement the reform. Note that this condition
is independent on whether the reform is a good one (µ < a − b), or not. Therefore the
boundary m̄1 ≡ µ − a is also the smallest punishment to introduce any reform (bad ones
included).
If µ > a − b, a benevolent reformer, implementing I1 , has to make sure the agents are
discouraged from switching, thus implying
¢
¡
(39)
b + t sB , ci , c−i , 0 ≥ −m̄ − ci
so that
m̄ ≥ −b − µ.
(40)
−a + µ > −b − µ.
(41)
m̄ ≥ m̄1 = −a + µ,
(42)
Note that as µ > a − b, this boundary is below −a + µ, in other words
Therefore,
B
Proofs for the Discrete Case
Proof of lemma 5. From IC, for any profile of announcements θ that call for the reform,
in order to convince the agents to play the desired action, the authority should set
¢
¡
(43)
m sA , θ, 1 = 0;
¢
¡
A
(44)
m B, s−i , θ, 0 = m̄;
¢
¢
¡
¡
(45)
= −u B, sA
t B, sA
−i , θ, 0
−i .
22
It implies, that
¡
¢
a − ci + t sA , θ, 1 ≥ −m̄ for all i.
(46)
In case the reform is not to be implemented based on the announced valuations the same
argument implies that the corresponding incentive constraint should be of the form
¢
¡
(47)
b + t sB , θ, 0 ≥ −m̄ for all i.
Note however, that this latter constraint always holds as long as m̄ ≥ 0, and another
constraint (RC) is satisfied, i.e.,
¢
¡
(48)
b + t sB , θ, 0 ≥ 0.
The conclusion then follows.
Proof of proposition 6. By lemma 5,
a − ci + t(sA , θi , θ−i , 1) ≥ −m̄.
(49)
t(sA , θi , θ−i , 1) ≥ ci − a − m̄,
(50)
It implies that
which has to be satisfied for any announcements and any cost ci .
In addition, truthtelling constraint should be satisfied, in other words, compensation
should be formulated in such a way that nobody has a motivation to lie about the costs
of his transition. As the decision rule is constant, i.e., independent of the profile of the
announced costs, so should the transfer, as otherwise every agent would announce the cost
corresponding to the highest transfer. This implies that t(sB , θi , θ−i , 1) should not vary
with θi ∈ R,
(51)
t(sB , θi , θ−i , 1) = t̂
Combining with (50) , it implies that
t̂ ≥ ci − a − m̄
(52)
for all ci ∈ [c̄, c]. To minimize the transfer while still satisfying the incentive compatibility
constraint for all ci ∈ [c̄, c], we set
t̂ = c̄ − a − m̄.
To balance the budget, the sum of the transfers has to be non-negative. That is,
m̄ ≥ c̄ − a.
23
Therefore, the minimum of m̄ is
m̄2 = c̄ − a.
Note that t̂ = 0 for all ci ∈ [c̄, c] is individually feasible, satisfying (RC) . It also satisfies
the rest of the constraints for m̄ ≥ m̄2 .
Proof of proposition 7. First, denote
Pr{A} = Pr{c1 = c} Pr{nL ≥ n∗ − 1} + Pr{c1 = c̄} Pr{nL ≥ n∗ }
(53)
as the ex-ante probability that sA should be enforced and
Pr{B} = Pr{c1 = c} Pr{nL < n∗ − 1} + Pr{c1 = c̄} Pr{nL < n∗ }
as the probability that sB should be enforced ex-ante. Note that Pr{A} can also be expressed as
Pr{A} = Pr{nL ≥ n∗ } + Pr{c1 = c} Pr{nL = n∗ − 1}.
This is because
Pr{nL ≥ n∗ − 1} = Pr{nL ≥ n∗ } + Pr{nL = n∗ − 1}.
Suppose that agent 1 has switching cost c1 . Let EA τ (c1 ) denote the expected transfer agent
1 receives conditional on that sA should be implemented. Similarly, let EB τ (c1 ) denote the
expected transfer agent 1 receives conditional on that sB should be implemented. Thus,
EA τ (c) = E{τ (c, c2 , ..., cN )|nL ≥ n∗ − 1},
EA τ (c̄) = E{τ (c̄, c2 , ..., cN )|nL ≥ n∗ },
EB τ (c) = E{τ (c, c2 , ..., cN )|nL < n∗ − 1},
and
EB τ (c̄) = E{τ (c̄, c2 , ..., cN )|nL < n∗ }.
There are two stages in this game. First, the government announces transfers to the agents
contingent on their report of the costs. Second, the government announces whether or not
a reform will take place and punishments for anyone who does not follow the instructions.
Due to the revelation principle, we concentrate on direct mechanisms. We characterize the
conditions for an efficient equilibrium to exist, and then select the minimum punishment
for such an equilibrium to exist.
First, we consider the incentive compatibility constraints for agent 1 to follow the instruction of whether or not to switch from B to A. Consider an efficient equilibrium. If
he receives c̄ and nL ≥ n∗ , he is required to switch to A and gets a − c̄ + EA τ (c̄), where
24
EA τ (c̄) is the transfer in this case. (As we can easily see below, having a constant transfer
of EA τ (c̄) helps to satisfy the incentive compatibility constraint.) If he refuses to switch,
all of his income will be taxed away plus he is punished to the most extend. Therefore, he
receives −m̄ in this case. That is, for c1 = c̄ and nL ≥ n∗ ,
a − c̄ + EA τ (c̄) ≥ −m̄.
(54)
For c1 = c̄ and nL < n∗ , no switch is required, and
b + EB τ (c̄) ≥ −m̄.
(55)
Similarly, for c1 = c and nL ≥ n∗ − 1, switching to A is required, and
a − c + EA τ (c) ≥ −m̄.
(56)
For c1 = c and nL < n∗ − 1, no switch is required, and
b + EB τ (c) ≥ −m̄.
(57)
Now consider the information revelation in the first stage. Suppose that agent 1’s switching
cost is c. Then the incentive compatibility constraint for him to report c is given by
[a − c + EA τ (c)] Pr{nL ≥ n∗ − 1} + [b + EB τ (c)] Pr{nL < n∗ − 1}
≥ [a − c + EA τ (c̄)] Pr{nL ≥ n∗ } + [b + EB τ (c̄)] Pr{nL < n∗ }
(58)
Suppose that agent 1’s switching cost is c̄. Then the incentive compatibility constraint for
him to report c̄ is given by
[a − c̄ + EA τ (c̄)] Pr{nL ≥ n∗ } + [b + EB τ (c̄)] Pr{nL < n∗ }
≥ [a − c̄ + EA τ (c)] Pr{nL ≥ n∗ − 1} + [b + EB τ (c)] Pr{nL < n∗ − 1}
(59)
The assumption that the government cannot tax more than one’s income implies that
EA τ (c̄) ≥ −a,
EA τ (c) ≥ −a,
EB τ (c̄) ≥ −b,
and EB τ (c) ≥ −b.
These inequalities imply that (55) and (57) are automatically satisfied.
Let τ̄ (c1 ) denote the expected transfer agent 1 receives when his reported switching cost
is c1 . That is,
τ̄ (c) = EA τ (c) Pr{nL ≥ n∗ − 1} + EB τ (c) Pr{nL < n∗ − 1}
and
τ̄ (c̄) = EA τ (c̄) Pr{nL ≥ n∗ } + EB τ (c̄) Pr{nL < n∗ }.
25
The two incentive compatibility constraints for truthful reporting (58) and (59) can then
be simplified as
(60)
τ̄ (c̄) ≤ τ̄ (c) + (a − c − b) Pr{nL = n∗ − 1} ≡ β
and
τ̄ (c̄) ≥ τ̄ (c) + (a − c̄ − b) Pr{nL = n∗ − 1}
(61)
We argue that EB τ (c̄) = −b and that EB τ (c) = −b. This is because EB τ (c̄) and EB τ (c)
cannot be lower than −b from the tax constraint. If we raise them while lowering EB τ (c̄)
and EB τ (c) to keep the expected transfers τ̄ (c̄) and τ̄ (c) constant, it would not affect (60)
and (61), but make (54) and (55) more difficult to hold.
We want to characterize the minimum m̄ such that the budget is balanced ex ante,
that is, E (τ̄ (c)) ≤ 0. In order to do so, we fix m̄ and characterize the minimum expected
transfer that still implement the efficient equilibrium outcome. This transfer is a decreasing
function of m̄. We then set the expected transfer to zero to obtain the minimum feasible
m̄.
Given EB τ (c̄) and EB τ (c), from (54) and (56), we have
τ̄ (c̄) = EA τ (c̄) Pr{nL ≥ n∗ } + EB τ (c̄) Pr{nL < n∗ }
≥ γ ≡ (c̄ − a − m̄) Pr{nL ≥ n∗ } − b Pr{nL < n∗ },
and
τ̄ (c) = EA τ (c) Pr{nL ≥ n∗ − 1} + EB τ (c) Pr{nL < n∗ − 1}
≥ δ ≡ (c − a − m̄) Pr{nL ≥ n∗ − 1} − b Pr{nL < n∗ − 1}.
At τ̄ (c) = δ, noting (60),
β = τ̄ (c) + (a − c − b) Pr{nL = n∗ − 1}
= (c − a − m̄) Pr{nL ≥ n∗ − 1} − b Pr{nL < n∗ − 1}
+(a − c − b) Pr{nL = n∗ − 1}
< (c̄ − a − m̄) Pr{nL ≥ n∗ − 1} − b Pr{nL < n∗ − 1}
+(a − c̄ − b) Pr{nL = n∗ − 1}
< (c̄ − a − m̄) Pr{nL ≥ n∗ } − b Pr{nL < n∗ }
= γ
because Pr{nL ≥
n∗
− 1} − Pr{nL = n∗ − 1} = Pr{nL ≥ n∗ }.
Therefore, to minimize the expected transfer, we raise τ̄ (c) such that β = γ, and set
τ̄ (c̄) = γ. The first condition becomes
τ̄ (c) + (a − c − b) Pr{nL = n∗ − 1}
= (c̄ − a − m̄) Pr{nL ≥ n∗ } − b Pr{nL < n∗ },
26
which gives us
τ̄ (c) = −(a − c − b) Pr{nL = n∗ − 1}
+(c̄ − a − m̄) Pr{nL ≥ n∗ } + (−b − m̄) Pr{nL < n∗ }
= −m̄ Pr{nL ≥ n∗ } − a Pr{nL ≥ n∗ − 1} − b Pr{nL < n∗ − 1}
+c̄ Pr{nL ≥ n∗ } + c Pr{nL = n∗ − 1}
(62)
Therefore, the minimum expected transfer is
E (τ̄ (·))
=
Pr{c1 = c}τ̄ (c) + Pr{c1 = c̄}τ̄ (c̄)
=
Pr{c1 = c}[−m̄ Pr{nL ≥ n∗ } − a Pr{nL ≥ n∗ − 1}
−b Pr{nL < n∗ − 1} + c̄ Pr{nL ≥ n∗ } + c Pr{nL = n∗ − 1}]
+ Pr{c1 =
=
∗
(63)
∗
c̄}[(c̄ − a − m̄) Pr{nL ≥ n } − b Pr{nL < n }]
−m̄ Pr{nL ≥ n∗ } − a Pr{A} − b Pr{B} + c̄ Pr{nL ≥ n∗ }
+c Pr{c1 = c} Pr{nL = n∗ − 1}
The ex ante budget balance E (τ̄ (c)) ≤ 0 implies
m̄ ≥ −a
Pr{A}
Pr{B}
Pr{c1 = c} Pr{nL = n∗ − 1}
−
b
+
c̄
+
c
.
Pr{nL ≥ n∗ }
Pr{nL ≥ n∗ }
Pr{nL ≥ n∗ }
By taking the minimum of m̄, we obtain
m̄1 = −a
Pr{c1 = c} Pr{nL = n∗ − 1}
Pr{B}
Pr{A}
−
b
+
c̄
+
c
.
Pr{nL ≥ n∗ }
Pr{nL ≥ n∗ }
Pr{nL ≥ n∗ }
We still need to verify that the tax constraints are satisfied when sA is implemented; that is,
no one is taxed more than his income. First note that m̄1 < c̄, since a > c. From τ̄ (c̄) = γ,
we have
EA τ (c̄) Pr{nL ≥ n∗ } + EB τ (c̄) Pr{nL < n∗ }
= (c̄ − a − m̄) Pr{nL ≥ n∗ } − b Pr{nL < n∗ }.
That is,
EA τ (c̄) = c̄ − a − m̄ > −a.
Therefore, the tax constraint is satisfied for the high cost agents.
From (62) and the definition of τ̄ (c), we have
EA τ (c) Pr{nL ≥ n∗ − 1} + EB τ (c) Pr{nL < n∗ − 1}
= −m̄ Pr{nL ≥ n∗ } − a Pr{nL ≥ n∗ − 1} − b Pr{nL < n∗ − 1}
+c̄ Pr{nL ≥ n∗ } + c Pr{nL = n∗ − 1}.
27
Note that EB τ (c) = −b and substitute m̄1 for m̄. We can easily show that EA τ (c) > −a.
Therefore, the tax constraint for the low cost agent is satisfied as well. So the characterization we obtained indeed satisfies all of the conditions.
Proof of lemma 8. By the assumptions n∗ = 12 N under any such spread. By definition
of m̄1 , (20), it is enough to show that Pr{nL ≥ n∗ } > Pr{c1 = c} Pr{nL = n∗ − 1}. Indeed,
¸
¶
µ
∙
(N − n∗ )
1
N −1
∗
n∗
N−n∗ −1
∗
(64)
Pr{nL ≥ n } >
ρ (1 − ρ)
= if n = N
n∗ − 1
n∗
2
µ
¶
¶
µ
∗
∗
N −1
N −1
n∗
N−n∗ −1
=
ρ (1 − ρ)
ρn (1 − ρ)N−n =
>
(65)
∗
∗
n −1
n −1
(66)
= Pr{c1 = c} Pr{nL = n∗ − 1}.
Proof of proposition 9. Rewrite (20) in Proposition 7, we have
Pr{nL ≥ n∗ }m̄1 = −b − (a − b) Pr{A} + c̄ Pr{nL ≥ n∗ }
+c Pr{c1 = c} Pr{nL = n∗ − 1}.
(67)
Applying Proposition 7 to the small reform (sB to sX ), we have
Pr{nL ≥ n∗ }m̄α1 = −b − (x − b) Pr{X} + c̄(x) Pr{nL ≥ n∗ }
+c(x) Pr{c1 = c(x)} Pr{nL = n∗ − 1}
= −b − α(a − b) Pr{A} + αc̄ Pr{nL ≥ n∗ }
+αc Pr{c1 = c} Pr{nL = n∗ − 1} =
= αm̄1 − (1 − α) b.
(68)
(69)
noting that Pr{X} = Pr{A}.
∗
Let α∗ be the α such that m̄α1 = 0 in (68). As b > 0, α∗ > 0.
Lemma 14 Assume
the costs are distributed D independently, with a − c > b and a − c̄ < b.
∗c
,
so
that n∗ is preserved if either of the other parameters change. Then
Let c̄ = N(a−b)−n
N−n∗
∂ m̄1
∂a
∂ m̄1
∂c
> 0,
< 0.
28
∂ m̄1
< 0,
∂b
Proof.
N (a − b) − n∗ c
Pr{nL ≥ n∗ }
N − n∗
+c Pr{c1 = c} Pr{nL = n∗ − 1}.
Pr{nL ≥ n∗ }m̄1 = −a Pr{A} − b Pr{B} +
Pr{nL ≥ n∗ }
∂ m̄1
∂a
N
Pr{nL ≥ n∗ }
N − n∗
−[Pr{nL ≥ n∗ } + Pr{c1 = c} Pr{nL = n∗ − 1}]
µ
¶
n∗
+ 1+
Pr{nL ≥ n∗ }
N − n∗
¶
µ
∗
∗
N −1
ρn −1 (1 − ρ)N −n
−ρ
∗
n −1
∙µ
¶
¸
n∗
N −1
n∗
N−n∗ −1
ρ (1 − ρ)
+
+ ···
n∗
N − n∗
¶
µ
∗
∗
N −1
ρn (1 − ρ)N−n
−
∗
n −1
¶
µ
n∗ N − n∗
∗
∗
N −1
ρn (1 − ρ)N−n −1
+
n∗ − 1
N − n∗ n∗
0.
= − Pr{A} +
=
=
>
>
Similarly, we can prove that
Pr{nL ≥ n∗ }
∂ m̄1
n∗
=−
Pr{nL ≥ n∗ } + Pr{c1 = c} Pr{nL = n∗ − 1} < 0.
∂c
N − n∗
It is clear that
Pr{nL ≥ n∗ }
N
∂m1
= − Pr{B} −
Pr{nL ≥ n∗ } < 0.
∂b
N − n∗
Proof of proposition 13. If we replace a − c, a − c̄, c1 , nL , n∗ , τ (c) and τ (c) in the
proof of Proposition 7 by aH − c, aL − c, a1 , ñH , ñ∗ , τ (aH ) and τ (aH ) respectively, then
the proof goes through perfectly. From equation (62), we have
τ̄ (aH ) = −(aH − c − b) Pr{ñH = ñ∗ − 1}
+(c − aL − m̄) Pr{ñH ≥ ñ∗ } − b Pr{nH < ñ∗ }
29
By setting τ̄ (aL ) at its minimum level γ, we obtain the minimum (ex-ante) expected transfer,
E (τ̄ (·)) , as follows:
E (τ̄ (·)) = Pr{a1 = aH }τ̄ (aH ) + Pr{a1 = aL }τ̄ (aL )
= Pr{a1 = aH }[−(aH − c − b) Pr{ñH = ñ∗ − 1}
+(c − aL − m̄) Pr{ñH ≥ ñ∗ } − b Pr{nH < ñ∗ }]
+ Pr{a1 = aL }[(c − aL − m̄) Pr{ñH ≥ ñ∗ } − b Pr{ñH < ñ∗ }]
−aL Pr{ñH
= −m̄ Pr{ñH ≥ ñ∗ } − aH Pr{a1 = aH } Pr{ñH = ñ∗ − 1}
≥ ñ∗ } − b Pr{B} + c Pr{A}
To balance the budget ex ante, we require E (τ̄ (c)) ≤ 0. By taking the minimum of m̄, we
obtain
m̄1 =
C
−aH Pr{a1 = aH } Pr{ñH = ñ∗ − 1} − aL Pr{ñH ≥ ñ∗ } − b Pr{B} + c Pr{A}
Pr{ñH ≥ ñ∗ }
Proofs for the Continuous Case
Proof of proposition 10. Let the net payoff of player i, who follows the equilibrium
strategy in the coordination game and has a cost of transition ci be
v (I) − ci I + τ i ,
(70)
where
I is the indicator function for reform, unity (reform) or zero (status-quo);
τ i is i0 s transfer (provided he chooses the action suggested by the reformer);
equilibrium payoffs in the coordination game to follow;
v (1) = a;
v (0) = b.
Consider a social choice function φ (θ) = (I (θ) , τ 1 (θ) , .., τ N (θ)) , where I is either unity
(reform) or zero (status-quo). Let
τ̄ (θi ) ≡ Ec−i (τ (θi , c−i )) ,
I¯ (θi ) ≡ Ec−i (I (θi , c−i )) ,
v̄ (θi ) ≡ Ec−i (v (I (θi , c−i )))
30
(71)
(72)
(73)
be expected payment of i, probability of reform, and expected “equilibrium" payoff (in
coordination game next period), conditional on i reporting θi , and all the rest are telling
the truth. Note that
¡
¢
(74)
v̄ (θi ) = I¯ (θi ) a + 1 − I¯ (θi ) b.
Let
V (ci ) ≡ v̄ (ci ) − ci I¯ (ci ) + τ̄ (ci ) .
(75)
By a standard argument, Bayesian Incentive compatibility implies that I¯ (ci ) is non-increasing
(weakly decreasing). Moreover,
(76)
V 0 (ci ) = −I¯ (ci ) ,
which implies
V (ci ) = V (c) −
Z
ci
I¯ (s) ds.
(77)
c
Let us incorporate additional constraints.
Recall that by (46, 47) ,
a − ci + τ i (ci , c−i ) ≥ −m̄, if
b + τ i (ci , c−i ) ≥ 0, if
N
X
j=1
N
X
j=1
cj ≤ ∆ = N (a − b)
cj > ∆ = N (a − b)
(78)
(79)
This implies that the (soft) interim individual rationality constraint should be satisfied:
¡
¢
(80)
V (ci ) = I¯ (ci ) (a − ci + τ A (ci )) + 1 − I¯ (ci ) (b + EB τ (ci )) ≥ −m̄I¯ (ci ) ,
where, as in the discrete case
⎛
EA τ (ci ) ≡ Ec−i ⎝τ i (ci , c−i ) |
⎛
EB τ (ci ) ≡ Ec−i ⎝τ i (ci , c−i ) |
N
X
j=1
N
X
j=1
⎞
cj ≤ ∆⎠ ;
(81)
cj > ∆⎠ .
(82)
Combining (77) with (80), we get for all i
Z ci
I¯ (s) ds ≥ −m̄I¯ (ci ) ,
V (c) −
c
Z ci
I¯ (s) ds − m̄I¯ (ci ) ≤ V (c) .
c
31
⎞
(83)
(84)
When is this constraint binding?
As the left hand side in increasing in ci , it is enough to verify that the constraint holds
for the highest possible realization of cost, ci = c̄ :
Z c̄
(85)
I¯ (s) ds − m̄I¯ (c̄) ≤ V (c) .
c
Inequality (85) provides a lower bound on m :
¶
µZ c̄
1
I¯ (s) ds − V (c) ¯
= m̄1
I
(c̄)
c
(86)
By definition
V (c) = I¯ (c) (a − b − c) + b + τ̄ (c) .
(87)
It implies that
I¯ (c̄) m̄1 =
Z
=
Z
I¯ (s) ds − V (c)
(88)
I¯ (s) ds − I¯ (c) (a − b − c) − b − τ̄ (c)
(89)
c
c̄
c
c̄
= c̄I¯ (c̄) −
Z
c
c̄
∂ I¯ (s)
sds − I¯ (c) (a − b) − b − τ̄ (c)
∂s
(90)
We can express τ̄ (c) using the ex-ante budget balancedness,
N
X
i=1
Ei [τ̄ (ci )] ≤ 0
(91)
Recall that by (77) τ̄ (ci ) can be represented as follows
¡
¢
τ̄ (ci ) = τ̄ (c) + (a − b) I¯ (c) − I¯ (ci ) − cI¯ (c) + ci I¯ (ci ) −
Let x =
PN
i=1 ci
Note that
Z
ci
I¯ (s) ds.
(92)
c
be distributed Q on [c, c̄] , with Q derived from {Fi }N
i=1 . Then
I¯ (θi ) = Q (∆|ci = θi ) ,
¡
¢
Ei I¯ (θi ) = Q (∆) .
Z
c
cj
I¯ (s) ds = ci I¯ (ci ) − cI¯ (c) −
32
Z
c
ci
∂ I¯ (s)
sds,
∂s
(93)
(94)
(95)
and
Ei
Z
ci
c
¸
∂ I¯ (s)
sds f (ci ) d (ci ) =
∂s
c
c
Z c̄
∂ I¯ (s)
=
sds
[1 − F (s)]
∂s
c
∂ I¯ (s)
sds =
∂s
Z c̄ ∙Z
ci
Combining, the three observations above with (92) , we get
Z c̄
¡
¢
∂ I¯ (s)
Ei τ̄ (ci ) = τ̄ (c) + (a − b) (I¯ (c) − Q (∆) +
sds.
[1 − F (s)]
∂s
c
(96)
(97)
(98)
Therefore, (91) implies
τ (c) ≤ −
Z
c̄
c
[1 − F (s)]
¡
¢
∂ I¯ (s)
sds − (a − b) (I¯ (c) − Q (∆)
∂s
(99)
Substituting into (88) , we get
Z
∂ I¯ (s)
sds − I¯ (c) (a − b) − b +
∂s
c
Z c̄
¡
¢
∂ I¯ (s)
+
sds + (a − b) (I¯ (c) − Q (∆)
[1 − F (s)]
∂s
c
Z c̄ ¯
∂ I (s)
¯
sF (s) ds
= c̄I (c̄) − b − (a − b) Q (∆) −
∂s
c
I¯ (c̄) m̄1 = c̄I¯ (c̄) −
Substituting
Z
c
c̄
c̄
∂ I¯ (s)
sF (s) ds = c̄I (c̄) −
∂s
I (c̄) m̄1 =
Z
c
c̄
Z
c̄
[F (s) + sf (s)] I¯ (s) ds,
¸
Z c̄
F (s)
¯
+ s I (s) f (s) ds − (a − b)
=
I¯ (s) f (s) ds − b
f (s)
c
c
¸
Z c̄ ∙
F (s)
+ s − a I¯ (s) f (s) ds − (1 − Q (∆)) b.
=
f (s)
c
33
(101)
(102)
(103)
c
[F (s) + sf (s)] I¯ (s) ds − b − (a − b) Q (∆)
Z c̄ ∙
(100)
(104)
(105)
(106)
It is left to check that feasibility constraints (RC) are satisfied. Recall that
Z ci
¡
¢
¯
¯
¯
¯
τ̄ i (ci ) = τ̄ i (c) + (a − b) I (c) − I (ci ) − cI (c) + ci I (ci ) −
I¯ (s) ds =
c
¡
¢
= τ̄ i (c) + (a − b) I¯ (c) − I¯ (ci ) − cI¯ (c) + ci I¯ (ci )
¶
µ
Z ci ¯
∂ I (s)
¯
¯
sds
− ci I (ci ) − cI (c) −
∂s
c
Z ci ¯
¡
¢
∂ I (s)
¯
¯
= τ̄ i (c) + (a − b) I (c) − I (ci ) +
sds
∂s
c
(107)
(108)
(109)
(110)
which implies that the (interim) payoff schedule is quasiconcave in ci ∈ [c, c̄], as
τ̄ 0i (ci ) = −I¯0 (ci ) (a − b − ci ) =
= g (∆ − ci ) (a − b − ci )
(111)
≥ 0, if ci < a − b
≤ 0, otherwise.
(112)
Therefore, it is sufficient to verify the constraint (RC) for ci = c̄ and ci = c.
Let us start with the former. Recall that the incentive constraint a−ci +EA τ (ci ) ≥ −m1
is satisfied as equality only for ci = c̄.
It implies that
I (c̄) (a − c̄ + EA τ (c̄)) = −
∙Z
c
c̄
¸
¯
[F (s) + sf (s)] I (s) ds − b − (a − b) Q (∆)
(113)
I (c̄) EA τ (c̄) = c̄ − a − M,
Z c̄
[F (s) + sf (s)] I¯ (s) ds − aQ (∆) − (1 − Q (∆)) b.
M ≡
c
But M ≤ c̄ − a. (This
R c̄ can be shown by employing the argument from proposition (12)
demonstrating that c [F (s) + sf (s)] I¯ (s) ds increases with the first order stochastic dominance, and then showing (as in footnote (12) that under a Rlimiting (Dirac) distribution
c̄
with all the mass on the highest realization c̄ the expression c [F (s) + sf (s) − a] I¯ (s) ds
becomes c̄ − a, which completes the argument, as
Z c̄
Z c̄
¯
[F (s) + sf (s) − a] I (s) ds =
[F (s) + sf (s)] I¯ (s) ds − aQ (∆) .)
(114)
M≤
c
c
So EA τ (c̄) ≥ 0 > −a, as I (c̄) ≥ 0, the former being in compliance with (RC) .
It is then left to verify that EA τ (c) ≥ −a. Note that it has to be the case that the net
payoff of the lowest cost type is strictly positive in case of reform, i.e., a − c + τ A (c) > 0.
34
Indeed, if it is not the case, then, provided this type gets the highest interim utility (which
also implies in this case highest interim utility conditional on reform, as it is always feasible
to set the tax in case of no reform, EB τ (ci ) , to be −b), a non-positive payoff a − c + EA τ (c)
for the lowest cost type will imply that everybody else gets negative payoff from the reform,
contradicting it being worthwhile in the first place (in the view of balanced budget). But if
a − c + EA τ (c) > 0, then, clearly, (RC) constraint, EA τ (c) ≥ −a, is satisfied.
Proof of proposition 12.
excluding i :
Let G be the distribution of the sum of N − 1 valuations
G (y) = Pr
⎧
⎨X
⎩
j6=i
θj ≤ y
⎫
⎬
⎭
.
(115)
Then I¯ (s) = G (∆ − s) . Let GF be the cumulative distribution of the sum of N − 1
independent random variables (costs), where each variable is distributed F.Note that first
stochastic dominance order is closed under convolution by theorem 1.A.3 in Shaked and
Shanthikumar (1994) (similar is true for the peakedness order by theorem 2.C.3 in the same
book. ). This implies that if H FOSD F then GH FOSD GF .
Integrating by parts the first expression in (104) , or recalling (102) it is possible to show
that
R c̄
b + (a − b) Q (∆)
c g (∆ − s) sF (s) ds
+
(116)
m̄1 = c̄ −
G (∆ − c̄)
G (∆ − c̄)
Q(∆)
Note that G(∆−c̄)
is probability of the sum of the valuations to be below ∆ conditional on
the sum of the rest N − 1 variables is below ∆ − c̄. No doubt, it is unity.
Note that G (∆ − c̄) is non-increasing under both transformations.18 Therefore, for both
statements we can concentrate on the following part of the punishment bound,
Z c̄
gF (∆ − s) sF (s) ds,
(117)
mF ≡
c
having to show that
mH ≥ mF .
(118)
Assume H FOSD F (so that H describes a more expensive transition than F ), so
18
It is an implication of (1) closedness of both orders under convolution and (2) the definition of the
corresponding order in each case.
35
F (t) ≥ H (t) , implying the first inequality below,
Z c̄
Z c̄
H (t) gH (∆ − t) tdt = −
H (t) tdGH (∆ − t) ≥
mH =
c
≥
Z
(119)
t=c
c̄
t=c
(−F (t) t) dGH (∆ − t) ≥
Z
c̄
t=c
(−F (t) t) dGF (∆ − t) = mF ,
(120)
while the second inequality is due to closedness of this stochastic order under convolutions,
which implies that GH puts more weight on values with lower t and −F (t) t decreases in t.
Now let us prove the second part of the proposition. Assume that F is more peaked
than H, so that H is “more spread" than F.
Let us define a random variable U = |X − d| . Then if X is distributed with a distribution
F on [c, c̄] symmetric around d = 12 (c + c̄) , U is distributed ZF (u) = 1 − 2F (d − u) ,
u = (d − x) ∈ [0, d − c] with the corresponding marginal zF (u) = 2f (d − u) .
By symmetry, letting
δ ≡ ∆ − d,
u ≡ |t − d| ,
(121)
(122)
we have
mF
=
Z
k
0
+
Z
(d − u) F (d − u) gF (δ + u) du +
0
χF
ωF
(123)
k
(d + u) (1 − F (d − u)) gF (δ − u) du
= χF + ωF ,
Z k
≡
dgF (δ − u) du
0
Z k
≡
u (1 − 2F (d − u)) gF (δ − u) du,
(124)
(125)
(126)
0
where k ≡ d − c =
1
4
(c + c̄) = c̄ − d.
As F is more peaked than H, ZH FOSD ZF , and the order “more peaked" is closed
under convolution for unimodal (symmetric) distributions with the same mean,19 we have
Z k
Z k
χF = 2
dgZF (u) du ≤ 2
dgZH (u) du = χH ,
(127)
0
19
0
see Shaked and Shanthikumar (1994).
36
where gZF (u) is the marginal distribution of N −1 independent random variables distributed
ZF , as before.
As for the second term,
ωF = 2
Z
0
k
uZF (u) gZF (u) du ≤ 2
Z
0
k
uZH (u) gZH (u) du = ω H ,
(128)
where the inequality is due to the first part of the proposition.
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