Laboratory federalism: Policy diﬀusion and
yardstick competition
Simon Schnyder ∗†
May 24, 2011
Abstract
1 Introduction
The concept of ”laboratory federalism”, coined by Oates (1999), states that
federations and more generally politically decentralized countries may beneﬁt
from better policies than centralized countries, thanks to a greater eﬃciency
in identifying the best policies. Indeed, multiple small scale experimentation
may foster the identiﬁcation of the best policies, and reduce the aggregate
risk of experimentation. The basic intuition behind this result is that the
implementation of a given policy in a region generate an informational externality
which allows the other regions to learn about the quality and/or
adequacy of this policy. This process of learning is likely to generate better
outcomes than if policy making were made by a single decision maker for all
regions. In this paper I propose to explore another possible driver of policy
learning in a federation. Indeed, I show that yardstick competition is an important
source of policy diﬀusion. However, it may also have perverse eﬀects,
∗
Department of Economics, University of Fribourg, 1700 Fribourg (Switzerland), si-
mon.schnyder@unifr.ch
†
University fo Rennes 1 and CREM-CNRS (France)
1
such as policy pandering or populism.
Formal models studying the link between federalism and policy innovation
are presented in Rose-Ackerman (1980), Kollman et al. (2000) and Strumpf
(2002), who study the experimentation of new policies in decentralized vs
centralized countries. They do not consider rational model of elections. More
recent work introduce some politics in the analysis, like Cai and Treisman
(2009) and Kotsogiannis and Schwager (2006a). These diﬀerent papers concentrate
on the quantity of experimentation in a federation compared to a
centralized state. The basic intuition is that there is more experimentation in
a decentralized system because each lower state is doing some experimentation.
But some problems may arise, the most important being the incentive
to free-ride on the neighbours’ experimentation as a way to reduce the risk
and costs of own experimentation. As a consequence we may observe in fact
less experimentation in a federation than in a centralized state. This eﬀect
can be conter-balanced if the most innovative local politicians are promoted
to the national level: this gives an inventive to experiment as a way to signal
his ability to innovate.
However one may ask if the matter of the quantity of experimentation
is really the whole story behind the concept of laboratory federalism. In
this paper the focus is on the quality of the experimentation. Indeed, I
think that one main idea behind the concept of laboratory federalism is
that a decentralized system allows to learn more eﬃciently about the best
policies, thanks to a larger sample of small scale experimentation and a larger
variety of the politicians’ ability, than in centralized states. Hence in the
following I do not modelize the process experimentation as the choice of a
new, untried policy over a well-known policy, but as the choice of a policy
in a set risky policies. Only the experience from former (domestic and in
the neighbourhood) choices and expertise may guide the choice among the
available policies.
Another feature of the current models of laboratory federalism is that
politicians are supposed benevolents. Indeed, politicians are supposed to
diﬀer in their ability, but have the same objectives as the voters. However,
2
conﬂicts of interest between the voters and their representants are usually
a reality, and may play a role in the way good policies are uncovered. I
then suppose that, although all politicians learn and experiment about. This
illustrates the idea that a more precise knowledge about the quality and/or
the eﬃciency of diﬀerent policies may also being used agains the voters’
interests.
2 The model
I ﬁrst expose the basics and framework of the model. I then solve the game,
for a purpose of exposition, in the case of a single jurisdiction.
2.1 Basic setting
I ﬁrst present the basic framework of the model, which I borrow to Maskin
and Tirole (2004). Consider a two-periods game (t ∈ {1, 2}) between a representative
voter and a politician. In each period, a policy Pt ∈ {a, b} has to
be implemented. One of the two policies is the best an gives a higher utility
to the voter. Lets call this best policy x ∈ {a, b}. x is determined by Nature
and is the same over the two periods and is unobservable. At the beginning
of the ﬁrst period the voter has no a priori information on which policy is
the best: Prob (x = a) = γ = 1/2.
Given a policy choice P, the utility of the voter is u (P, x) = 1+ε if P = x
and u (P, x) = ε if P = x. ε is a random shock with mean zero, pdf = f (.)
and cdf = F (.) 1
.
In each period, the choice and implementation of the policy is delegated
to a politician, who has a more precise information on x; at the begining of
the ﬁrst period the politician receives a signal s on x. The signal has the
following conditional distibution:
1
we suppose that f (.) satisﬁes the MLRP
3
state ↓ signal → s=a s=b
x=a q 1-q
x=b 1-q q
with q > 1/2 being the probability for the politician to get a signal
s = a (s = b) if x = a (x = b) (hence s = a (s = b) is “good news” regarding
state a (b)).
A politician can be of two types: either congruent or dissonent2
. The
ex ante probability that a politician is dissonent is π. π can also be seen as
the share of dissonent politicians in the population of candidate politicians.
A congruent politician share the same preferences as the voter: his prefered
policy is P = x. A dissonent politician a opposed preferences: his prefered
policy is P = x.
In addition to the implementation of his prefered policy, a politician also
derives utility from the exercice of power (this can be seen as an ego-rent).
The utility function of a politician is then V = G + E, where G is the utility
provided by the implementation of his prefered politicy and E is the utility
provided from the exercice of power. For a discount factor β, I deﬁne
κ =
G
β (G + R)
the degree of patience of the politicians3
. Hence a high κ (κ > 1) means that
politicians are impatient: they give more value to the current period than
to the future. A lower level of κ (κ < 1) means that politicians are patient.
I suppose hovewer that there is always a small share ξ of totally impatient
politians.
Timing of the game
The sequence of events goes the following
1. Nature chooses a type of politician and the best policy x ∈ {a, b}
2
I borrow this nomenclature to Besley (2006)
3
κ is the same for both types of politicians
4
2. The incumbent gets a signal s and chooses a policy P
3. The shock ε is realized
4. The voter observes u (P1, x) and P1
5. Given u, P and π the voter chooses whether to reelect the incumbent
or the pick a challenger (congruent with probability π)
6. The second period’s incumbent chooses a policy P
7. The voter gets utility u (P2, x) and the game ends.
2.2 Policy making and elections in a single jurisdiction
In this subsection I focuse on the case of a single jurisdiction. I ﬁrst describe
the strategies of the representative voter and the politicians and then characterize
the diﬀerent equilibria of the game.
The simplest way to solve this two-periods game is to reason backwards.
In the last period, since there is no more electoral incentives, the incumbent
simply chooses his favorite policy. The nature of x is indirectly observable
from the ﬁrst period policy choice and outcome; then a second period politician
gets utility G + E. 4
. The expected second period utility of the voter
is then a function of the reputation of the second term incumbent. If this
reputation is π2, the voter gets the expected utility π2 for the second period 5
In the ﬁrst period, a politician has to take into account the impact of his
policy choice on his probability of reelection. Lets write σI (P, µ1, s) = σI (P)
the probability of reelection of the incumbent conditionally on his policy
choice P, the public belief µ1 about x in t = 1 and the private signal s. The
objective of an incumbent in the ﬁrst period is then to maximize
4
a reelected politician knows x; a newly appointed politician can learn once in oﬃce
the policy choice and outcome of his predecessor
5
Recall that the voter has the expected utility u = 1 if P = x, which happens with
probability π2, and zero otherwise
5
Max
P
G + R + σI (P, µ1, s) β (G + R) (1)
To make an informed choice on P, the incumbent form a private belief
about x using the public belief µ1 and his private signal s. Since there is no
a priori information about the good policy (the public belief µ1 = 1/2) the
private belief depends only on the signal: ˜µ (s). Since
Pr (x = s | s) = q (2)
Pr (x = s | s) = 1 − q (3)
˜µ (s = a) = q > 1/2 and ˜µ (s = b) = 1 − q < 1/2.
The favorite policy of a congruent politician is then P = s. It is optimal
for a congruent incumbent to implement is favorite policy iif:
q G + E + σI(P=s)β (G + E) + (1 − q) E + σI(P=s)β (G + E) ≥
q E + σI(P=s)β (G + E) + (1 − q) G + R + σI(P=s)β (G + R) (4)
or iif
κ ≥
σI(P=s) − σI(P=s)
2q − 1
(5)
Similary, for a dissonent politician, playing his favorite policy P = s is
optimal iif:
q G + E + σI(P=s)β (G + E) + (1 − q) E + σI(P=s)β (G + E) ≥
q E + σI(P=s)β (G + E) + (1 − q) G + R + σI(P=s)β (G + R) (6)
or iif
κ ≥
σI(P=s) − σI(P=s)
2q − 1
(7)
6
The only decision the voter has to make in the game is to choose at
the end of the ﬁrst period whether to reelect the incumbent or to pick a
new, “untried”, politician. Since, as seen above, the voter’s expected second
period utility is a function of the second period incumbent’s type, the relevant
decision variable for the voter is the reputation of the incumbent compared to
the reputation of a challenger. The voter needs then to update the reputation
of the current incumbent given the observable variables at disposal, namely
the policy implemented and the utility derived from this policy.
Suppose that a congruent politician always chooses the policy the most
likely to be the best by implementing P = s and that the probability that
a dissonent politician decides to be behave in the interest of the voter by
implementing P = s is λ. Being rational, the voter update the reputation of
the incumbent using the Bayes rule and formulate the incumbent’s ex post
reputation Π. Hence for P = a and u, the likelihood ratio π
1−Π
can be written
as:
LR( Π
1−π |P =a,u) =
π qµf(u−1) + (1−q) (1−µ) f(u)
(1−π) (1−ξ) λ qµf(u−1) + (1−q) (1−µ) f(u) + ((1−ξ) (1−λ) + ξ) qµf(u) + (1−q) (1−µ) f(u−1)
(8)
The similar argument apply when P = b. The condition for the re-election
is obvious and is given by Π ≥ π: it is optimal to re-elect the incumbent is
at least as good as the reputation of a challenger, since the voter’s expected
second period is precisly a function of this reputation. Taking into account
that the public belief is uninformed: µ = 1/2, the condition for the reelection
can be stated as (both for P = a and P = b):
LR( Π
1−Π ) ≥
π
1 − π
(9)
Since f (.) satisﬁes the MLRP, it is easy to see that 9 is monotonically
decreasing in u. There is then a unique level of utility ¯u such that the
condition (9 is binding. Since the distribution function f is symetric, it is
easy to see that ¯u = 1/2. Then u ≥ 1/2 ⇒ Π (u) ≥ π and the incumbent is
7
reelected. u < ¯u ⇒ Π (u) < π and the incumbent is not reelected.
Given the parameters and the strategies described above, there is two
possible equilibria of the game:
Result 1
• A pooling equilibrium λ = 1, in which all incumbent choose the
policy prefered by the voter (P = s), except a small share ξ of politicians
who always pick their own prefered policy, exists if κ ≤ F (1/2) −
F (−1/2):
• A separating equilibrium (λ = 0) in which incumbent always play
their prefered strategy in the ﬁrst period. This equilibrium exists iif
κ > F (1/2) − F (−1/2). (the demonstration for a dissonent politician
is similar to the demonstration for a dissonent politician in the pooling
equilibrium)
3 Policy making in a federation
The litterature on yardstick competition shows that when the voters in one
jurisdiction can observe the policy making in the neighbouring jurisdictions,
they can partly soften the agency problem they face in their relationship
with their representatives (see Besley and Case (1995). The notion of “neighbourhood”
is not to understand only litteraly as a measure of geographical
proximity. What makes yardstick competition possible is in fact: 1) the
proximity in terms of political and economic institutions, and 2) a certain
level of correlation in the random shocks and unobservable variables aﬀecting
the economy and the outcome of the economic policies. Hence by observing
two or more signals, instead of only one, on some unobservable variable,
say the performance of an incumbent politician, a voter can increases the
amount and the precision of the information at his disposal. More information
means in turn a less severe agency problem between the principal-voter
8
and the agent-politician.
To study the impact of the yardstick competition on the political economic
outcomes, the litterature typically modelize a game with two jurisdictions
in which incumbents simultaneously make policy decisions. Representative
voters can then oberve policies and outcomes in both jurisdictions.
Given a correlation and similarity between the two jurisdictions, voters in
both region make comparisions of policies and outcomes to infer the level of
competences/eﬀort of their own incumbent, and make a re-election decision.
In the following I propose to study, in a related framework, the role of the
yardstick competition on the diﬀusion of policies across jurisdicions6
. The
litterature cited above focuses on the yardstick competition as an alternative
or complement to the mobility of (voice vs exit, to use the conceptual
framework of Hirschman (1970)) to constraint the Leviatan (Brennan and
Buchanan (1978)). Decentralization introduce generates mobility/yardstick
competition and provide incentives to representatives to work in voters in-
terests.
Another role of decentralisation is to encourage experimentation and generate
learning about new policies or which policies are the right policies. This
is the idea of “laboratory federalism” proposed by Oates (1999) (see RoseAckerman
(1980), Strumpf (2002), LeBorgne and Lockwood (2006),Kotsogiannis
and Schwager (2006b), Kotsogiannis and Schwager (2006a)).7
3.1 Policy making and learning
I now extend the framework of the prededing section to condider a federation
8
composed of N similar jurisdictions. The same sequence of event happens
in both region, but not simultaneously, unlike in the typical models presented
6
Across jurisdicions in a federation, but also across countries in a union like the EU for
instance
7
si moins d’experimentation dans les papiers cites dans le cas de la decentr. peutetre
parce que decisions simultanees; merite peut-etre une comparaison: partie normative:
centre vs decentre ?
8
deﬁnir, elargir
9
xi = a xi = b
xj = a ρ
2
1−ρ
2
xj = b 1−ρ
2
ρ
2
Figure 1: Correlation across regions
in the litterature on yardstick competition cited above. Indeed, I suppose
that decisions (policy choice, re-election) are ﬁrst made in one jurisdiction,
then in the second jurisdiction, and so on. Hence voters in the jurisdiction
i can observe the policy choices and outcomes in other jurisdictions before
making their voting decision. I consider that the order of regions in the policy
making is exogeneous.
The modelize the dependance across regions I suppose that the best policies
x are correlated between two jurisdictions i, j, where i states for the ith
region to implement a policy, and j states for the next region to implement
a policy, according to the joint distribution in table (3.1), where ρ > 1/2,
such that there is a positive correlation between the best politicies.
The incumbent and the voter in the region i make their decisions according
to the sequence described in the preceding section for a single jurisdicion:
the incumbent makes a policy choice Pi, a level of utility ui is provided to the
voter, who chooses whether to reelect or not the incumbent. This sequence
of events generate some information for the region j. This information is
incorporated in the public belief.
Evolution of the public belief
The public belief µ1
i incorporates all the public information about the best
policy xi in state i before any action is taken. In the preceding section
with a single jurisdiction this prior belief was uninformed and µ1
i was equal
to 1/2. This is not the case anymore. After the occurence of the policy
choice Pi and outcome ui, the public belief about xi is updated and
10
becomes µ2
i . This public belief µ2
i incorporates all the public information
about xi after the policy making and its outcome. Given the structure
of the game, It is equivalent to formulate µ2
i as a function of µ1
i and the
policy choice and outcome: µ2
i (µ1
i , Pi, ui), or as a function of the complete
history of past policy choices and outcomes in all regions, incorporated in
Ii = {{P1, u1} , {P2, u2} , ..., {Pi−1, ui−1}} and then µ2
i (Ii, Pi, ui). As we will
see, this equivalence is due to the martingale property of the evolution of the
public believes µ across regions.
The updating of the public belief µi is a function of all public knowlege
and according to the Bayes rule:
µ2
i | P =a, u =
µ1
i Pr (P = x) f (u−1)
µ1
i Pr (P = x) f (u−1) + (1 − µ1
i ) Pr (P = x) f (u)
(10)
and µ2
i | P =a, u ≥ µ1
i ⇔ Pr(P=x)
Pr(P=x)
≥ f(u)
f(u−1)
µ2
i | P =b, u =
µ1
i Pr (P = x) f (u)
µ1
i Pr (P = x) f (u) + (1 − µ1
i ) Pr (P = x) f (u − 1)
(11)
and µ2
i | P =b, u ≥ µ1
i ⇔ Pr(P=x)
Pr(P=x)
≤ f(u)
f(u−1)
The policy choice and its outcome in the region i reveal some information
on the best policy xi in this region. Due to the correlation between the best
policies in two succeding regions i and j, the observable variables Pi and ui
also reveal some information about the best policy xj in the region j. This
is where the interjuridisctional learning potentialy takes place. Indeed, given
the updated belief in region i and the correlation between the good policies
across the states i and j, the public belief about the good policy is:
Pr (ai) = Pr (ai, aj) + Pr (ai, bj)
µ1
j = ρµ2
i + (1 − ρ) 1 − µ2
i (12)
11
Equilibrium with yardstick competition
To understand the impact of this learning/informational externality, one
must examine if and under which condition the two equilibria in result (1)
exist when the public belief may be informed.
Upon receiving a signal s, the incumbent forms his private belief ˜µ about
x:
˜µ = Pr (x = a | s) =
Pr (s | x = a) Pr (x = a)
Pr (s | x = a) Pr (x = a) + Pr (s | x = b) Pr (x = b)
(13)
The favorite policy of the incument is then given by the relative size of
the public belief about x and the precision of the private signal. Hence a
congruent incumbent prefers the policy annonced by his signal if
q > Pr (x¬s) (14)
while the opposite is true for a dissonent incumbent9
. Pr (x¬s) stands for
the public belief about the other policy than that announced by the signal.
Since the goal of the voter is unchanged, the re-election rule keeps the
same caracteristics: the voter re-elect the incumbent if and only if Π > π.
However, the public belief being now possibly informed, the utility threshold
below which the incumbent is not re-elected has now to be dependant on
the policy choosed. Hence I call ua and ub the thresholds for reelection for
P = a respectively P = b. The values of ua and ub are uniquely deﬁned by
the following conditions:
9
Pr (x = a | s) > 1/2 ⇔ q > 1 − µ1
12
f (ua)
f (ua − 1)
=
µ1
1 − µ1
(15)
f (ub)
f (ub − 1)
=
1 − µ1
µ1
(16)
Given the hypothesis that the distribution function f is singled peaked
and symetrical (which in turn allows us to use the MLRP), it is easy to see
that both ua and ub are uniquely deﬁned. Futhermore one can see that:
ub = 1 − ua
The next result allows us to understand the impact of the yardstick comparison
on the behavior of the dissonent type:
Result 2
Pooling equilibrium with multiple jurisdictions: a dissonant
incumbent pools on the voter’s favorite policy iif κ ≤ F (1/2)−F (−1/2)
and:
µ ∈ µ; ¯µ (17)
where 0 < µ < ¯µ < 1
•• Separating equilibrium with multiple jurisdictions: alternatively
a dissonant incumbent plays his own favorite policy if κ > F (1/2)−
F (−1/2) or
µ /∈ µ; ¯µ (18)
Thus, when the public belief about one of the policies is strong enough (that
is below µ or above ¯µ, a dissonant incumbent choose his favorite policy. The
result (2) basically says that a more precise information about the best policy
may reduce the disciplining incentive of election. This eﬀect is a common
result of the litterature on the yardstick competition and even more generally
of the principal-agent litterature. This decrease in the pooling by dissonent
13
incumbent is of course compensated by an increase in the quality of the selection
process: by choosing his favorite policy a dissonent incumbent is likey
to reveal his type and to loose the re-election, especially if the public belief
about one particular policy is strong.
3.2 Pandering
In the preceding result I made the assumption that congruent incumbents
would play the voter’s favorite policy, since such a policy also is their favorite,
and since they would maximize their likelihood of reelection by maximizing
the voter’s expected utility. However, as we shall see, some particular situations
may arise in which a congruent incumbent chooses a policy against the
voter’s preferences.
If the private belief ˜µ of a congruent incumbent goes the same way as
the public belief about which policy is to be implemented, then of course
this policy will be choosen. It could be however that the incumbent’s private
belief about the best policy diverges from the public belief, and designates
another policy as the most likely to be the best. Two cases may arise:
• Suppose that µ > 1/2 and that the incumbent receive a signal s = b;
if µ < q, then ˜µ < 1/2.
• Suppose that µ < 1/2 and that the incumbent receive a signal s = a;
if µ > 1 − q, then ˜µ > 1/2.
Then, the situation ˜µ < 1/2 < µ may arise if the incumbent gets a signal
s = b and µ < q; the situation ˜µ > 1/2 > µ may arise if the incumbent gets
a signal s = a and µ > 1 − q. In such cases the question for the congruent
incumbent is therefore whether to choose the policy the most likely to be the
best as pointed by his private belief ˜µ, or whether to implement the policy
the most popular, as pointed by the public belief µ. Indeed, implementing a
policy the most likely to be the best but going against the public belief may
be costly in terms of re-election likelihood. Implementing the best policy
14
(that is following the private belief ˜µ) is rational for a congruent politician
as long as:
κ >
˜µ [F (ua − 1) − F (1 − ua)] + (1 − ˜µ) [F (ua) − F (−ua)]
2˜µ − 1
(19)
Lets take the case where µ > 1/2 > ˜µ. The condition (19) can be written
as:
κ >
(1 − q) µ [F (ua − 1) − F (1 − ua)] + q (1 − µ) [F (ua) − F (−ua)]
µ − q
(20)
When the public belief µ is close to 1/2, the condition (20) is respected.
However, as shown in the appendix, there is a threshold ˆµ such that when
q > µ > ˆµ then the RHS of equation (20) is positive. The same reasonning
applies to the case where µ < 1/2 < ˜µ: there is a threshold ˇµ such that
when ˇµ > µ > 1 − q the RHS of condition (19) is positive. As stated in the
following proposition, there is a parameters set for which the condition (19)
is not satisﬁed. In such cases, the behaviour of the congruent incumbent is
guided by the public belief.
Proposition 1 Pandering equilibrium
When the variance of the distribution function F (.) is high enough relative to
the level of patience κ, a congruent incumbent may pander to voter by choosing
the popular policy with no consideration for his private information about
the best policy. This happens whenever µ ∈ ]q; µp[ or µ ∈ ]1 − q; 1 − µp[,
where µp is a threshold for pandering, whose existence is deﬁned in the ap-
pendix.
The consequence of Proposition (1) is that, in some circonstances, when
the voter has a strong belief about a given policy, even a congruent politician
may have the incentive to follow the public belief and to neglect his own
private information. Of course this outcome is costly in terms of utility. But
another important consequence arise if we consider again the mechanism
15
of interjuridictional learning. Indeed, by pandering to the public belief, a
politician slows the process of learning, since he do not use his private signal.
As long as the shock ε is not too high, the public belief in the next jurisdiction
is unlikely to be largely updated and the wrong policy is more likely to be
implemented.
16
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17
Rose-Ackerman, S. (1980). Risk taking and reelection: does federalism promote
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18
Proof
All proof will be provided upon request.
Given µ2 and s = b, the dissonent type chooses P = b if µ2 < q, that is if
µp
< 1/2, and if
(1 − µp
) [κ + Pr (ε ≥ −¯u)]+µp
Pr (ε ≥ 1 − ¯u) ≥ µp
[κ + Pr (ε ≥ ¯u − 1)]+(1 − µp
) Pr (ε ≥ ¯u)
(1 − 2µp
) κ+(1 − µp
) [Pr (ε ≥ −¯u) − Pr (ε ≥ ¯u)]+µp
[Pr (ε ≥ 1 − ¯u) − Pr (ε ≥ ¯u − 1)] ≥ 0
(1 − 2µp
) κ + (1 − µp
) (2Pr (ε < ¯u) − 1) + µp
(2Pr (ε < ¯u − 1) − 1) ≥ 0
⇒ κ ≥ 1 −
2 ((1 − µp
) Pr (ε < ¯u) + µp
Pr (ε < ¯u − 1))
(1 − 2µp)
.
Trouble when: (example for s = b) µp
< 1/2 < µ < q: in that case,
the dissonent type would prefere P = b. He plays P = b, thus maximizing
voter’s welfare, if
• µp
< 1/2 → µ < q, and
•
(1 − 2µp
) κ ≥ 1 − (1 − µp
) 2Pr (ε < ¯u) − µp
2Pr (ε < ¯u − 1) (cond1)
The LHS of cond1 is monotonically decreasing in µ: 2∂µp
∂µ
< 0, whereas the
RHS is monotonically increasing: 2∂µp
∂µ
(F (¯u) − F (¯u − 1))−2∂¯u
∂µ
((1 − µp
) f (¯u) + µp
f (¯u − 1)) >
0
When µ → q: (1 − 2µp
) κ → 0, and 1−(1 − µp
) 2Pr (ε < ¯u)−µp
2Pr (ε < ¯u − 1) →
1 − F (u (q)) − −F (u (q) − 1), which is positive for q > 1/2.
Then
• For µ = q cond1 is not fullﬁlled.
• cond1 is monotonic in µ
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• There is then a unique ˜µ such that cond1 is binding
• There is an interval [˜µ, q] such that cond1 is not fulﬁlled
Proof pooling equ: Given Prob (x = a) = 1/2 and q, Prob (x = i | s = i) =
q, for i ∈ {a, b}. Then given the reelection rule ¯u, it is optimal for a dissonent
politician to choose politicy P = s in the ﬁrst period only if
q [R + Prob (1 + ε ≥ 1/2) β (G + R)]+(1 − q) [G + R + Prob (ε ≥ 1/2) β (G + R)] ≥
q [G + R + Prob (ε ≥ 1/2) β (G + R)]+(1 − q) [R + Prob (1 + ε ≥ 1/2) β (G + R)]
(21)
Since q > 1/2 and given the cdf F (.) the condition can be writen as
(1 − F (−1/2)) β (G + R) ≥ G + (1 − F (1/2)) β (G + R) (22)
or, given the deﬁnition of κ
F (1/2) − F (−1/2) ≥ κ (23)
Proof sep equ
It is easy to see that it is always optimal for a congruent incumbent to
implement his (and voter’s) prefered strategy. Indeed, choosing P = s would
make sense for a congruent politician only if
q [G + R + (1 − F (¯u − 1)) β (G + R)]+(1 − q) [R + (1 − F (¯u)) β (G + R)] <
q [R + (1 − F (¯u)) β (G + R)]+(1 − q) [G + R + (1 − F (¯u − 1)) β (G + R)]
which would be the case if κ < F (¯u − 1) − F (¯u) < 0. Indeed this case
can be excluded from the deﬁnition of κ.
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