Soc Choice Welf (2014) 42:477–502
DOI 10.1007/s00355-013-0737-z
ORIGINAL PAPER
Expert advising under checks and balances
Tao Li
Received: 7 September 2010 / Accepted: 29 March 2013 / Published online: 7 May 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract This paper attempts to compare the efficiencies of different political institutions (with or without checks and balances) in extracting external “cheap talk” information. We find that a political institution with checks and balances can extract more
credible information (measured by the number of signals) from an external partisan
expert because no policymaker can unilaterally exploit revealed expert information
to her own best advantage. However, there is a tradeoff between signal quantity and
distribution under checks and balances, as more signals tend to be distributed more
unevenly due to the existence of an “inertia region” where credible communication
is more difficult. Institutions using checks and balances are more efficient when conflict of interest between policymakers is relatively small, in which case the signal
distribution is less uneven.
1 Introduction
It is conventional wisdom that political institutions with more veto players have more
policy stability. In his extensive review of the topic, Tsebelis (2002) argues that “significant departures from the status quo are impossible when the winset (i.e. the set of
outcomes that can replace the status quo) is small, that is, when veto players are many,
(and) when they have significant ideological distances among them.” Policy stability
may not be desirable when political reality calls for change and adaptation. Dictators
Electronic supplementary material The online version of this article
(doi:10.1007/s00355-013-0737-z) contains supplementary material, which is available to authorized users.
T. Li (B)
Department of Government and Public Administration, Faculty of Social Sciences
and Humanities, University of Macau, Av. Padre Tomas Pereira, Taipa, Macau, China
e-mail: [email protected]
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often justify their rule based upon this idea. For example, Deng Xiaoping, the supreme
leader who ruled China after Mao’s death, repeatedly argued that China had an edge in
having a political system without checks and balances and saw absolute concentration
of power as highly efficient and advantageous (Deng 1993).
However, the ability to adjust policy based on real circumstances hinges not only
on policymakers’ power, but also on their capacity to gather information about the real
world. If checks and balances can enhance the information-gathering capacity of the
government, the conventional line of thinking about checks and balances would appear
to be flawed. There is indeed some evidence that democracies with checks and balances
are not at an informational disadvantage at all. For example, Jones et al. (2009) argue
that abrupt changes “in response to incoming information” are quite regular in western
democracies. However, the theory of how the same information could be processed
differently in different political institutions (with or without checks and balances) is
not clear.
This paper aims to evaluate the informational efficiencies of different political institutions. In particular, we focus on strategic information acquired externally through
lobbying or consultation. In our increasingly specialized modern society, policymakers usually rely upon third-party experts to provide policy-relevant information. For
example, lobbyists routinely provide information to lawmakers in America. Consulting non-government experts is also a common policy-making practice in China. In
both cases, the external experts are hardly disinterested. Expert consultation in this
situation is a strategic process, which is modeled here as a “cheap-talk” game.
Formally, we consider a stylized model in which an informed third party, for simplicity called “expert,” has some expertise that bears on a policymaking decision. The
government, a two-principal organization, is structured according to one of the following two authority allocation schemes: (1) concentrating authority in a single hand
(dictatorship); and (2) dividing agenda-setting authority and veto authority between
two different principals (checks and balances).
After characterizing the equilibria under different authority allocation schemes and
comparing their informational efficiencies, we find that checks and balances always
enjoy a communication advantage measured by the number of credible signals. The
key insight is that the expert is willing to provide more precise information because
she knows that neither policymaker can unilaterally exploit her advice in equilibrium.
However, there is a tradeoff between signal quantity and distribution under checks
and balances. The existence of the status quo policy induces an “inertia region,” where
credible communication is more difficult as the principals do not trust minor policy
adjustment advice from a biased expert. So more signals tend to improve communication efficiency outside of the inertia region. This causes revealed information to be
distributed more unevenly, which tends to reduce overall communication efficiency.
Checks and balances are found to have higher communication efficiency than authority concentration when conflict of interest between policymakers is small relative to
the bias of the expert, in which case the signals are few but distributed more evenly.
When the number of signals increases as the agenda-setting policymaker’s preference approaches that of the expert, the uneven signal distribution becomes a more
serious problem, and checks and balances lose informational advantage to authority
concentration.
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Our findings suggest that checks and balances are not necessarily at a disadvantage
in collecting information needed for policy adaptation. As a result, political institutions
using checks and balances could adapt to a changing environment more effectively
than a dictatorship. Our paper also makes a useful technical contribution to the cheaptalk literature, which only talks about the quantity of credible signals but has not yet
paid any attention to its distributive quality.
The advantage of checks and balances revealed in our paper is relatively significant
when the preference divergence between the principles is relatively small. This moves
away from the conventionally held view that large conflicts of interest are crucial for the
success of a system of checks and balances. For example, in the context of discussions
about bicameralism, Madison argued that “the improbability of sinister combinations
will be in proportion to the dissimilarity of the two bodies” (The Federalist Papers,
No. 62). Our conclusion fundamentally differs from the traditional view because we
are exploring a different mechanism from what was focused on in the past.
It is also important to recognize that adding a veto player into a particular decisionmaking body not only brings about a new principal with potentially different policy
objectives, but also fundamentally changes the decision-making rules. The latter point
has not received enough attention when we discuss checks and balances in the literature. Because the veto player can now only choose between the new policy and the
status quo policy, the status quo policy becomes an integral part of the institutional
commitment created by systems using checks and balances. This arrangement plays a
central role in our strategic information transmission process. In contrast, when a single player dictates the decision, she cannot use the status quo position as a commitment
device to assure the expert.
The plan of the paper is as follows. Section 2 discusses the related literature. Section 3 describes the model. Section 4 characterizes the equilibrium under different
authority structures. Section 5 compares informational efficiencies of these equilibria
and discusses the intuitions behind our main result. Finally, section 6 concludes by
discussing some implications of the paper.
2 Related literature
This article follows the tradition of the so-called informational theories of the Congress
(for a concise summary see Grossman and Helpman 2001). Even though public policies
are seldom set by a single hand, the existing literature almost exclusively assumes a
unitary policymaker (e.g. the median voter in a legislature), ignoring the possibility
that lobbying can be affected by internal decision-making process within a legislature.
To our best knowledge, only a few papers have analyzed the strategic information
transmission process between the lobbyist, committee, and the floor under a unified
framework. Austen-Smith (1993) is the first to evaluate the effectiveness of informational lobbying during the committee agenda-setting stage as well as the floor voting
stage. He finds that lobbying is mainly influential at the committee agenda-setting
stage. In his paper policymakers are uncertain not only of the policy world, but also of
the bias of the expert. Li (2007) studies the legislative committee as a strategic messenger that transmits information from the lobbyists to the floor. He finds that even
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preference outlier committees may not distort lobbyist information to the floor. All
messages are purely cheap talk (i.e. done under the “open rule”) in Li (2007), whereas
communication between the principals is done under the “closed rule” in the current
paper. None of these papers directly compares informational efficiencies of different
institutions, as we do here.
Dividing agenda-setting authority and veto authority between different principals
is a standard interpretation of the so-called “checks and balances” (The Federalist
Papers, No. 51; Persson et al. 1997).1 The benefit and cost of this regime is well
known. For example, Spolaore (2004) formalizes the conventional wisdom by showing that the dictator system (called cabinet system in his paper) may adjust too often,
while the checks and balances system may fail to adjust when adjustment is optimal.
We contribute to this classical topic by adding an informational dimension based on
cheap talk games. Our novel prediction that checks and balances may do a better job
adjusting to pressing big policy change because of informational advantage is quite
counterintuitive. In a paper with a different modeling framework, Stephenson and
Nzelibe (2010) reach a similar conclusion that checks and balances may not reduce
the probability that a new policy will be adopted. In their paper, checks and balances, by blocking some policies one biased principal would otherwise like to undertake, helps the voters to bear with potentially detrimental consequences of some new
policies.2
Our paper is also related to the literature on internal authority allocation under
incomplete contract. In the absence of a message-contingent contract, delegation
offers the expert incentives to collect/reveal information (Aghion and Tirole 1997;
Dessein 2002). To extract more information, more authority has to be delegated to the
expert. This belief is also shared by the burgeoning literature on delegation (Alonso
and Matouschek 2008; Mylovanov 2008; etc). Similar to these contributions, our
paper also emphasizes institutional solutions to improve upon the most informative
cheap-talk equilibrium. However, our institutional solution does not involve delegating
authority to the expert. Instead, we argue that an authority-sharing contract between
the principals can motivate an expert to supply more information even when she has
no say at all in the decision-making process. Our setup is more natural in many real
world situations. For example, Congress cannot delegate its law-making power to lobbyists who have policy expertise. On the other hand, it is entirely legal for Congress
to delegate agenda-setting authority to its committees.
3 The model
An organization of two principals (P1 and P2 ) must screen among a range of policies
that differ from each other on a one-dimensional parameter ω with uniform prior on
1 It should not be confused with the principle of “separation of powers.” Even though all Western democracies incorporate both principles to some degree, it is entirely plausible to imagine a political system
embodying only one of them.
2 Information transmission between the leader and the multiple experts in so-called stovepipe systems or
kitchen cabinets provides a natural counterpoint to our one-expert-with-multiple-principals setup.
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[0, 1]. There is an interested expert (E) who is privately informed of ω.3 We consider
two possible exogenous authority allocations.
• P1 or P2 dominance (authority concentration in P1 or P2 )
• Checks and balances (agenda power and veto power separated)
Under P1 (or P2 ) dominance, the expert sends a cheap talk message to P1 (or P2 )
before the policy is chosen by the principal who has power. Under checks and balances
the advising and policymaking proceeds as follows.
1. The expert sends a private cheap talk message to P1 (agenda-setting stage advising).
2. P1 proposes a policy recommendation y to P2 (agenda setting).
3. P2 chooses to accept or reject the agenda. In case of rejection, an exogenous status
quo policy yo will be chosen (veto-stage policymaking).
Denote player i’s utility function as U i , with i = E, P1 , P2 . We have the following
assumptions about the players’ preferences.
U E (y, ω, b) = − (y − (ω + b))2 b > 0
U P1 (y, ω, b1 ) = − (y − (ω + b1 ))2 0 ≤ b1 ≤ b
U P2 (y, ω) = − (y − ω)2
The optimal policy for the expert (and respectively P1 and P2 ) is ω + b (and
respectively ω + b1 and ω). They all want higher policy for larger state of the world.
Nevertheless, given b > 0, for any state of the world the expert always prefers a
higher policy than P2 does. We can use b (and similarly b1 ) to measure their fixed
conflict of interest. The expert bias b has to be small enough such that some credible
information can be transmitted under any authority allocation (Crawford and Sobel
1982). To make our setup general, we try to put little restriction upon the preference
divergence of P1 . For ease of exposition we limit 0 ≤ b1 ≤ b. Relaxing this technical
restriction later does not compromise our conclusions. For ease of exposition, we
assume b < yo < 1 − 4b. It can also be relaxed. For future work, we may consider
principals determining the final policy through bargaining.
4 The equilibria
We characterize the perfect Bayesian Equilibrium of the game under both authority
allocations. An off-equilibrium signal is believed to be a particular equilibrium signal
chosen by us. When there are multiple equilibria, we always focus upon the most
informative one we can find. For many cheap-talk applications it is usually difficult to
find the most informative equilibrium. These are the standard practices of cheap talk
applications. To conserve space, whenever there is no confusion, we do not explicitly
specify the players’ beliefs.
3 Adapted from the standard setup in Crawford and Sobel (1982), Gilligan and Krehbiel (1987), Krishna
and Morgan (2001), Dessein (2002), and Alonso and Matouschek (2008), among many others.
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4.1 Authority concentration
We first consider the case of P2 dominance. In equilibrium the number of signals
sent by the expert (and policies induced from the principal) will be finite, and higher
policies are generally associated with higher states of the world. This is the so-called
partition equilibrium in Crawford and Sobel (1982).
Suppose in equilibrium N policies are induced by N signals. We can sort the
equilibrium policies as y1 < y2 < · · · < y N . In correspondence we can divide
the unit space into N intervals, with partition points a0 = 0 < a1 < a2 < · · · <
a N −1 < a N = 1. It is implicit that in equilibrium the expert will send signal i when
ω ∈ [ai−1 , ai ), and induce policy yi .
The equilibrium policies of a two-partition equilibrium example (with b = 1/12)
is graphically represented in Fig. 1. In equilibrium, the expert sends only two signals
to the principal. When ω < 1/3, the expert sends signal 1, meaning that the state of
world is relatively “low.” When ω ≥ 1/3, the expert sends signal 2, meaning that the
state of world is relatively “high.” Accordingly, the principal will pick policy 1/6 (a
low policy) and 2/3 (a relatively high policy) to maximize his expected payoff given
this new information. Because of preference divergence between the expert and the
principal, only coarse information is transmitted. Given the equilibrium play, we can
check that the expert has no incentive to send out the wrong signal, which can never
lead to higher payoff for the expert. In particular, at ω = 1/3, the two policies in
equilibrium have equal distance to the most preferred policy of the expert (i.e. 5/12),
which guarantees that the expert is better off by sending out signal 1 and 2 to the left
Fig. 1 Equilibrium policies under P2 dominance (b = 1/12)
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and right of ω = 1/3, respectively. Intuitively, when ω is low, sending out signal 2 will
mislead the policymaker and induce a relatively high policy, which is not optimal for
the expert at all. The exact equilibrium partition structure is defined in the following
proposition.
Proposition 4.1 (P2 dominance) There exists a perfect Bayesian equilibrium under
P2 dominance as the following N-partition equilibrium. The number of partitions
(or
signals) N is defined as the smallest integer greater than or equal to − 21 + 21 1 + b2 .
The endpoints of the partition intervals are defined as
aj =
j
+ 2bj ( j − N ) j = 0, 1, 2, . . . , N
N
For ai−1 ≤ ω ≤ ai , with i = 1, 2, . . . , N (last equality holds only for i = N ),
the player E adopts the signaling strategy s ∗ (ω) = i. Define yi = ai−12+ai . When he
receives signal i, the player P2 updates his belief that ω ∈ [ai−1 , ai ] and chooses
y ∗ (i) = yi .
If the player E sends signal x ∈
/ {1, 2, . . . , N }, without loss of generality we assume
that the player P2 updates his belief that ω ∈ [a0 , a1 ] and chooses y1 .
The expected payoff for P2 is displayed in the Appendix (available upon request).
Proof See the main theorem and example in Crawford and Sobel (1982).
When expert bias b decreases, we can achieve a more informative equilibrium with
larger N . Therefore, the number of signals (or partitions) is a rough measure of the
information transmission efficiency.
The case of P1 dominance is analytically identical, and we only need to substitute
b with |b − b1 |, since the bias between the expert and P1 is now |b − b1 |. The case of
P1 and the expert sharing the same preference (b = b1 ) is analytically trivial.
4.2 Checks and balances
Expert advising under checks and balances also leads to a partition equilibrium since
information transmission between the expert and the principals as a whole is still cheap
talk. The following equilibrium is the most informative one we can find, but we do
not know whether there is a more informative one or not.
Proposition 4.2 (Checks and balances when 0 ≤ b1 < b) Assume b < yo < 1 − 4b.
When the bias of P1 (the agenda setter) is less than that of E (the expert), there exists
a perfect Bayesian equilibrium under checks and balances as the following:
The number of signals equals to (k1 + k2 + 2), with k1 and k2 to be defined below.
Player E’s signaling strategies in equilibrium can be separately described when the
state of the world is relatively low, intermediate, or high. Denote 2(b − b1 ) as δ.
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(i) When 0 ≤ ω ≤ yo − b1 , denote k1 as the largest integer less than or equal to
(yo − b1 )/δ; denote x1 = yo − b1 − k1 δ. For i = 1, 2, . . . , k1 ,
∗
s (ω) =
1
if ω ∈ [0, x1 )
i + 1 if ω ∈ [x1 + (i − 1)δ, x1 + iδ)
(ii) When yo − b1 < ω < yo + 2b + b1 ,
⎧
⎪
⎨k 1 + 1
∗
s (ω) = k1 + 2
⎪
⎩
k1 + 3
if ω ∈ [yo − b1 , yo )
if ω ∈ [yo , yo + 2b]
if ω ∈ (yo + 2b, yo + 2b + b1 )
(iii) When yo + 2b + b1 ≤ ω ≤ 1, denote k2 as the largest integer less than or
equal to [1 − (yo + 2b + b1 )]/δ; denote x2 = [1 − (yo + 2b + b1 )] − k2 δ. For
j = 1, 2, . . . , k2 ,
∗
s (ω) =
k1 + 2 + j
k1 + 2 + k2
if ω ∈ [yo + 2b + b1 + ( j − 1)δ, yo + 2b + b1 + jδ)
if ω ∈ [yo + 2b + b1 + k2 δ, 1]
When the player P1 receives a signal s ∗ , he correctly infers that ω lies on the
partition interval associated with s ∗ . Signal k1 + 1, k1 + 3, k1 + 2 + k2 are each
associated with two neighboring partitions defined above. The player P1 cannot distinguish between these two neighboring partitions. So he will infer that ω lies on the
union of these two partition intervals if he receives one of these three signals. His
agenda-setting strategies in equilibrium can be described as the following:
⎧
⎪
⎨x1 + b1 + (s − 1)δ
∗
y (s) = yo + 2b
⎪
⎩
yo + 4b + (s − k1 − 3)δ
if s = 1, 2, . . . , k1 + 1
if s = k1 + 2
if s = k1 + 3, . . . , k1 + k2 + 2
Note that each distinct partition is associated with a distinct signal s ∗ , which is
further associated with a distinct agenda y ∗ in equilibrium.
When the veto player P2 receives an agenda y ∗ , he correctly infers that ω lies on
the partition interval associated with y ∗ . His veto strategies in equilibrium can be
described as the following: the veto player P2 always accepts the proposed policy y ∗ .
If the agenda-setting player proposes an agenda not in the set of equilibrium strategy
y ∗ (s) as specified above, the veto player always believes that ω ∈ [yo − b1 , yo ) and
rejects the out-of-equilibrium proposal.
The expected payoffs for P1 and P2 are displayed in the Appendix.
Proof The proof (included as supplementary materials) only involves routine calculations.
The following three numerical examples help illustrate our equilibrium construction. I set b = 1/12 and yo = 1/2 (the optimal policy for the unbiased principal P2
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Fig. 2 Equilibrium policies of checks and balances (yo = 1/2, b = 1/12, b1 = 1/24)
given the prior belief, i.e., the default policy choice in the absence of any informative
communication), and consider three cases: b1 = 0, 1/24 or 1/12.
(1) b1 = 1/24. The equilibrium policies are graphically represented in Fig. 2. In
equilibrium there are 10 signals associated with 10 partition intervals. There are in
total 11 end points including 0 and 1. For example, for any ω ∈ [0, 1/24] (the first
partition interval), in equilibrium the expert sends signal 1, then P1 sets the agenda to
be y1 = 1/12, with 1/12 = b1 + 1/24, and P2 approves the agenda.
5
are constructed with the
The five non-zero partition end points to the left of 12
11
1
(which equals to
following procedure. We divide 24 (which equals to yo − b1 ) by 12
1
2b − 2b1 ), and get an integer 5, with a residual 24 . Then we define these five points as
1
1 1
1 24 + 12 k1 , with k1 = 0, 1, 2, 3, 4. The approved agenda yi in equilibrium is 12 + 12 k1 .
This construction meets the equilibrium requirements.
It is easy to check that the expert has no incentive to send signal j when she knows
that the true state lies on the ith interval ( j = i), as she is indifferent between any
1
1 + 12
k1 (no-arbitrage condition in
two neighboring equilibrium policies when ω = 24
1
Crawford and Sobel 1982). Take ω = 24 (i.e. k1 = 0) for example. We know that the
expert’s most preferred policy is 18 , which lies on the line y = ω + b. The induced
equilibrium policy has a jump around ω =
1
24
in Fig. 2. When ω →
1 −
24 ,
the induced
1 +
→ 24
, the induced equilibrium policy is 16 . Note
1
1
6 is 8 , the expert’s favorite policy. Because of the
equilibrium policy is
When ω
1
and
that the midpoint between 12
symmetric nature of the utility function, the expert is indifferent between these two
1
so he has no strong incentive to lie. When ω is
equilibrium policies when ω = 24
1
12 .
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1
1
smaller than 24
(the other case is similar), the induced equilibrium policy 12
is always
1
closer to the expert’s most preferred policy line y = ω + b than 6 is. Therefore, the
expert has no incentive to lie at all. The same argument would apply for k1 = 1, 2, 3, 4
as well, because the induced equilibrium policies are always symmetric around the
expert’s favorite policy when the true state of world ω lies at the partition end points
1
1 24 + 12 k1 .
Similarly given his updated belief that ω lies on the ith interval the agenda-setting
principal has no incentive to propose any y j other than yi when he hears signal i from
the expert. Take the second partition intervalfor example.
When the agenda-setter
1 1
, 8 . On average his most-preferred
receives signal 2, he would conclude that ω ∈ 24
policy is 1/8 = (1/24 + 1/8)/2 + b1 . The equilibrium agenda is 16 . If he proposed
any other equilibrium agenda larger than 16 , he would have become much worse off
because they are much further away from 18 (see Fig. 2). A slightly lower policy in
1
would have given him the same expected payoff, because the
equilibrium i.e. 12
1
midpoint between 12
and 16 is 18 . Thus, he has no strong incentive to propose a slightly
lower equilibrium policy either.
Finally the proposal yi is never worse than the status quo policy for the veto principal, given her similarly updated belief. For any partition interval in Fig. 2, the induced
policy is much closer to the most preferred policy line y = ω than the status
is.
19quo
, 78 , 1 .
For ω > 43 , the procedure is similar and we get the partition end points as 24
11
5
The approved agenda yi in equilibrium is 56 , 12
, 1 . For 12
≤ ω ≤ 3 , two end points
1 2 4
1 2
are 2 , 3 , and the approved agenda yi in equilibrium is 2 , 3 . These equilibrium
strategies are identical to the ones constructed by Krishna and Morgan (2001) (see
Fig. 3 and discussions below).
In total we have constructed 11 partition interval end points including 0 and 1 for our
ten partitions, and 10 associated yi in equilibrium for our numerical example shown
in Fig. 2.
(2) b1 = 1/12. When b1 = b, the agenda setter P1 and the expert E are perfect
substitutes. This multi-principal problem is reduced to the original principal-expert
framework, in which the expert has agenda-setting power, and the principal has veto
power. The equilibrium for such a game was first constructed by Gilligan and Krehbiel
(1987) and then modified by Krishna and Morgan (2001). The solution by Krishna
and Morgan (2001) (see Fig. 3) can be derived as the limit of Proposition 4.2 when b1
approaches b. The verification is included as supplementary materials. If we compare
Figs. 2 and 3, we can imagine that when b1 → b, the partitions on both ends in
Fig. 2 (i.e. those with ω < 5/12 and ω > 3/4) would become more and more refined,
eventually shrinking into the expert’s most preferred policy line of y = ω + b.
(3) b1 = 0. The equilibrium policies are graphically represented in Fig. 4. In
equilibrium there are six signals associated with six equal partition intervals. The
partition end points to the left of 21 are constructed through the standard procedure in
Proposition 4.2. We divide 21 (which equals to yo −b1 ) by 16 (which equals to 2b−2b1 ),
and get an integer 3, with a residual 0. Then we define these partition end points as
0 + 16 k1 , with k1 = 0, 1, 2, 3. The approved agenda yi in equilibrium is 0 + 16 k1 . The
rest of the partition intervals can be constructed similarly. It is just by coincidence that
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Fig. 3 Equilibrium policies of checks and balances when b1 = b (a special case of Fig. 2 when b1 → b,
the same equilibrium as in Krishna and Morgan 2001, yo = 1/2, b = 1/12)
all partition intervals are equal, because the residuals in this case happen to be 0. In
general the residuals do not equal to 0. And the first and the last partition intervals
tend to have different lengths, just as in Fig. 2. Because b1 = 0, non-end partition
intervals all have the same length 2b − 2b1 = 2b. If we compare Figs. 2 and 4, we can
imagine that when b1 → 0, the non-end partitions on either side of Fig. 2 (i.e. those
with ω < 5/12 and ω > 3/4) would become less and less refined until their lengthes
expand to 2b.
4.3 Applications and discussions
For a real world example, we may imagine that a hawkish national security advisor who
knows the underlying terrorist threat level ω is advising the President and Congress
in selecting an anti-terrorism budget level y. For simplicity, imagine ω as the number
of active terrorists, and y as the number of bullets to buy. Suppose each bullet can kill
one terrorist before he does something bad. While the hawkish advisor systematically
prefers buying more bullets for a given security threat (i.e. b > 0), the best budget
level for an unbiased politician is simply ω, if he knows the underlying threat level,
in which case the outcome X = y − ω would equal to 0—no bullet is wasted, and no
terrorist would survive. If the budget is too big for the threat, X would be positive, and
some bullets are wasted. If the budget is too small, X would be negative, and some
terrorists would survive and do something bad. Suppose the advisor has a bias b > 0,
Congress P2 has no ideological bias, and the President P1 has a bias b1 ∈ [0, b]. If all
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Fig. 4 Equilibrium policies of checks and balances when b1 = 0 (a special case of Fig. 2, yo = 1/2)
three players have a quadratic loss function over X , this scenario is essentially what
we set up for our model.
The politicians can re-arrange authority allocation between themselves but cannot
delegate authority to the advisor. When Congress has full authority, there is a twopartition equilibrium for the numerical example in Fig. 1. Only a security alert system
with two categories is credible in this case. The advisor only indicates whether the
threat is “big” (ω > 1/3) or “small” (ω ≤ 1/3), and the corresponding y is chosen to be
either 2/3 or 1/6. For almost all ω the outcome X turns out to be disappointing: in half
cases there are surviving terrorists doing bad things, and in the other half cases there
are some bullets bought but wasted. Nevertheless, on average the unbiased Congress
is in control—the expected X is always zero, suggesting that on average the number
of bullets matches the number of terrorists.
If Congress allows the President to set the agenda and sits as a veto player himself,
the situation would be different. When the President’s bias is in the middle of the
advisor and Congress, there is a ten-partition equilibrium for the numerical example
in Fig. 2. In other words, a security alert system with ten categories is now credible.
This is a more refined alert system. For example, when ω < 1/8, the biased advisor can
credibly reveal two security alert levels: ω ∈ [0, 1/24], and (1/24, 1/8]. The policies
proposed by the President and approved by Congress are 1/12 and 1/6, respectively .
It is important to recognize that on average these policies are more ideal to the
advisor than to any of the politicians. For example, when the expert sends signal 1
to the agenda setter in Fig. 2, the agenda setter (and eventually the veto
after
player
1
. Given
hearing the equilibrium agenda) both share the ex post belief that ω ∈ 0, 24
1
this updated belief, on average the agenda setter’s most-preferred policy is 16
, and
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Expert advising under checks and balances
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1
the veto player’s most-preferred policy is 48
. Both are lower than the equilibrium
1
proposal of 12 . It would appear to be in their mutual interest for the President to
propose a smaller y (e.g. 1/24 between 1/16 and 1/48), and for Congress to accept it.
However, such an incentive to collude ex post exists in other cheap-talk applications
in politics as well.4
However, because 1/24 is not an equilibrium agenda, we should
take it for
1 not
upon receivgranted that Congress can always form the correct belief that ω ∈ 0, 24
ing such an agenda. Please note that when an agenda deviates away from the equilibrium path, non-equilibrium belief for the veto principal is specified in such a way that
all agendas off the equilibrium path will get rejected by the veto principal (see Proposition 4.2). This standard assumption in the cheap-talk literature is needed to preserve
equilibrium structure—otherwise a rational security advisor will stop providing good
information in the first place.
Because the imminent terrorist threat is quite low when ω < 1/48, the status quo
policy of 1/2 leads to a big waste of bullets. Politicians desperately need advice in such
situations. It is in the interest of both principals to accept a more informed final policy
(slightly) in favor of the hawkish advisor. If both principals play equilibrium strategies
constructed here, none of them can unilaterally exploit revealed expert information
to her best advantage. This is a key intuition why checks and balances can assure the
expert to provide more information.
5 Comparing information structures
Different internal authority allocations offer the principals different equilibrium information structures. More refined structures are not always more informative, as the
distribution of information is also important. The tradeoff between these two forces
under checks and balances will be illustrated in the second subsection, following our
formal result concerning the most optimal/informative authority allocation scheme
derived under two extreme preference distributions (b1 → b and b1 → 0) in the first
subsection. Finally, more general specifications are discussed in the last subsection.
5.1 Most informative authority allocation
Note that we can write the outcome of the equilibrium policy y(ω) as X (ω) = y(ω)−ω.
We can use var X to represent overall informational losses, and use (E [X ])2 to represent overall distributional losses of the associated information structure, respectively
(Krishna and Morgan 2001). The random variable y(ω)—ω measures the deviation
4 Formally speaking, we may argue that a signaling (or cheap-talk) equilibrium is not robust if there exists
a self-signaling set as defined in Cho and Kreps (1987) and Farrell (1993). Farrell (1993) pointed out that
such a set exists for Crawford and Sobel’s (1982) leading example (ω ∈ (2/3, 1] in my Fig. 1). We can
show that it also exists for Krishna and Morgan’s (2001) closed-rule signaling game (deviating agenda of
3/4 for ω ∈ (15/24, 17/24) in my Fig. 3), and for my checks and balances equilibrium (deviating agenda
of 1/12 for ω ∈ [0, 1/6] in Fig. 4). This is a topic that deserves more future study. It is also important
to point out that from a dynamic or evolutionary perspective, the above off-equilibrium behaviors are not
desirable because they are inefficient in the long run.
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T. Li
between the final policy and the underlying information the policy is trying to match.
The deviation could be either positive or negative. The squared deviation measures the
degree to which the final policy fails to match the underlying information. So we can
use (E [X ])2 to represent overall distributional losses. Using the anti-terrorism advising example we have discussed, distributional losses would be big if the anti-terrorism
budget level significantly diverges away (on either directions) from the underlying
threat most or all the time. On the other hand, large variation in the deviation suggests
that information is not well utilized into the making of the final policy. So we can use
the variance of this random variable to measure the overall informational losses. In
our example, informational losses would be big if the anti-terrorism budget proves to
be too big for the threat sometimes, but too small at other times.
Because the expected payoff of the unbiased principal P2 is simply −E X 2 , it
can be written as −var X − (E [X ])2 , and we have
− var X = E[U P2 ] + (E [X ])2
(1)
It is analytically complicated to directly compare the informational losses under
different authority allocations. If we can directly compare E[U P2 ] and (E [X ])2 under
different authority allocations, we can achieve this objective indirectly.
When b1 → b, Lemma 7.2 (see Appendix) says that P1 dominance leads to strictly
highest expected payoff for P2 , while Lemma 7.3 (see Appendix) says that the difference in distributive losses between P1 dominance and checks and balances approaches
0. Then from Eq. (1) we conclude that P1 dominance leads to lowest informational
losses.5 Similarly, for b1 → 0, we conclude that checks and balances lead to lowest
informational losses. Formally we have:
Proposition 5.1 (Comparing informational efficiencies) ∀b < yo < 1 − 4b, ∀0 ≤
b1 ≤ b, ∀i = 1, 2, we have
lim −var [X |P1 dominance] > lim −var [X |checks and balances]
b1 →b
b1 →b
lim −var [X |checks and balances] > lim −var [X |P1 dominance]
b1 →0
b1 →0
In both cases, the informational efficiency under P2 dominance is always the lowest
among all these authority allocations.
Proof Use Eq. (1), Lemmas 7.2 and 7.3.
Combining Lemma 7.2 and Proposition 5.1, we conclude that the ex ante optimality of authority allocation from the perspective of P2 is driven by the difference
in informational efficiencies. It is not difficult to check that this claim also applies
to P1 . Stated simply, the most informative authority allocation scheme under these
two extreme preference distributions also yields the highest expected payoff for both
principals. One caveat of our result is that we cannot rule out the possibility that there
exists a more informative equilibrium when b1 → b under checks and balances which
can outperform P1 dominance in terms of expected utilities.
5 The distributive loss under P dominance is highest.
2
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Expert advising under checks and balances
491
5.2 Tradeoff of signal quantity and distribution
If we make a rough comparison between Figs. 1 and 2, we can find out that checks
and balances seem to be associated with more information measured by the number of
signals/partitions. However, the distribution of information under checks and balances
also seems to be more uneven, with more signals/partitions on two wings and less
signals in the middle. This subsection tries to formally show how this tradeoff drives
the result in Proposition 5.1.
First, we need to better understand how informational loss is measured and how it
is affected by both signal quantity and signal distribution. Consider any information
structure with N partitions. Write = [0, 1]. Write any partition interval [ai−1 , ai ]
2, . . . , N . Write |i | as the length of the interval. It is obvious that
as i , with i = 1,
N
N = and
∪i=1
i
i=1 |i | = 1.
With some algebra, var X , which represents informational loss of an equilibrium information structure {1 , 2 , . . . , N } on [0, 1], can be decomposed into
three terms I1 , I2 , and I3 , with the first term exclusively determined by the number of signals, and the second and third term determined by the inflated partition
length variation (related to the distribution of ω), and the average partition outcome variation (related to the distribution of X = y − ω on each partition interval),
respectively.
N 1
1 2
1 2
|i | +
|i | −
+
I1 + I2 + I3 =
12N 2
12
N
N
i=1
+
N
2
|i | E i X − E X
i=1
N
where N1 = N1 i=1 |i |. The second term is inflated because the squared deviation
is multiplied by |i | + 2/N instead of by |i |.
Consider two equilibrium information structures i1 and i2 . Suppose
1
other things being equal, i comes with more signals, which would trans late into a lower I1 , so the informational loss associated with i1 tends to be
lower.
However, it is important to understand that the number of signals alone cannot determine the overall informational efficiency of {i }. Even for two information structures
with the same number
losses could be quite
of signals, the associated informational
different. Suppose i1 = {[0, 1/2], (1/2, 1]} and i2 = {[0, 1/100], (1/100, 1]}.
Both have two signals, but the signal distribution is quite uneven under i2 . Even
though I1 is the same in both situations, I2 associated with i1 is 0, while I2 asso 2
2
ciated with i is much bigger than 0, which means that i tends to have higher
informational
losses. Intuitively speaking, for ω ∈ [0, 1/100], the revealed information
under i2 tends to be precise. However, for ω ∈ (1/100, 1], the revealed information
under i2 tends to be quite poor. The difference tends to increase the variability of
X under i2 . If we replace 1/100 with one out of a million, the inferiority of i2
attributed to I2 would become more dramatic.
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T. Li
Finally, suppose i1 = i2 = {[0, 1/2], (1/2, 1]}, the associated informational
losses could still be quite different because they
different
may correspond to very
final policy functions. If E[X |ω ∈ [0, 1/2], i1 ], E[X |ω ∈ (1/2, 1], i1 ], and
2
2
E[X |ω ∈ [0, 1/2], i ] all equal to 0, while E[X |ω ∈ (1/2, 1], i ] equals to
100. Even though I1 and I2 are the same under i1 and i2 , I3 associated with
1
2
i is 0, while I3 associated with i is much bigger than 0, which means that i2
tends to have higher informational losses. The intuition is that the average
deviation
between the final policy and the underlying information under i2 is 0 on [0, 1/2]
and 100 on (1/2, 1], which translates into a large average partition outcome variation.
This happens when the final equilibrium policy on (1/2, 1] is too big relative to the
underlying information.
So the equilibrium policy y affects var X through and only through I3 . It does not
affect I1 and I2 , which are only affected by signal quantity and distribution, respectively. However, the way y affects I3 is also deeply related to signal distribution in
our model. An average partition outcome deviation like 100 can never happen in our
equilibrium. Since y is an equilibrium choice, it will not deviate too far away from
ω, ω + b1 , and ω + b, the optimal policies of the three players. By definition, E[X ] on
each partition interval under authority concentration is always a constant. For example, E[X ] on each partition interval under P1 authority when b1 = 0 is always 0, as
the midpoint of each partition lies on y = ω (See Fig. 1). So I3 is always 0 under
authority concentration. E[X ] on each partition interval under checks and balances is
either b or something close to b, as the midpoint of most partitions lies on y = ω + b
(See Fig. 2). So I3 is also close to 0 under checks and balances. Because I3 is so small,
it is unlikely to be a significant factor in the tradeoff. Moreover, when signals are more
evenly distributed, which leads to a lower I2 , I3 also tends to be lower. For example,
the information structure in Fig. 2 is more unevenly distributed than that in Fig. 4, and
both I2 and I3 are positive in the former case, while both of them are 0 in the latter
case. It is as if I2 and I3 respectively measures the distributive informational losses
along the horizontal (i.e. ω) and vertical (i.e. y) axis under checks and balances. Both
losses tend to be bigger when signal distribution is more uneven.
In summary, the informational losses of an equilibrium informational structure can
be decomposed into three terms: I1 , I2 , and I3 , which measures the number of signals,
the distributive quality of ω, and the distributive quality of y, respectively. When the
number of signals increases, I1 tends to decrease. When signals become more evenly
distributed, I2 and I3 tend to decrease in our case. It is those information structures
with a large number of evenly distributed signals that tend to have lower informational
losses under either authority allocation.
Now we can use this intuition to explain the tradeoff behind Proposition 5.1. It is
easy to see from Proposition 4.1 and 4.2 that the number of signals under checks and
1−4b
+ 1, while the number of signals under P1
balances is at the magnitude of 2(b−b
1)
2
authority concentration is at the (lower) magnitude of 21 + 21 1 + b−b
. Lemma 7.1
1
in the Appendix shows that checks and balances always produce more signals. When
b1 increases from 0 to b, the number of signals under checks and balances increases
much faster. Given the negative association between N and I1 , checks and balances
clearly enjoy a “signal quantity” advantage over authority concentration.
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Expert advising under checks and balances
493
However, there is also a tradeoff between signal quantity and distributive quality
under checks and balances. For example, when the number of signals increases from
6 in Fig. 4 to 10 in Fig. 2, and then to the limit case of ∞ in Fig. 3 as b1 goes from
0 to b, the signals also become more unevenly distributed. The key observation is
that all the extra signals come from the interval [0, 5/12] and [3/4, 1]. The middle
region [5/12, 3/4] contributes a constant number of three signals throughout. We
call this an “inertia region.” More generally, it is the region of [yo − b, yo + 3b].
In our security advisor example, in the presence of a biased advisor who is known
to favor a few more bullets than the politicians for any ω, the politicians are understandably reluctant to follow his warning of buying a few more bullets (on top of
the status quo level). Only when the expert advises reducing y (a policy change
opposite to his own bias) or buying a great number of bullets (a fire-alarm-type
warning which makes sense only for a big terrorist threat) would his advice look
more credible in the eyes of the politicians. Because credible information transmission in the inertia region is difficult, the ability of checks and balances to generate a
large number of signals (outside of the inertia region) creates a formidable enemy for
itself – the more signals are created, the more uneven the overall signal distribution
becomes.
When b1 = 0, there are not that many signals created under checks and balances,
and the signal distribution is relatively even (Fig. 4). This helps to explain why informational losses are lower under checks and balances when b1 → 0 in Proposition 5.1.
When b1 increases, N goes up and the “signal quantity” advantage for checks and
balances becomes more significant, but at the same time its “signal distribution” disadvantage also begins to pile up (Fig. 2). When b1 → b, the quantity advantage of
checks and balances vanishes, as the number of signals under both authority allocations approaches ∞, and I1 goes to 0 in both cases. The fact that I1 goes to 0 much
faster under checks and balances does not help at all in the limit. Now the “signal distribution” disadvantage for checks and balances becomes fatal, and forces it to suffer
from higher informational losses compared with P1 dominance (Fig. 3). In a word,
the tradeoff between signal quantity and distribution under checks and balances drives
the result in Proposition 5.1.
To numerically illustrate the tradeoff,
3 we have calculated N , I1 , I2 , and I3 , together
Ii ) under different authority allocations in
with the total informational losses ( i=1
Table 1, for the three numerical examples discussed in Sect. 4, with b = 1/12; yo =
1/2; and b1 = 0, 1/24 or 1/12, respectively. From Panel D we can see that checks
and balances have the lowest informational losses when b1 is small (b1 = 0, 1/24),
and P1 dominance has the lowest informational losses when b1 = b. This confirms
our Proposition 5.1. We can also see from Panel A–C that this is driven by the tradeoff
between signal quantity and distribution under checks and balances. It is indeed true
that checks and balances always have more signals (Panel A) and lower I1 (Panel B).
However, when I1 drops as b1 (or N ) increases, I2 + I3 , which is associated with
uneven signal distribution, actually goes up for checks and balances (Panel C). There
is no such tradeoff for P1 dominance, which sees both I1 and I2 + I3 drop as b1 (or
N ) increases. So the tradeoff between signal quantity and distribution under checks
and balances helps to make it enjoy lower informational losses when b1 is small, but
suffer from higher informational losses when b1 approaches b.
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T. Li
Table 1 The tradeoff of signal
quantity and distribution
b1 = 0
1
b1 = 24
1
b1 = b = 12
Panel A: N
P1 dominance
2
3
∞
P2 dominance
2
2
2
Checks & balances
6
10
∞
Panel B: I1
P1 dominance
0.0208
0.0093
0
P2 dominance
0.0208
0.0208
0.0208
Checks & balances
0.0023
0.0008
0
Panel C: I2 + I3
P1 dominance
0.0069
0.0046
0
P2 dominance
0.0069
0.0069
0.0069
0
0.0005
0.0008
P1 dominance
0.0278
0.0139
0
P2 dominance
0.0278
0.0278
0.0278
Checks & balances
0.0023
0.0013
0.0008
Checks & balances
3
Panel D: i=1
Ii
yo = 1/2
b, b1 are biases of the expert and
P1
5.3 Extensions
Proposition 5.1 shows that, for b1 ∈ [0, b], checks and balances are better than authority concentration in terms of information transmission efficiency when b1 approaches
0 in the limit, while the opposite is true when b1 approaches b in the limit. If the
essence of this result holds true more generally, checks and balances should be better
than authority concentration when b1 is in a neighborhood around 0, while the opposite is true when b1 is in a neighborhood around b. In other words, the essence of
Proposition 5.1 is consistent with the view that checks and balances are more efficient
when conflict of interest between principals is relatively small, while concentrating
authority in the hands of the principal more aligned with the expert is more efficient
when the opposite is true. One purpose of this subsection is to show that the essence of
the above result holds true for more general specifications of our parameters. Another
purpose is to show that, from the perspective of citizens/voters, checks and balances
still appear to be a desirable authority allocation scheme.
The essence of our results can be extended to any b1 ∈ (0, b). We can find b1∗ ∈
(0, b) such that for all 0 ≤ b1 < b1∗ , our results for b1 → 0 in Proposition 5.1
also apply. Similarly, we can find b1∗∗ ∈ (b1∗ , b) such that for all b1∗∗ < b1 ≤ b, our
results for b1 → b in Proposition 5.1 also apply. The difference between b1∗ and b1∗∗
is very small, and shrinks to 0 when b1 → b.6 Relaxing the technical restriction on
the exogenous status quo policy (b < yo < 1 − 4b) will change the functional form
of Proposition 4.2. It does not change the essence of our results. Finally we can use
6 Because the number of signals is a discrete function of b and b, the analytical form is difficult to deal
1
with. We use numerical calculation for our results.
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Expert advising under checks and balances
495
three numerical cases to show that our model results can be extended to more general
player preferences.
(1) We can consider b1 > b, in which case the expert has a preference between
that of two principals. At least for b < b1 < 2b, it is easy to check that there
is an equilibrium under checks and balances which is almost the same as that of
b1 = 2b − b1 ∈ (0, b). The equilibrium under P1 dominance is symmetric with
b, which is easy to understand since what matters now is the absolute difference
between b1 and b. Therefore, we conclude that for b1 ∈ (b, 2b), the situation is almost
symmetric with that of b1 ∈ (0, b).
To appreciate the intuitions, consider our example depicted in Fig. 2, with b =
1/12, yo = 1/2, and b1 = 1/24. Call the first principal in this example as P1 . Now
consider another “first principal” whose bias is b1 = 1/8, and call this principal P1 .
Note that the bias of P1 and P1 are symmetric around the bias of the expert, with
b1 + b1 = 2b. If we draw a line representing the ideal policy of P1 , it would be
symmetric to the ideal policy line of P1 around the ideal policy line of the expert.
If we replace P1 with P1 , the equilibrium depicted in Fig. 2 largely remains to be
true. To check this claim, we only need to check the IC condition of the agenda setter
P1 . We do not need to worry about the incentives of the expert or the veto player.
Because no change occurs to them, they tend to stick to their equilibrium strategies.
Interestingly, the IC condition that holds for P1 also roughly holds for P1 . This is as
if a group of students sort into different schools based on their parents’ income levels.
When everyone’s income doubles, it is still possible to create an identical sorting
outcome. What we have here is something similar. For P1 , the equilibrium agendas
are systematically above his ideal agenda, while for P1 , the same equilibrium agendas
are systematically below his ideal agenda. Yet in both cases, the IC conditions hold
for the agenda setter. If we put the ideal policy line of P1 on Fig. 2, we can see that,
given the new information from the expert, the agenda setter always weakly prefers the
equilibrium agenda he is supposed to propose to any other equilibrium agenda. In most
cases, the neighboring equilibrium agendas are symmetric around either y = ω + b1
or y = ω + b1 , helping the agenda setter to satisfy his IC condition.
We say the equilibrium for P1 “largely” remains to be true for P1 because we
need to make minor adjustment for those partitions on both ends (i.e. those related
to the residuals x1 and x2 in Proposition 4.2). For the example in Fig. 2, we need to
merge the first two partition intervals and choose the second equilibrium agenda for
P1 as the first equilibrium agenda for P1 so that P1 has correct incentives to comply
with the equilibrium strategies. Otherwise, he has incentives to send out the second
agenda when he learns from the expert that ω ∈ [0, 1/24]. Because the adjustment is
minor and symmetric on both ends, it does not affect the informational efficiencies of
checks and balances equilibria when b1 is symmetric about b. And we can conclude that
authority concentration is better than checks and balances when b1 is in a neighborhood
around b.
(2) We can also consider b1 < 0 < b, in which case the veto principal P2 has a
preference between that of the expert and the agenda-setting principal P1 . We can still
apply Proposition 4.2 to construct the equilibrium. However, because δ = 2(b − b1 ) is
now bigger than 2b, we can check that the second case in Proposition 4.2 is no longer
incentive compatible for the agenda setter. So we only consider the first and third case
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T. Li
in Proposition 4.2. In other words, we divide the interval [0, yo − b1 ] and [yo − b1 , 1]
by δ, find the residuals, and then follow the standard procedure in Proposition 4.2 to
construct the equilibrium. In particular, the residual x1 should form a separate partition,
while x2 should be merged with the previous partition, just as in Proposition 4.2.
For example, suppose b = 1/12, yo = 1/2, and b1 = −1/12, the equilibrium under
checks and balances contains two partitions, with 1 = [0, 3/12], and 2 = (3/12, 1].
The corresponding equilibrium policy y1 = 1/6, and y2 = 1/2.
In Proposition 5.1 we have shown that when b1 → 0, the informational losses
under checks and balances are lower than authority concentration. We can show that
this result continues to hold for b1 < 0 when b1 is sufficiently close to zero. When b1 is
too much below zero, even though checks and balances still have lower informational
losses compared to P1 dominance, it loses informational advantage to P2 dominance,
as the preference of P2 is now much closer to the expert than P1 . Some numerical
examples are presented in Panel A of Table 2. When b1 = −1/48, −1/12, or −/10,
the informational losses of checks and balances are lower than that of both P1 and P2
dominance. When b1 = −1/5, the informational losses of checks and balances are
lower than P1 dominance but higher than P2 dominance. So checks and balances are
better than authority concentration when b1 is in a neighborhood around 0.
(3) We can also talk about welfare implications for citizens/voters. Because the
fundamental problem is to match the policy with the underlying information, it is quite
natural to assume that the average voter shares the same preference as the unbiased
principal P2 . Then our welfare analysis for P2 would apply for the voters as well,
namely checks and balances tend to be a more optimal/informative authority allocation
scheme when b1 → 0.
Table 2 The informational
losses for general biases
Panel A
b
1/12
1/12
1/12
1/12
b1
–1/48
–1/12
–1/10
–1/5
P1 dominance
0.0317
0.0486
0.0544
0.0833
P2 dominance
0.0278
0.0278
0.0278
0.0278
Checks & balances
0.0044
0.0108
0.0115
0.0550
Panel B
b2
0
1/12
0
1/12
b1
1/12
1/12
0
0
P1 dominance
0
0
0.0278
0.0278
P2 dominance
0.0278
0
0.0278
0
Checks & balances
0.0008
0
0.0023
0.0023
Panel C
yo = 1/2, b = 1/12 in Panel B
and C
b, b1 , b2 are biases of the expert,
P1 and P2
123
b2
0
–1/12
0
–1/12
b1
–1/12
–1/12
0
0
P1 dominance
0.0486
0.0486
0.0278
0.0278
P2 dominance
0.0278
0.0486
0.0278
0.0486
Checks & balances
0.0108
0.0108
0.0023
0.0061
Expert advising under checks and balances
497
In our previous discussions, the preferences do not change over time. This is more
natural for the example of one principal being Congress and the other a congressional
committee. If one principal is the President and the other is Congress, election cycles
tend to significantly change the preference distribution over time. So we want to
conduct some welfare analysis for the average voter when the preferences of the
principals changes over time.
I assume that the average voter has a bias of 0, and the expert has a bias of b. For
simplicity, I assume that b1 and b2 , the bias of principal P1 and P2 , are independent
random variables that may take values in the set B = {0, b} with equal probability.
I also consider an alternative set of B = {0, −b}. In both cases, our numerical example
results reported in Panel B and C of Table 2 (for b = 1/12, yo = 1/2) show that checks
and balances on average have lower informational losses than any sort of authority
concentration. When the biases of the politicians oscillate between the preferences of
the average voter and the biased expert (Panel B), the expected informational losses
under either authority concentration scheme is 0.0139, while the same expected losses
under checks and balances are 0.0014, which is much lower. When the biases of
the politicians oscillate between the preferences of the average voter and one that
opposes the expert (Panel C), the expected informational losses under either authority
concentration scheme is 0.0382, while the same expected losses under checks and
balances are 0.0075, which is also much lower. In summary, from the perspective of
citizens/voters, checks and balances appear to be a more efficient authority allocation
scheme given the assumptions we have made.
6 Conclusions
This paper discovers that dividing agenda-setting authority and veto authority between
two different principals (or checks and balances) has the benefit of assuring a partisan
expert to reveal more credible information (measured by the number of signals) to
the principals as a whole since neither principal can unilaterally exploit expert advice.
There is a central tradeoff between signal quantity and distributive quality under checks
and balances. Overall, checks and balances, compared with the alternative institution
of concentrating authority in a single principal, have higher communication efficiency
when conflict of interest between the principals is small relative to the expert, in which
case information distribution is more even.
Our theory can be applied in many authority allocation problems with different
political institutions. One area is congressional politics. A strong legislative committee
system with agenda-setting privilege (such as the one in the U.S.) is formally similar
to our checks and balances case in which authority is divided between the floor and
the committees. A legislature without such an arrangement is more like our case of
authority concentration. Lobbyists with policy-relevant information are experts in our
model. The authority allocation within a legislature can have a profound impact on the
informational lobbying efficiency. Our theory suggest that if the preference divergence
between the floor and the committees is relatively small with respect to the lobbyists
(so-called ally committees in the literature), it is better to have a strong committee
system with agenda-setting privilege, since this would produce better information
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T. Li
transmission efficiency overall. This tends to be true at least in American politics.
There is strong evidence that many lobbyists are more extreme advocates than all
legislators (Poole and Rosenthal 1997).
Our model can also be applied to study how authority allocation between a highlevel executive (P2 ) and a low-level executive (P1 ) can have an impact on their capacity
to communicate with a manager who has expertise on one division. The high-level
executive P2 can consider three organizational forms: delegating the authority to the
low-level executive P1 (P1 dominance, or full delegation), keeping the authority to
himself (P2 dominance, or no delegation), or giving the low-level executive agenda
setting power and keeping veto power for himself (checks and balances, or veto delegation in the organizational design literature, e.g. Dessein 2002). Our model can be
understood as an extension of the organizational design literature which only considers
two players (e.g. one informed manager, and one uninformed CEO).
The idea that internal institutions can affect external communication efficiency can
be explored in other theoretical researches. This seems obvious, but has not been
fully exploited. Internal authority allocation is more often argued to serve an internal purpose, namely providing incentives for some political institutions to invest in
information/expertise and then share with other political institutions. For example,
it is often argued that legislative committees exist because Congress needs them to
become specialized and collect information for the floor. The possibility that internal
authority allocation may serve an external purpose is often ignored. Sometimes it is
recognized that internal authority allocation may help reveal certain information to the
voters (as in Persson et al. 1997; Stephenson and Nzelibe 2010). However, the opposite
channel of using internal authority allocation to encourage information flow from the
third-party experts to the policymakers is hardly addressed. This paper provides some
research results along this line. At a minimum, this paper helps us realize that internal authority allocation may be designed to serve very un-intuitive agency problems.
Having a systematic evaluation of the information transmission process beyond the
organizational boundary would be useful.
Acknowledgements I have benefited from discussions with Scott Ashworth, David Austen-Smith, Daniel
Diermeier, Yuk-fai Fong, Jas Sekhon, the editors, two anonymous reviewers, and seminar participants at
Harvard and Kellogg School of Management (MEDS). I am especially thankful to Ken Shepsle for his
continual support and encouragement.
7 Appendix
The expected payoff for P2 under P2 dominance is
E U P2 |P2 dominance = −
b2 N 2 − 1
1
−
12N 2
3
The expected payoff for Pi under P j dominance (for i = 2 or j = 2) can be
calculated similarly.
The expected payoff for P2 and P1 under checks and balances is, respectively
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Expert advising under checks and balances
499
1
E U P2 |checks and balances = − [(x2 − b1 )3 + (x1 + b1 )3 ]
3
2(k1 + k2 )
−
[(b − b1 )3 + 3(b − b1 )b2 ]
3
1 3
−
b1 + 16b3 + (b1 − 2b)3
3
and
1
8(k1 + k2 )
(b − b1 )3
E U P1 |checks and balances = − (x23 + x13 ) −
3
3
1 3
2b1 + 8 (b1 − b)3 − 2 (b1 − 2b)3
−
3
Lemma 7.1 (Comparing number of signals) For 0 ≤ b1 < b, b < yo < 1 − 4b,
checks and balances produce more signals than P1 dominance, which produces more
signals than P2 dominance does (N C B > N P1 ≥ N P2 ).
Proof Note that in Proposition 4.2, we define k1 , k2 = 0, 1, 2, 3, 4, . . . and 0 ≤
x1 , x2 < 2(b − b1 ) such that x1 + 2k1 (b − b1 ) = yo − b1 and x2 + 2k2 (b − b1 ) =
1 − (yo + 2b + b1 ). Adding up these two equations we get
2(k1 + k2 − 1)(b − b1 ) + x1 + x2 = 1 − 4b
(2)
1−4b
Proposition 4.2 shows that N CB = k1 + k2 + 2. Then N C B > 2(b−b
+ 1. From
1)
2
. It is easy to check that for
Proposition 4.1 we can show that N P1 < 21 + 21 1 + b−b
1
1√
1−4b
2
b≤
, N P1 < N C B for any b1 ∈ [0, b), because 2(b−b1 ) +1 ≥ 21 + 21 1 + b−b
.
1
4+2 2
The assumption of b < yo < 1 − 4b implies that b < 15 . For
1√
4+2 2
< b < 15 , when
b1 = 0 we know that N P1 = 2 < 3 ≤ k1 + k2 + 2 = N C B , because k1 ≥ 0, k2 ≥
1. Since when b1 increases, N C B → ∞ at a much faster speed, we conclude that
N C B > N P1 for b1 > 0 as well. That of N P1 ≥ N P2 follows from Crawford and
Sobel (1982).
Lemma 7.2 (Comparing expected payoffs) ∀ b < yo < 1 − 4b, ∀ 0 ≤ b1 ≤ b, we
have
lim E U P2 |P1 dominance > lim E U P2 |checks and balances
b1 →b
b1 →b
lim E U P2 |checks and balances > lim E U P2 |P1 dominance
b1 →0
b1 →0
In both cases, the expected payoff for P2 under P2 dominance is always the lowest
among all these authority allocations.
Proof We prove these three claims one by one.
(Part 1) From Proposition 4.2, we can show that
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500
T. Li
4
lim E U P2 |checks and balances = −b2 − b3
b1 →b
3
< −b2
= lim E U P2 |P1 dominance
b1 →b
(Part 2) When b1 → 0, E U P2 |checks and balances approaches the following
8b3
1
− (x13 + x23 ) −
(k1 + k2 + 1)
3
3
(3)
From Eq. (2) we obtain, after taking the limit when b1 → 0,
2(k1 + k2 − 1)b + x1 + x2 = 1 − 4b
(4)
Since both x1 and x2 belong to [0, 2b), there exists x ∗ ∈ [0, 2b) and k ∗ ∈
{1, 2, 3, 4, . . .} such that only one of the following two constraints holds.
(i)
x1 + x2 = x ∗
k1 + k2 = k ∗
or (ii)
x1 + x2 = x ∗ + 2(b − b1 )
k1 + k2 = k ∗ − 1
(5)
Recall also that in both cases k ∗ equals to the largest integer smaller than or equal
to 1−4b
2b + 1. We want to ask the following constrained minimization problem. For
given b, which implicitly fixes x ∗ and k ∗ , what is the minimum of (3) w.r.t x1 , x2 , and
k1 + k2 , given the constraint of either (i) or (ii)?
It is easy to check that under the first constraint, the minimum of − 13 (x13 + x23 ) could
be achieved when x1 = x ∗ and x2 = 0. Under the second constraint, the minimum of
− 13 (x13 + x23 ) could be achieved when x1 = 2b − 2b1 and x2 = x ∗ . Substituting these
two sets of values into (3), respectively with the corresponding value for k1 + k2 , we
get two possible solutions of the constrained minimization problem, which are both
equal to the following
8b3 ∗
1
k +1
− (x ∗ )3 −
3
3
(6)
So taking this minimization problem of E U P2 |checks and balances as solved
by x1 = x ∗ , x2 = 0, and k1 + k2 = k ∗ is without loss of generality. Substitute these
∗
values into (4), we have k ∗ + 1 = 1−x
2b . Substitute this value into (6) we get
1
4b2 − (x ∗ )3 −
1 − x∗
3
3
(7)
We know that x ∗ ∈ [0, 2b). It is easy to check that, when x ∗ = 0, the minimum of
(7) is −4b2 /3. We conclude that, for N ≥ 2 (or equivalently requiring b < 41 ),
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Expert advising under checks and balances
501
4
lim E U P2 |checks and balances > − b2
b1 →0
3
(b)2 N 2 − 1
1
>−
−
12N 2
3
= lim E U P2 |P1 dominance
b1 →0
(Part 3) From Krishna and Morgan (2001), we know that when b1 = b,
E U P2 |checks and balances > U P2 |P2 dominance
It is obvious that when b1 = 0, P1 or P2 dominance makes no difference. In
particular, the expected payoff for P2 would be the same.
Combined with Part (1) and (2), we have shown that P2 dominance leads to lowest
expected payoff for P2 .
Lemma 7.3 (Comparing distributive efficiencies) For any b1 ∈ [0, b],
−E 2 [X |P1 dominance] ≥ −E 2 [X |checks and balances]
−E 2 [X |P2 dominance] ≥ −E 2 [X |P1 dominance]
The 1st equality holds strict if and only if b1 = b.
The 2nd equality holds strict if and only if b1 = 0.
Proof The second inequality is obviously true, since from Proposition 4.1 we can
show that E [X |P2 dominance] = 0, and E [X |P1 dominance] = b1 ≥ 0. We know
from Proposition 4.2 and Eq. (2) that
E [X |checks and balances] =
N
|i |E i X
i=1
x1 x2 + x2 b1 −
= x1 b1 +
2
2
+ 2 (k1 + k2 ) (b − b1 ) b + 2b2 + 2bb1
x1 − x2
+ b1 − b + b
= (x1 + x2 )
2
For b1 = b, obviously we have x1 = x2 = 0 since x1 , x2 ∈ [0, 2(b − b1 )). Then
E [X |checks and balances] = E [X |P1 dominance].
For b1 < b, E [X |checks and balances] > E [X |P1 dominance] is equivalent to
x2 − x1 <
2(b − b1 )
− 2(b − b1 )
x1 + x2
(8)
For both cases of (5) we have x ∗ ≤ 1 − 4b. For case i we also have x1 + x2 <
2(b − b1 ). Combining these two results we can easily prove Eq. (8). For case ii we
can use the fact that x2 − x1 ≤ 2(b − b1 ) − x ∗ .
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T. Li
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