PhD Course
Voting and Political Debate
Lecture 7
Francesco Squintani
University of Warwick
email: f.squintani@warwick.ac.uk
1
Information Aggregation and Optimal Structure of the Executive
Model
• There is a set K = {1, ..., K} of politicians and of policy choices.
• Each agent i ∈ K has ideology bi ; b = {b1 , ..., bn }, b1 ≤ ... ≤ bn .
• Decision making is assigned according to a : K → K.
ai is the number of choices assigned to agent i.
• In the common-state model, the state θk is common across k and equals θ, uniformly
distributed on [0, 1].
• In the policy-specific information model, θk are i.i.d. across k and uniformly
distributed on [0, 1].
2
• Each politician k receives a private signal sk ∈ {0, 1} about θk ,
with Pr(sk = 1|θk ) = θk .
• If conversations are private, each politician i sends a possibly different message
m̂ij ∈ {0, 1} to any other politician j.
• If meetings are public, any player i sends a public message m̂i .
• After communication, policies ŷk ∈ ℜ chosen by politicians a(k).
• The payoff of politician i facing action profile ŷ and states θk is
∑
ui (ŷ, θ) = −
(ŷk − θk − bi )2
k∈K
Politicians are ideological about all policy choices.
3
• A pure communication strategy is mi (si ) = {miN }N ∈Ni (g) .
When miN (si )
= si , i communicates truthfully to audience N .
• A communication strategy profile is m = {m1 , ..., mn }, where
mi : {0, 1} → {0, 1}|Ni | .
• The strategy m defines the equilibrium communication network c(m):
• A policy strategy by i is yk : {0, 1}|N̄i | × {0, 1} → [0, 1] for all k = a−1 (i), where
N̄i is the set of players communicating with i.
4
• Our solution concept is pure-strategy Perfect Bayesian Equilibrium.
• A pure strategy mij : {0, 1} → {0, 1} is either truthful, if mij (si ) = si , or
babbling, if mij (si ) is independent of si .
Given the strategy profile m, define the factional network
c with cij = 1 if and only if
mij is truthful — and cii = 1.
• By sequential rationality, the equilibrium decisions are:
yk = ba(k) + E[θk |da(k),k (c), la(k),k (c)]
1.
di,k (c) is i’s equilibrium information on θk :
the number of signals on θk held by i, including her own;
2. li,k (c) is the equlibrium number of realizations equal to one.
5
• Because of Beta-binomial updating:
li,k + 1
E[θk |di,k , li,k ] =
di,k + 2
• Our welfare function is ex-ante weighted Utilitarian
∑ ∑
−
γi
E[(yk − θk − bi )2 ]
i∈K
k∈K
where γi is the weight of player i.
• We let politician j ’s moderation be the distance between bj and the average ideology
∑I
i=1 γi bi .
6
Two Forces
• By a mean-variance decomposition, the ex-ante welfare is:
[
]
∑ ∑
1
2
−
γi (bi − ba(k) ) +
6[da(k),k (c) + 2]
k∈K i∈K
• The first term in the square parenthesis is the aggregate ideological loss, the second
term is the residual variance.
Proposition 1. The optimal selection of the assignemnt a involves considering two
forces: politicians’ moderation and information.
• Intuitively, moderate policiticians lower the aggregate ideological loss, whereas
informed politicians lower the residual variance.
7
Factional Network: Private Conversations
Lemma 1 (GGS, 2010). Suppose that the state θ is common, and communication
private. The profile m is an equilibrium if and only if, whenever i is truthful to j,
1
|bi − bj | ≤
2[dj (c) + 2]
• i’s capability to communicate is independent across players.
• i’s capability to communicate with j declines with:
– their bias difference and,
– how many other players communicate truthfully with j .
8
Optimal Executive – Private Conversations
Proposition 2. For generic biases b, any optimal assignment is centralized to a single
leader j, i.e., a(k)
= j for all k.
• Intuition:
∈ a(K), the maximal equilibrium in-degree dj (c) is independent of the
set of tasks assigned to player j.
– For any j
– Maximizing the welfare function is equivalent to finding the index j that minimizes
I
∑
i=1
1
γi (bj − bi ) +
.
6(dj (c) + 2)
2
9
Factional Network: Public Meetings
Lemma 2 (GGS, 2010). Suppose that the θ is common, and communication is
public.The strategy m is equilibrium if and only if for all i communicating truthfully
∑
∑
γj (c)
bi −
,
bj γj (c) ≤
j∈K\{i} 2[dj (c) + 2]
j∈K\{i}
where
aj /[dj (c) + 2]
γj (c) = ∑
j ′ ∈K\{i} aj ′ /[dj ′ (c) + 2]
• i’s capability to communicate depends on a weighted average of all the biases of the
players.
10
Optimal Executive – Public Meetings
Consider a simple example: There are 4 players, with b1
= −β, b2 = ε, b3 = β, and
b4 = 2β, where ε is small.
For β
< 1/24 or β > 1/18, it is optimal to select 2 as the leader.
For β ∈ (1/24, 1/21), 3 decisions are optimally given to 2 and one to 3.
For β ∈ (1/21, 1/18), 2 decisions are given to 2 and 3 each.
• In fact, for β < 1/24, all players are fully informed;
• for β > 1/8, there is no truthful communication;
• for β ∈ (1/24, 1/18), player 4 is willing to talk to 2 and 3 publicly, and not to 2
privately;
• for β ∈ (1/24, 1/21), player 4 talks even if 2 is assigned 3 tasks.
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Private Conversations vs. Public Meetings
Proposition 4. For generic ideologies b, the optimal assignment with public meetings
dominates all assignments with private conversations.
Intuition:
• Under the conditions of Proposition 2, the optimal assignment with private
conversations selects a single leader.
• When a single leader is selected, private conversations and public meetings coincide.
• Hence, public meetings cover private conversations.
12
Optimal Executive – Public Meetings: Large Governments
The key result is that, as the number of politicians becomes large, the fraction of
decisions assigned to politicians close to the most moderate politician becomes one.
Proposition 5 Suppose that biases bj , j
= 1, ..., I are i.i.d. and drawn from a
distribution with connected support. For every small δ > 0, there exists a possibly large
Iδ > 0 so that for all I > Iδ , the fraction of decisions in the optimal assignment that are
not concentrated to politicians with biases b such
that |b − mI | < δ is smaller than δ,
∑I
1
b
where mI = arg mini=1...I bi − I
j=1 j .
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Intuition:
• We first prove that, as the number of politicians becomes large, allocating all
decisions to the most moderate politician aggregates all information.
• We then show that any possible informational gain is dominated by an ideological
loss, if assigning decisions to politicians significantly distant from the most moderate
one.
Note that this does not necessarily imply that all decisions are assigned to the most
moderate politician, as the number of politicians explodes.
There may still be informational gains at small ideological loss, when assigning decisions
to politicians close to the most moderate one.
14
Public Meetings – Small Government: Numerical Simulations
We ran simulations with I
= 7 politicians, and biases b drawn i.i.d. from skew normal
distributions of variance σ 2 and skewness γ .
Authority is highly concentrated to the leader: at least 5.51 decisions out of 7 are given to
the leader.
Cabinet executive significantly improves upon Ministerial government. The fraction of
draws for which all decisions is allocated to the leader may be as low as 36 %.
Both quantities are U-shaped in the variance of the bias distribution. The skewness
seems not to have a major effect.
15
γ=0
γ = 1/4
γ = 1/2
γ = 3/4
γ=1
σ 2 = 10
7.00
7.00
7.00
6.99
7.00
σ2 = 1
6.91
6.93
6.89
6.91
6.88
σ 2 = 0.1
6.17
6.21
6.18
6.16
6.21
σ 2 = 0.01
5.58
5.51
5.53
5.68
5.67
σ 2 = 0.001
6.35
6.35
6.37
6.49
6.35
σ 2 = 0.0001
7.00
7.00
7.00
7.00
7.00
Average number of decisions assigned to the leader.
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γ=0
γ = 1/4
γ = 1/2
γ = 3/4
γ=1
σ 2 = 10
1.00
1.00
1.00
0.99
1.00
σ2 = 1
0.95
0.96
0.93
0.95
0.92
σ 2 = 0.1
0.57
0.60
0.58
0.61
0.62
σ 2 = 0.01
0.40
0.36
0.36
0.41
0.45
σ 2 = 0.001
0.70
0.70
0.74
0.78
0.69
σ 2 = 0.0001
1.00
1.00
1.00
1.00
1.00
Fraction of draws where all decisions are assigned to the leader.
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Factional Network – Policy Specific Knowledge
Proposition 5. Suppose that θk are i.i.d. across k. The profile (m, y) is an equilibrium
if and only if, whenever i is truthful to j
= a(i),
|bi − bj | ≤ 1/6
• Intuition:
– Independently of the form of communication, there is only one signal to
communicate to a single player.
– Hence there is no communication congestion.
– Nor the average bias in the audience matters.
18
Optimal Executive – Policy Specific Knowledge
Corollary 2. With policy specific knowledge, private and public communication coincide.
Proposition 6. When knowledge is policy specific, each decision k is assigned to either
the most moderate politician m∗ or to m(k), the most moderate politician with bias
within 1/6 of bi , depending on whether
(
)2
∑I
2
∗
[
b
−
b
]/I
−
[(b
−
b
)
]/I < (>)1/36.
i
i
m
m(k)
i=1
i=1
∑I
The tradeoff is between the most moderate politician m∗ and the most moderate
informed politician m(k).
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Proposition 7. Despite policy specific knowledge, full decentralization is never optimal
for generic ideologies b.
A leadership of the few naturally arises even in the presence of policy specific
information.
∑I
∗
Proposition 8. If there is a politician m with bm∗ =
i=1 bi /I, then each decision k
such that |bk − bm∗ | ∈ (0, 1/6] ∪ [1/3, ∞] is optimally assigned to m∗ .
This result uncovers an interesting non-monotonicity, that suggests that the most
moderate politician makes a large fraction of decisions.
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Policy Specific Knowledge – Simulations
γ=0
γ = 1/4
γ = 1/2
γ = 3/4
γ=1
σ 2 = 10
6.96
6.95
6.97
6.97
6.97
σ2 = 1
6.70
6.73
6.69
6.81
6.74
σ 2 = 0.1
5.97
5.90
5.90
5.86
5.75
σ 2 = 0.01
6.59
6.54
6.53
6.57
6.55
σ 2 = 0.001
7.00
7.00
7.00
7.00
7.00
σ 2 = 0.0001
7.00
7.00
7.00
7.00
7.00
The average number of decisions assigned to the leader is not smaller than in the
common state model.
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