PhD Course
Voting and Political Debate
Lecture 5
Francesco Squintani
University of Warwick
email: f.squintani@warwick.ac.uk
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Strategic Communication (Crawford and Sobel, 1982)
Model
• An expert privately observes a state t ∈ [0, 1].
• The expert sends a report r ∈ ℜ (a real number) to a decision maker.
• After receiving the report, the decision maker chooses an action y ∈ ℜ.
• The DM knows that t is uniformly distributed in [0, 1]: the p.d.f. is f (t) = 1 for
t ∈ [0, 1].
• The players payoffs are independent of r. (I.e., the report is cheap talk).
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• Both players care that the action matches the state.
But the expert is biased relatively to the DM.
• The experts payoff is uE (y, t) = −(y − (t + b))2 .
(Her optimal action given the state t is y
= t + b).
• The DM’s payoff is uDM (y, t) = −(y − t)2 .
(Her optimal action given the state t is y
= t).
3
Perfect Communication
• I first show there cannot be a PBE in which the expert accurately reports the state.
(I.e., an invertible strategy ρ cannot be part of a PBE.)
• By contradiction, say that ρ is invertible and PBE.
• Then, beliefs consistency implies that the DM believes (correctly) that the state is t,
when the report is r = ρ(t).
• Hence, upon receiving report r = ρ(t), the DM chooses the action y = t.
• Anticipating this, when the state is t, the expert deviates from the strategy ρ.
• Instead of sending the report r = ρ(t), she sends the report r = ρ(t + b),
which induces the action y = t + b.
• Hence, ρ is not sequentially rational, and cannot be part of a PBE.
4
Perfect Bayesian Equilibrium
• By the above result, every PBE is described by a partition of the state space [0, 1].
• Each partition is composed of K intervals, where the admissible set of K depends
on the bias level b.
• The K intervals are [t0 , t1 ), [t1 , t2 ), ..., [tK−1 , tK ], where t0 = 0 and tK = 1.
• For any k = 1, ..., K − 1, all expert types t ∈ [tk−1 , tk ) send the same report rk .
• Hence, upon receiving report rk , the DM knows that t is in [tk−1 , tk ), and chooses y
so as to maximize
∫
−
tk
tk−1
(y − t)2
dt.
Pr{t ∈ [tk−1 , tk )}
5
• Taking the first-order condition,
[
]
∫ tk
tk−1 + tk
0 = −2
(y − t)dt = −2 y −
.
2
tk−1
• So, upon receiving report rk , the DM chooses the action y = (tk−1 + tk )/2.
• In a PBE, the expert anticipates that if reporting r = rk , she induces the action
y = (tk−1 + tk )/2.
• For any k = 1, ..., K − 1, an expert of type tk must be indifferent between sending
rk and rk+1 .
[
]2
[
]2
tk−1 + tk
tk + tk+1
I.e., −
− (tk + b) = −
− (tk + b) ,
2
2
(1)
or tk+1 − tk = tk − tk−1 + 4b.
• If conditions (1) hold, then, for any k = 1, ..., K, every type of expert t ∈ [tk−1 , tk )
prefers to send report r = rk than any other report.
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• The conditions (1) yield a second-order difference equation, solved with boundary
conditions t0 = 0 and tK = 1.
• Solving the difference equation, we obtain that the interval [tk−1 , tk ) has length
tk − tk−1 = t1 + (k − 1)4b.
• By the boundary conditions, the sum of the [tk−1 , tk ) for k = 1, ..., K, equals 1:
t1 + (t1 + 4b) + ... + [t1 + (K − 1)4b] = 1,
or
i.e.,
Kt1 + 4b[1 + ... + (K − 1)b] = 1,
Kt1 + 2bK(K − 1) = 1.
(2)
• So, if b is small enough that 2bK(K − 1) < 1, then there is a value of t1 > 0 that
satisfies condition (2), and hence a PBE with K intervals.
• For larger b, the maximum number of intervals K ∗ that can be a PBE is larger.
Specifically, for b > 1/4, K ∗ = 1. For 1/4 < b < 1/12, K ∗ = 2, etc.
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• I conclude by ranking the PBE in ex-ante sense (i.e., before the state t is realized).
• The experts ex-ante equilibrium payoff is:
EUE = −
K
∑
∫
Pr{t ∈ [tk−1 , tk )}
tk
tk−1
k=1
[(tk−1 + tk )/2 − (t + b)]2
dt.
Pr[tk−1 , tk )
• The decision makers ex-ante equilibrium payoff is:
∫ tk
K
∑
[(tk−1 + tk )/2 − t]2
EUDM = −
Pr{t ∈ [tk−1 , tk )}
dt.
Pr[tk−1 , tk )
tk−1
k=1
• Expanding the experts ex-ante equilibrium payoff,
K ∫ tk
∑
EUE = −
{[(tk−1 + tk )/2 − t]2 + b2 − 2b[(tk−1 + tk )/2 − t]}dt.
k=1
tk−1
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• Because, for any k,
∫ tk
[(tk−1 + tk )/2 − t]dt = (tk−1 + tk )/2 − (tk−1 + tk )/2 = 0,
tk−1
we obtain:
EUE = −
K ∫
∑
k=1
tk
[(tk−1 + tk )/2 − t]2 dt − b2 .
tk−1
• In sum, the experts and the decision makers ex-ante equilibrium payoff differ only by
the constant b2 .
• The PBE that maximizes a player’s payoff, maximizes also the other’s.
• Further, it can be shown that, for any b, the PBE maximizing ex-ante payoff is the one
with K ∗ intervals.
• This is because the sum of the loss terms in the ex-ante payoff expression is larger,
the coarser is the equilibrium partition.
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Related Literature
• Communication Games Theory: Sobel (1985), Aumann and Hart (2003), Forges
(1986), Morgan and Stocken (2008), ...
• Political Economy: Gilligan and Krehbiel (1987, 1989), Austen-Smith (1990), Morris
(2001), Battaglini (2002), Horner, Morelli and Squintani (2015), ...
Political Debate: Penn (2015), Patty (2015), Dewan, Galeotti, Ghiglino and Squintani
(2015), Dewan and Squintani (2015), ...
• Financial Advice: Benabou and Laroque (1992), Ottaviani and Sorensen (2006a,
2006b), ...
• Behavioral Cheap Talk: Crawford (2003), Kartik, Ottaviani and Squintani (2007),
Kartik (2009), ...
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• Experiments: Dickhaut, McCabe and Mukherji (1995), Costa-Gomes, Crawford,
Broseta (2001), Wang, Spezio and Camerer (2010), ...
• Verifiable or Costly Talk: Milgrom (1981), Milgrom and Roberts (1986), Dewatripont
and Tirole (2005), Kamenica and Gentzkow (2011), ...
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Multi-player Communication (Galeotti, Ghiglino and Squintani)
Model
• State of nature θ uniformly distributed on [0, 1].
• N = {1, ..., n} players.
• Each player i has bias bi ; b = {b1 , ..., bn }, b1 ≤ b2 ≤, ..., ≤ bn .
• Each player i receives a private signal si ∈ {0, 1} about θ, with Pr(s = 1|θ) = θ.
• The exogenous communication network g ∈ {0, 1} describes “who might talk to
whom”: when gij = 1, i can send a message to j .
• The communication neighborhood of i is Ni (g) = {j ∈ N : gij = 1}.
• The audiences are a partition Ni (g) of Ni (g) : i must send the same message to
all players in audience J, for all J ∈ Ni (g).
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Model: First Stage
• A pure communication strategy is mi (si ) = {miJ }J∈Ni (g) , when miJ (si ) = si , i
communicates truthfully to audience J .
• m = {m1 , ..., mn } communication strategy profile, where
mi : {0, 1} → {0, 1}|Ni (g)| .
• g and m define the equilibrium communication network c(m|g):
cij (m|g) = 1 if and only if gij = 1 and mij (s) = s for every s ∈ {0, 1}.
• Clearly, c(m|g) is a subgraph of g.
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Model: Second Stage
• After communication, each player i chooses an action ŷi ∈ [0, 1].
• An action strategy is yi : {0, 1}
|Ni−1 (g)|
× {0, 1} → [0, 1], where Ni−1 (g) is the
set of players communicating with i.
• y is the action strategy profile.
• The payoffs of i facing y and θ is
ui (y|θ) = −
∑
(yj − θ − bi )2
j∈N
• Natural extension of Crawford and Sobel’s quadratic utility to a network.
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Equilibrium and Welfare Concept
• Our notion of efficiency is ex-ante Pareto optimality.
• The equilibrium concept is pure-strategy Perfect Bayesian Equilibrium.
• Given the received messages mNi−1 (g),i , by sequential rationality agent i chooses yi
to maximize his expected utility, which yields:
yi (si , mNi−1 (g),i ) = bi + E[θ|si , mNi−1 (g),i ]
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Equilibrium: Private Communication
• Given an equilibrium communication network c(m), let kj (m) be the in-degree of j,
i.e., the number of agents who are truthful to j.
Corollary 1. For any communication network g, a profile (m, y) is an equilibrium if
and only if, whenever i is truthful to j,
1
|bi − bj | ≤
2[kj (m) + 3]
• 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 .
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Intuition Corollary 1 (the ‘equilibrium congestion effect’)
• If i is truthful to j , then yj increases if mij = 1, while it decreases if mij = 0.
• The effect of i’s message on j ’s action depends on how well informed j is:
– it is large if j is poorly informed,
– while it is small if j is well informed.
• Now, suppose i has an incentive to bias j ’s action upward, i.e., bi > bj , and that i
has received signal si = 0.
• When j is well informed, if i mis-reports si = 0 and sends message mij = 1, then
j ’s action increases only slightly and it gets closer to i’s bliss point bi + E[θ|Ωi ].
• So, i deviates from truthful communication, which cannot be part of an equilibrium.
• When j is poorly informed, if i mis-reports si = 0 and sends mij = 1, then j ’s
action increases substantially, possibly overshooting i’s bliss point.
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Equilibrium: General Audiences
• Consider a player i with communication neighborhood Ni (g).
• The communication technology of i can be represented as a partition of his
communication neighborhood, Ni (g).
• Player i must send the same message to all j ∈ J , for each J ∈ Ni (g).
• With private communication, the partition Ni (g) is composed of singleton sets.
• The trivial partition Ni (g) = {Ni (g)} models public communication.
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Theorem 1. Consider a communication network g and a collection of audiences
{Ni (g)}i∈N . The strategy profile (m, y) is an equilibrium if and only if, whenever i is
truthful to an audience J ∈ Ni (g) ,
∑ b − b ∑
1
j
i .
≤
2
kj (m) + 3 j∈J 2 (kj (m) + 3)
j∈J
• Player i will be willing to communicate if the average bias difference with players in
the audience J is not too large.
• Within the audience J, the relevance of each player j ’s bias difference bj − bi in the
above average is inversely related to j ’s in-degree kj (m), i.e., to how informed j is.
• We call these insights the ‘equilibrium average bias effect’.
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Pareto Optimality
• Given an equilibrium (m, y), let the proportion of players with in-degree k be
denoted by P (k|m).
Theorem 2. Consider communication networks g and g′ , with associated equilibria
(m, y) and (m′ , y′ ) respectively. The equilibrium (m, y) Pareto dominates the
equilibrium (m′ , y′ ) if and only if
n−1
∑
1
P (k|m)
<
k+3
k=0
20
n−1
∑
1
.
P (k|m )
k+3
k=0
′
• We can therefore rank equilibria based on their distribution of truthful messages
Corollary 2 (First Order Stochastic Shifts). If P (k|m) First Order Stochastically
Dominates P (k|m′ ) then equilibrium (m, y) Pareto dominates equilibrium (m′ , y′ )
Corollary 3 (Mean Preserving Spread). If P (k|m′ ) is a Mean Preserving Spread
of P (k|m) then (m, y) Pareto dominates equilibrium (m′ , y′ )
• Redistributing truthful reports evenly across players leads to a Pareto improvement.
• We call this result the ‘welfare congestion effect’.
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Related Literature
• Multiple Senders - One Receiver: Battaglini (2001), Levy and Razin (2007), Ambrus
and Takahashi (2008), Meyer, Moreno de Barreda and Nafziger (2014).
• One Sender - Two Receivers: Farrell and Gibbons (1989).
• Communication in Organizations: Wolinsky (2002), Dessein (2002), Alonso, Dessein
and Matouscheck (2008), Migrow (2015).
• Organization Design: Sah and Stiglitz (1986), Radner (1993), Bolton and Dewatripont
(1994), Garicano (2000), Cremer Garicano and Prat (2007), ...
• Network Economics: Bala and Goyal (2000), Jackson and Wolinsky (1996), Ballester,
Calvo-Armengol and Zenou (2006), ...
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