Does Majority Rule Produce
Hasty Decisions?
Jimmy Chan1
1 Fudan
2 University
Wing Suen2
University
of Hong Kong
October 18, 2013—Ohio State University
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Introduction
Madison, The Federalist Papers, No. 58:
[The supermajority requirement] might have been an additional
shield to some particular interests, and another obstacle generally
to hasty and partial measures.
[A]n interested minority might take advantage of it to screen
themselves from equitable sacrifices to the general weal, or in
particular emergencies to extort unreasonable indulgences.
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Introduction
Collective decision-making is not just about casting a vote.
A standards setting committee involves engineers to test and
discuss the proposed standards.
Countries deciding on whether and how to cut greenhouse gas
emissions engage a panel of scientists to gather evidence to guide
their actions.
During jury deliberation, the jurors together examine and learn from
the evidence presented at trial to form an opinion for a verdict.
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Introduction
These examples share common features:
Information discovered at the deliberative stage is public knowledge
to the group members.
Information discovery is costly
Members of the group themselves can decide to keep deliberating
until they reach a decision.
Members have different preferences for alternatives as well as
different trade-offs between quality of decisions versus speed
patient vs. impatient agents
high-stake vs. low-stake agents
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Introduction
Use a collective version of the “sequential probability ratio test”
(Abraham Wald 1947) to study the trade-off between the cost of
collecting additional information and the benefit from making a
more informed decision.
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Related Literature
Collective search/experimentation: Albrecht, Anderson and
Vroman (2010); Compte and Jehiel (2010a; 2010b ); Strulovici
(2010): loss of control
Lizzeri and Yariv (2012): one-dimensional heterogeneity
Gul and Pesendorfer (2012): two political parties
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Model
2m − 1 agents are choosing between two alternatives: α and β.
In state A, agent i’s payoff to α is 1, and his payoff to β is 0. In
state B, agent i’s payoff to α is 0, and his payoff to β is evi .
θ0 = log(Pr[ω = A]/ Pr[ω = B]) represents the common initial
belief.
The immediate expected payoff from α is higher than β if and only if
θ ≥ vi .
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Model
At each time t, each agent independently votes for α or β, or
abstains.
Decision rule is k ∈ {m, m + 1, . . . , 2m − 1}: an alternative is
adopted at time t if it is supported by k agents or more; the
decision process continues if neither α nor β receives sufficient
votes.
k = m corresponds to majority rule.
Agent i discounts future payoff at a rate ri .
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Model
Information accumulation is represented by a Wiener process dS:
State A: dS = µdt + ρdW
State B: dS = −µdt + ρdW
Accumulated evidence St is a sufficient statistic. Log-likelihood
ratio of observing St = s is 2µs/ρ2 .
Let St0 = 2µSt /ρ2 . Common belief at time t is:
θt = θ0 + St0 .
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Model
Focus on Markov threshold strategies: votes for α when θt ≥ Gi ,
votes for β when θt ≤ gi , and abstains when θt ∈ (gi , Gi ).
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Model
Let G be the k -th smallest Gi among the group.
if θt > G, α is adopted
payoff to agent i is eθt /(1 + eθt )
Let g be the k -th largest gi among the group.
if θt < g, β is adopted
payoff to agent i is evi /(1 + eθt )
If θt ∈ [g, G], deliberation continues. Agent i’s payoff satisfies:
ui (g, G|θ) = e−ri dt E[ui (g, G|θ + dS 0 )]
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Strategic Best Response in Stopping Decisions
Φi (g) is the optimal G that maximizes ui (g, G | θ) subject to G ≥ g.
It is the one-sided best-response stopping boundary for taking
decision α if the lower boundary for taking β is fixed at g.
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Strategic Best Response in Stopping Decisions
(g∗i ,G∗i )
G
(vi,vi)
φi
Φi
g
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Strategic Best Response in Stopping Decisions
G
φi(.; ri)
φi(.; ri′)
φi(.; ri″)
vi
ri″ > ri′ > ri
g
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Strategic Best Response in Stopping Decisions
Waiting has no value if g ≥ vi .
Waiting has little value if g is near vi .
Non-monotonicity reflects “loss of control.”
The best-response waiting window can be arbitrarily narrow if
lower boundary is fixed at near vi or if ri is very high.
Φi (g) ≤ Gi∗ for all g ≤ vi .
There is a strategic difference between too much waiting and too
little waiting.
Φi (g) can be lower than vi : agent prefers to stop and take α even
though β is better than α at that point.
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Equilibrium Analysis
“Pivotal” best response functions: Φpiv (g, k ) is the k -th smallest
Φi (g) for each g.
(ĝ, Ĝ) is an equilibrium if and only if it is a fixed point of (Φpiv , φpiv ).
One-sided best-response suffices for the two-sided problem.
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Equilibrium Analysis
φ2
φ3
φ1
G
G″
v3
G′
v2
v1
φpiv
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Equilibrium Analysis
Equilibrium Ĝ is no greater than the k -th smallest Gi∗ .
If all agents have the same discount rate, then equilibrium is
unique.
If agents have different discount rates, multiple equilibria may
exist.
In the most patient and the least patient equilibrium, Ĝ − ĝ
decreases in ri .
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Rushing to a Decision
One vote is sufficient to tilt the decision from α to β (or vice versa)
under majority rule k = m.
When ri is sufficiently high, agent i will always be pivotal against
agent m under majority rule.
This impatient agent i is willing to wait very little before voting for
an alternative.
Strategic complementarity in the decision to stop early then
suggests that the other pivotal agent m will respond by cutting
short his stopping threshold as well.
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Rushing to a Decision
Called a hasty equilibrium, because, starting at any initial belief
θ0 ∈ (ĝ, Ĝ), the time it takes for the belief θt = θ0 + St0 to reach the
boundaries is short.
Whether the true state is A or B has little effect on the probability
of choosing α.
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Rushing to a Decision
(a) r1 = r2 = r3
φ2
Φ3
Φ2
(b) r1 > r2 = r3
φ3
φ2
(c) r1 >> r2 = r3
φ3
φ2
Φ3
.
P1
.
Φ2
Φ2
P2
v3
φ3
Φ3
.
v3
v3
P3
G
v2
v1
G
v2
G
v1
v2
v1
φ1
Φ1
g
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Rushing to a Decision
Think of an impatient agent as a “swing voter.” He may be biased
in favor of α, but because of discounting he is willing to settle for β
as long as the evidence swings a bit to the left.
These agents are easy targets of alliance.
To avoid some members of the group from capturing these
impatient voters to adopt one alternative, the other members who
are biased for the other alternative will cut short the deliberative
process by pushing forward the stopping threshold for their
favored alternative as well.
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Rushing to a Decision
The effects of very patient agents and very impatient agents are
not symmetric.
patient agents are less likely to be pivotal
patient agents induce strategic substitution by other agents
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Rushing to a Decision
(a) r2 = r1 = r3
φ1
Φ3
(b) r2 << r1 = r3
φ3
.
φ1
.
Φ3
φ3
P4
P1
v3
Φ1
G
v2
v1
v3
Φ1
G
v2
v1
φ2
Φ2
g
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Supermajority Rule is More Robust
Ĝ − ĝ increases in k .
We can avoid a hasty equilibrium using decision rule k if the
number of impatient agents does not exceed 2(k − m).
There are at least 2k − (2m − 1) = 2(k − m) + 1 agents who are in
the winning coalition for both α and β.
At least one agent who is in the winning coalition for both α and β is
not impatient.
Such a patient agent cannot be maximizing his utility in a hasty
equilibrium.
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Supermajority Rule is More Robust
If the number of very patient agents does not exceed 2m − k − 1,
we can avoid an equilibrium with excessively long deliberations.
Example: 1/4 very impatient agents, 1/4 very patient agents, 1/2
“normal agents.” A super-majority rule with between 5/8 to 3/4
super-majority requirement can avoid very short or very long
deliberations.
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Supermajority Rule Has its Drawbacks
Majority rule (k = m) always respects the static preferences of
agents: the alternative chosen is preferred by a majority of the
group.
Supermajority rule (k > m) does not have this property: there
exist preference profiles such that in equilibrium only 2m-k agents
favor the alternative chosen by the group.
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Supermajority Rule Has its Drawbacks
Supermajority rule can produce an extortive equilibrium: A
minority can hold out for its favored decision, with the remaining
majority agreeing to it simply because they don’t want to wait too
long.
Φi (ĝ) falls without bound as ĝ goes to minus infinity
this is an extreme manifestation of strategic substitution
supermajority rule lengthens deliberation but does not necessarily
promote consensus
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Thank you!
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