Learning While Voting:
Determinants of Collective Experimentation
Bruno Strulovici∗
December 15, 2009

Abstract
This paper combines dynamic social choice and strategic experimentation to study thefollowing question: how does a society, a committee, or, more generally, a group of individuals
with potentially heterogeneous preferences, experiment with new opportunities? Each voter
recognizes that, during experimentation, other voters also learn about their preferences. As aresult, pivotal voters today are biased against experimentation because it reduces their likelihood of remaining pivotal. This phenomenon reduces equilibrium experimentation below
the socially efficient level, and may even result in a negative option value of experimentation.
However, one can restore efficiency by designing a voting rule that depends deterministicallyon time. Another main result is that, even when payoffs of a reform are independently
distributed across the population, good news about any individual’s payoff increases otherindividuals’ incentives to experiment with that reform, due to a positive voting externality.

∗

Northwestern University. I am grateful to Meg Meyer, Paul Milgrom, Andrea Patacconi, and Kevin Roberts

for insightful conversations, as well as to the editor and three anonymous referees, Daron Acemoglu, Patrick
Bolton, Alessandra Casella, Eddie Dekel, Jeff Ely, Bard Harstad, Godfrey Keller, Marco Ottaviani, Andrea Prat,
John Quah, Phil Reny, Marzena Rostek, Nicolas Vieille, Marek Weretka, and seminar participants at Oxford
University, Gerzensee’s ESSET, Cambridge University, the London School of Economics, the Kellogg School ofManagement, Princeton University, Northwestern University, the California Institute of Technology, the Universityof Pennsylvania, Columbia University, Stanford Graduate School of Business, University of Arizona, Penn State
University, the University of Toronto, University of Texas at Austin, CESifo’s conference on Strategic Information
Acquisition and Transmission, the University of Washington at St Louis, the Winter Meeting of the EconometricSociety 2009, the joint seminar of the École Polytechnique and HEC, the University of Tokyo and CIRJE, NewYork University, the University of Chicago, and UCLA for numerous comments.

1

Introduction

Every reform has consequences which cannot be fully known until it has been implemented. For
example, the diverse effects of trade liberalization on a country’s industrial sectors (e.g., which
sectors will gain or lose from liberalization, and when) cannot be easily or fully anticipated.
Similarly, although economic liberalization in the form of more business-friendly laws and fiscal
policy can be expected to create opportunities for entrepreneurship and individual success, its
specific beneficiaries are also unknown a priori. Moreover, those individuals who are not amongits beneficiaries but rather “losers” in it its wake may experience such hardships/penalties as
reduced income redistribution or job security. Other contemporary examples of reforms whose
benefits and costs are both uncertain and heterogeneous across the population are reforms inhealth care, national security policies, or environmental regulations.
This paper studies incentives for collective experimentation when individual interests may bein conflict and are revealed gradually and at times which are random and may also vary acrossindividuals. It addresses the following questions: How do these incentives evolve as preferences
and heterogeneity get revealed? How do they vary with group size? How do they compare to
the utilitarian optimum? How do they vary with the voting rule chosen? Under which collective
decision mechanisms do they result in efficient experimentation? How are they affected by theparticular learning process?
The analysis is conducted in a two-armed bandit model in which a safe alternative yields a
constant, homogeneous payoff to everyone, while a risky alternative yields payoffs whose unknown distribution, or type, may vary across individuals. At each instant, society elects one of
the two alternatives according to some voting rule. Individuals learn their type only throughexperimentation with the risky alternative.1 In the benchmark setting the risky action is, foreach individual, either good or bad, and these types are independently distributed across individuals. Moreover, any news shock fully reveals to its recipient that the risky action is good for
him, i.e., he is a sure winner. By contrast, unsure voters are those individuals who have not yet
received any positive news about their type, and who become increasingly more pessimistic asexperimentation goes on. The benchmark setting focuses on simple majority voting, with other1

Focusing on two actions gets rid of Condorcet cycles and ensures the robustness of the equilibrium concept

used in the analysis. An example with three actions is studied section 7.

2

voting rules considered in later sections. Payoffs are initially assumed to be publicly observed,2
but section 6 considers the case of privately observed payoffs. In the benchmark setting, learning
occurs at the individual level only (section 7 discusses the case of correlated types). The factthat an individual becomes successful, however, changes the unsure voters’ expected payoffs,
since it makes it more likely that the reform will not be overturned.
The first result is that incentives for experimentation are always weaker when power is shared,
compared to the case of a single decision maker, or to a dictatorship. Two kinds of risk shape
incentives for collective experimentation, in addition to the well-known trade-off between exploration and exploitation arising in individual experimentation. The loser trap occurs whenreform winners have enough power to make the reform irreversible, in effect trapping reform
losers into this new course of action. In contrast, winner frustration occurs when reform losers(more precisely, unsure voters with a low enough belief) are powerful enough to effect a returnto the status quo, frustrating reform winners and whoever else still wishes to continue experimentation. Costly reforms and projects may thus be abandoned too early if they do not garnerenough support, even when they turn out to be ex post efficient. These risks, which are specificto decisions taken collectively and where interests may turn out to be heterogeneous, reduce
incentives for experimentation.
To illustrate, consider a community of N individuals with equal voting rights. Every month,these individuals must decide between a centralized production, where tasks are fixed and earnings are divided equally, and a decentralized one, where each individual chooses his task andkeeps his earnings. There are two types in this community: individuals with talent and those
without, where “talent” refers to an individual’s ability to find a successful task. If the community tries decentralization, individuals gradually find out whether or not they are talented.
As time elapses, two things can happen: A majority of talented people may form, in whichcase decentralization is imposed forever. Alternatively, if few talents are revealed under decentralization, voters who remain unsure can impose reversal to a centralized production.3 In
the first case, untalented people are trapped in a community that essentially abandons themeconomically. In the second case, talented people are frustrated by the collective constraint. If
these risks are severe enough ex ante, the community may prefer not to experiment at all withdecentralization, even if it is efficient to do so.
Loser trap and winner frustration have a systematic impact on welfare: experimentation in2
3

Voters care only about the number of sure winners at any time, not about their identity.
For the sake of this example, we suppose that a centralized production is better for individuals who are sureof being untalented, and yields deterministic and homogeneous payoffs.

3

centives are always too weak, compared to the utilitarian optimum. This result stems from two
effects. First, the utilitarian policy, which is the optimum of a single decision-maker problem(faced by a utilitarian social planner), is not subject to the control sharing effects describedearlier. The value of the information acquired through experimentation is thus maximal, whichmakes experimentation more valuable from an efficiency perspective. Second, unsure voters only
care about their own utility and thus sometimes impose the safe action even when including surewinners’ utility would make the risky action more efficient.
This social inefficiency remains for any fixed voting rule. For example, if the risky action requires
unanimity, the risk of loser trap disappears. However, this very fact also makes experimentation
less attractive: winners are less likely to enjoy the risky action in the long run, for this would
require that all society members turn out to be winners. Unanimity rule thus exacerbates
winner frustration. Similarly, if the safe action requires unanimity, the risk of winner frustration
disappears, but the risk of loser trap becomes maximal.
However, efficiency can be restored by a voting rule that depends deterministically on time.
To implement the efficient policy, the number of votes required for the risky action increases
deterministically over time, according to a schedule agreed upon at the outset. Intuitively,
the more time elapses, the more numerous sure winners should be, if the reform is efficient.
Therefore, one way to make sure that the reform continues only if it is efficient is to graduallyraise the voting threshold required for it to be implemented. As the paper shows, the threshold
can be set so as to exactly implement the utilitarian policy.
Another dynamic aspect of experimentation concerns the impact of incoming news on experimentation incentives. To return to the example given earlier, how do other voters react wheneversomeone discovers a talent? The answer is that, in equilibrium, good news for anyone increases
others’ incentives to experiment. Intuitively, individuals vote for experimentation because they
hope to be winners and, hence, to enjoy the reform in the longer run. The appearance of a newwinner makes it more likely that others too will be able to enjoy the reform and thus makes it
more valuable to experiment.
As group size gets arbitrarily large, voters behave myopically, as if there were no value inexperimentation. Indeed, individual control over future decisions becomes infinitely diluted, so
one’s ability to react to individual news vanishes in an arbitrarily large group. For small groups,
however, equilibrium incentives for experimentation do not monotonically decrease with respect
to group size. This is because the addition of new voters reduces the risk of winner frustration,a benefit that may locally dominate a higher risk of loser trap.
4

Several key results are extended to general experimentation environments, beyond the benchmark setting. Most importantly, the main result on weak incentives for experimentation holds
even when individuals i) never fully learn their types, ii) receive both positive and negative news,
and/or iii) have correlated types. The analysis is based on a collective version of the Gittins
index. However, it is not true any more that experimentation is inefficiently short. In particular, Section 5.1 shows that with negative shocks, experimentation may be inefficiently shortor long depending on initial conditions. The paper also introduces a nonadversity condition on
the collective decision process under which the value of experimentation is always positive. A
collective decision rule is nonadverse to a given individual if, at any time, it is more likely to
select the risky action if that individual is a winner than if he is a loser.
Surprisingly, however, even fair-looking decision rules, such as the simple majority rule, canviolate the nonadversity condition. The value of experimentation may even be negative, in thatsociety may reject a reform with a higher expected payoff than that of the status quo.4 Thismeans that the common intuition of a positive “option value,” which captures a decision maker’s
ability to react to news (e.g. financial options, real options, options of waiting in endogenous
bankruptcy models), may be inaccurate in settings with multiple decision makers. In contrast,
the value of experimentation is always nonnegative when voters use the unanimity rule.
The paper contributes to a developing literature on experimentation with multiple agents, inwhich conservatism may arise as a consequence of strategic information acquisition. Bolton andHarris (1999), Li (2001), Décamps and Mariotti (2004), and Keller, Rady, and Cripps (2005)
analyze an informational free-riding problem in settings where agents can experiment individually with some risky action to learn about its common value. By contrast, the present paper
considers a reverse setting, in which a single action taken collectively is made at any time, but thevalue of the action varies across individuals.5 In these papers, experimentation is inefficiently
4

The result does not rely on commitment ability or asymmetric information, but is due solely to control sharing

effects, as shown here and in Section 5.2.
5In this way, the paper is also related to a burgeoning literature analyzing collective search in various settings,
where a group must choose, at any time, between accepting some outstanding proposal or trying a new proposal
with i.i.d. characteristics. Compte and Jehiel (2008) show, in particular, that more stringent majority requirements select more efficient proposals but take more time to do so. Albrecht, Andersen, and Vroman (2007) findthat committees are more permissive than a single decision maker facing an otherwise identical search problem.
Messner and Polborn (2008) discuss correlation across the two periods of their setting. In contrast to those papers, the present work focuses on social and individual learning and experimentation when voter types for a given
action are lasting and permanently influence collective decisions. Callander (2009) also considers experimentation
in a political setting. His focus is on the experimentation pattern of a single decision maker, the median voter,facing a continuum of correlated policies. Although the median voter is myopic and non-strategic, the nature of5low due to positive information spillovers that are not internalized by agents. In contrast, thecontrol sharing effects in the present paper reduce experimentation due to the negative payoff
externalities that voters impose on one another, and which are decomposed and quantified in
Equations (4) and (5). The analysis of the benchmark setting owes conceptual and technicalclarity to the use of exponential bandits, building on Keller, Rady, and Cripps (2005).6
The paper is related to Fernandez and Rodrik (1991), who identify an asymmetry between ex
ante and ex post support for reforms, which is due to uncertainty about winners’ identity. Votersknow that if the reform is implemented once, it will surely be implemented afterwards. However,they do not know whether they are winners or losers under the reform, and hence bear the riskof loser trap. Their setting is similar to the case of immediate type revelation and an infinite
population in the benchmark setting presented here. In the present paper, individuals learn at
different points in time, and the efficient policy is unknown a priori. The evolution of informed
agents introduces some of the interesting strategic issues that were absent from the model ofFernandez and Rodrik (1991).
The paper is organized as follows. Section 2 below analyzes the benchmark setting under the simple majority rule. Section 3 considers the utilitarian optimum and compares it to the majorityvoting equilibrium . Section 4 takes a broader design approach to voting procedures, showing
which rules can restore efficiency. Section 5 extends the analysis to more general type and learning structures, where types are never fully revealed, news can be positive and/or negative, andtypes may be correlated. Section 6 considers the case of publicly observed payoffs, showing thatthe majority voting equilibrium of Section 2 is truthful. Section 7 discusses several assumptionsof the model, and Section 8 concludes.

2

Benchmark Setting

The benchmark setting embeds the exponential bandit model analyzed by Keller, Rady, andCripps (2005) into a setting with majority voting. Time t ∈ [0, ∞) is continuous and payoffs
are discounted at rate r > 0. There is an odd number N ≥ 1 of individuals who continually
decide according to the simple majority rule which of two actions to choose. The first action S is“safe” and yields a flow s per unit of time to all individuals. The second action R is “risky” anduncertainty in that model produces interesting experimentation patterns.
6
Exponential bandits have also been used in economics by Malueg and Tsutsui (1997), Bergemann andHege (1998, 2001), and Décamps and Mariotti (2004).

6

can be, for each player, either “good” or “bad.” The types (good and bad) are independently
distributed across the group. (The case of correlated types is discussed in Section 7.)
If R is bad for some individual i, it always pays him 0. If R is good for i, it pays him lumpsum payoffs at random times which correspond to the jumping times of a Poisson process withconstant intensity λ. The arrival of lump sums is independent across individuals. The magnitudeof these lump sums7 equals h . If R is good for i, the expected payoff per unit of time is thereforeg = λh. The assumption 0 < s < g rules out the uninteresting case in which either R or S is
dominated for all beliefs.
Each individual starts with a probability p0 that R is good for him. This probability is thesame for all and is common knowledge. Thereafter, all payoffs are publicly observed, so thateveryone shares the same belief about any given individual’s type (for privately observed payoffs,
see Section 7). In particular, the arrival of the first lump sum to a given individual i makes him
publicly a “sure winner.” At any time t, the group is therefore divided into k “sure winners” forwhom R is good with probability one, and N −k “unsure voters,” who have the same probability
p of having a good type. Unsure voters’ probability evolves according to Bayes’ rule and obeys
the dynamic equation dp/dt = −λp(1 − p) if no lump sum is observed, with pj jumping to 1
when some voter j receives a lump sum.8 Type independence implies that an unsure voter learns
only from his payoff stream but not from those of others.
When N = 1, the setting reduces to the optimization problem of a single decision maker. The
optimal experimentation strategy is Markov with respect to the current belief p, determined by
a cut-off pSD such that R is played if and only if p ≥ pSD . This cut-off is determined by the7

All results hold if these lump sums have random, independently distributed magnitudes with constant mean h.

More generally, what matters to decision makers are the expected payoff rates of each action and the probability
that the risky action is good or bad. See Section 5 for a general specification of payoff distributions and beliefs.
8
One way to derive this dynamic equation is to observe that pt is a martingale and jumps to 1 with probability
rate pλ; hence, 0 = E[dpt |pt ] = ηt dt + λpt (1 − pt )dt, where ηt is the rate of decrease of pt conditional on notobserving any lump sum, i.e., ηt = dpt /dt if no lump sum is observed, and where the factor (1 − pt ) in the second
term is the change in probability in case a lump sum is observed. This yields the equation for dp/dt. One mayalternatively use Bayes’ rule to directly show that pt = (p0 e−λt )/(p0 e−λt + (1 − p0 )), which yields the equation
by differentiation.

7

indifference condition9
pSD =

µs
,
µg + (g − s)(1)where µ = r/λ. Let pM = s/g denote the myopic cut-off, i.e. the probability below which
R yields a lower expected flow payoff than S. The previous formula implies that pSD < pM .
Indeed, experimentation really takes place only for all p ∈ [pSD , pM ], since the single decision
maker then chooses the risky action, despite its lower payoff, in order to learn more about itstrue value for future decisions. Choosing R in this range is optimal due to the option value ofexperimentation.
For a group using the simple majority rule, the formal analysis to follow in this section showsthat collective decisions are determined by nonincreasing cut-offs {p(k)}0≤k≤N such that therisky action is played at time t if and only if pt > p(kt ), where kt is the number of sure winners
at that time. The dynamics of collective decisions can thus be described as follows. Startingwith some (high enough) level p0 , R is elected until the threshold p(0) is reached, at which point
experimentation either stops if no winner has been observed by then, or continues until anotherthreshold p(1) < p(0) is reached, and so forth. These dynamics are qualitatively represented by
Figure 1 for the case of three voters. Here and throughout, experimentation means choosing(or voting for) the risky action when one’s type is unknown. (Thus, only unsure voters areexperimenting.) The option value of experimentation is formally defined after the equilibrium
concept is introduced.
A collective decision rule, or policy, is a stochastic process C = {Ct }t≥0 adapted to the filtration
generated by the arrival of voters’ lump sums and taking values in the action space {R, S}. Anycollective decision rule determines a value function for each agent i:
Z ∞

i,C
−r(τ −t)i
Vt = EtedπCτ (τ ) ,ti (τ ) = hdZ i or 0 depending on whether R is goodwhere the payoff rate is dπSi (τ ) = sdτ and dπRτior bad for i, and {Z }1≤i≤N is a family of independent Poisson processes with intensity λ. At

any given time, let K denote the set of sure winners. The number k of sure winners is thus the
cardinal of K. A Markov strategy for voter i is a function di : (K, p) 7→ {R, S}.10 For a given
profile d = (d1 , . . . , dN ) of Markov strategies, let C denote the resulting (Markov) collective
9

Intuitively, if the decision maker experiments, his instantaneous payoff rate is pg, and with a probability rateof λp, his value function jumps to g/r, from its current value. If he chooses S instead, his immediate payoff rate iss. When the decision maker is indifferent, his current value is s/r, so the cutoff p solves the indifference equation
pg + λp(g/r − s/r) = s, which is exactly (1). The result is derived formally and more generally in the proof of
Theorem 1.
10
We assume that, in the case of a jump at time t, the strategy depends only on the belief before the jump.

8

type probability p1p0
pM = s/gp(0)switch to S if no winner so farp(1)switch to S if only one winner so far

pSD
ptp(2), p(3)0

R forever otherwise

t1

t0

time tt0 : experimentation end time if no winner is observed before reaching p(0).
t1 : experimentation end time if only one winner is observed before reaching p(1).
p(2) = 0: R is elected forever if winners have the majority, no matter what pt for the remaining unsure voter.
pSD < p(1): a single decision maker always experiments more than a group with a majority of unsure voters.

Figure 1: Dynamics of Collective Experimentation with 3 Voters.
decision rule, i.e., such that C(K, p) = R if and only if |{i : di (K, p) = R}| > N/2. C is based on
the simple majority rule. V i,C (K, p) denotes i’s value function under policy C when the current
state is (K, p). Under any Markov rule, C(K, p) = S implies that S is played forever, since thestate (K, p) can evolve only when R is played. Therefore, V i,C (K, p) = s/r for all i wheneverC(K, p) = S. This, among other things, rules out strategies of the grim-trigger type. To avoid
trivial equilibria, the equilibrium concept used in the paper requires the elimination of weakly
dominated strategies, iterated in the following sense.
Definition 1 The profile d is a Markov Equilibrium in Undominated Strategies if for all(K, p, i),di (K, p) = R⇔

pi g + λp

X

j ∈K
/


∂V i,C
V i,C (K ∪ {j}, p) − V i,C (K, p) − λp(1 − p)
(K, p) > s,∂p(2)

where pi = 1 if i ∈ K and pi = p if i ∈/ K.11Other assumptions would yield the same outcome, since they affect payoffs only over a discrete time set, but do
not affect information, since the probability that two jumps occur at exactly the same time is zero.
11
Since p only decreases over time, here and throughout derivatives of value functions should be understood asleft derivatives.

9Thus, i votes at each instant as if his vote were pivotal, realizing that voting for S in any given
state (K, p) will result in a constant payoff flow s forever. The left-hand side is i’s payoff fromthe risky action, including the impact of incoming lump sums and Bayesian updating on i’s
immediate payoff and value function, as will be explained shortly. This equilibrium concept
entails an iterated elimination of dominated strategies, where the iteration proceeds backwards
with respect to the state (k, p). For example, the definition implies that if voter i, given othervoters’ strategies, prefers S over R at some state (k′ , p′ ), then i will indeed vote for S if thatstate is reached; it therefore also implies that, seen from any state (k, p) with k ≤ k′ and p ≥ p′from which the state (k′ , p′ ) may be reached, only strategies where i chooses S at (k′ , p′ ) should
be considered. This concept is closely related to the elimination of conditionally dominatedstrategies as defined in Fudenberg and Tirole (1991), except that the present case corresponds
to elimination of conditionally weakly dominated, rather than strictly dominated, strategies.12
The (option) value of experimentation of an unsure voter is the difference between his valuefunction and the maximum payoff he could get if he had to decide today on one action played
forever. Formally
X

i,C(K, p) = V

i,C(K, p) − maxs pi g
,
r r



.

(3)

This value is positive for a single decision maker, since choosing a fixed action forever is only
one out of many policies over which the decision maker optimizes. In fact, this value is positive
for a single decision maker for any news arrival process generated by the risky action. It is also
positive under the majority rule in the present setting. However, Section 5.2 shows that whenboth positive and negative news shocks are allowed, XCi can be negative, even under the simple
majority rule.
Finally, we quantitatively define loser trap and winner frustration. Both risks depend on theprobability of an action being imposed that is individually suboptimal, and on the magnitudeof this prejudice. These probabilities depend on the particular collective policy being used.
Therefore, loser trap and winner frustration depend not only on parameters p0 , g, s, λ, and r
but also on the voting rule (see also Section 4). Let C denote any arbitrary policy and D denotethe policy that i chooses if he is a dictator. Then, the expected loser trap under C for individual
i isi

L (C) = E

Z ∞
0

−rt

e

1(Ct =R ∧ Dt =S) ·idπR(t) − sdt





,

(4)

where expectations, here and in the following equation, are taken with respect to player types
12

The iterated elimination of dominated strategies, as described here, gets rid of the need to consider arbitrary,suboptimal strategies at future states. In the present setting, the concept yields the same solution as the majority
voting equilibrium of Roberts (1989).

10and the arrival times of lump sums (and, therefore, independently of players’ actions). Similarly,
the expected winner frustration under C for i is
Z ∞


i
W i (C) = E
e−rt 1(Ct =S ∧ Dt =R) · sdt − dπR
(t) .

(5)

0

Thus, the expected loser trap is the expected relative loss that i incurs from R being imposed
whenever he would have chosen S had he had full control of the decision process. The difference
between i’s value function under C and D is the sum of the two control sharing effects.
Theorem 1 shows that there exists a unique Markov equilibrium in undominated strategies,
and that this equilibrium is characterized by cut-offs. Equilibrium uniqueness comes from abackward induction argument on the number of winners. Here is some intuition for the proof.
At any time t, the state of the group can be summarized by kt and pt . Each of the two voter
categories (i.e., sure winners or unsure voters) consists of individuals with currently perfectlyaligned interests. If sure winners have the majority, they optimally impose R, since any policy
involving R is strictly better for them than having S played forever. This determines thecommon value function of unsure voters when winners have the majority. Since an unsure voter
can become a winner but a winner remains a winner forever, majority can only shift from unsure
voters to winners. Proceeding by backward induction on the number of winners, one can showthat unsure voters (or sure winners) always share a common voting strategy, after the iterated
elimination of weakly dominated ones.
Let u(k, p) and w(k, p) denote unsure voters’ and sure winners’ respective value function when
the state is (k, p). When there is a majority of unsure voters, decisions are dictated by theircommon interest unless and until they lose the majority. The goal is therefore to determine
unsure voters’ preferences. These preferences are determined by the following Hamilton-JacobiBellman (HJB) equation, which is a simplified formulation of (2):ru(k, p) = max {pg + λp[w(k + 1, p) − u(k, p)]∂u+λp(N − k − 1)[u(k + 1, p) − u(k, p)] − λp(1 − p) (k, p), s . (6)∂p
The first part of the maximand corresponds to action R, and the second to action S. The effectof R on an unsure voter i can be decomposed into four elements: i) the expected payoff rate pg;ii) the jump of the value function if i receives a lump sum, which occurs at rate λ with probability
p: his value function jumps to w and the number of winners increases by 1, iii) the jump ofi’s value function if another unsure voter receives a lump-sum: i is still an unsure voter, but
the number of sure winners increases by 1, and iv) the effect of Bayesian updating on the valuefunction when no lump sum is observed. The independence of the Poisson processes governing11

individual payoffs implies that only one lump sum can be received during any infinitesimal
period of time, so that no term involving two or more jumps appears in the HJB equation. In
comparison, if S is chosen, learning stops and i simply receives payoff rate s.
Since unsure voters have identical value functions, they unanimously decide to stop experimentation if p becomes too low, which occurs when the R part of (6) equals s. At this level p, thesmooth pasting condition implies that the derivative term vanishes, since the value function isconstant and equal to s/r, below that level (see, for example, Dixit, 1993). This determines
the equilibrium policy’s cut-offs as stated by Theorem 1, whose proof is in the appendix. The
theorem is proved for the simple majority rule, but the backward induction argument can also
be applied to other voting rules.
Let kN = (N − 1)/2. kN is the number of winners for which i) sure winners are in the minority
but ii) only one new winner is needed for the majority to change sides from unsure voters tosure winners.
Theorem 1 (Equilibrium Characterization) There exists a unique Markov equilibrium in
undominated strategies. This equilibrium is characterized by cut-offs p(k), for k ∈ {0, . . . , N },such that R is chosen in state (k, p) if and only if p > p(k). Furthermore, for all k ∈ {0, . . . , kN },pM > p(k) > pSD ,13 p(k) is decreasing in k for k ≤ kN , and p(k) = 0 for all k > kN . The valuefunctions u and w satisfy the following properties:• u(k, p) and w(k, p) are nondecreasing in p,• w(k, p) is nondecreasing in k for all p,• u(k + 1, p) ≥ u(k, p) for all p and all k < kN ,• u(kN + 1, p) < u(kN , p) for all p,• u(k, p) = pg/r and w(k, p) = g/r for all p and all k > kN .
Cut-offs are decreasing in k: the larger the number of winners, the more remaining unsure
voters are willing to experiment. This result is perhaps surprising: why would unsure voterswant to experiment more when the risk of losing their majority and having R be imposed onthem forever increases? The intuition is as follows. Suppose that p is below the myopic cutoff pM but above p(k) so that with k current winners, unsure voters choose to experiment. By
13

The strict inequality p(k) > pSD holds only if N > 1.

12

definition of pM , unsure voters get a lower immediate expected payoff rate with R than with S.
Therefore, their only reason for experimenting is their hope of becoming winners. Now supposeby contradiction that p(k+1) > p(k), and that p lies in (pk , pk+1 ). Then, as soon as a new winner
is observed, k jumps to k +1, which implies that S is imposed forever, since p < pk+1 . Therefore,
the very reason why unsure voters wanted to experiment - namely, the hope of becoming surewinners - becomes moot: as soon as one of these unsure voters becomes a winner, he sees the safe
action imposed on him forever, which prevents him from actually enjoying any benefit of being awinner.14 Theorem 1 also states that p(k) > pSD for all k ≤ kN ; that is, a single decision makeralways experiments more than a group whose majority consists of unsure voters. The reason isthe control-sharing effect mentioned in the introduction: a single decision maker knows that if
he turns out to be a winner, he will be able to enjoy the risky action, while if he turns out to bea loser, he can stop experimentation whenever he wants. In a group, even if a voter turns out tobe a winner, he is not guaranteed that the risky action will be played forever, as a majority ofunsure voters may block it. And if he turns out to be a loser, he may still have the risky action
imposed on him forever if experimentation lasts long enough to reveal a majority of winners.
This double risk of losing control prompts unsure voters to experiment less than any one of themwould if he alone could dictate decisions in the future. In fact, a result stronger15 than cut-off
monotonicity obtains: when a new winner is revealed, the value function of both winners andunsure voters jumps upwards, provided that k < kN . For sure winners, this result is intuitive:
a higher number of sure winners means a higher probability that a winning majority will beachieved. To be complete, this argument also requires that experimentation gets longer as thenumber of winners increases, which is guaranteed by cut-off monotonicity. More surprising is thefact that the revelation of a new winner results in an upward jump of the unsure voters’ valuefunction unless this new winner is the decisive voter who gives the majority to winners. The
intuition here is that new winners reduce the risk of winner frustration, a risk that dominates
as long as unsure voters keep control of the decision process. Another possible interpretation ofthis result is that the emergence of new winners increases the expected “pivotality” of unsure
voters, as it reduces the imbalance between the majority and the minority. Finally, the utilityof unsure voters jumps downwards when winners gain the majority (i.e., k jumps from kN to
kN + 1). This is true even if p is large. This may seem surprising since, when p is large, voters
are happy to experiment and could appreciate a priori that the opportunity to experiment will
not be overturned. However, this opportunity would have been overturned only after p became
sufficiently low (below the myopic cutoff), and now that option is no longer available.
14

That is, apart from receiving a lump sum at the time of the jump, but the possibility of that gain is alreadytaken into account in the computation of the immediate expected payoff, which is still less than s for p < pM .
15
This result is used to analyze the case of privately observed payoffs; see Theorem 10.

13

The simpler case where learning is extremely fast teaches something else. When types are immediately revealed as soon as R is tried, a single decision maker is always willing to experiment.16
However, this result does not extend to the case of collective experimentation, for even as thetime cost of experimentation vanishes, the risk of loser trap remains. If that risk is severe enough,society may prefer to shun the opportunity of immediate type revelation and hence of making
a perfectly informed decision (clearly what a utilitarian planner would choose!). Keeping otherparameter values fixed, non-experimentation will occur if the total number N of individuals
is large enough and the initial probability p is low enough; experimentation cut-offs then staybounded away from 0 as learning intensity λ goes to infinity, provided that N is large enough.
The proof is a direct consequence of equation (17) in the appendix.
Corollary 1 (Immediate Type Revelation) If N > 2g/s − 1,
lim p(kN ) =λ→∞(N + 1)s/g − 2
> 0.
N −1

If N ≤ 2g/s − 1,
lim p(kN ) = 0.

λ→∞Corollary 1 suggests that the total number N of individuals has an important effect on experimentation. In fact, the next proposition states that with independent types, individuals behave
myopically as group size becomes arbitrarily large, electing the risky action if and only if its expected payoff is higher than that of S. To state the result, let p(k, N ) denote the experimentation
cut-off when there are k winners and N overall individuals.
Proposition 1 (Group Size) p(kN , N ) is nondecreasing in N . Moreover, for all k, p(k, N ) →pM as N goes to infinity.
Proof. The first part of the proposition is an immediate consequence of (16) in the appendix.
For the second part, (16) also implies that p(kN , N ) → s/g = pM as N goes to infinity. Finally,
Theorem 1 implies that p(kN , N ) ≤ p(k, N ) ≤ pM for fixed k and for all N ≥ 2k + 1. Taking
the limit as N goes to infinity proves the result.

In general, cut-offs p(k, N ) are not monotonic with respect to group size N , as can be proved bynumerical counter-example. Such violations may seem counter-intuitive: As N increases, individual power gets more diluted. Shouldn’t this reduce the value of experimentation? However,16

Mathematically, this result comes from the single decision-maker cut-off equation (1): as the intensity λ goesto infinity, µ goes to 0 and so does the cut-off pSD .

14

0.08

κ=2
κ=3
κ=4
κ=5

0.07

Belief p

0.06

0.05

0.04

0.03

0.02

0.01

3

5

7

9

11

13

15

Group Size N

Figure 2: Cut-offs as a function of group size N and the switching number κ. Parameter values: r = 1, λ = 10,
s = 1, g = 10. N takes all odd values from 3 to 17. For N = 1, pSD = 0.01.

adding unsure voters increases the expected number of winners, and thus the expected durationof experimentation, for given cut-offs. The addition of voters thus reduces the risk of winnerfrustration, which sometimes increases the attractiveness of experimentation.17
Figure 2 shows the numerical computation of cut-off policies for different values of N and of
the switching number κ = kN + 1 − k of switches required for winners to gain the majority. For
κ = 4, the cut-off is not monotonic in N . For κ = 5, the cut-off is actually decreasing in N when
N is in the range [9, 17].

3

Utilitarian Policy

This section characterizes the optimal experimentation policy of a utilitarian social planner andshows that it lasts longer than majority-based experimentation. A social planner faces a single
decision-maker experimentation problem, the solution of which can be computed by backward
17

The expected length of experimentation is also not generally monotonic in N . To see this, it is easy to build

an example where the risky action is played forever when N is arbitrarily large (see Theorem 6 and the discussion
below it), whereas experimentation stops with positive probability when N is small. Similarly, it is also easy tobuild an example where experimentation stops immediately when N is large, but has positive duration when N
is small.

15

17

induction on the number of observed winners, and is characterized by monotonic cut-offs.
Theorem 2 Under the utilitarian criterion, the optimal policy is determined by cut-offs q(k)such that C(k, p) = R if and only if p ≥ q(k). These cut-offs are decreasing in k for k < k̄ andequal to zero for k ≥ k̄, where k̄ = gs N .
Proof. See the appendix.
The next result shows that the equilibrium level of experimentation under the majoritarian rule
is inefficiently short compared to the utilitarian optimum. This result is due to two concurring
reasons. First, the social planner can exploit information better than individual voters, to meetits objective, since he has full control over future decisions. That argument shows up in the proof
below (see (9)). Second, the social planner takes into account winners’ utility, while unsure voters
do not. This implies that when, under the majoritarian equilibrium, unsure voters decide tostop, a social planner would take into account winners utility which, other things equal, makes
experimentation more attractive (see (10) in the proof below).
Theorem 3 (Majoritarian vs. Utilitarian Rules) Let {p(k)}k be the cut-off values associated with the majority rule. Then q(k) ≤ p(k) for all k ≤ kN .
Proof. The utilitarian cut-off q(k) solves(k/N )g + (1 − k/N )pg + (N − k)λp
W (k + 1, p)− s/r = sN(7)where W is the utilitarian value function. The left-hand side is the sum of the per-capita
immediate expected payoff given state (k, N ) and of the per-capita jump of welfare following
the observation of a new winner, weighted by the probability rate of this event. The majorityvoting cut-off, p(k), solvesw̄(k + 1, p) N − k − 1
pg + (N − k)λp+ū(k + 1, p) − s/r = s
N −k
N −k

(8)

where w̄ and ū are the value functions obtained under the majority rule. (The left-hand side is
obtained from (6) simplified through value-matching and smooth-pasting conditions.) Optimality of the utilitarian policy implies that for all k, p,
W (k, p)kk
≥ w̄(k, p) + 1 −ū(k, p).
N
N
N

(9)

Since w̄ > ū, this also implies that
W (k + 1, p)
1
1>w̄(k + 1, p) + 1 −ū(k + 1, p)
N
N −k
N −k
16

(10)

and, hence, that the left-hand side of (7) is higher than that of (8), for each p. Therefore, theroot of the first equation must be lower than that of the second.

4



Design of Voting ProceduresThis section considers which mechanism can improve efficiency compared to the simple majority
rule.

4.1

Fixed Quorum

The first issue is to determine how changing the number of votes (hereafter, quorum) required
for the risky action affects the length and efficiency of experimentation. The simpler case of aconstant quorum is considered first.18 In that case, Theorem 4 shows that there is no systematicadvantage of one voting rule over another. As one moves across the entire spectrum of voting
rules, from requiring unanimity for the safe action to requiring unanimity for the risky action,
the risk of loser trap diminishes while the risk of winner frustration increases, with exactly oneof the two risks entirely vanishing at the ends of the spectrum. Depending on the parametersof the model, which determine the magnitude of these risks, the optimal rule can be any rulein the spectrum. For simplicity, the analysis starts with the case of immediate type revelation,
which is sufficient to show the lack of comparability of voting rules.
Suppose that learning is arbitrarily fast (i.e. λ → ∞). In that case, there is no time cost ofexperimentation hence no winner frustration. If one requires unanimity for the risky action, thisalso gets rid of loser trap so will always prompt society to choose immediate type revelation.
However, once types are revealed, unanimity requires that R is only implemented if all voters
are winners, which typically is inefficiently too restrictive. Indeed, the utilitarian optimum isto get immediate type revelation and then choose the risky action if and only kg > sN . Forν ∈ {1, . . . N }, define the ν voting rule as the rule requiring ν votes for the risky action. Lettingν U = (sN )/g, a ν rule with ν > ν U will never implement the risky action when it is sociallyinefficient to do so. Let ν̄ denote the smallest integer such that society is ready to experimentwith the ν̄ voting rule, and let ν ∗ = max{ν̄, ν U }. Then, social efficiency is decreasing in ν for18

For any q-rule one may, proceeding as in Section 2, prove the existence of a unique equilibrium characterizedby monotonic cut-offs contained in [pSD , pM ]. The analysis of this section, based on immediate type revelation,
does not require this proof.

17ν ≥ ν ∗ , because on this range ν is high enough to prompt experimentation and the probabilityof implementing the risky action if it is socially efficient ex post is decreasing in ν, while theprobability of implementing the risky action if it is inefficient is zero. As is easily checked, ν ∗can take any value between 1 and N (ν̄ decreases from N to 1 as p increases from 0 to 1).
To generate the reverse inefficiency ranking, suppose that, in addition to immediate type revelation, p is arbitrarily close to 1. In that case, society always wishes to experiment, since theprobability of loser trap is arbitrarily small. Social efficiency is increasing in ν for ν ≤ ν U :since p is close to 1, initial experimentation takes place anyway, and ex post the probability of
implementing the risky action if it is socially inefficient decreases in ν. Since ν U can take any
value between 1 and N , this implies the following result.
Theorem 4 For any voting rules ν 6= ν̃, there exist parameter values and an initial belief psuch that the ν voting rule is strictly socially more efficient than the ν̃ voting rule.
It would seem that, as the risk of loser trap becomes more salient compared to winner frustration,
it becomes more efficient to have a more restrictive voting rule, i.e. a higher ν. However, thisintuition may sometimes be wrong. For example, suppose that types are immediately revealed
if R is played. Owing to the risk of loser trap, there must exist a level ν ∗ , depending on initialbelief p0 , such that society experiments for all ν ≥ ν ∗ and sticks to S for ν < ν ∗ . Now supposethat s is decreased by a small amount ε so that ν ∗ stays unchanged. This increase of s canbe interpreted as the risk of loser trap becoming marginally less salient than the risk of winnerfrustration. The change reduces voters’ value function for ν < ν ∗ by ε/r, since they still choose
S, but it reduces their value function for ν > ν ∗ by a lower amount, since under experimentation
the discounted time spent playing s is strictly less than 1/r.19 This shows that, at least in somecases, reducing the risk of loser trap relative to winner frustration, does not make less restrictive
rules more desirable.
Efficiency depends not only on voters’ ex ante probability of falling in the loser trap but also on
the magnitude of the loser trap (more generally, the relative values of g and s and 0). With slower
learning, the risk and magnitude of winner frustration also influences voting rule efficiency inthe opposite direction. The impact of magnitude, already implicit in the above analysis throughν U , is illustrated below for the comparison of the simple majority rule and the unanimity rulefor R (i.e. ν = N ). Let {χ(k)}0≤k≤N denote the cut-offs characterizing to the unanimity-voting
policy.
19

Cut-offs are also affected by this change but this change is of second order, by optimality.

18

Example 1 Suppose that N = 3 and s ≪ g. Then, χ(1) > p(1).
Proof. Equation (16) in the appendix implies thatp(1) =µsµs∼µg + (g − s) − (s − pg)(µ + 1)g(11)

if g ≫ s. In particular, p(1) is arbitrarily close to zero if g ≫ s. With the unanimity rule andk = 1, unsure voters are indifferent when p satisfies
pg + λp[w(2, p) − s/r] + λp[v SD (p) − s/r] = s,(12)where w(2, p) is the value of a sure winner under unanimity rule if there are two sure winners(and N = 3), and v SD (p) is the value function of a single-decision maker. As can be easily
checked, v SD (p) ≤ pg/r + (1 − p)s/r, while w(2, p) ≤ pg/r + (1 − p)s/r. This and (12) imply
that χ(1) must satisfy the inequality
pg + 2λp2 (g/r − s/r) ≥ s,
or

µs
∼ s/g
µg + 2p(g − s)(13)

if g ≫ s. Comparing (11) and (13) shows that χ(1) > p(1).



p≥

4.2

Time-varying quorum

Suppose now that at each time t, R is elected if and only if it gets νt of the votes. The next
result shows that even if νt is deterministic, efficiency can be fully restored.
Theorem 5 (Deterministic Quorum) There exists a quorum function t 7→ νt such that theresulting unique Markov equilibrium in undominated strategies implements the utilitarian policy.
Moreover, νt is increasing in time and is entirely determined by the initial belief p0 and theutilitarian cut-offs q(k).
Proof.

Starting from a given belief p0 , let tk denote the time such that ptk = q(k), for eachk ∈ {0, . . . N }. Since pt decreases over time, the sequence tk is increasing, and tN = ∞. Fort ∈ [tk , tk+1 ), let νt = k. This entirely and uniquely determines the function νt . By construction,
if there are at least k winners, they can impose R whenever t < tk , i.e. whenever p > q(k).
Moreover, if there are exactly k winners, they can only impose R if t < tk , i.e. when p > q(k).
19From Theorem 3,20 unsure voters always want to impose S when it is socially efficient to doso, which guarantees implementation of the efficient policy whenever t ≥ tk and there are only
k sure winners or less. Proceeding by backward induction on k, as in the proof of Theorem 1,
one may therefore conclude that t 7→ νt yields a unique Markov equilibrium in undominatedstrategies, and that this equilibrium implements the utilitarian policy.
The quorum is not unique. For example, near time zero, everyone wants to experiment (assuming
p0 is high enough) so any quorum initially yields the efficient policy. In general however, thequorum must be low enough to allow R whenever unsure voters want to stop experimentation
while it is efficient to pursue it, and high enough to prevent winners from imposing R whenever
S is the socially efficient action. More generally, Theorem 5 suggests that, in settings where
news events amount to good news and no news is bad news, an efficient quorum should increaseover time: as more time elapses, society should require a higher number of winners in order for
R to be elected. Although the precise times of quorum change rely on the exact map between
pt and the utilitarian cutoffs and on p0 , the insight that a gradual increase cut-offs may improve
efficiency over constant quorum is does not depend on that map.
Theorem 5 has several corollaries and equivalent formulations. For example, increasing voting
weights of unsure voters, relative to sure winners, offers an alternative way of implementing
efficiency. Indeed, it is easy to determine a particular weighting scheme that mirrors the proofof Theorem 5, under which sure winners are in control of the collective decision process only if
R needs to be implemented. Naturally, history-dependent quorums, which contain deterministic
ones as a particular case, can also be devised to implement the efficient outcome. Such quorums– as well as constant ones – amount to a form of commitment, as opposed to having the stronger
side impose his choice at any given time. In a setting where winners achieve increasing politicalpower (for example, their higher payoffs might give them higher lobbying power), the resulting
variation in voting weights goes opposite to the one implementing efficiency. In that case, winners
may impose the risky action, owing to their higher political power, when it is inefficient to do
so. Anticipation of this potential outcome should result in even more conservatism ex ante. The
next section considers efficiency under various forms of commitment assumptions.
20

The proof is actually slightly different and simpler in the present case. Unsure voters choose between contin-uing with the efficient policy versus imposing S forever.

20

4.3

Commitment and Endogenous Quorum

Theorem 5 shows that it is possible to implement the efficient outcome as long as one can commit
to some time varying quorum. If voters are initially homogeneous and can only commit to an
anonymous policy at the outset, they share, initially, a common objective function.21 Since
expected payoffs are identical, the optimal policy also maximizes the sum of these expected
payoffs, i.e. utilitarian welfare. Therefore, if given the possibility, voters would like to commit
to the utilitarian policy, showing the following result.
Theorem 6 (Commitment) If voters can commit to an anonymous policy at time 0, theychoose the cut-off policy determined by cut-offs {q(k)}0≤k≤N .
If voters can only commit to a fixed action, such as imposing a new rule for the next five years
no matter how well that rule performs over that period, efficiency need not be restored. To
give an extreme illustration, suppose that voters must commit to an action for the entire time
horizon. In that case, the risky action is chosen if and only if its expected payoff is above themyopic cut-off. This extreme case of action commitment thus entirely annihilates the value ofexperimentation. Commitment to an action is formally equivalent to reducing the frequency
of decision making. For example, voting every five years amounts to a succession of five-year
commitments. The previous observation can therefore be reinterpreted as follows: if votes takeplace at a low enough time frequency, individual control over collective decisions is reduced tosuch extent that the resulting policy may be more inefficient. However, provided that aggregate
uncertainty is small enough and initial beliefs are optimistic enough, commitment to a fixedaction can restore efficiency.22
21

Anonymity means that individuals cannot commit to a policy that favors or harms particular voters, suchas imposing generous redistribution if some given individuals turn out to be poor and no redistribution if thesesame individuals turn out to be rich. This assumption is consistent with veil-of-ignorance arguments.
22
With an infinite population, the law of large numbers allows one to compute the socially optimal action:
starting with an individual probability p that the action is good, the risky action is the social optimum if and only
if pg > s, since there surely is a fraction p of winners. Suppose that, initially, pg > s. From Theorem 6, individuals
find it optimal to commit to the risky action over the infinite horizon. What happens without commitment? The
second part of Proposition 1 implies that unsure voters, if they have the majority, impose the safe action as soon
as pt hits the myopic cut-off pM = s/g. This situation will occur surely if one starts with p = pM + ε for ε
for small enough. Indeed, from Proposition 2 in the appendix, the probability that an unsure voter with initialprobability p receives a lump sum before pt reaches q < p equals (p − q)/(1 − q). This and the law of large numbers
imply that, when society starts at pM + ε, the fraction of remaining unsure voters when pM is reached equals
1 − ε/(1 − pM ), which is greater than 1/2 for ε < (g − s)/2g. This shows that commitment to the risky action isstrictly more efficient than no commitment.

21

One may wonder whether a hybrid form of commitment, where voters commit to a fixed action
over some horizon but also get to dynamically modify the quorum required for the risky action,
can restore efficiency. The following result, which extends to a much more general setting thanthe present one, answers negatively. Considers the following recursive procedure, which may becalled the “Endogenous Quorum” Procedure: at time 0, voters decide on two policy dimensions:an initial horizon of experimentation, T1 , and a quorum, ν1 , used at T1 for deciding whether or
not to continue experimenting and, if so, on a new horizon of experimentation, T2 , and on thequorum ν2 used at T2 to vote on T3 and ν3 , etc. The key of this procedure is that voters cannot
commit to the quorum function at the outset. Rather, they can, at any given election, mitigate
the law of the strongest by controlling the quorum used at the next election. This procedure
is a natural way to capture the idea that voting rules cannot be committed upon ex ante. As
soon as sure winners meet the quorum, they impose the risky action forever, by setting thenext experimentation period to infinity. If sure winners do not meet the quorum, the procedure
grants unsure voters the right to choose the next horizon and quorum. To be complete, theprocedure must specify an initial quorum, ν0 , at time 0. Since voters are ex ante homogeneous,
this initial quorum plays no role in the present setting. In principle there may exist multiple
equilibria. However, the endogenous quorum procedure must be inefficient for any equilibrium,
as the next result shows.
Theorem 7 (Endogenous Quorum) There exist initial belief p0 and group size N such that
the Endogenous Quorum procedure does not implement the utilitarian policy.
Proof.

Consider any policy C consistent with an endogenous quorum procedure: C is a rightcontinuous stochastic process taking values in {S, R} characterized by an increasing sequenceof (possibly infinite) random times Tj , such that C is constant on any interval [Tj , Tj+1 ], andby a quorum process νj such that at each Tj , unsure voters impose S forever or set νj+1 andTj+1 optimally if there are at least νj of them, otherwise sure winners set Tj+1 = ∞. With
positive probability there exist both sure winners and unsure voters at time 1. Suppose thatsure winners do not meet the quorum ν1 , so that unsure voters can choose the next quorum
and experimentation horizon. If Theorem 7 were false, then these unsure voters, starting fromtheir current belief p1 and group size N − k1 , could choose ν1 and T1 as part of a policy thatmaximizes their expected welfare, i.e. ignoring the utility of current sure winners. Since it would
be optimal for them to do so, they implement that policy, contradicting efficiency of C.

22

5

General News Arrival Processes

Section 2 assumed that individuals perfectly learned their types upon receiving some lump sum,that news events amounted to good news, and that types were independently distributed. Relaxing these assumptions, this section reconsiders in a very general setting whether i) collective
experimentation is shorter than the single decision maker equivalent, ii) collective experimentation is shorter than the utilitarian optimum,23 and iii) there is always some experimentation,i.e. a set of voter beliefs where R’s immediate payoffs is lower than S’s but society still elects R.
Suppose that, for any given individual, the risky arm has a payoff distribution, or “type”, θlying in some finite set Θ. At any time, that individual’s belief about his type is summarized
by a probability distribution or “state” γ ∈ Γ, where Γ = ∆(Θ) is the set of all probability
distributions24 over Θ. The safe arm still pays a constant rate s. For a single decision maker, theGittins index of the risky arm is the map G : Γ → R such that, given state γ, G(γ) is the smallest
value of s for which the single decision maker prefers the safe action over experimentation.
Mathematically, G(γ)solves
G(γ) = inf s : s/r = sup E
σ

Z ∞

−rt

e

0

dπσt (t)|γ, s

,where σ is any policy, and the expectation is conditional on the current state γ and on the rate sof the safe action.25
Now consider the case of N decision makers and let {Ft }t≥0 denote the filtration generated by
all voters’ payoffs. At any time, the state, known to all, is denoted γ. If types are independent,then γ = γ 1 , . . . , γ N ∈ ΓN . In general, γ may contain information about type correlation. A
policy is a process adapted to the filtration {Ft }t≥0 and taking values in {S, R}.
For any rate s, policy C, and voter i, necessarily
Z ∞

Z ∞

−rt
i
−rt
i
sup E
e dπσt (t)|γ, s ≥ E
e dπCt (t)|γ, s .
σ

0

(14)

0

The inequality obtains because C is an element of the policy set over which the maximization
23
24

That result is considered in the negative-news setting to follow.
In the benchmark model, the type θ is either “good” or “bad” and the state γ is the probability p that thetype is good.
25
The results of this section are easily adapted to discrete-time settings. In fact, Theorem 8 does not assume
anything about the time domain.

23

is taken.26 We may define a policy-dependent generalization of the Gittins index as

Z ∞

i
GiC (γ) = inf s : s/r = E
e−rt dπC(t)|γ,s.
t
0Inequality (14) implies that GiD (γ) ≥ GiC (γ) for all i, γ, and C, where GiD (γ) is i’s Gittins index
if he has dictatorial power over all decisions.
The definition of Markov equilibrium in undominated strategies is extended as follows. Let ν
denote any integer in {1, . . . , N }Definition 2 (Voting Equilibrium) C is a ν-voting equilibrium if, for any belief γ,C(γ) = S ⇔ |{i : GiC (γ) ≤ s}| ≥ ν.
This definition should be interpreted as follows. If a voter is pivotal, the Markov property
implies that imposing S at any time amounts to imposing S forever, since the state γ is frozen
whenever S is played. Therefore, i votes for S if and only if he prefers getting the constant
payoff s forever over pursuing policy C, a choice that is determined by i’s C-Gittins index at
belief γ. The following result shows that collective experimentation is shorter than dictatorial
experimentation, in the following sense: if there are at least ν individuals who, taken individually,
would prefer the safe action if given dictatorial power over future decisions, then society also
picks the safe action in any ν-voting equilibrium. This result is an extreme generalization of thefact that all equilibrium cut-offs in Section 2 were above the single decision-maker cut-off.
Theorem 8 Suppose that C is a ν-voting equilibrium. Then, C = S whenever |{i : GiD (γ) ≤s}| ≥ ν.
The proof is an immediate consequence of the general inequality GiD (γ) ≥ GiC (γ) for all i and,
C and γ.
When types are independent, GiD (γ) = G(γ i ) where G(γ i ) is the Gittins index of the single decision maker problem with (individual) belief γ i . In that case, i’s optimal policy is independentof other individuals’ types. As a corollary of Theorem 8, therefore, collective experimentationis shorter than in an equivalent single decision maker setting. If types are positively correlated,
however, collective experimentation can last longer than in a single-decision maker setting, as
positive type correlation increases learning speed and thus reduces the time cost of experimentation (see also Section 7). In contrast, collective experimentation is always shorter, even with
26

In general, C depends on all voters’ types and need not be anonymous.

24

positive correlation, than what any voter would like if he could dictate all decisions, because a
dictator benefits from the same learning speed as society, unlike a single decision maker.
Theorem 1 also stated that all cut-offs were below the myopic cut-off, meaning that there always
was some experimentation. How general is this result? Are there cases where society elects thesafe action even when the risky action yields a higher payoff? To answer this question, thefollowing definitions will be used. For any probability distribution γ i over the type space, let
i /dt|γ i ]. g(γ i ) is i’ immediate expected payoff rate with action R given typeg(γ i ) = E[dπR

distribution γ i . For any individual type θ i , let, slightly abusing notation, g(θ i ) = g(δθi ), whereδθi is the Dirac distribution concentrated on type θ i , denote i’s true immediate expected payoffrate with action R when his actual type is θ i . Say that i is a winner if g(θ i ) > s, and a loser
otherwise. Hence, i is a winner if R is optimal for him given his true type. Θ can thus bepartitioned into “good” (winner) types and “bad” (loser) types.
Definition 3 A policy C is adverse for Voter i if the set{t : P r[Ct = R|θ i good] < P r[Ct = R|θ i bad]}has positive Lebesgue measure.
Adversity means that R is more likely to be chosen if i is a loser, at least for some nonzero time
set. Adversity can occur, for example, if a voter’s type is perfectly negatively correlated with amajority of voters. The majority then blocks R whenever that voter is a winner and imposes it
when he is a loser.27
Theorem 9 Suppose that C is a voting equilibrium for voting rule υ. Then, GiC (γ) ≥ g(γ i ) for
all i for which C is non-adverse.
Proof. See the appendix.
27

In that case, however, majority would simply ignore i and proceed with experimentation. As a stronger caseof adversity, suppose that 10 individuals face the following problem. Either they elect the safe action foreveror they try R, in which case types are immediately revealed and a dictator is randomly, uniformly chosen, suchthat the dictator has an opposite type from all other voters (i.e. either R is good for him and bad for allothers or vice versa), with a 50% chance of being a winner. Ex ante, R yields an individual expected value of
π = 1/10 ∗ [pg + (1 − p)s] + 9/10 ∗ (1 − p)s = pg/10 + (1 − p)s (letting r = 1). On the other hand, a voter’s
probability of being a winner is p/10 + (1 − p)9/10 = 1/2. Choosing g = 3s, the myopic cut off is pM = 1/3, so
p is above the myopic cut-off and yet voters prefer to avoid R since π < s. Section 5.2 provides an example ofendogenous adversity.

25

It would seem a priori that in settings where types are independent or positively correlated,usual voting rules would be non adverse. However, this intuition is incorrect, as explained in
Section 5.2.

5.1

Negative News Shocks

Several potential applications concern a setting, symmetric to the benchmark one, where news
events amount to catastrophes and no news is good news. One model such applications byassuming that the risky arm pays a positive constant rate if it is good and, in addition, payssome negative lump sums according to some Poisson process if it is bad. One may assume
without loss of generality that the payoff rate of S is zero, since all payoffs can be translatedby the same constant without affecting voters’ decision problem. The state variables are thenumber k of sure losers and the probability p that the arm is good for unsure voters. It may
be shown that the policy is also determined by cut-offs ρ(k) such that unsure voters impose therisky action if and only if p ≥ ρ(k) provided k ≤ kN , and losers impose S when k > kN . In thissetting, pt increases over time since no news is good news for unsure voters. Therefore, the risky
action can only be stopped, if used at all, when enough sure losers are observed, either because
those obtain the majority, or because the cut-off ρ(kt ) jumps over pt upon the observation ofa new loser (cut-off variation is discussed below). Theorem 8 implies that, provided that types
are independent, ρ(k) ≥ ρSD for all k < N/2, where ρSD is the single-decision-maker cut-off.
One may prove that the equilibrium policy resulting from the majority rule is non-adverse to
any voter.
With negative news shocks, it is not true any more that experimentation is inefficiently short.
Unsure voters, ignoring losers, may push experimentation further than a utilitarian social planner. However, a social planner still has a higher value of experimentation than control–sharing
voters.28

For example, a social planner would always welcome immediate type revelation,

whereas voters may prefer playing S forever rather than learn their type, however fast, provided that the risk of loser trap is high enough. At the outset, a social planner may thus be
more willing to experiment than individual voters. As the number of observed losers increases,
28

More precisely, the value of experimentation of an agent, for a given policy, is the difference between thatagent’s value function if that policy is followed, and the value that the agent gets if the action that gives him
the highest expected payoff is played forever (see (3)). This definition captures the potential gain achieved byreacting to incoming news. The social planner, being the sole decision maker of his welfare maximization problem,chooses the policy that maximizes his value of experimentation. By contrast, the equilibrium policy differs fromthe optimum of any given agent, and thus provides him with a lower value of experimentation.

26

the first effect starts to dominate, with the social planner stopping experimentation before unsure
voters under majority voting.
In view of Theorem 1, one may wonder whether cut-offs are also monotonic in this negative news
setting. The answer is negative. Counter-examples can be observed numerically or constructedwith analytical results omitted here. Such violations can be explained as follows. Essentially, theloser trap is more severe with negative news shocks. In the benchmark setting, unsure voters canalways impose the safe action when they have the majority, and the only shock that may occur inthat case is to become a winner. With negative news shocks, in contrast, any unsure voter can,upon receiving a negative lump-sum, suddenly join the minority of sure losers and hence face theworst possible situation. Negative news is compounded by a sudden control loss. This explains
why the “insurance” effect resulting from the apparition of a new loser can, paradoxically,
encourage experimentation. Seen differently, in the negative-news setting, p simply increases
over time, which is enough to make experimentation more attractive. In contrast, in the positivenews setting, the apparition of news winners is necessary for experimentation to continue, forotherwise, p decreases until it causes experimentation to stop.29 Note however that although cutoffs need not be monotonic, it is always true that experimentation decreases with the number ofsure losers. Indeed, experimentation can only stop when a new loser is observed, since otherwise
unsure voters become more optimistic about their types and have no reason to switch to S.

5.2

Mixed Shocks and Negative Value of Experimentation

Suppose that the benchmark setting is modified as follows: if R is good, it pays positive lumpsums according to the jumping times of some Poisson process with intensity λg , and if it is bad,it pays negative lump sums according to the jumping times of a Poisson process with intensity
λb . Without loss of generality, also suppose that the payoff rate of S is zero. In this case, statevariables consist of the number kW of observed winners, the number kL of observed losers, andunsure voters’ probability p that R is good for them. Since the number of revealed winners
and losers can only increase over time, a backward induction argument on kW and kL showsthat there exists a unique majority voting equilibrium policy. If λg > λb , then no news is bad
news, since shocks are more likely to happen if R is good than if it is bad. This implies that,29

From a technical standpoint, another distinctive feature of the negative-news settings is that the smooth-pasting property does not hold any more. Indeed, as time elapses, p moves away from its threshold p(k), so thevalue function need not be smooth at that cut-off. Instead, cut-offs are determined by direct comparison of valuefunctions with and without starting experimentation.

27

under this assumption, unsure voters become more pessimistic over time30 , and that they stop
experimentation at some cut-offs p(kW , kL ), provided they are pivotal. Theorem 8 implies thatpSD ≤ p(kW , kL ), where pSD is single decision maker setting cut-off. This inequality holds
for all ν voting rules. If the risky action requires the unanimity rule, then Theorem 9 implies
that p(kW , kL ) ≤ pM , where pM is the myopic cut-off: unanimity guarantees at least someexperimentation.
Negative Value of Experimentation With other voting rules, non-adversity need not hold,due to the following perverse effect: if a loser is observed, this may prompt other voters toexperiment more by reducing their risk of the loser trap. The value of experimentation can be
negative, i.e. voters may prefer to elect the safe action even if the risky action has a higher
immediate expected payoff. Here is such an example. There are three unsure voters, voting at
the simple majority rule. If a loser is observed, the remaining two unsure voters are “protected”:
it is as if R required unanimity among them two. This increases their willingness to experiment.
If a winner is observed, the remaining two unsure voters are now on the brink: any winneramong them will impose the risky action to the other. This risk reduces their willingness toexperiment. Therefore, ex ante, the three voters know that if any of them turns out to be awinner, other voters will soon revert to the safe action, while if one of them receives a negative
lump-sum, others will experiment more. This endogenous adversity makes R unattractive even if
its expected payoff is higher than S’s. For the value of experimentation to be negative, it requiresthat i) the magnitude of loser trap be severe, and ii) learning be slow, so that experimentation
takes time and the adversity described above lasts long. An explicit example in the Appendix
This section highlights an important virtue of requiring unanimity for R: the unanimity ruleguarantees a nonnegative value of experimentation, whereas other voting rules may yield anegative value of experimentation.

6

Privately Observed Payoffs

This section shows that even when payoffs are privately observed, the equilibrium policy ofSection 2 can be implemented. Suppose that individuals can only observe their own payoffs
and, at each time, the aggregate number of votes for each alternative. Voters cannot condition
their voting policy on the current number of winners, since when everyone votes for R it is
impossible to tell apart sure winners from unsure voters. However, voters do learn the number
30

Precisely, one may show that dp/dt = −(λg − λb )p(1 − p).

28

of sure winners when it matters, i.e. when cutoffs are reached. Indeed, each time a cut-off isreached at which unsure voters would want to stop given the number of winners that was last
revealed, unsure voters vote for the safe action and sure winners vote for the risky action. Thatway, everyone observes the current number of winners, and unsure voters then decide whether topursue experimentation to the next relevant cut-off (everyone votes for the risky action), or tovote for the safe action if no new winner is revealed. With this protocol, voters know the current
number of winners only when p reaches particular cut-offs, but that suffices to implement thepolicy of the public-information setting.31
To understand why it is in everyone’s interest to follow this scheme, the intuition is as follows.
First, sure winners always benefit from revealing their type, because this increases the durationof experimentation by cut-off monotonicity (Theorem 1). Second, unsure voters cannot gainfrom manipulating the choice process because, conditional on being pivotal (i.e. k ≤ kN ), they
are already choosing their optimal action. For example, if an unsure voter voted for R at some
cut-off where he is supposed to vote for S, prompting other voters to believe that there are
more winners than there really is, he will manage to extend experimentation. However, this will
increase his risk of loser trap, since other voters may become sure winners during that time.
The benefits and costs of such an extension of experimentation are already incorporated in the
cut-offs derived under public information, making deviations unprofitable. The proof that theabove protocol implements the public-information policy is sketched in the appendix.
Theorem 10 The above protocol yields the same equilibrium as the experimentation policy basedon publicly observed payoffs.

7

Extensions

Correlated Types Positive correlation across types reduces risks of loser trap and winnerfrustration and thus increases experimentation, compared to the case of independent types.
Moreover, this also increases the speed of learning, reducing the time-cost of experimentation.
Results are shown formally in a technical appendix, which studies correlation for the case of two
voters and where unanimity is required for R. With positive type correlation, an advantage ofcollective decision making compared to individual experimentation is to get rid of the free-rider
problem identified in papers such as Bolton and Harris (1999), Li (2001), and Keller, Rady,and Cripps (2005). With perfect type correlation, voting on joint decision fully restores efficient31

In small committees, cheap talk would be another natural way for voters to truthfully reveal their payoffs.

29

experimentation, where when types are independent, letting each individual experiment on its
own is efficient. A natural conjecture here is that imposing a joint decision over individual ones
(assuming both forms of experimentation are possible) becomes more efficient as types get morepositively correlated.
If some voters have negatively correlated types, this may increase or reduce experimentation,
depending on the particular correlation structure. For example, suppose that there are only two
voters, that unanimity is required for the risky action, and that voters have perfectly negatively
correlated types. Then, as soon as one voter receives a lump, the other voter knows that
he is surely a loser and this imposes the safe action. This completely destroys the value ofexperimentation, and voters stop at the myopic cut-off. A similar argument holds if unanimity
is required for the risky action. Consider now the mixed-correlation case in which, say, two voters
have a perfectly negatively correlated type with a third voter, and decisions are made according
to the majority rule. In that case, the first two voters have perfectly positively correlated types
and so have fully control over the decision process: if the third voter receives a lump sum, thefirst two voters know that they are losers and thus impose the status quo. If one of the firsttwo voters gets a lump, these voters impose the risky action. Overall, negative correlation must
be seen more broadly as part of a more general description of potential alliances which may
be formed, may have a positive or negative impact depending on the context: if it reduces apowerful group’s expected power, negative correlation is likely to reduce experimentation. If
it creates a more powerful group, the minority wants to experiment less but has no say on
decisions, and so only the majority’s increased incentive to experiment matters.
Factions and Heterogeneous Voting Weights. If some voters have a greater decision
weight, they are less subject to control sharing effects and wish to experiment longer. For
example, consider a setting with four voters, where Voter 1 (only) can cast two votes, anddecisions are made according to the simple majority rule. If, say, Voter 4 is the only sure winnerso far, Voter 1 can impose experimentation by siding with Voter 4. As long as no other winner
is observed, Voter 1 can push experimentation up to the single decision maker threshold. If,
say, Voter 2 becomes a winner, Voter 1 becomes subject to the risk of loser trap, as further
experimentation may reveal that Voter 3 also is a winner, resulting in a decisive coalition of surewinners. Contrary to the benchmark setting, thus, experimentation can be interrupted by theoccurrence of a new winner.
Why not a two-period model? Some features of collective experimentation such as cut-off
monotonicity, the impact of news arrival and type correlation, and the possibility of a negativevalue of experimentation, rely on the impact of one’s experimentation on other voters’ future30

experimentation, and hence require at least three periods. Infinite horizon provides time homogeneity, guaranteeing that cut-offs only depend on beliefs and not on time. Some potential
applications, such as joint R&D projects, can be seen as stopping games, where the time dimension is an important feature of the model. Some results in this paper can be interpreted ascomparative statics pertaining to stopping times.32
Risk aversion. The analysis above does not require risk neutrality: it is enough that voters
have a von Neumann-Morgenstern utility function, where lump sums actually correspond to“lump utils”, or certainty equivalents thereof if the magnitude of these lump utils is random.
Side payments. Side payments may restore efficiency under some implementations. However,a free-rider problem occurs if multiple individuals are needed to influence some voters. For
example, if sure winners must coordinate to influence some unsure voter, then obtaining thisswitch amount to a public good provision problem for sure winners. Furthermore, transfers may
also be impractical or unethical.
Switching costs. With a safe and a risky action, switching costs are easily accommodated,because the equilibrium policy can only switch actions once, from the risky to the safe action.
Adding a cost there simply reduces the value of experimentation ex ante and, once the risky
action is started, modifies indifference cut-offs.
Multiple Risky Actions Adding a second risky action to the benchmark setting may decrease
experimentation. This will occur, for example, if the two risky actions are perfectly correlated(for each voter) and the payoff of the new action exacerbates the loser trap.33 In that case,unsure voters may be unwilling to experiment with R for fear that winners impose the new
action, causing an even lower payoff for losers. Depending on the payoff structure, even a small
probability that the second risky action becomes relevant can have a dramatic impact on theequilibrium policy.
Two risky actions. Using a safe and a risky action provides an ideal setting to analyze
conservatism: conservatism means choosing the safe action when the risky action would be more
efficient. With two risky actions, conservatism could still be interpreted as settling inefficiently
too early on one of the two risky actions when it would be more efficient to continue learning
about the other action’s value. In this spirit, Albrecht, Anderson, and Vroman (2007) show
in their model of search by committees that collective search settles earlier (i.e. acceptance32
33

Setting λ = ∞ reduces the model to two periods: before and after type revelation.
Perfectly correlation ensures that there are still only two types of voters at any time and, therefore, noCondorcet cycles.

31thresholds are lower) than in the equivalent single-decision-maker setting.
Voter heterogeneity. If voters start with different type probabilities, this heterogeneity may
increase experimentation. Indeed, heterogeneity concentrates more power in the hands of those
voters who are pivotal today, because they are more likely to be also pivotal in the future. To
illustrate with an extreme case, suppose that there are 9 voters, 4 of which are (almost) sure
to be winners and 4 of which are (almost) sure to be losers. The remaining voter has (almost)perfect control over collective decision today, but also in the future: he will be able to side withwhichever group corresponds to his preferred action.
Power Concentration The impact of group size on experimentation, as described by Proposition 1 and the discussion that follows, can be reinterpreted as a result on power concentration.
As a simple model of power concentration, define an oligarchy as a subset of O (odd) voters suchthat, at any time, the collective decision is the action chosen by the majority of that subset.
Experimentation cut-offs are therefore defined as before, replacing k by the number of winners
within the oligarchy, and the total number of voters by the cardinal of the oligarchy. With thisinterpretation, Proposition 1, conveys a sense in which experimentation lasts longer if power isconcentrated into fewer hands. In particular, a dictator sets the same experimentation cut-off
as a single decision maker.34

8Conclusion

This paper has introduced a framework to study collective decisions when individual preferences
evolve through learning. In this framework, experimentation with new alternatives is influenced
by the potential effect of learning on future preferences and votes. Control sharing effects arise,
which introduce a conservative bias compared to the case of a single decision maker setting ora utilitarian social planner. Equilibrium policy is influenced by group size, voting rule, voting
frequency, voter heterogeneity and correlation, the relative strength of loser trap and winnerfrustration, the ability to commit to an observation-dependent policy or to a fixed action, theamount of aggregate uncertainty, and the particular process of news arrival, among other factors.
In addition to the points developed in Section 7, there are several other important extensions
to consider. For example, the cost or benefit of experimentation relative to the safe action
34

This assumes type independence. With positively correlated types, a dictator would learn from others andset lower cut-off than the single decision-maker cut-off.

32

may be heterogeneous across voters. Voters may also have an outside option allowing themto leave the group. In political applications, there may be several subgroups with high intragroup correlation and low inter-group correlation, with different and possibly evolving voting
weights. Finally, some risky decisions may be (at least partially) irreversible. Such features may
be necessary to analyze realistic settings. For example, “experimenting” with gas emissions has
long-lasting effects, implying irreversibility. A country’s cost of reducing gas emissions muchdepends on its primary source of energy, which is a source of heterogeneity. It will be useful
to investigate the effect of such features on the equilibrium policy. When fully observationcontingent commitments are not available, what forms of commitment can improve efficiency
under such extensions?

9

Appendix

9.1

Proof of Theorem 1

Suppose first that k > kN = (N − 1)/2, i.e. sure winners have the majority. We show that
C(K, p) = R for all p. If not, there exists K̄ with |K̄| > N/2 and p̄ for which C(K̄, p̄) = S. In
this situation, S is played forever whenever p reaches p̄, resulting in a constant value function ofs/r for all voters. Suppose that sure winner i is pivotal. Then voting for R yields an immediate
expected payoff of g and a continuation value function that is weakly greater than s/r, since surewinners get a payoff rate of at least s no matter which action is played. This strictly dominatess/r. So the only undominated strategies, starting from (K̄, p̄), must start with i voting for R.
Since this is true of all sure winners, and |K̄| > N/2, necessarily C(K̄, p̄) = R. This means thatin any Markov equilibrium in undominated strategies, R is elected forever as soon as winners
gain the majority. The value function of unsure voters is easily computed in that case: if an
unsure voter’s type is good, which happens with probability p, he gets the same expected value
as winners, g/r. Otherwise, he gets 0 forever. Therefore, u(k, p) = pg/r for k > kN . Now
consider the case k = kN , in which unsure voters have the majority, but only one new winner
suffices for sure winners to gain the majority. If i is an unsure voter, (2) reduces to
pg + λp pg∂V i,C− V i,C (k, p) + (N − kN − 1)λp
− V i,C (k, p) − λp(1 − p)(K, ) > s. (15)
r
r
∂p

g

In any equilibrium, C(K, p) = R if and only if condition (15) holds. This condition is formally
identical to the HJB equation for the optimization problem of a single decision maker. The
solution is characterized by some indifference threshold p(kN ) determined by the smooth-pasting33

condition of the Hamilton-Jacobi-Bellman equation (6), which reduces to
pg + pλ(g/r − s/r) + pλ(N − kN − 1)(pg/r − s/r) = s,(16)using the relation ukN +1 (p) = pg/r. The left-hand side of (16) is increasing in p, equal to 0 if
p = 0 and higher than g > s if p = 1. Therefore, the equation has a unique root, which can bereexpressed asp(kN ) =µs.
µg + (g − s) + (N − kN − 1)(p(kN )g − s)(17)This shows that C(K, p) = R if and only if p > p(kN ). If p ≤ p(kN ), S is chosen by unsure
voters. Since no more learning occurs, p remains constant forever, hence S is played forever.
The above policy entirely determines the value functions w(k, p) and u(k, p) of sure winners andunsure voters, for k = kN , which are in fact computable in closed-form by integration of theirdynamic equation (a similar derivation is done by Keller, Rady, and Cripps, 2005):
g g−s
w(kN , p) = −
r
r



1−p
1 − p(kN )N −kN andpg s − p(kN )gu(kN , p) =
+
r
r



1−p
1 − p(kN )Ω(p)Ω(p(kN ))N −kN 

µ

,

(18)

µ

(19)

Ω(p)
Ω(p(kN ))for p ≥ p(kN ), where Ω(p) = (1 − p)/p. These functions are easily shown to be increasing in p,with u(kN , p) ≥ pg/r. Moreover, u(kN , p) = w(kN , p) = s/r for p ≤ p(kN ), since the status quo
is imposed forever.
Now suppose that k = kN − 1. Then, any new winner results in the case k = kN just analyzed.
Again, (2) is formally equivalent to the stochastic control problem of a single decision maker.
Using again the smooth pasting property in (6), which implies that the derivative of the valuefunction vanishes, any indifference threshold p(kN − 1), must solve
pg + pλ(w(kN , p) − s/r) + pλ(N − kN − 2)(u(kN , p) − s/r) = s.

(20)

Since the left-hand side is increasing in p, equal to 0 for p = 0 and above s for p = 1, theequation has a unique root p(kN − 1). The choice rule thus defined entirely determines valuefunctions u(kN − 1, ·) and w(kN − 1, ·).
To show that p(kN − 1) > p(kN ), suppose that the contrary holds. Then, u(kN , p(kN − 1)) =w(kN , p(kN − 1)) = u(kN − 1, p(kN − 1)) = s/r, and by the smooth-pasting property, ∂u∂p (kN −1, p(kN − 1)) = 0. Therefore, (20) becomes p(kN − 1)g = s, which contradicts the assumptionthat p(kN − 1) ≤ p(kN ) < pM .

34

Let us now show that u(kN − 1, p) is nondecreasing in p. Suppose that pt = p̃ > p̄ and that
unsure voters behave as if pt were equal to p̄, meaning that they will stop experimenting after thesame amount of time σS , unless a new winner is observed before. Let σW denote the (possibly
infinite) time at which a new winner is observed. Until σ = min{σS , σW }, unsure voters receivenothing since R is played and no new winner is observed. The value function of this strategy isthus equal to −r(σ−t)u(pt ) = Et eq
N − kNs
1(w(kN , pσ ) + h) +u(kN , pσ ) + (1 − q),
N − kN + 1
N − kN + 1
rwhere q = P rob[σW < σS |pt ]. We saw that u(kN , ·) and w(kN , ·) were increasing in p. Moreover,these values are above s/r. Indeed, s/r is the value achieved if voters chose the status quo,
which is suboptimal by definition of σS and given that p(kN ) < p(kN − 1). Also, pσ is increasing
in pt given the Bayesian updating dynamics. Finally, σW is decreasing in pt , since a higher pt
makes it more likely that a payoff will be observed.35 This also implies that q is increasing in
pt , by definition of q and by the fact that σS is independent of pt , by construction. Combining
the above implies that u(p̃) > u(p̄). Since unsure voters optimize their value function withrespect to σS , this yields u(kN − 1, p̃) ≥ u(p̃) > u(p̄) = u(kN − 1, p̄), which proves monotonicityof u(kN − 1, ·). w(kN − 1, ·) is also increasing in pt . Indeed, let σ1 < σ2 the arrivals times of
lump-sum to the next two new winners. As is easily shown, these stopping times are decreasing
in pt in the sense of first-order stochastic dominance. This, given the fixed experimentationthresholds p(kN ) and p(kN − 1), implies that the distribution of the (possibly infinite) stopping
time σS at which experimentation stops increases in pt in the sense of first-order stochastic
dominance. Finally, sincew(kN −1 , pt ) = Et

hg 

 s
i
1 − e−r(σS −t) + e−r(σS −t) ,rrthis shows that w(kN −1 , ·) is increasing in pt . The remaining of the proof proceeds by backward
induction on k, where the induction hypothesis is that i) for all k′ > k, C(k′ , p) = R if and only
if p > p(k′ ), where ii) p(k′ ) is non-increasing for k′ > k, and iii) the resulting value functionsu(k′ , ·) and w(k′ , ·) are non-decreasing in p. The general induction step is then proved exactly
as above.
We now show cut-off monotonicity. We have seen above that p(k) = 0 for k > kN . The factthat p(kN ) ≥ pSD with strict inequality if N > 1 comes from the comparison of (17) and (1).
Monotonicity of p(k) is part of the induction in the proof of Theorem 1. There remains to show
35

Conditional on pt , σW is the mixture of exponential variables with intensity λj, j ∈ {0, . . . , N − kN + 1}, with

mixture weights {ρj } corresponding to the binomial distribution B(N − kN + 1, pt ). Monotonicity is in the senseof first-order stochastic dominance.

35

that pM > p(0). The indifference condition for p(0) is
p(0)g + p(0)λ(w(1, p(0)) − s/r) + p(0)λ(N − 1)(u(1, p(0)) − s/r) = s.

(21)

Since p(0) > p(1), unsure voters strictly prefer experimentation at p = p(0) when k = 1.
Therefore, u(1, p(0)) > s/r. Since winners always get a higher expected payoff than losers nomatter what action is chosen, w(1, p(0)) ≥ u(1, p(0)). Therefore, the second and third termson the left-hand side of (21) are positive, which implies that p(0)g < s, or equivalently thatp(0) < pM .
Monotonicity of u and w with respect to p was shown as part of the induction hypothesis of theabove proof. If k > kN R is elected forever since winners have the majority. This determines
value functions for this case and yields the last claim. To show monotonicity in k of w for k ≤ kN ,we proceed by induction. Clearly, g/r = w(kN + 1, p) ≥ w(kN , p). Suppose that w(k, p) ≤w(k + 1, p). We need to show that w(k − 1, p) ≤ w(k, p). Let φ(p) = w(k + 1, p) − w(k, p) ≥ 0
and ψ(p) = w(k, p) − w(k − 1, p). Since p(k − 1) ≥ p(k), ψ(p) ≥ 0 for p ≤ p(k − 1). Recall the
dynamic equation of w for p ≥ p(k − 1) and k̃ ≥ k − 1:−rw(k̃, p) + λ(N − k̃)p(w(k̃ + 1, p) − w(k̃, p)) − λp(1 − p)∂w(k̃, p) + g = 0.
∂p

Taking the difference of the resulting equations for k̃ = k, k − 1 and rearranging terms yields(r + λp(N − k + 1))ψ(p) = λp(N − k)φ(p) − λp(1 − p)ψ ′ (p).
Suppose φ is nonnegative by induction hypothesis, the previous equation can be rewritten asψ ′ (p) ≤ α(p)ψ(p) for function α. A direct application of Gronwall’s inequality along withψ(p(k − 1)) ≥ 0 proves that ψ is nonnegative, completing the induction step.
To show monotonicity of u with respect to k ≤ kN , fix some k ≤ kN . The dynamic equation ofu for p ≥ p(k − 1) and k̃ ≥ k − 1 is−ru(k̃, p)+λp(w(k̃+1, p)−u(k̃, p))+λp(N −k̃−1)(u(k̃+1, p)−u(k̃, p))−λp(1−p)∂u(k̃, p)+pg = 0.
∂p

Let φ(p) = u(k + 1, p) − u(k, p), φw (p) = w(k + 1, p) − w(k, p), and ψ(p) = u(k, p) − u(k − 1, p).
Taking the difference of the previous equation for k̃ = k, k − 1 and rearranging terms yields:(r + λp(N − k + 1))ψ(p) = λp[φw (p) + (N − k − 1)φ(p)] − λp(1 − p)ψ ′ (p).

(22)

We already know that φw is positive. Therefore, if φ were also nonnegative, the argument we justused for w would also show that ψ is nonnegative. In particular, if one can show that u(kN , p) ≥36

u(kN − 1, p), a backward induction will prove the result for all k ≤ kN . Combining (18) and (19)implies that, for k = kN ,g − s − (N − kN − 1)(s − p(kN )g)
φ (p)+(N −kN −1)φ(p) =
r
w



1−p
1 − p(kN )N −kN Ω(p)Ω(p(kN ))µTherefore, the left-hand side has the sign of g − s − (N − kN − 1)(s − p(kN )g). From the cut-off
formula (16), this expression has the same sign as s − p(kN )g, which is positive. Therefore,
the first term in the right-hand side of (22) is nonnegative for k = kN , which implies that ψ isnonnegative for k = kN . This fills the missing step of the induction, concluding the proof thatu is increasing in k for k ≤ kN .
To show the last statement, observe that u(kN + 1, p) = pg/r from Theorem 1, and thatu(kN , p) > pg/r, from (19).

9.2

Proof of Theorem 2

The proof is similar to that of Theorem 1, proceeding by backward induction on the number kof winners. For k ≥ k̄, the utilitarian optimum is to choose R forever even if p = 0, since
sure winners’ gains from R outweigh the aggregate gain from S even if all unsure voters get
nothing from R. This fact can be expressed as q(k) = 0 for k ≥ k̄. The resulting welfare isCC
W (k, p) = k gr + (N − k) pg
r . Consider next k = k̄ − 1. Let w (k, p) and u (k, p) denote the valuefunctions of sure winners and unsure voters if policy C is used, given that R is played forever if
a new winner is observed, and let W C (k, p) = kwC (k, p) + (N − k)uC (k, p), denote utilitarian
welfare under policy C. Then, the utilitarian criterion C must solve
"Z
#
σ
X
kt ,C
Wt
= sup Ete−r(τ −tdπθi τ (τ ) + e−r(σ−t) Wσkt +1,C ,θtiwhere σ is the first (possibly infinite) time at which a new winner is observed, and whereWσkt +1,C = W (k̄, pσ ), the welfare that was computed earlier for k = k̄. This is a standard
control problem, whose solution is Markov. The indifference boundary must satisfy the smooth
pasting condition
kg + (N − k)pg N s
kg + (N − k)pg + (N − k)λp−
= N s,rr
which has a unique root q(k), since the left-hand side is increasing in p, greater than N s if p = 1
and less than N s for p = 0, by definition of k̄. Therefore, C(k, p) = R if and only if p ≥ q(k).
This entirely determines w(k, ·), u(k, ·) and W (k, ·), which are easily shown to be increasing in37.

p. The remaining of the proof proceeds by backward induction on k as in Theorem 1, where
the induction hypothesis is that i) for all k′ > k, C(k′ , p) = R if and only if p > q(k′ ), where ii)
q(k′ ) is non-increasing for k′ > k, and iii) resulting value functions w(k′ , ·), u(k′ , ·), and W (k′ , ·)are non-decreasing in p.

9.3

Probability of receiving a lump sum between p and q < p

Let pS denote the probability that an individual with initial probability p of being a winnerreceives a lump-sum by the time his belief has dropped to q < p.
Proposition 2 pS = (p − q)/(1 − q).
Proof.

From the Bayesian updating equation, pt = (pe−λt )/((1 − p) + pe−λt ).Therefore, q isreached at a time T such that e−λT = Ω(p)/Ω(q), where Ω(p) = (1 − p)/p. Conditional on
the individual being a winner, the probability of getting a lump-sum before time T is simply
1 − e−λT , since the arrival rate is an exponential random variable with parameter λ. Combining
the previous formulas concludes the proof.

9.4



Proof of Theorem 9

For any safe rate s and policy C, Voter i’s expected payoff with policy C is
Z ∞
 Z ∞
i
−rt
i
i
VC = Ee dπCt (t) =e−rt E[dπC
(t)],
t
0

(23)

0

where expectations are conditioned on γ.
i
i
E[dπC
(t)] = P r[Ct = S]sdt + P r[Ct = R]E[dπC(t)|Ct = R]tti (t)|C = R] > sdt for all t, then V i > s/r, implying that Gi (γ) > s.
Therefore, if E[dπCtCCt

Suppose that s < g(γ i ). Then, by definition of g(·) and by the fact that the probability of eachi (t)] = g(γ i )dt > sdt. Moreover, C’s non-adversity with respect totype is a martingale, E[dπR
i (t)|C = R] ≥ E[dπ i (t)] as will be shown shortly. This inequality implies
i implies that E[dπCtRtthat GiC (γ) > s for all s < g(γ i ), which concludes the proof. To show the inequality, observe that,by Bayes’ rule, C is non-adverse for i if and only if P r[θ i good|Ct = R] ≥ P r[θ i good|Ct = S]38

for almost all t.36 Moreover,
i
i
E[dπC
(t)|Ct = R] = P r[θ i good|Ct = R]E[dπC(t)|Ct = R, θ i good]tti+ P r[θ i bad|Ct = R]E[dπC(t)|Ct = R, θ i bad]. (24)tCombining these results yields the inequality.

9.5

Negative Value of Experimentation

Let g > 0 and b < 0 the expected payoff rates of the risky arm for sure winners and sure
losers respectively. Let pM , pSD , pL , pW , p3 , respectively denote the myopic cut-off, the single
decision maker cut-off, the two unsure voters’ cut-off when the third voter is a loser, the two
unsure voters’ cut-off when the third voter is a winner, and the experimentation cut-off when
all three voters are unsure. For the following parameter values g = .1, b = −1, s = 0, r = 1,
λb = .1, λg = .11, cut-offs have the following values:
pM

pSD

pL

pW

p3

.9091 .9001 .9016 .9083 .9095

The most important result is that p3 > pM : voters stop experimentation at a probability level
where R’s expected payoff is strictly above S’s. As explained above, pL is much lower37 thanpW , meaning that if a voter is a loser, experimentation lasts much longer than if he is a winner.
3

From (3), this implies that the value of experimentation is negative at p3 , since V (p3 ) = sr < prg .

9.6

Proof of Theorem 10 (Sketch)

For sure winners, voting R forever is optimal as it maximizes their immediate payoff as well as
the length of experimentation, due to the cut-off monotonicity established in Theorem 1. Under
the protocol described in Section 6, unsure voters only observe the state k when particular cutoffs are reached. Let l denote the number of winners that was last revealed. For p > p(l), unsure
voters only know that the number k̃ of current winners is greater than or equal to l. Unsure
voters are only pivotal if k̃ ≤ kN . By Theorem 1, u(k̄, p) ≥ u(l, p) for l ≤ k̄ ≤ kN . Therefore,E[u(k̃, p)|l ≤ k̃ ≤ kN ] ≥ u(l, p) > s/r for p > p(l). Therefore, it is optimal for unsure votersto choose the risky action whenever indicated by the protocol, conditional on being pivotal. If,
36

Precisely, we have for all t, P r[Ct = R|good] ≥ P r[Ct = R|bad] ⇔ P r[Ct = R|good] ≥ P r[Ct = R] ⇔P r[good|Ct = R] ≥ P r[good] ⇔ P r[good|Ct = R] ≥ P r[good|Ct = B].
37
Indeed, pL is close to the single decision maker cut-off, while pW is close to the myopic cut-off.

39

upon reaching p(l), it turns out that k = l, i.e. no new winner has been observed since the last
release of public information, then it is optimal for unsure voters to stop: their value function
is identical to the benchmark case, equal to s/r.

40

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