Voting and Experimentation
Bruno Strulovici∗
December 29, 2007

Abstract
This paper analyzes a dynamic voting model where individual preferences
evolve through experimentation. Individual votes reflect not only current preferences but also the anticipated effect of elected alternatives on future preferencesand votes. The analysis is conducted in a two-arm bandit model, with a safe (status quo) alternative and a risky alternative whose payoff distribution, or “type”,
varies across individuals and may be learned through experimentation. Underany voting rule, society experiments less than any individual would if he could
dictate future decisions, due to a control-loss effect. Depending on the nature of
uncertainty, majority-based experimentation also has a systematic bias comparedto the utilitarian policy. For large groups with independently distributed types,
this control-loss effect annihilates the value of experimentation, prompting individuals to vote myopically. However, even with independently distributed types,
a positive news shock for anyone raises everyone’s value function and incentive
to experiment. Efficiency increases with ex ante preference correlation. The paper also discusses the effect on experimentation of the ability to commit and ofasymmetric information.

Keywords: Experimentation, Dynamic Social Choice, Reforms, Conservatism.
∗

Nuffield College and Economics Department, Oxford University. I am very grateful to Meg Meyer,

Andrea Patacconi, and Kevin Roberts for insightful conversations, as well as to Daron Acemoglu,
Patrick Bolton, Alessandra Casella, Robert Evans, Emeric Henry, Godfrey Keller, Jeremy Large,
Rocco Macchiavello, Paul Milgrom, Roger Myerson, Volker Nocke, Andrea Prat, John Quah, Marzena
Rostek, Gilles Serra and other seminar participants at Oxford University, Gerzensee’s European Summer Symposium in Economic Theory, and Cambridge University for numerous comments. I am also
particularly grateful to Paul Milgrom for his general insights and encouragements about research,which had a significant influence on this project.

1

Introduction

The positive social choice literature usually assumes that individuals perfectly know
their own preferences. In reality, however, preferences may evolve through learning.
For example, a reform may have uncertain consequences, hence desirability, which can
only be learned by experimenting with that reform. Similarly, decisions in committees
are often analyzed under the assumption that payoff distributions are perfectly known
to committee members at the time of their decisions, although this is rarely the case inpractice. This difference is important if those decisions are made repeatedly, because
one’s vote at any given time must take into account the impact of the current decision
on everyone’s future preferences and votes.
This paper analyzes how experimentation - the fact that an alternative is chosen,
despite its comparatively lower immediate expected payoff, in order to learn more about
its value - is affected by the nature and amount of individual control over collective
decisions. This raises another, dual question: how does the possibility that individual
rankings of social alternatives evolve through learning, potentially resulting in majority
shifts, affect collective decisions? The analysis is conducted in a two-arm bandit model(settings with multiple risky actions are discussed at the end of the paper). The “safe”
alternative yields a constant, homogeneous payoff to all. The “risky” alternative yields
payoffs according to some distribution, or type, which varies across individuals. At any
time, only one of the two action is taken, according to some voting rule. Individuals maylearn their type through experimentation with the risky alternative. The paper first
analyzes a benchmark setting, in which the type is either “good” or “bad”, then extends
several results to general structures of preference uncertainty, which accommodate arich set of applications.
In the benchmark setting, if an individual’s type is good, the risky alternative pays him
some lump-sums whose arrival times are exponentially distributed. If his type is bad,the risky action pays him nothing. Therefore, an individual knows for sure that his
type is good as soon as he receives a lump-sum (he is then a “sure winner”). However,2

he remains uncertain about his type as long as he has not received anything, for this
could be either due to a bad type, or to a late lump-sum arrival for a good type (he isthen an “unsure voter”). At each instant, society chooses a single action according tosome fixed voting rule. When society experiments with the risky action, this results ina better assessment of individual valuations for that action, hence a better knowledgeof individual rankings of alternatives.
A key feature of the analysis is the feedback effect occurring between individual preferences and collective decisions. Indeed, not only does preference uncertainty affectsociety’s choices, but the reverse is also true. This social phenomenon has been described in the context of conservatism by Kuran (1988):

I believe that a complete model for the study of conservatism would have acircular dynamic structure, with individuals’ choices driven by their beliefs
and preferences; society’s choices generated by its members’ choices; and,completing the circle, these members’ beliefs and preferences influenced bysociety’s choices. It would thus incorporate three interactive processes: thatby which individuals’ seek and integrate information to form their beliefs
and preferences regarding the alternatives they face; that by which society combines these choices to select policies, institutions, and technologies;
and finally, that by which collective outcomes mold individuals’ beliefs andpreferences. I am suggesting that the aim of theoretical analysis on thesubject should be to elucidate these three processes with an eye toward
deriving propositions as to when, how, and to what extent individuals andcollectivities adapt to changes in environmental factors.

Conservatism resulting from the interaction between preference uncertainty and collective decisions has been studied, both empirically and theoretically, in the context oftrade liberalization.1 Fernandez and Rodrik (1991), in particular, are motivated by the
behavior of industry groups who lobby against trade reforms ex ante, but a majorityof which benefits from these reforms once they are implemented. They explain thisparadox by showing that reforms having a negative expected value ex ante may turn1

See Baldwin (1985), Bhagwati (1988) and Fernandez and Rodrik (1991).

3

out to be beneficial to the majority once implemented. This will be the case, for example, if a reform generates a few “losers,” whom it severely hits, while providing small
benefits to a majority of “winners”. In such a scenario, when the identity of winners
and losers is a priori unknown, the reform is initially opposed by all, but eventually
gains the support of a majority as the identity of winners and losers gets revealed.
This paper provides a general analysis of the above questions. It shows in particularthat the value of experimentation is reduced, compared to a single-decision-maker setting, by one’s having to share control over future decisions with other society members.
This control-loss may be decomposed into a loser-trap effect: an individual has the risky
action imposed on him even though it turns out to be bad for him, and a symmetricwinner-frustration effect: an individual is blocked from enjoying the risky action in thelong run, despite his benefiting from it. These effects, each of which occurs with someprobability when control is shared, are absent from single-decision-making problems.
They reduce the attractiveness of experimentation. The loser trap effect arises evenwhen learning is infinitely fast, so that experimentation entails no time cost: given thepossibility of immediately revealing everybody’s type, society may prefer to reject thispossibility and impose instead the status quo forever, provided the loser-trap effect isstrong enough.
Moreover, provided that types are independently distributed, control-loss effects lead tothe stark result that, as group size goes to infinity, everyone votes myopically. The valueof experimentation vanishes and each individual votes for the risky action only if its
expected payoff is higher than that of the safe action. Intuitively, this can be explainedby the fact that control over future decisions is infinitely diluted and individual power
vanishes.
In the benchmark setting, society also experiments too little compared to the utilitarianpolicy. Intuitively, whenever a majority of unsure voters imposes the safe action, it
ignores the utility of sure winners. Utilitarian welfare, however, would take sure winners’ utility into account, making the risky action more attractive overall than from thesole viewpoint of unsure voters. This result can be reinterpreted in terms of commitment. If individuals are ex ante identical (i.e. behind a veil of ignorance) and able tocommit at the outset to some collective policy, they choose the utilitarian policy, since4their expected utilities are identical and proportional to utilitarian welfare. Therefore,
society experiments less under majority voting than it would if it could commit to acollective policy behind a veil of ignorance. In the symmetric setting where bad news
(negative lump-sums) reveals sure losers, and remaining voters are unsure, the opposite bias occurs to some extent: as long as they have the majority, unsure voters pushexperimentation too far compared to the socially efficient level, as they ignore the welfare of sure losers (this case is more complex however, see Section 6.3). Therefore, forapplications that are appropriately modeled by either of these settings, it is possible
to identify systematic biases from majority-based experimentation, relative to sociallyefficient decision making.
Given these results, it is tempting to shift to a normative analysis and ask whether
another voting rule would be more beneficial to society than the simple majority rule.
For example, if the risky action requires unanimity, an individual cannot has the risky
action imposed on him if it turns out to be detrimental for him. Unanimity thus
completely gets rid of the loser trap effect. However, this very fact also makes experimentation less attractive. Indeed, any winner is less likely to enjoy the risky actionin the long run, for this would require that all other society members also turn out
to be winners. Requiring unanimity for the risky action thus exacerbates the winnerfrustration effect. Whatever control is gained from being able to veto the risky action
is balanced by a control loss for enforcing that same action. Examples indeed showthat with the unanimity rule, experimentation may last longer or shorter than underthe majority rule.
In contrast to the above results, the paper also shows for the benchmark setting that,even when preferences are independently distributed, so that no learning can occur
from the observation of others’ payoffs, good news for any individual is good news for
all, and prompts society to experiment more. This result may seem counter-intuitive,
as the occurrence of a new winner brings unsure voters closer to the brink, where risky
action is imposed on them forever. However, it also makes unsure voters more likely
to enjoy the risky action if they turn out to be winners, an effect that must dominate
whenever society experiments in the first place.
As should be expected, the severity of the control-loss effect diminishes if types are pos-5

itively correlated ex ante, so that interests are more likely to be aligned. Such positive
correlation also makes experimentation more attractive as individuals learn from boththeir and other individuals’ payoffs, which reduces the time cost of experimentation.
This observation is particularly relevant for large societies composed of groups withhigh intra-group correlation (see also Section 7).
For several applications envisaged in this paper, such as national security and global
warming (Section 2), a more general description of preference uncertainty is required.
The paper shows, assuming that types are independently distributed across voters,that collective experimentation is always shorter than experimentation in otherwise
identical single-person decision making problems, or than what any individual would
want if he had dictatorial power over decisions (without assuming type independence).
This result holds for all voting rules and for general specifications of the risky action.
Moreover, provided that collective decision rules are not “adverse”, in the sense thatfor any individual, the risky action is more likely to be implemented if that individual
benefits from it than otherwise, there is always some amount of experimentation. In
particular, as long as any given individual has some control over decisions and othervoters are not encouraged to elect an action whenever it turns out to be detrimental tohim, that individual wishes to experiment with the risky action, i.e. to try it even insome circumstances where its immediate expected payoff rate is below the safe action’srate.
The present analysis contributes to the literature on collective conservatism. In contrast
to earlier literature, it does not rely on arguments such as exogenous transaction costs
or sunk costs or bounded rationality, as surveyed by Kuran (1988). Some results ofthis paper are closely related to Fernandez and Rodrik (1991), who were the first tointerpret the status quo bias as a consequence of preference uncertainty. In their setting
however, there is no experimentation. Whatever individuals learn in the first period
has no impact on the collective decision of the second period, which is known from theoutset, as there is no aggregate uncertainty. Moreover, preferences are immediately
fully revealed to all agents, which rules out any dynamic analysis, such as understanding
how a positive or negative news shock for one individual affects other voters’ willingness
to experiment.

6

The paper is also related to a developing literature on games and experimentation inwhich conservatism may arise as a consequence of strategic information acquisition, as
described by Bolton and Harris (1999), Décamps and Mariotti (2004) and Keller, Rady,and Cripps (2005). Those papers identify a free-riding problem which may result in
lower experimentation. Li (2001) provides a theory of conservatism based on a similarargument. In those papers, agents make individual investments to acquire information
about the common value of some alternative. In particular, information acquisition
amounts to a public-good problem. In contrast, the present analysis considers thereverse setting, in which a single collective action is made at any time, but the valueof the action may vary across individuals. This paper therefore also contributes to theliterature on experimentation by analyzing voting and experimentation.
More generally, the paper contributes to the experimentation literature in economics,
started by Rothschild (1974), Jovanovic (1979), Weitzman (1979), and Roberts andWeitzman (1981). For a recent survey of this literature, see Bergemann and Välimäki (2006).
With respect to this literature, the present paper is the first to consider voting andexperimentation.
The analysis of the benchmark setting owes conceptual and technical clarity to theuse of exponential bandits. Exponential bandits were introduced by Presman and
Sonin (1990) and Presman (1990), and used in economics by Malueg and Tsutsui (1997),
Bergemann and Hege (1998, 2001), Décamps and Mariotti (2004) and Keller, Rady,and Cripps (2005). In contrast to the last and most closely related paper, where many
asymmetric equilibria may occur, the setting considered here has a unique equilibrium
which is solved by backward induction on the number of winners observed. This equilibrium is robust to the chosen equilibrium concept, whether it be majority voting
equilibrium used by, among others, Roberts (1989), or the weak-dominance solution
concept, pervasive in the literature on legislative bargaining.
Section 2 presents a simple example providing intuition for several results of the paper, and describes several applications. Section 3 analyzes the benchmark setting withmajority voting and independent types. Section 4 analyzes other decision processes:i) utilitarian policy, ii) commitment, delegation, and delays, and iii) experimentation
based on the unanimity rule. Section 5 analyzes the effect of voter heterogeneity and7ex ante type correlation on experimentation. Section 6 has two purposes: i) introduce
settings with negative news shocks, useful for several applications, and ii) extend several results to general structures of preference uncertainty. Section 7 discusses someassumptions and extensions of the main model, in particular: i) the case of privately
observed payoffs, ii) the case of multiple risky actions. Section 8 concludes.

2

Example and Applications

2.1

A Simple Example

Three friends, Ann, Bob, and Chris, go to a restaurant once every week-end. Eachweek-end, they choose their restaurant using the majority rule. A new restaurant has
just opened. Should the friends try it? Do they try it? Suppose the alternative is a
restaurant that gives utility 1 to all. For each voter, the new restaurant can be either
bad (yielding 0 utility) or good (yielding utility u > 1). Suppose that preferences, or“types” are independently distributed across friends (e.g. Ann is no more or less likelikely to appreciate the new restaurant if Bob likes it, etc.), with both types having an
ex ante probability of 1/2.
Immediate Full Type Revelation Suppose that, if they try this new restaurant,
all voters immediately learn their type. With probability 1/8, Ann and Bob will like it
but Chris won’t. In this case, Chris is trapped into always returning to that restaurant,
as Ann and Bob have the majority. This situation will be referred to as the “loser trap”.
Also with probability 1/8, Chris is the only one who turns out to like the restaurant,
but is blocked from exploiting this discovery for future dinners by Ann and Bob. Thissymmetric situation will be referred to as “winner frustration”. Overall, there is a
probability 1/4 that Chris loses control over the decision process, compared to thesituation in which he could choose the restaurant by himself in the future. Dependingon u and on how time is discounted, these control-loss effects may be such that Chris
and, by symmetry, all voters prefer not to try the new restaurant even though each ofthem would have preferred to try it if he had full control over future decisions.

8Gradual Type Revelation Now suppose that a voter likes the new restaurant if
and only if he finds a dish there that he really likes. In this case several visits to therestaurant may be needed to find out one’s type. This can lead to situations in which
friends experiment with that restaurant until either a majority of them likes it, or amajority of them judges unlikely that they will find anything like there. With thisassumption, suppose that, in their first try, only Chris discovers that he likes the newrestaurant. What effect does it have on Ann and Bob? Does this incite them to try
it more or, on the contrary, prompts them to block new experimentation? Good news
for Chris reduces the risk of winner frustration for Ann and Bob, but increases theprobability of the loser trap. It turns out that good news for Chris always makes Ann
and Bob more willing to experiment, as shown in Section 3.
Social Efficiency What would a social planner, wishing to maximize the sum of
utilities of the three friends, choose to do? Suppose that u is very close to 1, so that“winners” (those who like the new restaurant) appreciate it only slightly more than
the incumbent. Then, the only case in which a social planner would impose the newrestaurant in the long run is if all friends turn out to be winners. If there is a loser(i.e. someone who dislikes the new restaurant), the very small utility gain achieved
by winners does not compensate the disutility experienced by the loser. However, thispolicy is incompatible with majority voting, which would result in the two winners
imposing the new restaurant despite the much larger magnitude of the loss incurredby the third, losing voter. This difference may result in all friends voting against the
new restaurant when their preferences are still unknown, due to the loser-trap effect,
while a social planner will try it to see whether all friends like it. In fact, Section 4shows that, with positive news shocks, majority-based experimentation is always lessthan the socially efficient policy.

2.2

Applications

The effects described in this paper can arise whenever decisions are made collectively
and repeatedly. Although the examination of any particular application is beyond thescope of this paper, the reader may keep in mind the following contexts when thinking
about the phenomena analyzed in the following sections.
9

Reforms with Unknown Losers Reforms constitute a natural domain of application of the above analysis. Even when they benefit a significant fraction of the
population, reforms usually harm some individuals or groups. Whenever the identityof these losers is ex ante unknown, the “loser trap effect” becomes a source of conservatism, as this paper shows. Such reforms include the case of trade liberalization
studied by Fernandez and Rodrik. Uncertainty regarding the role that each individual will play in a more open society can also justify some lack of popular support fortransitions from rigid economic systems to more open ones.
Ex Ante Public Goods Section 6 considers a general specification of preference
uncertainty that encompasses a setting, symmetric to the benchmark setting, in which
negative lump-sums occur if the action is bad, while the status quo entails a small
cost compared to the payoff of the risk risky action if it turns out to be good. Thissetting captures applications in which a costly effort is required of group members toprevent catastrophes from happening, but the identity of those who may suffer fromthese catastrophes, if no preventive action is taken, is unknown.
In the context of global warming, for example, the safe action is immediate adoption of
drastic policies to cut emissions of greenhouse gases. The risky action is to “explore”
the effects of global warming. The safe action has an economic cost. The risky action
results in losers, which are the countries most adversely affected by global warming.
The “winners” are those countries which suffer least from global warming. Actual consequences of global warming are largely unknown in magnitude and nature. Despitesome degree of predictability, the identity of losers and winners is to a large extent also
unknown.2 The effects, whatever they are, are likely to be lasting. Moreover, fighting
global warming requires coordination: all significant polluters must simultaneously reduce gas emissions, as emissions emanating from any country affect all other countries.
From this viewpoint, the analysis of the paper yields several observations and predictions (see Sections 6.1, 6.3, 4.1, and 4.2). It predicts two possible regimes, dependingon parameter values (payoffs, learning speed relative to discount rate, group size, etc.),
for gas emissions policies when countries cannot commit to a coordinated long-term
policy. In the first regime, individual polluting countries wait to better understand2

To take one example, the impact of global warming on several northern European countries will

dramatically depend on its influence on thermohaline circulation, which is largely unknown.

10the consequences of global warming until, possibly, countries which become actuallyharmed by its consequences gain enough power to impose drastic emission reductions toall polluters. In the second regime, individual countries, considering the risk of entering the first regime, agree very early on to drastically reduce emissions of greenhousegases, without learning anything about the actual consequences of global warming.
Both regimes are socially inefficient. The first regime pushes experimentation with
global warming too far, as polluting countries ignore the woes of affected countries.
The second regime does not explore enough the consequences of global warming, as it
imposes drastic actions to avoid a phenomenon that remains completely unknown. The
paper shows that social efficiency would require a unanimous long-term commitment
to a gas emission policy that depends on the observed consequences of global warming. This may entail, for example, all polluting countries agreeing to drastically cut
emissions as soon as any one of them is significantly harmed, where this last event may
be assessed based on different environmental indicators. The paper also shows thatcommitment to an observation-dependent policy is very different from commitment toan action, such as an irrevocable imposition of drastic reduction of gas emissions toall polluters. Indeed, commitment to an action reduces efficiency even further thanequilibrium policies without commitment, as it adds even more rigidity to the decision
process. Furthermore, the paper shows that without policy commitment, coordinatedreduction of greenhouse gases i) is implemented earlier than what any country would
prefer if it could dictate current and future gas emission policies to all countries, and ii)
always involves some experimentation, in the sense that for all voting rules, countries
always wait to assess, to some extent, the effects of global warming before reducing gas
emissions. (See Section 6.)
As another example, some security policies benefit from coordination at an international level (more generally a global, rather than local, level). All countries concernedwith the policy must bear the cost of this coordination. Investing in efficient security
coordination amounts to taking a “safe” action. The risky action is not to implement
such security coordination. The losers are countries who suffer from attacks whichmight have been prevented or reduced by global coordination. The “winners” are thosecountries who do not get attacked despite the lack of global security policies. The identity of losers is ex ante unknown, at least to some extent. To fit the dynamic analysisof this paper, it is important that losers are affected in such a way that, after their first11

attack, they wish, more than unaffected countries, to implement global coordination.
In such case, this paper makes similar predictions as in the context of global warming,
namely i) policy coordination suffers the inefficiency of the two regimes discussed in
the context of global warming, ii) coordination is implemented earlier, others things
equal, than what any individual country would prefer if it could dictate to others thedegree, present and future, of coordination in security policies, and iii) there is always
some experimentation, in the sense that for all voting rules, countries always try toassess to some extent the risk of attacks before implementing security coordination.
Joint Investment As an illustration of the winner frustration effect, suppose thata firm considers a new, challenging project, which involves several development tasks.
Each task has some probability of success. Tasks are complementary: the project
can only succeed if all of the tasks are successful. In such a situation, the managersof the individual tasks are exposed to the risk of the project being abandoned due tofailures with other tasks of the project, even if they achieve success with their own task.
The winner frustration effect dominates this setting, as the gain from the risky action(here, undertaking the project) is large compared to the status quo. The loser trap
effect is small here, as losers from the risky action are not significantly harmed. Thishighlights an important feature of many applications: the tension between a necessary
complementarity in actions with a possible misalignment of preferences.

3

Benchmark Setting

As a benchmark, this section embeds the exponential bandit model of Keller, Rady,and Cripps “KRC” (2005) into a setting with majority voting. Time t ∈ [0, ∞) is
continuous and discounted at rate r > 0. There is an odd number N ≥ 1 of individuals
who continually decide at 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” and can be, for each player, either “good” or “bad.”
The types (good or bad) are independently distributed across the group (the case ofcorrelated types is considered in Section 5). If R is bad for some individual i, it always
pays him 0. If R is good for i, it pays him lump-sum payoffs at random times which12correspond to the jumping times of a Poisson process with constant intensity λ. The
arrival of lump-sums is independent across individuals. The magnitude of these lumpsums3 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 be good for him. This probability is the same for all and is common knowledge.
Thereafter, all payoffs are publicly observed, so that everyone 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, therefore, the group is divided into k “sure winners”
for whom R is good with probability one, and N − k “unsure voters,” who have thesame probability p of having a good type. Unsure voters’ probability evolves according
to Bayes’ rule and obeying the dynamic equation dp/dt = −λp(1 − p) if no lump-sum
is observed, with pj jumping to 1 if some voter j receives a lump sum (see KRC, p. 45).
Type independence implies that an unsure voter only learns from his payoff stream but
not from 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 equals
pSD =

µs
,
µg + (g − s)(1)where µ = r/λ (see KRC). Let pM = s/g denote the myopic cut-off, i.e. the probability
below which R yields a lower expected payoff than S. The previous formula implies
that pSD < pM . Indeed, experimentation really only takes place 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 its true value for future decisions. Choosing R in thisrange is optimal due to the option value of experimentation.
For a group, the first results of this paper show that, with the simple majority rule,3

All results hold if these lump sums have random, independently distributed magnitudes withconstant mean h. More generally, what matters for decision makers are expected payoff rates ofeach action and the probability that the risky action be good or bad. See Section 6 for a general
specification of payoff distributions and beliefs.

13

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
pt R forever otherwisep(2), p(3)
0

t0

t1

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.
collective decisions are determined by a vector of nonincreasing cut-offs p(k)0≤k≤N suchthat the risky action is played at time t if and only if pt > p(kt ), where kt is the numberof sure winners at that time. The dynamics of collective decisions can thus be describedas follows. Starting with some (high enough) level p0 that is common to all, R is electeduntil the threshold p(0) is reached, at which time two things can happen: either no one
received any lump-sum so far, and the safe action is then elected forever. Or at least
one winner has been observed by then, in which case experimentation continues until
at least another threshold p(1) < p(0), and so on. The dynamics of collective decisions,which is next formally analyzed, is qualitatively represented by Figure 1 for the caseof three voters. The intuition for cut-off monotonicity is presented before Theorem 2.
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 theaction space {R, S}. Any collective decision rule determines a value function for each14

agent i:
Vti,C = EtZ ∞−r(τ −t)etdπCi τ (τ ),where dπSi (τ ) = sdτ , and dπRi (τ ) = hdNτi or 0 depending on whether R is good or bad
for i, where {N i }1≤i≤N is a family of independent Poisson processes with intensity λ.
At any time t, within each subgroup of voters (sure winners or unsure voters), allvoters have the same value function since their payoffs are identically distributed. Let
w k,C and uk,C respectively denote the value functions of sure winners and unsure voterswhere superscripts indicate the current number k of sure winners and the rule C thatis followed. Letting kN = (N − 1)/2, winners have the majority if and only if k > kN .
Definition 1 C is a Majority Voting Equilibrium (MVE) if for all t, it satisfies thefollowing conditions:• if kt ≤ kN , C solves
Z σ
kt ,C
ut = sup Et
e−r(τ −t) dπθuτ (τ )
θ
t


1
(N − kt − 1) kt +1,C−r(σ−t)kt +1,C
+e
w
+
uσ
, (2)
N − kt σ
N − kt• if kt > kN , C solves
wtkt ,C = sup EtθZ σ−r(τ −t)etdπθwτ (τ ) + e−r(σ−t) wσkt +1,C



,

(3)

where σ is the first (possibly infinite) time at which a new winner is observed, and θ isany policy.
This definition means that at any time, the subgroup with the majority follows thepolicy that is optimal for itself, until a change occurs in the composition of the subgroups. When unsure voters have the majority, the conditional probability that any
given unsure voter be that new winner is simply 1/(N − kt ), since there are N − kt
unsure voters with identical payoff distributions. This explains the last term in (2).
This definition extends to a non-Markov setting the standard notion of majority voting
equilibrium for dynamic Markov policies (see for example Roberts (1989)). In particular, if one imposes at the outset that the collective decision rule only depend on the15

state (k, p), the above equations then reduce to the following Hamilton-Jacobi-Bellman(HJB) equations (to be explained in detail shortly):• If k ≤ kN , C = R if and only if
pg + λp[w C (k + 1, p) − uC (k, p)]+ λp(N − k − 1)[uC (k + 1, p) − uC (k, p)] − λp(1 − p)∂uC (k, p)> s (4)∂p• If k > kN , C = R if and only if
g + λp(N − k)[w C (k + 1, p) − w C (k, p)] − λp(1 − p)

∂w C (k, p)
>s
∂p

(5)

The equilibrium concept also corresponds, in a dynamic, continuous-time setting, tothe usual concept of weak dominance equilibrium: it is the outcome obtained if any
member of the majority chooses his optimal action, as if he were pivotal, given thatthe same will be true of majority members at any future date. The concept rules out
trivial Nash equilibria, such as equilibria in which all individuals vote for the sameaction. It also gets rid of some problems and subtleties specific to continuous games,such as those identified by Simon and Stinchcombe (1989)
The first result of this paper states that there exists a unique4 majority voting equilibrium, that this equilibrium has the Markov property, and that it is determined by
cut-off policies. Existence of Markov equilibria has been widely studied and is common
in dynamic stochastic games. However, the fact that any equilibrium of the present
dynamic game is Markov, and that there exists indeed a unique such equilibrium, israre and noteworthy. It owes to the particular structure of the model, which makes
possible the use of a backward induction argument on the number of winners, andworks despite the infinite horizon, continuous-time nature of the model.
Before formally stating and proving the result, some intuition for it may be helpful(assuming for now the Markov property). At any time t the state of the group can4

As usual in the continuous-time stochastic control literature, uniqueness of the optimal policy isunderstood up to a subset of times of measure 0 on which actions can take any possible values without
affecting value functions.

16

be summarized by kt and pt . Each subgroup (sure winners or unsure voters) consistsof individuals with perfectly aligned interests. Majority belongs to either sure winners
if k > kN or to unsure voters if k ≤ kN . The group with the majority can enforce
whichever action it prefers. For example, if sure winners have the majority, it is clearlyin their interest to impose R forever. Similarly, if unsure voters have the majority andp is equal (or very close) to 0, so that they are in fact (almost) sure that the action isbad for them, they will impose the status quo S forever, and no further learning occurs.
Since an unsure voter can become a winner, but the reverse is false, majority can only
shift from unsure voters to winners. Starting with a majority of unsure voters, decisions
are dictated by unsure voters’ interest until they (possibly) lose the majority. The
main question is therefore to determine unsure voters’ preferences. These preferences
are assessed by the following Hamilton-Jacobi-Bellman (HJB) equation: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, the second to action S. The
effect of action R can be decomposed into four parts: i) the expected flow rate pg,ii) the jump of the value function if i receives a lump-sum, which occurs at rate λ withprobability p: his value function jumps to w and the number of winners increases by
1, iii) the jump of i’s value function if another unsure voter receives a lump-sum: iis still an unsure voter, but the number of sure winners increases by 1, and iv) the
effect of Bayesian updating on the value function when no lump-sum is observed.
Independence of the Poisson processes governing individual payoffs implies that only
one lump-sum can be received during any infinitesimal period of time, so that noterm involving two or more jumps appears in the HJB equation. In comparison, if S
is chosen, learning stops, and i simply receives the flow rate s. Since unsure voters
have identical value functions, they unanimously decided to stop experimentation if
p becomes too low. They do so when the R part of (6) equals s. At such level p,
the smooth pasting condition implies that the derivative term vanishes since the valuefunction is constant, equal to s/r, below that level (see for example Dixit (1993)). This
determines the equilibrium policy’s cut-offs.
Theorem 1 (Existence and Uniqueness) There exists a unique MVE. This equilibrium is characterized by cut-offs p(k), 0 ≤ k ≤ N, such that R is chosen in state17

(k, p) if and only if p > p(k).
Proof.

Suppose that k = N, i.e. all voters are sure winners. Then, σ is necessarily

infinite, so (3) reduces to
wtN,C = sup EtθZ ∞−r(τ −t)etdπθwτ (τ ).

The (essentially) unique solution is Cτ = R for all τ , since it provides winners at any
time with the maximal possible expected payoff g. This gives them the constant valuefunction wtN = g/r. The value function of unsure voters is also easily computed: if
an unsure voter’s type is good, which happens with probability pt , he gets the same
expected value as winners, g/r. Otherwise, he gets 0 forever. Therefore, uN
t = pt g/r.
For k = N − 1, (3) reduces to
wtN −1,C = sup EtθZ σ−r(τ −t)etdπθwτ (τ ) + e−r(σ−t) g/r

,

where I use the fact that wtN = g/r. Again, the (essentially) unique solution is Cτ = R−1
for all τ , value functions still equal wtN −1 = g/r and uN= pt g/r. By the samet

induction argument, Cτ = R and wtk = g/r and ukt = pt g/r for all k > kN . Now
consider the case k = kN , in which unsure voters have the majority, but only one newwinner among them is needed for the majority to switch to sure winners. Then (2)reduces to
ukt N ,C = sup EtθZ σ−r(τ −t)edπθuτ (τ ) + e−r(σ−t)

t



g
 (N − k − 1) p g 
1
N
t
+h +
,
N − kN r
N − kNr(7)

using the relations w kN +1 = g/r and ukN +1 = pt g/r. The optimization problem (7) isformally identical to the optimization problem of a single decision maker, with known
termination values. The solution of such problem is well known (see for example
Fleming and Soner (1993)). The control is Markov in p, with any indifference threshold p(kN ) determined by the smooth-pasting 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,(8)using the relation ukN +1 (p) = pg/r. The left-hand side of (8) is increasing in p, equal
to 0 if p = 0 and higher than g > s if p = 1. Therefore, the equation has a unique18

root, which can be reexpressed asp(kN ) =µs.
µg + (g − s) + (N − kN − 1)(p(kN )g − s)(9)This shows that C(p, kN ) = R if and only if5 p > p(kN ). If p ≤ p(kN ), S is chosen by
unsure voters. Since no more learning occurs, p remains constant forever, hence S isplayed forever. The above strategy entirely determines the value functions w C (kN , p)
and uC (kN , p) of sure winners and unsure voters, which are in fact computable inclosed-form by integration of their dynamic equation:N −kN 
µ
1−p
Ω(p)
g g−s
,
w(kN , p) = −rr
1 − p(kN )Ω(p(kN ))andpg s − p(kN )gu(kN , p) =
+
r
r



1−p
1 − p(kN )N −kN Ω(p)Ω(p(kN ))

µ

(10)

(11)

for p ≥ p(kN ), where Ω(p) = (1 − p)/p. These functions are easily shown to be
increasing in p, with uC (kN , p) ≥ pg/r. Moreover, u(kN , p) = w C (kN , p) = s/r forp ≤ 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 asingle decision maker. Using again the smooth pasting property in (6), which implies
that the derivative of the value function 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.

(12)

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
value functions 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-pastingC
property, ∂u(kN − 1, p(kN − 1)) = 0. Therefore, (12) becomes p(kN − 1)g = s,∂pwhich contradicts the assumption that p(kN − 1) ≤ p(kN ) < pM . Thus, necessarily,p(kN ) < p(kN − 1).
5

As before, this is up to action changes on a time subset of measure 0.

19

Let us now show that u(kN − 1, p) is nondecreasing in p. Suppose that pt = p̃ > p̄and that unsure voters behave as if pt were equal to p̄, meaning that they will stop
experimenting after the same amount of time σS , unless a new winner is observed σW .
Until σ = min{σS , σW }, unsure voters receive nothing since R is played and no new
winner is observed. The value function of this strategy is thus equal to

 


1
N − kNs−r(σ−t)u(pt ) = Et eq(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 , ·) are increasing in
p. Moreover, these values are above s/r. Indeed, s/r is the value achieved if voterschose 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 beobserved.6 This also implies that q is increasing in pt by definition of q and the factthat σS is independent of pt by construction. Combining the above implies that u(p̃) >u(p̄). Since unsure voters optimize their value function with respect to σS , this yields
u(kN − 1, p̃) ≥ u(p̃) > u(p̄) = u(kN − 1, p̄), which proves monotonicity of u(kN − 1, ·).
w(kN −1, ·) is also increasing in pt . Indeed, let σ1 < σ2 the arrivals times of lump-sum tothe 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 firstorder stochastic dominance. Finally, sincehg
 s −r(σ −t) i−r(σS −t)Sw(kN −1 , pt ) = Et
1−e
+ e
,
r
r
this 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 functions u(k ′ , ·) and w(k ′ , ·) are non-decreasing in p. The general
induction step is then proved exactly as above.

The main result of the next theorem is cut-off monotonicity: the larger the number ofwinners, and the more remaining unsure voters are willing to experiment. This result
6

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 sense of first-order stochastic dominance.

20

is perhaps surprising: why would unsure voters want to experiment more when the riskthat they lose majority and be imposed R forever increases? The intuition is as follows.
Suppose that p is below the myopic cut-off pM but above p(k) so that with k current
winners, unsure voters choose to experiment. By definition of pM , unsure voters get, insuch situation, a lower immediate expected payoff rate with R than with S. Therefore,
the only reason why they choose to experiment is that they hope to become winners.
Now suppose by 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 the status
quo is imposed forever, since p < pk+1 . Therefore, the very reason why unsure voterswanted to experiment, namely the hope of being winners, 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 a winner.7
In fact Theorem 3 shows that not only does experimentation increase when a newwinner is observed, but the value function of unsure voters also increases, as long as
this new winner does not gives majority to sure winners (i.e. as long as k < kN ).
Another important result contained in the next theorem is that p(k) > pSD for allk ≤ kN , which means that a single decision maker would always experiment more thana group whose majority consists of unsure voters. The reason is the control-loss effectmentioned in the introduction: when a single decision maker ignores his type, he stillknows that i) if he turns out to be winner, he will be able to enjoy the high-payoff,risky action forever, and ii) if he turns out to be a loser, he can stop experimentation
whenever he wants. Neither of these facts are true for unsure voters in a group: even
if an unsure voter turns out to be a winner, he is not guaranteed that the risky action
will be played forever, since a majority of unsure voters may block it. And if he turns
out to be a loser, he may still be imposed the risky action forever if experimentation
lasts long enough to reveal that a majority of voters are winners. This twofold control
loss prompts unsure voters to experiment less than anyone of them would if he could
dictate decisions in the future.
Theorem 2 (Cut-Offs Relations) Equilibrium cut-offs satisfy the following relations:7

That is, apart from receiving a lump-sum at the time of jump, but the possibility of that gain isalready factored in the computation of the immediate expected payoff, which is still less than s forp < pM .

21

• pM > p(0).
• p(k) > p(k + 1) for k ≤ kN .
• p(kN ) ≥ pSD with strict inequality if N > 1.
• p(k) = 0 for k > kN .
Proof. Theorem 1 already shows that p(k) = 0 for k > kN . The fact that p(kN ) ≥ pSD
with strict inequality if N > 1 comes from the comparison of (9) and (1). Monotonicityof p(k) is part of the induction in the proof of Theorem 1. There remains to show thatpM > 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.

(13)

Since p(0) > p(1), unsure voters strictly prefer experimentation at p = p(0) whenk = 1. Therefore, u(1, p(0)) > s/r. Since winners always get a higher expected payoffthan losers no matter what action is chosen, w(1, p(0)) ≥ u(1, p(0)). Therefore, thesecond and third terms on the left-hand side of (13) are positive, which implies thatp(0)g < s, or equivalently that p(0) < pM .

When learning is extremely fast, a single-decision maker is always willing to experiment
until he learns his type (almost) perfectly. Mathematically, this result comes fromthe single-decision maker cut-off equation (1): as the intensity λ goes to infinity, µgoes to 0 and so does the cut-off pSD . However, this result does not extend to thecase of collective experimentation. In this case, the time cost of experimentation is
only one of two reasons for preferring the status quo. The other reason is the riskof being imposed the risky action forever by the majority while being a loser. If
the control-loss effect is strong enough, society may prefer to shun the opportunityof learning everyone’s type and make a perfectly informed decision (clearly what autilitarian planner would choose!), and stay in the dark, i.e. stick forever to the statusquo. Keeping other parameter values fixed, this will happen if the total number N
of individuals is large enough, as individual power gets more diluted. Mathematically,
this immediate revelation does systematically occur if cut-offs stay bounded away from0 as learning intensity λ goes to infinity. The next result, which provides condition
under which this happens, is a direct consequence of (9).
22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 plays an important roleon experimentation. Indeed, the next proposition provides a stark result as N getslarge. Still assuming type independence, it shows that, at the limit, individuals behave
myopically, choosing the risky action only if its immediate, expected payoff is larger
than that of the safe action. For large groups with independent types, therefore, true
experimentation, understood as the election of an action despite a lower immediate
payoff in order to learn more about it, completely disappears. To state this result, letp(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 (8). For thesecond part, (8) also implies that p(kN , N) → s/g = pM as N goes to infinity. To
conclude the proof, observe that from Theorem 2, p(kN , N) ≤ p(k, N) ≤ pM for fixed
k and all N ≥ 2k + 1. Taking the limit as N goes to infinity proves the result.

Figure 2 illustrates cut-off policies for different values of N and of the number κ =kN + 1 − k of switches required for winners to gain the majority. In general, cut-offsp(k, N) are not monotonic with respect to group size, as can be proved by numericalcomputation. Such violations may seem counter-intuitive: as N increases, individual
power gets more diluted, so shouldn’t this reduce the value of experimentation? However, keeping k fixed, increasing N also makes it more likely, for any given unsure
voter, that other winners will be observed, for any fixed cut-offs value. Therefore, theaddition of new unsure voters reduces the winner frustration effect, for fixed cut-off
level. For some parameter values, this may, locally, increase the attractiveness of experimentation. The result of Proposition 1 is therefore not as natural as it may initiallyappear.
23

0.49
0.48
0.47
0.46

p

0.45
0.44
0.43

N=7
N=9
N=11
N=13
N=15
N=17

0.42
0.41
0.4

4

3

2

1

Number of switches required to achieve a winning majority

Figure 2: Policy Cut-Offs. r = 1, λ = 1, s = 1, g = 2.
The previous proposition is also interesting to think about power concentration. Indeed, define an oligarchy as a subset of O (odd) voters such that, at any time, thecollective decision is the action chosen by the majority of that subset.
Proposition 2 (Oligarchy) With an oligarchy of O voters, there exists a unique
MVE. This MVE is defined by cut-offs such that C(p1 , . . . , pN ) = R if and only ifp ≥ p(kO , O), where kO is the number of sure winners in the oligarchy.
Proof. Decisions are entirely determined by the oligarchy. From their viewpoint, thestrategic situation is therefore equivalent to a society with only O individuals voting
at the majority rule.

Combined with Proposition 1, Proposition 2 conveys a sense in which experimentation
lasts longer if power is concentrated into fewer hands. In particular, a dictator sets,
unsurprisingly, the same experimentation cut-off as the single-decision-maker cut-offpSD .
We have seen in Theorem 2 that society experiments more as the number of winners
increases. It is actually possible to prove a stronger result, pertaining to the monotonicity of value functions. Indeed, even the value function of unsure voters increases
24with the number of sure winners. For the value function of sure winners, this result
is more intuitive: a higher number of sure winners means a higher probability that awinning majority will be achieved. However, to be complete, this argument also requires that experimentation cut-offs decrease in k, which is guaranteed by Theorem 2.
Cut-off monotonicity implies that, when a new winner is observed, not only do winners
get closer to gaining the majority, but experimentation lasts longer in any case. Moresurprising is the fact that the occurrence of a new winner is also good news for unsure
voters, meaning that their value function jumps upwards, unless this new winner isthe decisive voter that gives the majority to winners. The intuition here is that, fork < kN , new winners make experimentation more attractive to unsure voters: if theyturn out to be winners, they will be more likely to enforce their preferred action. Of
course, it also increases the risk of being imposed that risky action if one turns out tobe a loser. However, because unsure voters were already willing to experiment before
the new winner is observed, it means that this trade-off was already resolved in favorof experimentation. For p < pM , unsure voters were already in a situation in which
the only reason to play the risky action was their hope of being winners. Therefore,
the fact that experimentation is facilitated through lower cut-offs and more winners is
also good news for them. This argument is contingent on the fact that unsure votersstill are in control of the collective decision process, however. When k = kN and a newwinner is observed, remaining unsure voters lose the majority and their value function
suddenly drops to the imposed value of the risky action. Finally, as one would expect,value functions are also increasing in p. For unsure voters, this is explained by theirhigher likelihood of get lump-sums through action R, while their payoff from action S
is unchanged. The value function of sure voters is also increasing in p. Indeed, themore likely unsure voters are to be winners, and the longer experimentation will last,in expectation and, hence, the longer sure winners will be able to enjoy high payoffs.
These various monotonicity properties are illustrated by Figure 3.
Theorem 3 (Value Function Monotonicity) The following holds:• u and w 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 k < kN ,
25

2

g/r

1.8

1.6

values

1.4

1.2

1

w(4, ·)
w(3, ·)
w(2, ·)
w(1, ·)
w(0, ·)u(3, ·)u(2, ·)u(1, ·)u(0, ·)u(4, ·)

s/r

0.8

0.6

0.4
0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

probability p

Figure 3: Value Functions. r = 1, λ = 1, s = 1, g = 2, N = 7.

261

• 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 k > kN .
Proof. Monotonicity of u and w with respect to p was shown as part of the induction
hypothesis in the proof of Theorem 1. If k > kN R is elected forever since winners
have the majority. This determines value functions for this case and yields the lastclaim. 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 toshow 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 as ψ ′ (p) ≤ α(p)ψ(p) for function α. A direct application of Gronwall’s inequality
along with ψ(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 of u 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).

(14)

We already know that φw is positive. Therefore, if φ were also nonnegative, the argument we just used for w would also show that ψ is nonnegative. In particular, if one
27

can show that u(kN , p) ≥ u(kN − 1, p), a backward induction will prove the result for
all k ≤ kN . Combining (10) and (11) 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 ))Therefore, the left-hand side has the sign of g − s − (N − kN − 1)(s − p(kN )g). From thecut-off formula (8), this latter term has the same sign as s − p(kN )g, which is positive.
Therefore, we can apply the first term in the right-hand side of 14 is nonnegative fork = kN , which implies that ψ is nonnegative for k = kN . This fills the missing step of
the induction, concluding the proof that u is increasing in k for k ≤ kN .
To show the last statement, observe that u(kN + 1, p) = pg/r from Theorem 1, andthat u(kN , p) > pg/r, from (11).

4Other Decision Rules

This section investigates how previous results are affected by changes of decision rules,
keeping the setting otherwise identical. The optimal experimentation policy of a social
planner maximizing utilitarian welfare is first derived and shown to last longer thanmajority-based experimentation. The impact of commitment, delegation, and delays
in decision rules is then considered. In particular, the utilitarian policy is shown tobe equivalent to the policy to which all voters would want to commit at the outset
if placed behind a veil of ignorance. The distinction between committing to a policy
versus committing to an action has dramatic consequences on policy efficiency. Finally,majority-based experimentation is then compared to the unanimity-based experimentation, where it is shown that no voting rule dominates the other in terms of efficiency.
In brief, when unanimity is required for the risky action, the loser trap effect disappears, but the winner frustration effect is reinforced, which may reduce experimentation
further than the majority rule under circumstances illustrated in this section.

28

µ

.

4.1

Utilitarian Criterion

Theorem 4 Under the utilitarian criterion, the optimal policy is determined by cutoffs q(k) such that C(k, p) = R if and only if p ≥ q(k). These cut-offs are nonincreasing in k, with q(k) = 0 ifsk ≥ k̄ = N.
g
Proof. The proof is similar to that of Theorem 1, proceeding by backward induction
on the number k of winners. For k ≥ k̄, the utilitarian optimum is to choose R forevereven 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. Consider next k = k̄ − 1.
for k ≥ k̄. The resulting welfare is W (k, p) = k gr + (N − k) pgr
Let w C (k, p) and uC (k, p) denote the value functions of sure winners and unsure votersif policy C is used, given that R is played forever if a new winner is observed, and let
W C (k, p) = kw C (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, andwhere Wσkt +1,C = W (k̄, pσ ), the welfare that was computed earlier for k = k̄. This is astandard control problem, whose solution is Markov. The indifference boundary mustsatisfy the smooth pasting condition
kg + (N − k)pg Ns−= Ns,
kg + (N − k)pg + (N − k)λprr
which has a unique root q(k), since the left-hand side is increasing in p, greater thanNs if p = 1 and less than Ns 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 areeasily shown to be increasing in 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, andiii) resulting value functions w(k ′ , ·), u(k ′, ·), and W (k ′ , ·) are non-decreasing in p. 
29

The next result states that majority experimentation is inefficiently short compared tothe utilitarian optimum.
Theorem 5 (Majoritarian vs. Utilitarian Rules) q(k) ≤ p(k) for all k ≤ kN .
Proof. The utilitarian cut-off q(k) solves
W (k + 1, p)(k/N)g + (1 − k/N)pg + (N − k)λp
− s/r = s,Nwhile the majoritarian 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

(15)

(16)

where w̄ and ū are the value functions obtained under the majoritarian rule. Optimality of the utilitarian policy implies that for all k, p, W (k,p)≥ (k/N)w̄(k, p) + (1 −Nk/N)ū(k, p). Since w̄ > ū, this also implies that W (k+1,p)> 1/(N − k)w̄(k + 1, p) +N
(1 − 1/(N − k))ū(k + 1, p), and subsequently that the left-hand side of (15) is higher
than that of (16) for given p. Therefore, the root of the first equation must be lower
than that of the second.

4.2

Commitment, Delegation, and Delays

If voters are initially homogeneous and can commit to a policy at the outset, they will
choose a policy that maximizes their expected payoffs which are identical and the sumof their expected payoffs. This latter maximization is identical to the utilitarian policy
above. This shows the following result.
Theorem 6 (Commitment) If voters can commit to a policy at time 0, they choose
the cut-off policy determined by cut-offs {q(k)}0≤k≤N .
Theorem 6 suggests that social efficiency can be partially restored if voters can commit to a policy to some extent. However, such choice should not be confused withcommitment to an action. Indeed, a policy prescribes which action should be taken
depending on past observations. It adapts collective decisions to circumstances. Commitment to an action, in contrast, is harmful because too rigid. The intuition appears
30best in the extreme case where voters must commit to a once-and-for-all action at theoutset. Then, their preference is to choose the risky action if and only if it is above themyopic cut-off. This extreme case of action commitment thus entirely annihilates experimentation. This result can be reinterpreted as follows: if votes take place at a low
time frequency, this reduces everyone’s individual control over collective decisions evenfurther than in the benchmark case, which causes even less experimentation and thus
even more inefficiency. As another consequence, suppose that voters can temporarilytransfer the decision process to a delegate who makes decisions based on a mixture ofelectoral and welfare concerns. Such delegation can improve efficiency to the extent
that welfare enters the delegate’s objective, as he is able to adapt to incoming information. In contrast, commitment to an action (either safe or risky) reduces willingness
to experiment and increases inefficiency.

4.3

Unanimity Rule

Suppose now that R can be enforced only if everyone votes in its favor.
Proposition 3 Under the unanimity rule, there exists a unique MVE. This MVE isdefined by cut-offs χ(k) which are decreasing in k, and such that χ(N − 1) = χSD .
Proof.

The first part is proved similarly to that of Theorem 1. For the last part,

observe that if k = N − 1, the remaining unsure voter has full control over collective
decisions. His optimal policy is therefore the same as that of a single decision maker.
In general, χ(k) can be smaller or greater (even for k ≤ kN ) than the majoritariancut-off p(k). Here are examples illustrating both possibilities.
Example 1 For N large, p(kN ) ∼ pM , independently of λ, but χ(kN ) → 0 as λ → ∞for N fixed.
Proof.

From (8), if N is large, necessarily p(kN ) is close to pM , independently of λ(to see this, divide (8) by Nλ). With the unanimity rule however, fixing N large but
finite, and letting λ go to infinity, the experimentation cost for unsure voters goes to0, as they learn almost immediately their type, and they lose no power. Specifically,31

u(kN , p) → pN g/r + (1 − pN )s/r as λ goes to ∞, which is strictly greater than s/r for
all p > 0. Hence χ(k, N) → 0 as λ → ∞.



Example 2 Suppose that N = 3 and s ≪ g. Then χ(1) > p(1).
Proof. Equation (8) implies thatp(1) =µsµs∼µg + (g − s) − (s − pg)(µ + 1)g(17)

if g ≫ s. In particular, p(1) is very close to zero if g ≫ s. On the other hand,
indifference of unsure voters with unanimity the rule, and k = 1 obtains if p satisfies
pg + λp[w(2, p) − s/r] + λp[v SD (p) − s/r] = s,(18)where w(2, p) is the value of a sure winner under unanimity rule if there are two surewinners (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 (18) imply that χ(1) must satisfy the inequality
pg + 2λp2 (g/r − s/r) ≥ s,
or
p≥

µs
∼ s/g
µg + 2p(g − s)

if g ≫ s. Comparing (17) and (19) shows that χ(1) > p(1).

5

(19)


Correlation and Heterogeneity

This section considers the case of two voters, 1 and 2, who share a common belief
about the initial joint distribution of their types, although this distribution need not
be symmetric any more, and allows for correlation between the voter types. Let θidenote Voter i’s type, and let pt1 t2 = P r[(θ1 , θ2 ) = (t1 , t2 ), where ti ∈ {g, b} represent
the possible types (good or bad) of each voter. Also let pi = P rob[θi = g] for i ∈ {1, 2},and α = pgg /(p1 p2 ). α is a measure of the correlation between voter types. The usual
correlation measure and α have a one-to-one relationship for any given p1 and p2 . If32

α = 1, types are uncorrelated. In general, α takes values in R+ , although not all valuesof R+ are achievable for given p1 , p2 . For example, p1 = 1 implies that α = 1, since inthat case Voter 1’s type is deterministic hence uncorrelated with Voter 2’s type. Let ∆denote the set of (p2 , α) that are achievable as elementary probabilities vary over the
four-dimensional simplex. The following proposition is a simple exercise of Bayesian
updating, whose proof is easy and omitted.
Proposition 4 (State Dynamics) Beliefs are governed by the following dynamics
equations. When no lump-sum is observed,
gg• dpdt = −λpgg (2 − p1 − p2 )bb

• dpdt = λpbb (p1 + p2 )bg

gb

• dpdt = −λpgb (1 − p1 − p2 ), dpdt = −λpbg (1 − p1 − p2 )• dα
= −λα(1 − α)(p1 + p2 )dtWhen Voter 1 receives a lump-sum,
gb

bb

gb
bg
p
p
bb
• pbb
+ = 0, p+ = p1 , p+ = 0, p+ = p1

• α+ = 1, p1+ = 1, p2+ = αp2
where the subscript ‘+’ denotes values immediately after the lump-sum is observed,
and its absence denotes values immediately before the lump-sum. Symmetric formulas
if instead Voter 2 receives a lump sum.
With two voters, let us replace the majority rule by assuming that unanimity is required
to play R. Since voters may now be heterogeneous (i.e. p1 6= p2 even if none of them isa sure winner, the concept of equilibrium must be modified. In the spirit of Section 3,
and given the unanimity rule, it is natural to assume that the voter who is the lesslikely of being a winner (i.e. voter i if pi ≤ pj ), is in control: if that voter wants to play
the risky action, so should the player with a higher expected type. This notion is also
consistent with elimination of weakly dominated strategies, because the pivotal voter
33

is always the voter who wishes to stop experimentation. For simplicity, let us therefore
define a unanimity equilibrium (UE) as follows: at any time t, if pi ≤ pj , then j votes
for R whenever i does.
Theorem 7 There exists a unique UE. This equilibrium determined by a cut-off function δ : ∆ → [0, 1] such that C(p1 , p2 , α) = R if and only if p1 > δ(p2 , α) wheneverp1 ≤ p2 , with the reverse relation if p1 > p2 .
Proof.

First suppose that p2 = 1. Then voter 1 has full control over the collective

decision. He therefore imposes his optimal policy, which is that of a single decision
maker. This defines δ(1, 1) = pSD . This also fully determines the value functions ofboth voters in that case. Let p 7→ w(p) denote the value function of voter 2, where p
is voter 1’s probability of being a winner is p, and v SD is the value function of a single
decision maker, which is also voter 1’s value function in this case. More generally
suppose that at time 0, p10 ≤ p20 . It follows from Proposition 4 that p1t ≤ p2t for allt preceding the first arrival of a lump-sum. In particular, this implies that 1 has fullcontrol of the collective decision (under unanimity) over that period. Therefore, hechooses a policy θ that solvesZ σ−r(τ −t)1−r(σ−t)2
1
ut = sup Eedπθτ (τ ) + eqw(pσ+ ) + (1 − q)v(pσ+ ) ,θtwhere, letting σi denote the (possibly infinite) time at which i receives his first lumpsum, σ = min{σ1 , σ2 } and q = P rob[σ1 < σ2 ]. This is a standard control problem,
whose solution is known to be Markov. Voter 1 is indifferent between R and S atprobability level p, if p solves the equation
pg + λp[w(αp2 ) − s/r] + λp2 [v SD (αp) − s/r] = s.

(20)

The left-hand side is increasing in p, equal to 0 for p = 0 and greater than g > s if
p = 1. Therefore, it has a unique root δ(p2 , α). This shows that C(p1 , p2 , α) = R if
and only if p1 > δ(p2 , α). The case p1 > p2 obtains by symmetry.



The next theorem shows that a voter’s incentive to experiment increases both withthe other voter’s probability of being a winner and with voters’ type correlation. The
latter result is intuitive: if types are more positively correlated, the control-loss effectweakens: if a voter turns out to be a winner, he is less likely to be blocked by the other
34

1

0.9

0.8

0.7

Experimentation

p2

0.6

0.5

0.4

alpha = 1

0.3
alpha = 3
0.2

Status quo

0.1

0.1

0.2

0.3

0.4

0.5

p

0.6

0.7

0.8

0.9

1

1

Figure 4: Experimentation Boundary δ as a function of α. r = 1, λ = 1, s = 1, g = 2.

voter. The first result is also intuitive, given the unanimity rule, a higher probabilityby the other voter of being a winner makes it less likely that one’s decision be blocked.
In the extreme case in which, say, Voter 2 is a sure winner (i.e. p2 = 1), Voter 1 has fullcontrol over collective decisions, and can behave in effect as a single-decision maker.
In addition, positive correlation increases the speed of learning, since voters learn fromboth their own and the other voter’s payoff observation. In the extreme case of perfect
type correlation, the setting is equivalent to a single decision maker setting in which
the learning intensity is the double of individual arrival rates. Faster learning reduces
the time-cost of experimentation, hence makes it more attractive and reduces cut-offs.
These effects yield the next result, which is illustrated by Figure 4 representing theexperimentation domain as a function of the correlation measure α.
Theorem 8 δ is decreasing in both components.
Proof.

The left-hand side of (20), is increasing in p, p2 and α. Therefore, keeping α

fixed, the root δ(p2 , α) must be decreasing in p2 , and similarly keeping p2 fixed, δ(p2 , α)35

must be decreasing in α.

6Negative Shocks and General Bandits

The purpose of this section is twofold. First, it considers the opposite of the benchmark
setting, where news shocks are negative. This setting is useful for several applications,
as illustrated by Section 2. Second, it generalizes two results of the previous sections:i) a group always performs less experimentation than any of its member would if hecould dictate future decisions, and ii) there is always some experimentation, provided
that decision rules are not adverse, in the sense that the preferences of any voter are,at any time, positively taken into account by society. Finally, another setting, allowing
for both positive and negative shocks is considered, in which population at any time isdivided between sure winners, sure losers, and unsure voters.

6.1

Negative News Shocks

Several applications mentioned in Section 2 require a setting in which news events
amount to bad news. To accommodate such applications, suppose that the risky arm
pays a positive constant rate if it is good and, in addition, also pays some negative
lump sums according to some Poisson process if it is bad. One may assume without
loss of generality that the safe rate is zero, since all payoffs can be translated by thesame constant without affecting voters’ decision problem.
In that case, the state is composed of k sure losers and the probability p that the arm
be good for other voters. Moreover, p increases in time, reflecting the fact that, in thissetting, no news is good news for unsure voters.
It can be shown that the policy is also determined by cut-offs ρ(k) such that unsure
voters impose the risky action if and only if p ≥ ρ(k). Since unsure voters’ belief pt
increases over time, the risky action can only be stopped, if used at all, when enoughsure losers are observed, either because those obtain the majority and impose the safe
action, or because the cut-offs ρ(kt ) get high enough and jump over pt .
36

This setting is also useful to visualize the general theorems to follow. Moreover, eachtheorem has a corollary for the setting with negative news shocks, as summarized in
Section 6.3.

6.2

General Bandits

Suppose that, for any given individual, the risky arm has a payoff distribution or “type”
θ lying in some finite set Θ. At any time, that individual’s belief about his type issummarized by a probability distribution or “state” γ ∈ Γ, where Γ = ∆(Θ) is the setof all probability distributions8 over Θ. The safe arm still pays a constant rate s. Fora single decision maker, the Gittins index of the risky arm is the map G : Γ → R suchthat, given state γ, G(γ) is the smallest value of s for which the single decision makerprefers the safe action over experimentation. Mathematically, G(γ) solves

Z ∞

−rt
G(γ) = inf s : s/r = sup Ee dπσt (t)|γ, s ,σ0where σ is any policy, and the expectation is conditional on the current state γ and on
the rate s of the safe action.9
Now consider the case of N decision makers. Still assuming publicly observed payoffs,
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 more information, such as type correlation (see Section 5). 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

(21)

0

The inequality obtains because C is an element of the policy set over which the maximization is taken.10 We may define a policy-dependent generalization of the Gittins
8

In the benchmark model, the type θ is either “good” or “bad” and the state γ is the probability

p that the type be good.
9
Although not explicitly stated, the results of this section naturally apply to discrete-time settings.
10
In general, C depends on all voters’ types and need not be anonymous.

37

index as
GiC (γ) = infs : s/r = E

Z ∞
0

−rt

e

dπCi t (t)|γ, s.

Inequality (21) 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 voting equilibria is extended as follows. The ν-supermajority rule isthe map υ : {S, R}N → {S, R} such that υ = S if and only if the number of votes for
S is greater than or equal to some constant ν.
Definition 2 (Voting Equilibria) C is a voting equilibrium for voting rule υ if
for all γ, C = S if and only if the number of voters i such that GiC (γ) ≤ s is greaterthan or equal to ν.
The following result shows that collective experimentation stops earlier than individual
experimentation in some qualified sense.
Theorem 9 Suppose that C is a voting equilibrium for voting rule υ. 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.
Theorem 9 states that experimentation stops whenever there are at least ν voters, eachof which would prefer to stop if he had dictatorial control over the policy. When types
are independent, then GiD (γ) = G(γ i ), where G(γ i ) is the Gittins index of the single
decision maker problem with state γ i . In that case, i’s optimal policy is independent ofother individuals’ types. Therefore, Theorem 9 implies that, in the case of independent
types, experimentation stops whenever at least ν voters would want to stop if placedin a single decision maker setting, given their individual state. This distinction isimportant: if types are positively correlated, 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, as shown in Section 5. In
contrast, collective experimentation is always less, even with positive correlation, thanwhat any voter would like if he could dictate all decisions. In this case, not only does
he avoid the loser trap and winner frustration effects, but he can also exploit other38

voters’ payoffs to learn about his type, which he cannot do in a single decision makersetting.
Theorem 9 is illustrated for the case of negative and mixed shocks in Sections 6.1
and 6.4.
The next result requires the following definitions. For any probability distribution γ iover the type space, let g(γ i) = E[dπRi /dt|γ i]. g(γ i ) is i’ immediate expected payoffrate with action R given type distribution γ i . For any individual type θi , let, slightly
abusing notation, g(θi ) = g(δθi ), where δθi is the Dirac distribution concentrated ontype θi , denote i’s true immediate expected payoff rate with action R when his actual
type is θi .
Definition 3 i is a winner (resp. a loser) ifg(θi ) > (≤)s.

This generalizes the definitions of winners and losers of Section 3. Individual i is awinner if, given his actual type θi , the risky action is better than the safe one, and aloser in the opposite case. Therefore, if i perfectly knows his type, he prefers R if he isa winner and S if he is a loser. The set of types can thus be partitioned into “good”
types and “bad” types, such that a type is good if and only if a voter with this type isa winner.
Definition 4 A policy C is adverse for Voter i if the set{t : P r[Ct = R|θi good] < P r[Ct = R|θi bad]}has positive Lebesgue measure.
A policy is adverse for Voter i if the risky action is more likely to be chosen if his typeis bad than if his type is good, at least for some nonzero time set. This may be thecase, for example, if i’s type is negatively correlated with a decisive majority of voters.
In this case, these voters will block the risky action whenever it benefits i, and impose
it whenever it hurts him. If types are independently or positively distributed, one
expects policies not to be adverse to any voter since, all else equal, a voter’s influence
39

over collective decisions results in decisions that favor him. Non-adversity is not as
benign an assumption as it may appear. For example, in a setting with both positive
and negative news shocks (Section 6.4), unsure voters may, upon observing a sure loser,
want to push experimentation further as the risk of the loser trap is reduced, which
adversely affects this loser. However, it holds in important particular cases. First, itholds when unanimity is required for either of the actions. Second, it holds in thenegative news shock setting, as will be verified in Section 6.3, on which several of
applications are based. Finally, it also holds in the mixed-shocks setting of Section 6.4,
as long as learning is fast enough or, equivalently, voters are patient enough.
Theorem 10 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. For any safe rate s and policy C, Voter i’s expected payoff with policy C is
Z ∞
 Z ∞
i
−rt
i
VC = Ee dπCt (t) =e−rt E[dπCi t (t)],
(22)
0

0

where expectations are conditioned on γ.
E[dπCi t (t)] = P r[Ct = S]sdt + P r[Ct = R]E[dπCi t (t)|Ct = R]
Therefore, if E[dπCi t (t)|Ct = R] > sdt for all t, then VCi > s/r, implying that GiC (γ) > s.
Suppose that s < g(γ i ). Then, by definition of g(·) and by the fact that the probabilityof each type is a martingale, E[dπRi (t)] = g(γ i)dt > sdt. Moreover, C’s non-adversity
with respect to i implies that
E[dπCi t (t)|Ct = R] ≥ E[dπRi (t)],as will be shown shortly. This inequality shows that GiC (γ) > s for all s < g(γ i),which proves the theorem. To show the inequality, observe that, by Bayes’ rule, C is
non-adverse for i if and only if P r[θi good|Ct = R] ≥ P r[θi good|Ct = S]. Moreover,E[dπCi t (t)|Ct = R] = P r[θi good|Ct = R]E[dπCi t (t)|Ct = R, θi good]+ P r[θi bad|Ct = R]E[dπCi t (t)|Ct = R, θi bad]. (23)Combining these results yields the inequality.
40



6.3

Properties of the Negative-Shock Setting

Theorem 9 implies the following result for the negative-news setting.
Corollary 2 Suppose that types are independent. Then, ρ(k) ≥ ρSD for all k < N/2,
where ρSD is the single-decision-maker cut-off.
Moreover, the equilibrium policy resulting from the majority rule is non-adverse to
any voter. Indeed, suppose that the risky action is elected by unsure voters at sometime t. Then, the (possibly infinite) time at which the risky action is abandoned
necessarily decreases with the number of sure losers, since as time passes, the beliefof unsure voters gets better and better (see Section 6.1). Therefore, for any unsure
voter i, P r[θi good|Ct = R] ≥ P r[θi bad|Ct = R], which shows non-adversity of theequilibrium policy. Theorem 10 then implies that all cut-offs lie below the myopic
cut-off, hence that unsure voters always experiment to some extent.
Section 4 showed that majority-based experimentation is inefficiently short comparedto the utilitarian policy, in the benchmark setting where news events amount to positive
news. With negative news shocks, the reverse intuition holds to some extent: provided
unsure voters have the majority, they may be willing to experiment for a wider set ofstates than a utilitarian social planner who takes into account sure losers. However,the value of the risky action is higher for society as a whole when the utilitarianpolicy is followed than with the majority-voting equilibrium policy, by social optimalityof the utilitarian policy. This effect makes experimentation more valuable for thesocial planner, and may offset the first effect. For example, suppose that no loser has
been observed yet. Then, unsure voters may require a higher probability of successto start experimentation than a social planner would, because the latter can always
exploit information to improve social welfare, whereas unsure voters face loser trap
and winner frustration effects. As the number of sure losers increases, the first effectstarts to dominate, with the social planner stopping experimentation if too many losers
are observed. This will be case if i) some losers have been observed, who face large
negative lump sums, but ii) unsure voters have a very low probability to be losers. In
such circumstances, a social planner will not wish to push experimentation further,whereas unsure voters, ignoring the plight of revealed sure losers, will continue with41

the risky action.
In the negative-shock setting, cut-offs need not be monotonic in the number of losers.
Such violations can be observed numerically or in combination with analytical results
omitted here. It is interesting to understand why such violation occurs in the negativeshocks setting but not in the positive-shocks setting. The apparition of a new loser has
mixed effects for unsure voters: it reduces the risk of being imposed the risky action,
but it also reduces the value of experimentation. The latter effect is similar to thepositive-news setting. The main difference is that the risk of being imposed the risky
action is more severe in the negative-news setting. Intuitively, unsure voters’ control
over collective decisions is better in the positive-news case than in the negative-news
one. In the positive-news case, the only event that can happen to an unsure voter is
to become a sure winner. If this does not happen, unsure voters can impose the statusquo at any time (or until they lose the majority, but this only occur if k = kN ). In
contrast, with negative news, the worst that can happen to an unsure voter is to receivea negative lump-sum and suddenly join the minority of sure losers, which have nocontrol of collective decisions. Thus, negative news is compounded by a sudden control
loss. This explains why the “insurance” effect resulting from the apparition of a newloser can, paradoxically, encourage experimentation. Furthermore, if nothing happensin the negative-news case, p simply increases which is enough to push experimentation
forward. In contrast, in the positive-news setting, the apparition of news winners isnecessary for experimentation to continue, for otherwise, p decreases until it causes
experimentation to stop.11

6.4

Mixed Shocks

Now suppose that the benchmark setting is modified as follows: if the risky arm is good,it pays positive lump sums according to the jumping times of some Poisson process withintensity λg , and if it is bad, it pays negative lump sums according to the jumping times
11

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

42

of a Poisson process with intensity λb . Without loss of generality, also suppose that thesafe rate is zero (action payoffs can be translated so as to achieve this condition, without
affecting preferences). In this case, there are three categories of voters: sure winners(who received a positive lump sum), sure losers (who received a negative lump sum),and unsure voter. Starting with homogeneous beliefs and independently distributed
types, the state at any time can be summarized by three numbers: the number kW ofsure winners, the number kL of sure losers, and unsure voters’ individual probability
p that the risky action be good for any one of them. Proceeding as in Section 3,
one may show that the unique majority voting equilibrium policy, starting with N
odd voters, is determined by cut-offs p(kW , kL ), that equals 0 if kW > N/2, that
equals 1 if kL > N/2, and that lies in (0, 1) in the remaining case, for which unsure
voters are pivotal. Theorem 9 implies that pD ≤ p(kW , kL ), where pD is the cut-off
an unsure voter would use if he had full control over future decisions or, in a single
decision maker setting, provided types are independent. This inequality holds for all
supermajority rules. Moreover, if the risky action requires the unanimity rule, thenTheorem 10 implies that p(kW , kL ) ≤ pM , where pM is the myopic cut-off. That is,unsure voters always wish to experiment to some degree given the unanimity rule.
With the majority rule, Theorem 10 does not apply directly, because non-adversity
need not hold.12 However, as learning becomes faster, an argument similar in spirit to
non-adversity also implies that unsure voters always experiment to some extent.
Indeed, suppose that either λb or λg becomes infinite, so that unsure voters immediately
learn their type if they elect the risky action. Let g > 0 and b < 0 the expected payoff
rates of the risky arm for sure winners and sure losers respectively.13 Then, the expectedvalue of the risky action for any unsure voter i, unsure voters are ex ante pivotal (i.e.
max{kW , kL} < N/2), is+++
V i = P r[kW> N/2] g/rP r[θi good|kW> N/2] + b/rP r[θi bad|kW> N/2]
+ P r[kL+ > N/2] g/rP r[θi good|kL+ > N/2] + b/rP r[θi bad|kL+ > N/2] , (24)+where kW
(resp. kL+ ) denotes the number of winners after all types are revealed. Clearly,
12
13

Whether it holds in this case is an open question.
These payoff rates can have any magnitude, since they are the product of jump intensities andlump-sum magnitudes.

43+
P r[θi good|kW> N/2] > P r[θi good] > P r[θi good|kL+ > N/2]. This implies that

V i > g/rP r[θi good] + b/rP r[θi bad] = pg/r + (1 − p)b/r,
which is nothing by the myopic payoff. Therefore, Voter i is willing to experiment atleast until p drops below the myopic cut-off pM , defined by pM g/r + (1 − pM )b/r = 0(since s = 0).
Since only the ratios r/λb and r/λg of the discount rate and learning intensities matterfor the analysis, the result can be reinterpreted as follows: if voters are patient enough,the majority voting equilibrium always entails some experimentation.

7

Extensions and Discussion

Privately Observed Payoffs
Previous sections assumed that all payoffs were publicly observable. What happens if
payoffs are privately observed? The following analysis shows that, perhaps surprisingly,
this need not affect experimentation.
First consider the case of two individuals with independent types voting at the unanimity rule. For a winner, playing the risky action is always optimal as it maximizes
immediate payoff and prompts the other voter to experiment longer, as indicated by
Theorem 8. However, why wouldn’t an unsure voter want to wrongfully pretend that
he is a winner? Indeed, this would prompt the other voter to experiment more, stillby Theorem 8, giving more time to the former voter to check whether he is a winner,
while leaving him the possibility to stop at any time.
However, such manipulation has no value. Indeed, suppose that Voter 2 mistakenly
believes that Voter 1 is a winner. When would Voter 1 want to stop pretending that
he is a winner and impose the status quo? In such scenario, Voter 2 thinks he has fullcontrol over the decision process, hence experiments up to the single-decision maker
threshold. Therefore, Voter 1 can choose any level of experimentation up to the single
decision maker threshold at which Voter 2 will stop if he turns out be a loser. Voter 1
may wish to stop earlier however. His optimal cut-off p is determined by the indifference44equation
pg + λp[w(p) − s/r] + λp[v SD (p) − s/r] = s,where w(p) the level to which his value function jumps if he receives a lump-sum,and v SD is his new value function (i.e. that of a single decision maker) if Voter 2
receives a lump-sum (see Section 5). This indifference equation is identical to the caseof complete information: i) the payoff flow (pg) is the same, ii) the winner value w(p)is the same, since a lump-sum makes Voter 1 an actual winner and iii) a lump-sum to
Voter 2 prompts that voter to choose the risky action forever, independently of what
Voter 1 had pretended to be. This suggests that truthful type revelation is optimal.
With N voters, the intuition is the same. When unsure voters have the majority andpayoffs are publicly observed, they already impose the experimentation level that is
optimal to each of them, since their interests are perfectly aligned. Therefore, assumingthat other voters are truthful, an unsure voter cannot benefit from manipulating thelevel of experimentation. Mathematically, his indifference equation is unchanged. Thisagain suggests that truthful type revelation is optimal.
There remains to discuss how individuals communicate their types. An obvious way,
if the group is small enough, is to use cheap talk. If the group is large, the following
protocol is natural. Suppose that experimentation starts with a common type probability p for all voters. Voters are willing to experiment until the first threshold p(0),even if no one has received any lump sum by then. When p(0) is reached, supposethat voters who are still unsure at that point vote for S, while sure winners vote for R.
Upon observing the number of votes for R, unsure voters can deduce whether to continue experimentation or not. If no one has voted for R, the status quo is imposed
forever. If k > 0 voters have chosen R, then experimentation resumes (i.e. everyone
votes for R) until p reaches the cut-off p(k), at which point unsure voters vote for the
status quo while sure winners, whose number k ′ is greater than or equal to k, choose R.
If k ′ > k, experimentation continues at least up to p(k ′ ). If k ′ = k, experimentation
stops at p(k). Also assume that off the equilibrium path if one voter chooses the safe
action when he should be experimenting, he is believed to be an unsure voter, andvice versa. The above protocol is not information efficient in the sense that voters only
know the true state of the world when p reaches some particular cut-offs. Indeed, otherless natural protocols would be more efficient, for example if a new winner reveals his
45

type change by voting for S for an infinitesimal amount of time. However, even theless informationally efficient but more natural protocol suffices to implement the fullinformation policy, as stated in the next theorem. What matters is that the natural
protocol is sufficiently informative to exactly implement the policy of Section 3.
Theorem 11 The protocol defined above is an equilibrium of the dynamic voting gamewith privately observed payoffs.
Proof.

For sure winners, it is clearly optimal to follow the protocol, as it requires

them to vote R forever. Indeed, such strategy maximizes their immediate payoff aswell as the length of experimentation, due to the cut-off monotonicity established in
Theorem 2.
With the protocol, the benchmark policy is exactly implemented. Therefore, unsure
voters value function, given k and p is the same as in Section 3. However, under
the above protocol, unsure voters (and, less importantly, other voters) only know thetrue state k when particular cut-offs are reached. Let l denote the last such public
release of information. For p > p(l), unsure voters only know that the number k̃
of voters who have received lump sums so far is greater than or equal to l. The
symbol tilde is added to indicate that k is random from unsure voters’ viewpoint.
The first part of the proof is to verify that under the protocol, unsure voters wish toexperiment for p > p(l). This is indeed the case if their value function is greater thans/r, the value they get with the safe action. This value function, when the protocol
is followed by all, equals U(l, p) = E[u(k̃, p)|l]. However, crucially, unsure voters only
matter if they have the majority. Conditioning on unsure voters being pivotal, thesupport of k̃ lies in K(l) = {l, . . . , kN }. By Theorem 3, u(k̄, p) ≥ u(l, p) for k̄ ∈ K(l).
Therefore, U piv (l, p) ≥ u(l, p) > s/r for p > p(l), where the superscript piv indicatesthe conditioning on the event k̃ ≤ kN . Therefore, it is optimal for unsure voters tochoose the risky action whenever indicated by the protocol. Similarly, if, upon reachingp(l), it turns out that k = l, i.e. no new winner has been observed since the last releaseof public information, then it is optimal for unsure voters to stop. Indeed, if they
follow the protocol, their value function is identical to the benchmark case, because
the protocol policy is exactly the same as in that case. Therefore, their indifference
equation is identical to the benchmark case, yielding the same cut-off p(l).
46There remains to show that an unsure voter cannot benefit from misrepresenting his
type. This is done by backward induction on k. If k > kN , sure voters impose the risky
action forever, so that unsure voter’s value function is identical to the benchmark case,and he cannot manipulate the protocol. Now suppose that k = kN . Pretending to
be a sure winner prompts unsure voters to continue experimentation beyond the fullinformation level. From above, however, it is suboptimal from that voter’s viewpoint,
given his value function and his value function if other sure winners were observed: his
HJB equation is identical to the benchmark case by induction hypothesis. Therefore,
the unsure voter cannot benefit from deviating if k = kN , resulting in the benchmark
policy being implemented in that case, which yields the value function u(kN , p). By
backward induction, suppose that the unsure voter does not benefit from deviating
for all k strictly greater than k̄ hence that his value function is the same as in thebenchmark case for such k. Then, his HJB equation at p = p(k̄) and knowing k̄ is thesame as in the benchmark case, so he cannot benefit from deviations. The benchmark
policy is thus also implemented in that case, which yields him a value function u(k̄, p)
for all p, concluding the induction step.

Factions and Heterogeneous Voting Weights. The analysis of this paper extends
naturally to settings in which some voters weigh more in the decisions than others.
Given the results of this paper, it is natural to expect that voters with more decision
weight will be more inclined to experiment longer. For example, suppose that there are
four voters with Voter 1 weighing twice as much as other voters and decisions being
made at the simple majority rule. Suppose that Voter 4 is the only sure winner so far.
Then, Voter 1 can impose experimentation to the level that he desires since, by siding
with Voter 4, he creates a majority for the risky action. Therefore, as long as no new
winner is observed, Voter 1 can push experimentation up to the single decision maker
threshold. However, he will stop earlier if, say, Voter 2 becomes a winner. Indeed,
Voter 1 should then fear the possibility that Voter 3 receives a lump-sum, resulting ina winning coalition that imposes upon Voter 1. Contrary to the benchmark setting,thus, experimentation can be interrupted by the occurrence of a new winner. (In thiscase, though, experimentation still lasts longer than if Voter 1 were split into two
independent voters.)
Risk aversion. Although not mentioned so far, the analysis of this paper does not47require that voters be risk neutral. Indeed, voters could have any von NeumannMorgenstern 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. In another paper, work in progress, I show how social efficiency,
according to utilitarian welfare, can be “spontaneously” restored in the two-arm bandit
setting of this paper, if side payments are allowed.
Coordination Breakdown with Multiple Risky Actions
In settings with multiple, correlated risky actions, even the slightest risk of modificationin the preference ranking of some voter can result in the group choosing the mostconservative action. The fear of such ranking modification may result in coordination
breakdown among group members who would otherwise agree to impose some morelucrative action. The following setting provides a stark illustration of the control-loss
effect.
Consider a group three individuals (1,2, and 3) voting at the majority rule. First
suppose that there are two actions, R and S. S pays a constant rate s = 1 to allplayers. R pays a certain rate of 2 to Voters 1 and 2, but has an uncertain payoff
distribution for Voter 3: with probability p it gives off lump-sums to Voter 3, whose
corresponding expected payoff is g = 0.1. In this simple configuration, it is easy to seethat there is a unique equilibrium, in which Voters 1 and 2 impose R forever. Thisequilibrium holds independently of p, and goes against Voter 3’s interest: even if p = 1,
Voter 3 prefers S, which gives him a higher payoff than 0.1.
Now suppose that there is another risky action, X, such that i) X is good for Voters 2
and 3 if and only if R is good for Voter 3, ii) X surely pays −9 to Voter 1, and has
23
expected payoffs gX= 2.1 and gX= 1.1 respectively to Voters 2 and 3 if it is good.

Perfect correlation implies that any lump-sum observed by Voters 2 or 3 with action X
or by Voter 3 with action R causes the common probability p to jump to 1.
Since there are three actions, let us assume that if no action receives at least two votes,
then the status quo S is imposed. The impact on equilibrium of preference uncertainty
appears through the next propositions.

48

Proposition 5 If p = 0 or p = 1, there is a unique MVE: if p = 0 R is played forever,
if p = 1, X is played forever.
Proof.

If p = 0 there is no learning. Individual preferences are ordered as follows:R ≻1 S ≻1 X, R ≻2 S ≻2 X, and S ≻3 X ∼3 R. In particular, R is the uniqueCondorcet winner hence the unique MVE. Similarly if p = 1, there is no learning:
individual preferences are ordered as follows: R ≻1 S ≻1 X and X ≻2 R ≻2 S, andX ≻3 S ∼3 R. X is the Condorcet winner hence the unique MVE.



For the next result, suppose that µ = r/λ = 1.
Proposition 6 If p ∈ (0.1, 0.8), there is a unique MVE. In this MVE, S is played
forever.
Proof. Voter 1 is indifferent between R and S if p solves the equation

9 1
2 + λp − −= 1,r rwhich yields p = 0.1. Thus if p is greater than p, Voter 1 prefers the low payoff of S¯¯
rather than risking that Voters 2 and 3 discover that X is good for them and imposing
it forever, from Proposition 5. Voter 3’s indifference equation relative to actions X and
S isp1.1 + λp[1.1/r − 1/r] = 1,
which yields p̄ = µ/(0.1 + 1, 1µ) = 0.8 if µ = 1. Therefore, if p ∈ (.1, .8), S is thepreferred choice for 1 and 3, hence chosen forever.



To illustrate the content of these propositions, consider the following variation of therestaurant example of Section 2. Three friends, Chris, Ian, and Paul go the restaurant
once every week-end, and choose each time their restaurant according to the majority
rule. They start with the following preferences. Chris likes Chinese cuisine above
anything else. Ian likes Indian cuisine but not Chinese one. Paul does not know Asiancuisine, and is thus uncertain about his preferences. There are three restaurants in
town: i) a gourmet Chinese restaurant, clearly the best choice for those like Chinese
cuisine, ii) a Singaporean restaurant whose menu contains both Chinese and Indian49

dishes, and iii) a plain restaurant who has a well known common, relatively low value
to all friends.
For some parameter values of everyone’s tastes, the following paradox may occur: Ian
and Paul vote for the plain restaurant. However, if the gourmet Chinese restaurant
closes down, Chris and Ian vote for the Singaporean restaurant. Why did Ian change
his vote? In both cases, Ian prefers the Singaporean restaurant. However, if he agrees
with Chris to go there, he runs the risk that Paul discovers that likes Chinese cuisine,resulting in Chris and Paul to impose the gourmet Chinese restaurant in the future. If
this risk is high enough, Ian prefers to vote for the plain restaurant, which Paul alsoprefers if his expected value for Asian cuisines is low enough.

8

Conclusion

The analysis of this paper has shown that, in a dynamic setting, collective decisions
tend to be too conservative compared to what any individual would choose if he had fullcontrol over future decisions, other things equal. Moreover, when news shocks amountto good news, experimentation is also inefficiently short compared to the utilitarianoutcome, with a partial equivalent for the negative-shock setting. These phenomena
stem from a twofold control loss: the risk of being imposed the riskier actions whenthose turn out to be detrimental for oneself (“loser trap” effect), and the opposite risk ofnot being able to enjoy those riskier actions when one turns out to be a winner, if thoseare blocked by a conservative majority (“winner frustration” effect). For large groups,
these effects can entirely annihilate the value of experimentation, causing individuals tovote myopically, but are reduced if actual preferences types are positively correlated.
The analysis also shows how commitment to an observation-dependent policy andcommitment to an action have very different implications for efficiency.

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