Pervasive Spurious Normativity

Pervasive Spurious Normativity
Gillian K. Hadﬁeld Dylan Hadﬁeld-Menell
Abstract
This paper proposes a mathematical model for a simpliﬁed version of the game deﬁned in Hadﬁeld
and Weingast [2012] which proposes that legal order can be described as an equilibrium in third-
party decentralized enforcement coordinated by a centralized classiﬁcation institution. We explore
the attractiveness of joining a new group (which is assumed to have settled on an enforcement
equilibrium already) where groups differ in terms of the frequency of interactions in which norm
violation is possible (normative interactions) and thus punishment is called for. We show that groups
in which normative interactions are frequent but involve relatively unimportant rules may achieve
higher value for participants.
1 Introduction
One of the attributes that differentiates human from other animal societies is the presence of norms and in particular
norms over behaviors that do not appear to have a material impact on well-being apart from the fact that people will
punish the failure to adhere to a norm. In all known human societies there is a rich normative landscape, which
attaches normative valence to actions including many that have no immediate material implications for those who
are expected to help enforce them through social sanctions. Most societies for example have rules about what it is
appropriate to eat when, what tone of voice to use in what settings, how close to stand to someone else, how to
behave when waiting with others to get into a venue or access to a resource, what information should be shared with
whom and when, who can participate in particular trades or economic activities and so on. In many cases, particular
norms are arbitrary, even though violation of them is treated as worthy of punishment by others. We call this spurious
normativity. Hadﬁeld [1999], for example, reviews the anthropological literature on the sexual division of labor
across pre-industrial societies. While almost all societies categorize work as either women’s work or men’s work, the
particular classiﬁcations vary substantially cross-culturally: what is women’s work in one society is men’s work in
another. The classiﬁcation is arbitrary.1
Fessler and Navarrete [2003] call the process by which patterns of behavior are imbued with moral sentiments that
motivate sanctioning of violations of the pattern normative moralization . They use as an example the normative
moralization of handedness. Most people are right-handed but particularly in societies with few specialized tools,
whether someone is right- or left-handed often has no material consequences for others. Nonetheless, many cultures
treat using one’s right hand as a morally approved category–denoting purity or politeness–and one’s left hand as cause
1Hadﬁeld [1999] shows that there is functionality to an arbitrary classiﬁcation scheme of enforced norms: it coordinates invest-
ments in specialized learning by gender which raises the value of heterosexual matching in the marriage market.
1
for opprobrium–revealing weakness or evil. Fessler [1999] hypothesizes that, driven forward by the emotions of shame
and pride that are triggered by violation or compliance with cooperative norms, culture extends the set of actions that
are subject to normative moralization as a way of extending the set of actions that can be used as information about
cooperation beyond those actions that directly involve cooperation. A norm that says “it is wrong to fail to look
where you are going” generates direct cooperative beneﬁts, helping people to avoid crashing into one another. A
norm that says “it is wrong for a man to walk down the street wearing shorts” does not generate cooperative beneﬁts–
unless people in this society treat conformity to this norm as informative about a person’s likelihood of behaving in
conformity with norms that do generate cooperative beneﬁts. As Fessler [1999] (who identiﬁes the the “watch where
you are going” and “men don’t wear shorts out in public” norms in an Indonesian village) puts this:
Ego’s inclusion in cooperative activity is dependent on her ability to meet Others’ expectations, and
those expectations are, in turn related to shared standards for behaviors which are relevant to coop-
eration. As a consequence, the signiﬁcant adaptive advantage offered by participation in cooperative
activities generated selective pressure for an increase in the attention paid to these standards. Late
[evolving] second order emotions [emotions that arise in response to others’ ﬁrst order emotions]
were the vehicle through which this increase in attention was achieved. Moreover, because late sec-
ond order emotions entail a sensitivity to the reactions of all individuals, Ego must be concerned with
her performance vis--vis shared standards when interacting with any other member of her group. It
is only a small step from this situation to one in which the shared standards with which Ego is
concerned are not limited to the question of cooperative activity–once Ego is concerned with how
all Others evaluate her, it is not difﬁcult for shared standards governing other types of behavior to
become salient as well. This is because an Other may extrapolate from situations that do not involve
cooperation to those that do–an Other may think ”if that individual does not follow shared standards
in this context, how can I be conﬁdent that he will do so if I invite her to engage in cooperative
activity?”(pre-pub p. 34, emphasis added)
Fessler’s account focuses on the evolution of emotions in response to norm violation (in particular shame) to moti-
vate voluntary conformance with norms. But as Boyd and Richerson [1992] and many others have emphasized, and
Fessler’s own account of shame as a second order response to Ego’s actual or imagined experience of hostility or
criticism from Other triggered by norm violation implies, effective third-party punishment plays a signiﬁcant role in
supporting norm compliance. Indeed, Mathew et al. [2012] argue that even small-scale cooperation among kin and
close associates may require third-party punishment to achieve evolutionary stability.
In this paper we analyze mathematically the potential beneﬁts of extending normative moralization to behaviors that
are from a material perspective irrelevant or at least of small consequence to most people, that is, the value of per-
vasive and spurious normativity. We ask: does a community generate higher payoffs for participants if it punishes
violation of apparently meaningless rules or if it focuses more narrowly only on norms that are functional in the sense
of generating direct beneﬁts? Intuitively, one might expect that only functional norms that govern behaviors that
matter for payoffs would emerge and stabilize in equilibrium: punishment is costly and why would a society expend
punishment resources on ensuring conformity with rules that have no impact on material well-being? Most analyses
of norms in the law and economics literature assume that norms coordinate outcomes that improve welfare. Sugden
[1986], McAdams [2005] and Myerson [2004] propose, for example, that property rules emerge because they solve
the coordination problems that arise in costly contests over resources (Hawk-Dove games).
2
We consider the impact of extending normativity to apparently arbitrary actions on the prospect for sustaining welfare-
improving behavior in an equilibrium social order based on collective third-party punishment. We demonstrate that
communities that extend normativity in this way can generate higher value for participants than those that restrict the
range of normativity.
The intuition of our result is as follows. Communities with pervasive spurious normativity provide agents with plentiful
and cheap opportunities to observe punishment behavior by others. An individual’s willingness to cooperate in a
community–which requires foregoing safe non-cooperative options and exposing oneself to the risk of being exploited–
depends on that individual’s beliefs about the likelihood that the community effectively punishes conduct that harms
the individual. If you are going to risk exposing yourself to harm, you want to know if your community contains
enough people who will punish the perpetrator to give you conﬁdence that harm is reasonably deterred. Assume you
are a newcomer to a community or that the community was recently handed a new set of norms. (Think here of
rule-of-law building efforts in developing countries.) Assume that the community is in an equilibrium in terms of
punishment behavior but you do not know the likelihood of effective punishment of norm violations. Assume also
that the only way to learn about the likelihood of punishment is to observe punishment behavior. You can learn this
information more cheaply if you are given abundant opportunities to observe what happens when there are violations
if the violations that you have to expose yourself to don’t really matter very much. You don’t really care whether men
walk down the street in shorts but by taking a walk yourself you can see how others react to shorts-wearing men and
thus gain information about how they would react to violations you do care about–careless driving for example. Thus,
if you could participate solely as an observer except when your own interests were directly at stake, you would prefer
to live in a world with pervasive even if spurious normativity–abundant opportunities to observe reactions to norm
violations–than one that was narrowly focused on punishing just the stuff you care about. This will still be true even
if you are required to participate in the community–complying with and punishing spurious norms–so long as those
costs, which increase with the pervasiveness of norms, are not too great.
Our results have implications for the evolution and microfoundations of law.
Hadﬁeld and Weingast [2012] present a model that derives characteristic features of law–such as generality, stability,
uniqueness and universality–as attributes necessary to support an equilibrium in which behavior is patterned on the
classiﬁcations of behavior articulated by a centralized institution. Enforcement is assumed to come exclusively from
decentralized collective punishment of conduct classiﬁed as punishable by the centralized institution; there is no cen-
tralized enforcement apparatus such as the state. This theory of the microfoundations of law proposes, contrary to most
economic and positive political theories of law, which deﬁne law as a set of rules enforced by a centralized enforce-
ment apparatus (see Hadﬁeld and Weingast [2014]), that law is an innovation in the mechanism used to coordinate the
same enforcement mechanism that supports other normative social orders–decentralized collective punishment. This
has important implications for our understanding of how law developed and how it can be built in environments where
it is currently lacking.
The equilibrium legal order in Hadﬁeld and Weingast [2012] is supported by a particular speciﬁcation of beliefs.
Their model posits that an agent (call the agent Ego) treats other agents’ (Others’) failure to punish behavior classiﬁed
by the institution as punishable, including behavior that has no impact on Ego’s (or perhaps any agent’s) payoff, as
informative of the likelihood that Others will also fail to punish wrongful behavior that does have an impact on Ego’s
payoff. This belief structure creates an incentive for individuals to participate in collective punishment, potentially
mitigating the free-rider problem in collective punishment. Ego has no incentive to participate in collective punishment
3
(the model assumes standard preferences) except to maintain an equilibrium in which behavior that reduces Ego’s long-
term payoff is deterred by the threat of collective punishment. Effectively, an agent’s participation in punishment is
treated as information about that agent’s continued assessment that an equilibrium in which collective punishment is
coordinated by the classiﬁcations articulated by the central institution will beneﬁt that agent and hence that the agent
will be willing to incur costs to support the equilibrium.
The classiﬁcation institution in this model is serving as what Hadﬁeld and Weingast [2012] call an authoritative
steward of a very simple binary partition of behaviors into those that are punishable and those that are not. This can
be interpreted as the construction of a sparse labeling system: lawful and not lawful.2We can interpret the beliefs
underlying legal order as beliefs about whether people are punishers or not of behavior to which the constructed (that
is, not natural) label ”unlawful” is attached.
Our model here provides a basis for understanding how a simple binary classiﬁcation scheme that is comprehensive–
covering a wide variety of conduct–can emerge in a setting in which equilibrium depends on 1) voluntary participation
in punishment and 2) a belief structure in which punishment of an action labeled punishable is considered informative
about the likelihood of punishment of other actions labeled punishable, even when the assignment of the label is
potentially arbitrary. We suggest that understanding how such a classiﬁcation scheme and belief structure can emerge
is critical for understanding the emergence of law.
The strategy of our paper is as follows. We ﬁrst give an overview of the model and basic notation in Section 2, to-
gether with some technical background from the analysis of multi-armed bandit games and partially observed Markov
decision processes. To build intuition, we then present in Section 3 analytical results for the limiting case in which
Ego bears no cost of complying with spurious norms or punishing their violation. Because the games we analyze
quickly becomes analytically complex but relatively easy to compute once we introduce a positive cost of complying
with norms and punishing their violation, we turn to computational results in Section 4. Section 5 relates our results
to conjectures about the likely growth and stability of communities in which norms are more or less pervasive and in
which a legal institution that reduces ambiguity about norm violations and increases the informativeness of punishing
behavior (by linking disparate rules into a code such that punishment of one rule is informative about the likelihood of
punishment of another) exists.
2 Overview of Model
The basic idea of the model is as follows. Consider an inﬁnitely repeated game setting in which an agent Ego is faced
with the choice in each period of participating or sitting out. If choosing to sit out, Ego receives a payoff normalized
to 0. If Ego chooses to participate, she plays a randomly selected game gwith two randomly selected agents drawn
from a population (Others). We model these games in reduced form. In each game, one of the Others is randomly
selected and presented with an opportunity to choose between two actions, one which is classiﬁed by a classiﬁcation
institutionLas ”rule violation” and another which is classiﬁed as ”not rule violation.” If Other chooses ”rule violation”
the remaining Other and Ego each independently choose either to punish or not punish. Rule violations are deterred
by collective punishment, that is, punishment that requires more than one agent to punish. For example, in Hadﬁeld
2Cooter [1998] also suggests that the effectiveness of law can be understood as deriving from the classiﬁcation of behavior as
lawful or not. Cooter however presumes the existence of preferences based on this simple binary classiﬁcation, that is, that at least
some people inherently prefer to avoid actions labeled unlawful. This presumes that a category ”law”, which extends to potentially
arbitrary actions, exists.
4
and Weingast [2012] two buyers and a seller engage in repeated contract and performance games. Actions for the
seller are drawn from a set of possible contract performances, some of which are classiﬁed as breach and others which
are not breach. A decision by a buyer not to purchase from a third-party seller in one period constitutes punishment,
speciﬁcally a boycott. Breach is deterred when the seller expects both buyers to boycott in response to breach. Games
are distinguished by the rule that may be violated. For example, there could be a game in which rule ”watch where
you are going” may be violated and one in which ”men should not wear shorts in the street” may be violated.
We assume, and Ego believes, that the community of Others is playing an equilibrium in the super game that consists
of the sequence of repeated games. Others are of two types, t: punishers ( t= 1), who punish anyone who chooses
an action classiﬁed by Las a rule violation, and non-punishers ( t= 0), who never punish anyone. Let be the true
proportion of punishers in the equilibrium. We assume that an Other’s type is observable by the other participants in
any particular game, that is, only in the context of the opportunity for rule violation. We focus on sub-game perfect
equilibria in which the knowledge that two punishers are present deters rule violation.3(That is, on the off-equilibrium
path where a violation does occur in the presence of an Other of type t, punishment is carried out that imposes costs
on the violator that exceed the present value of beneﬁts from violation.) We do not model how this equilibrium is
generated or supported but we observe that the equilibrium is not destabilized by the presence of non-punishers. We
assume, however, that Ego plays as a punisher, bearing an expected cost cin each round. ccan be thought of as the cost
to Ego of signaling that she is a punisher. For simplicity we assume that Ego is never presented with an opportunity
for rule violation.4Ego’s participation in the game is assumed to be on the margin, with no impact on the equilibrium
played by the Others. Ego is able to observe rule violations, the types of Others and punishments in games in which
she participates.
3 Formal Model Speciﬁcation
Before providing a formalization of our model we provide a brief overview of the theory of Markov decision processes
and multi-armed bandits.
3.1 Technical Background and Notation
We deﬁne a Markov decision process (MDP),M, is a tuple:M=hS;A;P;R;i5.Sis a set of states.Ais a set of
actions.P:SAS! [0;1]is a function that assigns probability to state transitions for each state-action pair. If
Ego is in state, s, and selects action athe probability of transitioning to s0is given byP(s;a;s0).Ris a (bounded)
reward function that maps states, to an interval of R, w.l.o.g.,R:S ! [0;1].2[0;1)is a discount factor that
expresses Ego’s preference for current versus future rewards.
3For an example of such a game, see Boyd et al. [2010]. They present an evolutionary game model in which punishment is a
heritable strategy and deterrence requires multiple punishers. A population with a fraction of punishers can be stable in equilibrium
when punishers can signal that they are punishers at low cost and thus avoid the costs of punishment if there are too few punishers
present.
4It is straightforward to generalize our interpretation of cas the cost of complying with spurious rules to signal Ego’s support
for the equilibrium rules.
5In standard treatments, Tis typically used for the transition distribution, we use Phere to avoid confusion with our model
speciﬁcation.
5
A solution to Mis apolicy ,, that maps states to actions, :S!A . The value of a state,s, underis the sum of
expected discounted rewards received by starting in sand selecting actions according to :
V(s) =E"1X
t=0tR(st)js0=s;#
:
The optimal policy, , maximizes this value and we will write V=Vfor the optimal value function. Standard
results show that a unique optimal value function exists[Puterman, 1994].
In a partially observed Markov decision process (POMDP), we additionally deﬁne a distribution over observations
(O) for a each state. A policy now maps a history of observations to an action, as the agent does not know the true
state of the world. A POMDP can always be converted into a (continuous state) MDP where each state is a distribution
over states of the world and the transition distribution is deﬁned by Bayesian inference.
An interesting class of POMDPs are multi-armed bandits (MAB). In a multi-armed bandit, an agent is given access to
several distributions. At each time step, that agent must selects a distribution to sample from, and receives a reward
that is equal to the value of the sample.
A multi-armed bandit provides an analytically and computationally tractable model of exploration-exploitation trade-
offs that occur with practical agents. In particular, success in this class of problems requires explicit reasoning about
the impact of information on future decision making quality. Recent applicability to online banner advertising has
led to rapid progress in both theoretical and computational methods for MABs [Auer et al., 2002]. The optimal full
information policy (which knows the distributions of the arms) will always select the arm with the highest mean.
A key result from Lai and Robbins [1985] lower bounds the number of times an optimal (partial information) policy
selects a suboptimal arm in expectation. This result holds for the class of consistent policies: policies where the
probability that the optimal arm is chosen at time tapproaches 1 as t!1 . We additionally require a smoothness
constraint on the MAB distributions.
Proposition 1. [Lai and Robbins, 1985] Let be a class of MAB arms (i.e., a class of distributions) with parameter
. Let()be the corresponding mean. If
82;8>0;906=such that()(0)() +
then, for any consistent policy the expected number of times a suboptimal arm is selected in the ﬁrst nrounds is

(logn)6.
3.2 Model Description
We deﬁne our super game as a tuple: hG;T;;U;;ciwhereGis a distribution over games and Tis a distribution
over punishment types tin the population of Others. We will abuse notation somewhat as use TandGto refer to
the support of the corresponding distributions where the meaning is obvious. is Ego’s prior distribution over the
parameters of T, andU:GT!Ris a mapping from types and games to immediate payoffs for Ego. The
understanding is that this mapping represents the results of the Others playing their role in the equilibrium. is Ego’s
discount parameter for future rewards. cexpresses a participation cost. This can be understood as the expected cost of
to Ego of signaling to an Other that she is a punisher.
6If the function g(n)is
(f(n)), then there is a positive constant csuch thatg(n)cf(n)8n
6
Ego begins in period 1 with perfect knowledge of how actions are classiﬁed by L, all payoffs and the distri-
bution of games. Ego does not know the distribution of types of Others but holds a prior which we will specify
shortly. Ego updates her beliefs about the distribution of types using Bayes’ rule. The super game is deﬁned as follows:
An initial game g0Gis drawn. Then, for each period j:
1. Ego chooses whether to participate or not. If she opts out, then she collects 0payoff and the next round starts.
2. If Ego opts in, she incurs the cost of signaling that she is a punisher cand a type,tjTis for the remaining
Other is drawn. If a non-punisher is drawn, a rule violation occurs; if a punisher is drawn, no rule violation
occurs.
3. The agent observes whether a punisher is present and whether a rule violation occurs and collects payoff
U(gj;tj).
4. A gamegj+1Gis drawn for the next round.
We assume that in equilibrium and given the ruleset created by L, there are two types of games from Ego’s perspective:
those to which Ego is indifferent and those that Ego cares about. Games to which Ego is indifferent always generate
a reward of 0for Ego. Suppose, for example, that a game involves a rule requiring genuﬂecting by an Other. We
assume Ego realizes no costs or beneﬁts from the Other’s choice about whether to genuﬂect or not, other than the cost
of signaling that she is a punisher. We will thus use represent the type of Other with an indicator variable that idicates
if she is a punisher for this game.
Games Ego cares about are ones in which Ego receives a positive reward, Rif there is another punisher present in the
game and a negative reward Rif there is not. We call these important games. We formalize the set of important
games as follows:
G0=fg2GjU(g;)6= 0g
U(g;tjg2G0) = (2t1)Rc:
We will use EU=Eg;t[U(g;t)jg2G0]to denote the expected utility of an important game. We let sdenote the
sparsity of the process generating games: the probability that a game is unimportant.
s= 1P(g2G0);gG:
Critically, we assume that the sparsity of games does not alter the (expected) rate at which important games are
presented to Ego. To be concrete, we assume the expected discounted reward obtained from important games is
independent of s. This condition can be attained through a suitably modiﬁcation of as a function of s:
Proposition 2. Setting
s= 1(1s)(1)
ensures that the expected sum of discounted rewards from important games is independent of s:
8s;2[0;1)Egj;tj2
41X
j=0jU(gj;tj)gj2G03
5=Egj;tj2
41X
j=0I[gj2G0]j
sU(gj;tj)s3
57:
7I[ ]is the indicator function for the condition  .
7
Proof. We ﬁrst show that it is sufﬁcient to ensure that the expected value of j
sis the same given that jis a round with
an important game:
Egj;tj2
41X
j=0I[gj2G0]j
sU(gj;tj)s3
5=1X
j=0Egj;tj
I[gj2G0]j
sU(gj;tj)s
=1X
j=0j
sEgj;tj[U(gj;tj)js;gj2G0]Egj[I[gj2G0]js]
= (1s)EU1X
j=0j
s
Where the ﬁrst line holds by the linearity of expectation, and the fact that gj;tjare independent iid draws from a
stationary distribution. Substituting the form of the inﬁnite geometric series, we see that
EU
1=(1s)EU
1s(1)
is sufﬁcient to acheive our goal. Substituting the form for sin the theorem statement and reducing shows that this
condition is satisﬁed.
It can be easily shows that this model describes a class of MABs. If the parameters that describe equilibrium were
known, the decision problem would be trivial. The safe option corresponds to a constant arm, which is a degenerate
distribution. Optimal policies for bandits with constant arms exhibit clear structure: is that if it is optimal to choose
a known option in round j, it will be optimal to choose the known option in round j+ 1 as well (Bradt et al.,
1956). The argument is straighforward: if Ego reaches a point at which her estimate of the tradeoff between risking
a negative payoff and learning so as to improve future decisions leads her optimally to choose not to participate, then
her information state can never change and so her optimal choice can never be any different than the current opt-out
decision. Thus, in our game, if Ego ever retires in a round, then she will never participate again. We refer to the
decision not to participate at any point, then, as a decision to retire.
Futhermore, an MAB is an instance of a POMDP, so it the optimal policy maps a distribution over states, a belief
state , to decision between retirement and particpation. We give our agent a Beta prior over this parameter so that the
belief space for our agent is a two dimensional lattice equivalent to Z2
+. Initially, the belief state is (0;0)and can
be understood as the state an agent would be in if she had seen 0punishers and 0non-punishers. The conditional
probability that a punisher is present in the ﬁrst game is
p=
+:
Once the games begin, Ego updates the prior beliefs using Bayes’ rule, which in the Beta distribution means adding the
counts of punishers and non-punishers observed to the prior values. In the following, we will use i(i)to represent
the number of observed punishers (non-punishers) prior to round i.
A second useful result from the theory of multi-armed bandits is that the optimal policy is a function that maps a
sequence of observations to a decision about retirement. If we restrict to two types, punishers and non-punishers, this
problem is a partially observed version of a Markov decision process where the state is the probability, p, of drawing a
punisher. From the theory of partially observable Markov decision processes (POMDPs), this optimal policy can also
be represented as a mapping from a distribution over pto an action about retirement [Puterman, 1994]. This reduces a
partially observed process to a fully observed deterministic process in belief space .
8
4 The Value of Sparsity
Consider ﬁrst the case in which the participation cost, c, is zero. In this case, Ego only faces a risky choice in periods
in which she is presented with an important game. In all other periods, the per-period the expected payoff of playing
the risky arm is a constant 0. Thus we can also conclude that if Ego retires, she will retire in a period in which she is
playing an important game. In order to maintain the structure of a multi-armed bandit problem, we specify that if Ego
chooses not to participate in an important game then the next game in the sequence is also an important game. We
can think of this as a suspension of the game.8This rules out the possibility that Ego can simply choose not to play
an important game and then reenter to play unimportant games in the hope of learning more before the next important
game comes along. A decision not to participate is a decision to retire assuming a rational agent.
We leti= 1 represent the state in which there is a punisher present in the game. The value of a state is then
characterized by the following recursion. To simplify notation we let pi;ibe the probability that a punisher will be
present in the game in the belief state (i;i)and we abuse notation somewhat by letting V((i;i);s;)represent
the discounted expected value of the super-game with sparsity sand discount factor sin the state (i;i).
We now show a property of the value of perfect information (VPI) in our super-game. The VPI for a state in a decision
processis a measurement of the improvement in decision making as a function of information gathering actions [ ?]. It
is deﬁned as the amount a rational agent is willing to pay to remove all uncertainty associated with a particular random
variable.
Proposition 3. If the participation cost, c, is 0, then, for any belief state, (i;i), and discount rate , the correspond-
ing VPI goes to zero as the sparsity ratio goes to 1. That is
lim
s!1VPI ((i;i);s;) = 0 (2)
Proof. Given, it is easy to compute the value of participation:
V() =EU1X
t=0t= (21)1X
t=0t=21
1: (3)
The optimal full information policy 0will retire whenever V()<0:We useV+() = maxfV();0gdenote the
value of0as a function of . The VPI is computed as the difference between the expected value of V+and the value
of the optimal policy that only uses the history of observations:
VPI ((i;i);s;) =E[V+()j(i;i)]V((i;i);s;) (4)
We proceed by lower bounding V.Vis the value of the optimal policy so it is weakly lower bounded by any arbitrary
policy. A useful candidate is one-step greedy policy, g, that always participates for unimportant games and retires
in important games if the expected value of participation is negative (disregarding the beneﬁt of new information).
We letbe the random number of games played before an important game is drawn. is geometrically distributed
with success parameter 1s. We letnpbe the random number of punishment actions observed prior to drawing
an important game. The distribution over npwill be a binomial distribution conditioned on . Thus, the value of
8In any multi-armed bandit game, the decision to stop suspends the game: deciding to return to the game implies making the
risky pull that was rejected previously.
9
executing this policy ca be written as a joint expectation under andnp:
Vg((i;i);s;) =E;np[maxfE0[V(0)j(i+np;i+np)];0gj(i;i);s]: (5)
We will be interested in the limit of this value, as s!1. Before proceeding with that, we note that, from the law of
large numbers, we have E0[V(0)ja+np;b+np]will concentrate about V(). Thus,
E0[V+(0)j(i;i)] = lim
!1Enp[maxfE0[V(0)j(i+n;i+np)];0gj;(i;i)]: (6)
Note that E[V()j(i;i)]only depends on the ratio of (i;i), so the difference between the left and righthand sides
of 6 is caused by the fact that the maximum must be taken at ﬁnitely many ratios (for ﬁnite ). Furthermore these
ratios are evenly spaced out, so the lefthand side can only increase as increases. We can use this to lower bound the
limit ofVg:
lim
s!1Vg((i;i);s;) = lim
s!1E;np[maxfE0[V(0)j(i+np;i+np)];0gjs] (7)
lim
s!1P(c(s)) min
0c(s)Enp[maxfE0[V(0)j(i+np;i+0np)];0gjs;(i;i);0]
+P(c(s)) min
0<c(s)Enp[maxfE0[V(0)j(i+np;i+0np)];0gjs;(i;i);0]
(8)
lim
s!1P(c(s))Enp[maxfE0[V(0)ji+np;i+c(s)np];0gjc(s);(i;i)] (9)
Using the form of the cumulative distribution of a geometric variable, P(c(s)) = 1P( <c (s)) =sc(s). We
set
c(s) =log 1s
so that lims!1c(s) =1;andlims!1sc(s)= 1. Combining 6 and 9 with these facts allows us to deduce the following:
lim
s!1Vgi((i;i);s;)E[V+()j(i;i)] (10)
Thus, lims!1VPI ((i;i);s;)0:However, we have that, for any s,VPI ((i;i);s;)0by standard prop-
erties of VPI. This shows the result.
Proposition 4. If the participation cost, c, is zero, then for any (i;i)such thatV+(i
i+i)>0, VPI is strictly
positive fors= 0.
Proof. From Lai and Robbins [1985], we have that, for consistent and assymptotically efﬁcient policies, the expected
number of pulls of a suboptimal arm after nrounds is lower bounded by clogn, wherecis a positive constant that
measures the similarity of the reward distributions for the arms. This class contains the optimal policy. Thus, we have
that for any ﬁnite s,
VPI ((i;i);s;)>0: (11)
The combination of these two propositions shows that for any (i;i), the corresponding value will eventually increase
assgoes to 1. Thus, in the case where participation costs can be neglected, Ego will prefer an equilibrium with a
highers.
10
0.0 0.2 0.4 0.6 0.8 1.0
θ0.00.51.01.52.02.53.03.54.0P(θ|(α,β))Low Confidence Initial Beliefs
P(θ|(0.8,1.2))
P(θ|(1.0,1.0))
P(θ|(1.2,0.8))(a) Low Conﬁdence Initial Belief Distributions
0.0 0.2 0.4 0.6 0.8 1.0
θ0123456P(θ|(α,β))High Confidence Initial Beliefs
P(θ|(20.0,30.0))
P(θ|(25.0,25.0))
P(θ|(30.0,20.0)) (b) High Conﬁdence Initial Belief Distributions
Figure 1: Plots of a selection of initial belief distributions. Each curve corresponds to a different value of E[]:(a)
shows low conﬁdence initial beliefs. In these information states, Ego has seen very little data and so her belief is
very spread out. In these scenarios, we expect sparsity to be helpful because the gap between full information and
partial information is large. Conversly, (b) shows belief distributions after Ego has seen 50 effective samples. The
corresponding distribution is more concentrated, so we expect less positive impact from sparsity.
5 The Cost of Pervasive Normativity: Computational Results
Our results above show that an environment with lots of rules that an agent cares nothing about intrinsically is more
valuable for an agent contemplating participation than one in which the only rules are ones that matter on the merits–
altering Ego’s payoff directly. We assumed, however, that increasing the number of spurious rules is costless to Ego
and of course this is not generally likely to be true. If Ego is going to participate in a community with lots of spurious
rules, Ego is also likely to bear costs, speciﬁcally the cost of participating in collective punishment and the cost of
complying with spurious rules. In this section, we relax the assumption that c= 0. Doing so, however, increases the
analytical complexity. We therefore turn to computational methods to explore environments in which Ego enjoys both
costs and beneﬁts from an increase in the number of spurious rules.
To illustrate the effect of participation costs, we select six initial belief states and computed values as a function of
c. Our initial beliefs vary the expected value of and the variance of the belief about its mean. We selected initial
states to cover scenarios where the expected values of an important game is negative, positive and equal to zero. In
this work, we chose E[]2f:4;:5;:6g. We varied Ego’s conﬁdence in her current estimate by varying the effective
number of samples (+)in the initial belief. Figure 1 shows the corresponding beta distributions for our selection
of initial states.
The optimal policy is invariant to scaling of rewards, so we ﬁx P= 1 and letc
Pbe the independent variable in
our computations. We compute these values with a variant of value iteration that takes advantage of the structure
of the state space. A python script to generate these plots is included as appendix A. We set the parameters of our
computation to allow for at most 108of error in the computation.
11
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (0.8, 1.2)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9(a) Low Mean, Low Conﬁdence
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (20.0, 30.0)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9 (b) Low Mean, High Conﬁdence
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (1.0, 1.0)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9
(c) Mid Mean, Low Conﬁdence
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (25.0, 25.0)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9 (d) Mid Mean, High Conﬁdence
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (1.2, 0.8)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9
(e) High Mean, Low Conﬁdence
10-510-410-310-210-1100
c/P0.00.20.40.60.81.01.21.41.61.8ValueInitial state (30.0, 20.0)
s = 0.0
s = 0.15
s = 0.28
s = 0.4
s = 0.5
s = 0.59
s = 0.68
s = 0.76
s = 0.83
s = 0.9 (f) High Mean, High Conﬁdence
Figure 2: Plots of V((i;i);s;:9)for select initial states. The probability density functions that correspond to these
beliefs are shown in Fig. 1. The one step value of participating in an equilibrium with enforcement is 1 ( P= 1).The
increase in value for larer sparsity for low values ofc
Pshows conﬁrmation of the results in Prop. 3. In comparing
vertically, we can see that sparsity has a larger effects on states with low mean and low conﬁdence: (f) is essentially
unimpacted by sparsity for lowc
Pwhile for (a) and (c) participation is suboptimal unless sparsity is positive. The cost
of this sparsity is increased sensitivity to participation costs.
12
Figure 2 shows value as a function ofc
Pfor several pairs of sparsity and initial states. We can clearly see the costs
of pervasive normativity: value functions at higher sparsity decreases more quickly and for lower setting of the par-
ticipation costs. This occurs because the number of rounds per important game increases so more participation costs
are paid. In essence, increased participation costs force Ego to pay more for information and so she may prefer an
equilibrium with less sparsity.
As we might expect, the net gain in value from sparsity is larger in belief states with that are very uncertain. For
example, the value function for initial state (30, 20) is essentially invariant to sparsity for lowc
P. Similarly, gains
from increased sparsity are more pronouced with low means. Essentially all value in those states is due to improved
decision making abilities from information.
6 Implications for the Microfoundations of Law
Our results have surprisingly powerful implications for the structure of normativity in human communities. [LETheory
readers: we are only sketching these implications for this draft. We anticipate that at least some of them will be capable
of more formal analytic or computational demonstration.]
6.1 Gossip, Silly Rules and Durability
The value of participating in a super-game depends on the expected cost of punishing rule violations relative to the
reward Ego expects if violations of rules she cares about are deterred. The computations in Section 4 indicate that
when expected costs relative to rewards are sufﬁciently low, ceteris paribus , Ego enjoys higher value in environments
with more rules that she does not care about. This provides an interesting explanation for the observation from
ethnographic studies that simple societies are characterized both by pervasive and apparently spurious rules that are
effectively enforced by low-cost collective punishments. Wiessner [2005], for example documents the use of gossip,
group criticism and mocking as the principal means by which norms are enforced among the Ju/’hoansi Bushmen
of northwest Botswana. In her observations, violations of norms were punished by escalating criticism and rarely
got to the point of physical violence. Assuming that the rewards generated for individuals aggregate to raise group
well-being, our model thus can be read to predict that communities that succeed in securing an equilibrium with many
rules that impose low compliance costs and which are enforced by low-cost means will outperform communities with
fewer rules and more costly forms of punishment.9
6.2 Birds of a Feather
The rewards Ego enjoys when joining a community depend on the rules of that community. In comparing across
communities with comparable forms of punishment, and comparable numbers of rules, Ego will prefer a community
with rules for important games that, when effectively enforced, generate higher rewards, R. We have not modeled
the source of rules in a community but if we suppose that rules emerge that reﬂect the interests of the members of
a community, this suggests that Ego is more likely to ﬁnd rules that achieve higher rewards in communities with a
number of agents with similar preferences, that is in more homogeneous communities.
9Our model takes into account Ego’s willingness to bear the higher cost of punishment in more sparse environments. We
assume, but have not shown, that the willingness of Others to punish is not reduced with sparsity–that is, that Others have incentives
comparable to Ego’s.
13
6.3 The Emergence of Law
As communities grow more heterogeneous–which they naturally do as they generate value and support greater spe-
cialization through the division of labor–we expect ambiguity about rule violations to increase. Increasing ambiguity
increases Ego’s expected cost of participating in a punishment regime if errors in punishment–failing to punish when a
violation is perceived by an Other and punishing when the Other does not perceive a violation–are themselves treated
as rule violations which produce punishment. This is a feature of the punishment schemes of many communities:
members of the Flemish cloth merchants guild in the 13th Century, for example, included in their rules the provision
that any member who failed to observe a boycott of a buyer who had cheated another guild member was barred from
trading and no guild member was to ”house the goods” or ”keep company” with the non-punisher. Increasing am-
biguity also reduces the expected rewards Ego enjoys in important games because with some probability a violation
perceived by Ego is not perceived as such by Others and hence the probability of a violation increases. Thus the cost
of punishing relative to rewards decreases with decreasing ambiguity. Hadﬁeld and Weingast [2012] argue that the
central function of law is to serve as a unique classiﬁcation institution, capable of resolving ambiguity about what
counts as a rule violation. Moreover, they show that for a classiﬁcation institution to effectively secure an equilibrium
of legal order around a given set of rules enforced only by decentralized collective punishment, the institution must
possess legal attributes such as neutrality, openness, clarity, consistency and stability. Our analysis here predicts that
communities that introduce law in the form of a classiﬁcation institution with legal attributes that reduces ambiguity
will enjoy higher value and greater durability.
6.4 Hammurabi’s Code
One of the key assumptions of our model is that people who punish any rule violation are expected to punish all rule
violations. This is a distinctive feature of the labeling system generated by a legal regime: people are ”law breakers”
or not; they are ”law-abiding” or not. Cooter [1998] proposes that ”law” is a meaningful category and that people
have preferences over behaviors solely on the basis of whether they are labeled ”lawful” or not. Our model captures
this idea by treating observation of punishment behavior in the context of any rule as informative of the probability of
punishment in important games. The model can be interpreted as representing, for example, a community in which
legal order is coordinated around a single legal code. Hammurabi’s Code from ancient Babylon, for example, consisted
of 247 individual rules, such as ”If any one hire an ox or an ass, and a lion kill it in the ﬁeld, the loss is upon its owner”
(Rule 244) and ”If any one open his ditches to water his crop, but is careless, and the water ﬂood the ﬁeld of his
neighbor, then he shall pay his neighbor corn for his loss” (Rule 55). These rules likely emerged individually over
time. We can imagine that knowing whether someone punished Rule 244 may or may not have helped to predict
whether they would also punish Rule 55. But when Hammurabi placed all 247 together on a stone pillar and named
the collection as his Code, he created the possibility for the emergence of two types of people: those who punished
violations of the Code and those that did not. Our analysis suggests that the creation of collections of rules, rather
than disparate rules, can generate value. Suppose, for example, that Ego cares about ﬁve rules, enjoying rewards when
violations of each of them is deterred. Our model treats these ﬁve rules as integrated into a single super game in which
the observation of punishment behavior in any game is informative, and equally so, of the likelihood of deterrence
of violations in any of the ﬁve games Ego cares about. But suppose instead that these rules are not connected in this
way. Suppose that punishment behavior in each game Ego cares about is only predicted by punishment behavior in
non-overlapping subsets of unimportant games. We could then decompose our single super-game into ﬁve distinct
super-games, each one of which would be considerably less sparse than our original game. Our results predict that
14
Ego’s value in a community with distinct super-games–with disconnected rules and a belief structure about punishment
that limits the informativeness of observing punishment of any individual rule–will be lower than the value enjoyed in
a community with a comprehensive code.
6.5 Legal Pluralism and the Nation State
Finally, we can combine some of the above observations to shed light on the phenomena of legal pluralism and the
emergence of the nation state. Modern advanced legal systems are characterized by comprehensive systems of rules,
with the label ’law’ attributed to any rule generated by a government organized within the constitutional framework of
a state. Indeed, law is frequently equated with the rules generated by a government, as distinguished from those gener-
ated from other entities such as corporations or schools or that emerge organically from social interaction [Ellickson,
1991]. But as Hadﬁeld and Weingast [2013] emphasize, prior to the emergence of the nation state, there were many
institutions that coordinated legal order in different spheres, often in competition. Medieval Europe, for example,
was characterized by multiple legal orders, with rules generated by merchant guilds, towns, churches, local rulers and
more. Many societies, particularly those seen as struggling to establish the rule of law, are characterized by multiple
legal orders–some governing family relations, others governing commercial dealings for example. Our model suggests
a way of thinking about the tradeoffs Ego will face between participating in multiple legal communities, each of which
coordinates punishment over some subset of rules and participating in a comprehensive legal community. On the one
hand, rewards may be higher when a legal community is comprised of a relatively homogeneous group with shared
interests–such as a community of traders–who can select rules that serve Ego’s interests. But such a community will
also have lower sparsity. On the other hand a system with a single system of comprehensive rules may achieve less
alignment with Ego’s interests but may also, because of its higher sparsity, provide Ego with more information about
the value of continuing to participate.
References
Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine
learning , 47(2-3):235–256, 2002.
Robert Boyd and Peter J Richerson. Punishment allows the evolution of cooperation (or anything else) in sizable
groups. Ethology and sociobiology , 13(3):171–195, 1992.
Robert Boyd, Herbert Gintis, and Samuel Bowles. Coordinated punishment of defectors sustains cooperation and can
proliferate when rare. Science , 328(5978):617–620, 2010.
Robert Cooter. Expressive law and economics. The Journal of Legal Studies , 27(S2):585–607, 1998.
RC Ellickson. Order without law: how neighbors settle disputes. 1991.
Daniel MT Fessler. Toward an understanding of the universality of second order emotions. Beyond nature or nurture:
Biocultural approaches to the emotions , pages 75–116, 1999.
Daniel MT Fessler and Carlos David Navarrete. Meat is good to taboo. Journal of Cognition and Culture , 3(1):1–40,
2003.
Gillian K Hadﬁeld. A coordination model of the sexual division of labor. Journal of Economic Behavior & Organiza-
tion, 40(2):125–153, 1999.
15
Gillian K Hadﬁeld and Barry R Weingast. What is law? a coordination model of the characteristics of legal order.
Journal of Legal Analysis , 4(2):471–514, 2012.
Gillian K Hadﬁeld and Barry R Weingast. Law without the state: legal attributes and the coordination of decentralized
collective punishment. Journal of Law and Courts , 1:3–34, 2013.
Gillian K Hadﬁeld and Barry R Weingast. Constitutions as coordinating devices. Institutions, Property Rights, and
Economic Growth: The Legacy of Douglass North , page 121, 2014.
Tze Leung Lai and Herbert Robbins. Asymptotically efﬁcient adaptive allocation rules. Advances in applied mathe-
matics , 6(1):4–22, 1985.
S Mathew, R Boyd, and M van Veelen. Human cooperation among kin and close associates may require enforcement
of norms by third parties. In PJ Richerson and M. Christiansen. Str ¨ungmann Forum Report , volume 12, 2012.
Richard H McAdams. Expressive power of adjudication, the. U. Ill. L. Rev. , page 1043, 2005.
Roger B Myerson. Justice, institutions, and multiple equilibria. Chi. J. Int’l L. , 5:91, 2004.
Martin L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming . John Wiley & Sons,
Inc., New York, NY , USA, 1st edition, 1994. ISBN 0471619779.
R Sugden. The economics of righs, cooperation, and welfare. Palgrave Macmillam, 1986.
Polly Wiessner. Norm enforcement among the ju/hoansi bushmen. Human Nature , 16(2):115–145, 2005.
Appendix A: Computing V((i; i);s;c; )
"""
File: sparse_bernoulli.py
Usage:
$ python sparse_bernoulli.py n_sparsity_values x_fidelity
[--resultfolder path/to/results]
Ex: to compute with 10 sparsity values with 500 samples along the x axis and
store the results in folder path/to/figures/ do
$ python sparse_bernoulli.py 10 500 --resultfolder path/to/figures
Not specifying result_folder defaults to the current folder
"""
from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
import argparse
import sys
eps = 1e-8
16
def main():
parser = argparse.ArgumentParser()
parser.add_argument(’n_sparsity_values’, type=int)
parser.add_argument(’x_fidelity’, type=int)
parser.add_argument(’--resultfolder’, type=str, default="")
args = parser.parse_args()
# These are the values we’ll use to compute cost response curves
sparsity_values = np.log(np.linspace(np.exp(0),
np.exp(.9),
args.n_sparsity_values))
# The x-axis values, determines the accuracty of the plots. Uses
# log linear spaces because that seems qualitatively better responses
costs = np.exp(np.linspace(np.log(0.00001),
np.log(.9),
args.x_fidelity))
max_ret_sparsity_values = np.exp(np.linspace(np.log(0.001),
np.log(.95),
args.x_fidelity))
start_values = {(.4, 2) : ’low-mu-high-sigma’,
(.4, 50): ’low-mu-low-sigma’,
(.5, 2) : ’mid-mu-high-sigma’,
(.5, 50): ’mid-mu-low-sigma’,
(.6, 2) : ’high-mu-high-sigma’,
(.6, 50): ’high-mu-low-sigma’}
resultfolder = args.resultfolder
for ratio, effective_samples in start_values:
# compute the costs for different sparsity values
alpha = ratio *effective_samples
beta = (1-ratio) *effective_samples
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_xscale(’log’)
ax.set_xlabel(’c/P’)
ax.set_ylabel(’Value’)
for s in sparsity_values:
17
print "s: {}, ratio: {}, effective samples: {}".format(
s, ratio, effective_samples)
cost_response_curve = compute_cost_response_curve(
alpha, beta, s, costs)
ax.plot(costs, cost_response_curve,
label="s = {:.2}".format(s))
plt.title(’Initial state ({}, {})’.format(alpha, beta))
ax.set_ylim([0, 1.8])
plt.legend(loc=’best’)
plt.savefig(
resultfolder+start_values[(ratio, effective_samples)] + ".pdf")
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_xlabel(’s’)
# ax.set_xscale(’log’)
ax.set_xlim([.001, 1])
ax.set_ylabel(r’$\frac{c}{P}$’)
plt.title("Maximal Participation Cost vs Sparsity")
for ratio, effective_samples in start_values:
print "ratio: {}, effective samples: {}".format(ratio, effective_samples)
# compute the costs for different sparsity values
alpha = ratio *effective_samples
beta = (1-ratio) *effective_samples
#compute the largest c such that participation is optimal
max_c_values = []
for s in max_ret_sparsity_values:
max_c_values.append(largest_possible_c((alpha, beta), s, 0.9))
print "s = 0, max_c = {}".format(max_c_values[0])
best_s = np.argmax(max_c_values)
print "s = {}, max_c = {}".format(
max_ret_sparsity_values[best_s], max_c_values[best_s])
ax.plot(
np.r_[max_ret_sparsity_values, [1]],
max_c_values + [0], label="({}, {})".format(alpha, beta))
plt.legend(loc=’best’)
plt.savefig(resultfolder + "retirement_points.pdf")
def largest_possible_c(s0, s, delta):
18
c_min, c_max = (0, 1)
while np.abs(c_max - c_min) > eps:
c = (c_min + c_max)/2.0
V = sparse_bernoulli_value_iteration(s0, s, c, delta)
if V > 0:
c_min = c
else:
c_max = c
return c_min
def sparse_bernoulli_value_iteration((a, b), s, c, delta,
tol=eps, verbose=true):
"""
Takes a belief state (a, b) and computes the V((a, b); s, c, delta)
Computation is done with value iteration so that the error is less
than tol
"""
delta_s = 1 - (1-s) *(1-delta) # As is Prop 2
"""
Values are initialized to 0, so maximal the maximal error is the
maximal positive reward for all time. With probability (1 - s) Ego
gets value P with probability \theta. We upper bound by letting
\theta = 1 and then normalize by P to get an upper bound:
UB(c, s) = (1-s -c/P)/(1-delta_s)
This ammount decreases by delta_s each step of value iteration so
we need delta_sˆH UB(c,s) <= tol ===> H >= log(tol/UB(c, s)) / log(delta_s)
"""
log_V_ub = np.log(1 - s - c) - np.log(1 - delta_s)
H_lb = ( np.log(tol) - log_V_ub ) / np.log(delta_s)
H = int(np.ceil(H_lb))
if verbose:
sys.stdout.write(’\r s: {} c: {} H: {} ’.format(
s, c, H))
sys.stdout.flush()
# Vector of a counts
a_vals = np.linspace(0, H-1, H) + a
19
# Allocate vectors to store values
Vt = np.zeros(H)
Vt_minus1 = np.zeros(H-1)
# Take the horizion from H-1 to 0
for t in range(H-1, 0, -1):
# after cur_h rounds we will have seen cur_h heads or tails,
# and we incorporate the priors
cur_confidence = cur_h + a + b
# P[i] = i / Nt; i in [0,...,cur_h]
P = a_vals[:cur_h] / (cur_confidence)
# do a value iteration backup
Vt_minus1 = backup(Vt, Vt_minus1, P, s, delta_s, c)
# set up for the next round, reuse the preallocated memory
# to avoid unnnecessary realloc calls
tmp = Vt
Vt = Vt_minus1
Vt = tmp[:-1] # decrease size by 1
return Vt[0]
def backup(Vt, Vt_minus1, P, s, delta, c):
"""
Computes a value iteration back for the super game
V: vector of values at time t, in increasing order of the number of heads
Vt_minus1: vector to return values for time t-1 (avoids reallocating memory)
P: vector of transition proabilities encoding probability of heads at time t-1
s: sparsity level
delta: discount factor
c: participation costs
"""
# First compute expected value of important game
# Vt_minus1[i] = delta *(P(tails) *Vt[i] + P(heads) *Vt[i+1])
Vt_minus1 = delta *(Vt[:-1] *(1-P) + Vt[1:] *(P))
# Expected reward at next step is 2 *\theta - 1 - c
Vt_minus1 += 2 *P - 1 - c
# Ego decides whether or not to retire
# After this line Vt_minus1 = P(important game) *E[Rt + Vt| important game]
Vt_minus1 = (1-s) *np.maximum(Vt_minus1, 0)
# Same as before with different rewards but its repeated
20
# and we don’t want to allocate extra space
Vt_minus1 += s *np.maximum(delta *(Vt[:-1] *(1-P) + Vt[1:] *P) - c, 0)
return Vt_minus1
def compute_val(alpha, beta, s, c_vals, delta=0.9):
"""
computes [V((alpha, beta); s, c, delta) for c in c_vals]
"""
vals = []
for c in c_vals:
vals.append(sparse_bernoulli_value_iteration((alpha, beta), s, c, delta))
return np.asarray(vals)
if __name__==’__main__’:
main()
21


The underwriter and the models-solo dances or pas-de-deux? What policy data can tell us about how underwriters use models

msamlin.comThe underwriter and the models
- solo dances or pas-de-deux?
What policy data can tell us about
how underwriters use models
Stuart Armstrong, Mario Weick, Anders Sandberg,
Andrew Snyder-Beattie, Nick Beckstead
The underwriter and the models - solo dances or pas-de-deux?   1Executive Summary
Using a collection of data on some of the
catastrophe policies written since 2006 by a major reinsurance organisation, this paper explores how tightly underwriters follow the models and under what conditions they deviate from them.
The data included underwriter premium and LE (loss
estimate), and sometimes included LE from up to four different models (AIR, RMS ALM, RMS DLM, and IHM – the in-house model). We analysed the data in order to see what could be said about the relationship between model LE and premiums (as well as underwriter LEs). Mimicking a common procedure in machine learning, the data was randomly divided into a training set and a testing set, allowing many different theories to be investigated on the training set without risk of overfitting and detecting spurious connections.
ResultsThere were three main results, all statistically
significant and with large effect sizes:
1.  The models gave good predictions as to what
the underwriter premium and LE would be. In a regression test, 79% of the variance in the premium, and 88% of variance in the LE, was explained by variance in the models. In fact, most of this variance was explained by the mean LE of the four models – 78% and 87% respectively, corresponding to correlations of 0.88 and 0.93.
2.  As the modelled loses rose, underwriter estimates
moved closer to those of the models. This was evident through a variety of measures: the underwriters would report using more models than otherwise, the underwriter LE would become more strongly correlated with the mean model LE, and they would have less extreme premium/LE ratios (meaning that the LE information was being used more to fix the premium).
It should be noted that one effect that we expected to find - that underwriters were more willing to follow the models if these were more closely bunched together - was not present in the data. It seems that the underwriter does not make much use of model spread.The role of underwriters
As far as can be seen in the data, the underwriters
premiums (and LE) were strongly correlated with the models’ LE estimates. The higher the expected loss (as seen by model LE), the less likely the underwriters were to deviate from the models.
That high correlation may seem to suggest a limited
role to the underwriter. However, this conclusion is premature for several reasons. Most significantly, this dataset only included policies that the reinsurance organisation had actually written, so the role of underwriters in rejecting policies could have been very important. There are also issues of yearly variation and changing market conditions (2012, for instance, seems to be an outlier in many ways). This analysis also ignores the effect of underwriters negotiating and interacting with brokers – even if every good underwriter were to sign similar kinds of deals, this does not mean the underwriters were superfluous. It is significant that the models were more predictive of underwriter LE than of premium (which would be influenced by negotiation).
Underwriters may also play an important role in
correcting erroneous information in the policy, and making sure that the correct models were applied in the first place. Finally, there were no details of outcomes in the data (which policies led to pay-outs, and by how much?), limiting our ability to estimate underwriter expertise. Thus it is possible and likely that the underwriter played (or could play) a more synergistic role with models, focusing on quality control, market insights, and business relations.

2  The underwriter and the models - solo dances or pas-de-deux? The underwriter and the models - solo dances or pas-de-deux?   3This thesis can be strongly questioned. First, the social
role of the underwriter should not be underestimated: they are involved in negotiations with brokers, and require perceptiveness of the opinions and behaviour of the other market actors. Secondly, when the practical role of the underwriter was analysed in detail, some of the work involved coping with poor data quality and correcting model errors.
5 Though these tasks do not
appear on the list of bottlenecks to automation, they are certainly tasks that cannot be easily automated, as they represent a failure of the automation process itself. Currently (and for the foreseeable future), only humans possess the skills to apply these kind of corrections, which often involve deducing what kind of errors have occurred or what kind of extra data could improve the situation. Thirdly, underwriters may be using strategic intelligence to select models and to maintain the sort of overall vision that could resolve larger systemic risks.
6
Therefore it would be incorrect to see the underwriters as necessarily in direct competition with the models, but as occupying (potentially) different and complementary roles. This was one of the approaches advocated in the “autopilot problem” paper: change the “pilot’s” (or underwriter’s) role.
7 Merely relying on effective
automation technology can both weaken the skills of the human and make the whole endeavour vulnerable to situations where the automation fails for some reason. However, making use of particular human abilities to control or complement the automation allows for both greater performance and better robustness. For instance, though simple linear models outperform expert predictions in many domains, these simple linear models could only be constructed thanks to expert knowledge of the important factors.
8
Similarly, experienced underwriters are potentially aware of what information is relevant in a particular type of case – and whether such a case is at hand.
The quality of expertise is highly dependent on
features of the task,
9 rather than on features of
the expert, so it is important that the task the underwriter performs be well designed to make best use of their human qualities (especially as expertise tends to be quite specific to the task performed
10). If this can be achieved, the underwriter
and models of the future will amplify each other.
We thus need to understand how underwriters
currently interact with models. To this end, a major reinsurance organisation has made available several years’ worth of records on policies priced by its underwriters and by its models so that a comparison could be made and the role of the underwriter teased out. The most useful features of this data were the premiums that were actually charged, the underwriters’ Loss Estimate (LE), and the same LEs as given by the models. Using this data, this paper explores how tightly underwriters follow the models and under what conditions they deviate from them.
5  This was the judgement formed by three of the paper’s authors during several periods of immersion at the reinsurance organisation, which
involved detailed conversations with, and questioning of, employees in various roles within the company.
6  Sandberg, Anders. “Defining Systemic Risk” in “Systemic Risk of Modelling.” Joint Future of Humanity Institute-MS Amlin White Paper 1 (2014).
7  Armstrong, Stuart. “The Autopilot Problem” in “Systemic Risk of Modelling.” Joint Future of Humanity Institute-MS Amlin White Paper 2 (2014).
8 Dawes, Robyn M. “The robust beauty of improper linear models in decision making.” American psychologist 34.7 (1979): 571.
9 Shanteau, James. “Competence in experts: The role of task characteristics.” Organizational behavior and human decision processes 53.2 (1992): 252-266.
10 Weiss, David J, and James Shanteau. “Decloaking the privileged expert.” Journal of Management and Organization 18 (2012): 300-310.Introduction
The underwriters have traditionally been the key
players in the insurance industry, making the final decision on any particular policy. It is their responsibility to negotiate a price and ultimately accept or decline to insure the risk. But recent years have seen the emergence of another key insurance player: computerised models that give their own estimation of risk, exposure, and other critical features of a policy. In CAT (catastrophe) insurance, these models are now used extensively by insurers, re-insurers, and regulators, and have underpinned the risk-linked securities markets.
1 In this new, model-centric world, what is the current role of the underwriter? More usefully, what will the role of the underwriter become? Some studies suggest that the underwriter is soon to be replaced by automation.
2 The Frey and Osborne study analysed
current automation trends and concluded, based on O*NET data (an online job classification service developed for the US Department of Labor
3), that
underwriting involved none of the skills estimated to be hard to automate such as manual dexterity, strategic intelligence or socially dependent tasks
4 (see Table 1).
Computerisation
bottleneckO*NET Variable O*NET Description
Perception and ManipulationFinger Dexterity The ability to make precisely coordinated movements of the fingers of one or both hands to grasp, manipulate, or assemble very small objects.
Manual Dexterity The ability to quickly move your hand, your hand together with your arm, or your two hands to grasp, manipulate, or assemble objects.
How often does this job require working in cramped work
spaces that requires getting into awkward positions?
Creative Intelligence Originality The ability to come up with unusual or clever ideas about a given topic or situation, or to develop creative ways to solve a problem.
Fine Arts Knowledge of theory and techniques required to compose, produce, and perform works of music, dance, visual arts, drama, and sculpture.
Social Intelligence Social Perceptiveness Being aware of others’ reactions and understanding why they react as they do.
Negotiation Bringing others together and trying to reconcile differences.
Persuasion Persuading others to change their minds or behaviour.
Assisting and Caring for OthersProviding personal assistance, medical attention, emotional support, or other personal care to others such as coworkers, customers, or patients.
Table 1 O*NET variables that are bottlenecks to automation, according to Frey and Osborne.
1 US Gov’t Accountability Office, GAO-02-941, “Catastrophe Insurance Risks: The Role of Risk-Linked Securities and Factors Affecting Their Use” (2002).
2  Frey, Carl Benedikt, and Osborne, Michael. “The future of employment: how susceptible are jobs to computerisation?” Oxford Martin School Working Paper (2013).
3 http://www.onetonline.org/
4 Or that, when it did involve these skills, that they were not fundamental to the job.
4  The underwriter and the models - solo dances or pas-de-deux? The underwriter and the models - solo dances or pas-de-deux?   5Results
Three primary results were prominent in this
initial analysis. First, variance in the model’s LE explained the majority of the variance in premium (and in underwriter’s LE). Second, underwriters tend to be conservative in estimating losses, more often setting expected losses above those of the models rather than below them. Last, the premiums (and underwriters’ LE) moved closer to the models for more expensive policies.
The causes and implications of these results are
still uncertain – this preliminary analysis is only capable of identifying correlations, not causations.
Result 1: Heavy Use of Quantitative Models A variety of regressions supported the notion that
variations in model LEs explain most of the variance in premium (and underwriter LEs). Regressing
17
premium against the four model’s LE resulted in an R2 of 0.79, while regressing18 premium
against mean LE (of all four models) resulted in an R
2 of 0.78 (implying that 79% and 78% of the
variance in premium is explained by variance in model LE and mean model LE, respectively).
Note that the high R
2 need not mean that the
underwriters are explicitly using the mean LE in
their estimates – the models are highly correlated with each other, as can be seen in Table 2. Thus many linear or quasi-linear combinations of models will be highly correlated with the models, with their mean, and hence with premium.
Indeed, the correlation between premium and
models is quite comparable with the correlation the models have with each other. The premium is actually more highly correlated with the other models than the in-house model (IHM) is. Or, put another way, premium deviation with model LE is comparable to the noise
19 in the model LEs.
Premium AIR LE RMS ALM LE RMS DLM LE IHM LE Mean LE
Premium 1 0.825 0.808 0.796 0.852 0.881
AIR LE 0.825 1 0.909 0.864 0.878 0.966
RMS ALM LE 0.808 0.909 1 0.854 0.797 0.930
RMS DLM LE 0.796 0.864 0.854 1 0.758 0.915
IHM LE 0.852 0.878 0.797 0.758 1 0.937
Mean LE 0.881 0.966 0.930 0.915 0.937 1
Table 2 Correlations between premium, model LEs, and mean model LE.
17  Of the form Ui = β0 + β1MAIRi + β2 MRMS-ALMi + β3 MRMS-DLMi +β4 MGMi +εi where U is the underwriters’ LE, MX is the LE of the X’th model, and εi is the error term for the i’th policy.
18 Of the form Ui = β0 + β1 MMEANi +εi where U is the underwriters’ LE, Mi is the average LE of all four models, and εi is the error term for the i’th policy.
19 Using model LE variation as an informal measure of noise.Data and Methods
The reinsurance organisation provided a large collection of policies with appropriate premium and LE information. Data was taken from two sources: a reinsurance system used to record catastrophe model outputs; and an underwriting system used to record class of business and other risk details. The data were compiled over several years since 2006, with modellers recording model outputs and underwriting teams recording risk details as business was transacted. The data included premium charged, underwriter LE, general location, year, and potential LEs from up to four different models. LE is the loss estimate – the mean amount of money that the underwriter expects their company will pay out on that policy. The more usual “loss on line” is the LE for a particular “layer” of insurance, divided by the limit for that layer. The four models were AIR Catrader (AIR), RMS RiskLink Aggregate level model (RMS ALM), RMS RiskLink Detailed level model (RMS DLM), and the organisation’s in-house model (IHM).
After cleaning the data and restricting to US/
Canada policies (where the models are the most reliable), we were left with a collection of 660 policies where all four models were used
11 – no
further selection was applied to this set.It was decided to split the data into a training and a testing set. This allows “cross validation”, where hypotheses are formed on the training set and tested on the testing set.
12 This prevents overfitting, in which
hypotheses are tailored to narrowly to the data, modelling noise rather than signal.
13 The training set
would be thoroughly analysed to generate hypotheses; these would then be tested for statistical significance on the testing set. The ideal size of the testing set is 1/3 of the original set.
14  As the splitting into testing
and training was done before restricting down to the 660 policies, this subset ended up divided into 457 policies in the training set, and 203 in the testing set.
A total of 32 hypotheses were formed on the training
set, which were then tested on the testing set, and all were found to be significant at the 5% level
15, even
when accounting for multiple comparisons16. Most of
these hypotheses were linear regressions, but other comparisons were made as well (see detail of results).
11 Out of a collection of 6138 US policies which used four or less models.
12 Picard, Richard R, and R Dennis Cook. “Cross-validation of regression models.” Journal of the American Statistical Association 79.387 (1984): 575-583.
13 Hawkins, Douglas M. “The problem of overfitting.” Journal of chemical information and computer sciences 44.1 (2004): 1-12.
14 Dobbin, Kevin K, and Richard M Simon. “Optimally splitting cases for training and testing high dimensional classifiers.” BMC medical genomics 4.1 (201 1): 31.
15 And all but two were significant at the 1% level.
16 Benjamini, Yoav. “Simultaneous and selective inference: current successes and future challenges.” Biometrical Journal 52.6 (2010): 708-721.
6  The underwriter and the models - solo dances or pas-de-deux? The underwriter and the models - solo dances or pas-de-deux?   7Result 2: Conservative Pricing
Underwriters tended to set their LEs somewhat
conservatively. Underwriter LE were above the minimum model LE 94% of the time (as compared to below the maximum model 83% of the time). See Table 5 in next section for breakdown of this into highest and lowest quartiles.
22 This was especially interesting as the maximum value and the mean were both better predictors of underwriter LE than the minimum (see Figure 1 and Table 4). Thus, it seems the minimum value was a rough lower bound for underwriters, but that they used little information from it beyond this.
22  Note that we are using quartile in the sense of a set of data representing a quarter of the policies, not in the sense of the three values
(lower quartiles, median, upper quartile) that divide the ranked data into those four sets.This strong correlation could be a sign of an autopilot
problem20, if underwriters put too much trust in the
models.
In the meantime, we can analyse how these
correlations change over time. For example, Table 3 demonstrates how in 2012, the fit between premium and model LE declined, as compared to surrounding years. This was around the period when the controversial “RMS 11” windstorm model was in use. Though the controversy was mainly around European windstorms, not US/Canada ones, this could have
had an impact on underwriter trust in models. Alternatively, the large losses in 2011 could have played a similar role.
21
Year # of policiesAll Model LE to
Premium R2Mean Model LE to Premium R2All Model LE to Underwriter LE R2Mean Model LE to Underwriter LE R2
2010 74 91.32% 76.08% 91.65% 89.87%
2011 345 84.94% 82.01% 91.65% 90.94%
2012 145 75.69% 69.56% 80.13% 75.58%
2013 92 95.15% 83.22% 94.83% 88.18%
Mean 86.78% 77.72% 89.57% 86.14%
Combined 656 78.80% 77.55% 87.63% 86.95%
Table 3 R2 between premiums and Underwriter LE with models, by year.
20 Armstrong, Stuart. “The Autopilot Problem” in “Systemic Risk of Modelling.” Joint Future of Humanity Institute-MS Amlin White Paper 2 (2014).
21 Aon Benfield. “Reinsurance Market Outlook.” Aon Benfield Analytics (2013).
Figure 1 - Underwriter’s LE as a function of cat model LE. With a set of models, underwriters are unlikely to set
LE below the lowest model, and tend to stick fairly close to the mean model estimate.Table 4 Correlation coefficients between underwriter LE and the mean, maximum, and minimum of the model LEs.Mean Max Min
Underwriter LE 0.933 0.913 0.863
8  The underwriter and the models - solo dances or pas-de-deux? The underwriter and the models - solo dances or pas-de-deux?   9There were many indications that the
underwriters tended to move closer to the model estimate as the LE moved higher.
First of all was the simple fact that they used (or at
least recorded the use of) more models. The full US/Canada data (6138 policies) was decomposed into categories dependent on the number of models recorded
23, and the median premium and underwriter LE of each category was computed (see Figure 2). For illustration, the interquartile distance was added in the case of zero and four models. Though the ranges overlap, those policies which recorded two or less models clearly had lower LEs and premiums than those that recorded three models or four.
Figure 2 - Median underwriter LE and premiums, depending on how many models were reported for that policy . The inter-quartile range
for 4 models and 0 models is also plotted.
23  5071 policies did not formally record a modelled loss, 27 had one, 85 had two, 295 had 3 and 660 had all four models.020000400006000080000100000120000
0 50000 100000 150000 200000 250000ELC
PremiumPremium  and  ELC  medians  for  varying  nbr.  of  models
0  model  median 1  model  median 2  model  median 3  model  median 4  model  medianResult 3: Model Dependency for Biggest Risks
Premium and LE medians for varying number of modelsLEFurthermore, the correlation between underwriter
LE and model LE increased for higher model LE. To see this, the 660 policies with four models were separated into quartiles.
24 The correlation between the underwriter LE and mean, minimum, and maximum model LE was computed in each quartile (Table 5). The data suggests underwriters hewed closer to the model information in these quartiles.
A last way of looking at the difference in behaviour, is to consider what happens when the underwriter has established their LE, and compare this with the final premium. The data indicates that for high underwriter LE
25, the premium/LE ratio is likely to
be less extreme that for low ones. The linear anti-correlation is weak (-11%), but the effect is clearer graphically (see Figure 3). Thus the premium is more influenced by LE estimates when these are large.
Figure 3 - Ratio of premium to underwriter LE plotted against underwriter LE.Table 5 Correlation coefficients between underwriter LE and mean model LE, maximum model LE, and minimum model LE, separated onto the
four quartiles.1st quartile 2nd quartile 3rd quartile 4th quartile
Mean 0.449 0.233 0.598 0.876
Max 0.486 0.387 0.363 0.847
Min 0.231 -0.035 0.329 0.747
24  That is, the policies were ranked according to their mean LE, and they were split into four sets at lower quartile, the median, and the upper quartile. By an abuse of nomenclature, these four
sets are also called quartiles.
25 Since the underwriter would set their own LE before setting a premium, it is valid to use underwriter LE on the x-axis at this point.Premium/LE ratio vs underwriter LEPremium/LE ratio
Underwriter LE
The underwriter and the models - solo dances or pas-de-deux?   11 10  The underwriter and the models - solo dances or pas-de-deux?To establish the results overleaf, a total of 32
hypotheses were tested. The significance of each result was established using the testing set, with each p-value adjusted – increased
26 – using the Holm-
Bonferroni method. See table 6 (where “HBp-value” is the Holm-Bonferroni27 adjusted p-value). As can
be seen, all hypotheses were significant at the 5% level, and all but two were significant at the 1% level (these significance levels were marked in bold).
Hypothesis p-value HBp-value Hypothesis p-value HBp-value
LE model mean correlation 2.20E-16 7.04E-15 LE all model regression 2013 3.37E-12 5.39E-11
LE model mean
correlation 2010-20132.20E-16 6.82E-15Premium all model regression 20137.18E-12 1.08E-10
LE model mean correlation 20112.20E-16 6.60E-15Premium correlation model mean 20107.90E-12 1.11E-10
LE model mean correlation 20122.20E-16 6.38E-15LE model mean correlation 20103.11E-11 4.04E-10
LE all model regression 2.20E-16 6.16E-15mean quartile difference (2-tailed)2.60E-09 3.12E-08
LE all model regression 2010-20132.20E-16 5.94E-15 LE all model regression 2010 4.99E-09 5.49E-08
LE all model regression 2011 2.20E-16 5.72E-15Premium all model regression 20105.02E-09 5.02E-08
Premium all model regression 2010-20132.20E-16 5.50E-15Underwriter LE difference for number of models2.94E-06 2.64E-05
Premium all model regression 20112.20E-16 5.28E-15max quartile difference (2-tailed)6.08E-06 4.87E-05
Premium correlation model mean 2010-20132.20E-16 5.06E-15equivalence between mean and full regression6.08E-06 4.26E-05
Premium correlation model mean 20112.20E-16 4.84E-15min quartile difference (2-tailed)9.20E-05 5.52E-04
Premium correlation model mean 20122.20E-16 4.62E-15mean more important than min (1-tailed)1.86E-04 7.45E-04
LE all model regression 2012 6.93E-16 1.39E-14Premium difference for multiple models2.05E-04 8.22E-04
Premium all model regression 20121.00E-15 1.90E-14correlation between underwriter LE and premium ratio9.04E-04 2.71E-03
LE model mean correlation 20139.80E-15 1.76E-13max more important than min (1-tailed)1.59E-02 3.18E-02
Premium correlation model mean 20139.34E-14 1.59E-12correlation between model mean and absolute deviation %4.85E-02 4.85E-02
Table 6 p-values and Holm-Bonferroni corrected p-values (for multiple comparisons) for all 32 hypotheses considered.
26  p-values of the individual hypotheses were calculated separately, then the hypotheses were ordered by increasing p-value. These p-values were then multiplied by (32+1-r), where r was the
rank of the hypothesis on the list. Thus the hypothesis with the lowest p-value had this value multiplied by 32, that with the next-lowest had its value multiplied by 31, all the way down to the one with the highest p-value, which was multiplied by 1. As long as these adjusted p-values were below the criteria of significance, then the familywise error rate would be below that.
27 Holm, Sture. “A simple sequentially rejective multiple test procedure.” Scandinavian journal of statistics (1979): 65-70.Of these hypotheses, 22 were regression/correlations
with premium or underwriter LE as dependent variable, and either all model LEs or mean model LE as independent variables. We should expect these regressions to show scale invariance to some extent: if all the model LEs double, then, say, the underwriter LE should also double as well. This means that we expect the residuals (the deviations of the underwriter LE from its “theoretic value” as predicted by the regression model) to double as well. Thus we expect residuals to be higher for high expected losses and lower for low expected losses: the data should be heteroscedastic. And indeed it is (see Figure 4).
Excessive heteroscedasticity precludes the use of the
standard F-test to determine the p-values for the model. Instead, we took the logarithms of all the variables, expecting that this would remove the scale variations in the residuals. The residuals that resulted were much closer to being homoscedastic (see Figure 4).
Thus the p-values of these regressions and correlations were calculated using the log-log regression rather than
the standard linear regression. These p-values were sufficient to show that all regression models were significantly different from the null hypothesis. All regression data given in this paper (the R2 and the correlation coefficients), however, came from a standard linear regression, as a linear model is more likely to be closer to the underwriters’ behaviour than a logarithmic one. Generally, the logarithmic model had slightly lower R2 than the corresponding linear model, but the values were very close (and on occasion the logarithmic model had a slightly better fit than the linear model).
Figure 4 - Residuals for linear and log-log regression for underwriter LE versus all four model LEs.
Statistical testing and significance
12  The underwriter and the models - solo dances or pas-de-deux? The underwriter and the models - solo dances or pas-de-deux?   13Discussion
The high correlation between model-estimated LE
and premium (and underwriter LE) may seem to suggest a limited role to the underwriter. However, this conclusion is premature for several reasons. For a start, this dataset is likely incomplete, as it had to be put together specifically, and many policies failed to record model estimates.
28 Most importantly,
it only included policies that had actually been written; the role of underwriters in rejecting policies could have been very important. The overall result ignores the fact of yearly variation: in particular, 2012 fits very poorly into the general analysis. It is likely that underwriters were aware of changing market conditions (or changes to the models themselves) and were able to react to them accordingly in that year.
This analysis also ignores the effect of underwriters
negotiating and interacting with brokers. It is significant that the models were more predictive of underwriter LE than of premium
29 (which would
be influenced by negotiation). Underwriters may also play an important role in correcting erroneous information in the policy, and making sure that the correct models were applied in the first place
30. Finally,
there were no details of outcomes in the data (which policies led to payouts, and by how much?), limiting our ability to estimate underwriter expertise.
31
Conversely, there could have been a lot of wasted effort on the underwriter’s part. The four models are so highly correlated (see Table 4) that any attempts by the underwriters to strike a fine balance between them would likely have made little impact.Given all those caveats, the story that the data presents is clear. If this sample is taken to be representative, then there seems little difference between using models to estimate loss, and having the underwriters do the same. More importantly, a similar pattern is true to a lesser extent in premium, where around 80% of the variance in premiums is explained by the models in a completely linear fashion.
The remaining 20% is unlikely to represent perfect
performance on the part of the underwriter, free of noise and bias. Underwriters, like all humans, are subject to general biases that interfere with their performance
32, some of which could have a specific
impact on their job.33 The important question is
how biases balance against expertise within this 20%. We have done some preliminary exploration of these biases and counteracting expertise with an experimental pilot study.
34 In that study, the
underwriters appeared less model-bound than in this paper: the actual conditions of work may play a large role in decisionmaking. A full analysis of the underwriters’ work is needed if the aim is to develop expertise that complements models in a robust way.
We believe that data analysis such as this paper,
an understanding of the cognition of underwriting, and a systemic perspective on the inherent risks of outsourced cognition – whether from autopilot biases or model-induced correlations across the insurance market – can help develop practices that both reduce systemic risk and amplify human capacity. It will become more and more vital for insurance companies to record their own data as this one has, and to analyse it intelligently.
28  In many cases, it is likely that models were used but the data wasn’t recorded.
29 The models explain 88% of the underwriter LE variance in the 2010-2013 period, but only 79% of the premium variance.
30  From conversations with people in the industry, this last effect is more likely to be a factor in insurance modelling than in reinsurance modelling, which seems to be a more mechanical process.
31  And even loss data in the short term is not enough to estimate true underwriter expertise, as many risk as of low probability/high return period, and wouldn’t show up in the data.
32  The bias literature is vast, but Kahneman, Daniel. Thinking, fast and slow. Macmillan, 2011 provides a good overview, Gigerenzer, Gerd. “How to make cognitive illusions disappear: Beyond
“heuristics and biases”.” European review of social psychology 2.1 (1991): 83-1 15 provides a good critique, and Kahneman, Daniel, and Gary Klein. “Conditions for intuitive expertise: a failure
to disagree.” American Psychologist 64.6 (2009): 515 gives a good synthesis of some opposed views on the subject.
33 Beckstead, Nick. “Biased error search as a risk of modelling in insurance” in “Systemic Risk of Modelling.” Joint Future of Humanity Institute-MS Amlin White Paper 3 (2014)
34 Tilli, Cecilia, and Sandberg, Anders. “Pilot study of underwriter cognitive bias.” Joint FHI-MS Amlin White Paper, forthcoming 3 (2016).

Stuart Armstrong
Mario Weick
Anders Sandberg
Andrew Snyder-Beattie
Nick Beckstead