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\title{Numerosity and allocation behavior: Insights using the dictator game}

\author{
Sunaina Shrivastava\thanks{Department of Marketing, Tippie College of Business,
University of Iowa, Iowa City, IA 52240. Email: sunaina-shrivastava@uiowa.edu.}
\and
Gaurav Jain\thanks{Lally School of Management,
Rensselaer Polytechnic Institute}
\and
Dhananjay Nayakankuppam\thanks{Department of Marketing, Tippie College
  of Business, University of Iowa}
\and
Gary J. Gaeth\footnotemark[3]
\and
Irwin P Levin\footnotemark[3]\;\,\thanks{Department of Psychology, University of Iowa.}
}

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\newcommand{\jhead}{Judgment and Decision Making, Vol.~12, No.~6}
\newcommand{\jdate}{November 2017}
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  \jdate \hfill Numeracy and allocation \qquad}
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%\begin{latexonly}
\vspace{-3mm}
\begin{onecolabstract}
%\end{latexonly}
  This paper investigates how the numerosity bias influences
  individuals’ allocation of resources between themselves and others,
  using the backdrop of the traditional dictator game. Across four
  studies including both hypothetical and real exchanges of money, we
  show that the form of the numerical value representing the quantity
  of the resource (e.g., \$20 vs 2000 cents) systematically biases the
  decision-maker to perceive the quantity s/he is thinking of
  allocating as being ``less than adequate'' or ``more than
  adequate''. Essentially, such a biased perception of adequacy with
  respect to the quantity of the resource consequently influences the
  decision-maker’s final allocation decision. We attribute this
  systematic bias to the ``numerosity'' of the resource. We find that
  bigger numerical values representing quantity (e.g., 2000 cents)
  bias decision-makers to over-infer the quantity, thus inducing them
  to allocate less to the entities they are focusing on.
  % Our work not
  % only contributes to the literature on the dictator game, and
  % research studying determinants of allocation behavior, but also adds
  % to the numerosity bias literature which has garnered considerable
  % research interest recently.

% lots of editing here

\smallskip
\noindent
Keywords: numerosity, framing, dictator game, allocation, behavior, resource allocation
%\begin{latexonly}
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]
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{\renewcommand{\thefootnote}{}

\footnotetext{


Copyright: \copyright\ 2017.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

\section{Introduction}

We often have to decide how to allocate resources between ourselves
and others. For instance, individuals must decide how to allocate
available resources, be it money or time, between their own financial
needs and other social causes they support. The traditional Dictator
game is an excellent representation of such an allocation context and
is probably the most frequently used tool to study allocation
behavior. In the standard Dictator game, the ``dictator'' is allotted an
initial sum of money in the form of an initial endowment and is asked
to allocate a portion of it to another individual who must accept the
allocated amount (Forsythe, Horowitz, Savin \& Sefton 1994). On the
basis of self-interest alone, the dictator should ideally take all the
money for herself and allocate nothing to the recipient. However, past
research has demonstrated that many who are put in the position of the
dictator allocate a substantial share of their initial endowment to
the recipient, with the modal allocation being as high as 30\%
(Bolton, Katok \& Zwick 1998).

Thus, the Dictator game is used as an illustration of the impact of
social and moral demands over self-interested rationality. Different
explanations have been proposed to explain the dictator’s non-trivial
allocation including a preference for fairness among the dictators
(Forsythe et al 1994; Kahneman, Knetsch, Thaler 1986), altruism
(Andreoni \& Miller 2002), inequality aversion (Bolton \& Ockenfels
2000; Fehr \& Schmidt 1999) and a social concern for what other
observers think (Hoffman, McCabe, Shachat \& Smith 1994).

% Even economic theory does not deny other-regardind preferences. You
% are knocking down a straw man.

In addition to understanding the motivation behind the dictator’s
generous allocation, more recent research has sought to understand
several moderating factors that influence the level of allocation in
the dictator game. For instance, Hoffman, McCabe and Smith (1996),
and, Bohnet and Frey (1999) found that allocation increased with
social closeness, a sense of relationship between the dictator and the
counterpart. Similarly, Aguiar, Brañas-Garza and Miller (2008) found
that the allocation increased as moral distance decreased.  In another
vein, it has been found that allocation increased when an anonymous
recipient was replaced by an established charity (Eckel \& Grossman
1996) or a deserving/skillful participant (Ruffle 1998). Also,
dictators increased allocation when they scored high on self-control
(Martinsson, Myrseth \& Wollbrant 2012); when they perceived the
recipient’s expectations to be higher (Heintz, Celse, Giardini \& Max
2015); or when they thought the endowment was earned rather than
unearned (Cherry, Frykblom \& Shogren 2002). These studies are among a
very large collection showing the sensitivity of the allocation level
to outside influences. (See Engel 2011 for a summary.) In general,
these effects generally look at the moderating influence of various
forms of social norms. Less research has examined the qualities of the
endowment that has to be divided between the dictator and the
recipient. We examine one aspect of this level of analysis, namely the
``\textit{numerosity}'' of the resource.

\pagebreak

To this end, we find that the form of the numerical value representing
the quantity of the resource (e.g., \$20 vs 2000 cents) systematically
biases the decision-maker to perceive the quantity s/he is thinking of
allocating as being ``less than adequate'' or ``more than adequate''. Such
a biased perception about the quantity of the resource consequently
influences the decision-maker’s final allocation decision.  We
attribute this systematic bias to the ``numerosity'' of the resource.

\subsection{Numerosity Bias}
\begin{quote}
  ``Make six slices of the pizza, I can’t eat eight.''  (``not so
  hungry'' Yogi Berra)\vspace{-1ex}
\end{quote}
As the classic pizza joke illustrates, we may perceive a resource,
such as a pizza, to be greater in quantity if it is represented in
terms of the more numerous eight slices than the less numerous six
slices. This has been called the numerosity bias (Pelham, Sumarta \&
Myaskovsky 1994), because despite a difference in the expression of
amount, the size of the resource (i.e., the pie) remains the same. The
numerosity bias explains that individuals tend to over-infer quantity
when it is represented with higher numeric values or bigger numbers
(Pelham et al 1994).

Numerosity has been shown to affect judgments across consumer domains
(Bagchi \& Davis 2016) such as points in loyalty/reward programs
(Bagchi \& Li 2011; Nejad \& Onay 2014), quality perceptions (Burson,
Larrick \& Lynch 2009) and pricing (Coulter, Choi \& Monroe 2012;
Coulter \& Coulter 2010). Nejad and Onay (2014) found that
loyalty/reward programs are perceived more positively by consumers if
the rewards points are more numerous; for instance, 1000 rewards
points cumulatively worth \$10 are valued more by consumers than 100
reward points cumulatively worth \$10. Burson et al.\ (2009) found that
expanded (higher numerosity) scales led to greater discrimination
between options than contracted (lower numerosity) scales; for
instance, consumers will prefer product A more if it is represented as
20 points superior to product B on a 100-point scale when compared to
the case where product A is represented as 1-point superior to product
B on a 5-point scale. Numerosity is shown to bias consumers'
perception of price as well --- prices or discounts which are
phonetically bigger (i.e., have more syllables in their pronunciation)
will be perceived bigger; for instance, \textit{three hundred
  seventy-eight }dollars may be perceived to be more than
\textit{three hundred eighty }dollars (Coulter et al 2012; Coulter \&
Coulter 2010). Thus, there is ample research on the impact of
numerosity on judgments in terms of how numerosity can bias
individuals' perceptions toward a product or brand.

Outside of this consumption context, few studies have examined effects
of numerosity. One example is denominator neglect (a.k.a., ratio
bias), in which individuals judge a low probability event to be more
likely when represented as a large-numbered ratio, such as 30/100,
than as a smaller-numbered but equivalent ratio, such as 3/10
(Kirkpatrick \& Epstein 1992). We extend the numerosity bias research
to another area, allocation behavior. Across four experiments, we show
that numerical values representing the quantity of a resource
systematically bias decision-makers' perception of the quantity to be
less than, adequate, or more than adequate. As we will demonstrate,
this bias affects decision makers' final allocations.

Another stream of relevant research concerns the money illusion, in
which the face value of currency influences individuals' perceptions
of its real value.  Shafir, Diamond and Tversky (1997) showed how
individuals do not account for inflation while making judgments about
the utility they are getting from the money they are receiving. For
instance, in one of their studies, people estimate
that someone earning a 5\% raise when there is a 4\% inflation will be
happier than someone earning a 2\% raise when there is no
inflation. Raghubir and Srivastava (2002) showed that individuals tend
to spend more when the face value of foreign currency is a fraction of
an equivalent amount in home currency (e.g., a US consumer in Jordan,
where 0.7 Jordanian Dinar is equivalent to 1 US dollar, will
overspend). Alternatively, individuals tend to spend less when the
face value of foreign currency is a multiple of the equivalent amount
in home currency (e.g., a US consumer in Malaysia, where 4 Malaysian
Ringgits is equivalent to 1 US dollar, will underspend). Wertenbroch,
Soman and Chattopadhyay (2007) introduce the idea that individuals use
the face value of the difference between available budget and prices
while making the purchase decision. For instance, US consumers are
more likely to purchase a headphone in Singapore worth S\$135
(equivalent to USD 100) when they have S\$270 (equivalent to USD 200)
left in their budget, than, the same headphone in Jordan worth JOD 70
(equivalent to USD 100) when they have JOD 140 (equivalent to USD 200)
left in their budget. Our paper adds on to this research on
the money illusion by showing how individuals' decision to allocate money
to different entities can be influenced by the face value of money.

Initially, in order to document the impact of numerosity on allocation
behavior, we manipulate the decision-maker’s perception of quantity
using different denominations for the resource – for example 10 dollars
versus 1000 cents. Study 1 demonstrates the impact of numerosity on
allocation using a hypothetical resource while Study 2 extends the
finding to decisions with real money. To provide further insight into
the manner in which numerosity drives allocation, we manipulate the
decision-maker’s focus by framing the decision as “giving to the other”
versus “keeping for oneself”, and, manipulate the decision-maker’s
relative attention by drawing attention to either the numerical value
or the denomination (Study 3). Finally, we use a hypothetical currency
to validate our process account and show that numerosity biases the
decision-maker’s perception of ``adequacy'' with respect to quantity of
the allocated resource (Study 4).

\subsection{Allocation Behavior: Experimental Setup}

\renewcommand{\baselinestretch}{1.04} \large\normalsize

To investigate numerosity and allocation behavior, we use the standard
dictator game, where an individual referred to as the ``dictator'' (i.e.,
decision-maker) is allotted an initial sum of money in the form of an
initial endowment and is asked to allocate a portion of it to another
individual. In the most commonly used version of the Dictator game, the
other individual has no option but to accept the allocated amount
(Forsythe et al., 1994). 

As mentioned earlier, given our research question, we manipulate the
decision-maker’s perception of quantity of resources (which is money in
this case) by describing the initial endowment using
objectively equivalent amounts expressed in either dollars or
cents. Essentially, we posit that when a decision-maker contemplates
how much to allocate to the recipient in ``cents'', the numerosity bias
will occur. By that we mean that a certain amount in cents such as 300
cents, will be perceived to be greater in quantity or more abundant
than an objectively equivalent amount expressed as 3 dollars, as the
numerical representation of 300 is much bigger than 3. Consequently,
under this ``Cents'' condition, any amount in cents that the
decision-maker is thinking about allocating to the other person (e.g.,
300 cents) will be perceived to be greater than an objectively
equivalent amount that a counterpart decision-maker may be thinking of
giving in the ``Dollar'' condition (e.g., 3 dollars). Because the
decision-maker tends to over-infer the amount when
represented in cents, and thus perceives it to be more than adequate,
s/he will adjust the contemplated allocation in cents to a lower value
(for, e.g., 270 cents), when compared to an allocation decided by the
counterpart decision-maker in dollars. Accordingly, as a result
of the numerosity bias, we expect to see that the final allocation made
to the recipient in the ``Cents'' condition will be lesser in absolute
value when compared to the allocation made in the ``Dollar'' condition.
We test this intuition in Study 1.

\section{Study 1}

\subsection{Methods}
One hundred and eight Mturk participants from the U. S. participated
(69 females; M\textsubscript{age} = 35.73 years) in a single factor
between-subjects study. Participants engaged in a Dictator game where
money was allocated in \textit{dollars} versus \textit{cents}.
Participants were told to imagine that they were allotted an initial
sum of \$10 (in the Dollar condition) or 1000¢ (in the Cents
condition) for themselves and another participant in this
experiment. They were told that they were randomly assigned to make
the distribution decision and were asked to indicate the amount of
money they would like to ``give the other participant''.\footnote{As
  the Dictator game is essentially a one player decision task, each
  Mturk participant in both the conditions acts as the decision
  maker. There was actually no other individual or participant serving
  as the recipient.} We used slider scales for the responses to avoid
granularity issues. That is, since an amount in cents is more granular
than an amount in dollars, we utilized a slider scale to ensure that
those in the Dollar condition were able to allocate at the same level
of granularity as those in the Cents condition --- so, in the Dollar
condition, the participant could allocate \$4.57, the same as a
participant in the Cents condition could allocate 457 cents. Thus, the
responses could be in steps of 0.01 dollars in the Dollar condition
and 1 cent in the Cents condition. In each condition (cents or
dollars) the sliders had just two labels, one on each side: 0 dollars
(0 cents) and 10 dollars (1000 cents). As the decision maker navigated
the scale, a number on the right of the scale displayed the
corresponding amount (as per the position of a pointer on the slider
scale). The displayed amount, in dollars for the Dollar condition and
in cents for the Cents condition, depicted how much would be allocated
to the recipient.

\subsection{Results}
Results from a one-tailed t-test supported our
intuition. Participants' allocation to the other individual was
significantly lower in the Cents condition when compared to the
allocation made in the Dollar condition (M\textsubscript{allocation in
cents} = \$3.438 (SD = 1.904); M\textsubscript{allocation in dollar} =
\$4.693 (SD = .573), \textit{t}(106) = $-$4.639;
\textit{p}\textit{\textsubscript{one-tailed}} {\textless}.001). 

\subsection{Discussion}
In Study 1, we found that the other individual’s share was substantially
(more than 10\%) lower in the more numerous Cents condition when
compared to that in the Dollar condition. These results suggest that
the recipient can actually be better off if the decision-maker
allocates resources represented in less numerous units, such as
allocating money in dollars instead of cents, or, allocating time in
hours instead of minutes, or, allocating commodities in gallons/pounds
instead of cups/ounces.

The results of Study 1 were significant but were not incentive
compatible, as participants responded to a hypothetical scenario and
were asked to imagine making the allocation decision. Therefore, with
the aim of replicating the core results obtained in Study 1, we
administered Study 2 in the context of real money, where participants
would receive a monetary payoff based on the allocation decision they
made.

\renewcommand{\baselinestretch}{1.00} \large\normalsize

\section{Study 2}

\subsection{Methods}
All members associated with The University of Iowa as faculty, students
or staff were emailed a request to participate in the research study in
return of monetary compensation. Eighty-six of the people who had
registered showed up for the study (54 females; M\textsubscript{age} =
39.53 years). Participants entered the main lounge of the lab. Each
participant first got her/his picture taken in ``room A''. Then all
participants entered ``room C'' where they were seated on individual work
stations. Partitions were set up at the work station so that
participants could not look at other individuals or computer screens
during the study. 

Participants did an unrelated task for ten minutes and then saw the
following message “\textit{As you might be aware, the study you are
  undertaking is taking place in different rooms right now. You are in
  room C (C220D PBB) and some other participants are in room D (S327
  PBB) which is on the third floor of this building. Each participant
  in Room C has been randomly paired with another participant in room
  D for the next task}”. On the next screen, participants saw a
photograph of ``another participant'' and were told that they have
been paired with this ``other participant in room D''. Please note
that all participants were shown the same picture in order to keep the
``other participant'' uniform for all, a 22 years old Caucasian male
who was a member of our research team and was unknown to our study
participants.

On the very next screen participants were given the instructions for the
allocation decision task. They were told, “\textit{As you
are aware, at the end of this session, you are being given an amount
for participating in this study. In addition to this participation
compensation, you are being allotted 10 dollars} (or 1000 cents for the
cents condition) \textit{for yourself and the other participant in room
D whose picture you are currently seeing. In each experimental session,
it is randomly decided whether participants in room C or in room D will
make the distribution decision. In this particular session, it has been
decided that participants in room C will make the distribution
decision. Since you are in room C, you have been assigned the task of
deciding the distribution. There is one-way anonymity in this task.
This means that you can see the other participant's
picture, but the other participant cannot see your name or picture -
you are anonymous to him/her. Even though your picture is not being
used in this session (as you are anonymous to the other participant in
room B), your picture was taken at the beginning of the study to keep
the process consistent for all participants irrespective of the room
they are in. How much will you give the other participant in dollars
}(in cents for the ``cents'' condition)”. Participants provided their
responses on a slider scale which was similar to the one used in Study
1. 

At the end of the experimental session, participants walked into a
separate room one by one and were paid \$5 as their participation
compensation plus an amount based on their allocation decision. So, for
example, if a participant’s response indicated that the participant had
allocated \$3 to the recipient (leaving \$7 for herself), then the
participant received \$5 (participation fee) plus an additional \$7,
i.e., total of \$12.

\subsection{Results}
Results from a one-tailed t-test replicated the
results of Study 1. Participants' allocation to the other individual
was again significantly lower in the Cents condition when compared to
the allocation made in the Dollar condition (M\textsubscript{allocation
in cents} = \$4.801 (SD = 1.75); M\textsubscript{allocation in dollar}
= \$5.795 (SD = 2.06), \textit{t}(84)= $-$2.398;
\textit{p}\textit{\textsubscript{one-tailed}} = .01). 

\subsection{Discussion}
The results of Study 2 establish the validity of the phenomenon with
actual money, where individuals' allocation decisions had an impact on
their actual payoff.

While building our intuition for Studies 1 and 2, we argued that owing
to the numerosity bias, individuals tend to over-infer the amount they
are thinking of allocating when it is represented with bigger numbers
(i.e., in the cents denomination). Such a biased perception would in
turn cause participants to adjust their contemplated allocation to the
other person --- in cents --- to a lower value. Consequently, the
final amount allocated by the decision-maker to the other individual
was less in the Cents condition than in the Dollar condition (as
observed from the results of Studies 1 \& 2). 

Please note that in Studies 1 and 2, the decision-maker was asked how
much money from the initial endowment s/he would be willing to
\textit{give} \textit{the other individual, }which means that the
decision-maker’s focus for allocation was on the other individual. If
numerosity indeed works in the manner as delineated above, then the
same reasoning should apply even when the decision-maker is
alternatively asked how much money from the initial endowment she would
\textit{keep for herself} --- in this manipulation, the
decision-maker’s focus for the allocation is now on herself. In this
latter case as well, we predict that individuals will tend to
over-infer an amount in cents – that is, any amount in cents that the
decision maker is thinking of allocating to \textit{herself} will be
perceived to be greater than an objectively equivalent amount that a
counterpart decision-maker may be thinking of in the Dollar condition.
If our numerosity based reasoning is correct, such a bias should thus
cause the decision-maker to adjust her contemplated allocation to
\textit{herself} – in cents – to a lower value. Consequently, the final
amount allocated by the decision-maker to \textit{herself} will be
lower in the Cents condition when compared to the Dollar condition. 

\renewcommand{\baselinestretch}{1.08} \large\normalsize

To summarize, in Studies 1 and 2 where the focus for allocation was
on the other individual, we found that the decision-maker’s allocation
to the other individual was lower in the more numerous ``Cents''
condition. If numerosity is indeed driving such an effect, then in the
case where the focus for allocation is on the decision-maker herself,
we should see that decision-maker’s allocation to herself should be
lower in the more numerous ``Cents'' condition, and, \textit{therefore}
the other individual’s share (which in this case is the initial
endowment minus the decision maker’s allocation to herself) should be
higher in the ``Cents'' condition – note that the other individual’s
share serves as the dependent variable for the study. Study 3
investigates this ``focus of allocation (other individual versus
decision-maker herself)''  x ``numerosity (money allocated in dollars
versus cents)''  interaction. 

Moreover, we had speculated that individuals pay attention to the
numerical information (for example 300 versus 3) and thus perceive an
amount of 300 cents that they may be thinking of allocating to be
greater than 3 dollars. We are arguing that this attention/emphasis on
numbers enhances the reliance on the numerosity bias, thereby driving
the allocation effects. This reasoning can be tested by
\textit{manipulating the decision makers' attention} – if we make
``\textbf{10}'' (``\textbf{1000}'') bold in the display of initial
endowment of 10 dollars (1000 cents), then we encourage
decision-makers to enhance their attention on numerical
information. In this case, we expect to obtain the ``focus of
allocation'' x ``numerosity'' interaction effect as predicted
above. On the other hand, if we divert attention away from the number
and make the ``\textbf{dollars}'' (``\textbf{cents}'') bold,
decision-makers will probably still attend to the number but now since
more attention is paid to the denomination, they can see through the
``numbers'' game. There is probably still a sense of 1000 being more
than 10 but this is tempered by the fact that the decision-makers are
made more explicitly aware of the denomination. Thus, the ``focus of
allocation'' x ``numerosity'' interaction effect should be diminished
in this case. So, in essence, we predict a significant three way
``focus of allocation'' x ``numerosity'' x ``attention on (numerical
information versus denomination)'' interaction. The results in the
posited form and direction will provide support for our numerosity
based reasoning driving allocation behavior.  We check these
predictions in Study 3.

With respect to the main effects of the three factors (``focus of
allocation'', ``numerosity'' , and, ``attention on'' ), we predict
only the main effect of ``focus'' – the other individual’s share will
be more when the decision maker’s focus for allocation is on the other
individual, and, the other individual’s share will be less when
decision maker’s focus for allocation is on herself (hence allocating
more to herself, which in turn, leaves less for the other individual).

\renewcommand{\baselinestretch}{1.00} \large\normalsize

\section{Study 3}

\subsection{Methods}
We conducted a 2(Focus of allocation: other individual versus
decision-maker herself) X 2(Numerosity: money allocated in dollars
versus cents) X 2(Attention on: numerical information versus
denomination) between subjects study. Two hundred and forty-seven Mturk
participants (103 females; M\textsubscript{age} = 31.97 years) were
randomly distributed across the cells. Participants were made to
imagine that they were allotted an initial sum of \$10 (in the Dollar
condition) or 1000¢ (in the Cents condition) for themselves and another
participant in the experiment. They were told that they were randomly
assigned to make the distribution decision. In the ``focus of allocation
= other individual'' condition (identical to Studies 1 and 2),
participants were asked ``\textit{how much would you like to give the
other participant''. }Alternatively, in the ``focus of allocation =
decision-maker herself'' condition, participants were asked ``\textit{how
much would you like to keep for yourself'' }(the remainder was left for
the other player). We encouraged half of the individuals to enhance
their attention to the numerical information by making bold the
numerical value in the initial endowment (\textbf{10} dollars or
\textbf{1000} cents). For the remaining individuals, we diverted their
attention away from the numerical information by making bold the
denomination of the initial endowment (10 \textbf{dollars} or 1000
\textbf{cents}). Similar to Study 1, we used slider scales for
participants' responses.

\subsection{Results}
We ran a three-way ANOVA with the other individual’s share as the
dependent variable. Results revealed a significant main effect of
``focus'' as the other individual’s share was higher when the
decision-maker’s focus for allocation was on the other individual as
compared to the case where the decision-maker’s focus for allocation
was on herself (M\textsubscript{focus of allocation = other individual}
= \$4.452, M\textsubscript{focus of allocation = decision-maker
herself} = \$2.602; \textit{F}(1,239) = 31.308; \textit{p} = .000; $\eta
$\textsubscript{p}\textsuperscript{2} = .116). The main effect of
Attention on numerical information versus the denomination was almost
significant (M\textsubscript{numerical value in bold} = \$3.174, M
\textsubscript{denomination in bold} = \$3.881; \textit{F}(1,239) =
3.797; \textit{p} = .053; $\eta $\textsubscript{p}\textsuperscript{2}
= .016). We found an insignificant main effect of dollars versus cents
(M\textsubscript{allocation in cents} = \$3.396,
M\textsubscript{allocation in dollars} = \$3.649; \textit{F}(1,239) =
0.843; \textit{p} = .359; $\eta $\textsubscript{p}\textsuperscript{2}
= .004).

Central to the question at hand, the three-way interaction was almost
significant and was in the predicted direction and form
(\textit{F}(1,239) = 3.109; \textit{p }= .079;
$\eta $\textsubscript{p}\textsuperscript{2} = .013). Under the
``Attention on numerical information'' condition, we found that the
interaction of (focus of allocation) x (money allocated in dollar
versus cents) was significant as predicted (\textit{F}(1,121) =
12.166; \textit{p} = .001; $\eta $\textsubscript{p}\textsuperscript{2}
= .091). Follow-up analyses in this regard revealed that when the
focus of allocation was on the other individual (identical to Studies
1 and 2), we replicated the results obtained in Studies 1 and 2, and,
found that the other individual’s share was \textit{less} when the
allocation was made in cents than when it was made in dollars
(M\textsubscript{allocation in cents} = 3.436 (SD = 1.99),
M\textsubscript{allocation in dollars} = 5.021 (SD = 3.19),
\textit{t}(59) = $-$2.293;
\textit{p}\textit{\textsubscript{one-tailed}} = .011). On the other
hand, when the focus of allocation was on the decision-maker herself,
we found that this was not the case; the other individual’s share
(calculated in this case by subtracting the decision maker’s
allocation to herself from the initial endowment) was more when the
allocation was made in cents than when it was made in dollars
(M\textsubscript{allocation in cents} = 2.911 (SD = 2.25),
M\textsubscript{allocation in dollars} = 1.5 (SD = 2.04),
\textit{t}(62) = 2.629; \textit{p}\textit{\textsubscript{one-tailed}}
= .006).

%Fig.1. 
\begin{figure}[t!]\centering
\caption{Results of Study 3.}
\includegraphics[clip=true,width=\columnwidth]{fig1a.eps}\bigskip

\includegraphics[clip=true,width=\columnwidth]{fig1b.eps}
\end{figure}

Alternatively when individuals' attention was diverted away from the
numerical information, i.e., under the ``Attention on denomination''
condition, the interaction of (focus of allocation) x (money allocated
in dollar versus cents) was not significant (\textit{F}(1,118) = .413;
\textit{p} = .522; $\eta $\textsubscript{p}\textsuperscript{2} = .003).
Follow-up analyses revealed that, when the focus of allocation was on
the other individual, the other individual’s share was insignificantly
different when the allocation was made in cents or in dollars
(M\textsubscript{allocation in cents} = 4.286 (SD = 1.44),
M\textsubscript{allocation in dollars} = 5.136 (SD = 3.84),
\textit{t}(60) = $-$1.153; \textit{p}\textit{\textsubscript{one-tailed}} =
.127). Also, when the focus of allocation was on the decision-maker
herself, the other individual’s share (calculated by subtracting the
decision maker’s allocation to herself from the initial endowment) was
insignificantly different when the allocation was made in cents or in
dollars (M\textsubscript{allocation in cents} = 2.914 (SD = 2.50),
M\textsubscript{allocation in dollars} = 3.116 (SD = 2.87),
\textit{t}(58) = $-$0.281; \textit{p}\textit{\textsubscript{one-tailed}} =
.39).

The means are represented in Figure 1. These results provide
support for our numerosity based reasoning driving the observed
allocation behavior. Figure 1 reflects that the ``focus of allocation'' x
``numerosity'' effect is significant when attention is on numerical
information (i.e., the numerical value is made bold), but it is
insignificant when attention is on denomination (i.e., the denomination
is made bold).

\subsection{Discussion of Study 3}
From the pattern of data shown in Figure 1, it can be noted that, when
the decision-maker’s focus for allocation is on the other player, the
other player’s share is higher in the Cents condition when the
denomination is made bold than when the numerical value is made bold.
However, the other player’s share in the Dollar condition does not seem
to vary when the denomination or the numerical value is made bold.
Alternatively, in the case where the decision-maker’s focus for
allocation was on herself, the other player’s share was higher in the
Dollar condition when the denomination is bolded than when the
numerical value is bolded. However, the other player’s share in the
Cents condition does not seem to vary when the denomination or the
numerical value were bolded. This is an interesting pattern of
results and was not predicted. It opens an interesting avenue, as in,
how framing the allocation task in different objectively equivalent
frames (in this case the foci of allocation) can influence our core
numerosity based results. 

An important and final part of our process account that warrants
investigation is whether numerosity biases the decision-maker’s
perception of ``adequacy,'' with respect to the quantity of the
allocated resource. While developing our process account for the
impact of numerosity on allocation behavior, we posited that the
bigger numerical values representing an amount in ``cents'' bias
individuals to perceive the amount, they are thinking of giving to the
other entity, as being larger, and thus more than adequate. Such a
biased perception causes individuals to adjust their contemplated
allocation in cents to a lower value. Consequently, in the case where
the decision-maker’s focus for allocation is on the other entity
(i.e., ``\textit{how much will you give to the other person}''), the
final allocation to the other entity is lower in the Cents condition
than in the Dollar condition (as demonstrated previously). Thus, we
did Study 4, to directly test our ``perception of adequacy'' based
underlying mechanism for the demonstrated phenomenon.

If an individual decides on a final allocation denominated in
dollars --- \$3 for example – then this same allocation should be perceived to
be \textit{more than adequate} when s/he sees the same amount expressed
in cents (300¢) at a later time. Alternatively, if a certain
participant decides on a final allocation in cents – let’s say 300¢,
then this very allocation should be perceived to be \textit{less than
adequate} when s/he sees the same amount expressed in dollars (\$3 in
this example) at a later time. The fact that an individual perceives
the same amount of money to be more than adequate or alternatively less
than adequate, when shown in a different denomination, would indicate
that numerosity biases the decision maker’s perception of ``adequacy''.
We conducted Study 4 to test this prediction. We use a fabricated currency –
Tekkas and Yokes – in place of the familiar dollars and cents used in
previous studies. This procedure mimics real-world conditions where
monetary units vary greatly across nationalities. 

\section{Study 4}

One hundred and twenty-one Mturk participants participated (50
females; M\textsubscript{age} = 34.24 years) in a 2(money initially
allocated in: tekkas versus yokes) X 2(reference denomination: same
versus different) study. Participants were initially given a cover
story where they were asked to imagine that they were in Takaland
where the currency was primarily expressed in the form of ``Tekka''
and ``Yoke''. All participants were told that 1 tekka equaled 1000
yokes. Then participants in the ``money allocated in tekkas'' condition
(``money allocated in yokes'' condition) were told that they were
randomly chosen to be allotted an initial sum of 10 tekkas (10000
yokes) and had to decide the distribution between themselves and
another player in this experiment. Similar to Studies 1 and 2,
participants were asked how much they would like to ``give the other
player.'' Each participant’s response (allocation decision) was stored
as amount ``p'' by the software. After participants had made their
allocation decision, they undertook a couple of buffer tasks.

After the buffer tasks, each participant was shown a \textbf{message}
depending on the condition s/he was in. Half the participants were
shown the message in terms of the same denomination in which they had
made the allocation earlier. That is, participants who had allocated ``p
tekkas'' in the first part of the study were told that another
participant who had undertaken this same experiment earlier had given
``p tekkas'' to the other player. Similarly, participants who had
allocated ``p yokes'' in the first part of the study were told the exact
same message in yokes. 

The other half of the participants were shown a similar message but the
currency was converted to the other denomination from the one in which
they had made the allocation earlier. That is, participants who had
allocated ``p tekkas'' in the first part of the study were told that
another participant who had undertaken this same experiment earlier had
given ``(p X 1000) yokes'' to the other player. Similarly, participants
who had allocated ``p yokes'' in the first part of the study were told
that another participant who had undertaken this same experiment
earlier had given ``(p/1000) tekkas'' to the other player.

Thus, we had four different conditions based on the denomination
participants had used to make the allocation in the first part of the
study, and, the denomination participants saw in the message later:
tekkas-tekkas (reflecting consistent numerosity); yokes-yokes (also
reflecting consistent numerosity); tekkas-yokes (reflecting low to high
numerosity); yokes-tekkas (reflecting high to low numerosity). 

After receiving the message, all participants were asked to indicate
whether they felt the amount shown in the message was adequate or not
on a seven-point scale ranging from strongly disagree to strongly
agree.

To illustrate an example, a participant in the (tekkas-yokes) condition
who gave 3 tekkas to the other player in the first part of the study,
was shown a message saying that “Another participant who undertook this
study earlier had given 3000 yokes to the other player.” Then s/he was
asked, “Do you agree/disagree that this person has given an adequate
amount to the other player?” on the seven-point scale. 

\subsection{Results}
We first analyzed the amount each decision maker allocated to the other
payer. One tailed t-test revealed that, in line with previous studies,
decision makers allocated less money to the other player in the more
numerous Yokes than in Tekkas (M\textsubscript{allocation in yokes} =
3.003 (SD = 2.316), M\textsubscript{allocation in tekkas} = 3.961 (SD =
2.871); \textit{t}(119)= 2.042;
\textit{p}\textit{\textsubscript{one-tailed}} = .022).

%Fig. 2.
\begin{figure}[t!]
\centering
\caption{Results of Study 4.}
\includegraphics[clip=true,width=\columnwidth]{fig2.eps}
\end{figure}

Participants' response on the seven point ``adequateness'' scale, where
they express whether the amount shown in the message was adequate or
not, was the dependent variable. We ran a two-way ANOVA and found the
main effect of money initially allocated in ``tekkas versus yokes''
(M\textsubscript{tekkas}= 1.69 (SD = 1.336), M\textsubscript{yokes} =
.72 (SD = 1.932); \textit{F}(1,117) = 11.319; \textit{p} = .001; $\eta
$\textsubscript{p}\textsuperscript{2} = .088) to be significant, and,
that of ``reference denomination: same versus different''
(M\textsubscript{same denomination} = 1.48 (SD = 1.765),
M\textsubscript{different denomination } = 0.93(SD = 1.649);
\textit{F}(1,117) = 3.210; \textit{p} = .076; $\eta
$\textsubscript{p}\textsuperscript{2} = .027) to be almost significant.

However, central to the question at hand, we are interested in the
two-way interaction between ``money initially allocated in tekkas versus
yokes'' and ``reference denomination: same versus different''. We found
this two-way interaction to be significant (\textit{F}(1,117) = 13.280;
\textit{p} {\textless} .001; $\eta
$\textsubscript{p}\textsuperscript{2} = .102). Follow-up tests revealed
clear support for our proposition. When participants were shown the
message in the same denomination as the one they had used for making
the allocation decision in the first part of the study, their responses
on the adequateness scale did not differ significantly
(M\textsubscript{tekkas-tekkas} = 1.44 (SD = 1.585),
M\textsubscript{yokes-yokes } = 1.52 (SD = 1.745); \textit{t}(59) = $-$.187;
\textit{p}\textit{\textsubscript{one-tailed}} = .426). 

In contrast, when participants were shown the message in a denomination
different from the one they had used for making the allocation decision
earlier, we find that those individuals who made the allocation
decision earlier in yokes (more numerous representation) but were later
shown the message in tekkas (less numerous representation), perceive
the same objective amount in the message to be less adequate than those
individuals who made the allocation decision earlier in tekkas but were
later shown the message in yokes (M\textsubscript{yokes-tekkas} = $-$0.03
(SD = 1.816), M\textsubscript{tekkas-yokes} =1.97 (SD = 0.944);
\textit{t}(58) = 5.290; \textit{p}\textit{\textsubscript{one-tailed}}
{\textless} .001). 

Means are described in Figure 2. As shown in Figure 2, individuals’
perception of adequacy does not differ when there was consistent
numerosity (i.e., when the denomination in the message was the same as
the one they had used for making the allocation decision in the first
part of the study). However, as reflected in Figure 2, individuals’
perception of adequacy was lower when numerosity went from high to low
than from low to high.

These results indicate that an individual perceives the same amount of
money to be less adequate or more adequate when the amount is shown in
a different denomination varying in numerosity. Thus, the study lends
direct support to our position that numerosity biases the decision
maker’s perception of ``adequacy'' which underlies the allocation
decisions described in the earlier studies.

\section{General Discussion}

Previous research on the Dictator game has investigated the impact of
several factors, including social distance (Bohnet \& Frey 1999),
moral distance (Aguiar, Branas-Garza \& Miller 2008) and recipient’s
deservedness (Eckel \& Grossman 1996), on the amount allocated to
another entity. We contribute to this research by examining the
qualities of the endowment that has to be divided between the dictator
and the recipient. In this regard, we examine how the numerosity
of the resource affects allocation decisions. We find that
the numerical values representing quantity of the resource (e.g., 10
dollars versus 1000 cents) systematically change the allocation
decision. We hypothesize, and defend, the explanation that the
numerical values bias the decision-maker’s perception of ``adequacy''
with respect to the quantity, consequently impacting the
decision-maker’s final allocation decision.  We have argued that this
bias is due to numerosity, which is the notion that individuals
tend to over-estimate quantity when it is represented with bigger
numbers. Thus, our work not only contributes to research studying
determinants of allocation behavior but also adds to the numerosity
bias literature which has garnered considerable research interest
recently. As a result of the numerosity bias, across all our
studies, we find that bigger numerical values – for example
representing an amount in ``cents'' – can bias individuals to
over-infer the amount, thus inducing them to allocate \textit{less} to
the entities they are \textit{focusing} on.

Past research has often used dictator games to study charity behavior
(e.g., Bekkers 2007; Engel \& Grosman 1996). In the context of
charitable giving, the Dictator is in the role of the benefactor and
the recipient takes the role of the charity. Thus the present work,
which demonstrates the impact of numerosity on allocation behavior in
dictator games, can have implications for charity behavior as well. In
this vein, our paper paves the way for further research on how
numerosity affects pro-social behavior which is currently understudied
(Bagchi \& Davis 2016).

Due to the use of dictator framework, our results can be used to
explain the high stakes dictator games conundrum (Novakova \& Flegr
2007). It has been shown in the research pertaining to dictator games
that individuals allocate less money when stakes are very high (Engel
2011). Researchers have asserted that this decrease in allocation with
an increase in stakes is because individuals have a lower sense of
fairness when the stakes are high. Our results suggest a more nuanced
explanation for the lower allocation when the stakes are high. The
studies (e.g., Novakova \& Flegr 2007) showing this effect of stakes
on allocation typically use the ``give to the other person'' paradigm
(just like our Studies 1 and 2). These studies use the same
denomination for different treatment groups and as a result, smaller
stakes and allocations are represented by smaller numbers (e.g., \$ 5)
and bigger stakes are represented by bigger numbers (e.g., \$
5000). While this is a straightforward way of manipulating the
magnitude of the stakes involved in dictator games (which the
researchers were interested in), it confounds numerosity with size of
stakes. For instance, if individuals decide to allocate thirty percent
of the total money to the other person, then, thirty percent of a
bigger stake can be over-estimated or over-inferred due to the
numerosity bias. Based on the results we have shown in the current
paper, one can argue that due to this misjudgment, individuals end up
making a lower allocation to other people when the stakes are high
(just as we see in the ``Cents'' condition of our Studies 1 and
2). One way to apply our work to this conundrum would be to control
for numerosity while looking at the impact of varying stakes on
allocation size.

We also feel it would be interesting to extend our approach to other
dimensions that can vary on numerosity, such as time (minutes versus
hours) or size (pounds versus ounces). Would a volunteer offer fewer
minutes of support than hours, or a friend a smaller portion of a
watermelon in ounces rather than pounds. If our research holds, it
might be smart to ask your friend for an allocation of his/her
delicious dessert as a quarter pound rather than 4 ounces.

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\end{hangparas}

\end{document}
