Everyone Wants (at Least) One Chance:
Experiments on Lotteries Assigning Initial
Positions
Gianluca Grimalda, Anirban Karyand Eugenio Protoz
November 2013
Abstract
We investigate experimentally the modi…cation of initial chances
to acquire the proposer’s position in ultimatum games. In the baseline
case players have equal opportunities to acquire the advantaged position. Chances become increasingly unequal across three treatments.
We …nd: (1) Receivers are signi…cantly more willing to accept a given
split when they are assigned a 1% chance of occupying the advantaged
position than when they have no chance. (2) The more unequal initial
chances, the lower the acceptance rates of a given split; consequently
inequality decreases. Players also respond to the way opportunities
are distributed across rounds, not just within rounds.
JEL Classi…cation: C92, C78, D63
Universitat Jaume I of Castelló, Spain. Centre for the Study of Global Cooperation
Research, University of Duisburg-Essen, Germany; Institute for the World Economy, Kiel,
Germany. Corresponding author: [email protected]
y
Delhi School of Economics, University of Delhi
z
University of Warwick
1
Keywords: Procedural fairness; Equality of Opportunity; Experiments.
1
Introduction
The idea that individuals are concerned not just with the outcomes of a certain social interaction, but also with the process leading to that outcome,
has gained increased consensus in …elds as disparate as law studies (Thibaut
and Walker, 1975), political philosophy (Rawls, 1999), and social choice (Elster, 1989). Although the characteristics that can make a process judged as
fair depend on the speci…c domain, the opportunity of having a voice - with
or even without the power to in‡uence the …nal outcome-, and the impartiality of the procedure, can both be considered as key aspects to achieve
procedural fairness (Tyler and Blader, 2003; Tyler, 2006; Krawczyk, 2011).
Procedural fairness has been contrasted with outcome fairness, which concerns the equality in …nal allocations (Sen, 1973; Trautmann and Wakker,
2010), or the degree to which …nal allocations reward individual contributions
(Leventhal, 1980).
Experimental evidence shows strong individuals’preferences for fair procedures and that individuals are willing to accept more unequal …nal allocations the fairer the procedures determining such allocations (see e.g. Bolton,
Brandts and Ockenfels, 2005; BBO henceforth). Survey evidence demonstrates that individuals who believe that fair opportunities are available to
advance in their lives, also demand less redistribution from their governments
(Corneo and Gruner, 2002; Fong et al., 2005; Alesina and La Ferrara, 2005).
Procedural fairness is also vital to many other aspects of economic decisions,
such as …rms’wage structure and workers’productivity (Bewley, 1999; Erkal
2
et al., 2010; Gill et al., 2012), and institutional mechanisms to allocate scarse
resource (Anand, 2001).
In this paper we extend the study of fairness in two directions that have
not been explored so far. First, we analyse whether assigning individuals
a minimal chance of achieving an advantaged role is enough to make individuals willing to accept substantially more inequality. This hypothesis is
motivated by Nozick’s (1994) claim that individuals may attach a disproportionate value to merely symbolic changes in procedural fairness. As a result, individuals’utility over the fairness of the procedure may not be linear.
Rather, it may su¤er a discontinuity in the origin of the space, when we move
from no opportunity to a minimal opportunity1 . This may also be related
to individuals’tendency to overweight small probabilities, as Kahneman and
Tversky (1979) famously argued. Several scholars stress the importance of
symbolic actions, such as the right to voice, in real life (see e.g. Hirschmann,
1970; Tyler, 2006). In this study we are able to study experimentally their
impact on the distribution of resources, and to quantify the e¢ ciency gains
that can be achieved with them.
Second, we focus on procedures determining the initial roles in a game.
All the experimental literature has thus far focused on procedures determining the …nal payo¤ allocations of a game. If we can show that individuals
are sensitive to fairness in the allocation of the game’s initial roles, the importance of procedural fairness will be further strengthened. We also study
whether variations in procedures a¤ecting the expected value of opportunities
over time may a¤ect the willingness to accept more outcome inequality. We
compare two conditions. In the …rst condition an individual is disadvantaged
1
There is a symbolic utility to us of certainty itself. The di¤ erence between probability
.9 and 1.0 is greater than between .8 and .9, though this di¤ erence between di¤ erences
disappears when each is embedded in larger otherwise identical probabilistic gambles– this
disappearance marks the di¤ erence as symbolic Nozick (1994, p. 34).
3
in any round of the interaction. In the second condition, the positions of advantage and disadvantage are reassigned at each round, making advantage
and disadvantage fair in expected value terms.
We take the Ultimatum Game (UG henceforth) as our basic interaction.
This game has been used extensively to examine individuals’assessment of
the fairness of payo¤ allocations. Interactions in the UG take place from
asymmetric positions. The proposer in a UG has a …rst-mover advantage
over the receiver in that she can dictate the shares of the …nal allocations.
This position of advantage is normally conducive to a larger share of the
payo¤s accruing to proposers, who on average obtain more than 60% of the
pie (see e.g. Oosterbeek et al. 2004). Guth and Tiez (1986) show that when
subjects are asked to bid on the two roles of a UG before bargaining, they
o¤er twice as much to occupy the proposer’s role as they do for the receiver’s
role. Arguably, the proposer’s role is more desirable than the receiver’s. For
this reason, a lottery giving one player higher chances of being assigned the
proposer’s role than another player, can conceivably be seen as not being
fully fair.
The main novelty of our experimental design is to make the access to
the two UG roles subject to a lottery, and to manipulate the distribution
of probability of these lotteries. The baseline case is that both players have
equal opportunities, as the lottery assigns both individuals a 50% chance of
acquiring the proposer role. In the other treatments, the initial lottery is
biased in favour of one of the two players. We consider three treatments
in which one of the two players is favoured with respect to the other in
that she has, respectively, 80%, 99%, and 100% probability of becoming
the proposer, while the unfavoured player only has the residual probability.
We call p (1
p) the probability that the unfavoured (favoured) player has
4
of becoming the proposer, where p = f0; 1%; 20%; 50%g. This is the key
parameter of our design. In this way we are able to assess the impact of
increasing disparity in the distribution of initial chances on the individual
assessment of the procedure’s fairness.
We run 20 interactions of the stage game with random rematching of subjects at each interaction. In the …xed position condition (FPC) an unfavoured
player keeps the same probability p of being assigned the proposer’s role in
each of the 20 rounds. In the variable position condition (VPC), an unbiased
lottery is run at the beginning of each round assigning players the favoured
and unfavoured positions. Consequently, the VPC provides an expectation
of equal opportunities across the whole 20 rounds of the experiment, while
chances are unequally distributed within each round. The FPC opportunity
assignment is unequal both within and across rounds. This enables us to
study the e¤ect of varying the distribution of opportunities over time.
The paper is organised as follows. Section 2 describes the literature and
presents the theoretical background for our study. We illustrate the main
hypotheses and the experimental protocol in section 3. Section 4 reports the
results. Sections 5 and 6 discuss the results and conclude the paper.
2
Theoretical framework
The experimental economics literature on procedural fairness began with the
pioneering study of BBO showing that procedural fairness is a substitute for
outcome fairness. Other studies replicated this result, showing that equality of opportunity is not a full substitute for equality of outcomes (see e.g.
Becker and Miller, 2009; Krawczyk and Le Lec, 2010). Furthermore, many
people are available to sacri…ce money to reject allocations that are brought
5
about by procedures that are extremely biased (BBO; Karni et al., 2008). In
other words, people seem to dislike procedures not guaranteeing a fair distribution of opportunities. We are the …rst to analyse the e¤ect of variation
in procedure assigning the initial role on the …nal surplus allocation and the
…rst to investigate the existence of a discontinuity between no opportunity
and a small, symbolic, probability of having an advantaged role.
At the theoretical level, individual preferences have traditionally been
held to be consequentialist (Hammond, 1988; Machina, 1989; Trautmann and
Wakker, 2010). These models posit that individual preferences only depend
on …nal outcomes, disregarding the process leading to such outcomes. Note
that consequentialist models allow for preferences being either purely selfinterested, or other-regarding, such as those modelled in Fehr and Schmidt’s
(1999) (FS henceforth), Bolton and Ockenfels’s (2000) (BO henceforth), and
Charness and Rabin (2002).
These models, however, fail to explain BBO’s experimental results. An
alternative route has since been taken, with the development of models of procedural fairness. The general idea is that individuals’preferences are assumed
to depend on the impartiality of the procedure determining …nal outcomes,
as well as on the standard self-interest motivation. The higher the fairness of
the process, the higher individuals’utility. Karni and Safra (2002) o¤er an
axiomatic account of individuals’sense of fairness. This builds on Diamond’s
(1967) idea that individuals prefer fair procedures to biased ones, even when
these lead to unequal outcomes. BBO extend the original BO model by de…ning the "fairest" available allocation in the game as the closest possible - in
expected value terms - to the equal divide. Individuals experience disutility
as the distance between the actual allocation and the fairest possible allocation grows. Trautmann (2009) uses the expected payo¤ di¤erence between
6
individuals as a proxy for the unfairness of the procedure. He then models
aversion to advantageous and disadvantageous expected inequality in a model
that is formally similar to the FS model. Krawczyk’s (2011), too, measures
procedural unfairness through expected payo¤ di¤erences, and combines it
with outcome inequality aversion a là BO. He posits an assumption of negative interdependence between the two motives. Accordingly, the higher the
unfairness of the procedure, the higher an individual’s desire for low actual
payo¤ inequality. In the next section we use a qualitative extension of these
models to make predictions for our experiments2 .
3
Experimental design
3.1
The Stage Game
The tree of the stage game is displayed in Figure 1. $10 are at stake in
every round. First, two randomly matched players are assigned the position
of either Player 1 or Player 2 through an even random draw. We call this
initial lottery L1. In the second phase, players are informed of the result of
L1 and make an o¤er to their counterpart. An o¤er is a proposal on how to
divide the $10 sum between the pair. Formally, player i’s o¤er is a division
(xi ; 10
xi ), where xi is the amount player i demands for herself and 10
xi
is the residual being o¤ered to the counterpart, i 2 f1; 2g. At this phase
players do not know their counterpart’s o¤er.
In the third phase, one of the two o¤ers is selected at random through a
lottery that we call L2. The key aspect of the design is that treatments di¤er
according to the probability with which Player 2’s o¤er (Player 1’s o¤er) is
2
In Grimalda et al. (2012) we provide an alternative formal treatment and a calibration
exercise.
7
randomly selected. This is given by the probability p (1 p). Such probability
has a maximum at p = 0:5 for Player 2 in the 50% treatment, it goes down
to p = 0:2 in the 20% treatment, it goes further down to p = 0:01 in the
1% treatment, and …nally reaches a minimum of p = 0 in the 0% treatment.
Player 1 always has a complementary probability to Player 2’s. Since in all
treatments apart from the 50% treatment, Player 2 (Player 1) always has a
lower (higher) probability of having her proposal being selected, we also call
such player unfavoured (favoured).
Finally, in the fourth phase, the player whose proposal has not been
selected has to decide whether she accepts or rejects the other player’s o¤er.
Suppose it is player i’s o¤er that is selected. Player i is informed that her
o¤er has been selected, but does not receive any information about player
j ’s o¤er. Conversely, xi is communicated to player j, who can either accept
or reject that o¤er. If player j accepts, payo¤s are xi and 1
xi for player i
and player j, respectively. If player j rejects, both players’payo¤ is 0.
INSERT FIGURE 1 ABOUT HERE
The key di¤erence between our extended UG and a standard UG is the
introduction of lottery L2, which randomly selects the o¤er that becomes
relevant for the …nal allocation. Note that players are always informed of the
lotteries outcomes. In particular, at the top node of the decision tree, people
are aware as to whether they are favoured or unfavoured at the moment of
submitting their proposal. In the 0% treatments, when p = 0, we dispensed
Player 2s from submitting an o¤er, as this would have no possibility of being
selected. Both Suleiman (1996) and Handgraaf et al. (1998) follow a similar
strategy in not asking players to perform an action when this has a 0%
probability of being relevant to the game. We discuss the implications of this
feature in section 5. After L2 has been run, the interaction becomes exactly
8
like a UG. The player whose proposal has (not) been selected becomes the
proposer (receiver), and payo¤s are determined as in standard UGs. All
random draws in L1 and L2 were made by the computer.
Subjects played the game described above anonymously for 20 rounds
with random re-matching at the beginning of each round. We varied the
way opportunities are assigned over time in the FPCs and VPCs. This is
represented in Figure 2.
INSERT FIGURE 2 ABOUT HERE
In FPCs L1 is run only once at the beginning of the experiment, and
then 20 L2s are run in each round. In VPCs, both L1 and L2 are run in
each round. Consequently, in FPCs a player remains unfavoured (favoured)
throughout the 20 rounds, while in VPCs each player has an even chance in
each round to be assigned the favoured or the unfavoured position3 .
Final payo¤s were given by the outcomes of two randomly-selected rounds
out of the 20: We opted for random payments to limit income e¤ects as the
play developed. We preferred to pay subjects for the outcomes of two rounds
instead of just one because we feared that a payment based on only one
round, coupled with the relatively low show-up fee (£ 5), may have discouraged receivers from rejecting unfair o¤ers. After each round each pair was
informed of the outcome of the interaction. No information about the outcome of the other pairs’interactions was instead released. Experiments were
conducted with a sample of 426 Warwick University undergraduate students,
using a between-subject approach. Supplementary details on the protocol,
and the experiment instructions are reported in the online Appendix.
3
Note that we use the term "roles" to indicate whether a participant is a proposer
or a receiver in the UG played in the last phase of the Stage Game (see Figure 1). We
refer to the term "position" to refer to whether a player is Player 1 (favoured) or Player
2 (unfavoured) in the lottery assigning UG roles - that is, L2 in Figure 1.
9
3.2
Our Hypotheses
From the strategic point of view, the introduction of L1 and L2 is irrelevant for a consequentialist player. Hence, self-interested proposers should
o¤er as low as possible, and self-interested receivers should accept any o¤er.
Consequentialist players motivated by social preferences will o¤er more than
the minimum and reject o¤ers below their minimal acceptable o¤ers, but
their behavior should remain constant across our treatments. Only processconcerned players should di¤erentiate their behavior across treatments.
The …rst hypothesis we want to test is Nozick’s (1994) proposition that
individuals are highly sensitive to the symbolic aspect of procedures. Accordingly, our conjecture is that the act of making an o¤er with only a 1% chance
of it being relevant, may symbolise, for the unfavoured player, expressive
value independently of the intrinsic expected utility coming from having this
option. This power of "voice" (Anand, 1991; see also section 5) may give the
individual what Nozick calls an "expressiveness", that is, a source of "value"
that goes beyond the mere utility associated with the act itself. Alternatively,
players may magnify the assignment of a small probability (Kahneman and
Tversky, 1979).
We thus posit a "Symbolic Opportunity Hypothesis":
H 1 : Receivers’ acceptance rate decreases signi…cantly in the 0% treatments in comparison to the 1% treatments.
Second, we want to test for the hypothesis that a given o¤er is more
acceptable when it has been generated within a game where players had
fairer initial chances. We assume that players’preferences are de…ned over
the procedures that determine certain consequences - in our case, the initial
positions in the game. Drawing on the existing literature (see the previous
10
section), it is natural to assume that agents will prefer, ceteris paribus, procedures providing players with a less biased distribution of opportunities. Here
we make the key assumption that players take the probability p as an index
of the fairness of the procedure. This seems a natural assumption because
p determines, in both the FPCs and the VPCs, the probability of attaining
the advantaged position in the stage game. The next step is to draw on
the interaction e¤ect proposed by Krawczyk (2011: 116). This assumes that
the lower procedural fairness, the higher the aversion to outcome inequality.
This entails that a responder in our game who is faced with a less fair initial
procedure will be more inclined to reject a given allocation.
This leads to our "Monotonic Fairness Hypothesis":
H 2 : The higher p, the higher receivers’acceptance rates for a given split.
We further hypothesise that VPCs can be deemed as more procedurally
fair than FPCs. This rests on the following statements, for which we provide
formal proof in the online Appendix, section 3. First, unfavoured players in
the FPC have strictly fewer expected opportunities than favoured players to
access the proposer role, whereas favoured and unfavoured players have the
same expected opportunities in the VPC. Second, VPC unfavoured players
enjoy strictly higher expected opportunities than FPC unfavoured players
involved in corresponding treatments. We de…ne corresponding treatments
as the pairs of an FPC and a VPC whose L2 is characterised by the same p.
There are three pairs of corresponding treatments, which we denote p_V P C
and p_F P C, p 2 P
f0%; 1%; 20%g. Third, if we take an ex ante perspec-
tive and look at the whole game before the start of the …rst round, even if
opportunities in the VPC and the FPC have the same expected value, never-
11
theless their variance is strictly higher in the FPC than in the VPC. In FPC
the outcome of the initial - and only - role assignment is extremely unequal
- i.e. one player is favoured (unfavoured) for all 20 rounds. Conversely, in
VPCs opportunities are much more evenly distributed, with the most likely
outcome being that a player is advantaged half of the times. Although we do
not develop this argument in this paper, it is quite plausible that risk-averse
individuals prefer the setting with lower variance - namely, the VPC - even
if this has the same expected value as the FPC.
Consistently with the considerations above, we expect that procedural
individuals will be sensitive to the procedural di¤erence between the FPC
and the VPC. We thus posit a "Dynamic Opportunities Hypothesis":
H 3 : For any corresponding treatment, receivers’acceptance rate decreases
signi…cantly in p_F P C as compared with p_V P C, p 2 P .
In this paper we focus on receivers’behavior. The analysis of proposers’
behavior is reported in the online Appendix. There we show that proposers’
patterns of behavior mirror receivers’ behavior. This is not surprising, especially in a context of repeated interactions, because it is very likely that
proposers conditioned their strategies upon the behavior they observed from
receivers to maximize their payo¤s. However, in the online Appendix we also
show that proposers were able to predict the di¤erences in receivers’behavior
across treatments in the …rst round of the game (see online Appendix, section
2). In the paper we only examine proposers’behavior descriptively to assess
the overall inequality of the interactions across the various treatments.
12
4
Results
4.1
4.1.1
Results for Fixed Position Conditions
Descriptive Analysis
Table 1 reports descriptive statistics for proposers and receivers’behavior in
each treatment. First, we note that overall acceptance rate in the 0%_FPC is
77:58%, while the mean proposers’demand is equal to 62:8%. This is largely
in line with other UGs4 . The 0%_FPC is the treatment in our experiment
that is closest to standard UGs, so we have some assurance that our results
are not due to speci…c idiosyncrasies of our sample. Comparing the 50%
treatment (Table 1a) and the FPCs (Tables 1 b-d) brings out the existence
of a monotonic pattern consistent with H 1 and H 2 . As the bias of the initial
lottery increases, both the mean and the median values of rejected demands
decrease (see Tables 1a-d, Columns 1). This means that as the initial lottery
becomes more biased, receivers request larger shares of the pie to accept an
o¤er.
INSERT TABLE 1 ABOUT HERE
Second, the acceptance rates of low o¤ers decreases as the bias of the
initial lottery increases (see Tables 1a-d, Columns 3). Consistently with much
of the literature, we consider an o¤er as "low" when the proposer o¤ers 20%
or less to the receiver. The drop in the acceptance rate for high demands is
particularly pronounced between 1%_FPC and 0%_FPC, consistently with
H 1 . This monotonic pattern does not emerge in overall acceptance rate, but
this is likely due to the variation in the magnitudes of o¤ers across treatments.
The econometric analysis of the next section controls for this aspect. Similar
4
In their meta-analysis, Oosterbeek et al. (2004) report that the weighted average
acceptance rate from 66 UG studies is 84:25%, whereas average demands equal 59:5% of
the pie in 75 UG experiments.
13
patterns are found for proposers’ behavior. As the initial lottery becomes
more biased, both the mean and median o¤ers of favoured proposers follow
a decreasing pattern (see Tables 1a-d, Columns 4). The frequency of high
demands decreases with the bias of the initial lottery. As far as unfavoured
proposals are concerned, the same monotonic pattern observed for favoured
proposers emerges between 20%_FPC and 1%_FPC, as o¤ers are higher in
20%_FPC than 1%_FPC (see Tables 1b-c, Columns 5).
Figure 1 in the online Appendix o¤ers a graphical representation of receivers and proposers’behavior in each treatment by reporting histograms of
demands as well as acceptance rates for di¤erent classes of demands. Acceptance rates tend to decrease within each class as the initial lottery becomes
more biased. The distribution of demands tends to become more skewed
towards the left as the chances of the unfavoured player decrease.
4.1.2
Econometric Analysis
We pool all observations coming from FPCs and the 50% treatment together.
We model the repeated nature of the data through a random-e¤ects model.
This is a common method to analyse experimental data coming from repeated interactions (see e.g. Armantier, 2006). However, it is plausible that
as interactions went on, subjects updated their beliefs over receivers’minimum acceptable o¤er on the basis of the feedback they received, and modi…ed
their o¤ers accordingly. If there is such a learning e¤ect, it is interesting to
investigate the …rst round separately. However, our results of the …rst round
alone do not deviate substantially from our overall results (see online Appendix, section 2). Hence, here (as well as in the VPC) we report only the
analysis of pooled observations.
Given the dichotomic nature of the receiver’s variable, we …t the following
14
logit models:
ACCEP T AN CEi;t =
i
+ F AV OU REDi;t +
ACCEP T AN CEi;t =
i
(1)
+ CHAN CE + OF F ERi;t +
+
k ROU N Dk
j T REAT M EN Tj
+ F AV OU REDi;t +
i Zi
+ ui + "i;t
+ OF F ERi;t +
(2)
k ROU N Dk
+
+
i Zi
+ ui + "i;t
The dependent variable is the dichotomic variable ACCEP T AN CE where 1 (0) denotes acceptance (rejection) of a given o¤er. In model 1,
CHANCE takes the value of p (see section 1). CHANCE thus o¤ers a continuous measure of the bias in the initial lottery. We control for the size of the
o¤er assigned to the receiver through the variable OFFER. The dummy variable FAVOURED identi…es whether a subject has been assigned the favoured
role. Thus this dummy captures the di¤erence in behavior between favoured
subjects and unfavoured subjects when drawn as receivers in the game5 . Obviously this only applies to the 1% and the 20% treatments. Subjects initially drawn as favoured may have formed higher earnings expectations for
that particular round than if they were drawn as unfavoured, thus increasing
the likelihood of rejection. On the other hand, favoured players may have
taken into account that they were overall more likely to occupy the proposer
role, thus being able to aspire to higher payo¤s across the experiment. Such
fairness concerns may make them more lenient in accepting o¤ers when occupying the receiver role. ROU N Dk ; k = f2:::20g dummies are also included
5
Since in FPCTs a player kept the favored role throughout the 20 rounds of interaction,
the sub-index t is in this case redundant.
15
to control for time trend e¤ects or e¤ects associated with speci…c rounds. Finally, ui and "i;t are individual-speci…c and observation-speci…c error terms.
The indexes i and t denote the individual and the round of the interaction,
respectively. Note that all regressors are exogenous. Hence, the individualspeci…c e¤ects
i
must be uncorrelated with the other regressors, thus en-
suring that a between estimator is consistent. To check for the robustness
of our results, we also consider another speci…cation where some individual
demographic characteristics are included in the model through the vector Zi :
The inclusion of Zi entails a considerable loss of observations, so we present
both the results with and without Zi : Model (2) keeps the same speci…cation
as model (1) apart from replacing CHANCE with dummy variables identifying individual treatments - named T REAT M EN Tj ; j = f0%; 1%; 20%g.
The 50% treatment is the baseline. This enables us to study the di¤erential
e¤ects of pairs of treatments on propensity to accept, thus testing directly
H 1 as well as performing a more stringent test of H 2 .
Regression results are reported in Table 2. CHANCE has a strong and
positive e¤ect (P < 0:001), thus supporting H 2 (Table 2, Column 1). Receivers were more likely to accept o¤ers when these came after a less unbiased initial lottery. As expected, OF F ER has a positive and strong e¤ect
(P < 0:001). F AV OU RED also has a positive sign (P = 0:037). Favoured
subjects were more likely to accept o¤ers when drawn as receivers than unfavoured subjects. This gives support to the idea that their motivations may
be a¤ected by the assessment of overall fairness across the whole interaction
(see above). These results are virtually unchanged even after controlling for
individual characteristics (Table 2, Column 2). Probability of acceptance was
signi…cantly higher for students attending Economics degrees (P = 0:026),
women (P = 0:040), and UK students (P = 0:029).
16
Figure 3a reports the probabilities of acceptance in each treatment for
various o¤ers, as predicted by model (2), omitting Zi (see Table 2, Column
3). The diagram shows that for any o¤er, as the bias of the initial lottery
decreases, the probability of acceptance increases. Probabilities are close to
0 (1) for the lowest (highest) o¤er considered. For the other intermediate
o¤er values, sizable di¤erences emerge across treatments. For instance, for
o¤ers equal to 20% of the pie, the predicted probability of acceptance is equal
to 0:92 in the baseline case, drops to 0:85 in the 20%_FPC and to 0:69 in
the 1%_F P C, and drops to a mere 0:17 in the 0%_FPC. Di¤erences appear to be particularly pronounced comparing the 0%_F P C vis-à-vis other
treatments, but are considerable in other treatments, too.
INSERT FIGURE 3 ABOUT HERE
Results of two-tailed Wald tests over treatment di¤erences are reported in
Table 4a. The null hypothesis is H0 :
k
l
= 0 against H1 :
k
l
6= 0, for
each pair of T REAT M EN T coe¢ cients. Note that a positive (negative) sign
for the z-statistic means that the probability of acceptance was higher (lower)
in treatment k - entered in the Table rows - than in treatment l - entered
in the Table columns. Table 4a supports hypothesis H 1 of a symbolic value
of opportunity. The di¤erence between
0%_F P C
and
1%_F P C
is negative
and signi…cant (P = 0:017). Receivers in 1%_F P C had, ceteris paribus,
a signi…cantly higher probability of accepting a given o¤er than receivers in
the 0%_F P C: Hence, an assignment of even minimal opportunities seems to
matter a great deal for FPC receivers. All the signs in Table 4a are negative
and thus in line with H 2 . Pairwise comparisons are signi…cant in four out
of six cases. Results are virtually identical when demographic controls are
introduced in the regression (Table 2, column 4).
On the basis of this analysis, we conclude:
17
Conclusion 1 Descriptive and econometric analysis supports H1 in the FPC.
Conclusion 2 Descriptive and econometric analysis supports H2 in the FPC.
4.2
4.2.1
Results for Variable Position Conditions
Descriptive Analysis
Table 1 shows that the monotonic pattern linking bias in the initial lottery
and rejection rates still holds moving from 0%_VPC to 20%_VPC, but is
reversed between 20%_VPC and 50%. Looking at the mean and median
values of rejected demands, we note that receivers’ hostility decreases between 0%_VPC up to 20%_VPC, but it then rises again (see Tables 1a,e-g,
Columns 1). A similar trend can be detected with respect to the acceptance
rate of low o¤ers (see Tables 1a, e-g, Columns 3), as well as for mean and
median o¤ers (see Tables 1a,e-g, Columns 4). As far as being favoured in
the lottery is concerned, a striking di¤erence between VPCs and FPCs is
that unfavoured proposers demand less (more) than favoured proposers in
the former (latter) set of treatments (see Tables 1b-c, Column 5). This is the
case for both 20%_VPC and 1%_VPC 6 . Figure 1 in the online Appendix
further con…rms these patterns.
4.2.2
Econometric Analysis
We …t models (1) and (2) to analyse receivers’behavior in VPCs (see section
4.1.2). As far as H 1 is concerned, the Wald test rejects the null hypothesis that treatment dummies are equal in 0%_VPC and 1%_VPC, albeit at
weak signi…cance levels (P = 0:063). As for H 2 , the variable CHAN CE is
6
According to Mann-Whitney tests, the di¤erence between o¤ers by favored and unfavored players is statistically signi…cant in both 20%_FRC (P < 0:001) and 20%_VRC
(P = 0:06). See also Table 1 and 2 in the SOM.
18
no longer signi…cant (Table 3, column 1). However, after adding a squared
term to CHAN CE both the linear and the quadratic term have signi…cant
e¤ects (Table 3, column 3). The probability of acceptance shows an invertedU pattern, reaching a maximum for CHAN CE = 0:27. Even in this case,
these results are robust to the introduction of demographic controls. Women
are again more likely to accept o¤ers (P = 0:048), as well as Economic students (P = 0:040), while UK citizenship is no longer signi…cant (Table 3,
column 2 and 4). Pairwise comparisons of treatment coe¢ cient di¤erences
con…rm the existence of a non-linearity in how receivers reacted to variations
in p (see Table 4b):All the signs of the z-statistics are negative and statistically signi…cant, limitedly to the three treatments 0%_VPC, 1%_VPC, and
20%_VPC.
Figure 3b depicts the predicted probability of acceptance based on model
2, omitting Zi (see Table 3, column 5). The three treatments 0%_V P C;
1%_V P C; 20%_V P C follow a monotonic trend. For instance, for o¤ers
equal to 15% of the pie, the predicted probability of acceptance is equal to
0:45 in the 0%_VPCT, it rises to 0:74 in the 1%_V P C, and to 0:90 in
20%_V P C: However, the probability of acceptance drops to 0:72 in 50%.
We conclude:
Conclusion 3 Descriptive and econometric analysis weakly supports H1 .
Conclusion 4 Descriptive and econometric analysis supports H2 in the VPC
only limitedly to 0%_V P C through 20%_VPCs. The monotonic pattern
breaks between 20%_V P C and 50%:
19
4.3
Comparing the VPC and the FPC
First, we note that descriptive statistics from Table 1 support H3 . For each
pair of corresponding treatments (see section 3), the mean and median value
of rejected demands, and the acceptance rate of high demands, are all lower
in FPCs than VPCs. Second, we …t the econometric model (2) to the pooled
dataset (see online Appendix: Table 9). Table 5 reports the results of Wald
tests conducted over pairs of coe¢ cient di¤erences. Acceptance rates are
ceteris paribus signi…cantly lower in FPCs than in VPCs in all corresponding treatments. The di¤erence is highly signi…cant between 0%_FPC visà-vis 0%_VPC (P = 0:002), and signi…cant between 20%_FPC vis-à-vis
20%_VPC (P = 0:012), and 1%_FPC vis-à-vis 1%_VPC (P = 0:033).
O¤ers follow the same pattern (See online Appendix: Table 10). We thus
conclude:
Conclusion 5 Descriptive and econometric analyses support H3 .
As far as e¢ ciency is concerned, Figure 4a shows that this is generally
higher in VPCs than in other treatments. The overall acceptance rates is the
inverse of the output gone lost because of the "con‡ict" between receivers
and proposers. The two treatments where losses were lowest were 20%_VPC
and 0%_VPC, with an overall acceptance rate of 85%. 20%_FPC comes
third, and 50% is only fourth in this ranking, with an acceptance rate of 81%:
The treatments with highest e¢ ciency losses were 1%_FPC and 0%_FPC.
The same pattern occurs in the last …ve rounds of the game (See Figure 4b).
5
Discussion
Our results con…rm and extend previous results that individuals are sensitive
to procedures leading to outcomes, rather than just outcomes. Our compre20
hensive study has enabled us to uncover some speci…c characteristics of such
preferences. It is striking that most of the observed variation in behavior
takes place as we move from 0% treatments to 1% treatments. When L2
is unbiased, receivers reject on average o¤ers of £ 2.15, and when L2 gives
people no chances of being a proposer in the F P C, receivers reject on average o¤ers of $2:96. Put it in a di¤erent way, receivers would be available to
pay on average 81p - the di¤erence between $2:96 and $2:15 - to be in the
50% treatment rather than being in the 0%_F P C. By the same token, receivers would be available to pay 43p to have a 1% chance of being proposers
compared to none, and only 38p more to have equal chances compared to a
1% chance. In other words, the "demand for opportunity" seems to be very
steep near the origin of the scale, but considerably less so afterwards.
Such a result may be due to the purely procedural aspect of having a say
in the collective decision problem, or to the actual allocation of a 1% chance of
acquiring the advantaged position, or to a combination of both. Our current
design does not enable us to discriminate between these two interpretations,
because 0% treatments di¤er under both respects to 1% treatments. Even
so, we believe it is important to have uncovered such a sizable response
to marginal procedural changes. Future research could easily ascertain the
relative importance of the purely symbolic value of "voice" compared with
the allocation of a 1% chance of acquiring the advantaged position.
Our results call for the need to re…ne existing theoretical models of procedural fairness. The use of expected payo¤s di¤erences between players
as a proxy for procedural fairness make the predictions of both Trautmann
(2009) and Kracwzyk’s (2011) models not appropriate for lotteries applied to
initial positions7 . Karni and Safra’s (2002) model does not su¤er from this
7
In our experiments, the average expected payo¤s for receivers in the last …ve rounds
- seemingly an appropriate measure for "equilibrium" payo¤s - are highest ($3:16) in the
21
problem because preferences are de…ned directly over procedures. However
their "hexagonal condition" linking the strength of the fairness motivation
with that of the self-interested motivation is not speci…ed. In our setting the
variable p is a natural way to measure "how fair" the procedure is. In other
contexts such a clear-cut proxy for procedural fairness may not exist. Alternatives to expected payo¤s, such as the ex ante willingness to pay to enter
the game in a certain position (Stefan Trautmann, private communication),
may in these cases be considered.
The break of monotonicity we observe in VPCs (see section 4.2) is undoubtedly surprising. This is associated with each VPC treatment having
lower con‡ictuality rates - and thus greater e¢ ciency - than the baseline
case of equal opportunities. A possible explanation is that VPCs made more
salient to subjects the possibility of achieving some form of fairness, albeit
over the whole 20 rounds of interaction rather than within each round, thus
inducing subjects to become more lenient over proposed allocations. This
may be due to the establishment of a "convention", legitimizing favoured
players to demand larger shares of the pie than what we observe in the 50%
treatment. A "convention" has been de…ned as a situation in which players use an exogenously given characteristic of an interaction - such as the
random assignment to one of two colours - to solve a coordination problem
(Hargreaves-Heap and Vaourofakis, 2002). In our case, players may have
used the assignment to the favoured role in the random draw as a characteristic enabling them to demand a larger share of the pie - thus acting more
"hawkishly" - whereas players being assigned the unfavoured role accepted
0%_FPC, which is arguably the most unfair procedure of our experiments. The only
unbiased procedure of our experiments, i.e. the baseline 50% condition, only yields $2:47
to receivers and comes …fth in the ranking of expected receivers’payo¤s across treatments.
In our case "equilibrium" expected payo¤ di¤erences are thus a very imperfect proxy for
procedural fairness.
22
with higher frequency such demands - thus acting more "dove-like" - in comparison with FPCs. The behavior of unfavoured VPC (FPC) proposers, who
demand signi…cantly less (more) than their favoured counterparts, seems to
be consistent with this conjecture (see section 4.2.1). The absence of any
role salience in the 50% treatment may have prevented the emergence of any
convention.
6
Conclusions
The main novelty of our study has been the analysis of the discontinuity
between no opportunity and 1% probability of acquiring the proposer role,
and the introduction of lotteries over the assignment of initial positions in
UGs, rather than over …nal allocations.
First, we …nd clear support for the Symbolic Opportunity Hypothesis. In
both FPCs and VPCs, receivers act signi…cantly more leniently after having
been previously assigned a mere 1% initial chance of acting as proposers
compared to having no chance. Handgraaf et al. (2004) found signi…cant
variations in proposers’behavior when receivers had 10% more “power” in
reducing proposers’payo¤s (see also Suleiman, 1996). However, power had
in their setting a direct bearing on …nal payo¤s, thus their result cannot be
ascribed to procedural fairness per se. We thus believe to be the …rst to give
experimental support to the Symbolic Opportunity Hypothesis.
Our study validates experimentally other empirical and survey evidence
regarding the importance of "voice" for people. Frey and Stutzer (2005)
…nd support for the thesis that the mere right to participate in the political
process - rather than actual participation - increases individual satisfaction
- a phenomenon they refer to as "procedural utility". Anand (2001) reports
23
survey evidence supporting the importance people place on having the right
to have their opinion heard - or appropriately represented - in collective
decision processes. The relevance of this right to voice may be caused by the
desire to express one’s position, or to obtain respect for one’s worth.
Second we …nd robust support for the Monotonic Fairness Hypothesis in
FPCs. The greater the inequality in the distribution of initial opportunities,
the lower the acceptance rates of a given o¤er. Consequently, average o¤ers
increase. This pattern of behavior reproduces the insights coming from survey analyses. These stress that the more a society is deemed as granting fair
opportunities to their citizens, the lower the demand for redistribution (see
section 1).
Third, we …nd support for the Dynamic Opportunities Hypothesis. Acceptance rates are signi…cantly higher in VPCs than FPCs, and as a consequence, proposers’demands are also higher. As argued in section 3, this is
consistent with our claim that subjects see VPCs as a fairer procedure by
which to allocate initial opportunities. It appears that players are prepared
to accept even extreme levels of opportunity inequality within each round,
in exchange for overall equality of opportunity across the whole series of
interactions.
Several open questions remain. Granting full equality of opportunity does
not produce the most e¢ cient outcome, in terms of reduction of con‡ictuality rates. Rather, the settings where opportunities are distributed unfairly
within each round, but are fairly allocated in expected terms across rounds,
turn out as being the most e¢ cient. This observation, if generalized, may
have potentially important implications for the design of policies a¤ecting initial opportunities, as well as a¢ rmative action policies. Furthermore, more
research needs to be done to understand the external validity of the Sym-
24
bolic Opportunities Hypothesis. Clearly, many people who feel marginalized
in societies will believe that they are deprived of even a negligible chance to
succeed. Our research points to the need to understand better the point at
which the "playing …eld" is judged to be su¢ ciently "level" by individuals.
Our research shows that subjective perceptions may radically di¤er from the
objective distribution of chances.
References
[1] Alesina, A. & La Ferrara, E. (2005). ”Preferences for redistribution in
the land of opportunities,” Journal of Public Economics, 89(5-6), 897931.
[2] Anand, P. (2001). "Procedural fairness in economic and social choice:
Evidence from a survey of voters," Journal of Economic Psychology,
22(2), 247-270.
[3] Armantier, O. (2006). "Do Wealth Di¤erences A¤ect Fairness Considerations?," International Economic Review, 47(2), 391-429.
[4] Becker, A., Miller, L. (2009). "Promoting justice by treating people unequally: an experimental study," Experimental Economics, 12(4), 437449.
[5] Bewley, T. (1999). Why Wages don’t Fall during a Recession. Harvard:
Harvard Univ. Press.
[6] Bolton, G. E., Brandts, J.,Ockenfels, A. (2005). ”Fair Procedures: Evidence from Games Involving Lotteries” . Economic Journal, 115(506),
1054-1076.
25
[7] Bolton, G., Ockenfels, A. (2000), A theory of equity, reciprocity and
competition, American Economic Review, 90, 166-93.
[8] Charness, G., Rabin. M. (2002). Understanding social preferences with
simple tests, Quarterly Journal of Economics, 117, 817–869.
[9] Corneo, G., Gruner, H. M. (2002). Individual preferences for political
redistribution, Journal of Public Economics, 83, 83–107.
[10] Diamond, P. (1967), "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: A Comment", Journal of Political
Economy, 75(5), 765-766.
[11] Elster, J. (1989). Solomonic judgements: Studies in the limitation of
rationality. Cambridge University Press.
[12] Erkal, N., & Gangadharan, L. Nikos Nikiforakis (2010). “Relative Earnings and Giving in a Real-E¤ort Experiment.”. American Economic Review, 101(7), 3330-48.
[13] Fehr E.,Schmidt, K. (1999). A theory of fairness, competition and cooperation, Quarterly Journal of Economics, 114, 817-68.
[14] Fong, C., Bowles, S., Gintis, H. (2005). “Strong Reciprocity and the Welfare State.” in: Gintis, H., Bowles, S., Boyd, R., Fehr, E. (eds). Moral
Sentiments and Material Interests: The Foundations of Cooperation in
Economic Life, Cambridge, MA: The MIT Press.
[15] Frey, B., Stutzer, A. (2005). Beyond outcomes: measuring procedural
utility. Oxford Economic Papers 57: 90-111.
26
[16] Gill, D., Prowse, V., & Vlassopoulos, M. (2012). Cheating in the workplace: An experimental study of the impact of bonuses and productivity.
Available at SSRN 2109698.
[17] Grimalda, G., Kar, A., Proto, E. (2012). Everyone Wants a Chance:
Initial Positions and Fairness in Ultimatum Games, Warwick Economics
Research Papers No 989.
[18] Guth, W., Tietz, R. (1986). Auctioning ulitmatum bargaining
positions— how to act rational if decisions are unacceptable? In R.W.
Scholz (ed.), "Current Issues in West German Decision Research",
Frankfurt: Verlag Peter Lang, 173–185.
[19] Hammond, P. (1988). Consequentialist foundations for expected utility,
Theory and Decision, 25 (1), 25-78.
[20] Handgraaf, M., Van Dijk, E., Vermunt, R., Wilke, H., De Dreu, C.
(2008). Less power or powerless? Egocentric empathy gaps and the
irony of having little versus no power in social decision making, Journal
of Personality and Social Psychology 2008. 95 (5), 1136-49.
[21] Hargreaves Heap, S., Varoufakis, Y. (2002). Some Experimental Evidence on the Evolution of Discrimination, Co-operation and Perceptions
of Fairness, Economic Journal, 112, 679-703.
[22] Hirschmann, A. O. (1970). Exit, voice, and loyalty: Responses to decline
in …rms, organizations, and states (Vol. 25). Harvard university press.
[23] Kahneman, D. and A. Tversky (1979) "Prospect Theory: An Analysis
of Decision under Risk", Econometrica, Vol. 47, No. 2. (Mar., 1979), pp.
263-292.
27
[24] Karni, E., Salmon, T., Sopher, B. (2008). "Individual sense of fairness:
an experimental study," Experimental Economics, 11(2), 174-189.
[25] Karni, E., Safra, Z. (2002). Individual sense of justice: a utility representation. Econometrica, 70, 263–284.
[26] Krawczyk, M. (2010). A glimpse through the veil of ignorance: Equality
of opportunity and support for redistribution, Journal of Public Economics, 94, 131-141.
[27] Krawczyk, M. and F. Le Lec (2010). Give me a chance! An experiment
in social decision under risk, Journal of Experimental Economics, 13,
500–511.
[28] Leventhal, G. S. (1980). What should be done with equity theory? (pp.
27-55), Springer US.
[29] Machina, M. (1989). Dynamic Consistency and Non-Expected Utility
Models of Choice Under Uncertainty, Journal of Economic Literature,
27 (4), 1622-1668.
[30] Nozick, R. (1994). “The nature of rationality”, Princeton: Princeton
University Press.
[31] Oosterbeek, H., Sloof, R., van de Kuilen, G. (2004). Cultural Di¤erences
in Ultimatum Game Experiments: Evidence from a Meta-Analysis, Experimental Economics, 7 (2), 171-188.
[32] Rawls, J. (1999). A Theory of Justice, rev. ed. Cambridge, MA: Belknap,
5.
[33] Suleiman, R. (1996). Expectations and fairness in a modi…ed UG, Journal of Economic Psychology, 17, 531–554.
28
[34] Thibaut, J. W., & Walker, L. (1975). Procedural justice: A psychological
analysis. Hillsdale: L. Erlbaum Associates.
[35] Trautmann, S. and P. Wakker (2010). "Process fairness and dynamic
consistency," Economics Letters, 109(3), 187-189.
[36] Trautmann, S.T. (2009). A tractable model of process fairness under
risk. Journal of Economic Psychology, 30(5), 803-813.
[37] Tyler, T. R. (2006). Why people obey the law. Princeton University
Press.
[38] Tyler, T. R., & Blader, S. L. (2003). The group engagement model:
Procedural justice, social identity, and cooperative behavior. Personality
and social psychology review, 7(4), 349-361.
A
Acknowledgements
We thank Iwan Barankay, Dirk Engelmann, Enrique Fatás, Peter Hammond,
Andrew Oswald, Elke Renner, Blanca Rodriguez, Tim Salmon, Stefan Traub,
Stefan Trautmann for useful discussion, participants in the Workshop on
"Procedural fairness - theory and evidence", Max Planck Institute for Economics (Jena), the 2008 IMEBE conference, the 2008 European ESA conference, the 2010 Seminar on "Reason and Fairness", Granada, and seminar
participants at Nottingham, Royal Holloway, Trento, Warwick. We especially
thank Malena Digiuni for excellent research assistance. The usual disclaimers
apply. This project was …nanced by the University of Warwick RDF grant
RD0616. Gianluca Grimalda acknowledges …nancial support from the grant
ECO 2011-23634 by the Spanish Ministry of Science and Innovation and
P1 1A2010-17 from the Universitat Jaume I.
29
FIGURE
RES/TABLES
S PAPER
Figure 1: Game ttree of the basic
b
intera
action
ure 2: Experim
ment Interaction Dynam
mics
Figu
Rou
und 1 Rou
und k,
Round
R
20
k={{2, 19}
FPC
F
L1
1+S
S
S
VPC
V
L1
1+S
L
L1+S
L1+S
Table 1: Descriptive statistics of receivers and proposers’ behavior per treatment
Table 1a: 50%
Responses
Mean
St. Dev
Median
Obs
(1)
RD
7.85
0.85
8
115
Demands
(2)
AR (All)
81.45%
0.39
(3)
AR (Low)
53.7%
0.50
620
134
Table 1b:FPC 20%
Responses
Mean
St. Dev
Median
Obs
(1)
RD
7.61
0.75
7.6
99
(4)
FAV
6.97
1.07
7.00
1240
(5)
UNF
Table 1e: VPC 20%
Demands
(2)
AR (All)
84.03%
0.37
(3)
AR (Low)
52.7%
0.50
620
91
(4)
FAV
6.89
0.82
7.00
620
(5)
UNF
7.17
1.09
7.20
620
Responses
Mean
St. Dev
Median
Obs
(1)
RD
8.39
0.75
8.4
90
Table 1c:FPC 1%
Responses
Mean
St. Dev
Median
Obs
(1)
RD
7.47
0.71
7.5
135
(3)
AR (Low)
46.9%
0 .50
640
66
Mean
St. Dev
Median
Obs
(1)
RD
7.04
0.71
7
139
(4)
FAV
6.77
0.88
6.99
640
(5)
UNF
6.62
1.77
6.75
640
Mean
St. Dev
Median
Obs
(1)
RD
7.97
0.78
8
106
(3)
AR (Low)
21.7%
0.42
620
23
600
225
(4)
FAV
7.37
0.98
7.50
600
(5)
UNF
7.13
1.40
7.50
600
Demands
(2)
AR (All)
81.07%
0.39
(3)
AR (Low)
51.4%
0 .50
560
134
(4)
FAV
7.20
0.90
7.00
560
(5)
UNF
6.87
1.79
7.00
560
Table 1g: VPC 0%
Demands
(2)
AR (All)
77.58%
0.42
(3)
AR (Low)
66.6%
0 .47
Responses
Table 1d:FPC 0%
Responses
(2)
AR (All)
85%
0.36
Table 1f: VPC 1%
Demands
(2)
AR (All)
78.91%
0.41
Demands
(4)
FAV
6.28
0.91
6.17
620
Responses
(5)
UNF
Mean
St. Dev
Median
Obs
(1)
RD
7.56
0.91
7.33
90
Demands
(2)
AR (All)
85%
0.36
(3)
AR (Low)
47.1%
0.50
600
70
(4)
FAV
6.56
1.07
6.50
600
(5)
UNF
Note: RD= Rejected demands; AR (All) =Acceptance Rate with respect to all offers; AR (Low) =Acceptance Rate with respect to
low offers (less or equal to 20% of the pie); FAV=FAVOURED; UNF=UNFAVOURED.
Table 2: Regression Analysis of Logit model for probability of acceptance – 50% treatment & FPC treatments
DEP VAR
CHANCE
ACCEPT
(1)
(2)
5.872***
(1.552)
6.695***
(1.878)
20%_FPC
(3)
(4)
-0.782
(0.898)
-1.735**
(0.882)
-4.114***
(0.919)
3.438***
(0.214)
1.061
(1.062)
-7.422***
(0.831)
YES
3.525***
(0.237)
1.666*
(0.934)
1.757**
(0.786)
-0.0654
(0.203)
1.512**
(0.737)
1.646**
(0.752)
118.8
(402.5)
YES
-4.751***
(0.779)
YES
-1.438
(1.025)
-1.788*
(1.003)
-4.819***
(1.068)
3.545***
(0.238)
1.173
(1.069)
1.858**
(0.765)
-0.0405
(0.196)
1.438**
(0.717)
1.559**
(0.726)
72.50
(388.7)
YES
Observations
Number of individuals
2,500
189
2,165
159
2,500
189
2,165
159
Chi2
264.7
227.5
265.8
228.8
1%_FPC
0%_FPC
OFFER
FAVOURED
3.425***
(0.214)
1.870**
(0.897)
ECONOMICS
YEAR
GENDER
UK
Constant
ROUND DUMMIES
Percentage of correct predicted outcomes
81.9%
82.7%
81.7%
82.5%
Notes: Dependent variable equals 1 if accepted, 0 if rejected (see Table 1 for descriptive statistics). Numbers in parentheses are
standard errors. Round dummies have been included in all regressions. Stars denote significance levels as follows: * = Pvalue<0.1; ** = P-value<0.05; *** = P-value<0.01. Predicted outcomes are computed from the model predicted probability of
acceptance by assigning a predicted outcome of acceptance (rejection) whenever the predicted probability is greater (smaller or
equal) to 0.5. So a predicted outcome is correct when it matches the actual decision of the subject, i.e. when the subject accepted
(rejected) an offer and the model predicted a probability greater (smaller or equal) than 0.5.
Table 3: Regression Analysis of Logit model for probability of acceptance – 50% treatment & VPC treatments
DEP VAR CHANCE ACCEPT (1) 1.245 (1.199) (2) 2.615* (1.587) CHANCE SQUARED (3) 15.11*** (4.837) ‐27.85*** (9.391) (4) 14.33** (6.787) ‐23.83* (13.37) 20%_VPC FAVOURED ECO YEAR GENDER UK Constant ROUND DUMMIES Observations N_g chi2 Percentage of correctly predicted outcomes Note: See Table 2.
‐4.839*** (0.662) YES 2380 238 236.1 84.2% 1.274* (0.654) 0.108 (0.654) ‐1.134* (0.657) 2.880*** (0.189) 0.0746 (0.454) ‐5.378*** (0.688) YES
3.049*** (0.252) 0.248 (0.672) 1.401** (0.658) 0.279 (0.230) 1.185* (0.651) 0.132 (0.638) ‐561.6 (456.7) YES
‐4.836*** (0.716) YES
1610 161 153.1 2380 238 239.8 1610 161 154.3 2380 238 241.3 1610 161 154.9 85.6% 84.6% 86% 84.3% 85.7% 0%_VPC 2.845*** (0.188) 0.263 (0.443) (6) 0.691 (0.942) ‐0.776 (0.844) ‐1.589* (0.902) 3.061*** (0.253) 0.250 (0.672) 1.400** (0.654) 0.257 (0.230) 1.127* (0.650) 0.160 (0.635) ‐517.7 (456.4) YES
1%_VPC OFFER (5) 3.043*** (0.253) 0.402 (0.659) 1.365** (0.666) 0.368 (0.229) 1.301** (0.659) 0.124 (0.648) ‐738.8 (454.8) YES
2.861*** (0.188) 0.0704 (0.454) Figure 3: Predicted probability of acceptance applied to 50% and FPCs (Panel a) and 50% and VPCs (Panel b)
Figure 3a: 50% and FPCs
Figure 3b: 50% and VPCs
1.2
1.2
1
1
0.8
50%
0.8
50%
0.6
FRC_20%
0.6
VRC_20%
0.4
FRC_1%
0.4
VRC_1%
0.2
FRC_0%
0.2
VRC_0%
0
0
0.5 1 1.5 2 2.5 3 3.5
0.5
1
1.5
2
2.5
3
3.5
Note: Predicted probabilities for Figure 3a (3b) have been derived from the logit model (4) applied to 50% and FPCs (50% and
VPCs )- see Table 2, column 3 (Table 3, column 5). ROUND has been set equal to the last interaction, and FAVOURED is set at
the mean value of the sample. The horizontal axis reports point values for offers ranging from 5% to 35% of the pie. The estimated
probability of acceptance for each treatment is reported on the vertical axis.
Table 4: Results of Wald test relative to econometric analyses for probability of acceptance in 50% and FPCs (Panel a) and
50% and VPCs (Panel b)
Table 4a: Results of Wald test relative to 50% and
FPCs
Table 4b: Results of Wald test relative to 50% and VPCs
FPC ACCEPTANCES ALL ROUNDS
50%
20%
20%
-0.87
(0.384)
-1.97**
-0.97
(0.049)
(0.330)
-4.48*** -3.28***
(0.000)
(0.001)
1%
0%
VPC ACCEPTANCES ALL ROUNDS 1%
20%
1%
-2.39**
(0.017)
0%
50%
1.95*
(0.051)
0.17
(0.869)
-1.73*
(0.084)
20%
-1.76*
(0.078)
-3.56***
(0.000)
1%
-1.86*
(0.063)
Note: Tables 4a (4b) report z-statistics and p-values relative to Wald tests for the hypothesis Ho: βk-βl=0 against H₁: βk-βl ≠0. βk
and βl are the coefficients of treatment dummies determined in the specification of Table 2, column 3 (for Table 4a)- and in the
specification of Table 3, column 5 (for Table 4b). Rejections of H₀ at the 10% / 5% / 1% is denoted by one, two or three stars
respectively.
Table 5: Results of Wald test relative to differences in acceptance rates between FPCs and VPCs
VPC ACCEPTANCE 20%
FPC
ACCEPTANCE
20%
1%
0%
1%
0%
-2.52**
(0.012)
-2.13**
(0.033)
-3.14***
(0.002)
Note: See Table 4. The econometric specification from which the tests are drawn is reported in the SOM, Table 9, column 1.
Figure 4: Distribution of pie per treatment: All rounds (Panel a) and last five rounds (Panel b)
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
Panel b: Last five rounds
1
Panel a: All rounds
50%
20%_FPC 1%_FPC 0%_FPC 20%_VPC 1%_VPC 0%_VPC
Loss
Receiver Share
Proposer Share
50%
20%_FPC 1%_FPC 0%_FPC 20%_VPC 1%_VPC 0%_VPC
Loss
Receiver Share
Proposer Share
APPENDIX-NOT FOR PUBLICATION (AVAILABLE AS ONLINE MATERIAL ON THE AUTHORS’WEBPAGE)
The document contains supporting material, statistical analyses, a description of the recruitment procedures, and the instructions of the experiments
carried out at the University of Warwick in relation to the paper "Everyone
Wants (at Least) One Chance".
INSERT FIGURE 1 ABOUT HERE
1
Analysis of proposer behavior
In this section we want to examine the following hypothesis:
H 4 : Proposers’ demands mirrors receivers’ acceptance rates. In treatments
characterised by higher (lower) probability of acceptance, average o¤ ers decrease
(increase).
Figure 2 plots the evolution of demands over rounds in FPCs (Panel a) and
VPCs (Panel b). A clear gap between 0% demands and 1% demands is apparent in both FPC and VPC. This is consistent with the Symbolic Opportunity
Hypothesis H1 (see section 3.2 in the paper).
INSERT FIGURE 2 ABOUT HERE
1.1
Econometric analysis of FPC
We model the longitudinal characteristic of the data using a random e¤ects
model. We consider the following models, analogous to models 3 and 4 of
section 4.1.2 of the paper:
DEM AN Di;t
DEM AN Di;t
=
=
i + CHAN CE + U N F AV OREDi;t +
+ k ROU N Dk + i Zi + ui + "i;t
(1)
+ j T REAT M EN Tj + U N F AV OREDi;t + (2)
+ k ROU N Dk + i Zi + ui + "i;t
i
The dependent variable DEM AN D is how much proposers demanded for
themselves. All controls included in model (3) of the paper, suitably adjusted,
are included, too. In particular, UNFAVOURED identi…es subjects who were
in Player 2 role. The error structure includes an individual-speci…c error term
ui , and an observation-speci…c error term "i;t . To prevent the risk of error heteroschedasticity, we use robust estimates of the variance-covariance matrix of
the estimator, with errors clustered on individuals (Froot, 1989). Clustering
makes it possible to treat errors as independent across decisions from di¤erent
1
individuals, and arbitrarily correlated for decisions made by the same individual1 . Model (2) substitutes treatment dummies for the CHANCE variable in
model (1).
The results of the regression are reported in Table 12 . Columns (1) and
(2) show that CHAN CE has a positive and strongly signi…cant e¤ect. Hence
proposers’behavior, too, reacted markedly to the degree of unbiasedness in the
initial lottery. The higher the unbiasedness of the initial lottery, the higher
proposers’demands. A weak positive e¤ect for U N F AV OU RED also emerges.
Unfavored players, knowing they had a low probability of being selected as
proposers, might have sought to compensate for their overall disadvantaged
position by demanding more than favoured players. Speci…cation (2) shows
that the same results hold qualitatively if demographic controls are introduced.
However, this e¤ect disappears in both speci…cations of model 2. (See Table 1,
column 3, 4).
Speci…cation (3) introduces treatment dummies (see Table 1, column 3).
Table 3a reports in each cell (k,l) the z-statistics and the standard error for Wald
tests over the null hypothesis Ho: k
l = 0 against H1 : k
l 6= 0,where k
is the row entry and l the column entry. Table 3a supports the hypothesis of a
symbolic value of opportunity. The di¤erence between 0%_F P C and 1%_F P C
is negative and signi…cant (P = 0:024). Table 3a is also consistent with the
Monotonic Fairness Hypothesis. All the coe¢ cient signs are in accord with
this hypothesis, apart from 20%_F P C - 50% where the sign is positive but the
coe¢ cient is indistinguishable from 0. In all of the other …ve comparisons, the
lower the initial opportunity for the unfavoured player, the lower on average the
proposals in that treatment. The di¤erence across treatments is signi…cant at
the 1% level in three out of the six comparisons, and signi…cant at the 5% level
in two other comparisons.
Figure 1 also allows us to examine whether o¤ers made by proposers were
those maximizing expected earnings, given the acceptance rates for each treatment. The dotted line marked with asterisks shows the expected earnings for
each proposal category, given by the product of the acceptance rate for that category and the lowest extreme of the interval. One can notice that in two cases
out of four, the mode of the proposal lies in the same category as the payo¤
maximizing proposal. In the 0% _FPC the maximum is in the adjacent category and is very close to the payo¤ maximizing category. Only in the 1%_FPC
can a sizable di¤erence be detected. Apart from this case, proposers’behavior
seems to have converged toward the income-maximizing proposal.
1 Note that clustering on sessions instead of individuals would be inappropriate because a
necessary conditions for the validity of cluter-robust standard errors is that the number of
clusters tend to in…nity. Given the relatively small number of sessions, this assumption would
not be satis…ed. See Woolridge (2002).
2 Unless not otherwise speci…ed, all the references apply to Tables and Figures included in
the SOM.
2
1.2
Econometric Analysis for VPC
Table 2 reports the results of regressions using models (1) and (2) applied to
VPCs and 50%. This analysis, too, con…rms the patterns observed for receivers
behavior. CHAN CE is not signi…cant in either speci…cation (1) or (2) (see
Table 2, columns 1-2), but the inclusion of a quadratic term makes both coef…cients signi…cant predictors of proposers’behavior (see Table 2, columns 3-4).
The models predicts the maximum to be reached at CHAN CE = 0:27 (speci…cation 3) and CHAN CE = 0:29 (speci…cation 4). Very similar values were
found for receivers’ behavior (see section 4.2.2). The analysis of Wald tests
over di¤erences in is reported in Table 3b. The di¤erence between 20%_V P C
and 1%_V P C is not large enough to reach signi…cance levels (P = 0:22), but
0%_V P C is signi…cantly smaller than both 20%_V P C (P < 0:01) and 1%_V P C
(P < 0:01). Thus, the hypothesis of a symbolic value of opportunity is strongly
supported in VPCs. Conversely, 20%_V P C is signi…cantly greater than 50%
(P < 0:01), thus reverting the previous trend, whereas the hypothesis that
1%_V P C is the same as 50% cannot be rejected (P = 0:15) and 0%_V P C is
signi…cantly smaller than 50% (P < 0:01).
Even in this case, speci…cation (6) controlling for individual characteristics brings about qualitatively similar results to speci…cation (5) (see Table 2,
columns 5-6). It is noteworthy that unfavoured proposers demanded signi…cantly less than favoured ones, this result being strongly signi…cant in speci…cations (1), (3), and (5), and weakly signi…cant in Speci…cation (6). This result is
in contrast with what found in FPCs, and points to an e¤ect of the di¤erential
procedures used in the FPC vis-à-vis VPC on unfavoured proposers (see also
section 5 in the paper). No e¤ect for individual characteristics can be detected.
If we compare the results of Table 3 of the Appendix with those of Table 4 in
the paper, we can …nd an almost perfect correspondence between the coe¢ cient
signs. Only in one case out of 12 possible comparisons do sign di¤er. A Binomial
test strongly rejects the null hypothesis that positive and negative signs are
equally likely in the 12 possible comparisons (B(12; 0:5) = 0:0032).
Finally, the same di¤erences between FPCs and VPCs we observed for responders (see section 4.3 of the paper) also emerged with respect to proposers’
behavior (see Tables 9 and 10). That is, in each corresponding treatment, proposers were signi…cantly higher in the VPC compared to the FPC.
We thus conclude:
Conclusion 1 Proposers’ behavior mirrored receivers’ behavior.
INSERT TABLES 1, 2, and 3 ABOUT HERE
2
Results for First Round
Descriptive statistics for Round 1 are reported in Table 4. Patterns are very
similar to what we observed across the whole 20 rounds. The mean for o¤ers
and rejected demands in the …rst round of FPCs present an identical pattern
3
to that emerging in the whole 20 rounds. The patterns are similar in VPCs
as well. We can notice from Figure 2 that the gap between 0% and 1% is
clearly existent from the …rst round of interactions, and that demands follow a
monotonic pattern. This is consistent with H 1 and H 2 , respectively.
INSERT TABLE 4 ABOUT HERE
The econometric analysis reveals some clear similarities to the patterns detected over the whole 20 rounds. As far as acceptances are concerned, we
observe in VPCs a clear di¤erence between 0%_VPC compared to all other
treatments (see Table 7b), and with 1%_VPC in particular (P = 0:003). This
strongly supports H 1 . Signs of treatment coe¢ cient di¤erences always change
monotonically in FPCs, though di¤erences are never signi…cant (see Table 7a).
A binomial test, though, rejects the null hypothesis that positive and negative
signs are equally likely in the 6 tests in Table 7a (B(6; 0:5) = 0:016). This is
consistent with H2 : However, the variable CHANCE has the correct sign, but
is not signi…cant (see Tables 5, column 1 and 3). A quadratic term for CHANCE
is not signi…cant (not reported). We conclude:
Conclusion 2 H1 is strongly supported in the VPC, but not in the FPC, in
Round 1. H2 is supported in the FPC in Round 1, albeit using a less strong
test - a Binomial test - than what used for the 20-round analysis.
As far as proposers’ behavior is concerned, the variable CHANCE has a
positive and signi…cant e¤ect in FPCs (P = 0:010), (see Table 6, column 1),
but no e¤ect can be detected in VPCs (see Table 6, column 3). Looking at the
treatment coe¢ cient di¤erences, the pattern of signs in the Tables relative to
o¤ers exactly matches that of the acceptance tables in both FPCs and VPCs (see
Tables 7 and 8). A Binomial test strongly rejects the null hypothesis that signs
in the proposer Tables are equally likely given the signs observed in the receiver
Tables (B(12; 0:5) = 0:0002). It seems that, in all cases, proposers in Round 1
were able to anticipate the variation of receivers’probability of rejection across
treatments. The signs are always negative in FPCs, consistently with H 2 (See
Table 8a), and are signi…cant in four out of six comparisons. O¤ers in 0%_VPC
are considerably lower than in all other VPCs (see Table 8b), the di¤erence being
strongly signi…cant with respect to 1%_VPC (P = 0:009), but no monotonic
pattern can be detected. We conclude
Conclusion 3 Proposers behavior perfectly mirrored receivers’behavior in Round
1.
INSERT TABLES 5-8 ABOUT HERE
3
Propositions on dynamic procedural fairness
Let the variable X(k), k = f1; : : : ; 20g be the random variable de…ning the
number of times a player becomes proposer in any round r from k onwards,
4
r = fk; : : : ; 20g. The expected value of X(k), E(X(k)), is an obvious indicator
of the opportunities a player can expect from round k of the interaction onwards
to access the advantaged bargaining position.
It is straightforward to prove the following propositions:
Proposition 4 In FPCs, the di¤ erence in E(X(k)) between favoured and unfavoured players is positive and proportional to (1 2p), whereas it is equal to
0 in VPCs. This holds for any k > 1.
Proof. Suppose an agent is about to play the k-th round of the stage game.
Let us take an ex ante perspective, that is, in VPCs L1 is yet to take place. In
p_F P C, L1 has instead already taken place in Round 1 and a player knows her
role. If the player is favoured in p_F P C, then her distribution of X(k) follows
Bin(21 k; 1 p). This is the case because each lottery is independent, and the
favoured player is assigned the proposer role with probability 1 p. Similarly an
unfavoured player faces a distribution Bin(21 k; p). Conversely, in VPCs X(k)
follows Bin(21 k; 1=2) for both types of players. This is the case because in
each VPC round players are …rst faced with L1 that is an even lottery, and later
with L2 that assigns the proposer role with probability p if favoured and (1 p)
if unfavoured. Considering this compound lottery, the probability that a VPCT
player is assigned the proposer role is thus 1=2(p) + 1=2(1 p) = 1=2. Thus,
for any FPCT, E(X(k))F AV E(X(k))U N F AV = (21 k)(1 p) (21 k)p =
(21 k)(1 2p) > 0 because p < 1=2. Conversely, for any VPCT, E(X(k))F AV E(X(k))U N F AV = (21 k)1=2 (21 k)1=2 = 0 QED.
Thus, unfavoured players su¤er a clear disadvantage in expected opportunity
vis-à-vis favoured players in any FPCs, whereas this is not the case in VPCs.
The case of k = 1 is analysed in Proposition 6.
Let us now compare the perspective of two unfavoured players in corresponding treatments (see section 3.2 of the paper). In the current round these
two players are faced with the same probability of accessing the proposer role.
However, it is easy to show the following:
Proposition 5 E(X(k + 1)) is greater for a VPCT unfavoured player compared to an FPCT unfavoured player. This holds for any pair of corresponding
treatment and for any k < 20.
Proof. Suppose that an agent is about to play the k-th round of the stage game.
Let us now take an ex post perspective. That is, in both p_V P C and p_F P C,
L1 has already been run. Let us take pairs of corresponding treatments, and
let us consider a player who is unfavoured at the k-th round of p_V P C. Her
situation is exactly the same as an unfavoured player of p_F P C for the kth round. But afterwards she faces a distribution of X(k) that is Bin(20
k; 0:5), whereas an unfavoured player in p_F P C faces Bin(20 k; p). Since the
expected value of the former (latter) distribution is (20 k)1=2 ((20 k)p); and
p < 1=2, a currently unfavoured player in p_V P C has more chances to occupy
the proposer role in the future than an unfavoured player in FPC. QED
5
In other words, VPCs unfavoured player have higher expected opportunities
compared to FPCs unfavoured players over the course of the experiment.
One may object that opportunities are as fairly distributed in FPCs as in
VPCs, because before the initial role assignment each player had an even chance
of being assigned the advantaged position. In fact, if we consider k = 1 and we
take an ex ante perspective, that is, before L1 has been run, E(X(20) is indeed
the same for any player. Nevertheless, we believe that a a plausible assumption
is that individuals are not only sensitive to the expected value of X(k), but also
to its variance. Let us call V ar(X(k)) the variance of X(k). We can prove the
following:
Proposition 6 V ar(X(20)) in FPCs is greater than V ar(X(20)) in VPCs for
any pair of FPC and VPC treatments.
Proof. Let us consider players’ prospects when k=1 and no L1 has yet been
run. The distribution of X(20) is thus as follows: in any p_V P C, X(20) follows
Bin(20; 0:5), thus E(X(20)) = 10, V ar(X(20)) = 20p(1 p) = 5. In p_F P C,
X(20) follows {Bin(20; p) with prob 0:5, Bin(20; 1 p) with prob 0.5}. Hence,
E(X(20)) = 10, V ar(X(20)) = 0:5[n(n 1)[p2 + (1 p)2 ] + n] 0:25n2 , where
n = 20. Note that if p = 0:5 in the latter formula, V ar(X(20)) = 5, exactly the
same as V ar(X(20)) under p_V P C. If we di¤erentiate the variance expression
with respect to p, we obtain 0:5n(n 1)[4p 2], which is lower than 0 for all
p < 0:5. Since the second derivative is positive for all p, p = 1=2 is a point of
minimum. Thus the variance expression is decreasing in p, for all p < 0:5. Thus
V (X(20)) under p_F P C> V (X(20)) under p_V P C. QED
4
Experimental Procedures
Experimental sessions were run at Warwick University between April and June
2007. On average 60 students per treatment took part in our experiments. Only
subjects who had not been attending courses in Game Theory were allowed to
participate. We ran three sessions per treatment. Due to varying show-up rates,
the number of subjects per session was not constant across sessions but varied
from a minimum of 16 to a maximum of 24 subjects, with an average of around
20 subjects per session. Each subject only participated in one session. We took
care to balance the composition of the sessions in terms of gender and number
of people enrolled in Economics and Psychology courses with respect to the
total. Each session was organized as follows. Subjects were paid a show-up fee
of £ 5 upon their entering the experimental room, and were randomly seated
to a workstation in the room. After instructions were administered, a written
comprehension test was carried out. Subjects making a mistake in the test
were asked to retake the wrongly answered quiz. Experimenters went through
the instructions again for subjects who failed even this second attempt, until
their full comprehension was ascertained. Subjects were then involved in the 20
interactions of the stage game. At the end of the decisions subjects completed
6
a short questionnaire asking demographic and attitudinal questions, and …nally
received their earnings. The whole session lasted around an hour. The average
earnings - in addition to the $5 show-up fee - was $8:22. The game was
conducted using the z-tree software (Fischbacher 2007).
5
Instructions
Welcome to this research project. A team of researchers is looking at the way
in which people make decisions. If you pay close attention to the instructions
then you could make a signi…cant amount of money. The research team that is
here today includes myself, Gianluca Grimalda, and my assistants.
Before starting with the explanation of the decisions you are going to make,
please pay attention to some important information and recommendations.
In this project you are going to be asked to make decisions with other people
who are currently in this room. Your choices, and the choices of others, will be
matched with the help of a computer programme as we proceed. It is important
for you to note that all interactions are entirely anonymous. Firstly, we will not
know anything about your choices and your payment. We will just record your
choices through the ID number that you have just drawn, and the payments
will be made using that number as identi…cation. It is therefore important that
you do not lose the card you have drawn, because that is the only document
that enables you to be paid. You may collect your payments at the end of this
session. You will be required to sign a receipt, but there is no need for you to
print your name. University administration does require that you write in your
student number when signing this receipt. However, your student number will
be held con…dentially by our research group, and we will not make any attempt
to link your student number to the decisions you have made.
At the end of your decisions, while we prepare your payments, we would ask
that you complete a short questionnaire. You are required to state your Student
ID number. Even in this case, your responses to this questionnaire will be held
under con…dentiality rules by our research group.
Secondly, the decisions you are going to make involve interacting with other
people who are present in this room. However, you will not have to talk or
communicate directly in any way with anybody in this room. Instead, your
decisions will be processed through a computer programme that networks all of
the computers in this room. In this way, nobody will be able to identify with
whom s/he is actually making decisions. The interaction will proceed as follows:
You will receive some messages on the screen in front of you. This will either
include some information on the state of the decisions, or prompt you to make
certain choices. Once you are sure about your choice, you have to press the
button OK, which will take you to the next stages of the decisions. At times,
you will be asked to wait for further instructions, because it may take a bit of
time before the programme processes all your decisions.
If you are not clear on this or on other issues, please raise your hand.
You will be involved in 20 di¤erent interactions with other people in this
7
room. In each interaction, you will be paired with another person, and the two
of you will be making a decision together. Our programme will draw at random
the pairs at the beginning of each interaction. This means that with very high
probability you will be paired with a di¤erent partner at each interaction.
As you will see, the decisions involve money. In each decision there will be
£ 10 at stake. Unfortunately, we will not be able to pay you for each decision
you make, but only for TWO interactions out of the 20. These will be drawn
at random at the end of this session, and everyone will be paid according to
the outcome of those 2 rounds. In this way, you are required to pay maximum
attention to each decision you are going to make, because only at the end of the
session we will learn which ones determine your payments.
We are now going to look at the simple rules that will govern each of the
interactions:
[All treatments]: An amount worth £ 10 is to be divided between you and
the person you have been paired with.
[1%, 20%, 50%]: Both of you are asked to make a proposal. [0%]: One
of the two people is drawn at random, and both people are informed about
whether s/he has been selected or not. The person who has been selected is
asked to make a proposal. [All treatments] : The proposal is any amount X
less than or equal to £ 10 that the ’proposer’ wants to keep for him/herself.
The proposer may use any number up to the second decimal digit. The residual
amount (10-X) is to be assigned to the other person in the group (the ‘receiver’).
[1%, 20%, 50%]: Once you and the other person in your group have
submitted your proposals, one of them is drawn at random. [1%, 20%]: The
random selection works as follows. Half of the people in this room are favoured
with respect to the others in having their proposals selected. In particular, half
of the people in this room have a [1-p]% probability that their proposals will be
selected within their groups, whereas the others have a [p]% probability. [{p=
0.01, 0.2}] [50%]: There is a 50-50 probability that either proposal is extracted.
[0%]: Each group is composed of a ’proposer’and a ’receiver’. Whether
you will act as a proposer or as a receiver is determined by a random draw that
will occur [0%_FPC]: before the …rst round. Your role will remain the same
throughout the 20 rounds. [0%_VPC]: before each round. [1%, 20%]: Each
group will be made up of a person with a [1-p]% probability and another person
with a [p]% probability of their proposals being selected. [{p= 0.01, 0.2}] You
will be informed about which probability your proposal has of being selected
[1%_FPC, 20%_FPC]: before the …rst round, and this probability will remain
the same throughout all the remaining rounds. [1%_VPC, 20%_VPC]: before
submitting it.
[All treatments]: The person whose proposal has been selected (the
‘proposer’) is asked to wait for the decision of the other person in the group.
The person whose proposal has not been selected (the ‘receiver’), is informed of
the share allocated to him/her by the proposal of the other person. She is then
asked to either ACCEPT or REJECT this proposal.
[All treatments]: If the receiver accepts this proposal, then everyone
gets the share determined by this proposal. If the receiver rejects this proposal,
8
then both people in the group get £ 0 each.
[All treatments]: At the end of each interaction, a new random draw
will take place to determine your next partner. [For FPCs only]: This will be a
person from the half of the people in this room with a probability di¤erent from
yours of their proposals being selected. [All treatments]: It is therefore very
unlikely you will be paired with the same person again. Moreover, all decisions
are independent. What you do in a round does not in‡uence the next rounds
and is not in‡uenced by the previous rounds.
Examples and comprehension test follow.
References
[1] Fischbacher, U. (2007). z-Tree: Zurich toolbox for ready-made economic
experiments. Experimental Economics 102, 171–178.
[2] Froot, K. A. (1989). Consistent covariance matrix estimation with crosssectional dependence and heteroskedasticity in …nancial data. Journal of
Financial and Quantitative Analysis 24: 333-355.
[3] Woolridge, J.M. (2002). Introductory Econometrics: A Modern Approach,
Cincinnati, OH: South-Western College Publishing.
9
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
7-7.5
6.5-7
6-6.5
5.5-6
5-5.5
4.5-5
<4.5
0
Frequency of proposals
.1
.2
.3
50% Treatment
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
FIGURES AND TABLES FOR ONLINE APPENDIX-NOT FOR PUBLICATION
Figure 1: Histograms of demands and acceptance rates per treatment
Proposal categories
9-9.5
9.5-10
8.5-9
8-8.5
7.5-8
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
6.5-7
6-6.5
5.5-6
5-5.5
4.5-5
7-7.5
.3
Frequency of proposals
.1
.2
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
7-7.5
6.5-7
6-6.5
5.5-6
5-5.5
4.5-5
0
0
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
Acceptance Prob.
0%_VPC Treatment
<4.5
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
7-7.5
6.5-7
6-6.5
0
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
.3
Frequency of proposals
.1
.2
0
5.5-6
Proposal frequency
Expected Earnings (%)
Acceptance Prob.
0%_FPC Treatment
5-5.5
7-7.5
.3
Frequency of proposals
.1
.2
0
Proposal categories
Proposal categories
Proposal frequency
Expected Earnings (%)
4.5-5
Acceptance Prob.
1%_VPC Treatment
<4.5
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
7-7.5
6.5-7
6-6.5
5.5-6
5-5.5
4.5-5
0
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
Frequency of proposals
.05 .1 .15 .2 .25
0
<4.5
Proposal frequency
Expected Earnings (%)
Acceptance Prob.
1%_FPC Treatment
<4.5
6.5-7
Proposal categories
Proposal categories
Proposal frequency
Expected Earnings (%)
Proposal categories
Proposal categories
Proposal frequency
Expected Earnings (%)
6-6.5
5.5-6
5-5.5
<4.5
0
Frequency of proposals
.05 .1 .15 .2 .25
20%_VPC Treatment
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
Acceptance Prob.
4.5-5
9.5-10
9-9.5
8.5-9
8-8.5
7.5-8
7-7.5
6.5-7
6-6.5
5.5-6
5-5.5
4.5-5
<4.5
0
Frequency of proposals
.05 .1 .15 .2 .25
20%_FPC Treatment
0
.2
.4
.6
.8
1
Acceptance Prob. / Expected Earnings (%)
Proposal frequency
Expected Earnings (%)
Acceptance Prob.
Proposal frequency
Expected Earnings (%)
Acceptance Prob.
Notes: Horizontal axes report various classes of demands. These have length equal to 0.5 for all offers smaller than five, whereas
all offers greater than five have been grouped in one category. Bars report the frequency with which a particular demand range has
been observed; The solid line reports the corresponding probability of acceptance; The dotted line indicates proposers’ expected
earnings for a given class of demand s. This has been computed as the product of the probability of acceptance of a certain class and
the lowest value in that interval class - e.g. 5 for the [5; 5.5] interval.
8
Mean Proposal
6.5
7
7.5
6
6
Mean Proposal
6.5
7
7.5
8
Figure 2: Evolution of mean demands over rounds in (a) FPCs & 50%; (b) VPCs & 50%
Panel (a): FPCs & 50%
Panel (b): VPCs & 50%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Round
50%
1% _FPC
1
2
3
4
5
20% _FPC
0% _FPC
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Round
50%
1% _VPC
20% _VPC
0% _VPC
Table 1: Regression Analysis of demands –50% treatment & FPCs- All rounds
DEP VAR CHANCE (1) 0.995*** (0.274) 20%_FPC 1%_FPC 0%_FPC UNFAVOURED 0.258* (0.154) ECONOMICS YEAR GENDER UK Constant ROUND DUMMIES Observations Number of id R2 Between R2 Within R2 Overall 6.357*** (0.110) YES 4,380 219 0.0510 0.00855 0.0339 DEMAND (2) (3) 1.072*** (0.341) 0.0384 (0.157) ‐0.298* (0.154) ‐0.684*** (0.159) 0.273* 0.0649 (0.148) (0.165) 0.499*** (0.143) 0.0663 (0.0492) 0.0174 (0.132) 0.152 (0.140) ‐125.7 6.786*** (97.74) (0.117) YES
YES
3,740 187 0.143 0.0117 0.0906 4,380 219 0.0817 0.00855 0.0522 (4) ‐0.0937 (0.185) ‐0.274 (0.190) ‐0.810*** (0.179) 0.0910 (0.162) 0.527*** (0.151) 0.0665 (0.0476) 0.0330 (0.128) 0.161 (0.140) ‐125.7 (94.56) YES
3,740 187 0.174 0.0117 0.110 Notes: We model the longitudinal characteristic of the data using a random effects model. Round dummies have been included
in all regressions. Stars denote significance levels as follows: * = P-value<0.1; ** = P-value<0.05; *** = P-value<0.01.
Table 2: Regression analysis of demands –50% treatment & VPCs – All rounds
DEP VAR CHANCE (1) 0.277 (0.266) (2) 0.485 (0.345) CHANCE SQUARED DEMAND (3) (4) 4.711*** 3.545** (1.196) (1.552) ‐8.889*** ‐6.214** (2.281) (3.073) 20%_VPC 1%_VPC 0%_VPC UNFAVOURED ‐0.264*** (0.0956) ECONOMICS AGE GENDER UK Constant ROUND DUMMIES Observations Number of id R2 Between R2 Within R2 Overall Note: See Table 1.
6.714*** (0.113) YES 4160 238 0.00591 0.0381 0.00593 ‐0.137 (0.0908) 0.134 (0.144) 0.0499 (0.0597) 0.0808 (0.141) ‐0.0641 (0.142) ‐92.76 (118.8) YES
‐0.279*** (0.0961) 6.574*** (0.119) YES
‐0.144 (0.0911) 0.156 (0.146) 0.0276 (0.0604) 0.0522 (0.137) ‐0.0622 (0.136) ‐48.42 (120.1) YES
2824 161 0.0143 0.0360 0.0169 4160 238 0.0330 0.0382 0.0289 2824 161 0.0389 0.0360 0.0279 (5) (6) 0.434*** (0.151) 0.221 (0.154) ‐0.415*** (0.143) ‐0.292*** (0.0957) 6.710*** (0.124) YES
0.234 (0.215) 0.0676 (0.192) ‐0.523*** (0.177) ‐0.157* (0.0908) 0.162 (0.145) 0.0111 (0.0592) 0.00372 (0.136) ‐0.0453 (0.134) ‐15.40 (117.7) YES
4160 238 0.0820 0.0382 0.0493 2824 161 0.0911 0.0360 0.0499 Table 3: Results of Wald test relative to econometric analyses of demands for 50% and FPCs (Panel a) and 50% and
VPCs (Panel b)
Table 3a: Results of Wald test relative to 50% and
FPCs
Table 3b: Results of Wald test relative to 50% and
VPCs
FPC DEMANDS ALL ROUNDS
20%
1%
0%
50%
0.24
(0.807)
-1.93*
(0.054)
-4.29***
(0.000)
20%
VPC DEMANDS ALL ROUNDS
1%
50%
2.87***
(0.004)
1.44
(0.151)
-2.90***
(0.004)
20%
-2.04**
(0.041)
-4.17***
(0.000)
1%
-2.26**
(0.024)
0%
20%
1%
-1.22
(0.224)
-5.55***
(0.000)
-4.09***
(0.000)
Note: See Table 4 in the paper. The coefficients of treatment dummies are determined in the specification of Table 1, column
3 (for Table 1a)- and in the specification of Table 2, column 5 (for Table 3b).
Table 4: Descriptive Statistics of Responses and Demands per Treatment – Round 1
Table 4a: 50%
Responses
(1)
RD
7.9
Mean
St. Dev
Median
Obs
0.82
7.5
5
Demands
(2)
AR (All)
83.9%
0.37
(3)
AR (Low)
77.8%
0.44
31
9
Table 4b:FPC 20%
Responses
(1)
RD
Mean
St. Dev
Median
Obs
7.7 1.40
8
5
(4)
FAV
6,83
1,15
7
62
(5)
UNF
Table 4e: VPC 20%
Demands
(2)
AR (All)
83.9%
0.37
(3)
AR (Low)
57.1%
0.53
31
7
(4)
FAV
6,66
,94
6,5
31
(5)
UNF
7,01
1,27
7
31
Responses
Mean
St. Dev
Median
Obs
(1)
RD
8.87
0,74
9
8
Table 4c:FPC 1%
Responses
Mean
St. Dev
Median
Obs
(1)
RD
7,48
0,55
7,5
6
(3)
AR (Low)
50%
0.58
32
4
Mean
St. Dev
Median
Obs
(1)
RD
7,33
0,98
7,25
6
(4)
FAV
6,59
,97
6,625
32
(5)
UNF
6,30
1,92
6
32
Mean
St. Dev
Median
Obs
(1)
RD
8,48
0.41
8,46
4
(3)
AR (Low)
66.7%
0.58
31
3
Notes: See Table 1 in the paper.
30
9
(4)
FAV
6,77
1,08
6,5
30
(5)
UNF
6,86
1,55
6,78
30
Demands
(2)
AR (All)
85.7%
0.36
(3)
AR (Low)
42.9%
53.4
28
7
(4)
FAV
6,98
1,07
7
28
(5)
UNF
6,89
2,16
7,09
28
Table 4g: VPC 0%
Demands
(2)
AR (All)
80.6%
0.401
(3)
AR (Low)
22.2%
.44
Responses
Table 4d:FPC 0%
Responses
(2)
AR (All)
73.3%
0.45
Table 4f: VPC 1%
Demands
(2)
AR (All)
81.2%
0.40
Demands
(4)
FAV
6,21
1,19
6
31
Responses
(5)
UNF
Mean
St. Dev
Median
Obs
(1)
RD
7,64
1,38
7
7
Demands
(2)
AR (All)
76.7%
0.43
(3)
AR (Low)
0
30
2
(4)
FAV
6,10
1,37
6
30
(5)
UNF
Table 5: Regression Analysis of Logit model for probability of acceptance in Round 1
DEP VAR TREATMENTS CHANCE 20%_FPC 1%_FPC 0%_FPC ACCEPT FPC & 50% (1) (2) 1.952 (1.498) ‐0.394 (0.828) ‐0.785 (0.773) ‐1.150 (0.889) VPC & 50% (3) 2.089 (1.697) 20%_VPC 1.121*** (0.275) 0.533 (0.794) ‐2.144** (0.938) 125 16.95 1.123*** (0.280) 0.395 (0.870) ‐1.223 (0.831) 125 17.19 1.637*** (0.308) 0.666 (0.854) ‐3.360*** (0.896) 119 29.90 ‐0.982 (1.013) 0.00710 (0.992) ‐2.875** (1.214) 2.083*** (0.398) 1.325 (1.070) ‐3.170*** (1.022) 119 31.62 83% 82.4% 87.4% 88.2% 1%_VPC 0%_VPC OFFER FAVOURED Constant Observations Chi2 Percentage of correctly predicted outcomes (4) Notes: A logit model has been fitted. See Table 2 in the paper.
Table 6: Regression Analysis of Demands for Round 1
DEP. VAR. TREATMENTS CHANCE 20%_FPC 1%_FPC 0%_FPC DEMAND FPCs & 50% (1) (2) 1.051** (0.406) ‐0.00498 (0.225) ‐0.401* (0.225) ‐0.611** (0.259) VPCs & 50%
(3) 0.556 (0.453) 20%_VPC 1%_VPC 0%_VPC UNFAVOURED Constant Observations R‐squared 0.165 (0.223) 6.373*** (0.127) 219 0.026 0.0222 (0.239) 6.830*** (0.147) 219 0.035 Note: An OLS model has been fitted. See Table 1.
0.248 (0.265) 6.563*** (0.156) 208 0.009 (4) ‐0.0128 (0.246) 0.107 (0.251) ‐0.731** (0.289) ‐0.00397 (0.282) 6.830*** (0.147) 208 0.037 Table 7: Results of Wald test relative to econometric analyses of probability of acceptance for 50% and FPCs (Panel a)
and 50% and VPCs (Panel b)-Round 1
Table 7a: Results of Wald test relative to 50% and FPCs –
Round 1
Table 7(b): Results of Wald test relative to 50% and VPCs
– Round 1
FPC ACCEPTANCES ROUND 1 50%
20%
1%
0%
-0.48
(0.634)
-1.01
(0.310)
-1.29
(0.196)
20%
VPC ACCEPTANCE ROUND 1 1%
50%
20%
-0,53
(0.599)
-0,87
(0.383)
1%
-0,49
(0.624)
0%
-0.97
(0.332)
0.01
(0.994)
-2.37**
(0.018)
20%
1%
1.22
(0.224)
-2.04**
(0.042)
-2.98***
(0.003)
Note: See Table 4 in the paper. The coefficients of treatment dummies are determined in the specification of Table 5, column 2
(for Table 7a)- and in the specification of Table 5, column 4 (for Table 7b).
Table 8: Results of Wald test relative to econometric analyses of proposals for 50% and FPCs (Panel a) and 50% and
VPCs (Panel b)-Round 1
Table 8a: Results of Wald test relative to 50% and FPCs –
Round 1
Table 8b: Results of Wald test relative to 50% and VPCs
– Round 1
FPC DEMANDS ROUND 1 50%
20%
1%
0%
-0.02
(0.982)
-1.78*
(0.076)
-2.36**
(0.019)
20%
VPC DEMANDS ROUND 1 1%
50%
20%
-1.66*
(0.098)
-2.22**
(0.027)
1%
-0.77
(0.442)
0%
-0.05
(0.958)
0.43
(0.671)
-2.53**
(0.012)
20%
0.42
(0.675)
-2.26**
(0.025)
1%
-2.61***
(0.010)
Note: See Table 4 in the paper. The coefficients of treatment dummies are determined in the specification of Table 6, column 2
(for Table 8a) - and in the specification of Table 6, column 4 (for Table 8b).
Table 9: Regression Analysis of Probability of Acceptance and Demands (Comparison FPC-VPC)
DEPENDENT VARIABLE 20%_FPC 1%_FPC 0%_FPC 20%_VPC 1%_VPC 0%_VPC OFFER FAVOURED Constant ROUND DUMMIES Observations Chi2 (R2 for Column 2) Percentage of correctly predicted outcomes DEMAND (2) ACCEPT (1) ‐0.396 (0.689) ‐1.492** (0.754) ‐3.613*** (0.778) 1.315** (0.669) 0.135 (0.673) ‐1.181* (0.673) 3.057*** (0.145) 0.243 (0.413) ‐4.740*** (0.624) YES 4,260 452.7 ‐0.0700 (0.158) ‐0.193 (0.155) ‐0.684*** (0.159) 0.414*** (0.147) 0.223 (0.144) ‐0.415*** (0.143) 6.729*** (0.111) YES 4,880 0.108 83.26% Note: See Table 2 in the paper for column 1, and Table 1 for column 2.
Table 10: Results of Wald test relative to differences in demands in FPCs vis-à-vis VPCs
VPC PROPOSALS ALL ROUNDS FPC PROPOSAL
ALL ROUNDS
20%
20%
1%
0%
1%
0%
-2.96***
(0.163)
-2.63***
(0.158)
- 1.67*
(0.161)
Notes: See Table 4 in the paper. The coefficients of treatment dummies are determined in the specification of Table 9,
column 2.