THE DOUBLE-CHANNELED EFFECTS OF EXPERIENCE
IN INDIVIDUAL DECISIONS∗
P EIRAN J IAO†
Abstract
People often make decisions with both descriptive and experiential information available.
For example, investors in the financial markets typically have access to both descriptions from
brochures, financial reports, analysts, etc., as well as own experience. However, little is known
about the role of experience in individual economic decision-making with the presence of descriptive information. This paper investigates the issue by experimentally testing the effects of
experience on individuals using an investment task with choice feedback and varying levels of
descriptive information. In the experiment, participants observed prices of hypothetical assets,
made buying decisions and reported subjective beliefs. We document the double-channeled effects of experience: when elicited beliefs were controlled for, participants significantly relied
on experience regardless of the descriptions, behaving consistently with the law of effect, i.e.
repurchasing more assets on which they gained rather than lost; additionally, beliefs were also
distorted by experience, in that participants were more optimistic about assets from which they
gained, and pessimistic about previously unowned assets. In a calibration exercise, reinforcement learning significantly added predictive power to expected utility models. These results
can inform theories of individual decisions and the design of behavioral interventions.
Keywords: Description, Experience, Law of Effect, Biased Belief, Individual Choice.
JEL Classification Numbers: C91, D03, D83.
Word Count: 7515
∗
The author thanks Peyton Young, Vincent Crawford, Peter Wakker, Olivier l’Haridon, Aurélian Baillon, Joshua
Tasoff, Henrich Nax, John Jensenius III, Chen Li and participants at the CESS colloquia at Nuffield College, Learning,
Games and Networks Seminar in University of Oxford, ETH Zurich, Erasmus School of Economics, Experimental
Finance Conference 2015, 6th Annual Conference of ASFEE for helpful discussions, suggestions and comments on
this project, and Kaixin Bao, Taotao Zhou and Patrick Lu for excellent research assistance during early stages of the
project. This research was supported by the George Webb Medley/OEP Fund and the John Fell OUP Research Fund
under project code CUD09320.
†
Address: Department of Economics, Mannor Road Building, Mannor Road, Oxford, OX1 3UQ, UK; Fax: +44
(0)1865 271094; Telephone: +44 (0)1865 278993; E-mail: [email protected].
1
1
Introduction
Conventional economic models assume that individual decision makers update beliefs follow-
ing the Bayes’ rule, correctly weighting experienced and observed outcomes, as well as descriptive
information. However, evidence in the lab and field documents clear departures. For instance, in
games players tend to repeat the actions that brought experienced gains even when the environment has changed [Camerer and Ho, 1999]; in individual decisions when the Bayesian predictions
clashed with reinforcement learning subjects followed each rule almost half of the time [Charness and Levin, 2005]. Evidence in the financial market suggests that individual investors (but not
professionals) increase their subscription to IPO auctions subsequent to previous successful experience, as if they overweighted personal experience [Kaustia and Knüpfer, 2008; Chiang et al.,
2011]. People seem to have the tendency to overweight experienced outcomes relative to observed outcomes and descriptive information, compared with what a Bayesian agent would do.
The role of experience in individual economic decisions is not yet well-understood. Our goal is to
use an experiment to create clear tension between rational Bayesian strategy and the experiencebased learning rule in individual decision-making under uncertainty, in an investment context. By
comparing actual choices with the clear Bayesian benchmark, we test (1) whether participants
overweighted personal experience, even when such experience was information-free, i.e. bearing
no information value for a Bayesian updater, (2) whether their behavior was consistent with the
law of effect, i.e. the probability of taking an action increasing in its past reinforcement, and (3)
whether experience induced any belief distortion. Our results uncovered positive answers to all
three questions. A better understanding of these issues can inform individual decision theories,
and help design effective tools to debias individuals, such as in financial markets.
Individual choice models in conventional and even main stream behavioral economics do not
distinguish two cases: description-based decisions, where the decision maker possesses complete information about the incentive structure, and experience-based decisions, where the decision maker has no knowledge about the environment but can sample or experience the outcomes.
Psychologists documented discrepancies between description and experience-based decisions, i.e.
2
the description-experience gap:1 for example, Barron and Erev [2003] find subjects exhibited underweighting of rare events in experience-based decisions, instead of overweighting them as in
prospect theory; Jessup et al. [2008] and Lejarraga and Gonzalez [2011] find experience overwhelms descriptive information when both are available. The effect of varying information structures on behavior has been studied in repeated games,2 but these issues were not well-understood
by economists in individual decisions. To the author’s knowledge, attempts to incorporate the
description-experience gap, particularly the effects of experience, in an individual economic or financial decision environment limit to the following: Abdellaoui et al. [2011] estimate prospect theory parameters and find less overweighting of rare events under experience than under descriptionbased context; Ben-Zion et al. [2010] experimentally show that under-diversification can result
from providing feedback on both diversified and non-diversified investment alternatives; Kaufmann et al. [2013] and Bradbury et al. [2014] both use simulated experience to help investors
understand risk.
The reasons for using an investment task in our experiment are threefold. Firstly, investment
decisions, like most real-world problems, are relatively more complex than the clicking paradigm
adopted in most psychological studies.3 Admittedly they have some advantages, but whether their
findings can be generalized to more complex and realistic situations is questionable [see e.g. Hau
et al., 2008]. Secondly, the financial market is a natural environment where decision makers have
access to both descriptive and experiential information. Descriptions are available through investor
education, fund brochures, financial reports, financial analysts, etc., but are often ignored: individual investors lose from under-diversification [Kelly, 1995] and the disposition effect [Shefrin and
Statman, 1985], both clearly against the advices in mainstream investor education. First-hand
experience from own trading history is often overweighted, such as in the IPO market [Kaustia
1
See Rakow and Newell [2010] for a review of this issue in psychology.
In game theoretic experiments, the experimenter either provided players with only descriptions of the game
[Rapoport et al., 2002], only feedback [Mookherjee and Sopher, 1994; Nax et al., 2013; Friedman et al., 2015] or
both; theories have demonstrated the equilibrium property of adaptive learning dynamics under limited information
settings [see e.g. Foster and Young, 2006; Young, 2009].
3
See e.g. Erev and Haruvy [2013] for a review of studies using the clicking paradigm, where participants make
binary choices between two buttons on a screen, each associated with some binary lotteries, known or unknown to the
decision maker.
2
3
and Knüpfer, 2008].4 And thirdly, individual investors’ learning from experience is an intriguing topic. Larger quantity of experience has been shown to induce better performance [Nicolosi
et al., 2009] and reduce behavioral biases [Dhar and Zhu, 2006]. If investors do overweight experience, then good experience can be dangerous, leading to overconfidence[Gervais and Odean,
2001], over-participation such as in 401(k) portfolio allocation [Choi et al., 2009], and unjustifiable
repurchases [Strahilevitz et al., 2011].
In our experiment, participants observed price sequences of 4 hypothetical assets generated
by 4 equally-likely underlying processes with drifts and had two tasks each period: they made
buying decisions, knowing that shares bought would be automatically sold when the new prices
were announced, and they predicted the probabilities of price increases. The price sequences were
designed so that knowledge of the underlying processes and observation of historical prices would
suffice for an agent to adopt the simple Bayesian strategy that relies only on the price change
directions, rendering experienced gains/losses information-free, so that the predictions of Bayes’
rule clashes with that of reinforcement learning in some cases. We find participants fail to sufficiently account for the descriptive information, but instead overweight on their own experience.
Different treatment conditions varied the amount of descriptive information provided regarding
the price generation processes, but this did not alter the effect of experience on decisions, suggesting that descriptions were possibly underweighted or sometimes ignored. More importantly,
we do find behavior consistent with the law of effect: controlling for elicited beliefs, participants
were more likely to repurchase assets from which they gained than those from which they lost.
They also placed considerable weight on the most recent payoffs, discounting past experience
heavily. The experienced gains/losses had additional explanatory power than just observed price
increases/decreases.
Additionally, experience distorted beliefs: participants were more optimistic about the assets
that offered better obtained payoffs, and believed those with positive forgone payoffs would mean
revert. We also tested the effect of two other stimuli, namely, regret and surprise, both believed
4
Another source of experience from observing that of others [see e.g. Kaustia and Knüpfer, 2012] will not be
discussed here.
4
to influence behavior when choice feedbacks are available [see e.g. Zeelenberg et al., 2001; Nevo
and Erev, 2012], but both exerted limited impact on choices. A further evaluation of predictive
performance among models with and without learning reveals that although reinforcement learning alone did not fit the data satisfactorily, adding it to the expected utility models significantly
improved predictive power.
2
The Experiment
2.1
Design
Each session of the experiment contained three rounds, with T = 15 decision periods per
round. In each round, participants were faced with a different set of K = 4 hypothetical assets,
labeled W, X, Y and Z. The price-generating processes were similar to Weber and Camerer [1998]
and Jiao [forthcoming] in their studies of the disposition effect. The price of an asset in any period
was determined in two independent steps. First, the price change direction was determined by one
of 4 equally-likely underlying processes, which differed only in the probability (θ) of price going
up every period, with θ = 65%, 55%, 45%, 35% respectively. The price went down with probability
1 − θ and could not stay unchanged. Then in the second step, the price change magnitude was
randomly drawn from {1, 3, 5}. Prices of all assets in all periods were independently determined.
The underlying process that generated all prices of an asset remained the same throughout the
same round. The participants could see all 4 price sequences evolving on the same chart every
round. The price sequences were predetermined using a computer random number generation
program. For the ease of data pooling and comparison, all participants viewed the same set of
predetermined price sequences but in different random orders. Figure I shows a snapshot of the
experimental interface. Before starting the experiment, all participants needed to correctly answer
all comprehension questions regarding the experimental setup.
There were 4 treatment conditions differing in the tasks after observing the price sequences,
and the information revealed about the price-generating processes. In condition A, participants
5
were clearly instructed about how prices were generated. Then in the beginning of each round,
they received an endowment of 500 ECU.5 Before the start of period 1, they could see 6 periods of
price history, and in each of the 15 subsequent periods, before seeing new prices announced, they
had two tasks: the Prediction task and the Buying task. The Prediction task asked participants to
predict the probability of price going up for each asset, by indicating their guesses on a slider bar
from 0 to 100 percent with increments of 1 percent. The buying task was to purchase one share of
one asset at the last observed price, knowing that it was to be automatically sold when new prices
were announced. The reason for automatic selling is to give everyone equal opportunities to obtain
experience.6
After making decisions in a given period, participants saw new prices of all assets, as well as
their updated balances, and they knew the payoff that could have been obtained if a different asset
had been selected. They also saw their cumulative earnings from each asset. If the payoff from
asset k in period t is denoted by πk,t , the cumulative earning, Rk,t , was calculated as follows:
Rk,t = Rk,t−1 + I(k, k(t))πk,t ,
(1)
where I(k, k(t)) is an indicator function with the value of 1 when asset k is chosen in period t, and
0 otherwise. Once a round was concluded, the balance was not carried over to the next round. The
Prediction task was incentivized using the Quadratic Scoring Rule (QSR) [Offerman et al., 2009]
and elicited beliefs were then corrected for risk attitudes.7 Participants were given an illustrative
table and examples to understand the QSR, and were also told the payoff-maximizing strategy
5
ECU is the monetary unit in the experiment, with an exchange rate of 60 ECU = £1.
If free buying and selling were allowed, a participant could buy and hold an asset without realizing any gain or
loss. Although it may also be interesting to study the difference between experiencing capital versus paper gains and
losses, it is not pursued in this paper.
7
Suppose Belief represents the reported probability, then the prediction reward Q(Belief ) was calculated in the
following way, where m = 100 and n = 100 are parameters that determine the size of rewards:
(
m + 2n × Belief − n × [Belief 2 + (1 − Belief )2 ],
if the actual price increased
Q(Belief ) =
2
2
m + 2n × (1 − Belief ) − n × [Belief + (1 − Belief ) ], if the actual price decreased.
6
The beliefs below were corrected for risk attitudes if not otherwise mentioned. The correction procedure can be
found in Offerman et al. [2009], with the CRRA utility and a probability weighting function of following form:
ω(p) = exp(−(−β(−ln(p)))α ) where we fix α = 0.65 and β = 1.
6
was to always report one’s true beliefs. Once a participant completed the experiment, the ending
balance of a randomly chosen round determined the Buying task reward; and one randomly chosen
prediction determined the Prediction task reward.
The advantage of this experimental method is a simple Bayesian benchmark. With the knowledge of those underlying processes, the Bayesian strategy should be to count the number of ups and
downs in each sequence and to choose the asset with the most number of ups every period, so the
price change magnitudes, experienced or observed, should carry no information value. The three
price change magnitudes create clashes between the Bayesian predictions and reinforcement learning: the asset with the largest price increase does not necessarily coincide with the one with the
most ups. Therefore, departures from the Bayesian benchmark by overweighting one’s experience
can be cleanly detected. This experiment is complementary to empirical studies of reinforcement
learning in finance, because experience is not necessarily information-free in real-world settings,
making clear identification of experience overweighting relative to the Bayesian benchmark difficult.
Compared with Condition A, participants in Condition B Round 3 only had the Prediction
task, but not the Buying task. Two things were different in Condition C. On the one hand, in
Rounds 1 and 2, Condition C participants received reduced descriptive information: they were told
everything except the probabilities of price increase: with regard to price change directions, they
were only told that there were 4 equally-likely processes, HH, H, L and LL, with the following
order of their probability of price increase each period: θHH > θH > 50% > θL > θLL . Even
without specific probabilities, the ordering should be sufficient for a Bayesian updater to follow
the Bayesian strategy. On the other hand, in Condition C Round 3, participants obtained the
same information as in Condition A, but only had the Buying task. In Condition D Rounds 1 and
2, participants received even less information: they were not told how the price change directions
were determined, but only knew prices could increase or decrease and the price change magnitudes
were independently randomly drawn from {1, 3, 5}. This does not provide enough information for
Bayesian updating. Lastly in Condition D Round 3, participants were given full information. In all
7
rounds of Condition D, participants completed both Buying and Prediction tasks. With both buying
and prediction tasks present, the issue of spillover effect between these two decisions emerges. To
address this, we use two approaches: we let subjects completed these tasks in random orders to
mitigate the spillover effect, and we use the variation of tasks across treatments to test for it.8 The
reduced information treatments were for the purpose of testing whether reliance on experience was
sensitive to the completeness of descriptions.
2.2
Procedure
The experiment was conducted at the Centre for Experimental Social Sciences, Nuffield College, University of Oxford. There were a total of 4 sessions and 89 participants (excluding 1
participant who withdrew): 23 in each of Conditions A, B and C, and 20 in Condition D. Among
the recruited participants, 58 were students in the University of Oxford and the rest residents in
Oxford city, with 62% male and an average age of 31. Among the participants, 16 had trading
experience, mostly managing portfolios of less than £10000 for less than a year.
In each session, participants were first consented, and then seated in front of isolated computer
stations. They read instructions of the experiment on the computer screen and were allowed to ask
questions or to withdraw without sacrificing the show-up fee at any time during the session.9 They
were not allowed to use calculators or any computer program, but could use paper and pen.
Upon completing the experiment, participants answered a short questionnaire regarding their
demographic information, economics and mathematics background, real-world trading experience
and risk attitudes. The risk preference questionnaire was adapted from [Weber et al., 2002]. And
we also used a simple incentivized Multiple Price List to elicit risk attitudes. Upon finishing the
survey, they were paid privately one at a time in a separate room. The payment included a show-up
fee of £4, plus the payment to each part of the experiment. The average earning per subject was
8
We find that in Condition C Round 3 where participants only had the Buying task, only 25.22% of the decisions
were consistent with Bayesian, significantly different (p < 0.01) from 31.48%, the proportion of buying decisions
consistent with Bayesian in presence of the Prediction task. This suggests that belief elicitation also drove beliefs
towards Bayesian.
9
The experimental instructions are available upon request.
8
£20.70. Each session took around 2 hours.
3
Theories and Hypotheses
Psychologists have a long history of studying the effect of past experience on decisions. In
the hungry cat experiment of Thorndike [1898], a cat was placed in a box and could free itself by
either pressing a lever or pulling a loop; it initially experimented with many actions but gradually
took less time to obtain freedom in successive trials. This suggests that actions are reinforced
by their consequences: association of a response to a stimulus will be strengthened if followed
by a satisfying state, and weakened if followed by an annoying state. This theory, the law of
effect [see e.g. Herrnstein, 1970], predicts a positive correlation between choice probability and
rewarding past experience, requiring no prior knowledge regarding the environment or formation
of subjective beliefs about the future. This effect has been studied under reinforcement learning in
modern psychology, and found numerous empirical supports [see e.g. Suppes and Atkinson, 1960].
Economists have studied this type of learning mostly in game theoretic interactions. For example, reinforcement learning in games typically assume that reinforcements associated with an
action are updated by obtained payoffs, which is positively correlated with choice propensities, and
experiments demonstrated that these models outperform equilibrium predictions in a wide range
of games [see e.g. Erev and Roth, 1998; Camerer and Ho, 1999]. In the context of individual
decisions, instead of interacting with other players, the decision maker receive reinforcements on
actions by interacting with the environment. Empirical work, such as [Kaustia and Knüpfer, 2008],
has shown the plausibility of such behavior in complex financial contexts. However, empirical
studies cannot determine it underlying mechanism, because (1) investors’ choice sets in the field
cannot be easily determined, i.e. we do not know what assets they monitor and what their forgone
opportunities are, and (2) we cannot calculate the objectively correct weight that should be placed
on experience. In the experiment, if participants had placed too much weight on experience, we
would expect the law of effect to manifest: the probability of choosing asset k should be positively
9
correlated with the reinforcement on that asset; hence relative to the Bayesian benchmark, better
experienced outcomes on asset k increases the chance of it being repurchased. Reinforcement on
asset k in period t is updated as follows,
Rk,t = φRk,t−1 + I(k, k(t))πk,t ,
(2)
where φ, the discount rate of past reinforcements, captures forgetting: when φ = 1 the reinforcement updating converges to equation (1); φ = 0 implies extreme forgetting, so that only the most
recent payoff matters. The value of this parameter may vary across participants. An issue in most
reinforcement learning models is to determine the initial reinforcement value and choice propensity before the decision maker obtains any experience. The experimental setting here evades this
issue because participants observed some price history before starting, which could give them some
initial propensity to choose.
The probability that a participant chooses asset k in period t + 1 is a monotonically increasing
function of its reinforcement. This function can take many forms, including exponential (logit),
power, and normal (probit) functions. Here we use the logistic probability rule:
eλRk,t
,
Pk,t+1 = PK
λRh,t
h=1 e
(3)
where λ regulates the sensitivity of choice probability with respect to reinforcements, and the
decision maker will choose the option with the highest Pk,t+1 . To save a parameter, we assume
λ = 1. The advantage of this functional form over others is its validity even with negative payoffs.
Law of effect implies a win-stay-lose-shift strategy or trial-and-error learning [see e.g. Young,
2009; Callander, 2011]. It generates an asymmetry in the inertia propensity documented in many
experimental studies, i.e. a tendency for positive correlation between recent and current choices
[see e.g. Suppes and Atkinson, 1960; Erev and Haruvy, 2005; Cooper and Kagel, 2008]. In terms
of investment decisions, this means investors are more likely to repurchase assets on which they
gained than those on which they lost [Strahilevitz et al., 2011], although past and future perfor10
mances are not necessarily correlated.
Additionally, Jessup et al. [2008] compare the feedback and no-feedback treatments in the
presence of descriptions and find that feedback overwhelms the effect of descriptions, reducing the
overweighting of rare events and driving subjective beliefs toward objective probabilities. Lejarraga and Gonzalez [2011] further experimentally demonstrate that descriptions are neglected in the
presence of feedback regardless of task complexity. In voluntary contribution games, [Nax et al.,
2013] demonstrate that the asymmetric inertia learning pattern persists in voluntary contribution
games under a black box setting where only feedback on own actions were provided, and in a full
description environment. One possible reason for overweighting experience in real-world financial
decisions is the large quantity and complexity of descriptive information in the market. In our experiment, descriptions about the underlying processes should drive beliefs towards Bayesian, with
perfect descriptions in Conditions A and B. With reduced information in Condition C Rounds 1
and 2, a Bayesian updater still has enough information to follow the Bayesian strategy. In Condition D Rounds 1 and 2, the information setup does not allow a Bayesian updating, but could only
rely on experience. By contrast, if descriptions were neglected, we would expect the law of effect
to be robust under different information settings.
In other words, if following the law of effect is our natural tendency, then reliance on it should
be unconditional, independent from the information value of experience, knowledge about the
environment, or nature of the task. Hence the following hypotheses.
Hypothesis 1 Law of Effect: Pleasant experience with an asset increases the probability of the asset being repurchased subsequently, and unpleasant experience decreases it, even when experience
has no information value.
Hypothesis 2 Descriptions: Pleasant experience with an asset increases the probability of the
asset being repurchased subsequently, and unpleasant experience decreases it, regardless of the
descriptions provided.
The above hypotheses mainly deal with the effect of obtained payoffs. In the experiment,
since feedback on all assets were available, participants could see what payoffs would have been
11
obtained if a different asset had been selected. The effect of forgone payoffs from previously unselected alternatives is another important feature of learning, and also a very realistic consideration.
For example, the belief learning models in game theory, such as fictitious play, take into account
of forgone payoffs [see e.g. Brown, 1951; Fudenberg and Levine, 1998], assuming that players
calculate expected payoffs based on observations of past outcomes of all alternatives and choose
the one with maximum expected payoff. Many reinforcement learning models take into account
forgone payoffs as well: Erev and Roth [1998] model experimentation by adding a parameter for
the weight placed on unchosen alternatives according to their similarity to the successful ones;
the Experience Weighted Attraction (EWA) model [Camerer and Ho, 1999] considers the relative weights on forgone payoffs and obtained payoffs, without similarity judgment, hence the new
reinforcement updating rule:
Rk,t = φRk,t−1 + [δ + (1 − δ)I(k, k(t))]πk,t ,
(4)
where δ is the relative weight placed on forgone payoffs. If δ = 0, no attention is paid to unchosen
alternatives; if δ = 1, all alternatives, chosen or unchosen, are equally-weighted. Together with
the law of effect, this implies that assets with better payoffs are more likely purchased, especially
the gains are obtained. Camerer and Ho [1999] call this a ‘law of simulated effect’, as if decision
makers simulate outcomes of unchosen alternatives in their mind.
Forgone payoffs can influence behavior through the feeling of regret. For example, when the
forgone payoff is higher than the obtained payoff the decision maker might think that she could
have done better had she chosen another option. Psychologists find that experienced regret makes
the decision maker more willing to search for (even irrelevant) information [Shani and Zeelenberg,
2007], and that it provokes the feeling that one should have known better and the willingness to
correct one’s mistake Zeelenberg et al. [1998].10 Hart and Mas-Colell [2000] introduce a ’regret
matching’ adaptive procedure in games, in which the probability of choosing the current strategy
10
Post-decision experienced regret should be distinguished from anticipated regret. Economists have explicitly
modelled the latter, predicting that people make decisions to minimize anticipated regret [see e.g. Loomes and Sugden,
1982].
12
is decreasing in the size of regret, measured using the forgone payoff from unchosen strategies.
If this were true, larger regret should make participants less likely to repurchase, which poses a
limit to the positive recency resulted from law of effect: among assets offering the same obtained
payoff, the one associated with less regret should be more likely repurchased.
Hypothesis 3 Regret: A previously sold asset is more likely repurchased if it was associated with
a smaller regret.
Additionally, Nevo and Erev [2012] document another learning pattern in decision from experience, i.e. a tendency to change action after both a sharp increase and a sharp decrease of payoff,
dubbed surprise-trigger-change. Their study was motivated by [Karpoff, 1988] who find volume
spiked after both sharp price increases and decreases, suggesting increased willingness to sell after
both winning and losing. Nevo and Erev [2012] propose a model, called Inertia, Sampling and
Weighting (I-SAW), where they assume people have an innate tendency for inertia, and surprising
outcomes decrease their probability of remaining in the inertia state. If this were true, it implies
negative recency after a surprising positive obtained payoff, and positive recency, implied by law
of effect, after an unsurprising positive obtained payoff and a negative obtained payoff, hence an
asymmetry. With regard to forgone payoffs, they find negatively surprising forgone payoffs also
reduce inertia. They measure surprise using two gaps: the gap between the obtained payoffs from
action k in period t and in period t − 1, and the gap between the obtained payoff from action k in
period t and the average payoff from that action in all previous periods. In their clicking paradigm,
a surprise is just the rare event of a large positive (or negative) payoff between the two possible outcomes associated with a lottery. In our experiment, however, we define surprise as the difference
between realized payoff and expected payoff calculated from elicited beliefs, hence the following
hypothesis.
Hypothesis 4 Surprise Triggers Change: Positively surprising obtained payoffs and negatively
surprising forgone payoffs decrease the probability of repurchase.
13
4
Results and Discussion
4.1
Law of Effect
Law of effect predicts that participants should be more likely to choose an asset that brought
better payoffs in the past. In order to test this we use two different measures of experience with
an asset: the non-discounted cumulative obtained profits up to period t − 1 within a given round
(CumP rof it), or the most recent profit in period t − 1, i.e. RecP rof it. Out of 3402 total buying
decisions, 2278 (or 66.69%) were inconsistent with the Bayesian strategy.11 Among them, 818
(or 35.91%) were consistent with purchasing assets with the highest CumP rof it, and 1344 (or
59.00%) were consistent with purchasing assets with the highest RecP rof it. Another way to
evaluate this is to see how participants behaved when the prediction of Bayesian updating and law
of effect clashed. There were 261 cases where the Bayesian prediction clashed with buying at the
highest CumP rof it: 169 (or 64.75%) were consistent with Bayesian and 48 (or 18.39%) with
the latter. There were 397 cases where the Bayesian prediction clashed with buying the highest
RecP rof it: 193 (or 48.61%) were consistent with Bayesian and 176 (44.33%) with the latter.12
Thus experienced payoffs mattered, especially the most recent ones.
With each decision on an asset in a period as an observation, among the 13664 observations (not
including first periods and not including Condition B Round 3), given 4390 (3451) opportunities to
purchase at positive (negative) cumulative profit, 1053 or 23.99% (884 or 25.62%) were actually
purchased; given 1639 (1455) opportunities to purchase at positive (negative) recent profit, 906
or 55.27% (572 or 39.31%) were purchased. This comparison is graphically illustrated in Figure
II. The difference between proportions purchased was not significant when cumulative profit is
considered (p = 0.10), but highly significant with recent profit (p < 0.01).
[Insert Figure II about here.]
11
Period 1 decisions were excluded because the law of effect is only applicable once the participant obtained some
experience.
12
This is comparable to the finding in Charness and Levin [2005], that when predictions of Bayesian and reinforcement learning clashed in their experiment of individual decisions, subjects followed each in about half of the
cases.
14
We next test the law of effect in regressions controlling for beliefs and other variables, in order
to see whether experience adds additional explanatory power for buying decisions. At the decision
level, we use conditional fixed-effect logistic regressions grouped by participants, with robust standard errors. The dependent variable Buy is equal to 1 if the participant decided to buy the asset,
and 0 otherwise. The grouping controls for within-subject correlations. The independent variables
include the following: CumP rof itP os (RecP rof itP os) is a dummy variable for positive cumulative profits (recent profits) on an asset; Belief is the elicited probabilistic beliefs about price
going up next period, which is expected to be positively correlated with buying; and two additional
variables control for the observed price patterns. LastU p is equal to 1 if the recent outcome was
a price increase, and 0 otherwise; M oreU p is equal to 1 if there were more ups than downs in
the sequence, and 0 otherwise. LastU p controls for the effect of a recent observed price increase,
rather than experienced; M oreU p is the variable that a Bayesian decision maker cares about. Table
I reports the results.
[Insert Table I about here.]
All regressions clearly show significant positive correlation between belief and the buying decision, meaning that participants in general had little confusion about the task at least in terms of
making buying decisions consistent with their beliefs. Regression (1) in Table I shows that once
beliefs are controlled for, cumulative profits add no explanatory power. Regressions (2) and (4)
show that participants tended to buy an asset that brought them positive recent obtained profit, even
after controlling for the observed price patterns.
Both LastU p and M oreU p are significant in Regression (3), meaning that on the one hand
participants were, to some extent, momentum buyers chasing past returns, so that simply observing
good recent performance, without experiencing it, increased their willingness to buy; and on the
other hand they cared about the number of ups as a Bayesian does. In Regression (4), however,
recent experienced profit is still significant while LastU p is not, suggesting that participants cared
more about the experienced positive payoffs than observed ones. This is supportive evidence for
Hypothesis 1, suggesting the validity of law of effect even after controlling for beliefs, especially
15
when we consider most recent experience, and that direct experience of an outcome influences
decisions more than just observations.
Another piece of evidence for Hypothesis 1 is from asymmetric inertia, i.e. the probability
of maintaining one’s choice from last period (repurchasing an asset) after gains and losses.13 We
define a recent pleasant outcome in two ways. Firstly, it could be a larger absolute recent gain.
Panel A of Figure III illustrates a comparison of repurchases after gains and losses. The height of
the bars reflects the proportions repurchased out of opportunities to repurchase after experiencing
different levels of recent obtained profits, {−5, −3, −1, 1, 3, 5}. The proportion of repurchases
after a gain was generally larger than that after a loss, but the relationship seemed nonlinear in the
domain of losses.
Alternatively, a recent pleasant outcome could be an increase of experienced payoff from period
t − 2 to t − 1.14 There were 1317 opportunities to repurchase when payoff increased: in 716 (or
54.37%) of them the asset was repurchased; only 253 (or 35.34%) of those repurchases were
consistent with Bayesian. There were 1577 opportunities to repurchase when payoff decreased: in
689 (or 43.69%) of them the asset was repurchased; 292 (or 42.38%) of these repurchases were
Bayesian. The differences in proportions repurchased and proportions consistent with Bayesian
after payoff increases versus decreases were both significant (p < 0.01). That is to say participants
were more likely to repurchase (or remain in the inertia state) after a pleasant outcome, and that
they were more Bayesian in repurchase decisions after an unpleasant outcome. These results are
illustrated in Panel B of Figure III. The evidence using both assessors of recent experience suggests
the existence of asymmetric inertia implied by law of effect.
[Insert Figure III about here.]
The results in Table II further supports this. We use the conditional logistic regression grouped
by participants, but with decision in each period (not on each asset) as an observation. We separately estimate for the gain and the loss domains in order to detect asymmetry in behavior. The
13
The analyses above use a decision on each asset in each period as an observation, while here we treat each period
as an observation.
14
Note that the assets chosen in these two periods need not be the same.
16
dependent variable is a binary variable, Repurchaset , that is equal to 1 when in period t the decision maker repurchased an asset that was purchased in period t − 1. Beliefs are controlled for:
Belieft−1 is the period t − 1 elicited belief of price increase on the asset purchased. We create
two dummy variables in each of the gain and loss domains for the price change magnitudes of 3
and 5, so the coefficients represent their effects relative to 1 or −1. Increaset is equal to 1 if the
participant experienced a payoff increase from period t − 2 to period t − 1, and 0 otherwise. The
determinants of repurchases after a gain and a loss were different. Regressions (1) and (3) implies
asymmetric inertia: after a larger gain, participants were more likely to repurchase, and after a
larger loss, they were more likely to shift.15 However, once we consider relative payoff increase in
Regressions (2) and (4), the significance level on recent payoffs decreased in the domain of gains;
in the domain of losses, payoff increase had no effect, possibly because a loss was rarely associated with a payoff increase. Thus participants probably cared about whether there was a relative
increase in payoff, rather than just absolute gains.16 Another asymmetry lies in the effect of cumulative profits: positive cumulative profits increased the likelihood of repurchase after gains, but
had no significant effect after losses.
[Insert Table II about here.]
4.2
The Effect of Descriptions
The above regressions were pooling together treatments with different information structures.
Next we investigate the effect of differences in descriptions. In Treatment A and B with perfect
descriptions, 36.79% of the buying decisions were Bayesian, and in Treatment C with reduced
information, the number was 31.06% (difference: 5.73%, p < 0.01), although both groups of
participants should be able to use the Bayesian strategy with the information given.17 This suggests
that the descriptions were not completely neglected: more complete descriptions drove beliefs
15
The nonlinearity observed in Figure III Panel A manifests in the insignificant coefficient on recent profit −3.
The insignificance of Increase after a loss is not due to insufficient number of observations. There were 325
cases with payoff increases among 1576 cases of negative recent profits.
17
Treatment D is not shown here, because their information did not allow them to use the Bayesian strategy.
16
17
towards Bayesian.
Next we test whether descriptions influenced reliance on law of effect. To do this, we introduce binary variables C and D for Conditions C and D respectively, and interact them with
RecP rof itP os, so that significant coefficients on the interaction term would suggest differences
in the effect of experience in those conditions. The results are reported in Regression (5) of Table I.
The sign on the coefficients are not consistent with the conjecture, but they are insignificant. Being
in the conditions with less information did not significantly influence the extent to which participants relied on experience after beliefs are controlled for. This is consistent with Hypothesis 2 and
with findings in the description-experience gap literature. Therefore in the following analysis we
continue to pool all treatments together. We will show more supportive evidence for Hypothesis 2
in Section 4.4.
4.3
Regret and Surprise
Regrett−1 , the experienced regret from choosing asset k in period t − 1, is measured by the
absolute difference between obtained payoff from asset k in period t − 1 and the maximum realized payoff from unchosen assets in period t − 1. Computed from the 6 possible price change
magnitudes, the size of regret can be {0, 2, 4, 6, 8, 10}. First, if participants were more careful after
a larger regret as proposed by psychologists, we would expect more buying decisions consistent
with Bayesian. However, although there was such a tendency, the differences were not significant
at the 95% level, as shown in Panel A of Figure IV. On the other hand, according to regret theories,
larger Regrett−1 leads to a smaller probability of repurchasing, or a larger probability of shifting
to another asset. We calculated the proportion of repurchases out of opportunities to repurchase
after experiencing each value of Regrett−1 . Figure IV Panel B illustrates the results. Although
zero regret was associated with the highest repurchase rate, significantly different from those of
all other regret values, the trend was nonlinear, and the repurchase rates among all other regret
values were mostly not significantly different. Further tests in Regressions (1) and (4) of Table III
suggest regret had no significant effect on repurchases. This can emerge under two scenarios: (1)
18
based on equation (4), participants placed too little weight on forgone payoffs, or (2) their beliefs
were also distorted, i.e. they were pessimistic about unchosen assets that gave them the regret. The
regression results give some credits to the first possibility, because beliefs were already controlled
for. An analysis of beliefs in Section 4.5 will explore the second possibility.
[Insert Figure IV about here.]
Hypothesis 4 involves two types of surprise:
a positively surprising obtained payoff
(SObtained), defined as the difference between the obtained payoff from asset k in period t − 1
and the expected payoff calculated using elicited beliefs on asset k in period t − 2; a negatively surprising forgone payoff (SF orgone), defined as the absolute value of the largest negative surprise
from an unselected asset in period t−1 based on expectations elicited in period t−2. The surprisetrigger-change theory predicts larger values of both variables should lead to lower probability of
repurchase.
[Insert Table III about here.]
Regressions (2) and (5) of Table III test the effects of surprise. In the domain of gains, a larger
surprise from obtained payoff increased the probability of repurchase, which is contradictory to
Hypothesis 4; and in the domain of losses, the surprise from obtained payoff had insignificant effect. The negative surprise from forgone payoff had significant effect in none of the regressions.
Putting all variables together in Regressions (3) and (6), we find the above results still hold. In
our experiment, regret and surprise did not have the effects we expected, or the effect of reinforcement overwhelmed the effect of surprise. This was probably because the task was slightly more
complicated than in a typical clicking paradigm, involving belief updating and buying decisions.
Belief distortion could still be a plausible explanation: participants were optimistic about assets
from which they gained.
We also tested the effects of demographic variables by interacting them with independent variables in regressions (3) and (6) of Table III, but did not find significant differences. The only exception was that younger male participants had (sometimes marginally) significantly larger propensity
19
to rely on experience after gains.18
4.4
Simulation Test
The foregoing analyses use experience as an explanatory variable to establish correlations between experience and repurchasing decisions. We next conduct simulations to measure the predictive power of reinforcement learning. Six models are evaluated. The first is a baseline model
of risk-neutral expected utility with elicited beliefs, containing no parameter. The second is a
parsimonious reinforcement learning model with only one parameter, φ, the discount rate of past
reinforcements, and reinforcements update according to equation (2). The third is a two-parameter
reinforcement learning model with both φ and δ, whereas the latter is the weight on unchosen
alternatives, and reinforcements update according to equation (4). These two parameters capture
the main findings in the results above, namely the law of effect, asymmetric inertia and the effect
from forgone payoffs. For parsimony, we do not introduce parameters particularly for the effects
of regret or surprise because there was scant evidence for them in the experiment.
In the fourth model, we use CRRA expected utility, together with beliefs corrected for risk
attitudes.19 Before making buying decision for period t, the expected utility from buying asset k in
period t can be calculated according to the following:
1
1
E[Uk,t ] = Beliefk,t × [ (Wt−1 + E[∆P ])θ ] + (1 − Beliefk,t ) × [ (Wt−1 − E[∆P ])θ ],
θ
θ
(5)
where Beliefk,t is the probabilistic belief about the price of asset k going up in period t; 1 − θ is
the coefficient of relative risk aversion; Wt−1 is the ending balance in period t − 1; E[∆P ] = 3
is the expected price change, assuming that participants understood the random draw of price
change magnitudes. Then the decision maker would just choose the asset with the highest expected
18
This points to a plausible mechanism for the finding of Greenwood and Nagel [2009] that younger inexperienced
fund managers were more likely to follow trends during the tech bubble, possibly because they were more likely to
overweight recent personal experience.
19
The correction procedures can be found in Offerman et al. [2009]. Note that in the first model above, beliefs
were not corrected because of the assumed risk neutrality. In the second and third models, there was no need to correct
because beliefs were not included.
20
utility.20
The fifth model is a combination of the CRRA expected utility model and the one-parameter
reinforcement learning model. In this case, one extra parameter was added: the choice probability
of each asset will be a weighted average of the probabilities calculated from each model, with
α being the weight placed on reinforcement learning and (1 − α) on CRRA. Similarly, the last
model combines CRRA with the two-parameter reinforcement learning model, with α regulating
the relative importance of these two components.
The model parameters were estimated using nonlinear least square method. The model fits
were evaluated using the log likelihood (LL) calculated across all sampled periods.21 In order
to compare across models with different numbers of parameters, we also calculated the Akaike
Information Criterion (AIC) and the Bayesian Information Criterion (BIC).22 Table IV reports the
results. The whole sample contained 3094 observations. Reinforcement learning alone did not
generate very good fit, possibly because of its complete reliance on experience neglecting beliefs.
The CRRA model, even with only one parameter, performed much better than the baseline model.
Adding reinforcement learning to CRRA improved the fit. The estimated values of φ were small,
confirming that participants in this experiment discounted past reinforcements heavily, and placed
more weight on recent experience. The value of δ suggests that previously unchosen assets were
weighted less than 50% than experienced outcomes. This can partially explain why they did not
shift to the assets that offered high forgone payoffs.
[Insert Table IV about here.]
20
The risk attitude parameter was estimated from their asset choices, using choices in the MPL, and the questionnaire responses. But poor correlation was found among these different measures of risk attitudes, consistent with
previous evidence of the domain specificity of risk preference elicitation[see e.g. MacCrimmon and Wehrung, 1990].
21
For instance, for the two-parameter reinforcement learning model:
!
T X
N
K
X
X
i
LL(φ, δ) =
ln
(I(k i , k(t)i )Pk,t
) ,
(6)
t=1 i=1
k=1
where T denotes the number of sampled periods; N denotes the number of participants; K = 4 is the number of assets
to choose from; I(k i , k(t)i ) is an indicator function that is equal to 1 when participant i chose asset k in period t, and
i
0 for unchosen assets; Pk,t
is participant i’s probability of choosing asset k in period t calculated from the model.
22
If q represents the number of parameters and M represents the number of observations, then AIC is LL − q, and
BIC is LL − (q/2) × log(M ).
21
We then use two-thirds of the sample to calibrate parameter values and validate the model by
predicting into the rest of the sample, using the Mean Squared Deviation (MSD) to evaluate the
prediction performance.23 Even without elicited beliefs, the reinforcement learning models generated good prediction performance: the one-parameter reinforcement learning model performed
slightly better than the baseline; the two-parameter reinforcement learning model performed better
than the CRRA model. Adding the learning component to expected utility significantly improved
predictive performance. The best fit and predictive power both belong to the CRRA combined with
the two-parameter learning model.
We also conducted an additional test of the effect of descriptions, by comparing the estimated
values of α between Conditions A, B and Conditions C, D. There is no statistically significant
difference, reassuring that the differences in descriptions did not influence the extent to which participants relied on experience. Additionally, although there was considerable variation across individuals in the parameter values, there was no significant difference between demographic groups
based on gender, student status, math and economics background, and trading experience.
4.5
Experience-Induced Belief Distortion
So far our argument has been that experience added explanatory and predictive power even
after beliefs are controlled for, without considering the formation of beliefs per se. There could
be another channel, i.e. belief distortions, through which experience influenced decisions. In
order to test this, we start by investigating whether positive experienced payoffs, either cumulative
(CumP rof itP os) or recent (RecP rof itP os), influenced the participants’ optimism towards the
asset. Table V summarizes elicited beliefs, as well as the benchmark Bayesian beliefs, grouped by
positive and negative obtained payoffs.
23
This is how MSD was calculated:
M SD =
T X
N X
K
i
X
[Pk,t
− I(k i , k(t)i )]2
t=t i=1 k=1
where t is the starting period of the validation sample.
22
(T − t)N K
,
(7)
[Insert Table V about here.]
There were two important patterns. Firstly, participants tended to believe better outcomes for
assets on which they had better experience. This tendency is inconsistent with Bayesian updating. Particularly, we calculated the average elicited beliefs under positive and negative obtained
profits, and compared with Bayesian beliefs. Under both measures of experience, participants had
significantly higher beliefs about price increase (or were more optimistic) for assets on which they
gained than on which they lost. The gap was larger when we considered the most recent experience.
Bayesian beliefs were roughly the same for positive and negative experience groups.
Secondly, participants had a tendency to believe assets that offered positive forgone payoffs
to subsequently mean revert. To show this, we calculated a measure of belief in mean reversion,
called Signed Reaction Measure (SRM ).24 SRM is negative when the participant believed in
mean reversion, or that an asset whose price went up (or down) last period to be more likely to
go down (or up). We found belief in mean reversion on unchosen assets that yielded positive
forgone payoffs, and belief in continuation on assets that yielded negative forgone payoffs (difference=0.0985, p < 0.01). But Bayesian beliefs exhibited no such pattern. To put it another way, on
average, participants tended to weakly believe bad things would happen to unchosen assets. These
results suggest that previous ownership matters when it comes to belief formation.
The following regressions in Table VI confirm the above tendencies. The dependent variable, Belief , is the elicited beliefs corrected for risk attitudes. Experience is captured by two
binary variables for positive cumulative and recent profits respectively. Bayesian is the benchmark Bayesian belief. Elicited beliefs had a significant positive correlation with Bayesian beliefs.
Cumulative profits were not significant in explaining beliefs, but positive recent profits were significantly biasing beliefs upwards. In Regression (3) controlling for observed price increase, a recent
experienced price increase had a drop in its coefficient magnitude but was still highly significant,
suggesting experienced outcomes had additional influence than just observed ones. Regression
24
SRM is equal to the elicited belief minus 0.5, when the previous outcome was an up; and it is equal to the
elicited belief minus 0.5, multiplied by −1 when the previous period outcome was a down. This measure was adapted
from an experimental study of the law of small numbers in Asparouhova et al. [2009].
23
(4) introduced forgone payoffs, F orgoneHigh, a binary variable that is equal to 1 for a foregone
payoff higher than the obtained payoff. The negative sign on this variable indicates that participants tended to guess lower probability of price increase for assets that performed better than their
purchased asset. These results point towards an experience-induced belief distortion, that has not
been well documented or understood in the economics literature. They can potentially partially
reconcile the findings regarding regret and surprise as discussed above.25
[Insert Table VI about here.]
4.6
Discussion
In conventional theories of economics and finance, the past should only be relevant for future
decisions in its informational value, and thus for this purpose no distinction is made between experienced and observed past outcomes. However, empirical and experimental studies document clear
departures from these predictions. The cohort effect offers an example: people who experienced
the great depression exhibited different risk preferences and investment choices compared with
those who only learned from textbooks [Malmendier and Nagel, 2009; Malmendier et al., 2011].
Investors tend to place too much weight on their own experience for various reasons, such as the
difficulty in collecting and processing descriptive information, or simply emotions [Shiller, 2002].
In this paper, we find strong evidence for the law of effect in experimental asset buying tasks
with full descriptive information and complete feedback. This effect can potentially explain some
empirical puzzles in finance, such as investors’ unjustifiably erecting and eradicating certain investment styles [Barberis and Shleifer, 2003]. For another instance, Feng and Seasholes [2005]
find that individual investors’ sophistication and trading experience combined completely eliminated their tendency to hold losers, but only 37% of their tendency to sell winners, for which they
could not find a plausible explanation. This might be driven by naı̈ve reinforcement learning or
25
A comparison of beliefs in Condition B Round 3, where there was no Buying task, with the other rounds revealed
the following: in Condition B Round 3 the elicited beliefs had a correlation coefficient of 0.1297 (p < 0.01) with
Bayesian beliefs; and this correlation coefficient was only 0.0855 (p < 0.01) for the rest of rounds where participants
had both Buying and Prediction tasks. This is another piece of evidence for experienced outcomes to potentially drive
beliefs away Bayesian when participants had stakes involved in the Buying task.
24
asymmetric inertia: selling winners always leads to capital gains, which positively reinforces the
action, while holding on to losers leads to mixed results and the negative outcomes could make
it unattractive due to loss aversion. More empirical and experimental studies should address the
relation between experience-based learning and the role of experience in disposition effect.
We show that in presence of both descriptions and feedback, decision makers may overweight
experience regardless of the descriptive details. The relative importance of descriptions and experience can be exploited to help individual investors understand the risks associated with financial
products [Kaufmann et al., 2013; Bradbury et al., 2014]. Additionally, we find that people attached
considerable weight to recent experience. This can cause the reinforcement strategy to be lossinducing, because personal recent experience is just a small and possibly unrepresentative sample
of the underlying distribution. The management literature argues that reliance on experience may
lead managers to overlook long-term objectives, leading to failure [Levinthal and March, 1993].
Rare disastrous events may not exist in recent experience, leading people to under-predict a crisis,
hence the black swan argument in Taleb [2007], such as during the formation of an asset price
bubble.
Additionally, beliefs were distorted by experience in our experiment. In the literature, belief
distortions can be caused by the desirability of these events [see e.g. Mayraz, 2011], or by the
similarity with past cases in memory [Billot et al., 2005]. However, little has been done with
regard to belief distortion induced by different experienced payoffs associated with states in the
past. We uncovered some evidence that beliefs were more optimistic on assets that offered positive
obtained payoffs but more pessimistic on unchosen assets, and thus ownership mattered. This could
be driving our participants to repurchase assets from which they obtained positively surprising
payoffs, and not to shift to assets that offered high forgone payoffs. Cognitive dissonance can
also play a role in belief formation here. These patterns of beliefs call for further theoretical and
empirical works.
25
5
Conclusion
This paper uses an experimental investment task with full descriptive information and feed-
back to investigate the role of experience in individual investment decisions under uncertainty.
This issue has not been well-understood and has important implications for theories and empirical
studies of individual investor behavior. The experiment creates an environment with clear tension
between Bayesian strategy and the experience-based reinforcement strategy, so that experienced
outcomes has no value to a Bayesian agent with complete knowledge of the incentive structure.
In the experiment, participants made buying decisions and price predictions after observing some
price sequences. We document a double-channeled mechanism through which experience influence decisions. After beliefs are controlled for, participants made decisions consistent with the
law of effect, repurchasing assets on which they had good prior experience, especially placing
more weight on recent outcomes. Experienced outcomes had additional explanatory power than
observed ones. On the other hand, beliefs were also biased by obtained and forgone payoffs.
Additionally, we find that larger regret, positively surprising obtained payoff and negatively
surprising forgone payoff did not significantly decrease the propensity to repurchase in the subsequent period; a positive surprise from obtained payoff actually increased repurchases. It is possible
that in more complex settings, such as in financial context, certain learning patterns may be different from those documented in the clicking paradigm. The biased beliefs can potentially explain
these differences: after an asset had a recent value appreciation, beliefs about future price increase
tended to be biased upward when the gain was obtained, and downward when the gain was forgone. The experimental results in this paper call for more research efforts on learning in individual
decisions, especially in financial contexts, and on experience-induced belief distortions.
D EPARTMENT OF E CONOMICS , U NIVERSITY OF OXFORD
26
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30
purchased, and 0 otherwise. The independent variables include the following: Belief is the
elicited belief; CumProfitPos and RecProfitPos are binary variables that are equal to 1 for
positive cumulative profit or recent profit respectively; LastUp is a binary variable that is 1 if the
last outcome was up; MoreUp is a binary variable if the sequence contained more ups than
downs; C (or D) is a binary variable for Condition C (or D) Rounds 1 and 2. Robust standard
errors are in parentheses.
(*** p<0.01, **
* p<0.10)
Table I: D ETERMINANTS
OFp<0.05,
B UYING
D ECISIONS : L AW OF E FFECT
(1)
Belief
8.4898
***
(0.9376)
CumProfitPos
(2)
8.0263
(3)
***
(0.8864)
8.1389
***
(0.8919)
(4)
7.8456
***
(5)
7.8433***
(0.8728)
(0.8720)
0.8253***
0.8687***
(0.1083)
(0.1180)
0.1336
0.1328
(0.0940)
(0.1091)
(0.1098)
0.4443***
0.4397***
0.4381***
(0.1508)
(0.1436)
(0.1435)
0.1376
(0.0915)
0.8850***
RecProfitPos
(0.0987)
LastUp
0.3032
MoreUp
***
C×RecProfitPos
-0.2692
(0.2156)
D×RecProfitPos
-0.0784
(0.2371)
N
LL
Pseudo R
2
12956
12956
12956
12956
12956
-5563.16
-5438.35
-5611.40
-5393.70
-5392.24
0.2117
0.2294
0.2195
0.2357
0.2359
Note: This table reports the results of testing effect of experience and description on buying
decisions, using conditional logistic regression grouped by subjects, and a decision on each asset
in each period as an observation. The dependent variable is Buy, which is equal to 1 if the asset
is purchased, and 0 otherwise. The independent variables include the following: Belief is the
elicited belief; CumP rof itP os and RecP rof itP os are binary variables that are equal to 1 for
positive cumulative profit or recent profit respectively; LastU p is a binary variable that is 1 if the
last outcome was up; M oreU p is a binary variable if the sequence contained more ups than downs;
C (or D) is a binary variable for Condition C (or D) Rounds 1 and 2. Robust standard errors are
in parentheses. (*** p < 0.01, ** p < 0.05, * p < 0.10)
31
Belieft-1 is elicited probabilistic belief on the asset purchased in the previous period; CumProfit is
equal to 1 for positive cumulative profits on an asset up to period t-1, and 0 otherwise; Increaset
is equal to 1 when the payoff increased from period t-2 to t-1; and there are four binary variables,
one for each possible period t-1 profit, 3, 5, -3 and -5. Regressions (1) and (2) are in the domain
of gains; (3) and (4) are in the domain of losses. Robust standard errors are in parentheses. (***
p<0.01, ** p<0.05,
* p<0.10)
Table II:
D ETERMINANTS OF R EPURCHASES : A SYMMETRIC I NERTIA
After Gains
(1)
Belieft-1
0.0287
***
(0.0060)
CumProfitPos
0.9831
***
(0.1779)
RecProfit=3
0.3566
**
(0.1492)
RecProfit=5
0.5022
***
(0.1855)
After Losses
(2)
0.0290
***
(0.0059)
1.0197
***
(0.1811)
0.2809
(3)
0.0259
***
(0.0081)
(0.0082)
-0.2824
-0.2807
(0.2019)
(0.1998)
-0.0826
-0.0803
(0.1522)
0.3330*
(0.1957)
(0.1880)
RecProfit=-5
-0.5125
**
(0.2281)
0.4482
***
LL
Pseudo R
2
(0.1928)
-0.5073**
(0.2511)
0.0134
(0.1291)
N
0.0259***
*
RecProfit=-3
Increaset
(4)
(0.1766)
1540
1540
1108
1108
-756.95
-750.26
-430.90
-430.90
0.1137
0.1215
0.0539
0.0539
Note: This table shows the results of testing the effect of gains and losses on repurchasing,
using conditional logistic regression grouped by subjects, and a decision in each period as an
observation. The dependent variable is Repurchaset , which is equal to 1 if the asset bought in
previous period was repurchased, and 0 otherwise. Independent variables include the following:
Belieft−1 is elicited probabilistic belief on the asset purchased in the previous period; CumP rof it
is equal to 1 for positive cumulative profits on an asset up to period t-1, and 0 otherwise; Increaset
is equal to 1 when the payoff increased from period t−2 to t−1; and there are four binary variables,
one for each possible period t − 1 profit, 3, 5, -3 and -5. Regressions (1) and (2) are in the domain
of gains; (3) and (4) are in the domain of losses. Robust standard errors are in parentheses. (***
p < 0.01, ** p < 0.05, * p < 0.10)
32
previously defined, as well as the following: Regret is the difference between obtained payoff
and the largest forgone payoff in period t-1; SObtained is the difference between the obtained
payoff and the participants’ expected payoff from asset k in period t-1; SForgone is the
difference between the forgone payoff and the expected payoff from an unchosen asset k in
period t-1. Regressions (1) to (3) are in the domain of gains; (4) to (6) are in the domain of losses.
Robust standard
in parentheses.
p<0.01, ** :p<0.05,
* p<0.10)
Tableerrors
III: Dare
ETERMINANTS
OF(***
R EPURCHASES
R EGRET
AND S URPRISE
After Gains
(1)
Belieft-1
0.0597
***
(0.0083)
CumProfitPos
1.0834
***
(0.1878)
After Losses
(2)
0.0599
***
(0.0079)
0.9685
***
(0.1822)
RecProfit=3
(3)
0.0598
***
(0.0079)
0.9651
***
(0.1838)
(4)
0.0791
***
(5)
0.0793
***
(6)
0.0791***
(0.0180)
(0.0178)
(0.0181)
0.0636
0.0652
0.0283
(0.2368)
(0.2316)
(0.2332)
0.1079
(0.1597)
RecProfit=5
0.1163
(0.2026)
RecProfit=-3
-0.2091
(0.2339)
RecProfit=-5
-0.3491
(0.3414)
Increaset
0.3880
***
(0.1406)
Regret
0.6283
***
(0.1444)
-0.0108
(0.0354)
SObtained
0.0971
0.0273
0.0181
(0.1615)
(0.2201)
(0.1983)
(0.2585)
0.0278
0.0128
0.0435
(0.0475)
(0.0571)
(0.0345)
0.1142
SForgone
0.6504
***
***
0.1145
***
-0.0279
-0.0297
(0.0235)
(0.0232)
(0.0284)
(0.0299)
0.0502
0.0511
0.0084
0.0061
(0.0379)
(0.0381)
(0.0346)
(0.0345)
N
1442
1442
1442
917
917
917
LL
-648.55
-629.10
-628.69
-311.25
-310.78
-310.06
0.1841
0.2085
0.2091
0.1726
0.1738
0.1757
Pseudo R
2
Note: This table reports the results of testing hypotheses involving regret and surprise-trigger
change, using conditional logistic regression grouped by subjects, and a decision in each period as
an observation. The dependent variable is Repurchaset , which is equal to 1 if the asset bought
in period t − 1 was repurchased in period t, and 0 otherwise. Independent variables include those
previously defined, as well as the following: Regret is the difference between obtained payoff and
the largest forgone payoff in period t − 1; SObtained is the difference between the obtained payoff
and the participants expected payoff from asset k in period t − 1; SF orgone is the difference
between the forgone payoff and the expected payoff from an unchosen asset k in period t − 1.
Regressions (1) to (3) are in the domain of gains; (4) to (6) are in the domain of losses. Robust
standard errors are in parentheses. (*** p < 0.01, ** p < 0.05, * p < 0.10)
33
weight on unchosen alternatives; θ is the coefficient of relative risk aversion; α is the weight
placed on reinforcement learning. The parameter estimates using the whole sample are reported
first. Then we use two-thirds of the sample to calibrate the models, and predict choices in the rest
of the sample. For comparison of model fit with varying number of parameters, Log-likelihood
(LL), AIC, and BIC are reported. For evaluation of predictive power, Mean Squared Deviation
(MSD) is reported. (* Table
indicates
best fit.)
IV: the
M ODEL
C ALIBRATION AND VALIDATION
Models
(1) Baseline
(2) RL1
(3) RL2
(4) CRRA
(5) CRRRA
(6) CRRA
+RL1
+RL2
Parameter Estimates (Whole Sample, N=3094)
Parameters
0
ϕ
1
2
0.1211
(0.2490)
δ
1
3
4
0.1650
0.1887
0.2123
(0.3080)
(0.3140)
(0.3382)
0.2330
0.4418
(0.2844)
(0.3938)
θ
0.6240
0.5085
0.5882
(0.3965)
(0.4527)
(0.4337)
0.3195
0.3932
(0.2532)
(0.3222)
α
LL
-5116.41
-5867.66
-5621.28
-3799.55
-3542.18
-3245.08
AIC
-5116.41
-5868.66
-5623.28
-3800.55
-3545.18
-3249.08*
BIC
-5116.41
-5871.68
-5629.32
-3803.57
-3554.24
-3261.15*
Calibration (Sample Size=2078)
LL
-3547.12
-3954.82
-3798.36
-2593.74
-2421.84
-2217.76
AIC
-3547.12
-3955.82
-3800.36
-2594.74
-2424.84
-2221.76*
BIC
-3547.12
-3958.64
-3806.00
-2597.56
-2433.30
-2233.04*
Validation (Sample Size=1016)
LL
-1569.29
-1999.78
-1901.29
-1226.17
-1167.93
-1092.09
MSD
0.1982
0.1965
0.1824
0.1627
0.1533
0.1428*
Note: This table reports the results of the performance of six models: (1) baseline model: risk
neutral expected utility with elicited beliefs; (2) reinforcement learning with one parameter; (3)
reinforcement learning with two parameters; (4) CRRA expected utility with beliefs corrected for
risk attitudes; (5) CRRA expected utility with corrected beliefs, and reinforcement learning with
one parameter; (6) CRRA expected utility with corrected beliefs, and reinforcement learning with
two parameters. Parameters: φ is the discount rate of past reinforcements; δ is the relative weight
on unchosen alternatives; θ is the coefficient of relative risk aversion; α is the weight placed on
reinforcement learning. The parameter estimates using the whole sample are reported first. Then
we use two-thirds of the sample to calibrate the models, and predict choices in the rest of the
sample. For comparison of model fit with varying number of parameters, Log-likelihood (LL),
AIC, and BIC are reported. For evaluation of predictive power, Mean Squared Deviation (MSD) is
reported. (* indicates the best fit.)
34
(RecProfit). The table also compares Signed Reaction Measure (SRM) of beliefs on an asset
unchosen in the last period after its positive and negative forgone payoffs. SRM is defined as the
elicited belief minus 0.5, multiplied by -1 if the previous outcome was a price decrease, and not
if the previous outcome was a price increase. Negative SRM indicates belief in mean reversion.
In each case, the comparison results of Bayesian beliefs (Bayesian) are also reported, as a
benchmark. The p-values from t-tests
Standard
errors are in parentheses.
Table are
V: Sreported.
UMMARY
OF B ELIEFS
CumProfit
Positive
Negative
N
3584
2814
Bayesian
0.5044
0.5033
(0.0554)
(0.0561)
0.4530
0.4389
(0.2134)
(0.2172)
Elicited Belief
RecProfit
p-value
p=0.43
p<0.01
Positive
Negative
1954
975
0.5165
0.5153
(0.0555)
(0.0584)
0.5038
0.4562
(0.2326)
(0.2219)
p-value
p=0.59
p<0.01
Forgone Payoff
Positive
Negative
N
3104
4438
Bayesian
0.0154
0.0145
(0.0514)
(0.0521)
SRM of
-0.0429
0.0556
Elicited Belief
(0.2084)
(0.2321)
p=0.46
p<0.01
Note: This table reports the comparison results of elicited beliefs after positive and negative
cumulative profits on an asset (CumP rof it), after positive and negative last period profit on an
asset (RecP rof it). The table also compares Signed Reaction Measure (SRM ) of beliefs on an
asset unchosen in the last period after its positive and negative forgone payoffs. SRM is defined
as the elicited belief minus 0.5, multiplied by -1 if the previous outcome was a price decrease,
and not if the previous outcome was a price increase. Negative SRM indicates belief in mean
reversion. In each case, the comparison results of Bayesian beliefs (Bayesian) are also reported,
as a benchmark. The p-values from t-tests are reported. Standard errors are in parentheses.
35
Independent variables include the following: Bayesian is the benchmark beliefs calculated using
the Bayes’ Rule; CumProfitPos is equal to 1 if the cumulative profit on an asset was positive,
and 0 otherwise; RecProfitPos is equal to 1 if the recent profit on an asset was positive, and 0
otherwise; LastUp is equal to 1 if the last observed outcome was a price increase; ForgoneHigh
is equal to 1 if the forgone payoff from the asset was higher than the obtained payoff last period.
Standard errors clustered by subjects
(*** p<0.01, ** p<0.05, * p<0.10)
Tableare
VI:inBparentheses.
ELIEF D ISTORTION
(1)
Bayesian
0.3059
***
(0.1026)
CumProfitPos
0.0216
(2)
0.2473
***
(0.0920)
*
(0.0119)
RecProfitPos
(3)
0.1713
*
0.1016
(0.0100)
LastUp
0.2047**
(0.0939)
(0.0918)
0.0123
0.0121
(0.0120)
***
(4)
0.0770
***
(0.0093)
0.0519
***
(0.0125)
(0.0119)
0.0671***
(0.0092)
0.0534***
(0.0124)
-0.0246***
ForgoneHigh
(0.0070)
Constant
N
R
2
0.2823
***
0.3054
***
0.3166
***
0.3079***
(0.0517)
(0.0464)
(0.0476)
(0.0466)
13432
13432
13432
13432
0.0085
0.0305
0.0440
0.0465
Note: This table reports the results of testing the determinants of elicited beliefs, using OLS regressions. The dependent variable is Belief , the elicited belief about price going up on each asset.
Independent variables include the following: Bayesian is the benchmark beliefs calculated using
the Bayes Rule; CumP rof itP os is equal to 1 if the cumulative profit on an asset was positive,
and 0 otherwise; RecP rof itP os is equal to 1 if the recent profit on an asset was positive, and 0
otherwise; LastU p is equal to 1 if the last observed outcome was a price increase; F orgoneHigh
is equal to 1 if the forgone payoff from the asset was higher than the obtained payoff last period.
Standard errors clustered by subjects are in parentheses. (*** p < 0.01, ** p < 0.05, * p < 0.10)
36
W X
X Y Z W X
Y Z X Panel A
W X
X Y Z W X
X X
X Y Z Panel B
Figure 1 Experiment Screenshots
Note: This figure shows two Figure
example
from
the experimental computer interface.
I: screenshots
E XPERIMENT
S CREENSHOTS
Panel This
A is the
screen
participants
sawscreenshots
in Period 1from
of Round
1 in Stage 2computer
of the experiment,
Note:
figure
shows
two example
the experimental
interface. when
Panel
they
were
asked
to
make
the
buying
decision.
Panel
B
is
the
screen
that
announced
the
result
of
A is the screen participants saw in Period 1 of Round 1 in Stage 2 of the experiment, when they
Periodasked
1 of to
Round
after
that period
concluded.
were
make1the
buying
decision.
Panel B is the screen that announced the result of Period
1 of Round 1 after that period concluded.
37
.6
Proportions Repurchased
.3
.4
.5
.2
Negative
Positive
Recent Profits
Negative
Positive
Cumulative Profits
Figure 2 Proportions Purchased Given Positive and Negative Experience
Note:
is a barPchart
showing
the proportions
purchased
out ofE XPERIENCE
opportunities to
FigureThis
II: Pfigure
ROPORTIONS
URCHASED
G IVEN
P OSITIVE AND
N EGATIVE
purchase
an asset
at chart
(1) positive
recent
profit, out
andof(2)
positive and
negative
Note:
This figure
is a bar
showingand
the negative
proportions
purchased
opportunities
to purchase
cumulative
profit
in
all
previous
periods
in
a
given
round.
The
95%
confidence
interval
bars
an asset at (1) positive and negative recent profit, and (2) positive and negative cumulative profit
in
were
added.
all previous periods in a given round. The 95% confidence interval bars were added.
38
.7
Proportions Repurchased
.2
.3
.4
.5
.6
.1
0
-5
-3
-1
1
Recent Profit
Actual Values
Fited Values
3
5
95% CI of Actual
95% CI of Fitted
.3
.4
Proportions
.5
.6
Panel A
Payoff Decrease
Payoff Increase
Payoff Decrease
Repurchases/Opportunities
Payoff Increase
Bayesians/Repurchases
Panel B
Figure 3 Asymmetric Inertia Figure III: A SYMMETRIC I NERTIA
Note: Panel
PanelAAis aisbar
a chart
bar chart
showing
the proportion
repurchased
out of opportunities
to
Note:
showing
the proportion
repurchased
out of opportunities
to repurchase
repurchase
after each
possible
payoff
fromperiod,
the previous
period,
with 95%
after
each possible
payoff
from the
previous
{−5, −3,
−1, 1, {-5,-3,-1,1,3,5},
3, 5}, with 95% confidence
confidence
interval
bars;
theissolid
line is afit,quadratic
and the
lines
the 95%
interval
bars;
the solid
line
a quadratic
and the fit,
dashed
linesdashed
indicate
theindicate
95% confidence
interval
of the
fitted values.
shows Panel
two things:
(1) the
repurchased
out of
confidence
interval
of the Panel
fitted Bvalues.
B shows
twoproportions
things: (1)
the proportions
opportunities
to
repurchase
after
a
payoff
increase
and
after
a
payoff
decrease;
(2)
the
proportion
repurchased out of opportunities to repurchase after a payoff increase and after a payoff decrease;
of
decisions
the Bayesian
strategy
repurchases
after a out
payoff
(2)repurchase
the proportion
of consistent
repurchasewith
decisions
consistent
withouttheof Bayesian
strategy
of
increase
and
after
a
payoff
decrease,
with
95%
confidence
interval
bars.
repurchases after a payoff increase and after a payoff decrease, with 95% confidence interval
bars.
39
.7
.6
Proportions Bayesian
.2
.3
.4
.5
.1
0
0
2
4
6
Last Period Regret
Actual Values
Fited Values
8
10
95% CI of Actual
95% CI of Fitted
0
.1
Proportions Repurchased
.2
.3
.4
.5
.6
.7
Panel A
0
2
4
6
Last Period Regret
Actual Values
Fited Values
8
10
95% CI of Actual
95% CI of Fitted
Panel B
Figure 4 The Effect of Regret Figure IV: T HE E FFECT OF R EGRET
Note: Regret
Regret isis defined
definedasasthe
theabsolute
absolutevalue
valueofof
difference
between
obtained
payoff
Note:
thethe
difference
between
thethe
obtained
payoff
and and
the
the
maximum
forgone
payoff
in
the
previous
period.
Potential
magnitudes
of
regret
in
the
maximum forgone payoff in the previous period. Potential magnitudes of regret in the experiment
experiment
{0,2,4,6,8,10}.
A is showing
a bar chart
showing the
proportions
decisions
are
{0, 2, 4, 6,are
8, 10}.
Panel A is Panel
a bar chart
the proportions
decisions
consistent
with
Bayesian
after predictions
different magnitudes
of regret,
with 95%ofconfidence
interval
Panel
consistentpredictions
with Bayesian
after different
magnitudes
regret, with
95% bars.
confidence
B
is a barbars.
chartPanel
showing
of assets the
repurchased
out of opportunities
to repurchase
interval
B isthea proportions
bar chart showing
proportions
assets repurchased
out of
after
different
magnitudes
of
regret,
with
95%
confidence
interval
bars.
In
both
panels,
the
solid
opportunities to repurchase after different magnitudes of regret, with 95% confidence interval
line
a quadratic
fit, and
dashed
thefit,
95%
confidence
interval
the fittedthe
values.
bars.isIn
both panels,
thethe
solid
linelines
is a indicate
quadratic
and
the dashed
linesofindicate
95%
confidence interval of the fitted values.
40