There has been much reported on decisions from experience, also referred to as decisions in a complete ignorance fashion. This note lays out a Bayesian decision-theoretical framework that provides a computable account for decisions from experience. To make the framework more tractable, this note sets up and examines decisions in a incomplete ignorance fashion. The current discussion asserts that well-known behavioural effects, such as the hot stove effect, and the Bayesian framework may lead to different predictions

Fujikawa, Takemi

Kobayashi, Yohei

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
An incomplete ignorance state in
repeated-play decision making: A note on
Bayesian decision-theoretical framework
Yohei Kobayashi and Takemi Fujikawa
Universiti Sains Malaysia
1. December 2010
Online at https://mpra.ub.uni-muenchen.de/28265/
MPRA Paper No. 28265, posted 23. January 2011 21:30 UTC
A Incomplete Ignorance State in
Repeated-play Decision Making: A Note on
Bayesian Decision-theoretical Framework
Yohei Kobayashi¤ Takemi Fujikawa†
December 2010
Abstract
There has been much reported on decisions from experience, also re-
ferred to as decisions in a complete ignorance fashion. This note lays out
a Bayesian decision-theoretical framework that provides a computable ac-
count for decisions from experience. Tomake the frameworkmore tractable,
this note sets up and examines decisions in a incomplete ignorance fashion.
The current discussion asserts that well-known behavioural effects, such as
the hot stove effect, and the Bayesian framework may lead to different pre-
dictions.
Keywords: complete ignorance, decisions from experience, incomplete ig-
norance
¤Advanced Medical and Dental Institute, Universiti Sains Malaysia. Email:
kobapie@gmail.com.
†Graduate School of Business, Universiti Sains Malaysia. Email: takemi@usm.my.
1
1 Introduction
The Bayesian decision-theoretical framework is useful to examine behavioural
tendencies in decisions under ambiguity. A concept of the Bayesian framework
is one of normative framework that are ubiquitously used by behaviourists to
provide a computable account for behavioural tendencies in “decisions from
experience”. It asserts that the Bayesian decision maker’s ultimate goal is to
judge the likelihood of events by updating her/his subjective probabilities in the
face of new evidence as a result of sequential search process (Fujikawa, 2007).
An area of decisions from experience (also referred to as “decisions in a com-
plete ignorance fashion”) is fast moving. Many research based on laboratory
experiments has been presented (e.g., Barron and Erev, 2003; Hertwig, Barron,
Weber, and Erev, 2004; Weber, Shaﬁr, and Blais, 2004; Erev and Barron, 2005;
Yechiam and Busemeyer, 2006; Fujikawa, 2009; Barron and Yechiam, 2009). For
example, Table 1 shows two choice problems from Fujikawa (2009), where the
participants chose, at each period t (t = 1, 2, . . . , 400), between two unmarked
buttons that provided outcomes sampled from two distributions, “R” and “S”.
Let (v, p) denote a distribution, where the outcome v occurs with probability p
(otherwise zero). The right hand column shows the aggregated proportion of R
choices over 400 trials. The maximisation rate over the 400 trials were 0.28 in
Problem 2, for example. This result suggested that deviations from maximisa-
tion (i.e., the participants’ less selection of R) in a state of “complete ignorance”
were the consequence of the hot stove effect that could lead to a bias toward
S. The existence of the hot stove effect in decisions from experience was exam-
ined by analysing results of Fujikawa’s (2009) experiment, involving the state of
complete ignorance.
The participants did not receive prior information on payoff structure, but
2
Problem R S PR
1 (4, 0.8) (3, 1) 0.48
2 (32, 0.1) (3, 1) 0.22
Table 1: Three choice problems and aggregated proportion of R choices in Fu-
jikawa (2009). The notation (v, p) denotes a prospect that provides v with prob-
ability p and 0 otherwise.
received the feedback that was limited to obtained payoffs at each round t. That
is, the experiment was run on a state of complete ignorance, where they were
disclosed neither the possible payoffs nor its likelihoods. The apparatus in this
state, however, seems to have challenging in examining the existence of the hot
stove effect in light of the Bayesian framework that combines prior information
on payoff distributions with data to obtain a posterior estimate. The participants
in Fujikawa (2009) who were in the state of complete ignorance were likely to
fail to use Bayesian framework, as they were not provided with any prior infor-
mation on the payoff distributions.
This note extends Fujikawa (2009) by laying out the Bayesian decision-theoretical
framework that accounts for decisions from experience. Instead of a state of
complete ignorance employed by Fujikawa (2009), we shall employ in this note
a state of “incomplete ignorance” in which the decision makers (DMs) can ob-
tain posterior estimates calculated by the Bayesian framework. A state of in-
complete ignorance is deﬁned as one, where the DMs are disclosed possible
payoffs of available options, but not disclosed likelihood of the payoffs. Making
possible payoffs available to the participants could allow them to update data
through the Bayesian framework.
3
2 A State of Complete Ignorance
We now deﬁne a state of complete ignorance as the state, where the DM’s prior
probability distribution (i.e., the distribution at t = 0) is uniform. A state of
complete ignorance was experimentally manipulated in previous studies on de-
cisions from experience given above. The authors used pairwise choice prob-
lems, such as the following Problem X:
Problem X. Choose between
AX: a points with probability of P¤AX ; 0 otherwise
BX: b points with probability of P¤BX ; 0 otherwise
We let a, b > 0, P¤AX , P
¤
BX 2 [0, 1], aP¤AX > bP¤BX . Typically, the DM is not pro-
vided any prior information on the payoff structure, and repeatedly asked to
make decisions, relying on the obtained feedback in the situation in the past.
Thus, it is unknown to the DM that one selection of AX (BX) yields a (b) points
with probability of P¤AX (P
¤
BX ) and zero point with 1¡ P¤AX (1¡ P¤BX ). It is, how-
ever, known to her that one selection of each option yields certain payoffs with
unknown probabilities. Suppose that the DM is asked to choose either AX or BX
t (0 · t · T) times in Problem X. Given that each of the mutually exclusive and
exhaustive outcomes x is equally likely, the prior probability distribution of AX
is
f0(x) =
8>><>>:
1
x1+1
for 0 · x · x1,
0 for x < 0 or x > x1.
In case of a state of complete ignorance, x1 may largely vary among the DMs,
as they do not have any prior information on possible outcomes and probabili-
ties. For example, some participants in Fujikawa (2009) were likely to have high
4
x1, while others low x1. Thus, it seems that a prior probability distribution on
AX would largely vary among the participants.
3 A State of Incomplete Ignorance
We here aim at introducing the Bayesian framework to examine behavioural
tendencies in decisions from experience. For this aim, we here present a state of
incomplete ignorance that is concerned with the situation, where the DMs have
incomplete information on a payoff structure. Let us consider the following
Problem Y:
Problem Y. Choose between
AY: g points with probability of P¤AY ; 0 otherwise
BY: q points with probability of P¤BY ; 0 otherwise
We let g, q > 0, P¤AY , P
¤
BY
2 [0, 1]. The DM makes a choice between AY and BY
at each period t (t = 1, 2, . . . , T). She is informed of possible payoffs of each
option, but not informed of corresponding probabilities. That is, she knows that
one selection of AY (BY) yields g (q) points with an unknown probability. Thus,
the density function of her a priori formulated beliefs is
f (x) =
8>><>>:
0.5 if x = g or x = 0,
0 otherwise.
A goal of Bayesian DMs is to compute a posteriori probabilities from a priori
probabilities. Since the DM does not possess the available objective prior in-
formation on P¤AY , she is to compute a posteriori probabilities of P
¤
AY
from a priori
5
probabilities of P¤AY , and her experienced probability Pexp. Below, we shall apply
a theoretical framework to the case of P¤AY , as the same argument holds for the
case of P¤BY . For computing the a posteriori probabilities, we consider the case of
n events: We deﬁne an event Rk (k = 1, 2, . . . , n) as an event that P¤AY falls within
region k ( k¡1n · P¤AY < kn ) at period t. Each Rk has a probability Pt(Rk) =
St(k)
n
that deﬁnes the likelihood of Rk. Each event is exclusive to the all of the other
events. Thus,
8x 2 1, 2, ..., n, 8y 2 1, 2, ..., n, x 6= y : Pt(Rx \ Ry) = 0. (1)
Note that
Pt(
n[
1
Rk) =
n
å
k=1
Pt(Rk) =
ånk=1 St(k)
n
= 1 (2)
as P¤AY belongs to one of the regions. For example, if n = 2, there are two possible
events that are R1 (0 · P¤AY < 0.5) and R2 (0.5 · P¤AY < 1), and their probabili-
ties are Pt(R1) =
St(1)
2 and Pt(R2) =
St(2)
2 . The P
¤
AY
belongs to either of the region,
so the sum of the probabilities is Pt(R1) + Pt(R2) = Pt(R1 [ R2) = 1. By deﬁ-
nition, we can calculate the probability of the event Pt+1(Wt+1) at period t+ 1,
where Wt+1 is an event that the highest payoff, g, being realised at the period
t + 1. To calculate the probability Pt+1(Wt+1), we separate it to Pt+1(Wt+1jRk)
when an event Rk occurs. From eq.(1) and eq.(2), we can calculate Pt+1(Wt+1) as
follows:
Pt+1(Wt+1) = Pt+1(Wt+1j
n[
1
Rk)
=
n
å
k=1
Pt+1(Wt+1jRk)Pt(Rk)
=
n
å
k=1
Pt+1(Wt+1jRk)St(k)n . (3)
6
Now we estimate and calculate Pt+1(Wt+1jRk). We approximate this as fol-
lows:
Pt+1(Wt+1jRk) = 2k¡ 12n . (4)
Plugging this into eq.(3) gives
Pt+1(Wt+1) =
n
å
k=1
(2k¡ 1)St(k)
2n2
. (5)
Having observed the result of period t+ 1 sequence, the DM updates the pos-
sibility St(k)n of an event Rk by using the Bayesian framework. If an event Wt+1
occured, the possibility of an event Rk (i.e.,
St+1(k)
n ) is
St+1(k)
n
= Pt+1(RkjWt+1) =
Pt+1(Wt+1jRk)Pt(Rk)
Pt+1(Wt+1)
=
(2k¡ 1)
2n
St(k)
n
ånj=1
(2j¡ 1)St(j)
2n2
=
(2k¡ 1)St(k)
ånj=1(2j¡ 1)St(j)
. (6)
7
If an eventWt+1 is not realised, the possibility of an event Rk (i.e.,
St+1(k)
n ) is
St+1(k)
n
= Pt+1(RkjWt+1) =
Pt+1(Wt+1jRk)Pt(Rk)
Pt+1(Wt+1)
=
(2n¡ 2k+ 1)
2n
St(k)
n
ånj=1
(2n¡ 2j+ 1)St(j)
2n2
=
(2n¡ 2k+ 1)St(k)
ånj=1(2n¡ 2j+ 1)St(j)
. (7)
Thus, the possibility of an event Rk (i.e.,
St+1(k)
n ) is
St+1(k)
n
=
8>>>>>>>>>><>>>>>>>>>>:
(2k¡ 1)St(k)
ånj=1(2j¡ 1)St(j)
if Wt+1 is realised
(2n¡ 2k+ 1)St(k)
ånj=1(2n¡ 2j+ 1)St(j)
ifWt+1 is realised.
(8)
Eq.(8) is discrete, and we apply it to continuous representations. By so doing,
we get
St+1(k) =
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
lim
n!¥
(2k¡ 1)St(k)
ånj=1(2j¡ 1)St(j)
1
n
if Wt+1 is realised
lim
n!¥
(2n¡ 2k+ 1)St(k)
ånj=1(2n¡ 2j+ 1)St(j)
1
n
ifWt+1 is realised.
(9)
8
By using inﬁnity, we assume St(k)n to be a continuous probability distribu-
tion. Then, we deﬁne the continuous probability density function pt(x)with the
following properties:
lim
n!¥
Z k
n
0
pt(x)dx = limn!¥
k
å
j=0
pt
³
j
n
´
n
= lim
n!¥
k
å
j=0
St(j)
n
(10)
and
lim
n!¥
Z 1
k
n
pt(x)dx = limn!¥
n
å
j=k
pt
³
j
n
´
n
= lim
n!¥
n
å
j=k
St(j)
n
, (11)
where k = 1, 2, . . . , n, and x is the imaginary possibility of the P¤AY .
Thus, Eq.(9) is
pt+1
µ
k
n
¶
=
8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:
lim
n!¥
(2k¡ 1)
n
pt
³ k
n
´
ånj=1
(2j¡ 1)
n
pt
³ j
n
´ 1
n
ifWt+1 is realised
lim
n!¥
(2n¡ 2k+ 1)
n
pt
³ k
n
´
ånj=1
(2n¡ 2j+ 1)
n
pt
³ j
n
´ 1
n
ifWt+1 is realised.
(12)
9
Then, we obtain
pt+1
µ
k
n
¶
=
8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:
lim
n!¥
k
n
pt
³ k
n
´
ånj=1
j
n
pt
³ j
n
´ 1
n
ifWt+1 is realised
lim
n!¥
0B@1¡ kn
1CA pt³ kn´
ånj=1
0B@1¡ jn
1CA pt³ jn´ 1n
ifWt+1 is realised.
(13)
From eq.(11), we have
pt+1(x) =
8>>>>>>>>>><>>>>>>>>>>:
xpt(x)R 1
0 ypt(y)dy
ifWt+1 is realised
(1¡ x)pt(x)R 1
0 (1¡ y)pt(y)dy
ifWt+1 is realised.
(14)
As shown in eq.(11), pt(x) is a probability density of Pt(Rk), so that
Z 1
0
pt(x)dx = 1. (15)
Without losing much generality and accuracy, it can be said that the DMs are
updating the probability density pt(x) from experienced results. Simply put, we
10
deﬁne
ft(x) =
8>>>>>><>>>>>>:
fw(x) = x ifWt+1 is realised
fl(x) = 1¡ x ifWt+1 is realised.
(16)
Eq.(14) is transformed as
pt+1(x) =
ft(x)pt(x)R 1
0 ft(y)pt(y)dy
. (17)
11
Eq.(17) holds true at any time period t, thus
pt+1(x) =
ft(x)pt(x)R 1
0 ft(y)pt(y)dy
=
ft(x)
ft¡1(x)pt¡1(x)R 1
0 ft¡1(z)pt¡1(z)dzR 1
0 ft(y)
ft¡1(y)pt¡1(y)R 1
0 ft¡1(z)pt¡1(z)dz
dy
=
ft(x) ft¡1(x)pt¡1(x)R 1
0 ft(y) ft¡1(y)pt¡1(y)dy
=
ft(x) ft¡1(x) ft¡2(x)pt¡2(x)R 1
0 ft(y) ft¡1(y) ft¡2(x)pt¡2(x)dy
...
=
ft(x) ft¡1(x) ft¡2(x) ft¡3(x).... f2(x) f1(x)p1(x)R 1
0 ft(y) ft¡1(y) ft¡2(x) ft¡3(x).... f2(x) f1(x)p1(x)dy
=
p1(x)Õtk=1 fk(x)R 1
0 p1(y)Õ
t
k=1 fk(y)dy
. (18)
Letting r be a total number of times of the highest payoff realised, we have
t
Õ
k=1
fk(x) = fw(x)r fl(x)(t¡r) = x(tPexp)(1¡ x)(t(1¡Pexp)). (19)
Finally, we obtain
pt+1(x) =
p1(x)x(tPexp)(1¡ x)(t(1¡Pexp))R 1
0 p1(y)y
(tPexp)(1¡ y)(t(1¡Pexp))dy
. (20)
12
Figure 1: The process of probability dencity change in case of realisedWt+1 and
Wt+1. We label the axes: P¤AY on x-axis and pt(x) on y-axis.
Thus, pt+1(x) only depends on Pexp (i.e., the proportion of the highest payoff
realised) at period t and initial probability density of p1(x). If we assume the
initial probability density function has the constant value as p1(x) = 1 then
eq.(20) only depends on Pexp as
pt+1(x) =
x(tPexp)(1¡ x)(t(1¡Pexp))R 1
0 y
(tPexp)(1¡ y)(t(1¡Pexp))dy
. (21)
13
Figure 2: The temporal development of population distribution in the case of
P¤AY = 0.5.
4 Discussion and Concluding Remarks
Figure 1 shows a process of the earlier stage: It shows how the probability den-
sity changes as a result of an alternative case in each trial. At period t = 1, there
is no prior information on the probability of the highest payoff, so that its den-
sity is constant as p1(x) = 1. After the decision is made at t = 1, the DM will
face either of the two events in choosing an option at t = 2: (1) an event that
the highest payoff is realised (i.e., an eventW2); (2) an event that the lowest pay-
off is realised (i.e., W2). If the highest payoff is realised, the probablity density
function is updated as p2(x) = 2x by the result. If, on the other hand, the lowest
payoff is realised, the probability density function is updated as p2(x) = 2¡ 2x
by the result. At t = 3, the following three cases are possible: ﬁrst, p3(x) = 3x2
if eventsW2 andW3 are realised. Second, p3(x) = 6x(1¡ x) if either of the two
conditions are met: (i) a condition that W2 and W3 are realised; (ii) a condition
thatW2 andW3 are realised. Third, p3(x) = 3(1¡ x)2 ifW2 andW3 are realised.
Figure 2 shows the earlier stage of the population histogram for the case
of P¤AY = 0.5. In this case, the population of the DMs are divided into two
symmetrical situations: (1) a situation, where the lucky DMs face an event W,
and (2) a situation, where the unlucky DMs face an event W. Stochastically
14
Figure 3: The temporal development of population distribution in the case of
P¤AY = 0.1.
and approximately, the population is gathered to the center of the histogram,
predicting the probability density function as similar to Gaussian.
On the other hand, Figure 3 shows the earlier stage of the population his-
togram for the case of P¤AY = 0.1. In this case, the population of the DMs
are divided into the asymmetrical situation. Stochastically and approximately,
the population is gathered to Pexp = 0.1, predicting the the probability den-
sity function with its maximum value at x = 0.1. From these aspects, by using
the Bayesian framework, we can predict P¤AY intuitively by the mathematical
background without any previous information, and its coefﬁcient of conﬁdence
stochastically depends on its trial number which increases the value of it.
An attention is to be given to future laboratory experiments to show ro-
bustness of the Bayesian decision-theoretical framework developed in this note,
which could provide an alternative account for behavioural tendencies in deci-
sions from experience. For example, Fujikawa (2009) presented experiments on
decisions from experience, and discussed the existence of the hot stove effect,
the predictions of which are different from predictions implied by the Bayesian
framework. Future experimental work will help us document whether people
often make choices predicted by well-known behavioural effects (e.g., the hot
stove effect and payoff variability effect), or choices predicted by the Bayesian
15
framework.
References
Barron, G., I. Erev. 2003. Small feedback-based decisions and their limited corre-
spondence to description-based decisions. Journal of Behavioral Decision Mak-
ing 16 215–233.
Barron, G., E. Yechiam. 2009. The coexistence of overestimation and under-
weighting of rare events and the contingent recency effect. Judgment and De-
cision Making 4 447–460.
Erev, I., G. Barron. 2005. On adaptation, maximization, and reinforcement learn-
ing among cognitive strategies. Psychological Review 112 912–931.
Fujikawa, T. 2007. Perfect Bayesian vs. imperfect Bayesian in small decision
making problems. Behaviormetrika 34 27–44.
Fujikawa, T. 2009. On the relative importance of the hot stove effect and the
tendency to rely on small samples. Judgment and Decision Making 4 429–435.
Hertwig, R., G. Barron, E. Weber, I. Erev. 2004. Decisions from experience and
the effect of rare events in risky choices. Psychological Science 15 534–539.
Weber, E. U., S. Shaﬁr, A. R. Blais. 2004. Predicting risk sensitivity in humans
and lower animals: Risk as variance or coefﬁcient of variation. Psychological
Review 111 430–445.
Yechiam, E., J. R. Busemeyer. 2006. The effect of foregone payoffs on under-
weighting small probability events. Journal of Behavioral Decision Making 19
1–16.
16


An incomplete ignorance state in repeated-play decision making: A note on Bayesian decision-theoretical framework

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveAn incomplete ignorance state inrepeated-play decision making: A note onBayesian decision-theoretical frameworkYohei Kobayashi and Takemi FujikawaUniversiti Sains Malaysia1. December 2010Online at http://mpra.ub.uni-muenchen.de/28265/MPRA Paper No. 28265, posted 23. January 2011 21:30 UTCA Incomplete Ignorance State inRepeated-play Decision Making: A Note onBayesian Decision-theoretical FrameworkYohei Kobayashi¤ Takemi Fujikawa†December 2010AbstractThere has been much reported on decisions from experience, also re-ferred to as decisions in a complete ignorance fashion. This note lays outa Bayesian decision-theoretical framework that provides a computable ac-count for decisions from experience. Tomake the frameworkmore tractable,this note sets up and examines decisions in a incomplete ignorance fashion.The current discussion asserts that well-known behavioural effects, such asthe hot stove effect, and the Bayesian framework may lead to different pre-dictions.Keywords: complete ignorance, decisions from experience, incomplete ig-norance¤Advanced Medical and Dental Institute, Universiti Sains Malaysia. Email:kobapie@gmail.com.†Graduate School of Business, Universiti Sains Malaysia. Email: takemi@usm.my.11 IntroductionThe Bayesian decision-theoretical framework is useful to examine behaviouraltendencies in decisions under ambiguity. A concept of the Bayesian frameworkis one of normative framework that are ubiquitously used by behaviourists toprovide a computable account for behavioural tendencies in “decisions fromexperience”. It asserts that the Bayesian decision maker’s ultimate goal is tojudge the likelihood of events by updating her/his subjective probabilities in theface of new evidence as a result of sequential search process (Fujikawa, 2007).An area of decisions from experience (also referred to as “decisions in a com-plete ignorance fashion”) is fast moving. Many research based on laboratoryexperiments has been presented (e.g., Barron and Erev, 2003; Hertwig, Barron,Weber, and Erev, 2004; Weber, Shafir, and Blais, 2004; Erev and Barron, 2005;Yechiam and Busemeyer, 2006; Fujikawa, 2009; Barron and Yechiam, 2009). Forexample, Table 1 shows two choice problems from Fujikawa (2009), where theparticipants chose, at each period t (t = 1, 2, . . . , 400), between two unmarkedbuttons that provided outcomes sampled from two distributions, “R” and “S”.Let (v, p) denote a distribution, where the outcome v occurs with probability p(otherwise zero). The right hand column shows the aggregated proportion of Rchoices over 400 trials. The maximisation rate over the 400 trials were 0.28 inProblem 2, for example. This result suggested that deviations from maximisa-tion (i.e., the participants’ less selection of R) in a state of “complete ignorance”were the consequence of the hot stove effect that could lead to a bias towardS. The existence of the hot stove effect in decisions from experience was exam-ined by analysing results of Fujikawa’s (2009) experiment, involving the state ofcomplete ignorance.The participants did not receive prior information on payoff structure, but2Problem R S PR1 (4, 0.8) (3, 1) 0.482 (32, 0.1) (3, 1) 0.22Table 1: Three choice problems and aggregated proportion of R choices in Fu-jikawa (2009). The notation (v, p) denotes a prospect that provides v with prob-ability p and 0 otherwise.received the feedback that was limited to obtained payoffs at each round t. Thatis, the experiment was run on a state of complete ignorance, where they weredisclosed neither the possible payoffs nor its likelihoods. The apparatus in thisstate, however, seems to have challenging in examining the existence of the hotstove effect in light of the Bayesian framework that combines prior informationon payoff distributions with data to obtain a posterior estimate. The participantsin Fujikawa (2009) who were in the state of complete ignorance were likely tofail to use Bayesian framework, as they were not provided with any prior infor-mation on the payoff distributions.This note extends Fujikawa (2009) by laying out the Bayesian decision-theoreticalframework that accounts for decisions from experience. Instead of a state ofcomplete ignorance employed by Fujikawa (2009), we shall employ in this notea state of “incomplete ignorance” in which the decision makers (DMs) can ob-tain posterior estimates calculated by the Bayesian framework. A state of in-complete ignorance is defined as one, where the DMs are disclosed possiblepayoffs of available options, but not disclosed likelihood of the payoffs. Makingpossible payoffs available to the participants could allow them to update datathrough the Bayesian framework.32 A State of Complete IgnoranceWe now define a state of complete ignorance as the state, where the DM’s priorprobability distribution (i.e., the distribution at t = 0) is uniform. A state ofcomplete ignorance was experimentally manipulated in previous studies on de-cisions from experience given above. The authors used pairwise choice prob-lems, such as the following Problem X:Problem X. Choose betweenAX: a points with probability of P¤AX ; 0 otherwiseBX: b points with probability of P¤BX ; 0 otherwiseWe let a, b > 0, P¤AX , P¤BX 2 [0, 1], aP¤AX > bP¤BX . Typically, the DM is not pro-vided any prior information on the payoff structure, and repeatedly asked tomake decisions, relying on the obtained feedback in the situation in the past.Thus, it is unknown to the DM that one selection of AX (BX) yields a (b) pointswith probability of P¤AX (P¤BX ) and zero point with 1¡ P¤AX (1¡ P¤BX ). It is, how-ever, known to her that one selection of each option yields certain payoffs withunknown probabilities. Suppose that the DM is asked to choose either AX or BXt (0 · t · T) times in Problem X. Given that each of the mutually exclusive andexhaustive outcomes x is equally likely, the prior probability distribution of AXisf0(x) =8>><>>:1x1+1for 0 · x · x1,0 for x < 0 or x > x1.In case of a state of complete ignorance, x1 may largely vary among the DMs,as they do not have any prior information on possible outcomes and probabili-ties. For example, some participants in Fujikawa (2009) were likely to have high4x1, while others low x1. Thus, it seems that a prior probability distribution onAX would largely vary among the participants.3 A State of Incomplete IgnoranceWe here aim at introducing the Bayesian framework to examine behaviouraltendencies in decisions from experience. For this aim, we here present a state ofincomplete ignorance that is concerned with the situation, where the DMs haveincomplete information on a payoff structure. Let us consider the followingProblem Y:Problem Y. Choose betweenAY: g points with probability of P¤AY ; 0 otherwiseBY: q points with probability of P¤BY ; 0 otherwiseWe let g, q > 0, P¤AY , P¤BY2 [0, 1]. The DM makes a choice between AY and BYat each period t (t = 1, 2, . . . , T). She is informed of possible payoffs of eachoption, but not informed of corresponding probabilities. That is, she knows thatone selection of AY (BY) yields g (q) points with an unknown probability. Thus,the density function of her a priori formulated beliefs isf (x) =8>><>>:0.5 if x = g or x = 0,0 otherwise.A goal of Bayesian DMs is to compute a posteriori probabilities from a prioriprobabilities. Since the DM does not possess the available objective prior in-formation on P¤AY , she is to compute a posteriori probabilities of P¤AYfrom a priori5probabilities of P¤AY , and her experienced probability Pexp. Below, we shall applya theoretical framework to the case of P¤AY , as the same argument holds for thecase of P¤BY . For computing the a posteriori probabilities, we consider the case ofn events: We define an event Rk (k = 1, 2, . . . , n) as an event that P¤AY falls withinregion k ( k¡1n · P¤AY < kn ) at period t. Each Rk has a probability Pt(Rk) =St(k)nthat defines the likelihood of Rk. Each event is exclusive to the all of the otherevents. Thus,8x 2 1, 2, ..., n, 8y 2 1, 2, ..., n, x 6= y : Pt(Rx \ Ry) = 0. (1)Note thatPt(n[1Rk) =nåk=1Pt(Rk) =ånk=1 St(k)n= 1 (2)as P¤AY belongs to one of the regions. For example, if n = 2, there are two possibleevents that are R1 (0 · P¤AY < 0.5) and R2 (0.5 · P¤AY < 1), and their probabili-ties are Pt(R1) =St(1)2 and Pt(R2) =St(2)2 . The P¤AYbelongs to either of the region,so the sum of the probabilities is Pt(R1) + Pt(R2) = Pt(R1 [ R2) = 1. By defi-nition, we can calculate the probability of the event Pt+1(Wt+1) at period t+ 1,where Wt+1 is an event that the highest payoff, g, being realised at the periodt + 1. To calculate the probability Pt+1(Wt+1), we separate it to Pt+1(Wt+1jRk)when an event Rk occurs. From eq.(1) and eq.(2), we can calculate Pt+1(Wt+1) asfollows:Pt+1(Wt+1) = Pt+1(Wt+1jn[1Rk)=nåk=1Pt+1(Wt+1jRk)Pt(Rk)=nåk=1Pt+1(Wt+1jRk)St(k)n . (3)6Now we estimate and calculate Pt+1(Wt+1jRk). We approximate this as fol-lows:Pt+1(Wt+1jRk) = 2k¡ 12n . (4)Plugging this into eq.(3) givesPt+1(Wt+1) =nåk=1(2k¡ 1)St(k)2n2. (5)Having observed the result of period t+ 1 sequence, the DM updates the pos-sibility St(k)n of an event Rk by using the Bayesian framework. If an event Wt+1occured, the possibility of an event Rk (i.e.,St+1(k)n ) isSt+1(k)n= Pt+1(RkjWt+1) =Pt+1(Wt+1jRk)Pt(Rk)Pt+1(Wt+1)=(2k¡ 1)2nSt(k)nånj=1(2j¡ 1)St(j)2n2=(2k¡ 1)St(k)ånj=1(2j¡ 1)St(j). (6)7If an eventWt+1 is not realised, the possibility of an event Rk (i.e.,St+1(k)n ) isSt+1(k)n= Pt+1(RkjWt+1) =Pt+1(Wt+1jRk)Pt(Rk)Pt+1(Wt+1)=(2n¡ 2k+ 1)2nSt(k)nånj=1(2n¡ 2j+ 1)St(j)2n2=(2n¡ 2k+ 1)St(k)ånj=1(2n¡ 2j+ 1)St(j). (7)Thus, the possibility of an event Rk (i.e.,St+1(k)n ) isSt+1(k)n=8>>>>>>>>>><>>>>>>>>>>:(2k¡ 1)St(k)ånj=1(2j¡ 1)St(j)if Wt+1 is realised(2n¡ 2k+ 1)St(k)ånj=1(2n¡ 2j+ 1)St(j)ifWt+1 is realised.(8)Eq.(8) is discrete, and we apply it to continuous representations. By so doing,we getSt+1(k) =8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:limn!¥(2k¡ 1)St(k)ånj=1(2j¡ 1)St(j)1nif Wt+1 is realisedlimn!¥(2n¡ 2k+ 1)St(k)ånj=1(2n¡ 2j+ 1)St(j)1nifWt+1 is realised.(9)8By using infinity, we assume St(k)n to be a continuous probability distribu-tion. Then, we define the continuous probability density function pt(x)with thefollowing properties:limn!¥Z kn0pt(x)dx = limn!¥kåj=0pt³jn´n= limn!¥kåj=0St(j)n(10)andlimn!¥Z 1knpt(x)dx = limn!¥nåj=kpt³jn´n= limn!¥nåj=kSt(j)n, (11)where k = 1, 2, . . . , n, and x is the imaginary possibility of the P¤AY .Thus, Eq.(9) ispt+1µkn¶=8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:limn!¥(2k¡ 1)npt³ kn´ånj=1(2j¡ 1)npt³ jn´ 1nifWt+1 is realisedlimn!¥(2n¡ 2k+ 1)npt³ kn´ånj=1(2n¡ 2j+ 1)npt³ jn´ 1nifWt+1 is realised.(12)9Then, we obtainpt+1µkn¶=8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:limn!¥knpt³ kn´ånj=1jnpt³ jn´ 1nifWt+1 is realisedlimn!¥0B@1¡ kn1CA pt³ kn´ånj=10B@1¡ jn1CA pt³ jn´ 1nifWt+1 is realised.(13)From eq.(11), we havept+1(x) =8>>>>>>>>>><>>>>>>>>>>:xpt(x)R 10 ypt(y)dyifWt+1 is realised(1¡ x)pt(x)R 10 (1¡ y)pt(y)dyifWt+1 is realised.(14)As shown in eq.(11), pt(x) is a probability density of Pt(Rk), so thatZ 10pt(x)dx = 1. (15)Without losing much generality and accuracy, it can be said that the DMs areupdating the probability density pt(x) from experienced results. Simply put, we10defineft(x) =8>>>>>><>>>>>>:fw(x) = x ifWt+1 is realisedfl(x) = 1¡ x ifWt+1 is realised.(16)Eq.(14) is transformed aspt+1(x) =ft(x)pt(x)R 10 ft(y)pt(y)dy. (17)11Eq.(17) holds true at any time period t, thuspt+1(x) =ft(x)pt(x)R 10 ft(y)pt(y)dy=ft(x)ft¡1(x)pt¡1(x)R 10 ft¡1(z)pt¡1(z)dzR 10 ft(y)ft¡1(y)pt¡1(y)R 10 ft¡1(z)pt¡1(z)dzdy=ft(x) ft¡1(x)pt¡1(x)R 10 ft(y) ft¡1(y)pt¡1(y)dy=ft(x) ft¡1(x) ft¡2(x)pt¡2(x)R 10 ft(y) ft¡1(y) ft¡2(x)pt¡2(x)dy...=ft(x) ft¡1(x) ft¡2(x) ft¡3(x).... f2(x) f1(x)p1(x)R 10 ft(y) ft¡1(y) ft¡2(x) ft¡3(x).... f2(x) f1(x)p1(x)dy=p1(x)Õtk=1 fk(x)R 10 p1(y)Õtk=1 fk(y)dy. (18)Letting r be a total number of times of the highest payoff realised, we havetÕk=1fk(x) = fw(x)r fl(x)(t¡r) = x(tPexp)(1¡ x)(t(1¡Pexp)). (19)Finally, we obtainpt+1(x) =p1(x)x(tPexp)(1¡ x)(t(1¡Pexp))R 10 p1(y)y(tPexp)(1¡ y)(t(1¡Pexp))dy. (20)12Figure 1: The process of probability dencity change in case of realisedWt+1 andWt+1. We label the axes: P¤AY on x-axis and pt(x) on y-axis.Thus, pt+1(x) only depends on Pexp (i.e., the proportion of the highest payoffrealised) at period t and initial probability density of p1(x). If we assume theinitial probability density function has the constant value as p1(x) = 1 theneq.(20) only depends on Pexp aspt+1(x) =x(tPexp)(1¡ x)(t(1¡Pexp))R 10 y(tPexp)(1¡ y)(t(1¡Pexp))dy. (21)13Figure 2: The temporal development of population distribution in the case ofP¤AY = 0.5.4 Discussion and Concluding RemarksFigure 1 shows a process of the earlier stage: It shows how the probability den-sity changes as a result of an alternative case in each trial. At period t = 1, thereis no prior information on the probability of the highest payoff, so that its den-sity is constant as p1(x) = 1. After the decision is made at t = 1, the DM willface either of the two events in choosing an option at t = 2: (1) an event thatthe highest payoff is realised (i.e., an eventW2); (2) an event that the lowest pay-off is realised (i.e., W2). If the highest payoff is realised, the probablity densityfunction is updated as p2(x) = 2x by the result. If, on the other hand, the lowestpayoff is realised, the probability density function is updated as p2(x) = 2¡ 2xby the result. At t = 3, the following three cases are possible: first, p3(x) = 3x2if eventsW2 andW3 are realised. Second, p3(x) = 6x(1¡ x) if either of the twoconditions are met: (i) a condition that W2 and W3 are realised; (ii) a conditionthatW2 andW3 are realised. Third, p3(x) = 3(1¡ x)2 ifW2 andW3 are realised.Figure 2 shows the earlier stage of the population histogram for the caseof P¤AY = 0.5. In this case, the population of the DMs are divided into twosymmetrical situations: (1) a situation, where the lucky DMs face an event W,and (2) a situation, where the unlucky DMs face an event W. Stochastically14Figure 3: The temporal development of population distribution in the case ofP¤AY = 0.1.and approximately, the population is gathered to the center of the histogram,predicting the probability density function as similar to Gaussian.On the other hand, Figure 3 shows the earlier stage of the population his-togram for the case of P¤AY = 0.1. In this case, the population of the DMsare divided into the asymmetrical situation. Stochastically and approximately,the population is gathered to Pexp = 0.1, predicting the the probability den-sity function with its maximum value at x = 0.1. From these aspects, by usingthe Bayesian framework, we can predict P¤AY intuitively by the mathematicalbackground without any previous information, and its coefficient of confidencestochastically depends on its trial number which increases the value of it.An attention is to be given to future laboratory experiments to show ro-bustness of the Bayesian decision-theoretical framework developed in this note,which could provide an alternative account for behavioural tendencies in deci-sions from experience. For example, Fujikawa (2009) presented experiments ondecisions from experience, and discussed the existence of the hot stove effect,the predictions of which are different from predictions implied by the Bayesianframework. Future experimental work will help us document whether peopleoften make choices predicted by well-known behavioural effects (e.g., the hotstove effect and payoff variability effect), or choices predicted by the Bayesian15framework.ReferencesBarron, G., I. Erev. 2003. Small feedback-based decisions and their limited corre-spondence to description-based decisions. Journal of Behavioral Decision Mak-ing 16 215–233.Barron, G., E. Yechiam. 2009. The coexistence of overestimation and under-weighting of rare events and the contingent recency effect. Judgment and De-cision Making 4 447–460.Erev, I., G. Barron. 2005. On adaptation, maximization, and reinforcement learn-ing among cognitive strategies. Psychological Review 112 912–931.Fujikawa, T. 2007. Perfect Bayesian vs. imperfect Bayesian in small decisionmaking problems. Behaviormetrika 34 27–44.Fujikawa, T. 2009. On the relative importance of the hot stove effect and thetendency to rely on small samples. Judgment and Decision Making 4 429–435.Hertwig, R., G. Barron, E. Weber, I. Erev. 2004. Decisions from experience andthe effect of rare events in risky choices. Psychological Science 15 534–539.Weber, E. U., S. Shafir, A. R. Blais. 2004. Predicting risk sensitivity in humansand lower animals: Risk as variance or coefficient of variation. PsychologicalReview 111 430–445.Yechiam, E., J. R. Busemeyer. 2006. The effect of foregone payoffs on under-weighting small probability events. Journal of Behavioral Decision Making 191–16.16

English

https://mpra.ub.uni-muenchen.de/28265/1/MPRA_paper_28265.pdf

An incomplete ignorance state in repeated-play decision making: A note on Bayesian decision-theoretical framework

Abstract

Similar works

Full text

Available Versions

Munich Personal RePEc Archive

Munich RePEc Personal Archive