ISSN (Print) 0473-453X

Discussion Paper No. 1213

ISSN (Online) 2435-0982

AN EXPERIMENT ON
A MULTI-PERIOD BEAUTY CONTEST GAME

Nobuyuki Hanaki
Yuta Takahashi

Revised January 2024
September 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

An Experiment ona Multi-Period Beauty Contest Game∗
Nobuyuki Hanaki†

Yuta Takahashi‡

January 23, 2024

Abstract
We present and conduct a novel experiment on a multi-period beauty contest
game motivated by the canonical New-Keynesian model. Participants continuously provide forecasts for prices spanning multiple future periods. These
forecasts determine the price for the current period and participants’ payoffs.
Our findings are threefold. First, the observed prices in the experiment deviate
more from the rational expectations equilibrium prices under strategic complementarity than under strategic substitution. Second, participants’ expectations
respond to announcements of future shocks on average. Finally, participants
employ heuristics in their forecasting; however, the choice of heuristic varies
with the degree of strategic complementarity.

Keywords: Expectation formation, Learning-to-forecast experiment
JEL Code: C92, D84, E70∗

We thank Elena Asparouhova, Te Bao, John Duffy, Charles Noussair, Yoshiyasu Ono, Lube
Petersen, Marc Willinger, and the participants at 2023 CEFM workshop (Nagasaki), AP-ESA 2023
(Seoul), EF 2023 (Sofia), and 2023 ESA world meeting (Lyon) for their comments and helpful
discussion. This research has benefited from the financial support of (a) the Joint Usage/ResearchCenter (JJURC), the Institute of Social and Economic Research (ISER), and Osaka University, and(b) Grants-in-aid for Scientific Research Nos. 18K19954, 20H05631, and 23H00055 from the Japan
Society for the Promotion of Science. The design of the experiment reported in this paper wasapproved by the IRB of ISER (#20221102) in November 2022. The experiment is preregistered at
aspredicted.org (#128898), which is available at https://aspredicted.org/ix6th.pdf. We thankYuka Tamura and Yuta Shimodaira for their help in preparing software for the experiment, and Yuki
Hamada, Hiroko Shibata, Saori Shikibu, and Satsuki Yamada for their support in conducting theexperiment.
†
Corresponding author.
Institute of Social and Economic Research, Osaka University,and University of Limassol.
6-1 Mihogaoka, Ibaraki, Osaka, 567-0047, Japan.
E-mail:nobuyuki.hanaki@iser.osaka-u.ac.jp. TEL: +81 (6) 6879 8552 FAX: +81 (6) 6879 8583‡
Institute of Economic Research, Hitotsubashi University

1

1

Introduction

The assumption that individuals form rational expectations is pervasive and often
taken for granted in macroeconomic modeling. Over the last five decades, a substantial body of macro and experimental economics literature has evaluated the rational
expectations equilibrium (REE) hypothesis. One notable finding from the macroeconomics literature is that models with the strong form of rational expectations canlead to erratic implications, such as the forward-guidance puzzle. Consequently, researchers have become increasingly skeptical of this assumption, proposing alternative
frameworks (for example, Angeletos and Lian, 2018; Farhi and Werning, 2019; Garcı́aSchmidt and Woodford, 2019).
Another finding emerges from the literature on so called “learning-to-forecast”
experiments (Marimon and Sunder, 1993). This literature finds that the results ofnumerous experiments are inconsistent with the strong form of rational expectations(see, among others, Hommes et al., 2005; Heemeijer et al., 2009; Bao et al., 2017).
In these experiments, unlike an experiment where participants make their forecasts
about exogenously-generated series (e.g., Afrouzi et al., 2023), participants submit
an expectation about an aggregate outcome (e.g., prices), and the aggregate outcome
is endogenously determined through an aggregation of these individual forecasts.1
1

Note that unlike experiments in which participants actually trade in the market (see, e.g., Smithet al., 1988; Haruvy et al., 2007; Akiyama et al., 2017; Asparouhova et al., 2016; Crockett et al.,2019), set prices (see, e.g., Fehr and Tyran, 2008; Noussair et al., 2015; Orland and Roos, 2013,2019; Petersen, 2015) or quantities (see, e.g., Bosch-Domènech and Vriend, 2003; Huck et al., 1999,
2004; Offerman et al., 2002), in learning-to-forecast experiments, participants do not trade, set prices
or quantities, but only forecast. This eliminates the need for participants to trade or set prices (orquantity) optimally and thus allows us to focus on their expectation formation and its aggregate consequences. See, Bao et al. (2013), for a comparison with learning-to-forecast experiments with thosewhere participants need to decide on the quantities (which the authors call “learning-to-optimize”
experiments). The learning-to-forecast experimental framework has been used to investigate policyrelevant questions, such as the causal impacts of the central bank communication on the expectationand the aggregate outcomes (see, e.g., Kryvtsov and Petersen, 2021; Mokhtarzadeh and Petersen,2021).

2

This feedback effect is often referred to as an expectational feedback : past events canshape individual expectations which, in turn, affect the current aggregate market
outcome. As a result, the results of these experiments can deviate substantially,
depending on the nature and the strength of the feedback, from those predictedby the rational expectations hypothesis. Instead, they can be better explained bymodels with backward-looking expectation formation (see, for example, Anufriev
and Hommes, 2012; Evans et al., 2021).
Notably, most of these existing learning-to-forecast experiments are essentiallysingle-period.2 Participants only need to forecast the outcome of today or in one
future period (e.g., tomorrow), and the realized outcome is a function of submitted
forecasts regarding this single-period (see Bao et al., 2021, for a survey).3 As a result,
it is difficult to provide useful insights for macroeconomists who study expectation
formation in a multi-period environment. In a typical macroeconomic model andmore generally a dynamic game, individuals forecast not only actions of others in one
period, but also their actions in multiple future periods. And these forecasts about
multiple future periods determine the current outcome. The single-period framework
does not enable us to analyze how these series of expectations change in such a multiperiod environment, for example, in response to news about future shocks.
We contribute to the literature by filling this gap. We propose and conduct a novel
experiment using a multi-period beauty contest game motivated by the core part ofthe canonical New-Keynesian model (e.g., Woodford (2003) and Galı́ (2015), which is
initially developed by Calvo (1983)). In our theoretical model, there are many firms,
and each firm faces the same linear demand curve. The demand for this firm depends
2

There are several studies, mostly recent, who study multi-period experiments. We discuss themin detail later.
3
This approach stands in stark contrast to survey-based studies on expectation formation, which
elicit expectations of the same future periods multiple times and study their evolution (see, e.g.,
Coibion et al., 2018).

3

on the price of the firm and the aggregate price. The demand decreases as the firm
increases its price, but can increase or decrease as other firms increase their prices.
If firms can choose their prices freely, firms choose them in order to maximize theirprofit given the prices of others. A REE price, in this case, is a fixed point of thisprocess and often referred to as the optimal flexible price.
Now we introduce friction that prevents firms from choosing this flexible pricemotivated by Calvo (1983). We assume that they can only change their prices witha probability of 1 − θ. This friction compels them to take future prices into accountsince they affect profit. The market has this additional dynamic feedback effect since
expectations of future prices directly affect today’s action.
We transform our theoretical model into a multi-period version of the “learningto-forecast” game. In each period, participants submit their forecasts for prices in
multiple future periods. Each of these submitted forecasts can determine participants’
payoff as well as prices.
We undertake a 3 by 2 between-subjects experimental design. Our first focus is on
games that feature strategic complementarity, as well as weak and strong substitution.
Furthermore, during the course of the experiment, we introduce two upward shifts
in the demand curve, which we refer to as “shocks.” Our second focus is on whenthese shocks occur unexpectedly, as well as on cases where shocks are announced toparticipants two periods in advance.
Using this experimental framework, we begin by testing two hypotheses. In thefirst hypothesis, we investigate the effect of the strategic environment (Hanaki et al.,2019). We test whether observed prices deviate more from the REE prices understrategic complementarity than under strategic substitution. This hypothesis test
is mainly to confirm existing results of past learning-to-forecast experiments within

4

our new experimental framework.4 In the second hypothesis, we examine a nature of
forward-lookingness of expectation. Specifically, we assess whether participants adjust
their expectations for future prices in response to announcements of future shocks inenvironments where prices move along the REE in periods before the announcement.
Our proposed multi-period framework facilitates the testing of this latter hypothesis, a feat unachievable with single-period experimental designs. We show that ourexperimental results validate the two hypotheses.
We then proceed to analyze how participants form their expectation in our multiperiod beauty contest game. We uncover various facts. First, as observed in singleperiod experiments, a sizable fraction of forecasts has naive expectations. In otherwords, participants anticipate that today’s and future prices will mirror the price observed yesterday. Interestingly, the fraction of naive forecasts declines sharply following the shocks or their announcement, pointing to an increase in “eductive” reasoning.
Second, participants revise their predictions when they observe a discrepancy between
their past expectations and realized prices, but they do so sluggishly. For example,
even if yesterday’s price prediction was higher than the price that was realized, they
do not excessively lower their price forecast. Finally, participants consider the past
inflation rate in shaping their expectations only when the game they play exhibits
strategic complementarity. In such cases, prices are more likely to follow a nonstation4

The existing learning-to-forecast experiments, such as Heemeijer et al. (2009); Bao et al. (2017),
demonstrate that, on one hand, prices deviate substantially from those expected under REE when
there is a strong positive feedback between expectations and prices, and on the other hand, prices
converge quickly to REE prices when there is a negative feedback between the two. Such an effectof strategic environment has also been demonstrated in a price-setting game (Fehr and Tyran, 2008;
Funaki et al., 2023), a duopoly game (Potters and Suetens, 2009), and one-shot beauty contest
games (Sutan and Willinger, 2009; Hanaki et al., 2019). Recently, however, Evans et al. (2022) andAnufriev et al. (2022a) demonstrate that eliciting forecasts for a longer horizon stabilizes the pricedynamics under strategic complementarity, although not to the extent that prices converge to REE
prices. While the frameworks of Evans et al. (2022) and Anufriev et al. (2022a) are different fromours, we would like to investigate whether the effect of a strategic environment is observed in ourframework.

5ary process or deviate from REE prices persistently. Therefore, it would be optimal
for participants to incorporate past inflation rates in their predictions. In contrast,
under strategic substitution, realized prices tend to be closer to the steady-state REE
prices, motivating participants to base their predictions solely on the market pricelevel s, not its change.
From these findings, we draw the following conclusions. There is not a single,
universally correct model for expectation formation. Factors such as economic conditions, the presence of shocks, or their announcements seem to prompt participants
to adaptively choose their predictive models. Similarly, participants are not purely
backward-looking or forward-looking. Some participants only form forward-looking
expectations when the shocks or their announcements occur. The studies most closelyaligned with our paper’s conclusions are those by authors Anufriev and Hommes(2012) and Evans et al. (2021). In these models, participants endogenously choose aprediction model that enhances the accuracy of their forecasts.
To the best of our knowledge, only a few, mostly recent, papers elicit forecasts
for multiple future periods in the framework of learning-to-forecast experiments.5
Colassante et al. (2020) elicit a series of forecasts for multiple future periods, however,unlike our framework, the forecasts beyond that for the next period do not determine
the market outcome.
Evans et al. (2022) study the impact of forecasting horizon on the stability of
the aggregate outcome. Their experiment is based on Lucas’ asset pricing model(Lucas, 1978). Participants submit their forecast for the average price over multiple
future periods, instead of the forecasts about the prices for each of these future pe5

There are, however, several papers that elicit forecasts for multiple future periods in the asset
market experiments pioneered by Smith et al. (1988). In these experiments, participants do not only
forecast future prices, but actually trade the asset; see, e.g., Haruvy et al. (2007) and Akiyama etal. (2014, 2017).

6

riods, which determines the prices. They mix the short-horizon (those who forecast
only one period ahead) and long-horizon (those who forecast over one to ten periods
ahead) participants, and change the fraction of each across treatments. They find thatwhile markets only with short-horizon forecasters exhibit substantial and pro-longed
deviation from the REE, those markets with even a modest share of long-horizon
forecasters converge.
In Adam (2007), Rholes and Petersen (2021), and Petersen and Rhoes (2022),
they study a New Keynesian learning-to-forecast experiment. In their experiments,
the one- and two-period-ahead forecasts of inflation rates elicited from participants
determine the current outcome. These studies differ from our paper in how theymodel the effects of these forecasts on today’s equilibrium outcomes. In Rholes andPetersen (2021); Petersen and Rhoes (2022), two forecasts impact the outcome inthe opposite way. The one-period ahead forecast is positively related to the current
outcome, the two-period ahead forecast is negatively related.
The paper most closely related to ours is Lustenhouwer and Salle (2022). Theirexperiment is based on a New Keynesian model with an inflation-targeting interest
rate rule and a government sector. Participants submit their forecasts for output (interms of its percentage deviation from the “normal” level) in multiple future periods.
Like our study, these forecasts determine the current outcome. Furthermore, they
study the impact of the announcement regarding future policy changes on the expectations as we do in our experiment. In this study, participants receive only qualitative
information regarding how their forecasts, along with policy variables, collectively influence the output. In our study, however, participants do know how their forecasts
collectively affect the outcomes. Also, we aim to complement this study by studying
the strategic environment effects, which is not the focus of the paper.
The rest of the paper is organized as follows. Section 2 presents a model of a7multi-period beauty contest based on Calvo (1983). The design of the experiment,
as well as its procedure, are presented in Section 3. Section 4 shows the REE as thebenchmark. The results of the experiment are summarized in Section 5, including a
discussion of implications for modeling expectation formation. Section 6 concludes.

2

A Model of a Multi-Period Beauty Contest

We introduce our economic model, which serves as a basis for our experiment design.
Suppose that there is a continuum of firms that have monopolistic power uniformly
distributed over [0, 1]. Let pi denote the price chosen by individual firm i. For
notational simplicity, we drop subscript i. The demand function for an individual
firm isD (p; P ) ≡ [a − bp + cP ]+ ,(1)

where P is the aggregate price given byZ 1
pi di.

P =

(2)

0

We assume that a > 0, b > 0, and c ∈ R. We impose the following condition on b andc:−∞ < c ≤ 2b.
This assumption guarantees that the single-period version of this game has a REE
price. Note that parameter c can be positive or negative, which governs the degree ofstrategic interaction between the firms. The firms have an identical linear technology
function, and the unit cost of production is denoted by κ.
Firm i maximizes the present value of its profit in each period but is subject
to pricing friction. The friction prevents the firms from changing its price in every
8period and only allows the firm to do so with probability 1 − θ. We assume that if a
firm cannot change its prices T consecutive times, it can change the price next timewith probability 1. Moreover, we assume that with a probability of γ ∈ [0, 1), all
the firms are forced to exit from the market. This assumption is later used to mimic
the indefinite-horizontal nature in our actual experiments conducted by participants,
although this assumption plays no role in the theory.
The maximization problems in period t for firms which can reset their prices are
max
pt

T
X
((1 − γ)θ)s πt+s ,

(3)

s=0

where
πt+s = (pt − κ) D (pt ; Pt+s ) .
The demand function, D (pt ; Pt+s ), is given by equation (1). Recall that in every
period, firm i can re-optimize its price with probability 1 − θ. Since the continuation
payoffs in these events are not a function of pt , they do not appear in objective function(3).
We establish the following proposition:
Proposition 1. The optimal price for firms that can reset their prices at period t isgiven bypt =TX((1 − γ)θ)s(α + βPt+s ) ,PT
s
s=0 ((1 − γ)θ)
s=0

(4)

where
1
a
α=
κ+
2
b

β=

1c
.
2b

Moreover, the law of motion of the aggregate price (REE price) is
Pt = (1 − θ) pt + θPt−1 .
9(5)

Proof. Taking the first-order conditions of maximization problem (3), we obtain equation (4). To derive equation (5), recall that the aggregate price is given by the averageof individual prices (2) and that only fraction 1 − θ of firms can reset their prices.
Because they are randomly chosen, the average price among firms that cannot reset
their price today is Pt−1 . Firms that can reset their prices choose the same price level
given by equation (4). So, average price Pt satisfies equation (5).
It is important to note that α + βPt corresponds to the optimal flexible price inperiod t, if the firms can reset their prices. Therefore, the optimal price (4) constitutes
a weighted average of the optimal flexible prices. Equation (4) elucidates the dynamic
thought process within firms: the optimal price, pt , depends on current actions byothers, as well as future actions, encapsulated in P = (Pt+s )Ts=0 . In other words, thebest response function, given by (4), embodies dynamic feedback mechanisms. The
aggregate price today is influenced by future prices. This is the primary reason why
this model is frequently described as a dynamic beauty contest model (e.g., Angeletosand Lian, 2018).
In contrast, when firms can fully adjust their prices (θ = 0 or T = 0), the dynamic
component of the decision-making process is eliminated: each firm’s focus shifts solely
to predicting the present-day actions of the other firms. The experiment we propose,
inspired by this economic model, stands apart from the existing single-period one
because it necessitates that participants anticipate not only the actions of others inperiod t, but also in later periods t + 1, · · · , t + T .

3

Experimental Design

10

3.1

Setup

In our experimental setup, we employ a cohort of six participants.6 These participants
indexed by i engage in the game over a series of periods. As each period begins, theparticipants put forward their price forecasts for five subsequent periods, including thecurrent one.7 For example, in the first period, participants submit their price forecasts
for periods 1 through 5. In the second period, forecasts are given for periods 2 through6, and this pattern continues for subsequent periods. We denote each participant’si. Importantly, notforecast submitted in period ‘t’ for the price in period ‘k’ as ft,kall submitted forecasts affect the participants’ rewards, as will be explained below.
However, at the beginning of a period, when participants submit their forecasts, allfive of these forecasts potentially influence their rewards.
Let πti denote the payoff of participant i at period t. It is determined by
πti =

100
|Fti − Pt | + 1,

where Pt is the realized price in period t and Fti denotes the payoff-relevant forecast ofparticipant i at period t, which is recursively defined.8 In the first period t, the payoffi
relevant forecast Fti is set to F1i = f1,1
. In period 2, with probability 1 − θ = 1/2, theinewly submitted set of forecasts in this period is used for determining F2i : F2i = f2,2.

With probability θ = 1/2, the forecasts submitted in period 1 is used for determiningiF2i . In this case, F2i = f1,2
. The payoff-relevant sets of forecasts for t = 3 andonward are determined in the same manner. When a set of forecasts has been payoff6

One may consider a cohort of six to be too small for an experiment to have macroeconomic
implications. However, the main results of existing learning-to-forecast experiments do not changeeven if conducted with larger groups of 20 to 30 participants (Bao et al., 2020) or with close to 100
participants (Hommes et al., 2021).
7
All the forecasts are constrained to be integers. We provide a rationale for our choice of thehorizon, T = 4, in footnote 12 below.
8 i
πt is rounded to the nearest integer value. This way of incentivizing the forecast accuracy isalso used, for example, in Adam (2007); Assenza et al. (2021); Anufriev et al. (2022b).

11

Table 1: Hypothetical Submitted Forecasts
Period\Forecast Period

t=1

t=2

t=3

t=4

t=5

t=6

t=1
t=2
t=3
t=4

10

11
12

12
11
9

12
12
10
13

12
13
11
12

13
10
12

t=7

10
10

t=8

Fti

10

10
11
9
13

relevant for five consecutive periods, then with probability one, the new set of forecasts
submitted in the next period becomes payoff-relevant.9
This adjustment process is motivated by the multi-period beauty contest modelin Section 2. The probability of firms being given an opportunity to re-optimize
their prices in period t, 1 − θ, is translated into the probability of the new set of
forecasts becoming payoff-relevant for participants. And the horizon over which firms
optimize, T , in the model is equivalent to the number of future periods, in addition
to the current one, over which our participants forecast in each period. In the caseof our experiment, T = 4. We set the payoffs so that the equilibrium path of the
Nash equilibrium (subgame perfect Nash equilibrium) corresponds to the REE of themulti-period beauty contest game in Section 2.
To give a concrete example, consider hypothetical sets of forecasts in Table 1. Each
entry represents a submitted forecast. For example, in period t = 1, the participant
submits her forecasts of the prices of period t = 1 to 5 as follows:

i
i
i
i
i
f1,1
, f1,2, f1,3, f1,4, f1,5= (10, 11, 12, 12, 12) .
In t = 1, this set of forecasts becomes payoff-relevant: F1i = 10. Suppose that in
9This is because, in such cases, the set of forecasts that has been payoff-relevant does not contain the forecast for the next period. For example, if the set of forecasts submitted in period 1
i
i
i
i
i
, f1,4, f1,5has been payoff-relevant for five consecutive periods, i.e., between periods
f1,1, f1,2, f1,3
1 to 5, then as it does not contain a forecast for period 6 price, we need to make the new set of
forecasts submitted in period 6 payoff-relevant.

12

period t = 2, the new set of forecasts does not become payoff-relevant. So, F2i isdetermined based on the forecasts submitted in period t = 1: F2i = 11. In periods
t = 3 and t = 4, as the new set of forecasts becomes payoff-relevant, F3i = 9 andF4i = 13.10
Now we move on to explain how the aggregate price at period t, Pt , is determined.
Motivated by the model in Section 2, the aggregate price is given by
1
Pt =
6

!
j

(1
−
γθ)
i
pit−1 +
α + βft,t+j
,
P4
l
(1
−
γθ)
l=0
i cannot reset
i can reset j=0
X

X

4
X

(6)

where α and β are the parameters of the model, θ = 1/2, and γ = 0.05.11 Here, equation (6) is the empirical-counterpart of equation (2). Some of the participants cannotchange their prices, which are captured by the first term of the RHS in the above
equation. Moreover, the mapping between the submitted forecasts to the optimal
choice, the second term, is motivated by equation (4).12
On the screen in which participants submit their forecasts, the values of α and β
are presented clearly. On the same screen, participants are informed of the realized
Pt as well as the payoff-relevant forecast Ft in all the past periods. See Appendix C
for the screenshots.
In our experiment, we assume that the game ends with probability γ = 0.05 at the10iIf the new sets of forecasts in period 3 and 4 do not become payoff-relevant, then F3i = f1,3
= 12ii
and F4 = f1,4 = 12.
11
In the experiment, however, due to an oversight, the price was determined with γ = 0 insteadof γ = 0.05. The price determination equation with the wrong value of the weights was also
communicated to participants in the instruction. As we show in Appendix A, however, this oversightwould result in the REE price to differ only slightly (by less than 1). Furthermore, because Pt isrounded to the nearest integer in our experiment as the forecasts submitted by participants areconstrained to be integers, we believe this oversight would not change the experimental outcomes inany major way.
12
Equation (6) provides us a rationale for our choice for T = 4. Note that when θ = 1/2 andiion the optimal price is minimal. This is because the weight for ft,t+4
γ = 0.05, the impact of ft,t+4P
4is ((1 − γ)θ)4 / k=0 ((1 − γ)θ)k , which is approximately 2.7%. Therefore, allowing participants tomake longer forecasts is unlikely to change the results. This argument does not hold if we choose alower adjustment probability, θ.

13end of each period. Otherwise, the game continues to the next period. Participants
are rewarded based on the sum of points they earn from the beginning until the endof the game.
However, implementing such a probabilistic termination rule in a laboratory experiment is problematic because the game can end after only a few periods, thereby
not allowing us to study the evolution of the forecasts in response to shocks, or thegame may not end during the time for which participants are recruited. To circumvent the former problem, we employ the block random termination method Fréchette
and Yuksel (2017) commonly used in the experiments of indefinitely repeated games.
Under this method, participants are asked to play the game in a block of B periods.
Only at the end of the block are the participants informed of whether the game has
already ended during the block just completed.13 If the game has done so, they gain
the sum of πti up to that period. If the game has not ended, the game continues foranother block of B periods. For example, if the game has actually ended in periodPτ < B, they are rewarded based on τt=1 πti . They are also well informed, the gamecan continue beyond B periods, and if that is the case, they play the game for at least
another B periods. In our experiment, we set B = 20. Therefore, participants are
informed that they are going to play the game for at least 20 periods.
Furthermore, because the duration of a game may vary across groups, participants
are informed that this multi-period game can be repeated several times if all the groups
finish playing a game within 30 minutes from the start of the first game. When a
new game starts, new groups of six participants are randomly formed (and thus,
everyone in the same experimental session has to wait until all the groups in the samesession finish a game before starting a new one), and they play the game with the13

In our experiment, this information is communicated to the participants with the sequence ofrandom numbers that determine the termination of the game in each period displayed on the same
screen.

14

same parameter values.14 If the game is repeated several times, one of them will beselected randomly, and participants are paid according to the points they earn in thechosen game.
For example, if the last group to finish the first game does so within 15 minutes
from its start, the second game will be played. If the last group to finish the second
game does so at 35 minutes after the start of the first game, the experiment ends. In
this case, because the game has been repeated twice, either the first or the second
game is selected randomly, and the participants are paid based on their earnings in
the selected game. Note that it is also possible that the first game ends 30 minutes
after its start. If this is the case, only one game is played during the experiment, andthe participants are paid according to the points they earned in this game.
During the games, we introduce (a maximum of) two shocks to α, which captures
the size of the demand during the first two blocks of 20 periods (one shock in eachblock). The literature suggests, even under strategic substitution where prices often
converge to the steady state level, it may take several periods to do so. To allow for
the prices to stabilize before introducing a shock, we have introduced the first shockat the beginning of period 14, and the second shock, in case it is reached, at period
29. We choose α so that the initial flexible REE price is 65, and it becomes 85 and
110 after the first and the second shocks, respectively.15
14Because the game was repeated with the same parameter values, we have opted for random
re-matching to avoid participants coordinating on the realized prices in the earlier games.
15
One could consider introducing shocks differently, for example, to introduce a positive shockthat is followed by a negative shock. We have opted for two positive shocks to investigate whether
participants better adjust to the second shock, if they reach it, as they have gained experiencethrough the first shock.

15

3.2

Treatments

In our 3 by 2 between-participants experiment, we focus on modulating two primary
aspects of the games. The first aspect is the degree of strategic interaction. We
consider the games with β ∈ {0.9, −0.9, −1.8} . That is, the games exhibit strategic complementarity (positive feedback) or substitution (negative feedback).16 We
also explicitly consider a strong substitution case, β = −1.8. This is because NewKeyenesian models might exhibit such strong substitutability. Garcı́a-Schmidt andWoodford (2019) propose an expectation formation models allow this kind of strong
substitutability.
The second aspect is the anticipation of the shocks to α. We consider treatments with and without pre-announcement of shocks. In the treatment without preannouncement, participants are informed of the new value of α only at the occurrenceof the shock, that is, in period 14 for the first shock and in period 29 for the second
shock. In the treatments with pre-announcement, participants are informed of the
new value of α at period 12 for the first shock, and at period 27 for the second shock.
Thus, we can study the adjustment of participants’ forecasts, say for period 14 price,
before and after the pre-announcement.

3.3

Procedures

We use oTree (Chen et al., 2016), an open-source platform for web-based interactive tasks, to conduct the experiments online. Participation is from where they are,and not from our physical experimental laboratory. We used Zoom to manage the16

Because participants are only allowed to submit integer forecasts, there might be multiple equilibria for cases of strong strategic complementarity. It turns out that when β = 0.9, there are
multiple equilibria. However, the set of equilibria is not huge and this multiplicity does not affectthe results of the paper. Therefore, we do not address this issue explicitly.

16

experiments.17
Once ready and after a general instruction about the online experiment is given,
the prerecorded instruction video is played on the screen. Although participants are
not given a hard copy of the instruction slides, they are informed that they can go
through the same set of slides after the video finishes until they finish answering
the comprehension quiz. All the participants need to answer all six questions of
the quiz correctly for the first game to start. As noted, participants can review the
instruction slides before and while answering the quiz. The final rewards are paid
through Amazon Gift Cards (e-mail version).
Participants are students at Osaka University. They are recruited using ORSEE(Greiner, 2015). An English translation of the instruction slides, examples of thedecision screens, and the comprehension quiz are provided in Appendix C.

4

Rational Expectations Hypotheses

We take rational expectations as our benchmark. Given the initial conditions, P−1 ,REE prices pREE
, PtREE solve the following equations uniquely:t
REE
PtREE = (1 − θ) pREE
+ θPt−1
t

pREE
=
t

TX
−t−1
k=t


((1 − γ)θ)k−t
REEαt,k + βPt+k
,
PT
s
((1
−
γ)θ)
s=0

(7)
(8)

where αt,k is the expected value of α at period k from a period t perspective. Whenthe shocks are not announced, αt,k = αt,t for all k and t. Equation (7) describes
the law of motion of PtREE and pREE
is the optimal price subject to the pricingt
friction. Figure 1a demonstrates the REE price sequence PtREE t when the shocks
are pre-announced, and Figure 1b depicts the price sequence when the shocks are not
17

See Appendix B for details regarding how we have conducted our online experiment.

17

110

110

100

100

90

90

80

80

70

70

60

60
5

10

15

20

25

30

35

40

45

50

(a) With Announcement

5

10

15

20

25

30

35

40

45

(b) Without Announcement

Figure 1: Price Dynamics under Rational Expectations

pre-announced.
These figures are intuitively understood. When the game demonstrates strategic
complementarity (β > 0), the transition to new steady-state equilibrium prices isslower, a feature resulting from Calvo pricing friction. Participants know that others
might not be swiftly adjusting their prices due to the pricing friction. As individual
optimal prices are positively correlated with the pricing actions of others, they preferto adjust their prices slowly.
On the contrary, when the game displays strategic substitution (β < 0), theoperative mechanism is reversed. Given that some participants fail to adjust theirprices, the prevailing price becomes too low. This low price motivates individuals topost higher prices. Thus, the transition to the new steady-state equilibrium price isfaster.

18

50

Table 2: Number of Groups per Treatment

Treatments

β = 0.9 β = −0.9 β = −1.8

With Announcement

8

8a

8

Without Announcement

8

8

7b

Notes: Row “With Announcement” represents the results where the shocks were pre-announced
while row “Without Announcement” shows the results where the shocks were not pre-announced. a:
One group stopped at t = 33 due to a technical problem. b: One group stopped at t=39 due to atechnical problem.

5

Results of the Experiment

We conducted our experiments in April and May 2023. There was a total of 294
participants in the experiments.18 Table 2 summarizes the number of groups pertreatment. Each experiment lasted for 90 minutes on average, and participants earned
2482 JPY (approximately 18 USD based on the exchange rate at the time) including
the show-up fee of 500 JPY on average.19 The average payment varied across thevalue of β. It was lowest in the treatments with β = −1.8 (1778 JPY) followed by
2806 JPY and 2844 JPY in treatments with β = −0.9 and β = 0.9, respectively.
18

However, in one session with announcement of the future shock with β = 0.9, one participant
decided to leave the experiment while answering the comprehension quiz. The experimenter has
substituted this participant, and thus, the experiment could continue. We have dropped the dataof this group from the analyses. But because we did the same treatment in which 30 participants
showed up, we have a total of eight groups for this treatment. Furthermore, in one session without
announcement of the shock and β = −1.8, one of the participants lost his/her internet connection
around period 11 and switched to a different mode of connection. As a result, an error occurred
and the experiment could not continue for this group. In addition, in the same session, an error
occurred for a group in period 40. Thus, we only have data for seven groups for this treatment,with one group missing the price and forecasts submitted in period 40. Finally, in the treatment
with announcement of the shocks and β = −0.9, one group encountered a technical problem and theexperiment stopped in period 33. Thus, while we have data for eight groups for this treatment, one
lacks the data from period 34 onward.
19
Exchange rate between the points earned during the experiment and JPY was 1 point = 2 JPY.

19

5.1

Effect of the Degree of Strategic Interaction

Figure 2 shows the dynamics of prices observed in each treatment. One line corresponds to a group in each panel. As one can observe, regardless of the existence of theannouncement, the prices follow the REE prices quite closely when β = −0.9. When
β = −1.8, the prices are close to the flexible REE prices but fluctuate greatly aroundthem. These fluctuations are also in sharp contrast to the rational expectations outcome presented in Figure 1. When the game exhibits strategic complementarity,
β = 0.9, they persistently deviate from the REE prices. These findings are very similar to the results from the existing learning-to-forecast experiment such as those of
Heemeijer et al. (2009) and Bao et al. (2012). The equilibrium outcomes are morestable if the game exhibits strategic weak substitutability, β ∈ (−1, 0].
Now, let us quantify the degree of deviation from the REE prices. We computethe relative absolute deviation, RAD, and the relative deviation, RD, proposed by
Stöckl et al. (2010). For group g, RADg and RDg are calculated as follows:
1 X Pg,t − PtREE
RADg =K t
PtREE
1 X Pg,t − PtREE
RDg =
K t
PtREE
where Pg,t is the realized period t price for group g and PtREE is the REE price inperiod t.20 K is the total number of periods (40 except for the two groups that faced
a technical problem).
Figure 3 shows the empirical cumulative distribution (ECD) of RAD (top) andRD (bottom) in the treatments with (left) and without (right) announcement. In
each panel, the distributions for each β are shown. The top panels show that, regard20
Here we use the REE price based on the price determination equation used in the experiment,
namely, the one that set γ = 0. See, footnote 11.

20

300

250

250
Price Index

Price Index

300

200
150

200
150

100

100

50

50
0

5

10

15

20
Round

25

30

35

40

0

128

128

103

103

78

53

15

20
Round

25

30

35

40

78

53

28

28
0

5

10

15

20
Round

25

30

35

40

0

(c) β = −0.9 with announcement

5

10

15

20
Round

25

30

35

40

(d) β = −0.9 without announcement

150

150

125

125

100

100

Price Index

Price Index10(b) β = 0.9 without announcement

Price Index

Price Index(a) β = 0.9 with announcement

5

75
50
25

75
50
25

0

0
0

5

10

15

20
Round

25

30

35

40

(e) β = −1.8 with announcement

0

5

10

15

20
Round

25

30

35

(f) β = −1.8 without announcement

Figure 2: Realized Price Dynamics
Notes: The red dashed lines represent the steady-state levels of the aggregated prices.

21

40

(a) RAD (With announcement)(b) RAD (Without announcement)

ECD

ECD

1.0

1.0

0.8

0.8
β=-0.90.6

β=-1.8
β=0.9

0.4
0.2

β=-0.90.6

β=-1.8
β=0.9

0.4
0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

RAD

0.0

0.2

p = 0.0001

0.4

0.6

0.8

1.0

1.2

1.4

RAD

p = 0.0002(c) RD (With announcement)(d) RD (Without announcement)

ECD

ECD

1.0

1.0

0.8

0.8
β=-0.90.6

β=-1.8
β=0.9

0.4
0.2

β=-0.90.6

β=-1.8
β=0.9

0.4
0.2

-0.2 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

RD-0.2 0.0

p = 0.0437

0.2 0.4 0.6 0.8 1.0 1.2 1.4

RD

p = 0.0005

Figure 3: Distribution of RAD (top) and RD (bottom)
Note: P-values of Kruskal–Wallis (KW) test are reported.

less of the existence of announcement, RADs are positive for each β. Applying thesingle-rank test, the observed prices are significantly different from the REE prices.21
Furthermore, the observed RADs are largest under β = 0.9, followed by β = −1.8,
and then β = −0.9 regardless of the existence of the announcement. The differences between the three treatments are statistically significant both with and without
announcement.22 Therefore, the deviations of the observed prices from the REE
21

They are all significantly different from zero at the 5% level according to the signed-rank test.
P − values are 0.018 β = −1.8 without announcement and 0.012 for all the other five treatments.
22
P < 0.05 based on the Kruskal–Wallis (KW) test for both cases. For pairwise comparisons,p-values based on the Mann–Whitney (MW) test (two-tailed) are 0.0002 (β = 0.9 vs β = −0.9),
0.0022 (β = 0.9 vs β = −1.8), and 0.0093 (β = −0.9 vs β = −1.8) without announcement, and
0.0002 (β = 0.9 vs β = −0.9), 0.0047 (β = 0.9 vs β = −1.8), and 0.0003 (β = −0.9 vs β = −1.8)
with announcement.

22

prices are smaller under strategic substitution than under strategic complementarity,and furthermore, they are smaller under “weak” substitution (β = −0.9) than under“strong” substitution (β = −1.8).
The bottom panels show, however, that while RDs are larger under strategic
complementarity than under strategic substitution as in the case of RADs, there is
no longer a statistically significant difference between β = −0.9 and β = −1.8.23
Note that because RD takes the direction of the deviation from REE into account,it becomes close to zero when the prices fluctuate around REE as in the case of
β = −1.8.24

5.2

Forward-Lookingness of Expectations

Let us now turn to the way forecasts have responded to the announcement of future
shocks. To do so, we focus on the difference in the forecast adjustments at the timeof shock between treatments with and without announcement. Namely, we computethe average difference in the forecast price between the period of the shock and theperiod just before the shock (periods 14 and 13, respectively, for the first shock andperiods 29 and 28, respectively, for the second shock) submitted in the periods after
the announcement (periods 12 and 13 for the first shock and periods 27 and 28 forthe second shock) and those before the announcement (periods 10 and 11 for the firstshock and periods 25 and 26 for the second shock). That is, for each participant i,
the average forecast responses to the announcement of the first and second shocks,
23
For pairwise comparisons, p-values based on the MW test (two-tailed) are 0.0002 (β = 0.9 vs
β = −0.9), 0.0012 (β = 0.9 vs β = −1.8), and 0.1520 (β = −0.9 vs β = −1.8) without announcement,
and 0.0650 (β = 0.9 vs β = −0.9), 0.0379 (β = 0.9 vs β = −1.8), and 0.2345 (β = −0.9 vs β = −1.8)
with announcement.
24
In fact, RD is not significantly different from zero for β = −1.8 without announcement(p=0.6875, signed-rank test) while it is significantly different from zero at the 5% significance levelfor five other treatments (p=0.0391 for β = 0.9 with announcement and 0.0078 for the remaining
four treatments, signed-rank test.)

23

∆f1i and ∆f2i , are defined as follows:∆f1i =

13
11
 1X

1X i
i
i
i
ft,14 − ft,13−ft,14− ft,132 t=122 t=10∆f2i =

28
26
 1X

1X i
i
i
i
−
.
− ft,28
ft,29 − ft,28ft,29
2 t=272 t=25

We compute the same measure for the participants in the treatment without announcement as for the benchmark.
Figure 4 shows the empirical cumulative distributions of (∆f1i )i (left) and (∆f2i )i(right) with and without announcement for three values of β. P-values are from
MW and Kolmogorov–Smirnov (KS) tests. We also report the distributions between
treatments with and without announcement.
Figure 4 shows that, without announcement, both ∆f1i and ∆f2i are concentrated
around zero. This is especially so for β = −0.9 for both the first and second shocks
in which prices were very close to the steady-state equilibrium prices before the realization of these shocks. With announcement, however, ∆f1i and ∆f2i are distributed
more toward positive values (except for ∆f2i of β = 0.9), suggesting that participants adjusted their forecasts upward in response to the announcement. Except for
β = 0.9, there are significant differences in ∆f1i and ∆f2i between treatments with andwithout announcement. Thus, consistent with participants having forward-looking
expectations, they expect prices to go up after the announcement under strategic
substitution.25
25

While ∆f2i is significantly greater than ∆f1i for β = −0.9 (pM W = 0.041 and pKS = 0.034),
there is no significant difference between the two for β = −1.8 (pM W = 0.968 and pKS = 0.760) and
β = 0.9 (pM W = 0.311 and pKS = 0.531).

24

∆f1i , β = 0.9∆f2i , β = 0.9

ECD

ECD

1.0

1.0

0.8

0.8

0.6

With
Annoucement

0.6

With
Annoucement

0.4

Without
Annuncement

0.4

Without
Annuncement

0.2
- 20

0.2

- 10

10

20

30

40

50

ΔF

- 20

- 10

10

pM W = 0.052, pKS = 0.075

ECD

1.0

1.0

50

ΔF

0.8

0.6

With
Annoucement

0.6

With
Annoucement

0.4

Without
Annuncement

0.4

Without
Annuncement

0.2

0.2

- 10

10

20

30

40

50

ΔF

- 20

- 10

10

20

30

40

50

ΔF

pM W = 0.002, pKS < 0.001

pM W < 0.001, pKS < 0.001∆f1i , β = −1.8

∆f2i , β = −1.8

ECD
1.0

ECD
1.0

0.8

0.8

0.6

With
Annoucement

0.6

With
Annoucement

0.4

Without
Annuncement

0.4

Without
Annuncement

0.2
- 10

40

∆f2i , β = −0.9

ECD

0.8

- 20

30

pM W = 0.376, pKS = 0.176∆f1i , β = −0.9

- 20

20

0.2
10

20

30

40

50

ΔF

- 20

pM W = 0.067, pKS = 0.036

- 10

10

20

30

40

50

ΔF

pM W = 0.018, pKS < 0.001

Figure 4: Empirical Distributions of (∆f1i )i and (∆f2i )i

25

5.3

Expectation Formation in a Multi-Period Environment

In this section, we analyze how participants form expectations in our multi-period
experiments. Similar to the existing literature, we categorize price expectations into
two types: naive expectations and others. In Subsection 5.3.1, we define and analyze
naive expectation formation in our setting. Following that, in Subsection 5.3.2, weproceed to analyze non-naive forecasts.

5.3.1

Naive Expectation Formation

We commence by defining a naive expectation. These definitions extend the notionof naive expectation from single-period environments to multi-period environments.
i
We say that the forecasts of participant i in period t, ft,t+j, is naive if for allj∈Jj ∈ J,
i
ft,t+j= Pt−1 ,and ε-naive if for all j ∈ J ,ift,t+j− Pt−1 ≤ ε.

Figure 5 shows the fraction of the naive forecasts for ε ∈ {0, 2, 4} in the treatment
without the pre-announcements of shocks (left) and with the pre-announcements(right).26 In each panel, the data from three values of β are pooled. There arethree points to be made about which kind of expectation formation participants will
engage in. First, the number of participants forming naive expectations increases
over time.27 This is because there is a tendency for the economy to converge to
steady-state REE prices. As a result, a growing number of participants form naive
26

If we assume that prices had converged to a steady state, those exhibiting a deviation of approximately less than 5% from the steady state prices would be classified as naive when ε = 4.
27
Under the REE hypothesis, the fraction of naive forecasts is zero after the realization of theshocks, and becomes 1 when the price sufficiently converges to the steady-state REE prices.

26

Figure 5: Fraction of Naive Forecasts for Various Cutoffs(a) Without Announcement
1000Fraction of Naive Forecasts

Fraction of Naive Forecasts

100(b) With Announcement

2
75
4
50

25

0

0
2

75
4
50

25

0
0

10

20

30

40

0

10

20

30

40

Notes: The solid vertical lines represent the rounds at which the shocks occurred. The dashedvertical lines in the right figure represent the rounds at which the announcements of the shocks were
made. The difference in line types represents the difference in the value of ε. As indicated in thelegend, a solid line represents ε = 0, a dotted line indicates ε being 2, and a dashed line 4.

expectations following the shocks or announcements endogenously.
Second, despite the shocks or announcements, a significant proportion of participants remain inclined toward naive expectations. To see this point, consider thefraction of naive forecasts when ε = 2. 25 percent of the forecasts are naive even rightafter the shocks or their announcement (Figures 5a and 5b).
Finally, a major chunk of participants either do not stick to, or entirely abandon
naive expectation formation when faced with shocks or announcements. Instead, their
expectation pattern leans more toward “eductive” tendencies, which will be elaborated
on in the next section.

5.3.2

Non-Naive Expectation Formation

We now pivot to non-naive expectations, basing our analysis on experimental datathat is not ε-naive with ε = 4. While it is hard to identify how exactly each participant forms her forecast (e.g., level-1) due to the multi-period nature, we can still
understand how participants, on average, form their forecasts. Drawing inspiration27

from Anufriev and Hommes (2012), we structure the forecast formation around two
forecasting rules: adaptive heuristic and trend-following. If participant i adopts theadaptive rule, their belief adjustment would proceed as follows:iii− Pt−1 + Pt−1 ,+ (1 − ω) Pt−1 = ω ft−1,t−1= ωft−1,t−1
ft,t+j(9)where ω is the weight assigned to the forecast at period t − 1. When ω ∈ (0, 1), thisforecasting rule says that they only adjust their forecast gradually. When ω < 0,participant i adjusts their forecast downward if the forecast at period t − 1 is higher
than the realized price at period t − 1.
The trend-following rule says that participant i assumes that the aggregate trend
will persist:i
ft,t+j= Pt−1 + χ (Pt−1 − Pt−2 ) .

(10)

Parameter χ governs the strength of the extrapolation effect. When χ is negative,then we can interpret equation (10) differently. Namely, participant i expects theshort-run trend to be reversed.
We nest models (9) and (10) as follows and obtain the baseline regression specification:
ii
ft,t+j−Pt−1 = a∆αt +1At +γ ω ft−1,t−1− Pt−1 + (1 − γ)|{z}
Adaptive Forecastingχ (Pt−1 − Pt−2 )|{z}+FEi +εt,i,j ,Trend-Following Forecasting

where parameter γ assigns the weight to the adaptive forecasting rule, ∆αt is theshock, and 1At is the indicator function that is one if at period t, the announcement
to a shock is made. When γ = 1, then participant i only uses the adaptive forecasting
rule (9). If γ = 0, then they only use the extrapolation. By running the regression, we
estimate the parameters γω and (1 − γ) χ. The multiplicative γω reflects the effective“weight” assigned to the adaptive forecasting rule.
28

The regression results for each forecast horizon j are presented in Table 3. Fromthese regression results, four points regarding expectation formation can be discerned.
First, in almost all cases, participants’ expectations are influenced by the shocks.
Importantly, this reaction is not just limited to expectations at the time of the shock,iift,t, but also for future expectations, ft,t+j. Interestingly, in the case where β = 0.9,

the shocks have a greater impact on future time points than on the current time
point. Such predictions also hold when participants have rational expectations. On
the other hand, no such empirical patterns are observed when β < 0. This implies
that participants do not necessarily have qualitatively correct dynamic price forecasts
if β < 0.
Second, in many cases, participants’ predictions at the time of the announcement
do not significantly respond to the announcement itself. This observation is evident
from the fact that the coefficients for 1At when j = 0 are insignificant. However, starting from j = 2, there are cases where the coefficient becomes significant. Participants
anticipate that other participants will not immediately respond to the announcement
itself, but they anticipate that others will react when the shocks occur two periods
later. This finding is consistent with that in Section 5.
Third, it is evident that participants are employing adaptive-heuristic forecasting.
Across all cases, it can be interpreted that when making price predictions, participantsrefer to the discrepancy between their past predictions and actual prices. Notably,
the coefficients do not increase as the value of j becomes larger.
The final point is that participants employ a trend-following forecasting rule only
when β > 0. Also, in this case, if there was inflation in the past, they predicthigher prices in the future: the coefficient at j = 0 is 0.5, but at j = 4, it is 1.2,
which is significantly higher. In contrast, with β < 0, participants scarcely rely on
past inflation. This stark difference is thought to arise from whether the economy is29

Table 3: Regression Results

Panel A (β = 0.9)

j=0

j=1

Horizon
j=2

∆αt

0.789
(0.537)1.405(1.212)0.301∗∗∗(0.016)0.518∗∗∗(0.054)
3,168
0.162

1.084∗∗(0.479)2.100∗(1.081)0.255∗∗∗(0.014)0.644∗∗∗(0.048)
3,168
0.186

1.212∗∗(0.617)2.949∗∗(1.393)0.197∗∗∗(0.019)0.888∗∗∗(0.062)
3,168
0.117

2.917∗∗(1.186)5.060∗(2.679)0.244∗∗∗(0.036)1.331∗∗∗(0.120)
3,168
0.129

3.325∗∗(1.386)4.457(3.130)0.110∗∗∗(0.042)
1.249∗∗∗(0.140)
3,168
0.174

j=3

j=4

1At
γj ωj
(1 − γj )χj
N
R2

j=3

j=4

Panel B (β = −0.9)

j=0

j=1

Horizonj=2∆αt0.250∗∗∗(0.030)3.553∗∗∗(1.273)0.370∗∗∗(0.015)0.213∗∗∗(0.047)
3,120
0.244

0.257∗∗∗(0.023)1.856∗(1.002)0.228∗∗∗(0.011)
0.013(0.037)
3,120
0.212

0.294∗∗∗(0.029)4.507∗∗∗(1.231)0.265∗∗∗(0.014)0.050(0.045)
3,120
0.201

0.303∗∗∗(0.028)4.830∗∗∗(1.186)0.161∗∗∗(0.014)−0.055(0.043)
3,120
0.128

0.248∗∗∗(0.028)5.364∗∗∗(1.198)0.185∗∗∗(0.014)−0.018(0.044)
3,120
0.116

j=0

j=1

Horizon
j=2

j=3

j=4

0.183∗∗∗(0.032)3.280(1.999)0.331∗∗∗(0.017)0.015(0.041)
2,964
0.270

0.197∗∗∗(0.031)3.290∗(1.945)0.292∗∗∗(0.016)0.005(0.040)
2,964
0.257

0.173∗∗∗(0.038)7.498∗∗∗(2.427)0.303∗∗∗(0.021)−0.005(0.050)
2,964
0.189

0.207∗∗∗(0.037)8.841∗∗∗(2.345)0.286∗∗∗(0.020)−0.022(0.048)
2,964
0.617

0.204∗∗∗(0.041)6.550∗∗(2.572)0.216∗∗∗(0.022)−0.108∗∗(0.053)
2,964
0.575

1At
γj ωj
(1 − γj )χjN
R2

Panel C (β = −1.8)
∆αt
1At
γj ωj
(1 − γj )χj
N
R2

30

converging to the steady-state REE prices or not. When β > 0, the economy deviatesfrom the REE prices persistently, and the realized prices move in a nonstationary
manner. Therefore, forecasting prices becomes more difficult when β > 0. This would
be a reason why participants also use past information to predict prices. If β < 0, thepast information about prices becomes less relevant (because prices fluctuate aroundthe steady-state REE prices.)
Our results in this section can be summarized as follows. There exists a groupof participants who consistently stick to naive expectations, even when faced withdynamic and evolving scenarios. In contrast, a significant number rely on both theeductive and evolutive methods of expectation formation, as implied by Binmore(1987). Importantly, there does not exist the ideal model for nonrational expectations formation. The appropriate approach for modeling expectations largely hinges
on prevailing economic conditions that participants face. Specific factors, such as thedegree of strategic complementarity shown by β, the presence of shocks, ∆αt , andtheir announcement, play a pivotal role in influencing these expectation formations.
Thus, appropriate models should allow endogenous selection of an optimal economic
forecasting model tailored to the environment that participants face. Papers consistent with our findings include Anufriev and Hommes (2012) and Evans et al. (2021).
Extending this line of research would be promising.

6

Concluding Remarks

In this paper, we extend the conventional single-period beauty contest game to a
multi-period context and conduct experiments. This multi-period model is constructed based on the Calvo (1983) pricing model commonly used in macroeconomics.
Our new paradigm of “learning-to-forecast” extends the existing ones and allows us
31

to investigate how participants update their forecasts for the same future periods
over time in response to the arrival of new pieces of information. The experimental
results indicate that, on average, individuals react to announcements of future shocks(eductive reasoning), while a sizable fraction of participants forms naive expectations
even in the multi-period environment considered in our experiment. Additionally, theincorporation of trends, shocks, and their announcement into expectation formation
varies depending on economic conditions. Fitting the model in which decision makers
choose different forecasting models depending on the situation, as in Anufriev andHommes (2012) and Evans et al. (2021), to our data would be a fruitful next step.
Another approach is to conduct experiments to understand how participants selectthe variables to use for forecasting, and under which circumstances they are more
susceptible to shocks. This is important because changes in the way people form
expectations can impact the dynamics of the economy and potentially cause significant
fluctuations through expectational feedback loops.
To gain a deeper understanding of the above point, it might be beneficial to change
the magnitude of the shocks. When a game is with strategic complementary, evena small shock is magnified, exerting a considerable influence on equilibrium prices.
However, if individuals expect that others do not respond to the shock, its initial
impact on the economy is tiny, and the economy will slowly transition to the new
equilibrium price. This is markedly different from the case of strategic substitutes.
In that case, if people expect that others do not react to the shock, it can have abigger impact on the economy, which might explain why participants respond to theshocks in games with strategic substitutes. To explore the impact of shocks and theirannouncement, it is helpful to consider sufficiently large shocks in experiments ofgames with strategic complementarity. These remain as future research topics.

32

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39

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
10

20

30

40

50

(a) With Announcement

10

20

30

40

50

(b) Without Announcement

Figure 6: Comparison

A Comparison of REE Prices for γ > 0 and γ = 0
In our experiments, we inadvertently ignored the impacts of termination probability,γ, in determining the price. However, this oversight is not likely to result in significant changes to our study. This is because the impacts are minor as mentioned infootnote 11. In this subsection, we formally substantiate our claim. We calculate the
REE prices under the assumption that γ = 5%, and compare the prices with thosecomputed under the assumption that γ = 0. We find that the prices are sufficiently
similar between the two.
Let (pγt , Ptγ ) denote the equilibrium individual price and the equilibrium pricewhen γ ≥ 0.
Figure 6 depicts the absolute difference between Pt0.05 and Pt0 in case with andwithout announcement. Note that the differences are less than one in all the periods
including those around the shocks.

40

B

Detail of the Procedure of the Online ExperimentParticipants join our experiments via Zoom with their cameras and microphone turned
off. The camera of the experimenter is always turned on, but her or his microphone
is turned on only when necessary.
Upon connecting to a Zoom session, participants first wait in the waiting room.
We let participants enter the main room one by one to check their names and toverify whether they are indeed registered for our experiments. Then, each participant
is given participant ID in the form of “sub##”, where ## is the two-digit numberthat is valid during the experiment. Once their participant ID is given, they aresent back to the waiting room until the start of the experiment. By following thisprocedure, we ensure anonymity.
Once ready, participants re-enter the Zoom meeting room and are given general
instructions regarding the online experiment (for example, what to do, including which
number to call when their internet connection fails during the experiment). Then,the prerecorded instruction video is played first. Although participants are not givena hard copy of the instruction slides, they are informed that they can go through thesame set of slides after the video finishes until they finish answering the comprehension
quiz. All the participants need to answer all six questions of the quiz correctly forthe first game to start. As noted, participants can review the instruction slides before
and while answering the quiz. While participants are asked to communicate directlywith the experimenter using the chat function of Zoom when they have questions or
encounter problems, they could not communicate with each other via Zoom chat.

41

C

English Translations of the Instructions and Examplesof Screenshots

English translation of the instruction slides can be found at https://osf.io/gecv8/?view_only=005886a62c3148f6afe46f0406b2887d
Figures 7 to 10 show examples of the decision screen participants faced in our
experiment.

Note: The values of α and β are shown in red and with a large font in period 1 as well as when theychange (in periods 14 and 29) in all the treatments (see Figure 9). In other periods, they are shownin black with a regular sized font (see Figure 10).

Figure 7: Screen (in period 1) in which participants submit their five forecasts (common to
all the treatments)

42

Note: This is a demo screen in which the shock is introduced in period 5 (this is why the preannouncement is made in period 3). The text in the yellow box states that “This is information
regarding the future changes in the parameter values.
The values of the parameters that determine the price will be
α = xxx and β = yyy from period T ,” where xxx, yyy, and T depend on the treatment and whether
it is the first shock or the second shock. The current values of α and β are shown in black text below
the announcement.

Figure 8: Screen with a pre-announcement of future shock (only for the treatments withpre-announcement)

43Note: The values of α and β are shown in red with a large font when they change (in periods 14
and 29) in all the treatments.

Figure 9: Screen when the shock is realized (common to all the treatments)

Note: The values of α and β are shown in black with a regular sized font.

Figure 10: Screen for normal periods (common to all the treatments)

44

C.1

Comprehension quiz

Here is an English translation of comprehension quiz.
(1) In this experiment, every morning, you will be asked to predict prices up to
K periods ahead, including the current period. For the forecasts that will be
rewarded in period t (t > 1), please select all correct statements.
A) The latest price forecast entered in period t will be the subject of the
reward.
B) There is a 0.5 probability that the latest price forecast entered in period t
will be the subject of the reward.
C) If the latest price forecast entered in period t is not the subject of the
reward, then the price forecasts entered in period t − 1 or earlier will be
the subject of the reward.
(2) Let’s assume the price forecast that is the subject of the reward in period t is
10. Also, let’s assume the price that materialized in period t is 14. In this case,
how many points can you earn?
A) 20 points.
B) 25 points.
C) 50 points.
(3) Regarding how the price in period t is determined, which statement is correct?
A) The realized price is determined based on the latest price forecasts submitted by participants in the same group for period t.

45B) For participants in the same group, the realized price is determined basedon the payoff-relevant price forecast in period t.
(4) Please select all correct statements about the repetition period of a single game.
A) A game is repeated for at least 20 periods.
B) The game can end after just one period as there is a 0.05 probability of
the game ending at the end of each period.
C) The game is repeated only for 20 periods.
(5) Which statement is correct regarding the points that can be earned in a single
game?
A) Since the game is repeated for at least 20 periods, you can earn all the
points accumulated during at least 20 periods.
B) Although the game is repeated for at least 20 periods, the number of points
that can be earned might be less than the total points accumulated over
20 periods, because there is a probability of 0.05 that the game will end at
the end of each period.
(6) Please select all correct statements regarding today’s experiment.
A) The game is conducted only once, and the compensation is paid according
to the points earned in that game.
B) It is not known in advance how many times the game will be conducted.
C) One of the games conducted will be randomly selected to be the subject
for compensation.

46