How Concept Learning Changes Strategic Choice in Guessing Games?

Oleksii Ignatenko

1,2 a

Ukrainian Catholic University, 17 Sventsitsky Str., Lviv, 79011, Ukraine

Kyiv Academic University, 36 Vernadsky Blvd., Kyiv, 03142, Ukraine

Keywords:

Behavioral Game Theory, Guessing Game, K-Beauty Contest, Active Learning, R, Agent-Based Modeling.

Abstract:

This paper deals with the k-beauty contest game. The research question formulated in this work is how players

(mainly children and undergraduates) learn in complex strategic situations that they have never faced before

and how we can model this learning process. We examine data from different games played during popular

lectures about game theory and present ﬁndings of players’ progress in strategic learning while competing

with other players. The raw data gathered is available in an open repository for replication and analysis and an

R ﬁle with data manipulations, metrics calculations, and plots. Based on the conclusions from experimental

data, we create the agent-based model and launch ten thousand simulations for different setups. Then we apply

analysis from experimental data and discuss ﬁndings and similarities between agents and humans.

1 INTRODUCTION

Game theory is a ﬁeld of science investigating ratio-

nal players’ decision-making under uncertainty. The

source of the uncertainty can be strategic structure,

e.g., probability of certain events, lack of information

about future possibilities, or decisions of other agents

that can generate it. In the last case, we can talk about

the interdependence of strategic actions, when some

players’ decisions affect others’ payoffs. Such situa-

tions arise around us daily, and we, consciously or un-

consciously, participate in them. The success heavily

relies on our perception of the actions of other play-

ers.

The problem is how we can know the future ac-

tions of other players. We cannot, but we can start

with some assumptions that will help create a frame-

work, model, or theory of “mind”, which will predict

future (reasonable) actions. Game theory proposed

approach, which is now under questioning (especially

from the side of the experimental or behavioral econ-

omy). Nevertheless, we will start from standard no-

tions and proceed to experimental data.

One can expect that other players will play “rea-

sonably”, and this game theory means they will try

to achieve a better result in some agent’s sense. This

idea is grasped by the term rationality. Every rational

player must calculate the best possible result, taking

https://orcid.org/0000-0001-8692-2062

into account the rules of the game and the interests of

other participants. In other words, think strategically.

It is well known from theory that rational players will

play Nash equilibrium (NE) if there is any, which is

very useful in games where only one unique NE ex-

ists. The notion of rationality was indeed fundamen-

tal for the development of game theory. However, the

problems with this notion are also quite famous.

First, it is very demanding because it presupposes

that the agent has complete, transitive preferences

and is capable of computing equilibrium in a given

strategic situation. However, this is not feasible in

many natural situations (for example, we know about

NE in chess, but still no computer can compute it).

Secondly, probably more important, there are many

games where NE is the poor prediction of actual hu-

man behavior. This paper investigates some of the

data from such a game and discusses the difference.

All this makes decision making exciting problem

to investigate. This is a rich area of research, where

theoretical constructions of the game theory seem to

fail to work, and experimental data shows unusual

patterns. These patterns are persistent and usually

do not depend on age, education, country, and other

things. During the last 25 years, behavioral game the-

ory in numerous studies has examined bounded ratio-

nality (best close concept to the rationality of game

theory), cognitive distortions, and heuristics people

use to reason in strategic situations. For example

we can note surveys of Crawford et al. (Crawford

706

Ignatenko, O.

How Concept Learning Changes Strategic Choice in Guessing Games?.

DOI: 10.5220/0012067100003431

In Proceedings of the 2nd Myroslav I. Zhaldak Symposium on Advances in Educational Technology (AET 2021), pages 706-720

ISBN: 978-989-758-662-0

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

et al., 2013) and Mauersberger and Nagel (Mauers-

berger and Nagel, 2018). Also, there is a comprehen-

sive description of the ﬁeld of behavioral game theory

by Camerer (Camerer, 2003).

We will concentrate on the guessing games, which

are a signiﬁcant part of research because of their sim-

plicity for players and easy analysis of rules from a

game-theoretic perspective. In this paper, we present

the results of games played during the 2018-2021

years as part of popular lectures about game theory

(Ignatenko, 2020). The audience of these lectures was

quite heterogeneous, but we can distinguish the fol-

lowing main groups:

• children at schools (strong mathematical schools,

ordinary schools, alternative education schools);

• students (bachelor and master levels);

• mixed adults with almost any background.

• adults with a business background

• participants of Data Science School

• participants of summer STEM camps for children

We propose a framework of four types of games,

each presenting one idea or concept of game theory.

These games were introduced to players with no prior

knowledge (at least in the vast majority) about the

theory. On the other hand, games have simple for-

mulation and clear winning rules, making them intu-

itively understandable even for kids. This makes these

games the perfect choice to test the ability of strategic

thinking and investigate the process of understanding

complex concepts during the play, with immediate ap-

plication to the practice. This dual learning, as we

can name it, shows how players try and learn in natu-

ral conditions and react to interaction challenges with

other strategic players.

In this paper, we will concentrate on the ﬁrst

game – the famous p-beauty contest game. For this

game, we analyze data and try to formulate simple

rules which are plausible for an explanation of play-

ers’ behavior. In the next section, we created the

agent-based model using the Netlogo environment

and discussed the model’s main features. To investi-

gate the model and its properties, we performed sim-

ulations using the BehaviorSpace tool of Netlogo; as

a result, about 10000 games were simulated. Such

a volume of data is impossible to get using human-

based experiments. We recreated plots and metrics

developed for human data and analyzed its similarity

and differences. In the end, we formulate conclusions

and future work directions.

First, let us start with some deﬁnitions.

1.1 Game Theory Deﬁnitions and

Assumptions

Consider games in strategic or normal form in a non-

cooperative setup. A non-cooperativeness here does

not imply that the players do not cooperate, but it

means that any cooperation must be self-enforcing

without any coordination among the players. The

strict deﬁnition is as follows.

A non-cooperative game in strategic (or normal)

form is a triplet G = {N , {S

}

i=∈N

,{u

}

i∈N

}, where:

• N is a ﬁnite set of players, N = {1, . . . , N};

• S

is the set of admissible strategies for player i;

• u

: S −→ R is the utility (payoff) function for

player i, with S = {S

×· ··×S

} (Cartesian prod-

uct of the strategy sets).

A game is said to be static if the players take their

actions only once, independently of each other. In

some sense, a static game is a game without any no-

tion of time, where no player has any knowledge of

the decisions taken by the other players. Even though,

in practice, the players may have made their strate-

gic choices at different points in time, a game is still

static if no player has any information on the deci-

sions of others. In contrast, a dynamic game is one

where the players have some (full or imperfect) infor-

mation about each others’ choices and can act more

than once. In this work, we deal with repeated static

games, which means that the same game is played

twice (sometimes three times) with the same players.

Agents’ rationality is a signiﬁcant issue; some-

times, it is called full rationality (to differentiate it

from bounded rationality – a less restricting notion).

When a fully rational agent tries to ﬁnd the best ac-

tion, it usually depends on the action of other self-

interest agents. So the ﬁrst agent must form beliefs

about the second agent’s beliefs about the beliefs of

the ﬁrst agent, and so on. Such constructions seem

too complicated, but they are based on the predictions

of classical game theory, which assumes all agents to

be fully rational.

One quite famous result by Aumann (Aumann,

1995) is that for an arbitrary perfect-information

extensive-form game, the only behavior that is com-

patible with (1) common knowledge of rationality,

and in particular by (2) each agent best responding

to their knowledge is for each agent to play according

to the strategy, obtained by the backward induction.

Aumann and Brandenburger (Aumann and Branden-

burger, 1995) showed that common knowledge of ra-

tionality, the game payoffs, and the other agent’s be-

liefs are sufﬁcient conditions for Nash equilibrium in

an arbitrary game.

How Concept Learning Changes Strategic Choice in Guessing Games?

707

In this regard, the most accepted solution concept

for a non-cooperative game is that of a Nash equilib-

rium, introduced by John F. Nash (Kuhn et al., 1996).

Loosely speaking, a Nash equilibrium is a state of a

non-cooperative game where no player can improve

its utility by changing its strategy if the other players

maintain their current strategies. Of course, players

also use information and beliefs about other players,

so we can say that (in Nash equilibrium) beliefs and

incentives are important to understand why players

choose strategies in real situations.

The NE is the core concept of game theory, but

it differs from experiments and sometimes reality.

In some games, humans demonstrate convergence to

equilibrium, but in others do not. This gap between

similarly-looking games is slim and not easy to catch.

We will consider guessing games as a playground to

work with players’ behavior.

2 GUESSING GAMES HISTORY

In early 90xx, Rosemary Nagel started a series of ex-

periments of guessing games, summarized in (Nagel,

1995). She was not the ﬁrst to invent the games;

they were used during the lectures by different game

theory researchers (for example, Moulin (Moulin,

1986)). In recent work (Nagel et al., 2017) authors

provide extensive research of the origins of the guess-

ing game with unexpected links to the editor of one

french newspaper Alain Ledoux, who, as far as it is

known today, was the ﬁrst who used the rules and then

publish an article about unusual patterns observed

(Ledoux, 1981). The work of Nagel (Nagel, 1995)

was the ﬁrst experimental try to investigate the hid-

den patterns in the guessing game, and in this working

framework of k-level models was proposed.

Later, Ho et al. (Ho et al., 1998) gave the name “p-

beauty contest” inspired by Keynes’s comparison of

stock market instruments and newspaper beauty con-

tests.

The beauty contest game (BCG) has become an

important tool to measure the “depth of reasoning” of

a group of people using simple abstract rules. To be-

gin with, we should note that behavioral game theory

aims to develop models, which explain human behav-

ior in a game-theoretic setup more accurately, based

both on experiments and theory (Camerer, 2003).

There are two main approaches how to dealing with

the problem of replacing full rationality with bounded

rationality. The ﬁrst view is to consider bounded-

ness as an error. For example quantal response no-

tion (Camerer et al., 2002) or ε-equilibrium (Leyton-

Brown and Shoham, 2008) assume that agents make

an error by choosing not optimal strategy proﬁle.

They play near-optimal response because they do not

have the capacity to calculate the exact best action.

The second approach is to treat bounded rational-

ity as a structural property of an agent’s reasoning

process. One of the most prominent classes of mod-

els of this type is the iterative model’s scheme. They

include the k-level reasoning (Nagel, 1995; Costa-

Gomes et al., 2001), cognitive hierarchy (Camerer

et al., 2004) and quantal cognitive hierarchy models

(Wright and Leyton-Brown, 2017). All these models

consider boundedness as an immanent part of the rea-

soning. Each agent has a non-negative integer level

representing the degree of strategic reasoning (i.e.,

modeling of recursive beliefs) of which the agent is

capable. Level-0 agents are nonstrategic – they do

not model other agents’ beliefs or actions at all; level-

1 agents model level-0 agents’ actions; level-2 agents

model the beliefs and actions of level-1 agents; and

so forth (Wright and Leyton-Brown, 2016). In this

work, we support the latter idea, analyzing experi-

mental data to estimate changes in numbers of dif-

ferent levels in the learning and teaching process.

2.1 Learning Models

Recently, game theorists began to research the process

of reasoning toward equilibrium actively. Two promi-

nent simple learning models are reinforcement and

belief learning (e.g., ﬁctitious play). In reinforcement,

strategies have numerical attraction levels, which are

reinforced (increased) when a strategy is chosen, and

the result is good. Reinforcement is a good model of

animal learning but does not gracefully accommodate

the fact that people often choose strategies that have

not been directly reinforced.

In the ﬁctitious play, players form beliefs based

on a weighted average of what others have done in

the past, and best respond given their beliefs. Re-

markably, weighted ﬁctitious play is equivalent to a

generalized reinforcement model in which unchosen

strategies are reinforced by the forgone payoffs they

would have yielded.

There are a lot of other approaches; we will men-

tion the approach which enriches 0-level reasoning

(Wright and Leyton-Brown, 2017). Speciﬁcally, they

investigate general rules that can be used to induce

a level-0 speciﬁcation from the normal-form descrip-

tion of an arbitrary game.

Also, we can note work (Gill and Prowse, 2016),

where participants were tested on cognitive abilities

and character skills before the experiments. Then

authors perform statistical analysis to understand the

impact of such characteristics on the quality of mak-

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

708

ing strategic decisions (using a p-beauty contest game

with multiple rounds). In more recent work (Fe et al.,

2019) even more elaborate experiments are presented.

It is interesting that in the mentioned paper, experi-

ments are very strict and rigorous (as close to labo-

ratory purity as possible) in contrast to games played

in our research. However, at the end of the day, the

results do not differ very much.

As we know, there are not many works about game

theory experiments for children. In our previous work

(Ignatenko, 2020) we presented data from games with

participants 15-18 years old. There is a master thesis

(Povea and Citak, 2019), with the study of the behav-

ior of children aged 8-11 in a beauty contest game

with ten repetitions. The author found evidence that

children are able to play a beauty contest game using

not only cognitive skills but also empathy.

To deal with these problems, computer simula-

tion, mainly agent-based modeling ABM can be used.

Agent-based models are essentially a tool to discover

patterns in behaviors that emerge from simple rules –

micro behavior. Agent-based modeling for guessing

games is not a very developed area of research. For

example see paper (Nichols and Radzicki, 2007).

3 EXPERIMENTS SETUP

We claim that our setup is closer to reality than the

laboratory, and this is the point of this research: how

people learn under real-world situations. All games

were played under the following conditions:

1. Game was played during the lecture about the

game theory. Participants were asked not to com-

ment or discuss their choice until they submitted

it. However, this rule was not enforced, so usually,

they have this possibility if they wanted;

2. Participants were not rewarded for a win. The

winner was announced (so get some “good feel-

ings”), but no more;

3. During some early games, we used pieces of pa-

per, and we got some percentage of joking or

trash submission, usually very small. Later we

switched to google forms, which is a better tool

to control submission (for example, only natural

numbers are allowed).

4. Google forms give a possibility to make multiple

submissions (with different names) since we did

not have time for veriﬁcation, but a total number

of submissions allows us to control that to some

extent.

The aim of this setup was to free participants to

explore the rules and give them the ﬂexibility to make

a decision in an uncertain environment. We think it is

closer to real-life learning without immediate rewards

than laboratory experiments. Naturally, this setup has

strong and weak sides. Let us summarize both.

The strong sides are:

1. This setup allows to measure how people make

decisions in “almost real” circumstances and un-

derstand the (possible) difference with laboratory

experiments;

2. These games are part of an integrated approach to

active learning, when games are mixed with ex-

planations about concepts of game theory (ratio-

nality, expected payoff, Nash equilibrium), and

they allow participants to combine experience

with theory;

3. Freedom and responsibility. The rules do not reg-

ulate manipulations with conditions. So this setup

allows (indirectly) to measure the preferences of

players: do they prefer to cheat with rules, choose

random decisions without thinking or put effort

into solving the task;

4. During the 2020-2021 years, lectures were mainly

online. That fact brings new challenges for our

experiments, but since we initially rely on google

spreadsheets for gathering the answers, all rou-

tines remain mainly the same.

Weak sides are:

1. Some percentage of players made “garbage” de-

cisions. For example, choose the obviously worse

choice to spoil efforts for others;

2. Kids has (and often use) the possibility to talk out

decision with the neighbors;

3. Sometimes participants (especially kids) lost con-

centration and did not think about the game but

made a random choice or did not make decisions

at all;

4. Even for the simplest rules, sometimes partici-

pants failed to understand the game the ﬁrst time.

We suppose it is due to conditions of lecture with

(usually) 30-40 persons around;

5. Still, we should note that online lectures are less

involving. It is demotivating for a lecturer not to

see listeners and also demotivating for students to

listen to lectures online.

3.1 Rules

In this paper, we concentrate on the p-beauty contest.

The winning number is the closest to p of average.

As usual, p = 2/3, but we have used other setups as

well (for clarity, we omit data from that games in this

How Concept Learning Changes Strategic Choice in Guessing Games?

709

analysis). Participants are asked to choose an integer

number in the range 1 – 100, margins included. Note

that some setups investigated in references use a range

starting with 0. However, the difference is negligible.

To provide quick choice calculation, we have used a

QR code with a link to google form, where partici-

pants input their number. All answers were anony-

mous (players indicated nicknames to announce the

winners, but then all records were anatomized).

4 RESULTS AND DATA ANALYSIS

In this section, we present a summary of data gathered

during the games. A summary of the results of the

game is given in table 1. Columns descriptions are:

• id is the id of the experiment;

• type is the type of group. Alternative H and M

are for alternative schools (not in the governmen-

tal system) with humanitarian and mathematical

directions, respectively. Math lyceum also goes

for summer camps with participants from differ-

ent lyceums;

• age is the approximate age of participants, only

indicated for children, to distinguish possible bor-

derline between stages of strategic reasoning;

• round is the round of the game;

• average is the average of choices;

• winning number is the average * 0.66;

• zlevel is the percent of players, choosing numbers

bigger than 50. It is an estimation of 0-level play-

ers in this round. As one can expect, it is declining

with round;

• median is the median of choices (sometimes it is

more informative than average);

• count is the number of choices;

• irrationality is the percent of choices wider than

90.

First, we observe statistics of choices for different

types of participants.

As we can see from the plot (ﬁgure 1) some setups

have one round, some two and two cases have three

rounds. It was limited by format (popular scientiﬁc

lecture), so we can only safely compare two rounds.

Almost all winning numbers fall (roughly) in

the experimental margins, obtained in (Nagel, 1995).

With winning number no more signiﬁcant than 36 and

not smaller than 18 in the ﬁrst round. Two exceptions

in our experiments were Facebook online game (15.3)

when players could read information about the game

in, for example, Wikipedia. Moreover, Another is the

alternative humanitarian school (40.1), where partici-

pants seem did not get the rules the ﬁrst time.

4.1 Metrics and Analysis

The ﬁrst metric to observe is the percent of “irrational

choices” – choices that cannot win in (almost) any

case. Let us explain, imagine that all players will

choose 100. It is impossible from practice but not for-

bidden. In this case, everybody wins, but if only one

player deviates to a smaller number – he/her will win,

and others will lose. So playing numbers bigger than

66 is not rational unless one does not want to win.

Furthermore, here we come to an important point,

in all previous experiments, this metric drops in the

second round and usually is very low (like less than

5%) (Ho et al., 1998). However, in our case, there

are experiments where this metric becomes higher or

changes very slightly. Moreover, initially, values are

much higher than expected. So here, we should in-

clude the factor of unique behavior; we can call it “let

us show this lecturer how we can cheat his test!” what

is more interesting – is that this behavior is more ap-

parent in the case of an adult than in kids.

It is also interesting to see a distribution of choices

for different groups. We can summarize choices on

the histograms (ﬁgure 2). Using models of strate-

gic thinking, we will adopt the theory of k-levels.

According to this idea, 0-level reasoning means that

players make random choices (drawn from a uniform

distribution), and k-level reasoning means that these

players use the best response for the reasoning of the

previous level. So 1-level reasoning is to play 33,

which is the best response to the belief that the av-

erage will be 50, 2-level is the best response to the

belief that players will play 33, and so on.

As we can see from the diagram (ﬁgure 2), some

spikes in choices are predicted very well, but it de-

pends on the background of the players. The best

prediction is for the Data Science conference atten-

dees, which presume a high level of cognitive skill

and computer science background.

On ﬁgure 3, we can see boxplots deﬁned by sev-

eral players with different levels of perception for

different types of players. We compare here ordi-

nary schools and mathematical lyceums. Levels are

deﬁned in the following subsection, but we can see

a pattern of behavior. The number of “irrational”

(choices with big numbers) is decreasing, so as “next-

to-win-but-bigger” numbers. Several 2-level reason-

ing, especially after explaining the equilibrium con-

cept, is growing substantially, while some “too smart”

choices from [1, 5] are more or less the same.

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

710

Table 1: Summary of ﬁrst game for id of experiment and type of players. Explanation of columns is in the text

Id Type Age Round Average Winning Zlevel Median Count Irrationality

1 Alternative H 12-14 1 66.7 44.5 69.23 78 13 46.15

1 Alternative H 12-14 2 3.91 2.61 0 3.5 12 0

2 Alternative M 12-14 2 42.82 28.54 23.52 45.0 17 0

2 Alternative M 12-14 2 24.37 16.24 0 26.5 16 0

3 Adults 1 40.57 27.05 31.57 40.0 19 5.26

4 Alternative H 12-14 1 52.54 35.03 63.63 55 11 9.09

4 Alternative H 12-14 2 15.41 10.27 8.33 6 12 8.33

5 Adults 1 22.98 15.32 11.76 17.0 102 0

6 TechSchool 16-18 1 43.41 28.94 35.29 45.0 51 3.92

6 TechSchool 16-18 2 46.5 30.99 35.48 29.0 62 32.25

7 Math lyceum 16-18 1 30.58 20.38 16 27.5 50 2.0

7 Math lyceum 16-18 2 14.26 9.5 5.26 7 57 5.26

8 Math lyceum 15-16 1 37.06 24.71 20.68 33.0 29 3.44

8 Math lyceum 15-16 2 26.20 17.47 10.34 17.0 29 6.89

9 Math lyceum 14-16 1 42.0 27.99 44.44 42.5 18 11.11

9 Math lyceum 14-16 2 23.1 15.39 5.0 19.0 20 0

10 Ordinary school 14-16 1 48.69 32.46 46.15 46.5 26 0

10 Ordinary school 14-16 2 19.78 13.18 0 22.0 23 0

11 DS conference 1 37.25 24.83 28.33 33.0 60 8.33

11 DS conference 2 21.44 14.29 15.78 9.0 57 12.28

12 Students 1 42.40 28.27 33.33 40.0 27 3.7

13 Students 1 27.37 18.24 12.5 25.5 8 0

13 Students 2 8.62 5.74 0 8.5 8 0

14 Math lyceum 14-16 1 41.05 27.37 22.22 35.0 18 11.11

14 Math lyceum 14-16 2 17.23 11.49 5.88 13.0 17 0

15 Adults 1 34.32 22.88 20.73 30.0 82 1.21

15 Adults 2 12.48 8.32 2.19 8.0 91 2.19

16 Adults 1 43.05 28.70 33.96 40.0 53 1.88

16 Adults 2 14.69 9.79 1.88 11.0 53 1.88

17 Adults 1 50.33 33.55 41.66 50.0 12 8.33

17 Adults 2 13.50 8.99 0 12.0 46 0

18 Math lyceum 14-16 1 41.72 27.81 36.36 37.0 11 9.09

18 Math lyceum 14-16 2 26.36 17.57 0 30.0 11 0

19 Math lyceum 14-16 1 29.43 19.62 13.63 25.0 44 0

19 Math lyceum 14-16 2 27.25 18.16 20.45 9.5 44 20.45

20 Students 1 30 19.9 5.2 27 19 0

20 Students 2 24.9 16.6 20 11.5 20 15

21 Ordinary school 14-16 1 43 28.7 33.9 40 53 1.88

21 Ordinary school 14-16 2 14.7 9.7 1.88 11 53 1.88

Interesting hypotheses that need to be tested in de-

tail can be formulated: Higher number of choices

from [50, 100] in the ﬁrst round leads to the higher

number of choices from [1, 5] in the second round

and vice verse. We can support this hypothesis with

the following plot (ﬁgure 5).

Another metric (G

uth et al., 2002) is how much

winning choice in the second round is smaller than in

the ﬁrst. Due to multi-level reasoning, every player

in this game is trying their best to win but can’t do

all the steps to winning. So there are players who

have 0-level reasoning. They choose random num-

bers. First-level players choose 33, which is the best

response for players of 0-level and so on. Based on

the result of the ﬁrst round and, in fact, an explana-

tion of the Nash equilibrium, players must know that

it is better to choose much lower numbers. However,

the graph shows that the decrease is quite moderate.

Only students show good performance in this matter.

Moreover, the tech school shows a (small) increase in

winning number in the second round!

How Concept Learning Changes Strategic Choice in Guessing Games?

711

Figure 1: Histogram of choices for each round.

Figure 2: Histogram of choices for each round.

4.2 Levels of Reasoning Snalysis

Another point about the learning process in this game

is how players’ decisions are distributed over the

space of strategies. We claim that there is a distinct

difference in changes between the ﬁrst and second

rounds for different groups. To perform this analysis,

we apply the idea of k-level thinking.

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

712

Figure 3: Change in winning number for number of ordinary schools participants.

Figure 4: Change in winning number for number of mathematical lyceums schools participants.

To ﬁnd differences, we need to simplify this ap-

proach. First, we deﬁne b-level players players who

choose numbers from the range [50, 100]. Beginner

players who do not understand rules (play randomly)

or do not expect to win or want to lose intentionally

(for reasons discussed above). The substantiation for

such a range is that numbers above 50 did not win

in any game. Second level we call m-level, it is for

range [18, 50]. It is for players with middle levels of

reasoning. The ﬁrst-round winning number is usually

in this range (and in part of the second rounds). Third

level is h-level, it is for range [5, 18]. It is for high

level reasoning and ﬁnally inf-level ([1, 5] range) is

for “almost common knowledge” level of thinking.

By calculating the number of levels for each game,

we can estimate change (in the percentage of the num-

ber of players) in adopting different strategy levels.

What conclusions can we draw from this data?

There are no clear differences in changes, but at least

we can summarise a few points:

How Concept Learning Changes Strategic Choice in Guessing Games?

713

Figure 5: Choices number in [1, 5] range and in [50, 100] range by rounds and type.

Table 2: Summary of change in strategy levels

Type b-difference m-difference h-difference inf-difference

Alternative humanitarian -72 -8 0 72

Alternative mathematical -24 -6 30 -6

Alternative humanitarian -52 0 17 43

Math lyceum -9 -36 24 34

Math lyceum -10 -24 28 7

Ordinary school -49 12 20 4

DS conference attendees -14 -32 14 27

MS students -12 -50 50 12

Alternative mathematical -17 -34 23 23

DS conference attendees -17 -30 23 35

Business -32 -17 21 28

• Usually after ﬁrst round and equilibrium concept

explanation there is decrease in b-level and m-

level;

• Symmetrically, there is an increase in two other

levels, but sometimes it is more distributed, some-

times it is (almost) all for inf-level;

• the Last situation is more likely to happen in

schools, where kids are less critical of new knowl-

edge;

• Usually second round winning choice in the realm

of h-level, so groups with the biggest increase in

this parameter are the ones with better understand-

ing.

Another possible approach to the measurement

of irrationality is to calculate the percent of choices

wider than 50 (quite low chances for the win) and the

percent of choices wider than 90 (no chances to win).

We can see an interesting picture when we plot these

metrics for different types.

As we can see here, there is a nice direct line of

green dots of unknown nature. And all red dots are

gathered below. This is interesting dependence that

needs to be investigated in detail.

4.3 Size and Winning Choice

This game is indeed rich for investigation. Let us for-

mulate the last (in this paper) ﬁnding of this game.

Can we in some way establish the connection between

the number of players and the winning number (actu-

ally with strategies players choose during the game)?

To clarify our idea, see at ﬁgure 7. It is a scattered

plot of a two-dimensional variable; the x-axis is for

the number of participants in the game, and the y-axis

is for the winning choice per round. Different colors

are for different types of groups where games were

played.

As we can observe, the ﬁrst and second rounds

form two separate clusters. This situation is expected

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

714

Figure 6: Irrationality of participants.

Figure 7: Change in winning number for number of participants.

and informs us that players learn about the equilib-

rium concept between rounds and apply it to practice.

Also, there is a mild tendency for smaller groups to

have bigger winning numbers. At least the variation

is more signiﬁcant.

This is yet too bold to formulate a connection be-

tween the size of the group and the winning number,

but probably the reason is that when the size of the

group is bigger, the number of “irrational” players in-

creases. It can be due to some stable percentage of

such persons in any group or other reasons, but it is

an exciting connection to investigate.

4.4 Intentionally Irrational?

Another interesting ﬁnding is that after the ﬁrst round

ﬁnished, observing the result and listening explana-

tion about the NE number of players who choose over

90 (it is an non-winning choice) increases. It is not ac-

cidental; data from (Povea and Citak, 2019) also show

an increase in 2-5 rounds. We believe that this is quite

an important part of the play. This phenomenon is

evident in high school children with solid math back-

grounds (usually, they have more freedom and self-

conﬁdence in choosing non-standard strategies).

How Concept Learning Changes Strategic Choice in Guessing Games?

715

All experimental data and R ﬁles for graphs can

be accessed in the open repository (Ignatenko, 2021).

5 AGENT-BASED MODEL

The existence of irrational behavior challenges the ba-

sic game-theoretic assumption about self-interest and

the capability to calculate the best option. In other

words, real people do not think like machines or algo-

rithms. They form hypotheses or expectations using

simple rules. These rules are inﬂuenced by emotions

and social norms and can be changed depending on

feedback (reinforced). This use of inductive reason-

ing leads to two issues. First, what rules that people

follow? Second, suppose we know these rules. How

do we model the behavior of many interacting, het-

erogeneous agents in that situation? We start with the

deﬁnition of agents and the formulation of rules of

their behavior.

So we have agents of one type – players. Each

agent has three variables: level of thinking (cur-

rent level of reasoning), choice (a current number he

chooses), and Boolean variable irrational, which is

true or false.

Also, we deﬁne interface parameters, which de-

ﬁne the setup of players. First of all, it is the number

of players. We can also deﬁne the percentage of level

0 players, p of the game, and irrational setup, which

will be explained later.

The setup of the game is following:

1) creates several players;

2) deﬁnes the level of reasoning for each player us-

ing the formula: level-0 percent from the inter-

face, 100 – level-0 is divided into three parts. Two

parts are level-1 reasoning, and one part is level-2

reasoning;

One round of the game proceeds in the following

way:

1) each player chooses a number using their level

of reasoning and some randomization. Basically,

player with k level generates normal variable with

mean 50 ∗ p

if k > 0 and k ≤ 4. If k = 0, the

choice is uniformly random from 1 to 100. When

k = 5, the choice is 1 (this is an inﬁnity level of

reasoning);

2) if the player is irrational, he chooses 100. It is

to model irrational behavior that can be observed

from experiments;

3) the winner is calculated using the game formula;

4) players, with a choice wider than the winning

number, increase their understanding level by 1.

If the level is already equal to 5, it remains the

same;

5) if the player is not winner, he becomes irrational

with some small probability if boolean variable ir-

rational is on.

The model is available in COMses library of Net-

logo models. To measure experiments data we use

BehaviorSpace tool with following parameters:

["percent-level-0" [50 5 100]]

["num-players" [10 10 100]]

["irrational1" false true]

["p" 0.66]

Here [10 10 100] means we launch simulation for

10, 20, and so on several players. On each step, we

wrote to ﬁle the choices of players. 10 steps limited

each particular games. In total, we had 11000 runs.

Epstein (Epstein, 1999) deﬁnes following charac-

teristics of agent-based model:

1) heterogeneity; agents are different in some ways;

2) autonomy; each agent make own decisions;

3) explicit space; agents interact in a given environ-

ment;

4) local Interaction; agents generally interact with

their neighbors and immediate environment.

5) bounded Rationality; agents have limited infor-

mation and computing power. Agent behavior

is generated by simple rules that may adapt over

time;

6) non-equilibrium dynamics.

In this model, we consider only ﬁve levels, where

level 5 means common knowledge when a player

chooses 1. When no irrationality is in the model, we

can observe typical convergence to equilibrium (ﬁg-

ure 8) left, and this is a stable pattern.

But as we already know from the experiments, it

is not what we can observe in real life. So irrational

behavior was included to meet the pattern from the

data. Irrationality in our model is implemented as

’anger’ when a player who is currently a loser some-

times goes to irrational mode and chooses 100 in one

next round. This leads to an exciting pattern (ﬁgure 8)

right, when sometimes the winning number increases

in the second round, but convergence to equilibrium

is inevitable.

5.1 Data Analysis

In this section, we analyze data from simulations and

compare them to previous results. In the following

plot (ﬁgure 9) we can observe convergence towards

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

716

Figure 8: Plots of players win choice over round.

Figure 9: Plots of players win choice over round.

equilibrium. As expected volatility of choices is more

signiﬁcant in smaller groups.

Now let us plot (ﬁgure 10) winning choice in the

ﬁrst round compared to the winning choice in the sec-

ond round. As we can see, there is a big cluster around

(33,31). The color here is the initial percentage of

low-level players. High level means that almost all

players initially are randomizers, and then they learn

How Concept Learning Changes Strategic Choice in Guessing Games?

717

Figure 10: Plots of players win choice for ﬁrst and second rounds.

Figure 11: Irrationality of players.

from round to round. A trim level allows for more

efﬁciency when points are like (30,20). This is quite

similar to human experiments data.

The next plot is about the irrationality of players.

It seems that there is a connection between the percent

of players who chooses numbers bigger than 90 and

the percent of players who chooses numbers bigger

than 50. It is very similar to ﬁgure 8 from experimen-

tal data. The reason, as we can understand it, is in

the nature of irrational behavior in games like the k-

beauty contest. It was a common situation when play-

ers at some point (we have data for round 2) lost hopes

of winning and just chose 100 for fun. At the same

time, the majority of players are still thinking strate-

gically and choose numbers around winning numbers

(20 – 30 range). This gives a characteristic line of dots

because, at some point, all players who chose num-

bers bigger than 50 are the players who chose 100.

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

718

6 CONCLUSIONS

In this paper, we have presented data from experi-

ments on the k-beauty contest game. We provide ac-

cess to raw data ﬁles and ﬁles with data manipulation,

metrics calculation, and plot building. This will hope-

fully support reproducibility in this area of research.

Then we discuss the results of the experiments and

provide an analysis and explanation of patterns of be-

havior. It seems possible to conﬁrm the existence of a

pattern in decision-making – every group behaves al-

most the same way when dealing with an unknown

strategic situation. We can formulate ﬁndings in a

few short notes. First, participants have chosen not

winning moves (> 66) partly because of a new situ-

ation and trouble understanding the rules. However,

a high percentage of such choices was present in the

second round, when players knew what was going on.

This effect was especially notable in the cases of high

school and adults and almost zero in the case of spe-

cial math schools and kids below 9th grade. We can

hypothesizeesis that high school is the age of experi-

mentation when children discover new things ared do

not afraid to do so. Second, the winning number as

the decision of a group is decreasing in all cases, so

we can see that group is learning fast and steady. Even

if some outliers choose 100, the mean still declines

with every round. There seems to be an unspoken

competition between players that leads to improve-

ment in the aggregated decision even if no prize is at

stake. It is a plausible scenario when all participants

choose higher numbers. However, this did not hap-

pen in any experiment. Third, a a stable percent of

people choose about 100, and it is not about learning

how to play the game. We think this is something like

a -1 level of reasoning when the player intentionally

plays a “bad move”, and this is an essential part of the

model. If we neglect such persons and their motiva-

tion, our model will not be correct.

In the second part of the paper, we presented an

agent-based model using conclusions about human

behavior in the game. Simpliﬁcation is the key to

building good ABM, so in the model, agents have

only a few parameters: level and irrationality. Based

on these two parameters, agents choose a number

on each step. Depending on other players’ choices

model determine winners and losers: this state inﬂu-

ences agents’ future level and irrationality. We per-

form about ten thousand games (each ten rounds long)

and apply the same analysis as before. This approach

shows that in some aspects, agents’ behavior is close

to humans. Future investigation of this model will

concentrate on the following possible modiﬁcations:

• deterministic choices of agents;

• different numbers of reasoning levels;

• rewards and punishments as elements of learning;

• implementation learning models from current re-

search;

• build testing environment to automatically com-

pare different learning strategies.

REFERENCES

Aumann, R. and Brandenburger, A. (1995). Epis-

temic Conditions for Nash Equilibrium. Economet-

rica, 63(5):1161–1180. http://www.jstor.org/stable/

2171725.

Aumann, R. J. (1995). Backward induction and com-

mon knowledge of rationality. Games and Eco-

nomic Behavior, 8(1):6–19. https://doi.org/10.1016/

S0899-8256(05)80015-6.

Camerer, C. F. (2003). Behavioral game theory: Experi-

ments in strategic interaction. The Roundtable Series

in Behavioral Economics. Princeton University Press.

Camerer, C. F., Ho, T.-H., and Chong, J.-K. (2002). Sophis-

ticated experience-weighted attraction learning and

strategic teaching in repeated games. Journal of

Economic theory, 104(1):137–188. https://doi.org/10.

1006/jeth.2002.2927.

Camerer, C. F., Ho, T.-H., and Chong, J.-K. (2004). A cog-

nitive hierarchy model of games. The Quarterly Jour-

nal of Economics, 119(3):861–898. https://doi.org/10.

1162/0033553041502225.

Costa-Gomes, M., Crawford, V. P., and Broseta, B. (2001).

Cognition and Behavior in Normal-Form Games: An

Experimental Study. Econometrica, 69(5):1193–

1235. https://doi.org/10.1111/1468-0262.00239.

Crawford, V. P., Costa-Gomes, M. A., and Iriberri, N.

(2013). Structural models of nonequilibrium strategic

thinking: Theory, evidence, and applications. Journal

of Economic Literature, 51(1):5–62. https://doi.org/

10.1257/jel.51.1.5.

Epstein, J. M. (1999). Agent-based computational models

and generative social science. Complexity, 4(5):41–

60. https://cutt.ly/hKtpQa9.

Fe, E., Gill, D., and Prowse, V. L. (2019). Cognitive skills,

strategic sophistication, and life outcomes. Working

Paper Series 448, The University of Warwick. https:

//tinyurl.com/2p8jmnjn.

Gill, D. and Prowse, V. (2016). Cognitive Ability, Character

Skills, and Learning to Play Equilibrium: A Level-k

Analysis. Journal of Political Economy, 124(6):1619–

1676. https://doi.org/10.1086/688849.

uth, W., Kocher, M., and Sutter, M. (2002). Experi-

mental ‘beauty contests’ with homogeneous and het-

erogeneous players and with interior and boundary

equilibria. Economics Letters, 74(2):219–228. https:

//doi.org/10.1016/S0165-1765(01)00544-4.

Ho, T.-H., Camerer, C., and Weigelt, K. (1998). Iterated

Dominance and Iterated Best Response in Experimen-

tal “p-Beauty Contests”. The American Economic

How Concept Learning Changes Strategic Choice in Guessing Games?

719

Review, 88(4):947–969. https://www.jstor.org/stable/

117013.

Ignatenko, O. (2020). Guessing Games Experiments in

School Education and Their Analysis. In Sokolov,

O., Zholtkevych, G., Yakovyna, V., Tarasich, Y.,

Kharchenko, V., Kobets, V., Burov, O., Semerikov,

S., and Kravtsov, H., editors, Proceedings of the

16th International Conference on ICT in Educa-

tion, Research and Industrial Applications. Integra-

tion, Harmonization and Knowledge Transfer. Vol-

ume II: Workshops, Kharkiv, Ukraine, October 06-10,

2020, volume 2732 of CEUR Workshop Proceedings,

pages 881–892. CEUR-WS.org. https://ceur-ws.org/

Vol-2732/20200881.pdf.

Ignatenko, O. P. (2021). Data from experiments. https://

github.com/ignatenko/GameTheoryExperimentData.

Kuhn, H. W., Harsanyi, J. C., Selten, R., Weibull, J. W.,

Van Damme, E., Nash Jr., J. F., and Hammerstein, P.

(1996). The work of John Nash in game theory. Jour-

nal of Economic Theory, 69(1):153–185.

Ledoux, A. (1981). Concours r

esultats complets. Les vic-

times se sont plu

a jouer le 14 d’atout. Jeux &

Strat

egie, 2(10):10–11.

Leyton-Brown, K. and Shoham, Y. (2008). Essentials of

game theory: A Concise, Multidisciplinary introduc-

tion. Synthesis lectures on artiﬁcial intelligence and

machine learning, 2(1):1–88. https://doi.org/10.2200/

S00108ED1V01Y200802AIM003.

Mauersberger, F. and Nagel, R. (2018). Levels of rea-

soning in Keynesian Beauty Contests: a generative

framework. In Hommes, C. and LeBaron, B., editors,

Handbook of Computational Economics, volume 4 of

Handbook of Computational Economics, pages 541–

634. Elsevier.

Moulin, H. (1986). Game theory for the social sciences.

New York Univeristy Press, New York, 2nd edition.

Nagel, R. (1995). Unraveling in Guessing Games: An Ex-

perimental Study. The American Economic Review,

85(5):1313–1326. https://www.cs.princeton.edu/

courses/archive/spr09/cos444/papers/nagel95.pdf.

Nagel, R., B

uhren, C., and Frank, B. (2017). In-

spired and inspiring: Herv

e Moulin and the discov-

ery of the beauty contest game. Mathematical So-

cial Sciences, 90:191–207. https://doi.org/10.1016/j.

mathsocsci.2016.09.001.

Nichols, M. W. and Radzicki, M. J. (2007). An Agent-

Based Model of Behavior in “Beauty Contest” Games.

Working Paper 07-010, University of Nevada. https:

//www.researchgate.net/publication/23694542.

Povea, E. and Citak, F. (2019). Children in the beauty con-

test game: behaviour and determinants of game per-

formance. Master’s thesis, Norwegian School of Eco-

nomics. https://openaccess.nhh.no/nhh-xmlui/handle/

11250/2611651.

Wright, J. R. and Leyton-Brown, K. (2016). Models of

level-0 behavior for predicting human behavior in

games. CoRR, abs/1609.08923. https://doi.org/10.

48550/ARXIV.1609.08923.

Wright, J. R. and Leyton-Brown, K. (2017). Predicting

human behavior in unrepeated, simultaneous-move

games. Games and Economic Behavior, 106:16–37.

https://doi.org/10.1016/j.geb.2017.09.009.

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

720