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AIIDE’23: Experimental AI in Games, October 8, 2022, Salt Lake City, UT

[1]You can find all source code, dataset, and results at our Gitlab Repository
https://gitlab.com/humin-game-lab/artificial-psychosocial-
framework/-/tree/master/LLM%20Personality

<EMAIL_ADDRESS>url=http://ljklinkert.com/home.html, ]

<EMAIL_ADDRESS>url=https://stephbuon.github.io/, ]

<EMAIL_ADDRESS>url=https://humingamelab.com/, ]

# Driving Generative Agents With Their Personality

Lawrence J. Klinkert  Southern Methodist University, 6425 Boaz Lane, Dallas,
TX, 75205, United States  Stephanie Buongiorno  Corey Clark

(2023)

###### Abstract

This research explores the potential of Large Language Models (LLMs) to
utilize psychometric values, specifically personality information, within the
context of video game character development. Affective Computing (AC) systems
quantify a Non-Player character’s (NPC) psyche, and an LLM can take advantage
of the system’s information by using the values for prompt generation. The
research shows an LLM can consistently represent a given personality profile,
thereby enhancing the human-like characteristics of game characters.
Repurposing a human examination, the International Personality Item Pool
(IPIP) questionnaire, to evaluate an LLM shows that the model can accurately
generate content concerning the personality provided. Results show that the
improvement of LLM, such as the latest GPT-4 model, can consistently utilize
and interpret a personality to represent behavior.

###### keywords:

Large Language Models chatGPT psychometrics procedural content generation

##  1 Introduction

In the dynamic and rapidly evolving domain of the video game industry, the
exigent task of developing Non-Playable Characters (NPCs) that emulate human-
like behavior presents a formidable challenge. A significant obstacle is the
aspiration to create NPCs interacting with their environment and displaying a
rich tapestry of character depth and emotional complexity. The current state
of NPCs often fails to meet this ambitious objective, resulting in a gaming
experience that is less immersive and emotionally engaging.

The impetus for this study is rooted in the potential of Affective Computing
(AC) to bridge this existing gap. With its capacity to recognize, interpret,
and simulate human emotions, AC can supply the necessary psychometric
components to delineate NPCs’ emotional and personality traits. By
capitalizing on components from an AC system, we show to augment the emotional
intelligence of NPCs, thereby fostering a more immersive and emotionally
engaging gaming experience.

This paper hypothesizes that Large Language Models (LLMs) can be
synergistically used with AC systems to engender more human-like NPCs. This
research suggests that LLMs can be prompted with psychometric values derived
from an AC system, particularly personality, the LLM can generate content
aligning with the expected behaviors of the NPC. This process is validated by
repurposing the International Personality Item Pool (IPIP) questionnaire, a
human measuring evaluation, to evaluate the LLMs generated content. By
comparing the prompted personality against the questionnaire’s results, we
show that the latest LLM can consistently generate the results expected for
the initial personality. The demonstration and validity of the personality
with an LLM indicates the plausibility of utilizing more values from an AC
system, providing additional quantitative information for an LLM to generate
content aligning with an NPCs psyche.

The subsequent sections of this paper will delve deeper into this hypothesis,
exploring the potential of integrating AC psychometrics with LLMs. Section two
will elucidate the background information, encompassing topics such as video
game companies researching AC for their games, the latest scholarly research
in AC, LLM integrations with video games, and the rationale for integrating AC
with an LLM. Section three will delve into personality psychometrics,
specifically the Big Five personality model. Section four will discuss using a
personality test dataset and how its information can be used to benchmark
against an LLM. Section five will generate synthetic data from the LLM and
compare its results against the baseline dataset. Section six will outline a
use case of an LLM using personality for gameplay. The paper will conclude
with section seven.

##  2 Background Information

Prominent video game corporations increasingly endeavor to integrate AC
systems into their games. Corporations such as Square Enix, Worldwalker Games,
and Eidos Sherbrooke are exploring the possibilities of AC and its
assimilation into their gaming products [1, 2, 3]. These corporations employ
AC to empower Non-Playable Characters (NPCs) to perceive and project emotions
within their games. In executing this approach, these video game corporations
experimented with emotive simulations for NPCs so that the player can
recognize and empathize with the character. However, AC systems such as
Popescu et al.’s GAMYGDALA, Shirvani & Ware’s emotional narrative planning
system, and Klinkert and Clark’s Artificial Psychosocial Framework (APF) [4,
5, 6], are proficient in discerning, interpreting, processing, and simulating
human emotional states between player and NPCs. Enhancing this pursuit for
human-like NPCs is the integration of an LLM, which further improves the
sophistication of dialogue generation and decision-making processes not only
with the player but with other NPC interactions as well.

The advent of LLMs is paradigm shifting in the contemporary technological
landscape. Many companies, such as Microsoft, Google, Meta, and Nvidia, are
actively seizing opportunities to integrate LLMs into their products [7].
However, this transition is not without its challenges, which include issues
such as data hallucinations, memory limitations, and constrained accessibility
of these models [8, 9]. These issues become magnified within the context of
the video game industry. This sector is keen to employ LLMs to generate in-
game content but encounters additional complications, including the assurance
of correct behavior, potential ethical liabilities, and extensive retesting
debugging scenarios [10]. These complexities, coupled with the initial
challenges, have induced a level of hesitancy within the industry toward fully
embracing LLMs [11]. Nevertheless, the work conducted by Park et al., which
highlights the development of Generative Agents, serves as a remarkable
exemplar, demonstrating that the successful integration of an LLM system into
video games is indeed achievable [12]. With the integration of an LLM into a
simulated village, Park et al. show NPCs understand their surroundings, plan
their days with what they know, and spread information to share knowledge. To
further push these NPCs to be believable humans, the game would integrate an
AC system that provides psychometric values for the LLM to process alongside
their knowledge representation and dissemination.

The primary objective of this research is to investigate the potential
capability of an LLM to output the correct content of a character’s
psychological persona based on the input of psychometric values. AC systems
are instrumental in providing these psychometric values, encompassing facets
such as personality traits, emotional states, and the dynamics of
relationships between game entities and other characters in the environment
[4, 5, 6]. This paper lays out a crucial first step in this inquiry:
discerning whether an LLM can effectively interact with and interpret these
psychometric values, with an initial focus on personality information.
Personality traits, whether based on popular psychological models such as the
Big Five or other frameworks, can significantly impact a character’s
decisions, actions, and reactions within a game’s narrative. Therefore, the
capability of an LLM to properly utilize these traits can play a crucial role
in creating characters that are not only believable but also possess a
consistent behavioral pattern.

AC systems capitalize on personality models such as the Big Five, otherwise
referred to as the Five Factor or OCEAN model, to represent a character’s
fundamental psychological composition within a game [6]. This model comprises
five diverse factors: Openness to novel experiences, Conscientiousness in
tasks and interpersonal relationships, Extraversion in social contexts,
Agreeableness towards diverse viewpoints and mutual understandings, and
Neuroticism in interpreting circumstances [13].

We postulate the potential benefits of incorporating a personality model with
an LLM. These include optimizing prompt use, broad application of general
knowledge, and providing dynamic character transformations within the video
game. We hypothesize that an LLM can craft content consistent with the
specified personality when instructing it to roleplay as a character using a
well-researched personality framework, such as the Big Five. Representing the
Big Five as a 5-tuple to prompt the LLM, we use fewer tokens to explain the
expected personality for an NPC. Given the extensive array of internet-based
data used to train an LLM, it is plausible that research related to the Big
Five forms a portion of that data set, including examples of individuals
exemplifying each personality factor. Consequently, the LLM can generate
dialogue content aligned with a given personality. Moreover, it is vital to
acknowledge that a character’s personality will likely evolve as players
engage with games. Accordingly, the content generated by these characters
should mirror their evolving persona.

This research aims to facilitate game developers in creating NPCs that exhibit
a higher degree of human-like behavior and characteristics. It could provide
invaluable insights to game designers, aiding them in integrating and
effectively using AC systems within their design process. This not only
enhances the realism of the characters but also allows for the generation of
more diverse and nuanced in-game interactions.

Incorporating personality profiles into LLMs is just the start. Including
other aspects of AC systems, such as dynamic emotional states and social
relationship metrics, can further provide an LLM with the necessary context to
generate even more realistic, human-like behavior. This holistic approach can
truly bring NPCs to life, making them more believable and immersive.

##  3 Personality Representation

The Big Five breaks personalities into five categories: Openness,
Conscientiousness, Extraversion, Agreeableness, and Neuroticism, which are
represented on a continuous scale ranging from zero to one. A score of one
signifies the full expression of a particular trait, while zero implies the
presence of the opposing attribute [13].

To illustrate, a score of 1.0 on the Openness dimension indicates that an
individual exhibits high creativity, readily embraces novelty, is driven by
tackling new challenges, and engages in abstract thought. Conversely, a score
of 0.0 in the Openness factor signifies resistance to change, disinterest in
novelty, reluctance to welcome new ideas, and a deficit in imaginative
thinking. Therefore, the Openness factor can also be conceptualized as a
spectrum ranging from cautious/consistent to inventive/curious, corresponding
to zero and one, respectively.

However, it is important to note that the values assigned to any particular
trait need not be restricted to the extremes of this scale. Instead, they can
occupy any position within this range, providing an extensive, nuanced
representation of personality traits. Consequently, the model spans five
dimensions and is infinite along any axis, suggesting the potential for a
virtually limitless array of personality combinations across these five
factors.

Drawing on the work of Van Mensvoort, the Big Five can be discretized into 20
distinct personality profiles [14]. This subdivision encompasses a range of
behavioral characteristics, from negative behavioral disorders such as
Paranoid and Schizoid, to positive behavioral traits including Accommodating
and Laissez-faire. This fine-grained classification allows for a more precise
and detailed depiction of personality, accommodating its complex, multifaceted
nature.

These personality profiles discretize the Big Five to distinct points, shown
in Figure 1, which enables one to label a particular personality configuration
as one of the established profiles. This process identifies a specific
categorization and imbues additional context into what the given personality
configuration symbolizes. Moreover, these profiles, each denoting a general
region within the model’s domain, can serve as the foundation for a
hierarchical structure.

Employing these profiles as categorical labels, we leverage human data to
establish our Baseline in this research. This Baseline serves as a comparative
standard against which we evaluate the performance of our methodology and the
LLM’s capacity to embody and express various character personalities. Thus,
the defined profiles and our chosen baseline data inform our research approach
and assist in objectively evaluating our outcomes.

Figure 1: Van Mensvoort’s 20 personality Profiles. Plotted using CS vs CF, equations 1 and 2. [14]. Table 1: The statistical description of the evaluated test results from IPIP Big Five Factor Markers dataset. Baseline | Openness | Conscientiousness | Extraversion | Agreeableness | Neuroticism | Alpha | Beta
---|---|---|---|---|---|---|---
count | 50500 | 50500 | 50500 | 50500 | 50500 | 50500 | 50500
mean | 0.699831 | 0.584225 | 0.449248 | 0.566697 | 0.512838 | 1.638084 | 1.149079
std | 0.177709 | 0.205618 | 0.242912 | 0.219459 | 0.230343 | 0.42377 | 0.33324
min | 0 | 0 | 0 | 0 | 0 | 0.05 | 0.025
25% | 0.575 | 0.425 | 0.25 | 0.4 | 0.35 | 1.35 | 0.9
50% | 0.725 | 0.575 | 0.45 | 0.575 | 0.525 | 1.6 | 1.15
75% | 0.825 | 0.75 | 0.625 | 0.725 | 0.675 | 1.925 | 1.4
max | 1 | 1 | 1 | 1 | 1 | 3 | 2

##  4 Data Collection

We investigate the results from a comprehensive Big Five personality dataset
using the personality profiles. The dataset, derived from the Open-Source
Psychometrics Project, comprises personality test results from over a million
participants (n = 1,015,342) who responded to a 50-item Big Five personality
test [15, 13]. Each test item was a statement to which participants responded
on a Likert scale, with one indicating a strong disagreement with the
presented behavior and five denoting strong agreement. The dataset not only
includes the results of the personality test but also records auxiliary
information such as the time taken to complete the test, the number of
attempts made by the same participant (identified based on IP address),
geographical coordinates (latitude and longitude), race, and so forth.

We undertook a thorough data-cleaning process to optimize the dataset for our
research. We first ensured that each test response included answers to all 50
statements. We further filtered the data to include only those responses that
took more than 300 milliseconds to answer a question, as the average human
response time ranges between 150 and 300 milliseconds, thus mitigating bot
submissions to the analysis. Finally, we only retained the first submission
from each unique IP address. Following these preparation steps, a total of
596,956 test results remained for our analysis. This extensive and robust
dataset now forms the foundation for our subsequent investigations. You can
find the cleaned version of the dataset within the provided repository.

Figure 2: Nine different plots of evaluated test results using CS vs CF
schema, equations 1 and 2. The green squares represent the baseline results.
Each row is a different personality profile represented by the magenta X. Each
column is a different LLM test result, starting with davinci, turbo, then
gpt-4, represented by blue, lime, and purple circles, respectively.

The subsequent phase involved the evaluation of the personality test results.
This evaluation was conducted based on a scoring key included with the test,
accessed from the IPIP website [16]. Each statement in the test corresponds to
a particular personality factor. As a participant assigns a Likert value to a
given statement, that value is accumulated into a running total for the
respective personality factor. The assigned value may be inverted based on the
statement’s wording directionality. For instance, a statement reflective of
Extraversion, such as "Don’t talk a lot," possesses a negative direction,
thereby necessitating the inversion of the corresponding response value.
Consequently, a response of 5 for this statement would be translated into a
value of 1.

Following the completion of this evaluation, each personality factor scores
within a range of [10, 50]. These values are then mapped to a [0.0, 1.0] range
using linear interpolation to normalize the scores for comparative analysis.
The final output constitutes a 5-tuple vector representative of the values for
each personality factor, (O,C,E,A,N). This detailed evaluation process ensures
an accurate and comprehensive representation of an individual’s personality
configuration based on the Big Five model.

We executed the label assignment of a personality profile with the evaluated
test results using a nearest-neighbor approach. For every test result
represented as a 5-tuple vector, we calculated the Euclidean Distance to each
of the 20 personality profiles. The personality profile with the smallest
distance to the profile was identified as the corresponding label.

Figure 3: Nine different plots of the 50 test responses using LDA. The green
squares represent the baseline responses. Each row is a different personality
profile. Each column is a different LLM test response, starting with davinci,
turbo, then gpt-4, represented by blue, lime, and purple circles, respectively

The final preparatory procedure was conducting disproportionate sampling for
each personality profile. As the goal of this research is not to replicate the
profile distribution of the entire population but rather to examine the
distribution of test results per personality profile, we opted for an equal
representation across all profiles (m = 2525). Consequently, our baseline
dataset incorporated a total of 50,500 test results, and the statistical
description can be found in Table 1. This methodological approach ensures that
each personality profile is adequately represented in our analysis, enabling a
balanced and comprehensive investigation of how an LLM can interpret and
utilize psychometric values. The subset version of the cleaned dataset can
also be found in the provided repository.

From the works of Van Mensvoort’s research, we projected the various
personalities onto a two-dimensional plane by computing the values for
Cognitive Stability (CS) and Cognitive Flexibility (CF) [14] defined by
Equation 1 and 2 respectively. In Equation 1, A is Agreeableness, C is
Conscientiousness and S is Emotional Stability which is the inverse of N,
Neuroticism. In Equation 2, E is Extraversion and O is Openness.

$\text{Cognitive Stability}=Alpha=A+C+S=A+C+(1-N)$ (1)

$\text{Cognitive Flexibility}=Beta=E+O$ (2)

This methodology allows us to visualize the distribution and interrelation of
the various personality profiles within our baseline dataset. The resulting
two-dimensional representation of the Big Five offers a clear and intuitive
overview of the personality landscape, as illustrated in Figure 2. This
approach not only affords an understanding of individual profiles but also
elucidates the underlying relationships between various personality factors.

Figure 2 shows that each personality type forms a cluster around its
corresponding profile, represented by the central X within each cluster.
Beyond this evaluative visualization, we have also employed a dimensionality
reduction technique, Linear Discriminant Analysis (LDA), to plot the 50-item
personality test responses, shown in Figure 3. This method, once again,
yielded distinct clusters that effectively distinguished between different
personality profiles.

##  5 Generate Synthetic Data

The LLMs selected for this research comprised a suite of OpenAI models: text-
davinci-003, gpt-3.5-turbo-0613, and gpt-4-0613. Each LLM model was prompted
with a succinct description delineating the extremes of each personality
factor. This was followed by the 5-tuple representing the personality profile
and instructions to complete the personality test. Each model underwent this
process 128 times for each profile, a sample size ensuring statistical
significance.

The resulting test responses demonstrated variability, ranging from the
expected Comma-Separated Value (CSV) format to comprehensive paragraph-based
breakdowns encompassing each statement and the corresponding rationale. The
results provided in the CSV format were added to the analysis; however, the
comprehensive results were parsed and converted to the proper CSV format and
then added to the analysis. Note that the final results and the prompt
provided to the LLM are available for download from the provided repository.
An assessment determined the correspondence between the test results and the
assigned personality profiles. This assessment involved the computation of
accuracy using the same approach as in the baseline dataset, namely
determining the shortest Euclidian Distance from each personality profile. The
results exhibited significant disparities across the models, as shown in Table
2. The text-davinci-003 model performed the poorest, with an accuracy of
13.83% (+/- 1.34% at 95% confidence interval) , followed by gpt-3.5-turbo-0613
at 17.77% (+/- 1.4809% at 95% confidence interval). In contrast, the next-
generation model, gpt-4-0613, exhibited a substantial leap in performance,
achieving an accuracy of 73.98% (+/- 1.6995% at 95% confidence interval).

Another assessment method employed in this research is the Root Mean Squared
Prediction Error (RMSPE). This measurement assumes that if the labeling of the
personality profile is accurate, the error in the evaluated test results
should be marginal; thus, they should closely align with the respective
profile. Two variants of RMSPE were utilized: Euclidian Distance and Cosine
Similarity. In either case, treating the 5-tuple personality traits either as
a point in the 5D space or a vector within the space led to the same
conclusion. As long as the RMSPE values were close to zero, the error was
considered negligible, indicating a successful personality embodiment by the
model.

Table 2: LLM Accuracy of Personality Profile prompting and test evaluation. Model | Accuracy | 95% CI
---|---|---
text-davinci-003 | 13.8281% | +/- 1.3372%
gpt-3.5-turbo-0613 | 17.7734% | +/- 1.4809%
gpt-4-0613 | 73.9844% | +/- 1.6995%
Table 3: The RMSPE of each personality profile based on Euclidian Distance (ED) and Cosine Similarity (CS). The dark Red cells have the have the highest error, while the dark blue has the smallest error. RMSPE | Baseline (ED) | davinci (ED) | turbo (ED) | gpt-4 (ED) | Baseline (CS) | davinci (CS) | turbo (CS) | GPT-4 (CS)
---|---|---|---|---|---|---|---|---
Paranoid |  0.15121 |  0.05686 |  0.07514 |  0.05724 |  0.03752 |  0.01899 |  0.01463 |  0.01348
Schizoid |  0.14364 |  0.06812 |  0.06281 |  0.06424 |  0.05832 |  0.02825 |  0.01208 |  0.01714
Schizotypal |  0.12053 |  0.06933 |  0.06238 |  0.04932 |  0.02520 |  0.02071 |  0.00934 |  0.00728
Antisocial |  0.10042 |  0.08380 |  0.05742 |  0.05838 |  0.03055 |  0.02403 |  0.01190 |  0.00895
Borderline |  0.11230 |  0.05655 |  0.07902 |  0.04410 |  0.01625 |  0.01182 |  0.01756 |  0.00581
Histrionic |  0.09747 |  0.07189 |  0.04280 |  0.05646 |  0.01242 |  0.02003 |  0.00562 |  0.00469
Narcissistic |  0.12100 |  0.06959 |  0.05257 |  0.04399 |  0.02427 |  0.02516 |  0.01098 |  0.00405
Avoidant |  0.13601 |  0.05957 |  0.08602 |  0.05439 |  0.01870 |  0.01792 |  0.02363 |  0.00752
Dependent |  0.12703 |  0.05074 |  0.05185 |  0.04126 |  0.01370 |  0.01279 |  0.00975 |  0.00425
Obsessive-Compulsive |  0.10744 |  0.06192 |  0.04801 |  0.04660 |  0.01796 |  0.01836 |  0.01078 |  0.00692
Pronoid |  0.11304 |  0.05641 |  0.04385 |  0.04479 |  0.01478 |  0.00974 |  0.00463 |  0.00302
People-person |  0.11572 |  0.05542 |  0.04589 |  0.02738 |  0.01232 |  0.01634 |  0.00513 |  0.00229
Sensible |  0.11865 |  0.04355 |  0.04866 |  0.03344 |  0.00909 |  0.01026 |  0.00399 |  0.00314
Prosocial |  0.12594 |  0.05353 |  0.04408 |  0.03500 |  0.01702 |  0.01108 |  0.00402 |  0.00427
Straightforward |  0.13536 |  0.05110 |  0.04819 |  0.03792 |  0.02613 |  0.00637 |  0.00520 |  0.00316
Non-theatrical |  0.12259 |  0.06200 |  0.06121 |  0.05274 |  0.02716 |  0.02085 |  0.00680 |  0.01033
Unpretentious |  0.12516 |  0.05849 |  0.05433 |  0.04495 |  0.01867 |  0.01662 |  0.00510 |  0.00469
Accommodating |  0.13966 |  0.04904 |  0.04501 |  0.04639 |  0.01346 |  0.00932 |  0.00587 |  0.00360
Independent |  0.12653 |  0.06668 |  0.04477 |  0.05471 |  0.03466 |  0.01061 |  0.00512 |  0.00317
Laissez-faire |  0.10422 |  0.07214 |  0.04324 |  0.03760 |  0.02407 |  0.02335 |  0.01075 |  0.00358

Table 3 provides an overview of the RMSPE values for Euclidian Distance and
Cosine Similarity, respectively, for the Baseline and each of the three
models. Comparatively, the text-davinci-003 model performed the poorest again,
followed by gpt-3.5-turbo-0613. However, the gpt-4-0613 model outperformed its
predecessors and demonstrated a better RMSPE than the Baseline, highlighting
its superior ability to generate responses in line with given personality
profiles.

Table 4: The IRR of each personality profile with the Baseline and each LLM  IRR | Baseline | text-davinci-003 | gpt-3.5-turbo-0613 | gpt-4-0613
---|---|---|---|---
Paranoid |  0.29206 |  0.63894 |  0.45003 |  0.64559
Schizoid |  0.31384 |  0.57597 |  0.46754 |  0.61732
Schizotypal |  0.29647 |  0.62922 |  0.45777 |  0.65924
Antisocial |  0.26189 |  0.56691 |  0.50752 |  0.65480
Borderline |  0.30278 |  0.65931 |  0.47490 |  0.67507
Histrionic |  0.29312 |  0.52166 |  0.57820 |  0.70631
Narcissistic |  0.27433 |  0.52846 |  0.53288 |  0.67039
Avoidant |  0.34765 |  0.68141 |  0.41200 |  0.59760
Dependent |  0.29470 |  0.66391 |  0.45892 |  0.71050
Obsessive-Compulsive |  0.30529 |  0.60722 |  0.54664 |  0.75075
Pronoid |  0.31685 |  0.61189 |  0.57253 |  0.77202
People-person |  0.32219 |  0.56490 |  0.58915 |  0.75714
Sensible |  0.30756 |  0.64722 |  0.58296 |  0.76965
Prosocial |  0.30437 |  0.62679 |  0.55077 |  0.71128
Straightforward |  0.32797 |  0.65405 |  0.59831 |  0.79106
Non-theatrical |  0.30142 |  0.60615 |  0.50340 |  0.69410
Unpretentious |  0.30506 |  0.60481 |  0.51626 |  0.67787
Accommodating |  0.37616 |  0.61502 |  0.56838 |  0.74752
Independent |  0.28577 |  0.56754 |  0.54954 |  0.71682
Laissez-faire |  0.31103 |  0.51922 |  0.57253 |  0.73128

Visual representations of these findings can be seen in Figure 2, which shows
the clustering of test results according to the personality profiles. Since
text-davinci-003 is at the time considered a legacy model, we can clearly see
this by the clustering of evaluated test results and test responses outside
the clusters of the baseline. As the GPT model improves with the next
iteration of gpt-3.5-turbo-0613 and gpt-4-0613, the clusters of evaluated test
results and test responses move closer to their respective baseline clusters
to the point gpt-4-0613 clusters right above the personality profile location.
The degree of closeness within each cluster and the clear separation between
clusters further underscores the marked improvement in the performance of the
gpt-4-0613 model over the other models and even the Baseline.

Figure 4: Violin Plot of each personality profile (red point) against the
baseline (green), davinci (blue), turbo (lime), and gpt-4 (purple) generated
data evaluated test results along one of the personality factors.

Inter-rater reliability (IRR) is another key measure utilized in this
research. IRR assesses the extent to which different ratings of the same
entity agree with each other. This concept can be illustrated using the
example of judges scoring a competition, where ideally, the scores provided by
different judges for the same contestant should vary minimally, indicating
consistent evaluation criteria. In the context of this research, the intuition
behind applying IRR is that if 128 individuals with the same Paranoid
personality profile took the same personality test, their answers should be
fairly similar, reflecting their shared psychological characteristics.

Table 4 presents the IRR results for each model and the Baseline. The data
indicates that it is quite rare but possible for similar responses to be
observed in the Baseline test results. However, notable similarities were
evident in the responses generated by the text-davinci-003,
gpt-3.5-turbo-0613, and gpt-4-0613 models, with gpt-4-0613 once again
delivering the best performance in terms of consistent response generation.
The consistency of similar responses can also be seen in the same LDA plot. As
depicted in Figure 3, the visual representation underlines the congruity
within the responses generated by each model, most pronounced in the case of
gpt-4-0613. This underscores gpt-4-0613’s superior capacity to generate
responses befitting a given personality profile consistently.

The violin plots shown in Figure 4 provide a clear, visual representation of
the distribution of test results for each personality factor. These plots help
understand the variations in results for each LLMs and offer a basis for
comparison with the baseline. Each personality profile’s position, indicated
by the red circle, should ideally intersect within the baseline distributions,
represented by the green violin. This would imply that the generated test
results align closely with the actual personality profile. The distributions
for text-davinci-003 and gpt-3.5-turbo-0613, represented by the blue and lime
violins, respectively, seldom intersect with the actual personality profile,
suggesting that these models might not be accurately generating test results
consistent with the given personality. In contrast, the gpt-4-0613 results
intersect with the personality profile within the lower or upper quartile,
implying that the model is more successful at generating test results that
align closely with the given personality profile. This intersection suggests
greater accuracy and consistency with gpt-4-0613, making it a potentially more
effective tool for game developers in creating realistic, human-like
characters based on the Big Five. These findings underscore the potential of
utilizing advanced LLMs like gpt-4-0613 to generate more accurate and nuanced
character responses in video games, thereby improving the overall gaming
experience for players.

##  6 Use Cases

The potential of LLMs to generate content accurately reflecting a provided
personality opens up exciting possibilities for integrating AC systems and
LLMs for NPCs. This integration could be applied in various contexts,
enhancing the gaming experience by creating more dynamic and emotionally
nuanced characters. The following paragraphs will delve into three possible
scenarios where this integration could be utilized to enrich the narrative and
gameplay. Each scenario will explore how context clues can be used with an LLM
to generate unique and engaging content that aligns with the personality of
the NPC. From retelling stories based on experiences to improvising narratives
and piecing together the end of a story, the use of LLMs promises to
revolutionize the way NPCs interact within the gaming world.

In one scenario, context clues could be provided, prompting the LLM to retell
a story. The given personality would guide the model to focus on aspects
relevant to that personality, influencing the choice of semantics to reflect
the character’s disposition. This could be seen in games like "Tales of
Arabian Nights," "Grandpa’s Farm," or "This War of Mine," where NPCs
experience events and undergo narratives to advance the plot. The NPCs,
powered by an LLM, would interpret these experiences and share their stories,
offering additional information or different perspectives.

In another scenario, context clues could be generated on the spot to create an
active, evolving story. Similar to improvisation, the model would adopt a "yes
and…" approach, contributing to the communal story. The LLM, guided by its
assigned personality, would describe actions and aspects that align with its
psyche and how it would resolve a situation. This could be applied in games
like "Dungeons and Dragons," "Once Upon a Time," and "Zork," where players
need to describe what is happening and their actions in the moment.

Finally, context clues could be used to reveal the end of a story, with the
LLM tasked with explaining how the story reached that conclusion. The LLM
would need to link events together, with the assigned personality acting as a
heuristic to narrow down the choices. This could be seen in games like "Dark
Stories," "Sherlock Holmes Consulting Detective," or "Shadows of Doubt." The
NPC would use its personality-driven intuition to fill in the gaps in the
narrative and motivate the next steps to find the respective clues.

##  7 Conclusion

Our investigation of LLM’s proper use of personality information by
repurposing the IPIP questionary, a human measuring technique to determine
personality, has revealed promising potential for creating more engaging and
realistic NPCs. The continual improvement of LLMs, such as the current
dominating gpt-4-0613 model, will only further improve the output of proper
behaviors corresponding with psychometric values, particularly personality
traits, creating NPCs with consistent and believable behavioral patterns.
Incorporating personality models with LLMs enables dynamic character
transformations and leverages the broad application of general knowledge,
enhancing the overall gaming experience. Future research will continue to
explore this exciting frontier, looking at different psychometric values, such
as the emotional state of an NPC or their attitudes on objects, praises
towards actions, the outlook of events, and their relationship with other
NPCs. Additionally, finetuning an LLM basing the expected behavior on human
psychometric datasets can further improve the model’s content generation.
Confirming that an LLM can construct dialogue or make decisions with these
additional values can further necessitate the integration of an AC system for
NPCs, providing additional context clues to describe the nuances of believable
human behavior.

## References

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#  On coverings by Minkowski balls in the plane and a duality

Nikolaj M. Glazunov  Institute of Mathematics and Informatics Bulgarian
Academy of Sciences, 1113 Sofia, Bulgaria. Glushkov Institute of Cybernetics
NASU, Kiev, Ukraine Institute of Mathematics and Informatics Bulgarian Academy
of Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

Lattice coverings in the real plane by Minkowski balls are studied. We exploit
the duality of admissible lattices of Minkowski balls and inscribed convex
symmetric hexagons of these balls. An explicit moduli space of the areas of
these hexagons is constructed, giving the values of the determinants of
corresponding covering lattices. Low and upper bounds for covering constants
of Minkowski balls are given. The best known value of covering density of
Minkowski ball is obtained. One conjecture and one problem are formulated.

###### Key words and phrases:

Lattice covering, Minkowski metric, Minkowski ball, hexagon, covering
constant, covering density, thinnest covering

###### 2020 Mathematics Subject Classification:

Primary 11H06, 52C05; Secondary 49Q10, 90C25

The author was supported Simons grant 992227

##  1\. Introduction

A system of equal balls in $n$-dimensional space is said to form a covering,
if each point in space belongs to at least one ball of this system.

Let

(1.1) $D_{p}:\;|x|^{p}+|y|^{p}\leq 1,\;p\geq 1$

be (two-dimensional) Minkowski’s balls with boundary

(1.2) $C_{p}:\;|x|^{p}+|y|^{p}=1,\;p\geq 1.$

Lattice coverings in the real plane by Minkowski balls with inscribed convex
symmetric hexagons generated by admissible lattices of Minkowski balls are
studied. The covering problem is dual to the packing problem. In different
covering problems, this duality, apparently, can manifest itself in different
ways. In this paper, duality is understood as the duality between admissible
lattices having 3 pairs of points on the boundary $D_{p}$, and symmetric
convex hexagons (possibly a quadrangle) corresponding to these lattices and
determining the values of the determinants of the corresponding covering
lattices.

Geometrically, the set of Minkowski’s balls is a solid tube of variable
diameter with square sections at the ends $p=1$, and $p=\infty$ which we call
the limiting Minkowski’s balls. We use admissible lattices that have 3 pairs
of points on the boundary of $D_{p}$. We construct an explicit moduli space
(3.2) of areas of symmetric convex hexagons (possibly a quadrangle) which
determine values of the determinants of corresponding covering lattices .

This construction and its consequences make it possible to obtain the best
currently known lower bounds for covering constants of Minkowski balls. An
estimate from above is also given. These give the corresponding estimation of
the covering densities. The best known value of covering density of Minkowski
ball is obtained (Proposition 6.2).

Our investigations connect with Minkowski conjecture [1, 2, 3, 4, 5, 11] and
use results of its investigation. This conjecture was formulated by Minkowski
while studying a problem of Diophantine approximations for sums of powers of
linear forms [1]. Mordell tested and confirmed this conjecture for $p=4$ [2].
Davis adjusted Minkowski conjecture [3] for small values of $1<p\leq p_{0}$
(see theorems (5.3), (5.6)). Minkowski (implicitly) and Cohn [4] explicitly
defined an appropriate moduli space for computing determinants of critical
lattices. We follow these researchers, albeit in a slightly different
sequence. Our research on the packing problem [13, 14] was also used.
Corresponding results and conjectures are stated in the simpliest way in terms
of geometric lattices and covering lattices [1, 7, 8].

Based on these studies, one conjecture about the behavior of the covering
constants of Minkowski balls (see 6.3) and one problem on belonging of the
curve of maxima of covering constant to a family introduced in the work (see
6.4) are formulated.

##  2\. Two-dimensional Minkowski balls, admissible lattices and hexagons

######  Proposition 2.1.

The volume of Minkowski ball $D_{p}$ is equal
$4\frac{(\Gamma(1+\frac{1}{p}))^{2}}{\Gamma(1+\frac{2}{p})}$.

Proof. (by Minkowski) Let $x^{P}+y^{p}\leq 1,x\geq 0,y\geq 0.$ Put
$x^{p}=\xi,y^{p}=\eta.$

(2.1)
$V(D_{p})=\frac{4}{p^{2}}\iint\xi^{\frac{1}{p}-1}\eta^{\frac{1}{p}-1}d\xi
d\eta,$

where the integral extends to the area

$\xi+\eta\leq 1,\;\xi\geq 0,\;\eta\geq 0.$

Expression (2.1) can be represented in terms of Gamma functions, and we get

$V(D_{p})=4\frac{(\Gamma(1+\frac{1}{p}))^{2}}{\Gamma(1+\frac{2}{p})}.$

###  2.1. Admissible lattices

[7, 8].

Let $\mathcal{R}$ be a set and $\Lambda$ be a lattice with base

$\\{a_{1},\ldots,a_{n}\\}$

in ${\mathbb{R}}^{n}.$

A lattice $\Lambda$ is admissible for body $\mathcal{R}$
(${\mathcal{R}}-$admissible) if ${\mathcal{R}}\bigcap\Lambda=\emptyset$ or
$0.$

Let $\overline{\mathcal{R}}$ be the closure of ${\mathcal{R}}$. A lattice
$\Lambda$ is strictly admissible for $\overline{\mathcal{R}}$ (
$\overline{\mathcal{R}}-$strictly admissible) if
$\overline{\mathcal{R}}\bigcap\Lambda=\emptyset$ or $0.$

Let

(2.2) $d(\Lambda)=|\det(a_{1},\ldots,a_{n})|$

be the determinant of $\Lambda.$

###  2.2. Hexagons and coverings

Every admissible lattice of $D_{p}$ containing 3 pairs of points on the
boundary of $D_{p}$ defines a hexagon inscribed in $D_{p}$.

We denote such a hexagon as ${\mathcal{H}}_{p}$, and call it a hexagon of the
admissible lattice, briefly: al-hexagon.

######  Remark 2.2.

In the limiting cases $p=1$ and $p=\infty$ corresponding hexagons (following
Fejer Toth, see 3.1 ) ${\mathcal{H}}_{p}$ are quadrangles (see Proposition
3.3).

By specifying results from [8] to the case of Minkowski balles and al-hexagons
${\mathcal{H}}_{p}$ we have the next

######  Proposition 2.3.

Each Minkowski ball $D_{p}$ contains an al-hexagon (possibly a quadrangle)
${\mathcal{H}}_{p}$ of maximum area. If this area is denoted by
$\gamma_{h}(D_{p})$ (or for symmetric convex inscribed al-hexagons
${\mathcal{H}}_{p}$ by $a({\mathcal{H}}_{p})$) then

(2.3) $\Gamma(D_{p})=\gamma_{h}(D_{p}).$

For the density of thinnest lattice covering of $D_{p}$ we get

(2.4) $\vartheta(D_{p})=V(D_{p})/\gamma_{h}(D_{p}).$

######  Remark 2.4.

The expression (2.3) is the ${\bf covering\;constant}$ of Minkowski ball
$D_{p}$, and the expression (2.4) is the ${\bf density}$ of the ${\bf
thinnest\;lattice\;covering}$ within the framework of our considerations.

######  Example 2.5.

1\. The lattice

$\Lambda_{1}^{(1)}=\\{1,1),(2,0)\\}$

is the covering lattice by $D_{1}$. The limiting case of Minkowski balls with
$p=1$ gives an optimal covering (indeed a partition) of density $1$.

2\. The lattice

$\Lambda_{\infty}^{(1)}=\\{(2,2),(0,2)\\}$

is the covering lattice by $D_{\infty}$. Limiting case of Minkowski balls for
$p=\infty$ gives the optimal covering (indeed a partition) of density $1$.

3\. The lattice

$\Lambda_{2}^{(0)}=\\{(\sqrt{3},0),(\frac{\sqrt{3}}{2},\frac{3}{2})\\}$

is the covering lattice by $D_{2}$. The lattice gives the optimal covering of
density $\vartheta=\frac{2\pi}{3\sqrt{3}}\approx 1,19$.

##  3\. Moduli spaces, moduli problems and and areas of inscribed hexagons

Moduli spaces are spaces of solutions of geometric classification problems and
dynamics of varying objects in families [9, 10].

Example: moduli space for Legendre elliptic curve:

$y^{2}=x(x-1)(x-\lambda),\;\lambda\in{\mathbb{C}}$

Moduli problem - try to obtain solution for all objects of the moduli space or
for objects of some subspace of the corresponding moduli space

Currently, there is great interest in unifying problems of algebraic,
arithmetic and geometric classification and dynamics through moduli spaces of
object classes and studying related problems within such unifications. Various
types of moduli spaces were presented and studied in the famous works of
Eichler, Lazard, Shimura, Siegel, Igusa, Deligne-Mumford-Knudsen, Deligne-
Rapoport, Katz-Mazur, Faltings, Drinfeld and others. But the author is not
aware of applications of moduli spaces to covering problems.

In present our research the moduli space is real differentiable moduli space
with exceptional points.

###  3.1.  Solid Minkowski tube, Minkowski tube and metrics

######  Definition 3.1.

We will call (1.1) a solid Minkowski tube and (1.2) a Minkowski tube.

######  Remark 3.2.

Expressions

(3.1) $(|x|^{p}+|y|^{p})^{1/p}$

are metrics. They also form the tube of metrics for $1\leq p\leq\infty$. But
we won’t use it.

######  Proposition 3.3.

A Minkowski tube (1.2) is a real manifold which, for $p=1$ and $p=\infty$, has
exceptional points corresponding to the corners of the corresponding limit
quadrangles.

Proof. Follows from [1] and from the definition of Minkowski tube.

The use of moduli spaces and the study of moduli problems require the
adjustment of some well-known concepts.

Following Fejer Toth, a closed finite convex region bounded by at most six
straight line segments will be called a hexagon.

This definition of a hexagon was mentioned by Bambah, Rogers and other famous
covering researchers [6].

###  3.2. Inscribed hexagons of admissible lattices of Minkowski balls $D_{p}$
and their moduli space

######  Theorem 3.4.

The set of areas of the entire family of al-hexagons (possibly a quadrangle)
${\mathcal{H}}_{p},(1<p<\infty)$ inscribed in Minkowski balls $D_{p}$ is
parameterized by the function $A(\sigma,p)$ defining the corresponding moduli
space $\mathbf{A}:$

(3.2)
$A(\sigma,p)=3(\tau+\sigma)(1+\tau^{p})^{-\frac{1}{p}}(1+\sigma^{p})^{-\frac{1}{p}},$

In limiting cases we have the following areas of quadrangles:
$p=1,\;a({\mathcal{H}}_{1})=2;\;p=\infty,\;a({\mathcal{H}}_{\infty})=4$.

Proof. The area $a({\mathcal{H}}_{p})$ of an al-hexagon is given (see (2.2))
by

(3.3) $a({\mathcal{H}}_{p})=2\cdot d\left(\begin{array}[]{cc}x_{1}&y_{1}\\\
x_{1}-x_{2}&y_{1}+y_{2}\end{array}\right)+d\left(\begin{array}[]{cc}-x_{2}&y_{2}\\\
-x_{1}&-y_{1}\end{array}\right)=3(x_{1}y_{2}+x_{2}y_{1}),$

where $x_{1}>x_{2}\geq 0,y_{1}\geq 0,y_{2}>0$ and
$x_{1}^{p}+y_{1}^{p}=|-x_{2}|^{p}+y_{2}^{p}=(x_{1}-x_{2})^{p}+(y_{1}+y_{2})^{p}=1,\;1<p<\infty$.

The parameterization of admissible lattices (al) is implicitly given in [1]
and explicitly specified in [4, 5, 11, 12, 13].We specify the parameterization
of al-hexagons. Let

$0\leq\tau<\sigma,\;0\leq\tau\leq\tau_{p}.$

$\tau_{p}$ is defined by the equation
$2(1-\tau_{p})^{p}=1+\tau_{p}^{p},\;0\leq\tau_{p}<1.$

$1\leq\sigma\leq\sigma_{p},\;\sigma_{p}=(2^{p}-1)^{\frac{1}{p}}.$

For inscribed in $D_{p}$ a hexagon ${\mathcal{H}}_{p}$ with points
$(x_{1},y_{1})=((1+\tau^{p})^{-\frac{1}{p}},\tau(1+\tau^{p})^{-\frac{1}{p}})$,
$(-x_{2},y_{2})=(-(1+\sigma^{p})^{-\frac{1}{p}},\sigma(1+\sigma^{p})^{-\frac{1}{p}})$,
$(x_{1}-x_{2},y_{1}+y_{2})=(1+\tau^{p})^{-\frac{1}{p}}-(1+\sigma^{p})^{-\frac{1}{p}},\tau(1+\tau^{p})^{-\frac{1}{p}}+\sigma(1+\sigma^{p})^{-\frac{1}{p}})$.
This gives the formula (3.2). Limiting cases follow from the proof of
Minkowski conjecture (Theorem 5.3) (see also 1 and 2 of Example 2.5).

######  Definition 3.5.

Moduli space (3.2) is the set of areas of the entire family of al-hexagons
(possibly a quadrilateral) ${\mathcal{H}}_{p},(1\leq p\leq\infty)$ inscribed
in Minkowski balls $D_{p}$. Briefly we will call (3.2) the moduli space of
hexagons.

######  Remark 3.6.

The function domain of the function (3.2) is

${\mathcal{M}}:\;\infty>p>1,\;1\leq\sigma\leq\sigma_{p}=(2^{p}-1)^{\frac{1}{p}}$

${\mathcal{M}}$ has two limit points $p=1$ and $p=\infty$.

######  Remark 3.7.

The function (3.2) is upwardly convex (concave) in the real space with
Cartesien coordinates $(\sigma,p,A)$.

##  4\. Sections of moduli space of hexagons

We will solve the problem of the thinnest covering of the real plane by
Minkowski balls with hexagons ${\mathcal{H}}_{p}$ if we define a curve of
maxima on an upwardly convex (concave) surface (3.2). Unfortunately, to date
the author knows only a few points of maxima on this curve. In this regard, we
present here a family of surfaces, give some intersections of (3.2) with these
surfaces and formulate a (rough) hypothesis about the curve of maxima on
(3.2).

######  Definition 4.1.

Let

(4.1) $\sigma_{\alpha,p}=(2^{p}-1)^{\frac{1}{\alpha p}},\;\alpha\geq 1$

be a family of curves parameterized by ${\alpha}$ with $p$ varying from $1$ to
$\infty$. in real plane $(\sigma,p)$.

######  Remark 4.2.

$\sigma_{\alpha,p}\neq\sigma_{p,\alpha}.$

######  Remark 4.3.

Curves of the family (4.1) define family of curves

(4.2) $A(\alpha,p):3(\tau+(2^{p}-1)^{\frac{1}{\alpha
p}})(1+\tau^{p})^{-\frac{1}{p}}(1+((2^{p}-1)^{\frac{1}{\alpha
p}})^{p})^{-\frac{1}{p}}$

on the function range (3.2). These curves are parametrized by $\alpha$. For
each such $\alpha,\;p$ varies from $1$ to $\infty\;(1<p<\infty)$.

##  5\.  Estimates and Bounds on Covering Constants

Here we estimate covering constants of Minkowski balls from above and from
below. According to the results of Sas and Dovker (see [8]) in our case of
Minkowski balls we have

######  Proposition 5.1.

Let ${\mathcal{H}}_{p}$ be the hexagon of maximuml area inscribed in Minkowski
ball $D_{p}$.
Then $V({\mathcal{H}}_{p})=\gamma_{h}(D_{p})\geq\frac{3\sqrt{3}}{2\pi}\cdot
V(D_{p})$

######  Remark 5.2.

The trivial estimate from above of $V({\mathcal{H}}_{p})$ is given by the area
$V(D_{p})$ of Minkowski ball:

$V({\mathcal{H}}_{p})\leq V(D_{p})$

.

Let estimate covering constants of Minkowski balls $D_{p}$ from below on the
base of results of the proof of Minkowski conjecture. To do this, we will use
the results of the proof of the Minkowski conjecture (now theorem [11]) and
the constructed moduli space (3.2). In this regard recall at first some
definitions and results.

The infimum $\Delta(\mathcal{R})$ of determinants of all lattices admissible
for $\mathcal{R}$ is called the critical determinant of $\mathcal{R};$ if
there is no $\mathcal{R}-$admissible lattices then puts
$\Delta(\mathcal{R})=\infty.$

A lattice $\Lambda$ is critical if $d(\Lambda)=\Delta(\mathcal{R}).$

######  Theorem 5.3.

[11]

$\Delta(D_{p})=\left\\{\begin{array}[]{lc}\Delta(p,1),\;1\leq p\leq 2,\;p\geq
p_{0},\\\ \Delta(p,\sigma_{p}),\;2\leq p\leq p_{0};\\\ \end{array}\right.$

here $p_{0}$ is a real number that is defined unique by conditions
$\Delta(p_{0},\sigma_{p})=\Delta(p_{0},1),\;2,57<p_{0}<2,58,\;p_{0}\approx
2.5725$

######  Remark 5.4.

We will call $p_{0}$ the Davis constant.

######  Corollary 5.5.

From theorem (5.3) in notations [11, 12] we have next expressions for critical
determinants and their lattices:

  1. (1)

${\Delta^{(0)}_{p}}=\Delta(p,{\sigma_{p}})=\frac{1}{2}{\sigma}_{p},$

  2. (2)

${\sigma}_{p}=(2^{p}-1)^{1/p},$

  3. (3)

${\Delta^{(1)}_{p}}=\Delta(p,1)=4^{-\frac{1}{p}}\frac{1+\tau_{p}}{1-\tau_{p}}$,

  4. (4)

$2(1-\tau_{p})^{p}=1+\tau_{p}^{p},\;0\leq\tau_{p}<1.$

For their critical lattices respectively
$\Lambda_{p}^{(0)},\;\Lambda_{p}^{(1)}$ next conditions satisfy:
$\Lambda_{p}^{(0)}$ and $\Lambda_{p}^{(1)}$ are two $D_{p}$-admissible
lattices each of which contains three pairs of points on the boundary of
$D_{p}$ with the property that

  * •

$(1,0)\in\Lambda_{p}^{(0)},$

  * •

$(-2^{-1/p},2^{-1/p})\in\Lambda_{p}^{(1)},$

(under these conditions the lattices are uniquely defined).

We specify the results of the Theorem 5.3 for the moduli space (3.2).

######  Theorem 5.6.

(5.1) $\min({\bf A}({\mathcal{H}}_{p}))=\left\\{\begin{array}[]{lc}3\cdot
4^{-1/p}\frac{1+\tau_{p}}{1-\tau_{p}},\;1\leq p\leq 2,\;p\geq p_{0},\\\
\frac{3}{2}{\sigma}_{p},\;2\leq p\leq p_{0};\\\ \end{array}\right.$

To estimate the covering constants of Minkowski balls $D_{p}$ from below, we
define the inverse minimum $i-\min({\bf A}({\mathcal{H}}_{p}))$ for ${\bf
A}({\mathcal{H}}_{p})$.

######  Definition 5.7.

(5.2) $i-\min({\bf A})=i-\min({\bf
A}({\mathcal{H}}_{p}))=\left\\{\begin{array}[]{lc}\frac{3}{2}{\sigma}_{p},\;1\leq
p\leq 2,\;p\geq p_{0},;\\\ 3\cdot
4^{-1/p}\frac{1+\tau_{p}}{1-\tau_{p}},\;2\leq p\leq p_{0}.\\\
\end{array}\right.$

######  Example 5.8.

In the case $p=3$ for moduli space $\mathbf{A}$ (see 3.2) we have
$\min({\bf A}({\mathcal{H}}_{3}))\approx 2.859$
$i-\min({\bf A})({\mathcal{H}}_{3}))\approx 2.870$
$\min({\bf A}({\mathcal{H}}_{3}))<i-\min({\bf A})({\mathcal{H}}_{3})).$

From theorems 3.4, 5.6 with definition 5.6 we have the next estimation from
below of the covering constant $\Gamma(D_{p})$.

######  Proposition 5.9.

(5.3) $\Gamma(D_{p})\geq i-\min({\bf A}).$

######  Remark 5.10.

For $p>2$ the estimate (5.3) is better than the estimate of the Proposition
5.1.

##  6\.  Estimates and calculation of Covering Density

At first give the estimation of the thinnest lattice covering of $D_{p}$ from
above on the base of results by Sas and Dovker ([8]).

######  Proposition 6.1.

The density $\vartheta$ of any covering of ${\mathbb{R}}^{2}$ by Minkowski
balls $D_{p}$ satisfies

(6.1) $\vartheta\leq\frac{2\pi}{3\sqrt{3}}\approx 1,19;$

so

(6.2) $\vartheta(D_{3})\leq 1,19;$

Proof. By Proposition 2.3 $\Gamma(D_{p})=\gamma_{h}(D_{p})$; hence follows (
6.1).

Let us now examine the curves of the family (4.2) from the point of view of
the coverage densities they determine. Here we do this only for one point on
one curve. But even such example gives earlier unknown minimum density for
Minkowski ball.

Take the curve from the family (4.2 ) defined by the function

(6.3)
$A(2,p)=3(\tau+(2^{p}-1)^{\frac{1}{2p}})(1+\tau^{p})^{-\frac{1}{p}}(1+((2^{p}-1)^{\frac{1}{2p}})^{p})^{-\frac{1}{p}}.$

######  Proposition 6.2.

The convex symmetric hexagon defined by the point $p=3$ on the curve (6.3)
gives a covering density $\vartheta(D_{3})\approx 1.0567$ by the Minkowski
ball $D_{3}$, the minimum for known densities of Minkowski balls at
$1<p<\infty$.

Proof.
${\sigma}_{2,3}=(2^{3}-1)^{\frac{1}{2\cdot 3}}\approx 1.383;$
$\tau_{3}\approx 0.20406,0\leq\tau\leq\tau_{3},$ for curve (6.3) take
$\tau=0.12;$
$V(D_{3})\approx 3.52,\;\gamma_{h}(D_{3})\approx 3.331;$
So $\vartheta(D_{3})=V(D_{3})/\gamma_{h}(D_{3})\approx 1.0567;$

We state here one conjecture and one problem which arise naturally from our
work.

######  Conjecture 6.3.

The curve of maxima of covering constants increases from $p=1$ to $p=2$ and
decreases from $p=2$ to $p=\infty$.

######  Problem 6.4.

Does the curve of maxima of covering constants belong to the family (4.2)?

Acknowledgments The author would like to warmly thank Prof. P. Boyvalenkov for
his support.

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