# Audio Retrieval for Multimodal Design Documents: A New Dataset and
Algorithms

###### Abstract

We consider and propose a new problem of retrieving audio files relevant to
multimodal design document inputs comprising both textual elements and visual
imagery, e.g., birthday/greeting cards. In addition to enhancing user
experience, integrating audio that matches the theme/style of these inputs
also helps improve the accessibility of these documents (e.g., visually
impaired people can listen to the audio instead).

While recent work in audio retrieval exists, these methods and datasets are
targeted explicitly towards natural images. However, our problem considers
multimodal design documents (created by users using creative software)
substantially different from a naturally clicked photograph. To this end, our
first contribution is collecting and curating a new large-scale dataset called
Melodic-Design (or MELON), comprising design documents representing various
styles, themes, templates, illustrations, etc., paired with music audio. Given
our paired image-text-audio dataset, our next contribution is a novel
multimodal cross-attention audio retrieval (MMCAR) algorithm that enables
training neural networks to learn a common shared feature space across image,
text, and audio dimensions. We use these learned features to demonstrate that
our method outperforms existing state-of-the-art methods and produce a new
reference benchmark for the research community on our new dataset.

Index Terms—  Music Retrieval, Multimodal processing, cross attention.

##  1 Introduction

With increasing proliferation of on-demand web/mobile-based graphics design
softwares111https://www.adobe.com/express, 222https://www.canva.com,
333https://www.sketch.com, designing creative documents has become very easy
for any occasion, e.g., greeting cards, event invitations/flyers, social media
infographics etc. In most cases, such design documents tend to be multimodal,
i.e., they comprise some visual imagery aspects and some textual elements (see
Fig 1(right)). For such documents, adding an additional modality in the form
of relevant audio/music files will not only enhance the consumption experience
of users but also improve document accessibility for visually impaired users.
To this end, our first contribution is the consideration and proposal of a new
problem involving the retrieval of relevant audio files given a multimodal
design document. While much work in the past [1, 2] has focused on audio
retrieval for natural images, there has not been any work in the context of
the kind of design documents referred to above, and this paper takes a step
towards bridging this gap in the literature (see Fig 1).

Fig. 1: Natural images (left) vs. design documents (right).

As the problem is unexplored, the existing datasets [3, 2] for audio retrieval
contains only natural images and are not suitable for our proposed problem. To
this end, our second contribution is the collection and curation of a new
paired design-audio dataset that comprises multimodal design documents,
scraped from publicly available data from Adobe Stock
444https://stock.adobe.com/, paired with relevant audio files collected from
the MTG-Jamendo [4] repository. With $\approx 500k$ design documents paired
with over $\approx 7.5k$ audio files, this is a first-of-its-kind dataset that
we believe will help advance research in multimodal design understanding.

Finally, our third contribution is a novel multimodal cross-attention
algorithm that enables training neural networks to learn a shared
representation among the image, text, and audio modalities present in our
problem setting. In particular, given paired design-audio samples, we extract
individual modality features and learn per-pair as well as overall weights to
learn a unified design-audio embedding. With extensive experiments on our
proposed new dataset, we demonstrate our algorithm substantially outperforms
the existing state-of-the-art audio retrieval methods.

##  2 Related Work

As noted in Section 1, works in multimodal audio retrieval are mainly focussed
on natural images. In particular, in Image2Song [3], the shuttersong dataset
was used to map images and song lyrics to the same feature space, which was
then used for downstream tasks like retrieval. Using the same dataset, Liang
et al. [1] proposed a method to jointly learn a feature space by fusing visual
and acoustic features. Similarly, even datasets like VGGSound [2], and MUGEN
[5], while having a video modality, also focus on either naturally occurring
human actions or videos generated by game engines. In contrast, our
contribution is unique by proposing a new dataset solely focused on multimodal
creative design documents like greeting cards, infographics etc., that are
commonly created using creative design software. Our large-scale dataset
comprising hundreds of thousands of design documents paired with audio files
provides a challenging testbed for advancing retrieval research.

##  3 Melodic Design (MELON) - A new dataset

As discussed in Section 1, given a multimodal design document like the ones
shown in Fig 1, our problem is one of retrieving a short list of audio files
that go well with the various elements of the input. For example, the first
row/second column in Fig 1 shows an adventure-themed design document
containing both images and text. Different elements include the background
image/color, text fields (e.g., “Mountains”), and decorative elements and
shapes. Note that each of these elements forms a layer in the design document
(e.g., background image is the background layer, and the textual greeting is
the foreground layer), giving a multi-layered multimodal design document. As
noted in Section 1, and also from Figure 1, existing datasets focus solely on
natural images, whereas our problem entails design documents. To bridge this
clear gap in the literature, we collect and curate a new dataset comprising
pairs of multimodal design documents and corresponding audio files, and we
call our dataset “MELOdic desigN” (MELON).

Fig. 2: Comparison of time-frequency chroma features of audio of happy, sad,
upbeat and calm moods. Colour (dark to light) represent the energy content
(low to high) in each time-frequency bin.

Dataset |  | Visual
---
Content
Modalities | #Images | #Audio | Tasks
VGGSound [2] | Action vid. | I+A | 199k | 199k | C
Audio set [6] | Human vid. | I+ A | 2.1m | 2.1m | R + G
MUGEN [5] | Game vid. | I+A+T | 233K | 233K | R+G
Shuttersong [3] | Images | I+A+T | 17k | 17k | MR + C
IMEMNet [7] | Images | I+A | 25k | 1.8k | MR
| MELON
---
[Proposed]
Various | I+A+T | 488k | 7.7k | MR + C

Table 1: Melodic-Design vs. other datasets. I, A, and T corresponds to image,
audio, and text respectively. “Various” for Melodic-Design covers
illustrations, vectors, template, and background designs. C, R, G, MR
represents classification, retrieval, generation, and music retrieval
respectively.

###  3.1 Collecting Raw Dataset Samples

We use the publicly available MTG-Jamendo [4] database that encompasses a
variety of mood/theme categories, instruments, and genres as our source of
audio files. We use various time-frequency features like intensity, timbre,
pitch, tempo, and rhythm [8] to identify the mood of audio. For example, the
pitch varies from very high to very low as we move from the “happy” to “sad”
mood. Similarly, the intensity and tempo of mood “upbeat” is very high,
whereas that of “calm” is very low (see Fig. 2). For mapping/associating audio
to design documents below, we use music files corresponding to 50 mood
categories in MTG-Jamendo. Since MTG-Jamendo also has audio files labelled
with multiple mood categories, we only retain those data samples labelled with
only one mood for simplicity.

We use publicly available data from Adobe Stock as our source for collecting
multimodal design documents comprising image and text content. To scrape
images, we built a software utility that can query Stock with any mood
category along with data types as part of the input. For instance, one such
query would involve fetching illustrations, vectors, templates, and background
images for the adventure mood. By restricting the search-page limit to 10, we
obtain about $10,000$ images across all the above document types for every
mood category. Note that Adobe Stock also provides image metadata which
contains manually generated captions describing the image elements in detail.

Fig. 3: Block diagram of the proposed music retrieval model using image and
text pair.

###  3.2 Establishing Correspondence & Dataset Statistics

In our dataset, images and text are already paired since each downloaded image
comes with a ground-truth caption. We use the common mood categories across
the MTG music dataset and our proposed design document dataset to form image-
caption-audio pairs. Specifically, given a mood category, we first extract the
CLIP [9] features for an image-text sample. For each audio file tagged with
the same mood, we extract Wav2CLIP [10] embeddings and compute cosine
similarities between the image-audio embeddings (denoted $s(i,a)$) and text-
audio embeddings (denoted $s(t,a)$). We use the weighted sum
$\lambda_{1}s(i,a)+\lambda_{2}s(t,a)$ to retain the audio files corresponding
to the top-N similarity scores. We repeat this process for all images in each
mood category to curate our audio-design dataset.

Our proposed MELON dataset consists of 488,510 images and corresponding
captions and 7,737 music audios belonging to 50 moods/themes. Each mood
category consists of $\approx 10k$ images. A distribution plot of the audio
samples per category as well as per-mood word clouds to demonstrate the
diversity and variability of our dataset are provided in the supplementary
material555Link to supplementary document: https://shorturl.at/hsDU5. We plan
to make materials publicly available at this link.. In Table 1, we
quantitatively compare the proposed MELON dataset with existing datasets. One
can note that while existing datasets are focused on natural images and
videos, our proposed dataset is unique in the sense it contains creative
illustrations, vectors, templates, and background designs with complete
descriptions of the image content as part of the caption. With $\approx 500k$
images, our dataset will help the community build robust models for both music
retrieval (MR) and classification (C) tasks.

##  4 Multi-modal Cross-attention Audio Retrieval (MMCAR)

Here, we describe our proposed algorithm for retrieving audio files given an
input design document. Our key algorithmic novelty is a multi-modal cross-
attention module that operates on feature vectors from all three input
modalities (image, text, and audio) and learns a common representation space
for the downstream retrieval task.

Figure 3 visually summarizes our proposed algorithm. During training, given
input triplets from our MELON dataset, we first use per-modality embedding
extractors to compute feature vectors $\mathbf{i}$, $\mathbf{t}$, and
$\mathbf{a}$ for the image, text, and audio modalities respectively. For
$\mathbf{i}$ and $\mathbf{t}$, we use the CLIP [9] model to obtain
512-dimensional embeddings each. For $\mathbf{a}$, we train a Resnet-18 model
(for audio classification) on the publicly available VGGSound dataset.

Given the $\mathbf{i}$, $\mathbf{t}$, and $\mathbf{a}$ embeddings, we propose
a multi-modal cross-attention operation to learn a unified multi-modal design-
audio embedding. Given the three feature vectors, we perform pairwise cross-
attention pooling taking any two modalities
${\mathbf{x}},\boldsymbol{y}\in\\{\boldsymbol{i},\boldsymbol{t},\boldsymbol{a}\\}$
such that ${\mathbf{x}}\in\mathcal{R}^{d}$ is the query and
$\boldsymbol{y}\in\mathcal{R}^{d}$ is the key, resulting in an output
$\hat{x}$. Similarly, by interchanging ${\mathbf{x}}$ and $\boldsymbol{y}$, we
compute the output $\hat{y}$. These two outputs are then used to compute a
common embedding $\mathbf{u}_{xy}$ for this particular pair of $x$ and $y$. We
repeat this for all the possible pairs ($(x=i,y=t),(x=i,y=a),(x=t,y=a)$), and
use the corresponding outputs to obtain the proposed unified embedding
$\mathbf{u}_{\text{all}}$ as:

$\begin{split}\boldsymbol{C}_{xy}&={\mathbf{x}}\boldsymbol{y}^{T}\in\mathcal{R}^{dXd}\quad
and\quad\boldsymbol{C}_{yx}=\boldsymbol{y}{\mathbf{x}}^{T}\\\
\boldsymbol{S}_{x}&=\sigma(\boldsymbol{C}_{xy}*\boldsymbol{W}+\boldsymbol{B})\in\mathcal{R}^{dXd}\\\
\hat{{\mathbf{x}}}&=diag(\boldsymbol{S}_{x}\boldsymbol{C}_{xy}^{T})\\\
\end{split}$ (1)
$\begin{split}\boldsymbol{S}_{y}&=\sigma(\boldsymbol{S}_{yx}*\boldsymbol{W}+\boldsymbol{B})\in\mathcal{R}^{dXd}\\\
\hat{\boldsymbol{y}}&=diag(\boldsymbol{S}_{y}\boldsymbol{C}_{yx}^{T})\\\
\end{split}$ (2)
$\begin{split}\boldsymbol{u}_{xy}&=\hat{{\mathbf{x}}}\oplus\hat{\boldsymbol{y}}\in\mathcal{R}^{2dX1}\\\
\boldsymbol{u}_{all}&=\boldsymbol{u}_{it}\oplus\boldsymbol{u}_{ia}\oplus\boldsymbol{u}_{ta}\in\mathcal{R}^{6dX1}\end{split}$
(3)

This unified embedding $\mathbf{u}_{\text{all}}$ is then passed to a fully
connected neural network unit which generates, with a sigmoid operation, a
scalar score $\hat{z}$ in range $\in[0,1]$. We compare this with the ground
truth score $z=1$ (if the input is a correct pair) and $z=0$ otherwise,
resulting in a binary cross-entropy training objective

$L=\frac{1}{B}\sum_{i=1}^{B}z\text{log}(\hat{z})+(1-z)\text{log}(1-\hat{z})$
(4)

where B is the batch size. During inference, given a design document input and
a database of $n$ audio samples from which to retrieve relevant files, our
model computes the similarity scores $\hat{z}_{1},\hat{z}_{2},...,\hat{z}_{n}$
for the input image-text pair with all the $n$ audio samples. Given these
scores, we pick the audio files corresponding to the top-$k$ highest scores as
the retrieval results (see Fig 3 (right)).

Fig. 4: Confusion matrix of proposed MMCAR vs. baseline JTAV and Wav2CLIP.

adventure | advertising | drama | funny | love
---|---|---|---|---
fun | commercial | dramatic | groovy | romantic
game | corporate | movie | happy | nature
holiday | ambiental | dream | hopeful | summer
horror | calm | emotional | motivational | retro
space | relaxing | heavy | melodic | background
sport | soft | melancholic | children |
upbeat | mellow | sad | christmas |

Table 2: Lists of moods/themes used for training and evaluation. Fig. 5: t-SNE
plots of baselines vs. proposed approach.

##  5 Experiments and Results

Since our proposed problem, data, and algorithm is centered around audio
retrieval given image-text design documents, the closest baselines in the
literature include JTAV [1] and Wav2CLIP [10]. While JTAV operates on an
image, its caption, and the textual lyrics of an audio to learn features,
Wav2CLIP uses the CLIP [9] image encoder and an audio autoencoder to map audio
and image features close.

To benchmark the performance of these algorithms and compare them to our
proposed method on our new dataset, we use rank-based evaluation metrics
proposed in prior work [1]. In particular, we use the Med r $\in[1,M]$ metric
that represents the medium rank of the ground-truth retrieved audio, where $M$
is the maximum number of classes considered. A lower value of Med r indicates
better performance. We also use recall@k ($k=\\{1,5,10,15,20\\}$) that is the
fraction of ground-truth audios retrieved in the top-k ranked items across all
test cases, and higher values indicate better performance.

For training and evaluating all models, we selected 38 moods in our dataset
based on maximum uniqueness among all the 50 moods (see Table 2 for a list).
We construct the training split using $60\%$ of the image-caption pairs and
audio samples. Each triplet comprises the image, caption, and audio along with
a $1/0$ label based on the correct audio mapping. To form positive triplets,
we select $10$ audio samples for each design based on mood as discussed in
Section 3.2. The negative triplets are formed by randomly selecting $5$
different moods from the mood of the image and selecting $2$ audio samples
from each mood. The remaining $40\%$ of the data is equally split to obtain
validation and test splits with $20\%$ of the samples each.

We next present our evaluation results. Since we seek to retrieve the best
matched audio file in terms of mood, we first compute a mean feature vector
for each mood from our audio repository. Then, given the feature vector for an
input design document, we generate a score vector
$\boldsymbol{\hat{z}}\in\mathcal{R}^{M}$, where $M=38$ as noted above, for the
mean features of all the audio files. We then compute the recall@k and Med r
metrics using the reference mood label.

In Table 3, we compare the performance of our proposed MMCAR algorithm with
JTAV and Wav2CLIP baselines on both val and test sets. One can note that our
proposed MMCAR gives the lowest $Med$ $r$ of $3.9$ and the highest recall
accuracy at all ranks, e.g., $79\%$ accuracy at k=5 on the test set compared
to JTAV’s $30\%$ and Wav2CLIP’s $39\%$, accounting for more than $100\%$
relative improvements. For an even more fair comparison, we also reimplemented
our MMCAR with Wav2CLIP’s features (noted MMCAR* in table). While MMCAR* leads
to performance degradation when compared to our original, end-to-end-learned
MMCAR model, it is still substantially better than the baseline Wav2CLIP
approach. This provides evidence for our multimodal cross-attention module’s
discriminative capabilities in the shared image-text-audio feature space.

To provide additional evidence, we show t-SNE plots of the learned design
document embeddings in Figure 5 where one can see a clearer clustering,
compared to baseline approaches, of the features according to the mood using
the proposed MMCAR approach. In Figure 4, we compare MMCAR’s confusion matrix
with the baseline ones for a random selection of 14 moods, where one can note
while baseline predictions are biased towards a few specific moods, the
proposed method has a close-to-diagonal matrix as expected.

Methods | $Med$ $r$ | Recall@k
---|---|---
k=1 | k=5 | k=10 | k=15 | k=20
Val
JTAV [1] | 17.1 | 0.12 | 0.30 | 0.35 | 0.46 | 0.62
Wav2CLIP [10] | 9.2 | 0.12 | 0.37 | 0.66 | 0.83 | 0.92
MMCAR [Ours] | 3.8 | 0.42 | 0.80 | 0.92 | 0.96 | 0.98
MMCAR* | 7.0 | 0.23 | 0.59 | 0.77 | 0.87 | 0.93
Test
JTAV [1] | 16.9 | 0.12 | 0.30 | 0.35 | 0.45 | 0.62
Wav2CLIP [10] | 9.2 | 0.11 | 0.39 | 0.67 | 0.83 | 0.91
MMCAR [Ours] | 3.9 | 0.42 | 0.79 | 0.92 | 0.96 | 0.98
MMCAR* | 7.2 | 0.22 | 0.56 | 0.76 | 0.89 | 0.93

Table 3: Proposed MMCAR vs. baselines.

##  6 Summary

We considered and proposed a new problem of retrieving relevant audio files
given multimodal design documents as input. In the absence of any relevant
datasets in the literature, we built and presented a first-of-its-kind
multimodal design-audio dataset called MELON comprising hundrends of thousands
of design files with mapped audio files. We then proposed a multimodal cross
attention algorithm that enables training neural networks to learn a joint
image-text-audio feature space for design documents and used it to retrieve
relevant audios given a certain design input at test time. We benchmarked our
algorithm against the existing state of the art on our new dataset and hope
that this will spur further research in this area.

##  7 ACKNOWLEDGEMENTS

The authors would like to thank Dr. Sriram Ganapathy of LEAP Lab, Indian
Institute of Science, Bangalore, for his valuable input and help in offering
the required resources to run the experiments.

## References

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  * [7] Sicheng Zhao, Yaxian Li, Xingxu Yao, Weizhi Nie, Pengfei Xu, Jufeng Yang, and Kurt Keutzer,  “Emotion-based end-to-end matching between image and music in valence-arousal space,”  in Proceedings of the 28th ACM International Conference on Multimedia, 2020, pp. 2945–2954.
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# Forward-Forward Mean Field Games in mathematical modeling with application
to opinion formation and voting models. ††thanks: This work was partially
supported by the Haute-Normandie Regional Council via the M2NUM project and
the project GO 1920/7-1 by the German Research Foundation (DFG).

Adriano Festa Politecnico di Torino, Dipartimento di Scienze Matematiche
”Giuseppe Luigi Lagrange”, Corso Duca degli Abruzzi, 24, 10129 Torino
<EMAIL_ADDRESS>Simone Göttlich University of Mannheim,
Department of Mathematics, A5-6, 68131 Mannheim, Germany. goettlich@uni-
mannheim.de Michele Ricciardi KAUST (King Abdullah University of Science and
Technology), Computer Electrical and Mathematical Science and Engineering
(CEMSE) Division, Thuwal, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract

While the general theory for the terminal-initial value problem in mean-field
games is widely used in many models of applied mathematics, the modeling
potential of the corresponding forward-forward version is still under-
considered. In this work, we study the well-posedness of the problem in a
quite general setting and explain how it is appropriate to model a system of
players that have a complete knowledge of the past states of the system and
are adapting to new information without any knowledge about the future. Then
we show how forward-forward mean field games can be effectively used in
mathematical models for opinion formation and other social phenomena.

##  1 Introduction

Mean-field games (MFGs) are models for large populations of competing rational
agents who seek to optimize an individual objective function. They were
introduced for the first time in 2006 in [25]. The most investigated form has
a backward-forward structure that can be sketched by the following system of
partial differential equations:

$\left\\{\begin{array}[]{ll}-u_{t}(t,x)+H(x,\nabla
u(t,x))=\operatorname{div}(D(x)\nabla u(t,x))+F(t,x,m(t,x)),\\\
m_{t}(t,x)-\mathrm{div}(H_{p}(x,\nabla
u(t,x))m(t,x))=\operatorname{div}(D(x)m(t,x)).\end{array}\right.$ (1.1)

Here, a density of individuals $m$ moves in the direction determined by the
potential function $u$. In the system above, $D$ is a diffusion matrix,
$\nabla$ is the usual space gradient and $H_{p}$ is the gradient w.r.t. the
variable $p$ of the Hamiltonian operator $H(x,p)$.

To give a hint of the idea behind the model, while the Hamilton-Jacobi
equation (HJ) contained in the first line of the system solves, backward in
time, an optimal control problem relative to the strategic choice of any
individual, this information is used, forward in time, to move the density $m$
accordingly to the second equation of (1.1).

The existence and uniqueness of equilibrium solutions of 1.1 can be found, for
example, in [6, 17]. Other researches on this subject are [20], where strong
solutions for parabolic problems were considered, or [32], for weak solutions
of parabolic problems with Dirichlet and Neumann boundary conditions. In [7]
the authors discuss the existence of weak solutions for first-order MFGs. The
stationary case was also investigated in detail and it was first considered in
[25]. The uniqueness of solution is obtained (both for stationary and time-
dependent MFGs) via a monotonicity argument introduced in [25].

Another model strictly related to what we are going to discuss, is the Hughes’
model [22, 8], where the optimization process of every player uses information
only about the _actual_ state of the system without any forecasting of the
future states of the system. This means that the density evolution will be
paired with a stationary HJ equations containing only the actual configuration
of the density. The latter model, has been proved to be appropriate effective
to effectively simulate scenarios where the agents _may change_ their
strategy, due to some unpredicted events, but they are _prone to change
instantaneously_ strategy, without any relevance given to their past
behaviour. The model is used in particular to simulate large crowd fluxes of
pedestrians.

A version of a MFG system, in a forward-forward form, has been proposed in
[2], as effective way to approximate the stationary state of the system, and
in [18, 19], in its one-dimensional form, mostly with the purpose of studying
the qualitative properties of the system.

Our paper wants to explore from the model point of view the forward-forward
MFGs (FF-MFGs), and showing as they are not only an effective tool to
approximate stationary problems, or a mere theoretical curiosity, but an
effective and in some cases more adequate model to simulate a collective
process. Here, the agents are driven by the past states of the system in the
sense that they have no access to information about the future while their
collective past configurations play a key role in their current strategic
choices. This features can be effectively exploited to model some collective
learning phenomena where tradition and stubbornness play a key role as, in
particular, opinion formation and voters’ tendencies.

We give a brief summary of the content of the paper: in Section 2, we derivate
formally a class of FF-MFG of our interest, giving a stochastic interpretation
depending on a backward in time SDE. Some general results of existence and
well-position are derived in Section 3. We stress on the fact that the well-
posedness results are very general, since we prove existence and uniqueness of
solutions for the Cauchy problem and the case of Dirichlet and Neumann
boundary conditions. The case of periodic solutions, namely where the state
space $\Omega=\mathbb{T}^{d}$, is not investigated here, but it can proved in
an easier way than the other cases. Two FF-MFGs models for social processes
are proposed and discussed in Section 4, with various numerical simulations
that show in practice their features. The numerical tools that we used along
the paper are briefly explained in A.

##  2 Derivation of the evolution process

Let us denote by $X_{s}\in\mathbb{R}^{n}$ the quantity of interest (e.g.
opinion, ideological position on a political spectrum, etc.) at time $s$,
which follows the stochastic differential equation (SDE)

$\left\\{\begin{array}[]{ll}dX_{s}=\alpha(s,X_{s})ds+\sqrt{2}\sigma(X_{s})\,dB_{s},&s\in(t,T),\\\
X_{t}=x,&\end{array}\right.$ (2.1)

for some $x\in\mathbb{R}^{n}$. Here, $\sigma(X_{s})$ is a $n\times n$ matrix,
and $B_{s}$, $s\in[0,T]$ is a standard $n$-dimensional Brownian motion. The
control $\alpha$ contributes to the motion of $X$, depending on time and the
value. We assume $\alpha$ being the minimizer of

$v(t,x)=\inf_{\alpha}\;\left\\{\mathbb{E}\Big{[}\int_{t}^{T}\left[\ell(X^{x,t}[\alpha](s),\alpha(s,X^{x,t}[\alpha](s)))\right.\right.\\\
\left.\left.+F(T-s,X^{x,t}[\alpha](s),m(T-s,\cdot))]ds+u_{0}(X^{x,t}[\alpha](T))\right]\right\\},$
(2.2)

where $X^{x,t}[\alpha](s):=X_{s}$ with $s\in[0,t]$ is assumed to be the unique
strong solution to (2.1),
$\ell\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ and
$F\colon[0,T]\times\mathbb{R}^{n}\times\mathcal{P}(\mathbb{R}^{n})\to\mathbb{R}$
are running cost functions. Defining an Hamiltonian function $H$ as

$H(x,p):=\max_{\alpha\in\mathbb{R}^{n}}\left\\{-\alpha\cdot
p-\ell(x,\alpha)\right\\},$ (2.3)

the optimal solution of the optimization problem in (2.2) is given in feedback
form by $\alpha(t,x)=-H_{p}(x,\nabla v(t,x))$ and consequently the optimal
trajectory $X^{x,t}$ is a solution of

$\left\\{\begin{array}[]{ll}dX(s)=-H_{p}\left(X(s),\nabla
v(s,X(s))\right)ds+\sqrt{2}\sigma(X(s))\,dB_{s},&s\in(t,T),\\\
X(t)=x.&\end{array}\right.$ (2.4)

Moreover, a classic Dynamic Programming Principle (see [3]) applies to the
value function $v(t,x)$ in (2.2) bringing us a differential representation
formula for $v(t,x)$ in HJ form as

$-v_{t}(t,x)+H(x,\nabla v(t,x))=F(T-t,x,m(T-t))+\mbox{div}(D(x)\nabla v(t,x))$

provided with the boundary condition $v(T,x)=u_{0}(x)$ and with
$D(x):=\sigma(x)\sigma^{\mathsf{T}}(x)$. Let us assume that we want to reverse
the time evolution stating $v(t,x)=u(T-t,x)$: we would obtain (with the
additional change of variables $t^{\prime}=T-t$)

$\left\\{\begin{array}[]{ll}u_{t}(t,x)+H(x,\nabla
u(t,x))=F(t,x,m(t))+\mbox{div}(D(x)\nabla u(t,x))\\\
u(0,x)=u_{0}(x).&\end{array}\right.$ (2.5)

We can interpret the function $u(x,t)$ as the value function for an agent who
plays optimally, facing, at each time $t\in[0,T]$, a running cost $\ell$
depending by its state and choice, as well as a criterion $F$ that depends on
the moving distribution $m$ of all the players in the interval $[0,t]$. We
underline as the optimization problem at time $t$ no information comes from
the future distribution of $m$.

Now, we define the quantity $m$ as the probability density of players moving
accordingly to the SDE

$dY(t)=-H_{p}\left(Y(t),\nabla
u(t,Y(t))\right)dt+\sqrt{2}\sigma(Y(t))\,dB_{t}$

and initial distribution $m(0,x)=m_{0}(x)\in\mathcal{P}(\mathbb{R}^{n})$.
Classical arguments in diffusion theory imply that the measures $m$ of the
initial distribution $m_{0}$ will be characterized by the dynamics given by
the Fokker-Planck (FP) equation

$\left\\{\begin{array}[]{ll}m_{t}(t,x)-\mbox{div}\big{(}m(t,x)\,H_{p}(x,\nabla
u(t,x))\big{)}-\mbox{div}(D(x)\nabla m(t,x))=0,\\\
m(0)=m_{0}(x)&\end{array}\right.$ (2.6)

The latter has to be understood in the weak sense, see, e.g., [6]. Coupling
(2.6) and (2.5), we obtain a Forward-Forward MFG system.

Clearly, the motion of the density and the optimal control are not done on the
same trajectories, therefore, as it happens for the Hughes’ model, this
framework cannot be properly called a _game_. Essentially this is exactly the
key point that makes our model to simulate possible changes of strategy and
less rational behaviours of the players. We keep the term FF-MFGs to be
consistent with the already existing literature.

We observe again that the model uses, in its optimization part, only
information about $m$ in its past states: this makes this framework suitable
to model some _evolution processes_ where tradition and established believes
are more influential with respect to rationality. Essentially an agent,
starting from an _initial guessed potential_ $u_{0}$ (due to some kind of
prior knowledge of the environment or just neutral, as in the case of a
constant function) improves it _continuously_ with the incoming data available
to any time of the process $t\in[0,T]$.

The model can be defined in a limited open subset of $\mathbb{R}^{n}$; in such
a case, some additional boundary conditions are needed. A stochastic
interpretation in case of Dirichlet and Neumann boundary conditions can be
given: in the case of Dirichlet boundary condition, the trajectory of the
player has to be stopped if he reaches the boundary. In the case of Neumann
boundary conditions, a reflecting process at the boundary is added in the SDE
(2.1), see [26]. We do not deal with these interpretations here, which are a
readaptation of the backward-forward MFG developed in [12, 33].

##  3 Some results about Forward Forward Mean Field Games

The forward-forward model was introduced in [2] to approximate stationary
MFGs. The key insight is that the parabolicity in (3.1) should imply the long-
time convergence to a stationary solution.

In this section, we want to prove existence and uniqueness of classical
solutions for the forward-forward MFGs. In particular we discuss the existence
and the uniqueness of the FF-MFG general framework which includes the models
that we are going to discuss in the rest of the paper. We consider the
forward-forward MFG problem described by the system

$\left\\{\begin{array}[]{ll}\displaystyle u_{t}-\mathrm{div}(D(x)\nabla
u)+H(t,x,\nabla u)=F(t,x,m)\,,&(t,x)\in(0,T)\times\Omega\\\ \displaystyle
m_{t}-\mbox{div}(D(x)\nabla m)-\mathrm{div}\big{(}mH_{p}(t,x,\nabla
u)\big{)}=0\,,&(t,x)\in(0,T)\times\Omega\\\ \displaystyle
u(0,x)=u_{0}(x)\,,\qquad m(0,x)=m_{0}(x)\,,&x\in\Omega\,.\end{array}\right.$
(3.1)

Here, $\Omega\subseteq\mathbb{R}^{n}$ is an open domain. We distinguish
between three cases, giving rise to three different kind of problems. Namely,
either

$\Omega=\mathbb{R}^{n}\,,\mbox{ no boundary conditions prescribed,}$ (C)

which is known as _Cauchy problem_ , or we require $\Omega$ to be a bounded
domain, complementing the system with _Dirichlet_ or _Neumann_ boundary
conditions, i.e.

$u(t,x)_{|x\in\partial\Omega}=0\,,\qquad m(t,x)_{|x\in\partial\Omega}=0\,,$
(D)

or, for $x\in\partial\Omega$,

$D(x)\nabla u(t,x)\cdot\nu(x)=0\,,\qquad\big{[}D(x)\nabla m+mH_{p}(t,x,\nabla
u)\big{]}\cdot\nu(x)=0\,,$ (N)

The existence and the long-time convergence of the forward-forward model have
not been addressed, except in a few cases, see [20] and [21]. In [21], the
forward-forward problem was examined in the context of eductive stability of
stationary MFGs with a logarithmic coupling. In [20], the existence and
regularity of solutions for the forward-forward, uniformly parabolic MFGs with
subquadratic Hamiltonians was proven. Except for these cases, the question of
existence and regularity is open in all other regimes. In the case of forward-
forward MFGs without viscosity, these questions are particularly challenging.
Moreover, the long-time convergence has not been established even in the
parabolic case. Nevertheless, numerical results in [1] indicate that
convergence holds and that the forward-forward model approximates well
stationary solutions.

Let $\mathcal{P}(\Omega)$ be the set of probability measures on $\Omega$ with
finite first moments. Then, $\mathcal{P}(\Omega)$ is a compact topological
space endowed with the weak*-convergence and metrizable with respect to the
_Wasserstein distance_ , which is defined as the distance between two
probability measures $m_{1},m_{2}\in\mathcal{P}(\Omega)$ as

$\displaystyle\mathbf{d}_{1}(m_{1},m_{2})$ $\displaystyle:=\sup_{g\in
Lip_{1}(\Omega)}\left\\{\int_{\Omega}g(x)d(m_{1}-m_{2})(x)\right\\},$

where with $Lip_{1}(\Omega)$ we mean the space of Lipschitz continuous
functions with Lipschitz constant equal to one. The same definition applies in
case of (D), with $\mathcal{P}(\Omega)$ replaced by
$\mathcal{P}^{sub}(\Omega)$ and test function $g$ with $Lip(g)\leq 1$ and
$\left\Arrowvert g\right\Arrowvert_{\infty}\leq 1$, see [30]. This is called
_generalized Wasserstein distance_.

We observe that, in case of (N) and (C), we can restrict to test functions $g$
with $g(x_{0})=0$, for a fixed $x_{0}\in\Omega$. In particular, in case of
Neumann condition we have $\left\Arrowvert g\right\Arrowvert_{\infty}\leq C$,
where $C=diam(\Omega)$, and in case of the Cauchy problem we have
$|g(x)|\leq|x|$.

Henceforth, when there is no possibility of mistake, we use the notation
$\mathbf{d_{1}}$ to refer to both Wasserstein distance and generalized
Wasserstein distance.

We start giving the basic assumptions that we need for a well-posedness
result.

#### Hypotheses:

Suppose that, for a certain $\gamma\in(0,1)$ and $C>0$,

  1. (H0)

$u_{0}\in\mathcal{C}^{2+\alpha}(\Omega)$, $m_{0}\in\mathcal{P}(\Omega)$, or
$\mathcal{P}^{sub}(\Omega)$ in the case (D), with a $\mathcal{C}^{2+\gamma}$
density, called also $m_{0}$. Moreover $D(\cdot)\in\mathcal{C}^{2}(\Omega)$,
$D=\sigma\sigma^{\mathsf{T}}$ with $\sigma\in Lip(\mathbb{R}^{n\times n})$ and
for certain $\mu>\lambda>0$,

$\lambda|\xi|^{2}\leq\langle
D(x)\xi,\xi\rangle\leq\mu|\xi|^{2}\qquad\forall\,\xi\in\mathbb{R}^{n};$

  2. (H1)

$H:[0,T]\times\Omega\times\mathbb{R}^{n}\to\mathbb{R}$,
$F:[0,T]\times\Omega\times\mathcal{P}(\Omega)\to\mathbb{R}$ (or
$F:[0,T]\times\Omega\times\mathcal{P}^{sub}(\Omega)\to\mathbb{R}$ in the case
(D)) are continuous functions and $\mathcal{C}^{\frac{\gamma}{2},\gamma}$ in
$(t,x)$ variable, with $H$ differentiable and convex in the last variable, and
$H(\cdot,\cdot,0)$ is globally bounded. Moreover, the growth of $H$ is at most
quadratic:

$|H(t,x,p)|\leq C(1+|p|^{2})\,,\qquad|H_{p}(t,x,p)|\leq
C(1+|p|)\,,\qquad|H_{pp}(t,x,p)|\leq C\,;$

  3. (H2)

For $\mu_{1},\mu_{2}\in\mathcal{P}(\Omega)$ or $\mathcal{P}^{sub}(\Omega)$,
the following conditions hold:

$\begin{split}|F(t,x,\mu_{1})-F(t,x,\mu_{2})|&\leq
C\mathbf{d}_{1}(\mu_{1},\mu_{2})\,,\\\
&\int_{\Omega}\left(F(t,x,\mu_{1})-F(t,x,\mu_{2})\right)d(\mu_{1}-\mu_{2})(x)\geq
0\,;\end{split}$

  4. (H3)

In case of boundary conditions, we require compatibility conditions for the
initial data: namely, in the setting (D), we require

${u_{0}(x)}_{|x\in\partial\Omega}=0\,,\qquad{m_{0}(x)}_{|x\in\partial\Omega}=0\,,$

whereas in the setting (N)

${D(x)\nabla
u_{0}(x)\cdot\nu(x)}_{|x\in\partial\Omega}=0\,,\qquad\big{[}D(x)\nabla
m_{0}+m_{0}H_{p}(0,x,\nabla u_{0})\big{]}\cdot\nu(x)=0\,.$

In the setting (C), we require instead a finite first order moment for
$m_{0}$, namely

$\int_{\mathbb{R}^{n}}|x|\,m_{0}(dx)\leq C\,.$

From now on, $C$ indicates a non-negative constant, that can change from line
to line.

###### Theorem 3.1

Let Hypotheses $(H1)-(H3)$ hold true. Then there exists a unique solution
$(u,m)\in\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}([0,T]\times\Omega)\times\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}([0,T]\times\Omega)$
for the FF-MFG system (3.1).

Proof. For $M>0$ which will be chosen later, we consider the following set:

$X=\left\\{\mu\in\mathcal{C}([0,T];V)\quad|\quad d_{V}(\mu(t),\mu(s))\leq
M|t-s|^{\frac{1}{2}}\right\\}\,,$

where in case of (C) or (N), $V=\mathcal{P}(\Omega)$ and
$d_{V}=\mathbf{d}_{1}$, otherwise in case of (D) $V$ is the dual of
$\left\\{\phi\in\mathcal{C}^{1}(\Omega)\,|\,{\phi}_{|x\in\partial\Omega}=0\right\\}$,
with $d_{V}$ the classical distance induced by the duality norm:

$d_{V}(\mu,\theta)=\left\Arrowvert\mu-\theta\right\Arrowvert_{V}=\sup\limits_{\begin{subarray}{c}\left\Arrowvert\phi\right\Arrowvert_{1}\leq
1\\\ \phi_{|\partial\Omega=0}\end{subarray}}\langle\mu-\theta,\phi\rangle\,.$

Note that $X$ is closed and convex in the topology inherited by
$\mathcal{C}([0,T];V)$. In order to apply Schauder’s Theorem, we define a
functional $\Phi:X\to X$ in the following way: for $\mu\in X$, we take
$u=u_{\mu}$ as the solution of

$\begin{cases}u_{t}-\operatorname{div}(D(x)\nabla u)+H(t,x,\nabla
u)=F(t,x,\mu(t))\,,\\\ u(0)=u_{0}\,,\end{cases}$

with conditions (C), (D) or (N).

In all the three cases, a solution $u$ is unique and regular. In case of
boundary conditions (D) and (N), from the comparison principle we have
$\left\Arrowvert u\right\Arrowvert_{\infty}$ uniformly bounded. Then from
_Theorem V.4.1_ and _Lemma VI.3.1_ (resp. _Theorem V.7.2_) of [24], we have
$u$ globally Lipschitz. Hence, we can linearize the problem of $u$, which
turns out to be the solution of

$\begin{cases}u_{t}-\operatorname{div}(D(x)\nabla u)+V(t,x)\cdot\nabla
u=F(t,x,\mu(t))+H(t,x,0)\,,\\\ u(0)=u_{0}\,,\end{cases}$

with conditions (D) or (N), and with

$V(t,x):=\int_{0}^{1}H_{p}(t,x,\lambda Du(t,x))\,d\lambda\,.$

Then we can apply the Corollary of _Theorem IV.9.1_ of [24] to obtain, for all
$0\leq\gamma<1$,

$\begin{split}\left\Arrowvert
u\right\Arrowvert_{\frac{1+\gamma}{2},1+\gamma}\leq C\Big{(}\left\Arrowvert
F\right\Arrowvert_{\infty}+\left\Arrowvert
H(\cdot,\cdot,0)\right\Arrowvert_{\infty}+\left\Arrowvert
u_{0}\right\Arrowvert_{2}\Big{)}\,.\end{split}$

For the Cauchy problem (C), we consider, for $k\in\mathbb{N}$, the solution
$u^{k}$ of the following problem

$\begin{cases}u^{k}_{t}-\mathrm{div}(D(x)\nabla u^{k})+H^{k}(t,x,\nabla
u^{k})=F(t,x,\mu(t))\,,\\\ u^{k}(0)=u_{0}\,,\end{cases}$

where

$H^{k}(t,x,p)=\frac{kH(t,x,p)}{k+|p|^{2}}$

With this choice, $H^{k}(\cdot,\cdot,p)$, $H_{p}^{k}(\cdot,\cdot,p)\in
L^{\infty}([0,T]\times\mathbb{R}^{n})$ and $H^{k}\to H$ pointwise. From
_Theorem 3.4_ and _Theorem 3.6_ of [23] we have $u^{k}(t,\cdot)$ bounded in
$W^{1,\infty}$ uniformly in $t$ and $k$, and so the same estimate holds for
$u$. Then we can apply _Theorem V.8.1_ of [24] and obtain
$u\in\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}([0,T]\times\mathbb{R}^{n})$.
Finally, we observe that $u$ solves

$\begin{cases}u_{t}-\operatorname{div}(D(x)\nabla u)=G(t,x)\,,\\\
u(0)=u_{0}\,,\end{cases}$

with $G(t,x)=F(t,x,\mu(t))-H(t,x,\nabla u)\in L^{\infty}$. Hence, from
_Theorem 5.1.2_ of [28] and the hypotheses on $H$ we have, for all
$0\leq\gamma<1$,

$\left\Arrowvert u\right\Arrowvert_{\frac{1+\gamma}{2},1+\gamma}\leq
C\Big{(}1+\left\Arrowvert F\right\Arrowvert_{\infty}+\left\Arrowvert\nabla
u\right\Arrowvert_{\infty}^{2}+\left\Arrowvert
u_{0}\right\Arrowvert_{2}\Big{)}\leq C\,,$

where $C$ does not depend on $\mu$.

Moreover, from this estimate we get $G$ bounded in
$\mathcal{C}^{0,\gamma}([0,T]\times\mathbb{R}^{n})$, uniformly in $\mu$.
Hence, from _Theorem 5.1.9_ of [28] we get

$\begin{split}\left\Arrowvert u\right\Arrowvert_{1,2+\gamma}\leq
C\left(\left\Arrowvert u_{0}\right\Arrowvert_{2+\gamma}+\left\Arrowvert
G\right\Arrowvert_{0,\gamma}\right)\leq C\,,\end{split}$ (3.2)

where $C$ does not depend on $\mu$.

Then we choose $\Phi(\mu)=m$, where $m=m_{\mu}$ is a solution, in the
distributional sense, of the FP equation

$\begin{cases}m_{t}-\mbox{div}(D(x)\nabla
m)-\mathrm{div}\big{(}mH_{p}(t,x,\nabla u)\big{)}=0\,,\\\
m(0)=m_{0}(x)\,,\end{cases}$ (3.3)

with conditions (C), (D) or (N).

For the distributional formulation, $m$ satisfies, for all $0<s<t<T$,

$\int_{\Omega}m(t,x)\psi(x)\,dx+\int_{s}^{t}\int_{\Omega}m(r,x)f(r,x)\,dxdr=\int_{\Omega}m(s,x)\phi(s,x)\,dx\,,$

where $f\in L^{\infty}$ and $\phi\in L^{\infty}$ satisfies

$\begin{cases}-\phi_{t}-\mathrm{div}(D(x)\nabla\phi)+H_{p}(s,x,\nabla
u)\cdot\nabla\phi=f\,,\\\ \phi(t)=\psi\,,\end{cases}$ (3.4)

with conditions (C), (D) or (N). We note that the results obtained for the HJB
equation can be applied also for the backward equation (3.4), with the change
of variable $(t,x)\to(T-t,x)$.

Taking $f=0$ and rearranging, we find

$\begin{split}\int_{\Omega}\big{(}m(t,x)-m(s,x)\big{)}\psi(x)\,dx=\int_{\Omega}m(s,x)(\phi(s,x)-\phi(t,x))\,dx\end{split}$

In case of conditions (C) or (N), we choose $Lip(\psi)=1$. As already said,
for (N) we can also take $\psi\in W^{1,\infty}$ and thanks to _Lemma 3.2_ of
[34] we have

$|\phi(t,x)-\phi(s,x)|\leq C|t-s|^{\frac{1}{2}}\,,$ (3.5)

which implies, for a $C$ not depending on $\mu$,

$\mathbf{d}_{1}(m(t),m(s))\leq C|t-s|^{\frac{1}{2}}\,.$ (3.6)

We argue in a similar way for the case (C). Here, thanks to _Theorem 3.6_ of
[23], we have $\nabla\phi$ uniformly bounded in $L^{\infty}$. Hence, $\phi$
solves

$\begin{cases}-\phi_{t}-\mathrm{div}(D(x)\nabla\phi)=f-H_{p}(s,x,\nabla
u)\cdot\nabla\phi\in L^{\infty}\,,\\\ \phi(t)=\psi\,,\end{cases}$

and applying again _Theorem 5.1.2_ of [28], (3.5) remains true, which implies
once again (3.6).

As regards Dirichlet boundary conditions (D), we take
$\psi\in\mathcal{C}^{1}(\Omega)$ with ${\psi}_{|x\in\partial\Omega}=0$ and
$\left\Arrowvert\psi\right\Arrowvert_{1}\leq 1$. From _Lemma 4_ of [12] we get

$\left\Arrowvert\phi\right\Arrowvert_{\frac{1}{2},1}\leq
C\left\Arrowvert\psi\right\Arrowvert_{1}\implies d_{V}(m(t),m(s))\leq
C|t-s|^{\frac{1}{2}}.$

Hence, if we choose $M$ sufficiently large, we have in all cases $m\in X$, and
$\Phi:X\to X$ is well-defined.

To apply Schauder’s theorem, we have to check that $\Phi(X)$ is relatively
compact in $\mathcal{C}([0,T];V)$ and continuous. For the relatively
compactness, let $\\{\mu^{k}\\}_{k}\subset X$, and let $u^{k}$ and $m^{k}$ be
the solutions of the HJB and the FP equation related to $\mu^{k}$.

As already seen, we have $\|u^{k}\|_{\frac{1+\alpha}{2},1+\alpha}\leq C$,
where $C$ does not depend on $k$. Hence, up to subsequences, we can apply
Ascoli-Arzelà’s Theorem and, for a certain
$u\in\mathcal{C}^{\frac{1+\gamma}{2},2+\gamma}$,

$u^{k}\to u\qquad\mbox{in }\mathcal{C}^{0,1}([0,T]\times\Omega)\,,$

at least for boundary conditions (D) or (N). In those cases, take $\phi^{k}$
and $\phi$ the solution of (3.4) related to $u^{k}$ and $u$, with $\psi\in
W^{1,\infty}$, satisfying ${\psi}_{|x\in\partial\Omega}=0$ in case of (D).
Then, arguing as before, $\phi^{k}$ is uniformly bounded in
$\mathcal{C}^{\frac{1+\gamma}{2},1+\gamma}$ and, up to subsequences,
$\phi^{k}\to\phi$ in $\mathcal{C}([0,T]\times\Omega)$. Then, subtracting the
distributional formulations of $m^{k}$ and $m$ with test functions $\phi^{k}$
and $\phi$ we get

$\int_{\Omega}\psi(x)(m^{k}(t,x)-m(t,x))\,dx=\int_{\Omega}(\phi^{k}(0,x)-\phi(0,x))\,m_{0}(dx)\to
0\,,$

which implies $m^{k}\to m$ in $\mathcal{C}([0,T];V)$.

In case of boundary conditions (C), the convergence of $u^{k}$ towards $u$ is
true only in $\mathcal{C}^{0,2}_{loc}([0,T]\times\Omega)$, thanks to (3.2),
with $u\in\mathcal{C}^{1,2+\gamma}([0,T]\times\Omega)$. In this case we note
that $m^{k}$ is the density of the following process

$\begin{cases}dX_{t}^{k}=b^{k}(t,X_{t}^{k})\,dt+\sqrt{2}\sigma(X_{t}^{k})\,dB_{t}\,,\\\
\mathscr{L}(X_{0}^{k})=m_{0}\,,\end{cases}$

where

$b^{k}(t,x)=-H_{p}(t,x,Du^{k}(t,x))-\Big{(}\sum\limits_{j=1}^{n}\frac{\partial
D_{ji}(x)}{\partial
x_{j}}\Big{)}_{i}\,,\qquad\sigma(x)\sigma^{\mathsf{T}}(x)=D(x)\,.$

In the same way we define the drift $b$ and the process $X_{t}$ related to
$m$, solution of the FP equation (3.3). Then, we have

$\mathbb{E}\left[|X_{t}^{k}-X_{t}|^{2}\right]\leq
2\,\mathbb{E}\left[\int_{0}^{t}\left|b^{k}(s,X_{s}^{k})-b^{k}(s,X_{s})\right|^{2}\,ds\right]\\\
+4\,\mathbb{E}\left[\left|\int_{0}^{t}\big{(}\sigma(X_{s}^{k})-\sigma(X_{s})\big{)}\,dB_{s}\right|^{2}\right]\,.$

The second integral is bounded using the Itô isometry and the Lipschitz bound
on $\sigma$:

$\mathbb{E}\left[\left|\int_{0}^{t}\big{(}\sigma(X_{s}^{k})-\sigma(X_{s})\big{)}\,dB_{s}\right|^{2}\right]\leq
C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{k}-X_{s}|^{2}\,ds\right]\,.$

For the first term we get

$\begin{split}&\mathbb{E}\left[\int_{0}^{t}\left|b^{k}(s,X_{s}^{k})-b^{k}(s,X_{s})\right|^{2}\,ds\right]\leq
C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{k}-X_{s}|^{2}\,ds\right]\\\
+C\,&\mathbb{E}\left[\int_{0}^{t}\big{|}H_{p}(s,X^{k}_{s},\nabla
u^{k}(s,X_{s}^{k}))-H_{p}(s,X_{s},\nabla
u(s,X_{s}))\big{|}^{2}\,ds\right]\,.\end{split}$

For the last integral we use the following strategy (see also _Theorem 6.3_ of
[23]):

$\begin{split}&\mathbb{E}\left[\int_{0}^{t}\big{|}H_{p}(s,X^{k}_{s},\nabla
u^{k}(s,X_{s}^{k}))-H_{p}(s,X_{s},\nabla u(s,X_{s}))\big{|}^{2}\,ds\right]\\\
\leq 2\,&\mathbb{E}\left[\int_{0}^{t}\big{|}H_{p}(s,X^{k}_{s},\nabla
u^{k}(s,X_{s}^{k}))-H_{p}(s,X^{k}_{s},\nabla
u(s,X^{k}_{s}))\big{|}^{2}\,ds\right]\\\
+2\,&\mathbb{E}\left[\int_{0}^{t}\big{|}H_{p}(s,X^{k}_{s},\nabla
u(s,X_{s}^{k}))-H_{p}(s,X_{s},\nabla u(s,X_{s}))\big{|}^{2}\,ds\right]\\\ \leq
C\,&\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla(u^{k}-u)|^{2}\,m^{k}(s,x)\,dxds+C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{k}-X_{s}|^{2}\,ds\right]\,.\\\
\end{split}$

This means

$\begin{split}\mathbb{E}\left[|X_{t}^{k}-X_{t}|^{2}\right]\leq
C\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla(u^{k}-u)|^{2}\,m^{k}(s,x)\,dxds+C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{k}-X_{s}|^{2}\,ds\right]\,.\end{split}$
(3.7)

The space-time integral is handled in the following way: for a compact
$E\subset\mathbb{R}^{n}$, we have

$\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla(u^{k}-u)|^{2}\,m^{k}\,dxds=\int_{0}^{t}\int_{E}|\nabla(u^{k}-u)|^{2}\,m^{k}\,dxds\\\
+\int_{0}^{t}\int_{E^{c}}|\nabla(u^{k}-u)|^{2}\,m^{k}\,dxds\leq\left\Arrowvert
u^{k}-u\right\Arrowvert^{2}_{\mathcal{C}^{0,1}([0,T]\times
E)}+C\left\Arrowvert m^{k}\right\Arrowvert_{L^{1}([0,T]\times E^{c})}\,.$

The first term goes to $0$ for each compact $E\subset\mathbb{R}^{n}$. For the
second term, we note that $m^{k}$ has finite first order moments, uniformly
bounded in $k$. Actually, if we take the solution $\phi$ of (3.4) with $f=0$
and $\psi=|x|$, from standard regularity results (see [31]) we have

$\left\Arrowvert\phi\right\Arrowvert_{\infty}\leq C(1+|x|)\,.$

Using hypothesis _(H3)_ , we obtain in the weak formulation of $m^{k}$

$\int_{\mathbb{R}^{d}}|x|m^{k}(t,x)\,dx=\int_{\mathbb{R}^{d}}\phi(0,x)\,m_{0}(dx)\leq
C\left(1+\int_{\mathbb{R}^{d}}|x|m_{0}(dx)\right)\leq C\,.$

This implies that $\left\Arrowvert m^{k}\right\Arrowvert_{L^{1}([0,T]\times
E^{c})}$ can be chosen arbitrarily small, if we take $E$ sufficiently large.
This implies that

$\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla(u^{k}-u)|^{2}\,m^{k}\,dxds=\omega(k)\,,\qquad\mbox{where
}\lim\limits_{k\to+\infty}\omega(k)=0\,.$

Coming back to (3.7) and using Gronwall’s inequality, we finally obtain

$\mathbb{E}\left[|X_{t}^{k}-X_{t}|^{2}\right]\leq
C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{k}-X_{s}|^{2}\,ds\right]+\omega(k)\implies\\\
\mathbb{E}\left[|X_{t}^{k}-X_{t}|^{2}\right]\leq e^{Ct}\omega(k)\to 0\,.$

Since

$\mathbf{d}_{1}(m^{k}(t),m(t))=\sup\limits_{Lip(\phi)=1}\mathbb{E}\left[\phi(X_{t}^{k})-\phi(X_{t})\right]\leq
C\mathbb{E}[|X_{t}^{k}-X_{t}|^{2}]\,,$

we have proved that $m^{k}\to m$ in $\mathcal{C}([0,T];V)$ (up to
subsequences) for all the boundary conditions (C), (D) or (N).

The continuity argument can be proven in the same way of the compactness. This
concludes the existence result.

Observe that, since $u$ is smooth, we can split the divergence term in (3.3),
obtaining for $m$ the equation

$\begin{cases}m_{t}-\mathrm{div}(D(x)\nabla m)-\mathrm{div}(H_{p}(t,x,\nabla
u))m-H_{p}(t,x,\nabla u)\nabla m\,,\\\ m(0)=m_{0}\,,\end{cases}$

with conditions (C), (D) or (N). This implies, arguing as in the HJB equation,
$m\in\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}$.

For the uniqueness, let $(u_{1},m_{1})$ and $(u_{2},m_{2})$ be two solutions
of (3.1), and set $(z,\mu):=(u_{1}-u_{2},m_{1}-m_{2})$. Then, called

$V(t,x):=\int_{0}^{1}H_{p}\big{(}t,x,s\nabla u_{1}(t,x)+(1-s)\nabla
u_{2}(t,x)\big{)}\,ds\,,$

the couple $(z,m)$ solves the following system:

$\begin{cases}z_{t}-\mathrm{div}(D(x)\nabla z)+V(t,x)\cdot\nabla
z=F(t,x,m_{1}(t))-F(t,x,m_{2}(t))\,,\\\
\mu_{t}-\mathrm{div}(D(x)\nabla\mu)-\mathrm{div}(\mu H_{p}(t,x,\nabla
u_{1}))=\mathrm{div}\big{(}m_{2}(H_{p}(t,x,\nabla u_{1})-H_{p}(t,x,\nabla
u_{2}))\big{)}\,,\\\ z(0)=0\,,\qquad m(0)=0\,,\end{cases}$

coupled, in the case of Dirichlet boundary conditions, with conditions (D) for
$(z,\mu)$. In the case of Neumann boundary conditions, we have for the couple
$(u,m)$ and for $x\in\partial\Omega$ the conditions

$D(x)\nabla z(t,x)\cdot\nu(x)=0$ $\big{[}D(x)\nabla\mu+\mu H_{p}(t,x,\nabla
u_{1})+m_{2}\big{(}H_{p}(t,x,\nabla u_{2})-H_{p}(t,x,\nabla
u_{1})\big{)}\big{]}\cdot\nu(x)=0\,.$

We start with the two cases (D) and (N). Multiplying by $\mu$ the Fokker-
Planck equation of $\mu$, we obtain

$\begin{split}\frac{1}{2}\int_{\Omega}\mu(t)^{2}\,dx&+\frac{\lambda}{2}\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds+\int_{0}^{t}\int_{\Omega}\mu
H_{p}(t,x,\nabla u_{1})\nabla\mu\,dxds\\\
&\leq\int_{0}^{t}\int_{\Omega}m_{2}\big{(}H_{p}(t,x,\nabla
u_{2})-H_{p}(t,x,\nabla u_{1})\big{)}\nabla\mu\,dxds.\end{split}$

Since $\left\Arrowvert\nabla u_{1}\right\Arrowvert_{\infty}\leq C$, we get
from Young’s inequality

$\int_{0}^{t}\int_{\Omega}\mu H_{p}(t,x,\nabla
u_{1})\nabla\mu\,dxds\geq-\frac{\lambda}{4}\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds-C\int_{0}^{t}\int_{\Omega}\mu^{2}\,dxds\,.$

In the same way, since $\left\Arrowvert m_{2}\right\Arrowvert_{\infty}\leq C$
and $H_{p}$ is Lispchitz in the last variable, we get

$\int_{0}^{t}\int_{\Omega}m_{2}\big{(}H_{p}(t,x,\nabla u_{2})-H_{p}(t,x,\nabla
u_{1})\big{)}\nabla\mu\,dxds\\\
\leq\frac{\lambda}{8}\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds+C\int_{0}^{t}\int_{\Omega}|\nabla
z|^{2}\,m_{2}\,dxds.$

This means

$\frac{1}{2}\int_{\Omega}\mu(t)^{2}\,dx+\frac{\lambda}{8}\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds\leq
C\int_{0}^{t}\int_{\Omega}\mu^{2}\,dxds+C\int_{0}^{t}\left\Arrowvert\nabla
z\right\Arrowvert^{2}_{L^{\infty}([0,s]\times\Omega)}\,ds$

which implies from Gronwall’s Lemma

$\begin{split}\sup\limits_{s\in[0,t]}\int_{\Omega}\mu(s)^{2}\,dx+\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds\leq
C\int_{0}^{t}\left\Arrowvert\nabla
z\right\Arrowvert^{2}_{L^{\infty}([0,s]\times\Omega)}\,ds\,.\end{split}$ (3.8)

To estimate the last integral, we study the equation of $z$. Since $V\in
L^{\infty}$ and, for all $s\in[0,T]$,

$\left\Arrowvert
F(t,x,m_{1}(t))-F(t,x,m_{2}(t))\right\Arrowvert_{L^{\infty}([0,s]\times\Omega)}\leq
C\sup\limits_{r\in[0,s]}\mathbf{d_{1}}\big{(}m_{1}(r),m_{2}(r)\big{)}\,,$

we have from the Corollary of _Theorem IV.9.1_ of [24],

$\begin{split}\left\Arrowvert\nabla
z\right\Arrowvert^{2}_{L^{\infty}([0,s]\times\Omega)}\leq
C\left[\sup\limits_{r\in[0,s]}\mathbf{d_{1}}\big{(}m_{1}(r),m_{2}(r)\big{)}\right]^{2}\,.\end{split}$
(3.9)

Since in case of (D) and (N) the domain $\Omega$ is bounded and the test
functions for $\mathbf{d_{1}}$ are bounded, we have

$\left[\sup_{r\in[0,s]}\mathbf{d_{1}}\big{(}m_{1}(r),m_{2}(r)\big{)}\right]^{2}\leq\left[\sup\limits_{r\in[0,s]}\sup\limits_{\begin{subarray}{c}Lip(\phi)\leq
1\\\ \left\Arrowvert\phi\right\Arrowvert_{\infty}\leq
C\end{subarray}}\int_{\Omega}\phi(x)\,\mu(r)\,dx\right]^{2}\\\ \leq
C\sup_{r\in[0,s]}\int_{\Omega}\mu(r)^{2}\,dx\,.$

Coming back to (3.8), we have

$\sup\limits_{s\in[0,t]}\int_{\Omega}\mu(s)^{2}\,dx+\int_{0}^{t}\int_{\Omega}|\nabla\mu|^{2}\,dxds\leq
C\int_{0}^{t}\sup_{r\in[0,s]}\int_{\Omega}\mu(r)^{2}\,dx\,ds\,,$

which implies, thanks to Gronwall’s Lemma, $\mu\equiv 0$, and so $z\equiv 0$.

For the Cauchy problem (C), we consider two Cauchy processes $(X_{t}^{1})_{t}$
and $(X_{t}^{2})_{t}$ with $\mathscr{L}(X_{t}^{1})=m_{1}(t)$,
$\mathscr{L}(X_{t}^{2})=m_{2}(t)$ and
$\mathbb{E}[|X_{t}^{1}-X_{t}^{2}|]=\mathbf{d_{1}}(m_{1}(t),m_{2}(t))$. Arguing
as in (3.7), we have

$\mathbb{E}\left[|X_{t}^{1}-X_{t}^{2}|^{2}\right]\leq
C\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla(u_{1}-u_{2})|^{2}\,m_{1}(s,x)\,dxds+C\,\mathbb{E}\left[\int_{0}^{t}|X_{s}^{1}-X_{s}^{2}|^{2}\,ds\right]\,.$

Applying Gronwall’s Lemma, we have

$\sup\limits_{s\in[0,t]}\mathbf{d_{1}}\big{(}m_{1}(s),m_{2}(s)\big{)}^{2}\leq\mathbb{E}[|X_{t}^{1}-X_{t}^{2}|^{2}]\leq
C\int_{0}^{t}\int_{\mathbb{R}^{n}}|\nabla z|^{2}\,m_{1}\,dxds\\\ \leq
C\int_{0}^{t}\left\Arrowvert\nabla
z\right\Arrowvert^{2}_{L^{\infty}([0,s]\times\Omega)}\,ds\,.$

Using once again (3.9), we get

$\sup\limits_{s\in[0,t]}\mathbf{d_{1}}\big{(}m_{1}(s),m_{2}(s)\big{)}^{2}\leq
C\int_{0}^{t}\sup\limits_{r\in[0,s]}\mathbf{d_{1}}\big{(}m_{1}(r),m_{2}(r)\big{)}^{2}\,ds\,,$

which implies $\mu\equiv 0$, and so $z\equiv 0$. This concludes the proof.

##  4 Two Forward Forward models

In this section we introduce two models which have the structure of a FF-MFG.
The framework is particularly appropriate when we want to model a continuity
of agents which are dynamically retrieving information from the past
environmental scenarios. We focus on an opinion formation model set in a 1D
space and a 2D voting model which share the same formal structure.

###  4.1 A 1D opinion formation model

In recent years, the process of opinion formation, like other sociological and
economical phenomena, has attracted a remarkable attention from physicists and
mathematicians, since it was pointed out that some of the tools used in
statistical mechanics could be successfully applied to model them. We cite
e.g. [15, 39, 11] among other works. In this context, quickly emerged two very
active new fields, usually called sociophysics or, by the mathematical
community behavioral social systems, devoted to the description of these
phenomena from the physicists point of view. We suggest the monographs [9, 16,
36, 4] for an overview and up-to-date references, while some more
mathematically related works are [5], where the authors discuss the use of
partial differential equations in modelling for social sciences and [14],
which is one of the first works that uses Fokker-Planck equations to model
socio-economic phenomena.

In the work [37] the authors propose an opinion dynamic model, where they
describe the evolution of an opinion in a 2-groups population of pliant and
stubborn agents where the second ones act as leaders. The model has a
structure of a multi-population backward-forward MFG. We borrow from [37] some
of the standard notation and concepts considering a population of homogeneous
agents, each one characterized by the opinion $X(t)\in\Omega\subset\mathbb{R}$
at time $t\in[0,T]$, where $[0,T]$ is the time horizon window. We consider
also the presence of some advertised opinions, as stated more clearly below.
As it is clear for the case of a FF-MFG, there is no modeling nor theoretical
need to bound $T$, as long as we do not have to use some numerical tools to
approximate the solution.

We assume that the opinion $X(t)$ is a solution of (2.1), where the dynamic is
led by the control variable $\alpha$. The presence of the stochastic term in
(2.1) is motivated by the effect of uncertainty in the opinion evolution. We
consider a probability density function
$m:\Omega\times[0,T]\rightarrow\mathbb{R}$ representing the percentage of the
agents in state $x$ at time $t$, which satisfies

$\int_{\Omega}m(t,x)dx=1,\hbox{ for any }t\in[0,T].$

The basic principle that we are going to use in the model is that every agent
adjusts his opinion based on some average opinion of its surrounding. In
particular, an agent is more sensitive and attracted by some high density
opinions which are not far from his, while he considers less the opinions too
far. This reflects a typical crowd-seeking behavior which was considered in
its non local variation in [37] and justified by a kinetic framework in [40].
Therefore, we consider a fixed probability distribution function $g(x)$ (for
example, $g$ may be the density of a normally distributed random variable) and
we define, for any fixed $t\in[0,T]$ the local average opinion as

$\bar{m}(t,x):=\mathbb{E}\left[\left(ym(t,\cdot)\ast
g(\cdot)\right)(t,x)\right]=C(t,x)\int_{\Omega}ym(t,y)g(y-x)dy,$

where $C(t,x)$ is a normalization variable, i.e.
$C(t,x)=1/\int_{\Omega}m(t,y)g(y-x)dy$. We model the running cost terms in
(2.2), from one side to penalize the distance from the average opinion of a
surrounding, determined by the shape of the distribution $g$. Thus,

$F(t,x,m)=a_{1}\left|\bar{m}(t,x)-x\right|.$ (4.1)

At the same time, we imagine that there exist a collection of points
$\\{\bar{x}_{i}\\}_{i=1,...,N}$ which are some special opinions where some
advertisement is spent, therefore, they are some special points of attraction,
whose strength depends on the parameters $k_{i}$. Therefore,

$\ell(x,\alpha)=\frac{1}{2}\left[a_{2}\alpha^{2}+a_{3}\min_{i=1,...,N}k_{i}\left(\bar{x}_{i}-x\right)^{2}\right].$
(4.2)

We notice that here, we penalized also the control, with the meaning that the
agents are stubborn and they want to change their opinion as little as it is
possible. The parameters $a_{1},a_{2},a_{3}\geq 0$, balance the influence of
these various principles leading the evolution of the agents’ choices.

We finally consider an initial distribution of the opinions $m_{0}(x)$ and an
initial cost $u_{0}(x)$. We may suppose either that $u_{0}(x)\equiv 0$ if the
agents start the game without any prior information about the system or

$u_{0}(x)=\min_{i=1,...,N}k_{i}\left(\left(\bar{x}_{i}-x\right)^{2}\right)$

if we suppose that there have been already some advertisement prior to the
start of the game so, they is already the presence of some polarising
opinions.

The final FF-MFG model that we obtain is, for $(t,x)\in[0,T]\times\mathbb{R}$

$\left\\{\begin{array}[]{ll}u_{t}(t,x)+\max\limits_{\alpha\in\mathbb{R}}\left\\{\alpha
u_{x}(t,x)-\ell(x,\alpha))\right\\}=F(t,x,m)+D(x)\,u_{xx}(t,x),\\\
m_{t}(t,x)-\left(u_{x}(t,x)m(t,x)\right)_{x}=D(x)\,m_{xx}(t,x),\\\
m(0,x)=m_{0}(x),\quad u(0,x)=u_{0}(x)\end{array}\right.$ (4.3)

We observe that using the standard Legendre-Fenchel transform, the control
depending part of the hamiltonian is

$H(x,p)=\max_{\alpha\in\mathbb{R}}\left\\{\alpha
p-a_{2}\alpha^{2}/2)\right\\}=\frac{1}{a_{2}}\,\frac{p^{2}}{2}.$

Therefore the standard hypotheses on $H(x,p)$ are verified.

###### Lemma 4.1

Let $D>0$ and $(m_{0},u_{0})$ satisfying H3 in the setting (C). Then all the
hypotheses H0-H3 are verified and therefore the system (4.3) is well posed.

Proof. The hypothesis H1 is obviously verified. For the hypothesis H2, observe
that for any $t\in[0,T]$

$\left|F(t,x_{1},m_{1})-F(t,x_{2},m_{2})\right|\leq
a_{1}\left|\left|\bar{m}_{1}(t,x_{1})-x_{1}\right|-\left|\bar{m}_{2}(t,x_{2})-x_{2}\right|\right|\leq\\\
a_{1}\left(|x_{1}-x_{2}|+\max{|C_{1}(t,\cdot)|,|C_{2}(t,\cdot)|}\left|\int_{\Omega}y(m_{1}(t,y)g(y-x_{1})-m_{2}(t,y))g(y-x_{2})dy\right|\right)\\\
\leq
a_{1}\left(|x_{1}-x_{2}|+\max{|C_{1}(t,\cdot)|,|C_{2}(t,\cdot)|}\max(\|g\|_{\infty},L_{g})\mathbf{d}_{1}(m_{1},m_{2})\right)\\\
\leq K(|x_{1}-x_{2}|+\mathbf{d}_{1}(m_{1},m_{2}))$ (4.4)

for a $K$ big enough. Therefore we can apply Theorem 3.1 and conclude the
proof.  It is interesting to remark also that the model is much closer to [40]
and to other kinetic based frameworks than to [37], since we suppose that the
agents do not have access to any information about future states of future
changes in the dynamics of the system. Nonetheless, the model has a steady
state formulation which assumes the typical stationary MFG form, so we can
guess that after a certain time the solutions $(u,m)$ will converge to
$(\bar{u},\bar{m})$, i.e. solutions of the system

$\left\\{\begin{array}[]{ll}\lambda+\max\limits_{\alpha\in\mathbb{R}}\left\\{\alpha\bar{u}_{x}(x)-\ell(x,\alpha))\right\\}=F(x,\bar{m})+D\,u_{xx}(x),\\\
Dm_{xx}(x)+\left(\bar{u}_{x}(x)\bar{m}(x)\right)_{x}=0,\end{array}\right.$
(4.5)

where $\lambda\in\mathbb{R}$ is a constant of compatibility between the two
systems to be defined. Proving the long time convergence of this class of
problems is still an open issue (cf. [1]).

###  4.2 FF-Opinion formation: Tests

In this section, we show some of the features of the model proposed. The main
goal of the following tests is to underline the differences from the
correspondent forward-backward system and to illustrate the qualitative
characteristics of the model. The simulations shown are obtained using semi-
lagrangian schemes (see A and the references within for details).

#### Test 1. A Forward-Backward/Forward-Forward comparison.

The purpose of our first test is to compare our opinion formation model to its
forward-backward respective. We underline that in the latter case, the agents
forecast all the future states of the system optimizing their policy on them.
In our case, instead, every agent decides its policy depending on the past
states of the system. The forward-backward coupling presents some
computational additional difficulties: due to the nature of its evolution, we
cannot use an explicit numerical scheme on both the equations. We solve the
issue as in [8], using a fixed point argument till convergence to equilibrium
of the discrete system.

We fix the opinion domain as $\Omega=[-4,4]$, the initial value of the
potential and density function as

$u_{0}(x)=a_{3}\min_{i=1,2}k_{i}\left(\left(\bar{x}_{i}-x\right)^{2}\right)$

with $(x_{1},k_{1})=(0.8,1)$ and $(x_{2},k_{2})=(0.2,3)$,

$m_{0}(x)=\left\\{\begin{array}[]{ll}1&\hbox{ if }x\in[0,1]\\\
0&\hbox{otherwise}\end{array}\right.,$

In other words, we assume that, initially, the opinion is uniformly
distributed, but the agents are aware of two polarizing opinions
$\\{0.2,0.8\\}$ for which, for example, some prior advertisement have been
spent. In particular, the attraction capacity of the pole $x_{2}=0.2$ is
stronger ($k_{2}=3$), but less spread around the opinion domain. Vice versa,
the pole $x_{1}=0.8$ has a weaker attraction capacity ($k_{1}=1$) but a larger
attraction base. The impact of this initial polarization is tuned by the
parameter $a_{1}$ which is set $a_{1}=1$. The same attraction factor appears
in the running cost $\ell(x,\alpha)$. In this case the strength of this factor
is tuned by the weight $a_{3}=2$ while the penalization of the control, i.e.
the resistence to change an opinion is set to $a_{2}=1$.

The probability distribution $g(x)$ is set to
$g(x)=\frac{1}{\sqrt{0.2\pi}}e^{-{\frac{x^{2}}{0.2}}}$ modeling the influence
on every agent of its surrounding. The weight of this tendency to conform to
the local mean is tuned by $a_{1}=1$.

Forward-Forward Model
---
Figure 1: Test 1. Forward-Forward model: evolution of the potential function
$u$ and density distribution $m$ (top), control map $-\nabla u$ and mean
opinion $\/|\Omega|\int_{\Omega}m(t,x)dx$.

Finally, the diffusion is set to $D(x)\equiv 0.01$. In Figure 1, we show the
results of this first test implemented with discretization parameters $\Delta
x=0.05$ and $\Delta t=0.02$. We can observe as the density distribution of the
agents, initially constant in $[0,1]$ and null elsewhere, splits in two
clusters, under the influence of the initial choice of $u_{0}$. During this
first phase, the right hand side cluster acquires progressively a bigger
attraction capacity and tends to absorb the second cluster. Around $t=10$ this
process ends: only one cluster is present although it is not centred in the
advertised pole $x_{1}$. The latter happens during the last phase of the
evolution of the system. The cluster slowly moves toward $x_{1}$ that will
become the center of its distribution. Clearly, there is no formation of
Dirac’s deltas due to the presence of a strictly positive diffusion parameter
$D$. In Figure 1 are also displayed the evolution of the control map
determined by the potential function $u(t,x)$ and the mean opinion of the
agents, i.e. $1/|\Omega|\int_{\Omega}m(t,x)dx$. We can observe as the crowd
reaches consensus around the pole $x_{1}$.

Forward-Backward Model
---
Figure 2: Test 1. Forward-Backward model: evolution of the potential function
$u$ and density distribution $m$ (top), control map $-\nabla u$ and mean
opinion $\/|\Omega|\int_{\Omega}m(t,x)dx$.

Let us compare the latter, to the forward-backward respective model. We keep
all the parameters of the system untouched, only inverting the evolution of
the potential equation. The results are shown in Figure 2. We observe as the
evolution of the system is very different. First of all, there is not the
formation of two competing clusters since the agents are aware of all the
future states of the system, therefore only one attraction point emerges. In
this case the ’winner’ of this competition is the pole $x_{2}$ and the average
opinion of the agents monotonically aligns to this final state. A more evident
comparison is reported also in Figure 3 where the evolution of the density of
opinions are shown side by side.

|
---|---
Forward-Forward Model | Forward-Backward Model
Figure 3: Test 1. Evolution of the density of opinion in the FF and FB model.

#### Test 2. Cluster formation and control.

In our second test, we want to show two distinct features of our opinion
model. In the first one, we show how our model can produce some clusters
sufficiently stable in time. To do so, we consider the same setting of the
previous test, with the only exception of the parameters $a_{1}=1$, $a_{2}=4$
and $a_{3}=1$. In practice, we penalize the possibility to change opinion,
facilitating the ’stubbornness’ of our density of agents. The two attraction
poles are again $0.2$ and $0.8$ with equal attraction capacities, i.e.
$(x_{1},k_{1})=(0.8,1)$, $(x_{2},k_{2})=(0.2,1)$.

Forward-Forward Model
---
Figure 4: Test 2. Forward-Forward model: evolution of the potential function
$u$ and density distribution $m$ (top), control map $-\nabla u$ and mean
opinion $\/|\Omega|\int_{\Omega}m(t,x)dx$.

The results of the test are displayed in Figure 4, where we can observe a
symmetric behaviour around the two attraction points, where the density of
agents, initially uniformly distributed, aggregates around the two advertised
poles. Due to the clear symmetry of the problem, the average opinion of all
the agents is stable, unless some numerical oscillations, on the middle point
between the two clusters $x=0.5$. We wonder which is, in a quite stable
situation like the latter, the best advertisement choice for one of the poles.
In particular considering a limited possibility of advertisement, is it better
to be the first one to advertise one pole or being able to do it for a longer
amount of time? We answer to this question performing the same test as before
setting

$\displaystyle\hbox{for
}t\in[0,0.5],\quad(x_{1},k_{1})=(0.8,3),\quad(x_{2},k_{2})=(0.2,1);$
$\displaystyle\hbox{then
}t\in(0.5,10],\quad(x_{1},k_{1})=(0.8,1),\quad(x_{2},k_{2})=(0.2,3).$

All the other parameters of the test are left as the previous case. Here, the
agent interested of advertising position $0.8$ acts _before_ and _for a
shorter amount of time_ , while its competitor has a much longer advertisement
span but it is less responsive at the beginning of the simulation. In
particular the advertisement parameters $k_{1}$, $k_{2}$ switch their roles at
$t=0.5$ which is $5\%$ of the total evolution of the system.

Forward-Backward Model
---
Figure 5: Test 2. Controlled FF model: evolution of the potential function $u$ and density distribution $m$ (top), control map $-\nabla u$ and mean opinion $\/|\Omega|\int_{\Omega}m(t,x)dx$. |
---|---
Forward-Forward Model | Controlled FF Model
Figure 6: Test 2. Evolution of the density of opinion in the uncontrolled and
controlled model

The results of this simulation are shown in Figure 5 and 6. We can observe how
in the long run, the position advertised with more anticipation, $0.8$ becomes
predominant, even if at $t=0.5$, when its advertisement advantage ends, it is
actually in a minority position, if we look at the average opinion of the
agents. Therefore we can claim, at least experimentally, that in our model is
much more relevant the advertisement in the first evolution moments of the
system, when essentially the first split in clusters is produced. This first
moments produce some non-trivial long time effects of polarisation as it has
been already observed in literature [27].

###  4.3 A dynamic 2D vote model

In this section, we adapt the previously introduced opinion formation model,
to a 2D political spectrum, often used to give a more detailed ideological
collocation of the voters. A remarkable example of this kind of 2D
representations is the Nolan Chart [38], where the axes are personal
freedom/economical freedom.

We assume that each voter assumes a position
$X(t)\in\Omega\subset\mathbb{R}^{2}$ as solution of the SDE dynamics (2.1)
controlled by the variable $\alpha$. Once again, the density function
$m:\Omega\times[0,T]\rightarrow\mathbb{R}$ is the percentage of the voters in
state $x$ at time $t$, which satisfies

$\int_{\Omega}m(t,x)dx=1,\hbox{ for any }t\in[0,T].$

Adopting the same principles of the previous 1D model, we states that every
agent adjusts his vote opinion based on some average of its surrounding:
therefore, we introduce a fixed probability distribution function $g(x)$ and
we define, for any fixed $t\in[0,T]$ the local average as

$\bar{m}(t,x):=\mathbb{E}\left[\left(ym(t,\cdot)\ast
g(\cdot)\right)(t,x)\right]=C(t,x)\int_{\Omega}ym(t,y)g(y-x)dy,$

where $C(t,x)$ is a normalization variable, i.e.
$C(t,x)=1/\int_{\Omega}m(t,y)g(y-x)dy$.

The probability distribution $g(x)$ is chosen as the isotropic multivariate
normal distribution
$g(x)=g(x1,x2)=\frac{1}{\sqrt{\mu\pi}}e^{-{\frac{x1^{2}+x2^{2}}{\mu}}}$. More
complex choices for modeling the influence on any voter of its surrounding are
possible.

The other terms in (2.2) are

$F(t,x,m)=a_{1}\left|\bar{m}(t,x)-x\right|.$ (4.6)

(which penalizes the distance from the average opinion of a surrounding) and,
assuming that there exist a collection of points
$\\{\bar{x}_{i}\\}_{i=1,...,N}\in(\mathbb{R}^{2})^{n}$ of ”advertised
political opinions” or policy-seeking behaviour of the various parties of
strength $k_{i}$,

$\ell(x,\alpha)=\frac{1}{2}\left[a_{2}\alpha^{2}+a_{3}\min_{i=1,...,N}k_{i}\|\bar{x}_{i}-x\|^{2}\right].$
(4.7)

The parameters $a_{1},a_{2},a_{3}\geq 0$ models, respectively _conformism_ (to
its political environment), _stubborness_ and _advertisement-influenced
behavior_ of each voter. As initial scenarios we consider an uniform initial
distribution of the opinions $m_{0}(x)$ and a constant initial cost
$v_{0}(x)$, i.e.

$m_{0}(x)=\frac{1}{4}\chi_{[-1,1]^{2}}(x),\quad u_{0}(x)=0.$

where $\chi_{A}(x)$, is the usual indicator function of the set $A$. Setting a
constant diffusion parameter $D(x)\equiv D$, we get the following FF-MFG
model, for $(t,x)\in[0,T]\times\Omega$

$\left\\{\begin{array}[]{ll}u_{t}(t,x)+\max\limits_{\alpha\in\mathbb{R}^{2}}\left\\{\alpha\cdot\nabla
u(t,x)-\ell(x,\alpha))\right\\}=F[m(t,\cdot)]+D\,\Delta u(t,x),\\\
m_{t}(t,x)-\operatorname{div}\left(\nabla u(t,x)m(t,x)\right)=D\,\Delta
m(t,x),\\\ m(0,x)=m_{0}(x),\quad u(0,x)=u_{0}(x)\end{array}\right.$ (4.8)

Also in this case, the standard hypotheses on $H(x,p)$ are verified.

###  4.4 FF-dynamic vote model: Tests

We show some simulations obtained using the same schemes presented in A in a 2
candidates, 2D opinion of voting setting. The main purpose of these tests is
to show the effectiveness of the model proposed to simulate some complex
phenomena as cluster formation or the adoption of some classic strategies to
improve the electoral performances of one candidate.

#### Test 3. A stable cluster formation between two candidates.

We show, in this 2D setting, a basic cluster formation, where a polarization
of the voters between two candidates is formed. We need to underline that our
model, since it is based on the presence of attraction poles, tends naturally
to the formation of clusters in a long run. In this test and in the following
one, the model parameters $a_{1}$, $a_{2}$ and $a_{3}$ are all set to one. The
constant $\mu$, modelling the size of the area of the region related to the
conformism tendency of a voter is set to 0.2, while the diffusion parameter
$D=0.01$. We suppose the presence of two candidates $C1$ and $C2$,
respectively on the positions $(0.8,0.8)$ and $(-0.8,-0.8)$. We suppose that
the two candidates perform a political campaign with a constant strength and
efficiency, setting $k_{1}=k_{2}=1$.

---
Figure 7: Test 3. Evolution of the density of voters at
$t=0,0.38,0.78,1.18,1.66,10$. The green squares are the position of the two
candidates, the red cross is the median voter position and the dotted white
line is the separation between the victory region of each candidate.

Since the voters are initially equally distributed on all the political
spectrum $[-1,1]^{2}$ and the potential $v_{0}$ is initially constant, no
previous knowledge of the candidates position is assumed. We recall that
following the median voter theorem [35] victory of one candidate on its
opponent is archived when the average of all the voters position (i.e. the
median voter) falls on the respective victory region, in this case,
respectively

$\Omega_{1}:=\\{x=(x1,x2)\in\mathbb{R}^{2}\hbox{ s.t. }x2>-x1\\}$ $\hbox{ and
}\Omega_{2}:=\\{x=(x1,x2)\in\mathbb{R}^{2}\hbox{ s.t. }x2<-x1\\}.$

We approximate the solution using semi-Lagrangian schemes with $\Delta x=0.04$
and $\Delta t=0.02$. In Figure 7 we report the outcome. We may observe some
interesting features of the model: first of all, even starting from no prior
knowledge about the candidates’ position and with no previous advertisement of
any position, still the first moments of evolution of the system ($t\in(0,1)$)
play a major role in the final results. We can see as immediately the density
of voters splits, due to the symmetry of the data, equally between the two
candidates. We can also notice as the conformism tendency creates various
clusters that progressively migrates toward the advertised positions of the
candidates. During such a migration, multiple clusters may fuse together.
Clearly, in this scenario we suppose an infinite advertisement strength from
each of the two candidates (in particular constant in time), therefore, the
distribution of the density of voters for $t\gg 0$, cannot be different from
one (or more) clusters on the advertised positions of the candidates. In this
case, there is a perfect balance between the two candidates and as it is clear
looking to the position of the median voter (the red cross in Fig. 7); no
candidate prevails on the opponent.

#### Test 4. Overcoming the impasse via different political strategies.

We ask to ourselves if it is possible to emulate some typical electoral
strategies in our model to overcome the impasse caused by the symmetric
setting of the previous test. One possible strategy is to get some electoral
alley. How does the presence of a third candidate $C3$, of position
$(0.6,-0.2)$, change the evolution of the system? We suppose, for this third
candidate, a very limited attraction strength setting $k_{3}=0.2$.

---
Figure 8: Test 4.a. ’Having a moderate ally’ strategy. Evolution of the
density of voters at $t=0,0.38,0.78,1.18,3.32,20$. The green squares are the
position of the three candidates, the red cross is the median voter position
and the dotted white line is the separation between the victory region of each
major candidate.

We display the results in Figure 8. We can notice how the presence of the
additional candidate $C3$ plays a crucial role in the evolution of the crowd
of voters. In fact, it facilitates the creation of a big central cluster of
voters which is slightly more in the victory region $\Omega_{1}$, region where
both $C1$ and $C3$ are placed. This big central cluster, despite initially it
does not affect consistently the position of the median voter, will attract
even the voters initially aggregating toward $C2$. After this, the cluster
will slowly move passing before on the candidate $C3$ position, then reaching
in a very long time horizon ($t=20$) the more extreme position of the
candidate $C1$.

---
Figure 9: Test 4.b. ’Advertising a more moderate position’ strategy. Evolution
of the density of voters at $t=0,0.38,0.78,1.18,1.66,10$. The green squares
are the advertised position of the two candidates (we note that one candidate
changes progressively its position), the red cross is the median voter
position and the dotted white line is the separation between the victory
region of each candidate.

We finally explore another possible strategy: what if a candidate advertises a
position which is not its final goal? In this intentionally expositive
example, we suppose that the position of the candidate $C1$ initially lays on
the point $(0.2,0.2)$ but switches to $(0.4,0.4)$ at $t=2$, to $(0.6,0.6)$ at
$t=4$ and finally to $(0.8,0.8)$ (which is the final goal of the candidate) at
$t=6$. We see in Figure 9 how this strategy immediately affects the median
voter position, effectively pushing the crowd of voters to move consistently
toward the final goal of $C1$. This happens even with a minor cluster
formation before reaching the final position of the candidate $C1$. The
tendency of politicians to gravitate toward the position occupied by the
median voter, or more generally toward the position favored by the electoral
system is sometimes called Hotelling’s law [10]. This test is also in some
sense viewed as a continuous version of the McKelvey–Schofield theorem [29],
where the authors observe that any position of a multidimensional political
spectrum can be reached by a sequence of majority votes.

##  5 Conclusions

In this work, we discuss the basic features of a general FF-MFG, showing how
such a framework can be used to create mathematical models in the case where a
large population of rational individuals are driven by their past strategies
and choices. This kind of framework is particularly appropriate in the case of
social mathematical models, being able to include non-local interaction terms,
diffusion effects, complex strategical behaviours, stubbornness of the agents
and so on.

##  Appendix A A semi-lagrangian scheme for FF-MFG

In this appendix we provide more details about the numerical schemes that we
used for approximating the FF-MFG system. To simplify a bit the presentation
we consider the case where $\Omega\subset\mathbb{R}^{2}$, $D(x)$ is a
constant, i.e. $D(x)\equiv\varepsilon$ and the Hamiltonian has the form (2.3).
The general case is straightforward and does not change the treatment of the
non-linear part of the equations.

Our numerical approach relies on the relation between the HJ framework and the
corresponding adjoint FP equation. Given a semi-discrete (discrete in space)
numerical scheme for the HJ part of (3.1), the same scheme can be used to
construct a consistent approximation for the FP equation.

Before proceeding, we define additional notation. Let $\Omega_{\Delta x}$ be
an uniform grid on $\Omega$ with constant discretization parameter $\Delta
x>0$. Let $x_{i,j}$ denote a generic point in $\Omega_{\Delta x}$. The space
of grid functions defined on $\Omega_{\Delta x}$ is denoted by
$\mathcal{G}(\Omega_{\Delta x})$, and the functions $U$ and $M$ in
$\mathcal{G}(\Omega_{\Delta x})$ (approximations of respectively $u$ and $m$)
are called $U_{i,j}$ and $M_{i,j}$, when evaluated at $x_{i,j}$.

We utilize a semi-discrete numerical scheme $N(x,p):\Omega_{\Delta
x}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ monotone and consistent to
approximate the operator $H(x,p)$, such that $U$ is the solution of the ODE

$U_{t}=N(x,\mathcal{D}U),$ (A.1)

where $\mathcal{D}U$ is a discretization of the gradient operator on $U$.
Thanks to the adjoint structure, we modify this scheme to approximate the FP
part. The discrete approximation, $M$, is the solution of the following system
of ODE

$M_{t}=K(x,\mathcal{D}U,M),$

where

$K(x,\mathcal{D}U,M):=(D_{p}N(x,\mathcal{D}U))^{T}M-\varepsilon\Delta_{d}M.$
(A.2)

Here, the nonlinear part of the operator corresponds to the discrete operator
$D_{p}N(x,\mathcal{D}U)$; $\Delta_{d}M$ is a discretization of the Laplacian.
We note that this operator depends on the monotone approximation scheme used
to discretize the HJ equation, and can be computed numerically or using a
symbolic differentiation operator. We stress that the features of positivity
and mass conservation are still holding also at the discrete level.

#### Semi-Lagrangian scheme.

To describe a semi-Lagrangian scheme appropriate to approximate (3.1), we
introduce the operator

$\mathcal{D}^{\alpha}U_{i,j}:=\max_{\alpha\in\mathbb{R}^{2}}\frac{\mathcal{I}[U](x_{i,j},\alpha)-U(x_{i,j})}{h}-\ell(x_{i,j},\alpha),$
(A.3)

where $h$ a parameter of the same order of $\sqrt{\Delta x}$ and

$\mathcal{I}[U](x_{i,j},\alpha)=\frac{1}{2}\sum_{i=1}^{2}\left(\mathbb{I}[U](x_{i,j}+\alpha
h+e_{i}\sqrt{2\varepsilon h})\right.\\\ \left.+\mathbb{I}[U](x_{i,j}+\alpha
h-e_{i}\sqrt{2\varepsilon h})\right).$ (A.4)

Here, $\mathbb{I}[u](x)$ is an interpolation operator on the matrix $U$, and
$e_{i}$ is the $i$ unitary vector of an orthonormal basis of the space. The
discrete operator is then simply

$N(x,\mathcal{D}U^{\alpha}_{i,j})=F(t,x_{i,j},M)+(\mathcal{D}^{\alpha}U_{i,j})^{\alpha}.$

We take the adjoint of the linearized of $N$ and we use it into (A.2),

$K(x_{i,j},(\mathcal{D}^{\alpha}U]_{i,j},M_{i,j})=\\\ \frac{1}{\Delta
x}\left(M_{i,j}\frac{\partial N}{\partial
p_{1}}(x_{i,j},[\mathcal{D}^{\alpha}U]_{i,j})-M_{i-1,j}\frac{\partial
N}{\partial p_{1}}(x_{i-1,j},[\mathcal{D}^{\alpha}U]_{i-1,j})\right.\\\
+M_{i+1,j}\frac{\partial N}{\partial
p_{2}}(x_{i+1,j},[\mathcal{D}^{\alpha}U]_{i+1,j})-M_{i,j}\frac{\partial
N}{\partial p_{2}}(x_{i,j},[\mathcal{D}^{\alpha}U]_{i,j})\\\
+M_{i,j}\frac{\partial N}{\partial
p_{3}}(x_{i,j},[\mathcal{D}^{\alpha}U]_{i,j})-M_{i,j-1}\frac{\partial
N}{\partial p_{3}}(x_{i,j-1},[\mathcal{D}^{\alpha}U]_{i,j-1})\\\
\left.+M_{i,j+1}\frac{\partial N}{\partial
p_{4}}(x_{i,j+1},[\mathcal{D}^{\alpha}U]_{i,j+1})-M_{i,j}\frac{\partial
N}{\partial p_{4}}(x_{i,j},[\mathcal{D}^{\alpha}U]_{i,j})\right).$

We note that the operator $N(x,p)$ is for contruction monotone. For more
details cf. [13].

## Acknowledgments

AF was partially supported by PRIN 2020 project “Integrated Mathematical
Approaches to Socio-Epidemiological Dynamics” (No. 2020JLWP23, CUP:
E15F21005420006).

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