MD Kévin Premat
email: kevin.premat@aphp.fr

MD Carole Azuar

MD, PhD Damien Galanaud

MD, MSc Alice Jacquens

MD, PhD Didier Dormont

MD, PhD Vincent Degos

MD, PhD Frédéric Clarençon

Acom Study Group

Eimad Shotar

Stéphanie Lenck

Nader Sourour

Aurélie Funkiewiez

Vincent Perlbarg

Grégory Torkomian

Pathomechanisms Behind Cognitive Disorders Following Ruptured Anterior Communicating Aneurysms: A Diffusion Tensor Imaging Study Mis en forme : Centré

Keywords: Subarachnoid Hemorrhage, MRI, Cognition, Aneurysm Highlights

 Cognitive disorders are underestimated in the aftermath of an aneurysmal rupture of the anterior communicating complex  When morphological MR sequences are normal, diffusion tensor imaging might still show signs of microscopic injuries  Axonal injuries seem to be diffuse, but appear more severe in frontal areas, close to the site of rupture Declarations

ABSTRACT Introduction

After the rupture of anterior communicating aneurysms, most patients experience debilitating cognitive disorders; and sometimes even without showing morphological anomaly on MRI examinations. Diffusion Tensor Imaging (DTI) may help understanding the pathomechanisms leading to such disorders in this subset of patients.

Methods

After independent assessment, we constituted a population of patients with normal morphological imaging (ACOM group). Then, a case-control study comparing volumetric and voxel-based DTI parameters between the ACOM group and a control population was performed. All patients underwent the full imaging and neuropsychological assessments at 6 months after the aneurysm rupture. Results were considered significant when p<2.02. 10 -4 .

Results

Twelve patients were included in the ACOM group: 75% had at least one disabled cognitive domain. Significant differences in DTI parameters of global white matter were noted (average Fractional Anisotropy: 0.915 [±0.05] in ACOM group versus 0.943 (±0.03); p=1. 10 -5 ) and in frontal white matter tracts (superior fronto-occipital fasciculus and anterior parts of the corona radiata) as well as in the fornix.

Conclusion

Cognitive disorders are under-estimated, and DTI confirmed that, even when conventional MRI examinations were normal, there were still signs of diffuse neuronal injuries that seemed to dominate in frontal areas, close to the site of rupture.

INTRODUCTION

Aneurysmal subarachnoid hemorrhage (aSAH) is the most frequent cause of spontaneous SAH and therefore, its prognosis is extremely unfavorable [START_REF] Connolly | Guidelines for the management of aneurysmal subarachnoid hemorrhage: a guideline for healthcare professionals from the American Heart Association/american Stroke Association[END_REF]. In the following months after initial management, two types of patients can be identified: patients who had a severe aneurysm rupture that required heavy care and prolonged hospital stays, therefore more willingly exposed to complications; and who subsequently evolve towards severe disability; and patients for whom the initial presentation and evolution were favorable, and could be discharged free of any objective neurological symptoms. Within this latter group, we commonly noticed during the follow-up, that, although they feature no morphological anomaly on conventional brain Magnetic Resonance Imaging (MRI), they frequently present with neuropsychological and neuropsychiatric disorders that may have disastrous consequences on their quality of life [START_REF] Wong | Cognitive domain deficits in patients with aneurysmal subarachnoid haemorrhage at 1 year[END_REF][START_REF] Al-Khindi | Cognitive and Functional Outcome After Aneurysmal Subarachnoid Hemorrhage[END_REF][START_REF] Wong | Evaluation of cognitive impairment by the Montreal Cognitive Assessment in patients with aneurysmal subarachnoid haemorrhage: prevalence, risk factors and correlations with 3 month outcomes[END_REF].

It was particularly noted in series of aneurysm rupture of the anterior communicating complex, possibly related to the immediate vicinity to the frontal lobes, the corpus callosum as well as the frontal part of the circuit of Papez [START_REF] Hong | Injuries of the cingulum and fornix after rupture of an anterior communicating artery aneurysm: a diffusion tensor tractography study[END_REF][START_REF] Martinaud | Anatomy of executive deficit following ruptured anterior communicating artery aneurysm[END_REF].

In this domain, Diffusion Tensor Imaging (DTI), could be valuable to explore such disorders that cannot be explained on basic morphological MRI sequences. Accurately detecting the underlying lesions by means of modern imaging technique could be of great use for such patients, as swift management has been proven to promote functional recovery and quality of life [START_REF] Powell | Psychosocial outcomes at 18 months after good neurological recovery from aneurysmal subarachnoid haemorrhage[END_REF].

The aim of the study was to compare volumetric and DTI parameters in a population of patients with normal morphological MRIs at 6 months after an aneurysm rupture of the anterior communicating complex, to a healthy control population.

METHODS

The manuscript was prepared according to the STROBE guidelines.

ACOM study

The ACOM study is a monocenter, prospective registry that consecutively included patients with ruptured aneurysms of the anterior communicating complex from 2013 to 2015. Main inclusion criteria were: 1) Age ≥18 years 2) modified Rankin Scale 0-2 and Glasgow Outcome Scale Extended 4-5 at 6 months. Six months after the aSAH, patients included underwent an extensive MRI workup with 3D-T1, Fluid Attenuated Inversion Recovery and DTI sequences as well as full neuropsychological and psychiatric assessments. Key exclusion criteria included contraindication to MRI, surgical clipping and inability to undergo neuropsychological assessment for any reason.

Exploration protocol of the ACOM study

Demographics, clinical and radiological data were prospectively collected. Glasgow Coma scale, World Federation of Neurological Surgeons (WFNS) score were evaluated at admission, GOSE and mRS were both performed at 3 and 6 months All patients underwent the same standardized imaging protocol 6 months after the aneurysm rupture. All examinations were performed on the same [START_REF] Al-Khindi | Cognitive and Functional Outcome After Aneurysmal Subarachnoid Hemorrhage[END_REF]

Present study

The protocol for the presented study was divided in two parts:

1) Constitution of the incident population (ACOM group) after independent reviewing by two neuroradiologists (K.P and F.C). Both investigators reviewed the MRIs of all patients included in the ACOM study and were asked to exclude patients featuring one of the following criteria: 1) Cumulated volume of parenchymal FLAIR hyperintensities > 5 ml 2) Major artifacts on DTI sequences.

2) Case-control (1:2) study comparing T1 volumetric and DTI parameters in the ACOM group in comparison to a control group of healthy volunteers. The control population was composed of 24 healthy adult volunteers who all underwent the exact same imaging protocol.

Figure 1. summarizes the flow of patients included in the study

Statistical analysis

Statistical analysis was performed using EPI Info Software (Version 7.2.3.1, Center for Disease Control and Prevention, Atlanta, USA). Inter-observer correlations kappa coefficients with their 95% confidence intervals were estimated using Cohen's test. Results were considered significant when p<0.05.

For comparative analysis, equality of variances was tested using Levene's test. Then, a nonparametric Mann-Whitney U test was performed for comparisons. Adjustment for multi-testing using Bonferroni method was performed and results were considered significant when p<2.02.10 -4 .

Ethical Statement

The protocol of the ACOM study was approved by the local Institutional Review Board (N°040114). All patients included gave written informed consent.

Results

After independent reviewing, inter observer correlation was perfect for patients to exclude from analysis (κ=1

Discussion

In this study, we were able to show that even in the most favorable population of patients with aneurysm rupture of the anterior communicating complex, who did not sustain major macroscopic brain damages, there were still signs of neuronal injuries on DTI imaging. These microscopic injuries seemed to affect the whole white matter but preferentially the frontal axonal bundles, near the site of rupture.

Cognitive disorders following an intracranial aneurysmal rupture are well-documented and were described to affect between 30 and 55% [START_REF] Powell | Psychosocial outcomes at 18 months after good neurological recovery from aneurysmal subarachnoid haemorrhage[END_REF][START_REF] Hillis | Cognitive impairments after surgical repair of ruptured and unruptured aneurysms[END_REF] of patients. In the highly selected population featured in this study, 75% of them had at least one disabled cognitive domain, half of them had depression and 25% suffered from post-traumatic stress disorder. These rates are higher than those previously reported in the body of literature, suggesting that the actual prevalence of cognitive impairment is probably underestimated, hence the importance of a comprehensive neuropsychological assessment to properly identify these patients that are sometimes loosely followed because of the apparent normality of their clinical and imaging evaluations.

After comparative analysis with control subjects, we noted no significant differences in brain volumes, confirming the absence of macroscopic sequalae; whereas the entirety of white matter sustained deep alterations of FA. Previously published data that focused on recovered cardiopulmonary arrests [START_REF] Velly | Use of brain diffusion tensor imaging for the prediction of long-term neurological outcomes in patients after cardiac arrest: a multicentre, international, prospective, observational, cohort study[END_REF] and traumatic brain injuries [START_REF] Galanaud | Assessment of white matter injury and outcome in severe brain trauma: a prospective multicenter cohort[END_REF] showed that FA could be a powerful marker of clinical outcome, which could also potentially be the case for cognitive disorders after aSAH. In regional analysis, alterations of DTI parameters mostly involved frontal white matter tracts such as the SFOF, the corpus callosum and both ACRs, which is also concordant with recent data from the literature [START_REF] Hong | Injuries of the cingulum and fornix after rupture of an anterior communicating artery aneurysm: a diffusion tensor tractography study[END_REF][START_REF] Fragata | Prediction of clinical outcome in subacute subarachnoid hemorrhage using diffusion tensor imaging[END_REF]. As previously shown in a case-control study including patients with anterior communicating ruptured aneurysms [START_REF] Hong | Injuries of the cingulum and fornix after rupture of an anterior communicating artery aneurysm: a diffusion tensor tractography study[END_REF], we also found alterations of MD in parts of the fornix, which is known for its implication in the emotional and memory pathways.

Two main mechanisms have been proposed to explain brain injuries following aSAH [START_REF] Al-Khindi | Cognitive and Functional Outcome After Aneurysmal Subarachnoid Hemorrhage[END_REF]: 1) Immediate injuries consecutive to jet damages, whose severity decreases with the distance from the site of rupture 2) Indirect diffuse disorders due to secondary insults (intracranial hypertension, vasospasm, inflammation and ischemia) that possibly induce lesions remotely from the aneurysm site. The results extracted from our analysis suggest that both mechanisms may be involved in the genesis of white matter injuries. As previously shown, additional factors such as pituitary dysfunction [START_REF] Khajeh | Pituitary dysfunction after aneurysmal subarachnoid haemorrhage: course and clinical predictors-the HIPS study[END_REF] may also play a role in cognitive and psychiatric disorders following aSAH. This work entails several limitations. First and foremost isThe main limitation of this study is the small volume of patients included, which is a direct consequence of the highly selective inclusion criteria. We were not able to perform exploratory subgroup analyses by lack of statistical power. The heterogeneity of the population is another limit for interpretation, as many confounding factors (SAH grade, delayed cerebral ischemia or length of hospital stay) may have impact on cognitive disturbances. One could argue that by selecting only patient with favorable clinical outcome and normal brain imaging, we could at least manage the extent of heterogeneity. Also, due to the large number of parameters that were extracted and compared, the multiplicity of statistical tests that needed to be performed decreased the validity of the results, even with Bonferroni correction. Nevertheless, we were still confident in our results considering they were reproduced on several parameters and in matching locations. This is also the first study to prospectively include and follow this subgroup of patients with normal conventional MRIs and the only one to confirm that DTI alterations are actually consecutive to microscopic neuronal lesions and not to parenchymal sequelae. Even though both groups can be considered as young patients, one may note that healthy controls were younger than their ACOM counterpart. Aging-related white matter integrity has been previously studied using DTI imaging. Most convincing data showed modifications in white matter DTI parameters with normal aging during adulthood, but these mainly occurred later in life, beyond 60 years old [START_REF] Beck | White matter microstructure across the adult lifespan: A mixed longitudinal and cross-sectional study using advanced diffusion models and brain-age prediction[END_REF]. As most of ACOM patients and controls were less than 60, the impact of normal aging in the results should be minimal.

This study does not allow to conclude whether these damages are specifically consecutive anterior communicating aneurysm SAH or only due to aSAH. Future researches will include other aneurysm locations, which may confirm a distance-related impact of aneurysmal rupture.

Conclusion

Cognitive disorders are very common in the aftermath of anterior communicating complex aSAH. When conventional imaging examinations are normal, these troubles might easily be overlooked despite having terrible consequences on the patients' quality of life; hence the value of a systematic neuropsychological assessment, in order to early detect and optimize their management. DTI confirmed that these patients sustained deep white matter damages, and seemed to associate immediate contiguity damage, mainly affecting the frontal tracts; to diffuse neuronal injuries. Right panel: Fifty-nine-year-old ACOM patient, also showing diffuse decreases in white matter FA. This case also illustrates the antero-posterior gradient of FA modifications.

Figure legend Figure 1 .Figure 12 .

 legend112 Figure legend

Figure 3 .

 3 Figure 3. Additional illustrations of Fractional Anisotropy changes on DTI in two ACOM patients severely impaired on neurocognitive evaluation

Table 1 .

 1 There was no difference in global or regional volumes that reached the level of significance.

	). Twelve patients (ACOM group) were finally included (mean age: 53.2 ± 10.4
	years) and were compared to 24 healthy volunteers (mean age: 37.0 ±11.1 years). Baseline
	characteristics of patients and flow of inclusions in the ACOM group are available in
	Supplementary Filefeatured in 900 [±0.06]
	versus 0.965 [±0.05]; p=2.10 -4 ), both anterior parts of the corona radiata (ACR) (right ACR
	mean FA: 0.873 [±0.10] versus 1.029 [±0.06]; p=1.10 -5 ; left ACR mean FA: 0.848 [±0.04]
	versus 1.000 [±0.07]; p<1.10 -5 ), both superior fronto-occipital fasciculus (SFOF) (left SFOF
	mean FA: 0.847 [±0.11] versus 0.965 [±0.06]; p= 1.10 -4 ; right SFOF mean FA: 0.894 [±0.09]
	versus 0.997 [±0.06]) and finally in the striae terminalis (Fx-ST) of the fornix (right Fx-ST
	mean FA: 0.914 [±0.08] versus 0.990 [±0.05], p= 1.10 -5 ; right Fx-ST mean MD: 1.095 [±0.04]
	versus 0.984 [±0.05]; p=2.10 -4 ; left Fx-ST mean FA: 0.894 [±0.09] versus 0.964 [±0.05], p=
	1.10 -5 ; left Fx-ST mean MD: 1.095 [±0.07] versus 1.024 [±0.06]; p= 2.10 -4 ).

Of over 240 comparisons of MRI parameters (Volumetric and DTI scalar), TTable 12.

summarizes the statistically significant differenonces in DTI parameters. On global white matter analysis, differences were found in: global white matter mean Fractional Anisotropy (FA) was significantly lower in the ACOM group (0.915 [±0.05] versus 0.943 [±0.03]; p=1.10 - 5 ) which was confirmed in hemispherical analysis. On white matter tracts analysis, there were significant differences in the corpus callosum (body of corpus callosum mean FA: 0.

Figures

12 and 3

. illustrates these findings in two several ACOM patients.

Table 1 .

 1 Baseline characteristics of the incident population (ACOM group)

	Parameter
	Age (years ± S.D)	53.2 ±10.4
	WFNS grade (n, %)
	-I	4 (33.3%)
	-II	4 (33.3%)
	-III	3 (25%)
	-IV	1 (8.3%)
	-V	0
	Fisher Score (n, %)
	-I	2 (16.7%)
	-II	1 (8.3%)
	-III	2 (16.7%)
	-IV	7 (58.3%)
	Hydrocephalus at admission (n, %)	2 (16.7%)
	Intra-ventricular haemorrhage (n, %)	7 (58.3%)
	External ventricular drain (n, %)	7 (58.3%)
	Neuropsychological impairement (n, %)
	-None	3 (25%)
	-Moderate (1 or 2 domains concerned)	6 (50%)
	-Severe (>2 domains)	3 (25%)
	Psychiatric disorder (n, %)
	-Depression	6 (50%)
	-Post-traumatic stress disorder	3 (25%)
	S.D: Standard Deviation; WFNS: World Federation of Neurological Surgeons

Table 12 .

 12 Summary of the main significant differences (p<2.02.10 -4 ) in DTI parameters

	Parameter	ACOM	group	Control group	p-value
		(n=12)		(n=24)
	Global analysis
	Global white matter
	-FA (±S.D.)	0.915 (±0.05)	0.943 (±0.03)	1.10 -5
	Left hemisphere white matter
	-FA (±S.D.)	0.915 (±0.05)	0.947 (±0.03)	2.10 -4
	Right hemisphere white matter
	-FA (±S.D.)	0.915 (±0.06)	0.939 (±0.03)	2.10 -4
	Regional analysis
	Corpus callosum
	 Body
	-FA (±S.D.)	0.900 (±0.06)	0.965 (±0.05)	2.10 -4
	-Lt (±S.D.)	1.141 (±0.10)	1.048 (±0.08)	3.10 -5
	 Splenium
	-FA (±S.D.) -Lt (±S.D.)	0.941 (±0.03) 1.166 (±0.11)	0.950 (±0.03) 1.083 (±0.07)	3.10 -5 2.10 -4
	Corona radiata
	 Anterior part
	o Left
	-FA (±S.D.)	0.848 (±0.04)	1.000 (±0.07)	<1.10 -5
	-MD (±S.D.)	1.113 (±0.08)	1.012 (±0.03)	8.10 -5
	-Lt (±S.D.)	1.193 (±0.11)	1.012 (±0.06)	1.10 -5
	o Right
	-FA (±S.D.)	0.873 (±0.10)	1.029 (±0.06)	1.10 -5
	-MD (±S.D.)	1.100 (±0.07)	1.007 (±0.03)	2.10 -4
	-Lt (±S.D.)	1.163 (±0.11)	0.992 (±0.04)	6.10 -5
	 Posterior part
	o Left
	-MD (±S.D.)	1.095 (±0.05)	1.008 (±0.03)	2.10 -4
	-Lt (±S.D.)	1.116 (±0.08)	1.057 (±0.05)	1.10 -5
	o Right
	-MD (±S.D.)	1.093 (±0.05)	0.997 (±0.04)	<1.10 -5
	-Lt (±S.D.)	1.106 (±0.07)	1.990 (±0.05)	2.10 -5
	Posterior thalamic radiations
	o Left
	-FA (±S.D.)	0.909 (±0.04)	0.940 (±0.04)	1.10 -5
	-MD (±S.D.)	1.074 (±0.05)	1.020 (±0.04)	8.10 -5
	-Lt (±S.D.)	1.124 (±0.09)	1.058 (±0.06)	1.10 -4
	o Right
	-FA (±S.D.)	0.909 (±0.04)	0.937 (±0.05)	1.10 -5
	-MD (±S.D.)	1.075 (±0.05)	1.024 (±0.04)	2.10 -5
	-Lt (±S.D.)	1.145 (±0.09)	1.069 (±0.07)	<1.10 -5
	Internal capsule

Acknowledgements: None

Funding: None

The protocol of the ACOM study was approved by the local Institutional Review Board (N°040114).

Data are available upon reasonable request Code availability : Not applicable

Competing Interest: None


V S Varma

L Salahaldin

S E Elayoubi

Investment games in Crowdfunding platforms -Extended version

Keywords:

We analyze the behavior of investors who have to decide on which projects to support in a crowdfunding platform. We model the behaviour of different sets of funders, as reported by crowdfunding literature, and also the behavior of utility-maximizing funders as a stochastic game. The founders' behaviour is also modeled as a Stackelberg game. Our results are illustrated by a numerical performance analysis which provides several insights into the choice of investment strategies.

I. INTRODUCTION

Crowdfunding allows entrepreneurs who are willing to overcome classical funding channels such as venture capital and credit loans to address directly the crowd via Internetbased Crowdfunding platforms (CFP) for collecting financial resources either in the form of donation or in exchange for the future product or some form of reward [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF].

The literature on crowdfunding is already rich and covers wide aspects, from the profile of funders and their behavior to the characteristics of projects that make them more successful. [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF] compares two forms of CF, reward-based and equity-based and shows that the choice made by the entrepreneur to select between these two forms depends essentially on the amount of required capital and that equitybased CF is more suitable for large projects. [START_REF] Parker | Crowdfunding, cascades and informed investors[END_REF] models investors choices when facing asymmetric information about the projects quality (good or bad) and showed that higher proportions of informed investors do not always lead to more good projects being funded, while higher proportion of bad projects can paradoxically increase the number of good projects that end up funded. Other literature works give an overview on the key factors that contribute to the success of the project. For instance, [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] shows that the size of the network of the project holder is positively correlated with the success and that the quality signals are also success factors. These signals have been identified as the advertisement videos, the update frequency and even the absence of typos, for reward-based crowdfunding [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF], [START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF], and the clarity of the financial roadmap, the transparency with respect to the risks, the professional experience and the education level of the funders for equity-based crowdfunding [START_REF] Ahlers | Signaling in equity crowdfunding[END_REF].

Most of the crowdfunding literature is based on empirical observation from data collected on different crowdfunding {vineeth.satheeskumar-varma}@univ-lorraine.fr.

L. Salahaldin, is with ESCE International business school, Paris, France , {linda.salahaldin}@esce.fr.

S.E Elayoubi is with the Laboratoire des Signaux et Systemes (L2S, CNRS-CentraleSupelec-Univ. Paris Sud), Gif-sur-Yvette, France. platform, and these observations lead sometimes to contradictory conclusions. For example, concerning the impact of the duration of the funding campaign, [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] shows that a longer duration of the campaign reduces the probability of success, while [START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF] observes a positive correlation between duration and success.

This illustrates the need for a theoretical model that is able to give insights on the impact of different parameters on the funders' choices and our aim in this paper is to fill this gap by developing a theoretical model for the choice of funders among several projects of interest. In order to be realistic, we base our model on the empirical observations about funders' behavior that have been reported in several papers. For instance [START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF] observes that CF has reinforcement characteristic in the sense that contributions generate additional contributions.

[6] studies the choice of donors toward funding a project early or wait until the end of the funding window and shows that if the donor decides to fund immediately, he can signal to the crowd the potential of the project; he can also decide to wait in order to see whether other donors fund the project. [START_REF] Kuppuswamy | Crowdfunding creative ideas: The dynamics of project backers[END_REF] show that donors mostly fund projects at the beginning or the end of a campaign. This tendency can be explained by the observations of other empirical works such as [START_REF] Agrawal | The geography of crowdfunding[END_REF] who have identified friend and family (F&F) as a source of early funding for projects that account for 15 to 20% of raised funds for equity-based campaigns and 30 to 40% for reward-based campaigns. Other potential funders tend to procrastinate and the late funding peak can be explained by the well-known deadline effect [START_REF] Ariely | Buying, bidding, playing, or competing? value assessment and decision dynamics in online auctions[END_REF] that pushes them to support the project near the campaign end.

The contribution of this work resides in its game theoretical formulation of the funder's strategies that includes the behavioral observations. We conduct the equilibrium analysis of the proposed game, which includes solving the issues of existence, uniqueness, and determination of a Nash equilibrium (NE).

The rest of the paper is organized as follows. Sec. II formulates the problem under study and introduces the main notation and concepts necessary for the model description. The game-theoretic analysis of the resulting model for two projects is presented in Sec. III and closed-form expressions for the Nash Equilibrium are derived. The general case of K projects is analyzed in section IV as a stochastic game and subgame perfect equilibria are derived using dynamic programming. Section V models the behaviour of founders who see their projects threatened by other projects on the platform and try to influence their campaign by injecting some money in it. The paper ends with some concluding remarks.

II. PROBLEM STATEMENT

A. Platform and funders

We consider a crowdfunding platform based on the threshold pledge model (or the all-or-nothing principle(AON)) [START_REF] Cumming | Crowdfunding models: Keep-it-all vs. all-or-nothing[END_REF], meaning that if the project is not 100% funded, the funding will be returned to investors without penalty. This platform proposes a set of projects K = {1, 2, . . . , K} that are of interest to a set of potential investors, henceforth referred to as agents, A = {1, 2, . . . , N }, with N ∈ Z ≥0 . We assume that each investor desires to invest in one project at most. At any discrete-time instant t ∈ Z ≥0 , each project has an accumulated promised investment denoted by x k (t) ∈ R ≥0 for all k ∈ K, which starts the crowdfunding campaign an initialization time t k ∈ Z ≥0 and ends at some T k ∈ Z ≥0 , with t k < T k . Without any loss in generality, we order the projects based on the end times, i.e., we consider

T 1 ≤ T 2 ≤ • • • ≤ T P .
Each project k ∈ K requires a certain threshold of investors τ k ∈ Z ≥0 in order to succeed by the end time, i.e. project k succeeds if and only if x k (T k ) ≥ τ k . Additionally, each agent makes an investment of δ k while investing in the project. For the ease of reading and without any loss of generality, we normalize δ k 's to 1. When successful, each agent that invested in k has a net gain denoted by R k ∈ R >0 . If an agent invests in project k at time t k ≤ t ≤ T k and the project fails, the agents the amount he invested is returned to him, but the agent still loses due to the discount factor γ < 1.

We use y n (t) ∈ K ∪ {0} to denote the investment choice of agent n ∈ A at time t ∈ R ≥0 . y n (t) = 0 indicates that the agent has not invested in any project at time t and we initialize this variable with y n (0) = 0 for all n ∈ A. At any time t, if y n (t) = 0, the agent is free to change is choice to any k ∈ K such that t ≤ T k . However, once a project is chosen, this choice can not be changed unless the project fails. We can therefore describe the constraints on investment choices as follows

y n (t+1) ∈     {0} ∪ {k ∈ K|T k ≥ t + 1} if y n (t) = 0 {y n (t)} if y n (t) = k ∈ K & t < T k {0} if y n (t) = k ∈ K, t = T k , x k (T k ) < τ k {y n (t)} if y n (t) = k ∈ K & x k (T k ) ≥ τ k
(1) and any investment profile must satisfy [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF].

The population of agents is divided into several classes, as indicated by the empirical literature [START_REF] Lin | How different are crowdfunders? examining archetypes of crowdfunders and their choice of projects[END_REF], denoted by the following sets 1) A set C ⊂ A of casual investors who aim at maximizing their utility and observe the states of funding x k (t), but are not well informed on {R k , k ∈ K}. 2) A set of agents P ⊂ A who comprise the players set, they are aware of all the project parameters and therefore can estimate the net gain R k they generate in case of success. They try to maximize their expected gain based on the current state of investments x(t).

3) Sets F k ⊂ A of Friends and Family (F&F) who have personal interest in specific projects k. While the first two sets of agents make rational decisions in order to maximize their gain, the funders in F k set invest only in project k. They will invest in the project as soon as possible to encourage further investment [START_REF] Kuppuswamy | Crowdfunding creative ideas: The dynamics of project backers[END_REF]. 4) Set N of naive investors, also known in the literature as fools to complete the 3F group, i.e., Friends, Family and Fools [START_REF] Ang | Small business uniqueness and the theory of financial management[END_REF]. They randomly invest in projects at random times.

The presence of the last two sets results in x k (t) > 0 for all k ∈ K and for all t. The latter results in an increment of the project investment at each stage modeled by a Gaussian variable at each step with mean λ and variance σ 2 0 (which can be interpreted as the sum of a large number of Bernoulli variables). Finally, we can write the accumulated investment for any project k ∈ K as

x k (t) ∼ x k (0)+ n∈C∪P 1 {k} (y n (t))+N(λ(t-t k ), σ 2 0 (t-t k ))
(2) where 1 S (y) is the indicator function which takes the value 1 when y ∈ S and 0 otherwise, and N(µ, σ 2 ) denotes the normal distribution with mean µ and variance σ 2 .

The primary objective of this work is to characterize the behavior of the agents in classes C and P using Game theory.

B. Temporal strategy

We denote by u n ∈ R, the net utility or gain of agent n ∈ A.

We first study the behavior of the agents in C. Indeed the casual set agents only need to decide on when to invest in a project and not on which project since they are not aware/not interested in the precise values of R k . We assume that any agent n ∈ C invests in project k with some probability π k , i.e. Pr(y

n (T k ) = k) = π k . If the agent n ∈ C decides to invest in project k at time t ≤ T k , then its net utility is u n (t ) = R k Pr(x k (T k ) ≥ τ k )γ T k -t -δ k (1 -γ T k -t ) Pr(x k (T k ) < τ k ) (3)
with R k > 0 and Pr(x k (T k ) < τ k ) ∈ [0, 1] unknown to the casual agent. However, it is clear that for any value of these parameters, the utility is maximized by picking t = T k as the loss term will vanish due to the discount factor. Therefore, we can predict that the set of casual investors invest in this project at T k :

1 k (y n (t)) = 0, (4)
for all t < T k , n ∈ C. At T k , each casual player decision can be modeled as a Bernoulli variable (invest in project k with probability π k and wait otherwise). We can therefore use the central limit theorem when |C| is large to write

n∈C 1 k (y n (T k )) → d N(π k |C|, σ 2 k |C|) (5)
where → d indicates convergence in distribution and

σ 2 k = (1-π k ) 2 +π 2 k 2 .
The agents n ∈ P are aware of the behavior of casual users, i.e., [START_REF] Ahlers | Signaling in equity crowdfunding[END_REF], and also of the project parameters. They make their decision at t based on x(t). Due to the lack of signaling or communication channel between the players, there is no way that player n ∈ P can observe the specific action of another player m ∈ P . In fact, the players must base their decision solely on the state vector x(t). However, the actions of the players affect the transition probability of x(t), with the actions constrained based on [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF].

In general, the interaction can be seen as a stochastic game if we quantize x(t) and consider a discrete process instead of a Gaussian distribution to model the random investments. From (3) and the condition x k (T k ) ≥ τ k , we can model the state and utility as a stochastic process depending on the investment profile y(t). Indeed, in this case, the existence of a Nash equilibrium (mixed or pure) can be shown as the set of actions and states are both finite [START_REF] Osborne | A Course in Game Theory[END_REF]. However, it is very hard to obtain useful insights in this case due to the complexity of the problem and this case is left as an extension.

Each player at time t will make a decision solely based on x(t) due to the lack of signaling. As a first simplification, using the same reasoning as in (3), we suppose that these players will only invest in project k at T k (if it picks project k) formally stated in the following assumption, inspired by the behavior of the agents in the set C.

Assumption 1: Any player n ∈ P , will invest in a project k only at T k , i.e. y n (t) = k, ∀t < T k .

III. PLAYER BEHAVIOR WHEN K = 2

A. Static game formulation

Under Assumption 1, when K = 2, each player n ∈ P only needs to decide if they invest project 1 or wait for 2, at time T 1 . This problem can be seen as a static oneshot game. There are exactly two investment strategies (or actions) a n ∈ {0, 1} available for n ∈ P , 1 : Invest in project 1 at T 1 . If it succeeds, all future opportunity in project 2 is lost and if it fails, invest in project 2 at T 2 . 0 : Ignore project 1 and invest only in project 2 at T 2 .

Therefore, the strategy of any player for the game in the case of K = 2 is decided once and for all at T 1 because if it invests in 1, a n = 1 and a n = 0 otherwise. This simplifies the stochastic game structure and the interactions can therefore be modeled as a static game. We can write the strategic form of the resulting static game as

G S := (P, {0, 1} |P | , {u n } n∈P ),
where {0, 1} |P | is the set of strategies for all players, and u n (a n , a -n ) is the expected gain for n ∈ P calculated as

u i (a i , a -i |x(T 1 -1)) = R 1 if x 1 (T 1 ) ≥ τ 1 & a i = 1, γ T2-T1 R 2 Pr(x 2 (T 2 ) ≥ τ 2 |x 2 (T 1 -1), a -i ) otherwise. (6)
Here, a -n is a vector of the strategies by all players other than n, i.e., for all i ∈ P \{n}. In the following, we explicitly find the Nash equilibrium when K = 2.

As a first step, we will define the concept of a pure Nash equilibrium (NE). Definition 1: A profile of actions a * is a pure NE if

∀a i ∈ A i , u i (a * i , a * -i ) ≥ u i (a i , a * -i ) (7)
for all i ∈ P .

Before looking at the NE of the game G 1 , let us define the function

Φ(f ) = Erf |C|σ 2 (x 2 (T 1 -1) + f + π 2 |C| -τ 2 ) . (8
) for f ∈ Z ≥0 , where Erf is the Gaussian error function. This function is increasing in f and is sigmoidal. We also define the two thresholds on the values of collected funds by project 2:

x 2 = 1 |C| Erf -1 ( R 1 R 2 γ T1-T2 ) -1 -π 2 |C| + τ 2 (9)
and

x 2 = 1 |C| Erf -1 ( R 1 R 2 γ T1-T2 ) -|P | -π 2 |C| + τ 2 (10)
Theorem 1: The game G S has at least one and at most 2 pure Nash equilibria, specifically,

• If R 2 Φ(1)γ T2-T1 > R 1 , i.e. if x 2 (T 1 -1) > x 2 , the
only pure NE is when a * n = 0 for all n ∈ P , i.e. all follower agents will wait for the second project.

• If R 2 Φ(|P |))γ T2-T1 < R 1 , i.e. if x 2 (T 1 -1) < x 2 , the
only pure NE is when a * n = 1 for all n ∈ P , i.e. all follower agents will invest in the first project at T 1 .

• Otherwise, i.e., if x 2 ≤ x 2 (T 1 -1) ≤ x 2 , there are two pure NE at a * n = 1∀n ∈ F and a * n = 0∀n ∈ F .

Proof: At time T 1 , for a given a, a player i ∈ P will estimate the probability that project 2 will succeed, given project 1 succeeded, as

Pr (x 2 (T 2 ) ≥ τ 2 |x 1 (T 1 ) > τ 1 , x 2 (T 1 -1)) = Erf |C|σ 2 (x 2 (T 1 -1) + |P | -n∈P \{i} a n + π 2 |C| -τ 2 ) = Φ(f ) (11) where f = |P | -n∈F \{i} a n .
Let us call α(a) to denote the probability that project 1 succeeds and β to denote the probability that project 2 will succeed given project 1 failed (with all players investing and therefore independent of a). We can now calculate the expected utility for player i corresponding to actions 0 and 1 if project 1 succeeded as

u i (1, a -i |x 1 (T 1 ) ≥ τ 1 ) = R 1 ( 12
)
and

u i (0, a -i |x 1 (T 1 ) ≥ τ 1 ) =R 2 γ T2-T1 Φ   |P | - n∈P \{i} a -i   (13)
On the other hand, if project 1 failed, both actions will lead to the same payoff as all players will invest in project 2. Therefore, we can calculate the gain by picking action 0 over action 1 as

R 2 γ T2-T1 Φ(f ) -R 1 ,
where f is as before. Since α(0, a -i ) > 0 as the probability of project 1 succeeding is never 0, this gain is positive when R 2 γ T2-T1 φ(f ) ≥ R 1 and negative otherwise. Now, we observe that 1) This gain is always strictly positive if R 2 γ T2-T1 Φ(1) > R 1 as φ(f ) is monotonically increasing. Therefore, the action a i = 0 for all i ∈ P is a pure NE. Additionally, for any player i ∈ P , u i (1, a -i ) < u i (0, a -i ). Therefore, this is also the only pure NE. 2) Using similar arguments, if

R 2 γ T2-T1 Φ(|P |) < R 1 ,
then the action 1 is always dominant implying that a * i = 1 is the only pure NE. 3) In the final case, when

R 2 Φ(0)γ T2-T1 ≤ R 1 ≤ R 2 Φ(|P | -1)
, we can easily verify that both a * i = 0 and a * i = 1 are pure NE as when n∈P \{i} a n = 0 or n∈P \{i} = |P | -1, the strategy of a i = 0 or a i = 1 satisfies [START_REF] Kuppuswamy | Crowdfunding creative ideas: The dynamics of project backers[END_REF]. Moreover, no other pure NE exists. This can be proven by contradiction by assuming that there exists a NE a * such that n∈P \{i} a * i = f , with f ∈ {1, . . . , |P | -2}. In this case, there is at least one player i ∈ P such that a * i = 1 which implies R 2 γ T2-T1 Φ(f ) ≤ R 1 . There also exists at least one player j such that a * j = 0, which implies that R 2 γ T2-T1 Φ(f -1) ≥ R 1 . However Φ(•) is a strictly increasing function, leading to a contradiction. Theorem 1 characterizes the possible NE for the static game G S . Note that in the third case of the theorem, both equilibria of waiting for project 2 or not are possible. In this case, the equilibrium where all players wait for the second project offers a higher expected payoff, but unless the players can cooperate and communicate with each other, there is no way of guaranteeing that this equilibrium will be reached. Player behavior in games with multiple NE are in general unpredictable and depends on the existence of communication channels. In practice, investors who are particularly interested in project 2 might encourage others to join them using public forums or other methods of communication.

B. Numerical illustrations

To illustrate how R 2 and x 2 (T 1 -1) impact the NE profiles, we provide Figure 1 We now turn to the evaluation of the NE for different sets of parameters of project 2, namely the amount of funds to be collected τ 2 and the amount of funds already collected at decision time T 1 -1, while fixing R 2 to 1.5 * R 1 . It can be observed in Figure 2 that the only possible NE when the amount of remaining funds to be collected is low is to wait for project 2. When the requested funds increases with respect to the already collected amount, there are two possible Nash equilibria (wait or invest in 1). When the objective is too ambitious, all users invest in project 1 without waiting. Figure 3 illustrates the impact of the remaining duration and the discount factor on the equilibrium. We can observe that a high γ (a low discount rate) encourages waiting. For a fixed discount factor, a longer remaining campaign duration increases the probability that project 2 succeeds to collect its required amount and pushes agents to wait. This ambiguity about the role of the campaign duration matches empirical observations. For instance, while [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] reports a negative correlation between the campaign duration and its success, [START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF] describes the inverse tendency with a positive correlation. The game theoretical results indicate that the design of the campaign duration is not an easy task, as a too short duration prevents from collecting sufficient funds and a too long one may lead to negative impacts due to funders' impatience.

IV. GENERAL CASE OF K ≥ 2

Under Assumption 1, the general case of K projects can be seen as a stochastic game of K -1 stages. At each stage k ∈ {1, . . . , K -1}, player n ∈ P must decide at T k , if he invests in project k or wait for a future project. This decision is based on the last seen state of the vector of collected funds, i.e. x k (T k -1), x k+1 (T k -1), . . . , x K (T k -1). The action at each stage T k is to invest in project k or to wait for a future project. Indeed, some of these actions may not be possible if a previous project succeeded or the player had invested in an ongoing project. Nevertheless, at each stage, we can denote the action set of a player as being represented by a n (k) ∈ {0, 1} for n ∈ P .

First, we use P k to denote the set of remaining players at stage k with P 1 = P , following which, the set is updated as per

P k+1 = {n ∈ P |a n (k) = 0} if x k (T k ) ≥ τ k P k otherwise (14)
Correspondingly, we can define the utility at stage k as

u n,k (a n (k), a -n (k), P k |x(T k -1)) = R k if x k (T k ) ≥ τ k & a n (k) = 1, γ T k+1 -T k E |x(T k -1) [u n,k+1 (a(k + 1), P k+1 |x(T k+1 -1))] otherwise. ( 15
)
for all n ∈ P and k < K -1. Note that the utility at the following stage depends on P k+1 determined as a function of a -n (k) and P k . At the final stage, we can explicitly include

x K (T K-1 -1) as a parameter of the function Φ:

u n,K-1 (a n (K -1)|x(T K-1 -1)) = R K-1 if x K-1 (T K-1 ) ≥ τ K-1 & a n (K -1) = 1, γ T K -T K-1 R K Φ(|P K |, x K (T K-1 -1) otherwise. ( 16
)
These set of utilities, transition probabilities of x k (t) and actions a n (k) define a stochastic game G. We can define a sub-game G k for G with the utility at stage k and the action taken at that stage once the strategies of future stages are fixed, as this will let us evaluate the expected utility of future stages. This lets us define Definition 2: A NE of G, denoted as a(1) * , . . . , a(K -1) * is said to be sub-game perfect if a * (k) is a NE of the subgame G k for all k ∈ {1, . . . , K -1}.

At the last state, for a given x K (T K-1 -1), the sub-game G K-1 has at least one pure NE as shown in the previous section as the sub-game G K-1 is mathematically equivalent to the static game G S defined previously. We can therefore find a sub-game perfect NE for the stochastic game G using a dynamic programming based approach. This involves taking the expectation over x k+1 (T k+1 ) given x k+1 (T k -1) while calculating the expected utility for k + 1 while at stage k, and assuming that a pure NE is played when at stage k + 1.

Since the NE at stage K -1 is known, we can find the NE at stage K -2 by taking an expectation of future x(T K-1 -1) and assuming the NE is played at K -1 for all of these realizations.

Since the expectation of the future utility u k+1 will be monotonically increasing w.r.t P k+1 , we can use similar arguments as in Theorem 1 to conclude that the pure NE of any subgame G k will be that when all players either invest in the project k or wait. However, this result is not mathematically proven in this version of the paper and is left for future extensions. This is nevertheless shown in the following numerical examples illustrating the sub-game perfect NE. We now take the parameters

λ 0 = 1, |C| = 100, σ 2 0 = 0.2, γ = 0.999, π 2 = π 3 = 20, σ 2 2 = σ 2 3 = 10, τ 2 = τ 3 = 100, R 1 = 1, R 2 = 1.5, R 3 = 2, T k = 10k for all k ∈ {1, 2, 3}.
In Figure 4, we plot the possible NE compared to x 2 (T 1 -1) and x 3 (T 1 -1) when K = 3. These equilibria were calculated using Monte-Carlo simulations for realizing the value of x 3 (T 2 -1) in the future for all possible values of P 2 , which is used to calculate the NE of the sub-game G 2 using Theorem 1. These equilibria and a realization of x 2 (T 2 ) are then plugged in to obtain the future decisions and the utility of not picking project 1. By repeating this evaluation over 10 5 realizations of x 2 (T 2 ) and x 3 (T 2 -1), we are able to compute the pure NE of the sub-game G 1 . For any realization, playing this NE for G 1 and the NE for G 1 constitutes a sub-game perfect equilibrium of G.

V. FOUNDER GAME

In the previous section, the funders in P set were playing a game between in order to choose the project that maximizes their utilities. While the project founders (holders) were considered as passive, they have been reported in the literature as taking some actions that aim at influencing the funders or to change the issue of the funding campaign. In practice, some founders inject some amount of money in their project, especially when it is close to success, using other classical sources of funding. As this practice is forbidden on most of platforms [START_REF] Ahlers | Signaling in equity crowdfunding[END_REF], they may activate their network of funders, or may coordinate amount founders of projects in other platforms in order to exchange support [START_REF] Zvilichovsky | Playing both sides of the market: Success and reciprocity on crowdfunding platforms[END_REF]. Let y max be the maximum amount of funding the project older is able to place urgently. y max is usually limited to a small amount. At T k -1, in order to incite funders to skip project k, founders of project k + 1, ..., K may place an amount of money, limited to y max , in their respective project. This corresponds to a Stackelberg game that we detail in the following for the case K = 2.

Let the expected utility of founder of project k be U k > 0 if he gets its project fully funded on the platform. If the founder has to complement the funding by an amount y, we assume that his utility become U k -y. We also assume the U k -y max > 0. We have the following result:

Theorem 2: The Stackelberg equilibrium for founder of project 2 that corresponds to a subgame perfect equilibrium with respect to subsequent funders actions is the following:

1) if x 2 -y max < x 2 < x 2 , invest an amount y = x 2x 2 (T 1 -1) + 1 2) otherwise, if x 2 -y max < x 2 < x 2 , invest an amount y = x 2 -x 2 (T 1 -1) + 1. 3) if any of the above conditions is not verified, do not invest any amount.

Proof:

We do not provide a complete proof due to lack of space but only some hints. For proving item 1, consider the case where the threshold x 2 is not reached. There is either one NE for funders to invest in project 1 or two NE that consist in funding project 1 or waiting. The expected utility of project 1 founder is thus either equal to 0 or to an unpredictable value. If he invests an amount of money equal to y so that x 2 (T 1 -1) > x 2 and as he makes the first move, funders in P who take their decisions at T 1 will observe an amount of collected funds larger than x 2 and their NE will be to wait for project 2. The utility of project 2 becomes equal to (U k -y)Φ(y + |P |) > 0. Any positive value of y that makes x 2 +y > x 2 increases the utility of the founder compared to a situation where he does not intervene. To complete the proof, we can show that the resulting utility is decreasing with y, for small value of y. Item 2 is proven similarly, having in mind that an unpredictable utility (0 or (U k -y)Φ(y + |P |)) is better than 0.

Note that the founder of project 1 may also invest at T 1 -1 in his project in order to increase its probability of success1 , but this does not change the outcome of the game as the NE does not depend on x 1 .

VI. CONCLUSION

In this paper, we introduced a novel game theoretical framework to model investor behavior in crowdfunding platforms. We considered a series of K projects that are of interest to common audience sets that aim at maximizing their utilities. We modeled the behaviour of well-informed agents using game theory. We defined a stochastic game in the general case and provide explicit expressions for the Nash equilibria in the special case of 2 projects of interest. For more than 2 projects, we define sub-game perfect equilibria and compute them using dynamic programming. We then considered a Stackelberg equilibrium where project founders try to influence funders decisions.

  which marks the regions where there are unique pure NE and the region with two pure NE as stated in the three cases of Theorem 1. The other parameters are held constant with values R 1 = 1, γ = 0.999, T 2 -T 1 = 50, |C| = 100, π 2 = 0.20, σ 2 2 = 10, λ = 1, σ 2 0 = 0.2 and |P | = 50.

Fig. 1 .

 1 Fig. 1. Regions corresponding to the three cases mentioned in Theorem 1 formed by varying R 2 and x 2 (T 1 -1).

Fig. 2 .

 2 Fig. 2. Impact of the parameters of project 2 on the NE.

Fig. 3 .

 3 Fig. 3. Impact of the discount factor and the remaining campaign duration.

Fig. 4 .

 4 Fig. 4. NE profile based on x 2 (T 1 -1) and x 3 (T 1 -1) when K = 3.

it has been reported in[START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] that many projects collect exactly 100% of their requested amounts and this may be due to the founders making up the difference at later stages


V S Varma

L Salahaldin
email: linda.salahaldin@esce.fr.

S E Elayoubi

† L Salahaldin

‡ S E Elayoubi

A real options model for crowdfunding platforms: optimal project promotion strategies

We study in this paper crowdfunding from the platform perspective. We consider a platform where several project campaigns are running and competing on fund collection. The platform manager's objective is to increase the proportion of successful projects and he uses the lever of project promotion on the platform website to influence the funders' and balance the potential future funds between projects. We first consider the platform policy derivation as an optimization problem and show to derive the optimal project promotion policy. We then consider the more realistic setting where the platform manager adapts his policy online when observing the status of fund collection. We model this online strategy as a real option and develop a dynamic programming algorithm that finds the optimal strategy depending on the observed project campaign status.

Introduction

Crowdfunding is becoming a popular funding channel among entrepreneurs as it allows them to address directly the crowd via Internet-based Crowdfunding platforms (CFP). A wide pan of the literature on crowdfunding deals with the parameters that make project campaigns successful and aim at guiding project founders in designing their campaigns. For instance, [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF] compares two forms of CF, reward-based and equity-based and shows that the choice made by the entrepreneur to select between these two forms depends essentially on the amount of required capital and that equity-based CF is more suitable for large projects. [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] shows that the size of the network of the project holder is positively correlated with the success and that the quality signals are also success factors. These signals have been identified as the advertisement videos, the update frequency and even the absence of typos, for reward-based crowdfunding [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF][START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF], and the clarity of the financial roadmap, the transparency with respect to the risks, the professional experience and the education level of the funders for equity-based crowdfunding [START_REF] Ahlers | Signaling in equity crowdfunding[END_REF]. Another set of works focused on the behaviour of funders and its impact on the project success. [START_REF] Agrawal | The geography of crowdfunding[END_REF] identify friend and family (F&F) as a source of early funding for projects that account for 15 to 20% of raised funds for equity-based campaigns and 30 to 40% for reward-based campaigns. Other potential funders tend to procrastinate and [START_REF] Solomon | Don't wait!: How timing affects coordination of crowdfunding donations[END_REF] studies the choice of donors toward funding a project early or waiting until the end of the funding window. [START_REF] Salahaldin | A duration-based model of crowdfunding project choice[END_REF] modeled the behaviour of the well informed funders and derived their optimal choice among the available projects using a dynamic programming approach.

The above cited body of work, and many others, gives a clear view about the success parameters of CF campaigns and the dynamics of fund raising. However, to the best of our knowledge, the perspective of platform managers is not yet covered. Many questions are open in this context, especially how to select candidate projects for the platform and how to accompany them during their fund raising campaign for maximizing their chance of success. While the first aspect (selection of candidate projects) may be partially addressed by identifying the success factors related to the campaign and project structures, as advocated by [START_REF] Lukkarinen | Success drivers of online equity crowdfunding campaigns[END_REF], the strategy of platform managers wit respect to pre-seleted projects is yet to be studied. This paper fills a gap in this domain, by considering the role of platform managers in promoting some projects during the lifetime of their campaigns, with the objective of maximizing the number of successful projects. The original contributions of the paper are as follows:

• We model the dynamics of fund raising on the platform based on empirical evidence from the literature.

• We consider the promotion strategy of the platform as a real option.

• We derive the optimal strategy using a dynamic programming approach and illustrate using numerical applications the impact of different paramaters on this strategy.

2 Problem statement

Platform and project model

We consider a crowdfunding platform based on the threshold pledge model (or the all-or-nothing principle(AON)) [START_REF] Cumming | Crowdfunding models: Keep-it-all vs. all-or-nothing[END_REF], meaning that if the project is not 100% funded, the funding will be returned to investors without penalty. This platform proposes a set of projects K = {1, 2, . . . , K} that are of interest to a set of a potential set of funders1 . The projects start at time t k and end at T k . Without loss of generality, we assume that

T 1 ≤ T 2 ≤ • • • ≤ T K .
We also consider that t k < 0 for all k ∈ K.

We define the next-closing project index by

N (t) = min{k ∈ K|T k ≥ t} (1)
The pledge made in project k at time t is denoted by X k (t) with t, T k , t k ∈ Z (discrete-time system). If a project k collects sufficient funds M k by the end of its duration, it is said to be successful, i.e., if

X(T k ) ≥ M k .
Funders typically belong to one of the three groups:

• Friends and family: who invest at the start of the project, and contribute to X(t k ) > 0.

• naive investors (referred to as fools): who invest in projects at all time.

The average amount of investment made by this group at any time is given by µ C .

• Rational investors: who invest in project k only at T k , so as to avoid loss due to interest during the time the investment is locked. The average amount of investment made by this group at any time is given by µ R .

The actual amounts of pledge made by the naive and rational investors are assumed to a Gaussian random variable with averages µ C and µ R and variances σ 2

C and σ 2 R , respectively. It is well known that the design of the project campaign (presentation, videos, pitches, etc.) has an important impact on its attractiveness and, eventually, on its success chances [START_REF] Cordova | The determinants of crowdfunding success: evidence from technology projects[END_REF]. This is modeled by a parameter α k > 0 for project k, equal to the fraction of the naive and rational investors it attracts when it is active, without any intervention from the platform (projects with larger α's attract more users).

Promotion strategy and its impact on attractiveness

The CF platform cannot be regarded as passive in the process of fund raising, as it can make strategic choices of highlighting some of them so that their success chances are increased. We'll see afterwards how the project to be promoted is highlighted and what is the objective that the platform pursues.

At any time t, if the platform highlights certain projects, they will attract a larger investment to that project at this time. We use γ(t) ∈ {1, G} K-N (t)+1 to denote the visibility allocation at time t with G > 1, K -N (t) + 1 being the number of remaining projects. If a project k is highlighted in the platform at time t, then γ k (t) = G and γ k (t) = 1 otherwise.

As a result, for any t ∈ {t k + 1, . . . , T k -1}, the expected investment in project k is given by

X k (t) -X k (t -1) ∼ α k γ k (t) K j=N (t) α j γ j (t) N µ C , σ 2 C ( 2
)
On the other hand, at the closing time, we have

X k (T k ) -X k (T k -1) ∼ α k γ k (T k ) K j=N (t) α j γ j (T k ) N µ C , σ 2 C + α k γ k (T k )N µ R , σ 2 R
(3) This latter equation takes into account the additional funds received at the campaign closing time from procrastinating funders, and that may be boosted if the project is promoted by the platform at that time.

Objective

Given X(0), design an optimal promotion strategy, i.e. γ(1), . . . , γ(T K ) in order to maximize the number of successful projects. Then the OP can be formalized as Maximize

γ K k=1 Pr(X(T k ) ≥ M k ) where X k (t) satisfies (2), (3) (4)
This maximization strategy is a natural choice for a platform as it helps increasing its attractiveness for future projects and maximizes its revenues from current ones as platforms usually receive some amount of money from successful project campaign.

Some definitions

Before analysing the optimal strategy, we start by some definitions and some preliminary, yet useful, results. Definition 1 A sequential policy is a policy that promotes a given project for a unique compact interval of time.

Definition 2 An offline policy is a policy that is defined at time 0 and that determines a priori the switching times between projects. An online policy is based on the observed amount of pledges collected by each project and is updated dynamically when time goes.

These notions of offline versus online policies is of utmost importance for platform managers. The former are useful for understanding the potential of the available projects at a given time, while the latter is more suitable for dynamically updating the marketing strategy, based on observed campaign outcomes.

Lemma 1 If there is an optimal offline strategy, there exists at least one sequential offline strategy that is optimal (maximizes the average number of successful projects) while minimizing the switching rate between marketing strategies.

Proof. For any given strategy, we can find a sequential strategy that induces the same amount of money for each project, and that highlights each project on a compact set of length equal to the sum of lengths of the highlighting periods of the original strategy. This is true because the amount of collected funds is Markovian. This policy minimizes the number of switching times. Definition 3 A One-at-Time (OAT) policy is a policy where the platform increases visibility of a unique project at a given time.

While such an OAT policy may be sub-optimal, it is of practical interest as it is suitable for a simple design of the platform and for maximizing the visibility of the selected project. Note that this does not mean that the platform highlights a unique project on its main page, but that it highlights a unique project per domain (technology, arts, video games, etc.).

Optimal offline policy

We start by a simple case of an OAT policy where the platform operator decides offline (a priori) the promotion strategy. We first derive the optimal policy when there are only two projects in competition, and then move to the general case.

Optimal OAT policy for 2 projects

Theorem 1 An optimal sequential offline OAT strategy for the case of two competing projects is given by the switching time τ * ≤ T 1 that verifies:

τ * = argmax τ [erf c( M 1 -X 1 (0) -a 1 (τ )µ C -b 1 (τ )µ R (a 1 (τ )σ C ) 2 + (b 1 (τ )σ R ) 2 ) +erf c( M 2 -X 2 (0) -a 2 (τ )µ C -α 2 Gµ R (a 2 (τ )σ C ) 2 + (α 2 Gσ R ) 2 )] (5)
where

a 1 (τ ) = α 1 Gτ α 1 G + α 2 + α 1 max(T 1 -τ, 0) α 1 + α 2 G (6) b 1 (τ ) = α 1 (1 + (G -1)1 τ =T1 ) (7)
a 2 (τ ) = α 2 τ α 1 G + α 2 + α 2 G max(T 1 -τ, 0) α 1 + α 2 G + α 2 G(T 2 -T 1 ) (8)
We use 1 C to denote the indicator function which takes the value 1 if condition C is verified and 0 otherwise.

Proof. In order to obtain the optimal policy, we first characterize the amount of funds received by project k during an interval of time [τ 1 > 0, τ 2 ≤ T k ]. This is the weighted sum of two Gaussian variables corresponding to the funds collected during the interval [τ 1 , τ 2 ] from naive investors plus the funds received at the end of its campaign from rational investors:

τ2 t=τ1 α k γ k (t) K j=N (t) α j γ j (t) N µ C , σ 2 C + α k γ k (T k )1 τ2=T k N µ R , σ 2 R (9)
Consider now two projects and an OAT policy that starts by promoting project 1 before switching to project 2. For a switching time τ < T 1 , the accumulated amount of funds for project 1 at its closing time T 1 is the sum of Gaussian variables as follows :

X 1 (T 1 ) = X 1 (0)+ α 1 G α 1 G + α 2 τ + α 1 α 1 + α 2 G (T 1 -τ ) N µ C , σ 2 C +α 1 N µ R , σ 2 R
(10) while the accumulated funds of project 2 are :

X 2 (T 2 ) = X 2 (0)+ α 2 (τ + G(T 1 -τ ) α 1 G + α 2 + α 2 G(T 2 -T 1 ) N µ C , σ 2 C +α 2 GN µ R , σ 2 R
(11) Note that, if the operator promotes project 1 during its whole campaign period, the term multiplying T 1 -τ disappears and project 1 receives G times more funds at its closing time, leading to the general expressions:

X 1 (T 1 ) = X 1 (0) + a 1 (τ )N µ C , σ 2 C + b 1 (τ )N µ R , σ 2 R (12) X 2 (T 2 ) = X 2 (0) + a 2 (τ )N µ C , σ 2 C + α 2 GN µ R , σ 2 R ( 13
)
The probability of success of project k being the probability that X k (T k ) exceeds M k , we obtain:

Pr(X 1 (T 1 ) ≥ M 1 ) = 1 2 erf c( M 1 -X 1 (0) -a 1 (τ )µ C -b 1 (τ )µ R 2(a 1 (τ )σ C ) 2 + 2(b 1 (τ )σ R ) 2 ) (14)
and

Pr(X 2 (T 2 ) ≥ M 2 ) = 1 2 erf c( M 2 -X 2 (0) -a 2 (τ )µ C -α 2 Gµ R 2(a 2 (τ )σ C ) 2 + 2(α 2 Gσ R ) 2 ) ( 15
)
Maximizing the overall number of successful projects, we obtain the switching time of equation ( 5).

For the first numerical applications, we consider the parameters of projects given in Table 1 (considering only projects 1 and 2). As of the platform parameters, the average and standard deviation of collected funds at any time interval are given by µ C = 3 and σ C = 0.5, plus an additional peak at the closing time with µ R = 5 and σ R = 0.5. If the platform promotes a project, its attractiveness increases by 30% (G = 1.3). We illustrate in Figure 1 the success probabilities when the switching time τ increases from 0 to T 1 . Increasing τ increases the chance of success of project 1, but decreases that of project 2; there is an optimal timing τ * = 9 in this case.

Optimal OAT policy in the general case

In the general case, where there are K ≥ 2 projects, a similar approach can be followed for obtaining the optimal strategy. We denote a strategy as a vector of switching times τ = (τ 2 , ..., τ K ), where τ k+1 ≤ T k is the time where the platform switches from promoting project k to promoting project k + 1. We denote by p(t, τ ) the project that is promoted at time t when policy τ is followed. The amount of funds collected by project k at its closing date is a random variable as follows: The sum success rate that is to be maximized is thus :

X k (T k ) = X k (0) + a k (τ )N µ C , σ 2 C + b k (τ )N µ R , σ 2 R (16)
K k=1 Pr(X k (T k ) ≥ M k ) = 1 2 K k=1 1 2 erf c( M k -X k (0) -a k (τ )µ C -b k (τ )µ R 2(a k (τ )σ C ) 2 + 2(b k (τ )σ R ) 2 ) (19)
The optimal platform strategy is that the vector τ * = (τ * 2 , ..., τ * K ) that maximizes (19).

We illustrate in Figure 2 the impact of changing τ on the sum success probability for 3 projects whose parameters are given in Table 1. There are two parameters to set, τ 2 and τ 3 , and there is a clear optimal strategy that consists in choosing τ 2 = 5 and τ 3 = 16 for the considered parameters.

Figure 1 :

 1 Figure 1: Individual and sum success rates for different timing strategies. Two projects case. The optimal policy is indicated.

Figure 2 :

 2 Figure 2: Success rates for three projects for different policies.

  ) α j + (G -1)α p(t,τ ) (17) and b k (τ ) = α k (1 + (G -1)1 τ k =T k )(18)

Table 1 :

 1

	Project parameters
	Project	1	2	3
	Initial X k (0)	0	5	0
	Target M k	30 30 40
	Attractiveness α k	1 0.5 0.5
	Closing time T k	15 20 25

Note that we focus here on a set of projects that can be regarded as competitors, i.e. that attract funders with common interest, e.g. technology, arts, etc. A platform may propose different types of projects but we focus here on projects belonging to the same domain.

Online policy as a real option: Dynamic programmingThe optimal policy of the previous section helps the platform manager understanding the impact of the platform strategy on its portfolio of projects. However, an offline optimisation is not the most adequate policy as the manager may have the opportunity, in case of a unexpected evolution of the funding campaign of a project , to adjust online its policy.

Real options formulation

This opportunity can be modeled as a real option, as it has the following characteristics:

• Irreversibility: Once the manager decides to switch from promoting project k to project k + 1, its decision is partially irreversible as the project has to be promoted at least for a certain interval of time. In the case of a sequential policy, this decision is completely irreversible (the future opportunities for promoting project k are lost).

• Uncertainty: The uncertainty here is included in the future evolution of collected funds over time. This uncertainty is resolved upon time and the manager decision can be made contingent upon the resolution of the uncertainty, creating the option.

• Flexibility: This means that the manager can, function of new information, change its decision where the decision is contingent of an event. In our case, the manager has a complete or partial flexibility, depending on the structure of the strategy (OAT or completely dynamic).

Dynamic programming approach

In order to derive the optimal decision for the general case, we adopt a dynamic programming approach. This method breaks a whole sequence of decisions into two components: the immediate decision, and a valuation function that encapsulates the consequence of all subsequent decisions (the continuation value). The platform manager has the opportunity to stop promoting project 1 and switch to project 2, lowering thus the future amount of collected funds for project 1 and increasing that of project 2. Otherwise, he may wait one period of time and then decide whether to switch or to wait another period before making the same decision. This process will depend on the levels of collected funds at each period of time.

The state space at time t is described by X(t) = (p, x 1 , ..., x K ), where p is the index of the project currently promoted at time t, and x k is the already collected amount of funds. Indeed, we have to keep track of the project that is promoted. The dynamic programming tree is then split into branches determined by the currently promoted project. We can jump between branch p and branch p + 1 at any time. In order to make the analysis tractable, we discretize the space of collected funds during one time slot into a finite number of possible values. Let J be the number of intervals for fund collection, c j being the jth value (c 1 = 0 and c J = X max , chosen so that the probability of exceeding the maximum for any project is very small). The step size is equal to = c J J and at any time t, the additional amount of collected funds by project k may take any value c j , j ∈ [1, J].

For illustration purposes, we detail the analysis for the case of two projects. We start by time T 1 when no decision is expected from the platform manager. We compute the utility at time T 1 knowing that the system is in state X, given by the expected success rate of project 2 (whose evolution is still uncertain), plus 1 or 0, depending on the final state of fund collection of project 1:

We then move backwards step by step and calculate the utility iteratively. For any time t < T 1 , as there are only two projects, the only branch where the manager still has a decision to make corresponds to p = 1, we may then drop p from the state space. If the decision is to keep the same strategy one more step and decide later ("waiting" strategy), the expected sum utility is given by:

where q k (t, i, n) is the probability that the project k goes up from state i to state n ≥ i during the interval [t, t + 1], knowing that the strategy of the platform is to promote project 1. These can be approximated by:

Note that the transition probabilities are independent of the time t, except for project 1 at the last time slot where there are the rational investors that contribute to the fund raising.

On the other hand, if the manager decides to "switch" its strategy from project 1 to project 2, he loses the future decision opportunities and the expected sum utility is calculated by: The utility of the platform at time t is thus the maximum of the utilities in cases of "waiting" and "switching":

Numerical applications

In order to illustrate the online policy, we implement the dynamic programming approach for the two projects case with the parameters of table 1. As the decision of the manager depends on the observed state, we illustrate in Figure 5 the values of the two different policies of the platform manager at a given time (t = T 1 /2) depending on the amounts of already collected funds by the two projects. If the value of waiting is higher than the switching value, the decision is to maintain the current policy at least until the next time period. We observe that, for the same X 1 (t), the value of switching decreases when X 2 (t) increases, making waiting a better decision. When both projects have collected a large amount of funds, both strategies have the same value, as they are likely to succeed. While the manager implements an online dynamic policy, the decisions may be predicted at time 0 based on the estimation of the future state evolution. The Gaussian assumption allows predicting this evolution. We plot in Figure 4 the evolution of the state probabilities with time, supposing that the policy is to maintain promoting project 1. Naturally, the average amount of collected funds increases with time. If we observe simultaneously figures 5 and 4, we can see that the platform is likely to be, at t = T 1 /2, in a state around X 1 ∈ [START_REF] Cumming | Crowdfunding models: Keep-it-all vs. all-or-nothing[END_REF]11] Figure 4: Evolution of state probabilities with time. and X 2 ∈ [START_REF] Lukkarinen | Success drivers of online equity crowdfunding campaigns[END_REF]10]. In these states, the waiting value is higher than the switching value, leading to a low switching probability. Figure 4 shows that this switching probability increases with time. We also illustrate in Figure 4 the impact of the initial collected funds on the strategy. A lower amount of initial funding for project 2 leads to an earlier switching in order to increase the overall success rate.

Conclusion

In this paper, we consider the perspective of a crowdfunding platform manager whose objective is to maximize the success ratio of his platform. We develop an optimization framework for the project promotion strategy that aims at balancing funds between projects and increasing the expected campaign success rate. We then developed an online strategy where the manager observes the project status evolution and adapts its strategy. We model this dynamic policy as a real option and developed the corresponding dynamic programming algorithm. The work of this paper is a first step towards a decision-making framework for crowdfunding platforms to pick projects and accompany them during their funding campaigns.