Kevin Primicerio
email: kevin.primicerio@centralesupelec.fr

Damien Challet
email: damien.challet@centralesupelec.fr

Stanislao Gualdi
email: stanislao.gualdi@gmail.com

Wisdom of the institutional crowd

published or not. The documents may come

Collective rationality and functional wisdom of the crowd in far-from-rational institutional investors

Introduction

The collective ability of a crowd to accurately estimate an unknown quantity is known as the "Wisdom of the Crowd" [START_REF] Surowiecki | The wisdom of crowds[END_REF] (WoC thereafter). In many situations, the median estimate of a group of unrelated individuals is surprisingly close to the true value, sometimes significantly better than those of experts [START_REF] Galton | Vox populi (The wisdom of crowds)[END_REF][START_REF] Hill | Expert stock picker: the wisdom of (experts in) crowds[END_REF][START_REF] Landemore | Why the many are smarter than the few and why it matters[END_REF][START_REF] Nofer | Are crowds on the internet wiser than experts? The case of a stock prediction community[END_REF]. WoC may only hold under some conditions [START_REF] Surowiecki | The wisdom of crowds[END_REF][START_REF] Davis-Stober | When is a crowd wise?[END_REF]: for example social imitation is detrimental as herding may significantly bias the collective estimate [START_REF] Lorenz | How social influence can undermine the wisdom of crowd effect[END_REF][START_REF] Muchnik | Social influence bias: A randomized experiment[END_REF]. WoC is a reminiscent of collective rationality without explicit individual rationality: when it applies, it is a consistent aggregation of possibly inconsistent individual estimates [START_REF] Hogarth | A note on aggregating opinions[END_REF]. This is to be contrasted with the mainstream economic paradigm which takes a short-cut by assuming that collective rationality reflects individual rationality, where only a "typical" decision maker -the representative agentis considered [START_REF] Hartley | The representative agent in macroeconomics[END_REF] or team reasoning where the individual agents explicitly optimize the collective welfare [START_REF] Colman | Collective rationality in interactive decisions: Evidence for team reasoning[END_REF]. Aggregation of quite diverse individual actions, especially in a dynamic context where expectations are continuously revised, is still an open problem [START_REF] Kirman | Whom or what does the representative individual represent?[END_REF].

Although almost all known examples of WoC are about a single number or coordinate, there is no reason why WoC could not be found for whole functional relationships between several quantities. For example, Haerdle and Kirman analyse the prices and volume of many transactions in Marseille fish market: while the relationship between these two quantities is rather noisy, the market selforganises so that when more fish are sold, prices are lower, as revealed by a local average [13]. More generically, many simple relationships found in Economics textbooks may only hold on average, but not for each agent or each transaction.

Asset price efficiency is an obvious instance of WoC in Finance: it states that current prices, determined by the actions of many traders, are the best possible estimates and fully reflect all available information [START_REF] Malkiel | Efficient capital markets: A review of theory and empirical work[END_REF][START_REF] Malkiel | The efficient market hypothesis and its critics[END_REF][START_REF] Fama | Market efficiency, long-term returns, and behavioral finance[END_REF]. Another WoC candidate is portfolios. While many market participants, especially investment funds, strive to build optimal portfolios, each following its own criteria and constraints (performance objective, risk, tracking error, etc.), the question here is whether their collective behaviour may be related to a rational benchmark. Fortunately, this implies that we do not need to understand the minute details of all the portfolios and can focus on average quantities instead.

Wisdom of crowd

Let us define some necessary quantities to be more precise. At time t, fund i has capital W i (t) which is invested into n i (t) securities among M (t) existing ones. As a result, each security α, whose capitalization is denoted by C α (t), is found in m α (t) portfolios. The explicit time dependence is dropped hereafter.

The only quantity defined above which depends on asset allocation strategies of fund i is n i , the number of securities it chooses to invest in. Our main hypothesis is thus that WoC is found in the average relationship between n i and W i . A simple rational benchmark is proposed by [START_REF] De Lachapelle | Turnover, account value and diversification of real traders: evidence of collective portfolio optimizing behavior[END_REF] : when a fund with capital W i is able to invest the same amount in each of the n i chosen securities and if the transaction cost does not depend on the security, then the optimal n i is such that

W i ∝ n µ i . (1)
where the exponent µ is determined by the transaction costs fee structure; for example, proportional transaction costs lead to µ = 1, while a fixed cost per transaction corresponds to µ = 2 (see [START_REF] De Lachapelle | Turnover, account value and diversification of real traders: evidence of collective portfolio optimizing behavior[END_REF] for more details). Allowing for individual fluctuations, Eq. (1) becomes log W i = µ log n i + i , where i has zero average. Denoting local average of x i by x, the local average of Eq. (1) yields

W ∝ n µ .
(

) 2
Flat fee per transaction (µ = 2) is a popular request of large clients of broker. [START_REF] De Lachapelle | Turnover, account value and diversification of real traders: evidence of collective portfolio optimizing behavior[END_REF] find indeed that for wealthy individual investors and asset managers, exponent µ = 2 within statistical uncertainty. We will thus test the occurrence of WoC from the value of exponent µ. More precisely, our hypothesis is that if (i) the effective transaction cost per transaction is the same for all assets and (ii) funds are able to build equally weighted portfolios, then Eq. ( 2) holds and that µ = 2, which is a sign of WoC.

Both conditions must cease to hold for larger investment funds. Indeed, condition (i) cannot be true for them since large trades (even when split into meta-orders) have a price impact which grows with their size and depends on volatility and average turnover [START_REF] Bouchaud | Price impact. Encyclopedia of quantitative finance[END_REF]. Condition (ii) ceases to hold for large funds which spread their investments on many securities: because the capitalization of assets and their average daily turnover are very heterogeneous, large funds cannot invest enough money in assets with a small capitalization so as to build an equally-weighted portfolio. As a result, on average, the local average W is expected to increase more slowly as a function of n in the large n region; equivalently, the exponent µ is expected to be smaller than 2. In summary, two different regimes should emerge: one with µ < = 2 for small enough n and µ > < 2 for larger n.

Figure 1 plots W i versus n i in logarithmic scale: a cloud of point emerges, with a roughly increasing trend. The large amount of noise confirms the great diversity of fund allocation strategies. II Figure 1: Total mark-to-market value W i as a function of the number of investments n i , with a robust locally weighted regression fit (yellow line) and two linear fits (blue dashed lines) for two different ranges of n. Robust locally weighted regression fit for the simulated data (in green).

WoC may only appear in some average behaviour. This is why we computed a locally weighted polynomial regression [START_REF] Cleveland | Local regression models[END_REF]. As expected, two distinct regions appear. In each of them, the local regression follows a roughly linear behaviour.

The cross-over point n * between the two regions is algorithmically determined for each quarterly snapshot (see S.I.); it is stable as time goes on (see Fig. 11 in S.I.). The two exponents µ < and µ > are quite stable as a function of time as well (see Fig. 11 in S.I.); their time-averages µ < 2.1 ± 0.2 and µ > 0.3 ± 0.1 are markedly different, which points to distinct collective ways of building portfolios in these two regions.

So far, µ < is compatible with the WoC hypothesis. Let us check the validity of conditions (i) and (ii) above. When condition (ii) is not satisfied, then condition (i) also must cease to hold, thus we can focus on the former. Condition (ii) says that the diversity of investment fractions p iα = W iα /W i for W iα > 0 must be very small among α. This may be summarized in a single number by the scaled Shannon Entropy S i = -1 log 2 ni α p iα log 2 p iα , which equals 1 and is maximal when all the non-null p iα are equal. Figure . 2 reports the scaled entropy S i of all the funds for a given time snapshot, together with the local average S. The latter increases up to about n n * and then decreases. The fact that S < 1 is due in part to price fluctuations: even if fund i builds an equally weighted portfolio at time t (thus S i,t = 1), S i,t+1 < 1 at a later date. The importance of this mechanism is confirmed by Monte-Carlo simulations: the red line of Fig. 2 shows the effect of natural asset price evolution on perfectly equally weighted portfolios after three months, using asset price volatility measured in our dataset between the time of the snapshot and the three previous months: the resulting scaled entropy S M C increases as a function of n, mirroring the local average of S i in the same figure for n < n * . Thus, the decrease for n > n * is due to impossibility for larger funds to build equally-weighted portfolios. A further argument supporting our claim that investment funds strive to build equally weighted portfolios (on average) is provided by the entropy measured on the set of common positions between two consecutive snapshots multiplied by S M C (n i )/S M C (n i,restricted ) in order to account for the dependence of S on n; the local average of the resulting entropy corresponds to the dashed blue line: it is clearly smaller than the entropy of III Figure 2: Scaled Shannon entropy S i as a function of the number of investments n i for all the funds on 2013-03-31 (circles) and robust local weighted regression fit, for all positions (blue line), numerical simulation of the effect of price fluctuations on the entropy on initially equally weighted portfolios (red line) , where a volatility similar to the observed volatility in the real data, is applied; robust local weighted regression fit restricted to the unchanged portfolio positions from the previous time step, multiplied by the ratio between the simulated entropy for the full portfolio and the restricted portfolio (dashed blue line); the new portfolio, hence new positions purposefully bring S i closer to equally weighted portfolios. Therefore, condition (ii) is valid when µ = 2; conversely, µ = 2 when condition (ii) ceases to hold.

Quite tellingly, the same exponent was found for large private investors and asset managers (with much smaller amounts of money under management). Thus the collective behaviour of large investment funds is essentially the same one. Since one finds the same exponent µ over many decades of portfolio values for a wide spectrum of market participants, and since µ = 2 corresponds to a realistic transaction cost per transaction, we argue that WoC is a plausible explanation of the average portfolio structure. Note that µ = 2 does not imply that funds really face constant transaction cost per transaction, only that their population acts as if it does. Finally, we stress that WoC holds for a whole functional relationship over many decades of n and W , not only for a single number, which considerably extends its reach.

Asset selection model

So far, bringing to light WoC in the µ = 2 region only required to focus on the number of securities in a portfolio, not on how funds select securities. This implicitly assumed that funds could invest in all securities they wished, which is clearly not the case in the large diversification region: the fact that the exponent µ is much smaller in this region implies that funds need on average to split their investments into many more securities. This is most likely due to liquidity constraints: large funds cannot invest as much as they wish in some assets because there are simply not enough shares to build a position larger than a certain size without impacting too much their prices. Each fund has its own way to determine the maximal amount to invest in a given security α; a common criterion IV is to limit the fraction W iα /C α . Fig. 8 in S.I. strongly suggests that each fund fixes its upper bound

f (max) i ≥ max α f iα where f iα = W iα C α . (3)
It turns out that f max i is highly heterogeneous among funds log 10 (f max i ) -3.0 ± 1.0 (see Fig. 9), which reflects both the heterogeneous ways of portfolio construction and also the confidence of a fund in its abilities to execute large trades without too much price impact. The existence of such limits implies that portfolios are less likely to be equally weighted in the large diversification region, as seen indeed in the decrease of the average portfolio weights scaled entropy for n ≥ 70 (blue line in Fig. 2).

Funds, however, do not invest in a randomly chosen security, even in the low diversification region. Figure 3 displays a scatter plot of the capitalization C α of each security α versus m α , the number of funds which have invested in this security, together with a local non-linear fit. Similarly to W vs n, one finds a power-law relationship

log C α = γ log m α + α (4)
for large enough m (see S.I.). Hence in local average notations, C ∝ m γ . Exponent γ is stable during the period 2007-2014 (see Fig. 11 in S.I.) and its average γ 2.2 ± 0.1.

In short, one needs to introduce a model of how funds choose to invest in securities to reproduce the average behaviour of both Eqs ( 4) and [START_REF] Surowiecki | The wisdom of crowds[END_REF]. Since one sees a cross-over between two types of behaviour rather than an abrupt change, we create logarithmic bins of the axis n i and denote the bin number of fund i by [n i ]. Two mechanisms must be specified: how a fund selects security α and how much it invests in it. The latter point is dictated by Fig. 8 in the large n i region where fund i invests W iα = f (max) i C α ; for the sake of simplicity, we approximate f [ni] . In the small diversification region, we assume that n i = n opt i , thus W iα = W i /n opt i to be consistent with our previous results. We choose a security selection mechanism that rests on the market capitalization C α of a security α (see S.I.) which is a good proxy of the liquidity (Fig. 10). We perform Monte-Carlo simulations from the empirical selection probabilities and f (max) [ni] and display the resulting W vs n and C vs m in Figs 1 and3 (continuous green lines), in good agreement with the local averages (continuous orange lines). One notices a discrepancy in the relationship C vs m for large n, which mainly comes from funds in the large diversification region. (See Fig 12 S.I).

The large diversification region illustrates how constraints may considerably modify the rational benchmark. While the above mechanism of security selection is able to reproduce adequately the behaviour of well diversified funds, we could not find a rational benchmark for the dependence of f max and n i . Thus, the case for WoC in the large diversification region is not entirely closed.

Data

Our dataset consists of an aggregation of the following publicly available reports (in order of reliability): the SEC Form 13F, the SEC's EDGAR system forms N-Q and N-CSR and (occasionally) the form 485BPOS. Our work focuses on the period starting from the first quarter of 2005 to the last quarter of 2013. V Figure 3: Market capitalization of securities as a function of the number of investors in logarithmic scale. From the local non-linear robust fit (yellow line) we observe a linear relationship for assets with more than about 100 investors. The blue dashed line corresponds to a linear fit on that group of asset. Hence W α ∝ m γ α , with γ 2.1. Robust locally weighted regression fit for the simulated data (in green).

These forms are filled manually and are thus error prone. We partially solve this issue by crosschecking different sources (which often contains overlapping information) and by filtering data before processing (see details in S.I.).

The main limitation of this dataset is that it provides accurate figures for long positions only. The other positions (short, bonds, ...) are most of the time only partially known. The frequency of the dataset is also inhomogeneous: data for most of the funds are quarterly updated (depending on regulations), hence we decided to restrict ourselves to 4 points in a year only. Such frequency is probably too low for investigating the dynamics of individual behaviour but is not a problem for we focus on an aggregate and static representation of the investment structure.

Discussion and conclusion

While WoC is commonly applied to a population collectively guessing a single number, we investigate here a fundamentally different situation and provide evidence for a collective functional optimization of the asset ownership structure. What the reference function should be is dictated by optimality arguments. In the case of financial markets, the rational benchmark was not related to the efficient market hypothesis, but to the way a large population of professional fund managers build their portfolios. Whereas each fund has its own benchmark with respect to which the fund performance may be assessed, this, fortunately, has no discernible influence on the average structure of their portfolio. In addition, WoC is often meant as a collective guessing of non-experts; one thus may conclude that the population investigated here has decidedly more expertise than the subjects of other WoC studies. What kind of expertise the typical fund manager has is not obvious, at least when one looks at their pure performance (see e.g. [START_REF] Barras | False discoveries in mutual fund performance: Measuring luck in estimated alphas[END_REF]). In addition, the optimal relationship between the number of assets in a portfolio and the value of the latter is clearly not broadly known in these circles, as shown by the very large deviations from the ideal case in Fig. 1, and the collective VI expertise only appears when their decisions are suitably averaged. The presence of WoC when the subjects face strong constraints, as those of highly diversified funds, is more conjectural, and more work will be needed in that respect.

At a higher level, our results suggest that, while individuals may deviate much from the rational expectation theory, standard economic theory may hold at a collective level, without need for micro-founded individual decisions: the average decision may in some cases be approximated by a rational, representative agent. Our results however only hold on a snapshot of the system, for which individual fluctuations may be averaged out. In a dynamic setting, the very large deviations from the rational benchmark may not be neglected in the presence of feedback loops [START_REF] Gualdi | Tipping points in macroeconomic agent-based models[END_REF]. In other words, the dynamics of these fluctuations are worth investigating in their own right.

Supporting Information (SI) 4 Filtering

In order to remove inconsistencies in the dataset, we applied the following filters

Country of origin

Our dataset is sparse and heterogeneous. Indeed, the quality of the sources of data is directly related to each country's disclosure regulations. For these reasons we decided to keep only the entities which use an US based mail address. About 60% of the total market capitalization of the dataset is concentrated in US based securities. Figure 4 shows two large clouds of dots, each of them corresponds to a different region of origin: green (resp. orange) cloud corresponds to non-US (resp. US) based securities. The origin of this large difference between these two regions are not clear: it could for example come from IX Figure 5: Temporal evolution of the number of funds N i and securities N α in the database. Unfiltered in dashed lines and US based only in solid lines. differences in regulations in non-US countries. It turns out that the ratio of the investment values in US and non-US assets varies little as a function of time (see Fig. 4), which does not affect the exponent µ in Eq. 1. As a consequence we focused on US securities.

Frequency

Large funds are requested to report their positions at a frequency which depends on the applicable regulation. As a result, reporting frequency ranges from monthly to yearly, most funds filing quarterly reports. We therefore focused of the latter.

Penny Stocks

The "penny stocks", i.e., usually securities which trade below $5 per share in the USA, are not listed on a national exchange. Since they are considered highly speculative investments and are subject to different regulations, we filtered them out.

Size

We also filtered out small founds and securities and applied the following filters: fW i > 10 5 USD, C α > 10 5 USD, n i ≥ 5, m α ≥ 10.

Output

We restricted our study to 36 quarterly snapshots starting from the first quarter of 2005 and ending with the last quarter of 2013. Figure 5 reports the evolution of the number of securities and funds in the database before and after filtering.

X

The framework we introduce in this paper follows a series of a few elementary steps described below. The aim is for the model to be sensitive to the different constraints which dominates the portfolio selection of a fund.

Finding n *

For date t, we define the cross-over point n * between the two regions which appear in the local polynomial regression. We determine this point value with a likelihood maximization of the model

W = µ < n + (µ > -µ < )(n -n * )θ(n -n * ), (5)
where θ(x) is the Heaviside function. We use a recursive method to find parameters µ < , µ > and n * [START_REF] Muggeo | Estimating regression models with unknown break-points[END_REF]. Figure 11 shows that n * is stable as a function of time.

Asset selection in the small diversification region n i < n *

In this region, we consider the equally weighted portfolio hypothesis to be true. Each position has a size Wi n opt i , where n opt i is the optimal number of position computed with eq 1. The funds select their asset randomly with a probability proportional to C α . Also, in order to construct an equally-weighted portfolio, a position is valid only if it is of size Wi n opt i .

Asset selection in the large diversification region n i ≥ n *

In this region, the liquidity constraints make it harder for funds to keep an equally weighted portfolio and portfolio values are thus spread on a larger number of assets. We propose here a stochastic model of asset selection based on two main ingredients: first that the selection probability of asset α by fund i depends on the diversification of a fund n i and on the scaled rank of the capitalization of asset α, and that the investment is bounded by an hard constraint on the fraction of market capitalization of asset α.

We chose a security selection mechanism which rests on the scaled rank of capitalization of security α, defined as ρ α = rα M where r α is the rank of capitalization C α and M the number of securities at a given time. The selection probability P (W iα > 0|ρ α ) is then obtained by parametric fit to a beta distribution in each logarithmic bin. Note that we do not use the same rank-based selection mechanism in the low-diversification region because in this case it is harder to have a good fit with the beta distribution. This is however only a minor point since the capitalization is approximately power-law distributed and the two selection mechanisms are basically equivalent (the rank is proportional to a power of the capitalization) and indeed results are very similar in both cases.

Figure 6 shows that the distribution of the ranks in which a fund is invested is sensitive to its diversification n i for 2013-03-31. The Beta distribution, which is limited to a [0, 1] interval, is flexible enough to describe the asset selection mechanism of a fund.

f (x; a, b) = 1 B(a, b) x a-1 (1 -x) b-1 , (6)
where a and b are the shape parameters of the distribution, and B is a normalization constant.

XI

Maximum investment ratio

The funds limit their investment in a given asset. They seem to follow a simple rule: defining the investment ratio f i,α = Wiα Cα , one easily sees in Fig. 8 that each fund has a maximum investment ratio

f max i = max α W iα C α (7)
Since the average exchanged dollar-volume of an asset is proportional to its capitalization (Fig. 10), the existence of f max i is a way to account for the available liquidity. Although that limit is clear for an individual fund, there is a large range of empirical values f max i Fig. 9.

Simulation

The simulation is done in a few simple steps:

1. Compute n * using the segmented model Eq. 5.

2. Select a fund i, with a number of assets n i .

3. If n i < n * : (a) Compute its optimal portfolio value using Eq. 1. The fund will invest (b) Select assets randomly following a Beta probability distribution Fig. 6 with the parameters found in Fig. 7.

By iterating those steps we obtain Fig. 1 Since the simulation outputs a portfolio for every fund, we can directly infer the number of investors m α of every security. XV Figure 10: Market capitalization as a function of the daily exchanged volume dollar. We find a slope close to 1 for all the dates in our database, confirming the hypothesis that the daily exchanged volume dollar of an asset is proportional to its market capitalization.

  i in the bin [n i ], denoted by f (max)

Figure 4 :

 4 Figure 4: Top: Market capitalization as a function of the number of investors for all securities. Bottom: Temporal evolution of the aggregated market capitalization of US over the total market capitalization.

Figure 6 :Figure 7 :

 67 Figure 6: Top: Empirical probability density function of investing in a security of scaled capitalization rank ρ given the diversification n i of the fund. Bottom: Probability density function of investing in a security of scaled capitalization rank ρ given the diversification n i of the fund, given by the model.

Figure 8 :.Figure 9 :

 89 Figure 8: Fraction of the market capitalization of a security held by a fund. Each color represent a different fund. Top: Funds with a large diversification (n i > 800). We can clearly see a delimitation for most of the funds, which correspond to the maximum fraction f max i . The value of f max i widely differs from one fund to another. Bottom: Funds with a low diversification (n i <60), f max i doesn't appear.

Figure 11 :Figure 12 :

 1112 Figure 11: Top: Temporal evolution of the coefficients µ < , µ > and γ. Bottom: Temporal evolution of the value of the cross-over point n * between the two regions as a function of time. It reaches a global minimum after the 2008 crisis.

Acknowledgements

S. Gualdi acknowledges support of Labex Louis Bachelier (project number ANR 11-LABX-0019)


Isabelle Piot-Lepetit
email: isabelle.piot-lepetit@inrae.fr

Fatih Ozogul

Marwa Moumni

Cengiz Gokbulut

Frontiers in Sustainable Food Systems 01 frontiersin

Keywords: sustainability, digitalization, servitization, open innovation, ambidexterity, startups, digitainability, agrifood value chains

By developing a conceptual analysis, this research explores the possibility of fostering a convergence between digitalization and sustainability, also named digitainability. This convergence is often seen as an opportunity to solve the various challenges the agrifood sector faces nowadays, especially those of feeding more people without deteriorating the environment and creating social divides and, at the same time, managing specific risks related to climate change, biodiversity loss, and hunger. The research describes how open innovation is becoming an imperative and explains the contribution of startups in the development of these new innovative initiatives. It also explains how sustainability considerations are closely linked to the development of servitization in the business context. Finally, it explores how these changes impact established companies, forcing them to develop ambidextrous innovation strategies to maintain and support their competitive advantage. In doing so, the paper illustrates some of the well-known effects of this ongoing change, while identifying the opportunities created and the challenges for which more research is still needed.

Introduction

The beginning of the use of digital technologies in the agrifood sector can be considered as having appeared with the creation of web pages [START_REF] Bellon-Maurel | Digital agriculture in Europe and in France: which organisations can boost adoption levels?[END_REF]. It started by presenting information, before considering the dematerialization of declarations, such as those related to fertilization practices, or developing new distribution channels, such as online platforms. Nowadays, digitalization is becoming an integral part of the development and evolution of the agrifood sector and agrifood value chains. These developments have been made possible due to two main important changes. The first one is the switch from a principle of product ownership to a logic of service access, which can be illustrated by the creation of platforms for sharing agricultural machinery or accessing local and cultural food. The second one relates to the establishment of startup ecosystems, characterized by a technological environment, a culture of innovation and risk, and an ability to push forward the limits of what already exists [START_REF] Florez | How do French Agritech start-ups contribute to the sustainability of food value chains?[END_REF]. At the same time, the agrifood sector is being asked to become more sustainable (UN-United Nations, 2015). Thus, the agrifood sector is looking for innovative solutions to deal with the multiple challenges that it has to manage, such as feeding a growing global population or dealing with climate change, resource depletion, or food waste [START_REF] Klerkx | Dealing with the game-changing technologies of agriculture 4.0: how do we manage diversity and responsibility in food system transition pathways?[END_REF], while generating social impacts. Digitalization is as a way currently explores to meet meeting these sustainability requirements (FAO-Food and Agriculture Organization of the United Nations, 2021).

The emergence of digital technologies is fostered by the outburst of the startup phenomenon in every place in the world. Among these high-potential and high-growth companies, dedicated startups are seizing all kinds of opportunities offered by both digital technologies and sustainability objectives. They imagine new uses, develop new products, and even create new needs, by placing digitainability innovations-as a mix of digitalization and sustainability initiativesat the core of their value propositions. By doing so, they are changing markets but also affecting how value chain stakeholders behave. As a result, startups are also influencing the way established companies embrace innovations. More specifically, they push for the development of ambidextrous innovative processes, where large companies rely on both internal and external innovations to engage in digitalization and open innovation initiatives [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF].

This research relies on an extensive literature review and proposes a conceptual analysis, aiming at gaining insights on the ongoing changes occurring in the agrifood sector in regards to digitainability and open innovation. After describing what the sustainability and digitalization transformative trends imply for the agrifood sector, the paper explores how open innovation is contributing to the development and implementation of either digital sustainability, sustainable digitalization or digitainability. It also highlights the contribution of startups to these trends and the ambidextrous innovation process well-established companies developed.

2 Digitainabilty: bringing together the two powerful transformative trends of sustainabilty and digitalization 2.1 Sustainability as a driving trend of the undergoing changes of the agrifood sector Sustainability is a very large concept integrating many aspects of the world and life. Indeed, sustainability does not only focus on the protection of the environment. It is also concerned with economic and social aspects [START_REF] Ford | Additive manufacturing and sustainability: an exploratory study of the advantages and challenges[END_REF]. That is why sustainability is most frequently described by means of the well-known pillars of economic, environmental and social sustainability (UN-United Nations, 2015) and is considered as a process to improve human wellbeing, while contributing to the resilience of the economic, environmental, and social systems [START_REF] Munasinghe | Sustainable development: Basic concepts and application to energy[END_REF].

However, sustainability is becoming a moral and economic imperative due to the intertwined connections between nature, society, and business activities [START_REF] Del Rio Castro | Unleashing the convergence amid digitalization and sustainability towards pursuing the sustainable development goals (SDGs): a holistic review[END_REF]. Regarding the agrifood sector, sustainability is of real importance. Indeed, enabling the pursuit of current level of consumption, especially in developed economies, while allowing it elsewhere in the world, has been found incompatible with the required level of resource utilization [START_REF] Lichtenthaler | Digitainability: the combined effects of the megatrends digitalization and sustainability[END_REF]. In a context of growing population worldwide and increasing environmental concerns, sustainability covers a large set of issues, such as negative footprints, climate change, biodiversity loss, environmental pollution, inequality and social tensions. It is also concerned by changes in behaviors to deal with all these issues [START_REF] Del Rio Castro | Unleashing the convergence amid digitalization and sustainability towards pursuing the sustainable development goals (SDGs): a holistic review[END_REF]. At the same time, sustainability encompasses considerations related to scale and time aspects, as it is nowadays a worldwide concern with potential important impacts on future generations. Therefore, the concept of sustainability is a very complex one. In effect, the concept of sustainability is based on both a holistic perspective and a normative dimension. It encompasses both means and ends and can be seen as both a process and a destination [START_REF] Johnston | Reclaiming the definition of sustainabilty[END_REF]. That is why sustainability is so difficult to define. As most of the sustainability boundaries are not clear or explicit enough [START_REF] Köhler | An agenda for sustainability transitions research: state of the art and future directions[END_REF], it is really difficult to evaluate or describe it without a context-based approach [START_REF] Waas | Sustainable development: a bird's eye view[END_REF].

From a business perspective, [START_REF] Gartner | The digitalization sustainability matrix: a participatory research tool for investigating digitainabilty[END_REF] proposes to define sustainability management as the focus on "products, services, the enterprise and supply chain, " while seeking to "optimally balance organizational performance and outcomes across economic, environmental, and social criteria over all time scale. " In this context, economic sustainability can be considered as the capacity of optimizing, in an efficient and responsible manner, the use of available resources, while maximizing profitability over time. Environmental sustainability is more focused on the natural ecosystems' capacity to be preserved for future generations. Social sustainability is concerned with the society's aspirations related to community well-being, equity, or justice but also trust and ethical behaviors [START_REF] Osburg | Sustainability in a digital world needs trust[END_REF].

As sustainability is becoming an important concern worldwide, companies are embracing this new demand to develop their competitive advantage. Sustainability is considered by most of them as an opportunity to exceed competitors, in order to not only generate higher economic benefits, but also to develop environmental and social values [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF]. Many firms have already established strategic initiatives aiming at exploring major opportunities for innovating through sustainability [START_REF] Lichtenthaler | Digitainability: the combined effects of the megatrends digitalization and sustainability[END_REF]. To enhance their sustainability outcomes, they are developing a triple bottom line approach, encompassing financial, environmental and social outcomes [START_REF] Elkington | 25 year ago I coined the phrase "triple bottom line[END_REF]. All the new initiatives developed by companies, which aim to achieve financial returns, while having a positive impact on both the environment and the society, are looking at the Sustainable Development Goals (UN-United Nations, 2015) to find guidance and directions. These initiatives foster different forms of innovations by focusing on either developing new products, services, processes, or business models, having potential positive impacts on their business activity [START_REF] Pfitzer | Innovating for shared value[END_REF]. Even though sustainability raises many challenges and uncertainties for individual entrepreneurs and companies, it represents a truly driving trend that needs to be embraced, so they can maintain and develop their competitiveness.

Servitization as a powerful sustainability strategy in the agrifood sector

The focus on sustainability is absolutely not a new thing for companies. However, it nowadays requires to undertake significant transformations to contribute to more of it worldwide [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF]. That is why developing strategies and executing business development plans able to explore and exploit sustainability opportunities is becoming crucial to secure companies' competitive advantage [START_REF] Lahti | A definition and theoretical review of the circular economy, value creation and sustainable business model: where are we now and where should research move in the future?[END_REF]. To embrace the sustainably challenge, companies need to develop new ways of working and doing their business activities (Gupta et al., 2020).

One way to deploy sustainability initiatives is to move from a product-centric to a service-centric approach (Lenka et al., 2018a), meaning a transformation toward servitization [START_REF] Vandermerwe | Servitization of business: adding value by adding service[END_REF]. Servitization is a concept based on the service-dominant logic approach [START_REF] Vargo | Service-dominant logic: continuing the evolution[END_REF], which emphasizes the importance of developing solutions for the customers and not just focusing on offering products (Lenka et al., 2018b). Thus, developing servitization requires companies and their stakeholders to switch from product-oriented business models toward either result-or outcomeoriented business models [START_REF] Reim | Product-service system (PSS) business model and tactics: a systematic literature review[END_REF]. A result-oriented business model business involves setting specific goals, measuring progress toward those goals, and adjusting strategies when needed. An outcome-oriented business model focuses on solutions to customers by developing more personalized and customers' specific offers. For instance, moving from unrenewable to renewable energy or allowing recycling and reuse can be implemented by means of measurable and verifiable goals to be achieved, so that progress toward these goals can be assessed. Improving maintenance commitments or providing on-demand services to customers develop a market culture that deeply changes the business models of previously product-oriented companies [START_REF] Tukker | Rare earth elements supply restrictions: market failure, not scarcity, hamper their current use in high-tech applications[END_REF] and call to develop a better knowledge of their partners and customers [START_REF] Vargo | Service-dominant logic: continuing the evolution[END_REF].

The servitization transformational challenges, resulting of either goal setting or service provision, force practitioners, managers, entrepreneurs, or corporate leaders to rethink their organizational priorities and reorganize their business operations [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF]. Indeed, servitization is not just a strategy of post-sales involvement that can become a source of additional revenues. It rather means that companies develop more integrated commercial offers [START_REF] Reim | Implementing sustainable product-service systems utilizing business model activities[END_REF], based on the provision of a mix of products and services answering to specific customer demands regarding not only economic aspects, but also environmental and social ones [START_REF] Tukker | Product-services for a resource-efficient and circular economy: a review[END_REF]. These new offers have greater potential for value creation, since they become able to fulfill unique customer needs. That is to personalize commercial offers by focusing more on the functionality of the solutions offered rather than on the commodities sold. Consequently, it implies that companies invest in developing new competencies and capabilities to enhance their integrated productservice offers [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF]. If these transformations are well managed within and outside companies, especially with their network partners, companies can gain long-term competitiveness, mainly due to a higher customization of their offers. In doing so, companies are expected to become able to meet the always evolving needs of their customers [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF] or to design and develop new roles in traditional value chains [START_REF] Reim | Mitigating adverse customer behavior for product-service system provision: an agency theory perspective[END_REF].

Consequently, servitization involves that producers take full responsibility of their offers. It implies a complete redefinition of customer-supplier relationships. It thus implies a new way for companies and their stakeholders to engage with customers through the development of more fidelity and trust. Doing so is really challenging. Indeed, designing and selling product-service combinations are not an easy task. It needs an alignment of all the stakeholders' interests, a change of mindset of all the participants, and the management of the behavior of less careful customers in order to avoid any rebound effect [START_REF] Ferreira | The transition from products to solutions: external business model fit and dynamics[END_REF]. Not all companies have the necessary capabilities and skills to implement successfully servitization. Main challenges most often result from immature service business models, misaligned internal processes, or underdeveloped service delivery networks. That is why many companies are struggling to develop and deliver servitization solutions effectively. To develop servitization, exploring digitalization capabilities is becoming a trend, companies are entering into.

Digitalization as a transformative trend in the agrifood sector

Digitalization is the result of the explosion of computation power and data-driven solutions (WEF-World Economic Forum, 2019) that has pushed forward the possibility to access new information and foster knowledge creation and sharing. Nowadays, digital technologies are developing across all segments of the economy and society, due to better connectivity, new capabilities to collect huge amount of data, and increased possibilities for generating, processing, analyzing, using, and transferring data and all kinds of information [START_REF] Katz | Measuring digitalization: a growth and welfare multiplier[END_REF]. As data are processed by intelligent algorithms and increasing computational capacities [START_REF] Duan | Artificial intelligence for decision making in the era of big-data-evolution, challenges and research agenda[END_REF], they provide unprecedented opportunities to explore, understand, predict, optimize, and innovate [START_REF] Del Rio Castro | Unleashing the convergence amid digitalization and sustainability towards pursuing the sustainable development goals (SDGs): a holistic review[END_REF]. Besides, digitalization refers to the social transformation triggered by the mass adoption and use of digital technologies. It refers to the changes in habits and ways of doing business due to the development of highly interconnected and more complex networks allowed by the development of digital technologies [START_REF] Letouze | Towards a human artificial intelligence for human development[END_REF].

Digitalization has been at the top of the strategic agenda of numerous companies that are transforming by means of new opportunities offered by digital solutions, advanced data analytics, and artificial intelligence applications [START_REF] Boldosova | Storytelling, business analytics and big data interpretation: literature review and theoretical propositions[END_REF]. Digitalization helps companies optimize their business operations, while enabling them to create new commercial offers for their customers or developing new business activities [START_REF] Haefner | Artificial intelligence and innovation mamagenet: a review, framework, and research agenda[END_REF]. For instance, numerous digital projects explore the potential of digitalization to enhance transparency, assess ESG -Economic, Social and Governance-performance [START_REF] Kiron | The convergence of digitalization and sustainability[END_REF], foster discovery (WEF-World Economic Forum, 2020), improve resource efficiency, generate evidence-based decision-making and prediction (GeSI-Global e-Sustainability Initiative, 2019), facilitate financial access [START_REF] Tchamyou | Inequality, ICT and financial access in Africa[END_REF], and even reduce inequality [START_REF] Asongu | How enhancing information and communication technology has affected inequality in Africa for sustainable development: an empirical investigation[END_REF], among others. However, digitalization also faces strong controversies and uncertainties, especially about the relationship between digitalization and sustainability [START_REF] Saunila | Smart technologies and corporate sustainability: the mediation effect of corporate sustainability strategy[END_REF], the creation of powerasymmetries, or the generation of digital divides (ITU-International Telecommunication Union, 2017). Indeed, digitalization can have unintended environmental or social effects [START_REF] Scholz | Unintended side-effects of the digital transition: European scientists' messages from a proposition-based expert round table[END_REF], like the disposal of raw materials affecting biodiversity (WEF-World Economic Forum, 2020), the emission of pollutants, an increasing use of energy and water (ITU-International Telecommunication Union, 2017), and even heath issues (WEF-World Economic Forum, 2020), implying that digitalization can create unsustainable outcomes. At the same time, technological impacts, derived from the digitalization process, are expected to be remarkable and have started to impact everyone in the world at an unprecedented pace [START_REF] Bernstein | The great decoupling: an interview with Erik Brynjolfsson and Andrew McAfee[END_REF].

Even though, no consensus has been achieved yet regarding the definition of digitalization, it can be defined, from a business perspective, as "the use of digital technologies to change a business model and provide new revenue and value-producing opportunities" [START_REF] Gartner | The digitalization sustainability matrix: a participatory research tool for investigating digitainabilty[END_REF]. As companies are becoming more digitalized, their ability to create data-driven activities increases. By allowing their products to collect data or be connected, more and more data are made available and can be used to improve analytics and generate new information and knowledge. That is why it becomes really important for companies to be aware of the consequences and impacts of digitalization, either positive or negative, when entering this trend [START_REF] Porter | How smart, connected products are transforming competition[END_REF]. Security, privacy and digital ethics are aspects that need to be considered, since they imply new specific requirements that have to be taken into account when they develop innovative initiatives for the digital business environment. For instance, consumer privacy and transparent use of data are nowadays becoming a prerequisite for entering new markets or developing new commercial offers.

Even though the economic and social impacts of digitalization are under the fire of controversy, it must be acknowledged that digitalization has already started transforming many business domains and is a of great opportunities to create and capture value [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF]. However, only a few companies are really able to truly benefit from the opportunities of digitalization, due to the transformational aspects it involves [START_REF] Porter | How smart, connected products are transforming competition[END_REF]. Indeed, some of them still lack the tools, processes, knowledge, skills, or competences to generate useful information from data. But, at the same time, not considering digitalization can have a huge impact on the competitiveness and efficiency of businesses. For instance, working with systems that do not communicate to each other, leading to the replication of data collection or generating process bottlenecks, can create unnecessary delays [START_REF] Parida | Why and how to compete through sustainability; a review and outline of trends influencing firm and network-level transformation[END_REF], which in turn can reduce the competitiveness of companies due to lower reaction time than their main competitors. Although digitalization can be, at the very beginning, a source of important challenges for entrepreneurs and all types of companies [START_REF] Malmström | The digitalization of banks disproportionately hurts women entrepreneurs[END_REF], it is also a genuine transformative trend. It may have important impacts on the future of companies' competitive advantage. That is why more and more companies have integrated it in their innovation strategies.

Digitainability: innovating by bringing together digitalization and sustainability

Digitainability is a concept created by Gupta et al. (2020) as the merging of digitalization and sustainability. Gupta et al. ( 2020) consider digitalization as a source of unique capacities that can significantly support robust societal and environmental disruption and describe digitainability as the cross fertilization between the digitalization and sustainability strategies of companies. Using an innovation-based theoretical perspective, Lichtenthaler (2021) introduces a conceptual framework allowing the positioning of companies depending on their level of digitalization and sustainability development and then identifies the best innovation trajectory they should embrace to capture the benefits of both the sustainability and digitalization trends.

In his conceptual framework, Lichtenthaler (2021) identifies four types of companies: (1) those that have established businesses with limited levels of either digitalization and sustainability. The core of their business model does not specifically address, either separately or jointly, the strategies of digitalization and sustainability; (2) those with established sustainability initiatives, but with a low level of digitalization. Sustainability is at the core of their business model, with some initiatives specifically focusing on providing solutions to address the Sustainable Development Goals (UN-United Nations, 2015); (3) those involved in digitalization initiatives, with a low focus on sustainability. The core business of these companies is mainly related to digital solutions and information technologies; and lastly (4) companies where digitainability opportunities are explored, with a core business aiming at leveraging the unutilized potential of high levels of digitalization in combination with high levels of sustainability (Gupta et al., 2020).

Depending on their initial position, each type of companies faces different challenges and opportunities toward innovating in both the sustainability and digitalization trends and thus has to develop very specific strategies [START_REF] Lichtenthaler | Digitainability: the combined effects of the megatrends digitalization and sustainability[END_REF]. Indeed, for companies with established business models that have not really explored the potentialities of either sustainability or digitalization in their core business, a way to innovate is to investigate new initiatives and opportunities in one or another of the two trends or fostering the combination of both. For instance, starting new projects thought and developed, from the very beginning, for enhancing sustainability by means of specific digital tools and solutions can create new sources of value creation. Some digital tools aiming at increasing the efficiency and effectiveness of business activities are also designed to help companies improve their environmental and social results and can be considered to develop new innovation strategies. For companies with established sustainability initiatives, a digitalization strategic focus could be used to transform, extend, or complement their sustainability initiatives. By developing digital sustainability innovation initiatives, these companies can expand their activities in a way that meets the specific needs and expectations of their customers, through more personalized offers. For instance, the use of specific digital solutions, such as sensors, QR codes or blockchain can really support more transparency and communication on the sustainability activities of these companies. For companies involved in digitalization initiatives, an important strategy to strengthen their competitiveness can be to focus on enabling sustainable digitalization initiatives. That means finding ways to adapt or extend their digitalization programs or solutions, so they can contribute to more sustainability outcomes. For instance, intelligent digital solutions, based on big data analysis or artificial intelligence algorithms, can be developed and used for optimizing energy or water consumption, but also for better connecting producers and consumers through more information sharing. The main advantage that can be expected here is a reduction of the environmental impacts of digital technologies and solutions, while at the same time improving social interactions. Finally, for companies where digitainability opportunities are already explored, meaning that they are developing initiatives that consider both sustainability and digitalization, the main challenge they face with their innovation strategy is to find the right balanced between both trends. Indeed, balancing both trends adequately is a way to ensure that they are able to generate financial returns, while at the same time creating sustainable outcomes. When well-managed, this specific strategic orientation can become the source of high level of value creation and capture. However, if mismanaged, companies

Frontiers in Sustainable Food Systems 05 frontiersin.org cannot really be described as having a digitainability strategy, as one trend dominates the other, and thus, they have to face the same strategic challenges than the two previous groups, which either focus on sustainable digitalization or digital sustainability.

3 Open innovation: developing innovative initiatives in partnership

Startups as powerful open innovation organizations

Startups play a key role in the innovation process [START_REF] Spender | Startups and open innovation: a review of the literature[END_REF], as they focus on developing new ideas that, when they are carried out and launched on the market, can transform them into economically sustainable companies [START_REF] Blank | Why the lean start-up changes everything[END_REF]. Startups can be defined as very small or small enterprises working in a dynamic and agile environment, where the possibility of developing breakthroughs in conventional innovation processes or disruptive products and services that impact value chains and dominant market positions is made possible [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF]. Most often, their value propositions are driven by human motivations, such as the development of socially responsible behaviors, the provision of products and/or services aiming at achieving sustainability goals, or the improvement of the quality of life [START_REF] Santos | Understanding how and when personal values foster entrepreneurial behavior: a humane perspective[END_REF]. However, for startups to develop their innovations and become scalable, they need a specific supportive environment that can be found in open innovation initiatives and partnerships.

Open innovation (OI) can be described either as a partnership process between new enterprises, as startups, and science-based innovation entities, such as universities or research institutions [START_REF] Van Beers | R&D internationalization, R&D collaboration and public knowledge institution in small economies: evidence from Finland and the Netherlands[END_REF] or as the joint development between market-based partners, through collaboration with suppliers, customers, consumers, and even competitors [START_REF] Hensen | Hierarchical business value of information technology: toward a digital innovation value chain[END_REF]. Thus, developing networks is the core of the success of OI processes. It allows OI to create channels enhancing and fostering the acquisition of resources, such as knowledge or funding, but also facilitating the introduction of new products on the market [START_REF] Lundberg | Triple helix in practice: the key role of boundary spanners[END_REF]. Working in networks is not without consequences, as networks mutually shape each participant's contribution and offer. Indeed, all main decisions are becoming intertwined as all the participants of the network contribute to defining the requirements for new products or services, producing new technological tools and solutions, which as a consequence modify the relationships among the network's participants (La Rocca and Snehota, 2014). At the same time, socio-economic environmental factors affect the strength and direction of networks [START_REF] Spender | Startups and open innovation: a review of the literature[END_REF]. Depending of the structure, institutions, and organizational relationships within, around, and outside the network, IO processes and initiatives do not have the same scope, magnitude, and impact on the different participating parties.

For startups, OI implies an involvement in developing relationships with a large range of actors and connecting to external initiatives. For instance, the creation of venture capital funds is considered as partnerships between two types of partners: the startup managers and the investor advisors. It helps startups raise funds for innovative business models and develop their networks in several dimensions, related either to financial, commercial, or technologybased aspects. The launch of sectoral incubators or accelerators is another possibility to help startups develop, especially by supporting the emergence of their innovative projects and providing services, such as training or office spaces [START_REF] Spender | Startups and open innovation: a review of the literature[END_REF]. The establishment of startup studios focuses on the simultaneous launch of several projects, while providing many strategic and operational skills, such as business, legal, design, prototyping, development… Both the environment they are in and the mentoring they have access to allow startups to gain in agility but also accelerate the time to market of their projects by boosting their growth and facilitating their access to funding and the implementation of their commercial and operational strategies. The participation in private or public acceleration programs, most often around a specific theme, allows startups to develop their strategic intelligence, together with their access to financial and human resources. The contribution of innovation lab is very different. It covers services such as foresight (anticipation of the future in 5-10 years), agility and incubation, open innovation, facilitation of collaborations… Among the other OI initiatives, hackathons, reverse pitches, competitions, and other calls for projects facilitate the sourcing of external ideas and become research-action devices of the innovation ecosystem, where participants are not simply end-users but become actors and collaborators. It is particularly true for farmers and consumers, when they are placed at the center of the innovation process [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF]. These different approaches of OI are rarely exclusive, as some of them can be implemented in parallel, creating what is often called startup ecosystems. Those ecosystems are built upon the interconnections between different groups of organizations, including new ventures, universities, and public administrations, with startups and aim at supporting and fostering the success of these innovative organizations.

Among the most important systems that need to interact with startups, [START_REF] Spender | Startups and open innovation: a review of the literature[END_REF] identify the financial system, the knowledge creation and diffusion system, and the governance system. They highlight not only their importance on the functioning and performance of startup ecosystems, but also their contribution to the facilitation of formal and informal relationships among various types of actors. More specifically, developing collaboration with different external innovation partners allows startups increase their exposure to "new ideas, gain access to a broader range of market information and complementary resources, benefit from the co-development of new products, or win new markets" [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF]. Indeed, research shows that the capability of startups to develop a large network, create a central position, and balance long-and short-term relationships with their network's partners has a very positive impact on their innovativeness. At the same time, the strength and complexity of the relationships within the network can have either be positive or negative, depending on the degree of uncertainty of the business environment there are in [START_REF] Spender | Startups and open innovation: a review of the literature[END_REF]. Thus, the network, they are developing IO with, may have a very important impact on the innovation performance of startups [START_REF] Wang | The moderating effect of environmental uncertainty on the relationship between network structures and the innovative performance of a new venture[END_REF], as it is able to foster or hinder their innovative capabilities.

In the face of the development of these OI initiatives, larger companies are adopting different approaches, such as corporate venture capital, internal incubators, strategic alliances, joint ventures with startups, new ventures, new spinoffs, or spinouts. However, the innovativeness, creativity, and growth of startups, as well as their ability to launch innovative-disruptive solutions, have been a real challenge for larger and well-established companies and have created the need to develop more agile and rapid means to engage with startups [START_REF] Weiblen | Engaging with startups to enhance corporate innovation[END_REF]. Indeed, larger companies, which henceforth Frontiers in Sustainable Food Systems 06 frontiersin.org face an increased competition coming from startups, are forced, in one way or another, to take new risks, launch new initiatives, and develop a strategy of open innovation. These changes involve important internal transformations, related to either the corporate culture, working environment, or organizational management. It also requires increased capabilities to continuously digitize their activities in order to sustain their resilience and support the development of new activities. At the same time, companies need to improve their positioning on their current markets, while entering the new markets created by the unfolding of digital technologies and sustainability. This is most often done by either integrating promising startups, creating them, or co-developing innovations with them. However, whatever the options chosen, two main challenges remain. The first one is a financial and economic issue concerning the return on investment of OI initiatives, as it can take several years for startups or new activities to become profitable. The second challenge deals with the cultural and organizational transformation needed to move from a conventional innovation process to OI. The conventional innovation process mostly relies on specific research and development (R&D) programs, focusing on time-limited projects or supporting corporate incubation. At the opposite, OI initiatives require a continuous and more integrated approach of innovation (Piot-Lepetit and Florez, 2022).

Open innovation and organizational ambidexterity

As many well-established companies face an increased competition from innovative startups [START_REF] Svahn | Mastering the digital innovation challenge[END_REF], they need to develop their ability to offer products in a new way, invent new products, and create or participate to new markets, so they can avoid being overtaken by the competition. Thus, companies are reconsidering their innovation strategies and investigating open innovation (OI) initiatives [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF]. This requires accelerating either investments to gain expertise, the acquisition of promising startups, or the display of innovations in their value proposition focusing on either sustainability or digitalization. Consequently, innovation, which was previously based on products and developed internally, now emerges from multiple horizons and leads corporations to move from inside-out innovation processes to outside-in OI approaches, allowing them to be confronted with new ideas and, at the time, fostering their ability to renew [START_REF] Weiblen | Engaging with startups to enhance corporate innovation[END_REF].

The strategic responses developed by established companies to preserve or expand their market shares, while facing unexpected innovations coming from startups, are varied. While a few companies choose to ignore the challenge and continue to invest in their business activities as usual, some companies embrace the new competition and try to adapt. Others not only adapt to the new competitive environment but also counter attack by developing new disruptive innovations [START_REF] Alpkan | Disruption and ambidexterity: how innovation strategies evolve?[END_REF]. However, most companies implement their new innovation strategy in two main directions. On the one hand, they pursue interval innovations within the company, the inside-out option, in the form of R&D projects, corporate incubators, or startup platforms [START_REF] Weiblen | Engaging with startups to enhance corporate innovation[END_REF]. On the other hand, they develop outside-in programs by sponsoring or partnering with startups through OI, or by sharing knowledge inputs and resources with external partners [START_REF] Du | Managing open innovation projects with science-based and market-based partners[END_REF]. By doing so, they are developing an environment where innovations derive from both internal and external sources. From an organizational perspective, they also transform themselves for developing more agility, so they can detect and seize market opportunities with speed and surprise. The transformative programs, they are carried out, aim to create a real culture of innovation, move toward a less hierarchical model with more horizontal structures helping promote collaborative work, develop multidisciplinary teams, and encourage individual initiatives. The main consequence of these changes is that their innovation processes are becoming more and more ambidextrous [START_REF] Lewin | Microfoundations of internal and external absorptive capacity routines[END_REF]. Indeed, companies try to stay aligned and efficient, while becoming adaptable and flexible to future changes [START_REF] Tushman | The ambidextrous organization: managing evolutionary and revolutionary change[END_REF]. Their long-term success depends on their ability to exploit their current capabilities, while simultaneously exploring new competencies, products, technologies, and markets [START_REF] Levinthal | Myopia of learning[END_REF]. Therefore, both inside-out and outside-out innovation processes are important for large companies. They allow them to achieve higher internal performance and develop both internal and external value-creating mechanisms, with as a constraint, the need to find an appropriate balance between their internal and external innovation components.

Some opportunities and challenges of open innovations fostering digitainability in the agrifood sector

The use of digital technologies is considered as a very relevant solution to empower the agrifood sector and meet sustainability objectives. It has the ability to enhance productivity and resilience, while reducing environmental and social impacts (FAO-Food and Agriculture Organization of the United Nations, 2020). By consequences, the way food is produced, processed, transported, distributed, and consumed is changing through the development and implementation of digital technologies and solutions as well as the request of meeting the sustainable development goals (SDGs) by 2030 (UN-United Nations, 2015). More specifically, digital technologies allow the collection, analysis, and utilization of data by means of sensors or imagery, advanced analytics or smart algorithms, while digital solutions focus more on providing software or platforms integrating different but complementary digital technologies. The SDGs are influencing the way current practices need to be thought about in order to develop more responsible approaches in value chains, such as reducing chemicals and food waste through changes in behaviors. Thus, digitalization and sustainability do not only affect on-farms productive processes but also changes entire agrifood value chains [START_REF] Klerkx | A review of social science on digital agriculture, smart farming and agriculture 4.0: new contributions and a future research agenda[END_REF]. Indeed, stakeholders of value chains, focusing on providing more traceability and information sharing, not only contribute to the development of new business models but also develop trust with their customers [START_REF] Porter | How smart, connected products are transforming competition[END_REF].

The development and deployment of these initiatives can come out from either micro enterprises, innovative startups, small and medium sized enterprises (SMEs) and of course large companies [START_REF] Birner | Who drives the digital revolution in agriculture? A review of supply-side trends, players and challenges[END_REF]. However, startups can be looked at as key actors of digitalization [START_REF] Kergroach | Giving momentum to SME digitalization[END_REF] in the agrifood sector. More specifically, [START_REF] Florez | How do French Agritech start-ups contribute to the sustainability of food value chains?[END_REF] found that the digital technologies provided by startups are most often using artificial intelligence (AI), internet of things (IoT), big data, blockchain, cloud computing, robots and e-commerce. But, with a dominance of IoT and e-commerce in the production and retail/consumption segments of agrifood chains, while AI is deployed in all segments. At the same time, new startups are more and more directing the development of their digital technologies and solutions in a way that supports sustainability initiatives [START_REF] Liguori | Rising to the challenge: entrepreneurship ecosystems and SDG success[END_REF]. Most often, their initiatives aim at achieving higher levels of productivity through resource efficiency, promoting local culture and products, or developing access to banking, insurance, and financial services in order to support the economic dimension of sustainability. Indeed, some startups focus on the environmental dimension of sustainability by means of more efficient use of natural resources, optimized application of chemicals, reduction of food waste, increased recycling and reuse, and improved resilience and adaptive capacities to climate hazards. Other startups provide solutions for social outcomes such as improving transparency, changing in consumption, production, and distribution behaviors, reducing food losses and wastes [START_REF] Florez | How do French Agritech start-ups contribute to the sustainability of food value chains?[END_REF], fostering inclusiveness, or developing the sharing economy [START_REF] Fauzi | The digitalization of micro, small, and mediumsized enterprises (MSMEs): an institutional theory perspective[END_REF]. In doing so, startups are becoming important providers of inclusive and sustainable innovations using digital technologies. They are finding ways to create and capture value in very specific markets where digitainability initiatives are welcome, but also in every segment of agrifood value chains, where new digitainability solutions are expected to resolve complex sustainable issues. However, these startups also face important challenges. Most often, they lack of tangible and intangible resources [START_REF] Wymer | Factors influencing e-commerce adoption and use by small and medium businesses[END_REF], involving very often financial and human issues. These issues hinder their possibilities of developing and supporting their new innovation processes. These limitations most often result from both their newness and smallness [START_REF] Bogers | The open innovation paradox: knowledge sharing and protection in R&D collaborations[END_REF]. That is why to overcome these constraints, startups are adopting open innovation practices. Meaning that they are partnering with different stakeholders and integrated networks with the aim of supporting their development and growth.

At the same time, well-established companies confront the emergence of these new competitors by moving their value proposition toward more digitalization and sustainability innovation. This implies modifying their market positions and creating new services, sales models, or distribution and communication channels for all the participants of value chains [START_REF] Piot-Lepetit | Start-ups and digital innovation in the Agrifood sector[END_REF]. Consequently, innovating, by means of the incorporation of digital services in products, the development of new services, or the generation of initiatives fostering sustainability, is becoming strategic for many wellestablished companies, as they face an increased competition from innovative startups [START_REF] Svahn | Mastering the digital innovation challenge[END_REF]. Thus, it strongly reinforces the importance of developing new innovation strategies helping them to embrace the ongoing digitainability transformation, as an important source of competitive advantage. However, it involves managing an ambidexterity innovation process and implementing the undergoing necessary organizational changes it requires, which remain an important challenge not to be overlooked.

Conclusion

The research developed a conceptual analysis examining the concept of digitainability, as the convergence amidst digitalization and sustainability, and the role of open innovation initiatives to support digitainability innovations in the agrifood sector. More specifically, the exploration of the literature shows that the digitainability concept is closely related to the aspects of servitization companies are trying to develop to meet customers' specific needs. It also shows that digitainability is more easily developed by startups. Due to their smallness and innovativeness, they are more prone to take risks on new ideas and value propositions. Established companies are challenged by this new competition, implying a transformation of their innovation strategies toward open innovation and as a consequence their innovation processes that are becoming more and more ambidextrous, as they try to develop both inside-out and outside-in innovation initiatives. Besides, those changes do not only affect companies, whatever their size, they also have an impact on their overall value chains and network arrangements.

Concerning the agrifood sector, digital solutions have been identified as a potential answer to sustainability issues by many international organizations and governments. All over the world, initiatives are developed to create powerful and creative startup ecosystems able to explore the solutions that are needed to support and develop sustainability solutions relying on digital technologies. The analysis of the literature, conducted in this research, shows that this is not without impact on the way business activities are designed and developed, relationships among agrifood value chains' stakeholders are organized, and companies' innovation strategies are designed and implemented. However, digitainability expectations and open innovation initiatives are currently reconfiguring agrifood chains in ways that are not well explored yet. Both concepts are two important topics for which the current conceptual analysis provides new insights, but where additional research is still needed to gain a better understanding on the ongoing transformations digitainability and open innovation are generating in the agrifood sector.

Frontiers in Sustainable Food Systems

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Acknowledgments

The author would like to thank her colleague Fatima El Hadad-Gauthier (CIHEAM-IAMM) for our fruitful discussions and her support all along the development and realization of this research project.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by a government grant managed by the Agence Nationale de la Recherche under the France2030 program, reference ANR-22-PEAE-0002.

Author contributions

IP-L: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Validation, Writing -original draft, Writing -review & editing.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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