Damien Besancenot
email: damien.besancenot@parisdescartes.fr.

Radu Vranceanu
email: vranceanu@essec.edu.

The generosity spill-over e¤ect of pledges in a two-person giving game

Keywords: Experiments, giving, communication, pledges, social in ‡uence. JEL Classi…cation: C92, D64, D83

This paper analyses the equilibrium of a two-person "pledge and give" game and tests its predictions in a lab experiment. Each person's endowment is private information. In the …rst stage, each agent informs the other about the amount he/she intends to give. In the second stage, each agent makes a contribution to the joint donation. We show that a linear pledge function can be an equilibrium of the game. Furthermore, if agents have a strong taste for conformity, the equilbrium donation is positively related to one's own endowment and to the pledge of his partner. In the lab, subjects pledged on average 60% of their income and donated on average 50% of their income. High pledges foster the generosity of the partner: an agent will increase the donation by 20 cents on average if the partner pledges one extra euro.

Introduction

A notable story in the history of philanthropy is the 1997 pledge by US media billionaire Ted Turner to give one billion dollars to the United Nations over the following ten years. In the giving literature, the pledge is de…ned as an unbinding promise to give money at a later time. Turner kept his promise, and ended his payment to the UN in 2016. 1 If his action could have been driven by several motives, he stated once that one main purpose of his generous pledge "was putting other rich people on notice that I would be calling on them to be more generous."2 Many large fundraising campaigns, such as the French Téléthon, which collects funds to …ght myopathic disorders, rely on the pledge mechanism (Sutan et al. 2018). Several …eld experiments revealed that donors to charity are subject to social in ‡uence; individuals can be induced to give more if they receive information that their peers made generous gifts (Martin and Randal, 2008; Shang and Croson, 2009; Sarah et al., 2015; Sasaki, 2019).

A large pledge is a double edge sword. On the one hand, a potential contributor who observes a high pledge might be prompted to give more, in particular if he/she has a strong taste for conformity. On the other hand, the same large pledge can adversely a¤ect the donation of the pledger, because the latter can now rely on the generosity of the others. To study these contradictory e¤ects, this paper introduces a simple "pledge and give" game with imperfect information about the income of the donors, and implements an original lab experiment to test its predictions.

The game features two agents with identical preferences who must make a joint donation to a charity. Agents are assumed to be altruistic, as the bene…t of the charity enters into their utility function next to their private consumption. They can have a di¤erent endowment, and the latter is private information. We explicitly model the preference for conformity and the cost of breaking a promise. To isolate the e¤ect of the pledge on the behavior of the partner from other motives to pledge such as improving self-image over time, the giving decision follows immediately after the communication. The decisions are simultaneous; both agents make pledges at the same time, and then decide on their contribution to the joint donation. The main implications of the theoretical analysis, which also constitute key hypotheses to test, is that a pledge proportional to the income is an equilibrium of the communication game. Because the signal is correct, agents with a strong preference for conformity will adapt their donation to the pledge of the partner. A numerical example shows that, depending on the parameters, agents might systematically over-state their e¤ective donation.

In the second part, the paper reports the results of a lab experiment designed to test these predictions. Participants were matched in pairs, and asked to pledge, then to make a joint donation to a charity. Our empirical results corroborate the linear relationship between the pledge and the individual's endowment on one hand, and that between the donation of one agent and the pledge made by his partner on the other hand. We thus provide evidence that pledges can be a vector of social in ‡uence, with donors being tempted to align their own donation to the gift promises of other donors in the group.

Our analysis is connected to a substantial body of literature on social in ‡uence. Scholars in social psychology have documented the strong tendency of human beings to imitate the actions of others (Asch, 1951; 1955; Nook et al., 2016), behavior also referred to as pro-social conformity. In Sugden (1984), individuals'conformism or reciprocation of minimum investment in the group can explain the production of higher than rational (Samulesonian) amounts of public good. Bernheim (1994) The literature on giving and philanthropy emphasizes the commitment and signaling e¤ect of pledges (see Vesterlund 2006 for a survey). Andreoni et al. (2015, 2016) assume that people dislike saying "no" when solicited to give; they analyze the donor's decision problem that compares the immediate gain from the pledge with the discounted cost of executing the promise at a later time (see also Damgaard and Gravert, 2017 and Schulz et al., 2018). This cost bene…t analysis reveals that actual donations are positively related to the strength of the commitment device used to enforce the pledge. On the other hand, pledges made in public make sense if donors care about their social image. In general, large donations boost the social image of the donors and signal their wealth (Glazer and Konrad, 1996; Bénabou and Tirole, 2006; Andreoni and Bernheim, 2009; Grossman, 2015). In these models, the observers'beliefs about the type of donor enter into the utility function of the latter. In this context, donors can use pledges as a signal for future donations, and bene…t from the enhanced social image until the time for donation arrives, when they might behave opportunistically or not. In our model, experimental wealth is attributed at random; subjects would not experience pride from this. Also, pledges and giving follow at a very short interval, thus the bene…t of a positive image is ephemeral, the essential element being the donation itself, and the value of the promise for the sender. Finally, Kartik (2009) analyses a sequetial sender-receiver communication game with quadratic lying costs (for the sender), and proves that in ‡ated language is almost always an equilibrium.

The paper is organized as follows. The next section introduces a simple pledging game between two identical agents, and solves it for the equilibrium pledge function. Section 3 presents the experiment used to test the predictions of the model. Section 4 is our conclusion.

2 The model

Main assumptions

This section analyzes the "pledge and give" game between two players with identical preferences.

First players receive their endowment or income, then simultaneously make reciprocal pledges, and then make a joint donation to a charity.

The two players are denoted by the index z; with z = fi; jg and where z denotes the partner of agent z: At the onset of the game, each one receives an income r z (respectively r z ) drawn at random from the same income distribution. Subjects know that the income is a random variable drawn from the same income distribution, but are not informed what this distribution is. This assumption allows for a simple rule to guess the income of the other. If the distribution is unknown and one does not know his/her own income, the income of the other could be anything. However, when one knows his own income, and the other obtained a draw from the same distribution, his/her best guess is to assume that the other got the same income, since, at least, this number exists in that distribution. Ex-post, these beliefs can be correct or not.4

In the next stage, players must simultaneously indicate how much they intend to give, or make a pledge; the pledge of player z is denoted by a z with z = fi; jg. This promise is unbinding.

Then, players simultaneously receive the information about the partner's pledge, and decide on their contribution to the pair's gift. Denoting by g z the donation of each agent (with g z < r z ), the pair will donate (g z + g z ) :

At the end of the game they learn the gift of the pair (they can thus infer the true gift of the partner).

We assume that, at the outset of the game, agent z seeks to maximize net utility:

Vz = U (ĉ z ; ĝz + ĝ z ) (ĝ z ĝ z ) 2 k(a z ĝz ) 2 ; with z = fi; jg (1)
where the "hat" represents an expected variable. The use, at the …rst stage, of expected variables conditional on the information set (i.e., his/her own income) is required because the agent does not know the income of the other player, and, as a consequence, does not know what his/her own actions will be once that this income will be revealed to him/her. At the last stage, the ex-post utility depends on the income of the player and the income of the partner.

In general, the net utility of the individual z includes three components: a) U (c z ; g z + g z ) is a standard utility function with altruistic agents; individuals prefer more consumption to less, @U=@c > 0; and care about the charity, @U=@(g z + g z) > 0: A standard assumption in the analysis of giving is that these utility curves entail convex indi¤erence curves, making possible the existence of interior solutions for the utility maximization problem (Glazer and Konrad, 1996; Martin and Randal, 2008; Andreoni and Bernheim, 2009; DellaVigna et al., 2012).

To keep the analysis tractable, in the following we use the multiplicative form:

U (c z ; g z + g z ) = c z (g z + g z ) ; with c z = r z g z : b) k(a z g z ) 2
is the cost related to the unful…lled promises, with k as a positive parameter; this preference for keeping a promise can be grounded in how individuals view and perceive themselves with respect to their own norms or express guilt when not living up to their partner's expectations. 5 While the pledge is not legally binding, the lying cost is a light form of commitment.6 c) (g z g z ) 2 is the cost of diverging from the donation of the other, related to the taste for conformism of the agents as documented in the introduction (DellaVigna et al., 2012; Barett and Dannenberg, 2016), with as a positive parameter.

This utility function discards the "warm-glow" motive of giving (Andreoni 1989, 1990). The implications of the model would not change if this motive is included in complement to the altruistic motive as an utility premium related to the agent's own donation.

Solution and equilibrium

An equilibrium of this game is characterized by the equilibrium donation and pledge functions, de…ning the strategy of an agent depending on his/her type.

In this game, the only di¤erence between the two agents, which can justify di¤erences in their pledges, is their income. Thus, the optimal pledge can be written as a function of the income of the subject, a z = a(r z ); and the donation is a function of both the subject's income and of the pledge of his partner, g z = g z (r z ; a z ) with z = fi; jg:

The complete resolution is presented in the Online Appendix. We here present only the basic steps.

Because of the complexity of the game, it would be di¢ cult to characterize the general class of pledge functions that are the equilibrium of the game. We instead follow a method frequently used in the auction literature: we make the assumption that individuals use a linear pledge function a z = r z with > 0 and then prove that this linear function is one possible equilibrium of the game. 7 The linear function has the bene…t of simplicity; this makes it an intuitively appealing choice for subjects in a lab experiment.

Given the sequential structure of the game, we …rst determine the optimal donations at the last stage of the game, taking the …rst stage pledges (a z ; a z ) as given. At that time the uncertainty about the revenue and donation of the partner is removed (so no hat is needed for the variables).

The net utility can be written as:

V z = (r z g z ) (g z + g z ) (g z g z ) 2 k(a z g z ) 2 (2)
Because the utility of an agent depends on the gift of the other, individual donations (g z ; g z ) appear as solutions of an elementary Nash equilibrium (both donations depending on the income of the players and their announcement at the …rst stage), with: g z = g z (r z ; r z ; a z ; a z ), for z = fi; jg:

If, according to our assumption, individuals use the linear pledge function a z = r z ; the gift g z can also be de…ned as a linear function of the player'own income and of the pledge of the other player (see Appendix Eq. 15):

g z = 1 r z + 2 a z ; (3)
where 1 and 2 are related to the taste parameters k and : We can ceck that 1 > 0, and 2 > 0 if > 1=2 (i.e., players exhibit a high taste for conformity).

We can then move to the …rst stage of the game. At the …rst stage, agents must decide the pledge. Each subject will chose the optimal announcement, arbitrating between two e¤ects: …rst the pledge will in ‡uence the gift of the other player and will have an indirect impact on his own utility; on the other hand the lying cost puts a break on how much one is tempted to in ‡ate the signal to in ‡uence the other player.

As shown in the Appendix, the …rst order condition d Vz =da z = 0 leads to:

a z = a z (ĝ z ; ĝ z ) (4)
Note that for each of the players Eq. ( 3) involves that ĝz = 1 r z + 2 â z : Therefore, the optimal pledge relies on the expected values of both the income and the pledge of the other player. At this stage of the game, players only know their own income and that incomes are drew from the same unknown distribution. Given this information, the best estimation by player z of income r z is simply : r z = r z and, given the pledge function, the expected pledge can be de…ned as â z = r z :With these de…nitions, Eq (4) can be expressed as a linear form (see Eq. 22 in the Appendix) :

a z = r z G( ) (5)
where G( ) is a function of : The linear pledge function a z = r z with z = fi; jg can be an equilibrium of this game if there is an > 0 such that:

= G( ) (6)
Proposition 1 A solution exists under the su¢ cient, not necessary condition > 0:5:

Proof. See Appendix.
Finally, the equilibrium gift function, given the income r i and r j ; is determined by Eq. ( 3) with 1 and 2 evaluated at the equilibrium : More precisely:

g z = q 1 r z + q 2 r z ; (7)
with q 1 = 2(1+ +k)(1+2k ) 4(1+ +k) 2 (1 2 ) 2 and q 2 = (2 1)(1+2k ) 4(1+ +k) 2 (1 2 ) 2 :

We can easily check that the gift increases with the individual's income (q 1 > 0). The e¤ect of the other player's income relies on the strength of the individual's taste for conformity. We can state:

Proposition 2 For > 0:5; the equilibrium gift increases in both the income of the player and the income of his partner.

Proof. By de…nition of q 1 and q 2 :

The assumed strong taste for conformity ( > 0:5) has as a main consequence a positive relationship between one donor's gift and the income of his partner.

In the Appendix we provide the proof of existence of ; but could not provide an explicit expression for : A numerical example allows us to provide additional intuition about the result.

Solving numerically the …xed point equation (24, in the Appendix) for k = 0:50 and = 2 we obtain the coe¢ cient of the equilibrium pledge function: = 0:52: Such an agent would pledge half of his/her income: His/her equilibrium donation function is: g i = q 1 r i +q 2 r j = 0:27r i +0:11r j :

If matched with a partner with the same income (r j = r i ); this player would give 0:38r i , which is less than the pledged amount (0:52r i ). In other words, the equilibrium of this game is consistent, at least for these parameters, with a situation where subjects are systematically overestimating the gift to be provided in a symmetrical (same income) pair. This form of deception driven by a good cause can occur despite the lying costs.

If < 0:5 (i.e., the taste for conformity is weak), the problem still can have solutions (with < 1=3), but with a coe¢ cient q 2 < 0 : there is a negative link between the income of the partner and the donation. 8

Whether the taste for conformity is weak or strong is ultimately an empirical issue. The next section reports the results of a simple lab experiment that replicates the main features of this game. As we will show, in this experiment the e¤ect of the pledge of the partner on the donation is positive, which suggests that the taste for conformity is relatively strong.

4 The experiment

Experimental design

All subjects were recruited from the student population of the ESSEC Business School (France), speci…cally those who answered to an advertisement for paid decision experiments. 9 Six sessions were organized at the ESSEC Experimental Lab in May and September 2018. Subjects were seated in cubicles and made their decisions on a computer screen; instructions are provided in the Appendix 3. 10

The experiment comprises two independent parts. In part one, subjects received an endowment (either high or low) and could make a donation to a charity. We committed to make the payment to the charity shortly after the experiment. Participants were provided with a list of 15 main charities in France with di¤erent scopes of social action. 11 At the end of the session, the 8 In the simplest case where the two subjects play Nash in a perfect information framework without lying costs (k = 0) and without a preference for conformity ( = 0), the optimal gift is negatively related to the endowment of the other g i = 2 3 r i 1 3 r j and g j = 2 3 r j 1 3 r i: 9 As "French Grande Ecole" students, this group is relatively homogenous in terms of computing and intellectual abilities, age and educational background. 1 0 The computer program was developed using z-Tree (Fichbacher, 2007).

1 1 These charities sit on the top of the list of the 60 largest charities with respect to collected funds in France 8 computer randomly matched one gift to one and only one charity. Under this design, the amount of the donation is not related to the intensity of the preference of the donor for a speci…c charity, but to a "generic" charity encompassing a general commitment for the common good. Part two is distinct from part 1: it allows us to measure altruism and other personal characteristics of the subjects.

Our main treatment included 86 subjects. At the onset of the experiment, subjects were randomly assigned to pairs of anonymous subjects. Each of them received an endowment, drawn at random by the computer from the binomial distribution B(5;10jp=0.5); the computer chose either 5 euros or 10 euros with a 1/2 probability. If a pair has the income (r i ; r j ); the 86 subjects made for: 20 pairs (5 euro, 5 euro), 21 pairs (5 euros, 10 euros), 21 pairs (10 euros,5 euros) and 24 pairs (10 euros, 10 euros).

Participants did not know this statistical distribution. They only knew that the endowment was a random variable, and that the partner would receive an endowment drawn from the same yet unknown statistical distribution.

They were informed that the pair will donate to a charity selected at random from the list of charities. After receiving the endowment, each participant was asked to record the amount he/she intended to give, knowing that this piece of information would be delivered to the partner in the next stage, and that they would receive the same information on behalf of the partner.

The message was unbinding. Instructions referred to the pledge as a "gift promise". The stated intention was recorded simultaneously by the two participants on the computer screen. In the next step, both participants simultaneously received the pledge information provided by their partner.

At the last step, given this information, they had to decide on the amount they wanted to give, and their decision was irrevocable. At the same time, they also had to make an incentivized guess about the gift of the other. 12 The payo¤s for this …rst part were not communicated before the end of the experimental session (to contain the possible income e¤ect in the second task). in 2016, as indicated in the independent report "La Générosité des Français" (Bazin et al., 2017). The amount of collected funds is inferred from National Income Tax administration data.

1 2 They are paid 2 extra euros if the guessing error is lower than 10%.

In the second part of the experiment, we collected additional measures to help us interpret the results from the …rst part. First, subjects answered a survey about their gender, age and admission track. They were also asked whether they had a satisfactory knowledge about the action of the charities on our list; they could answer from 1, if "none of them", to 5, if "all of them".

Then subjects participated in a basic dictator game aimed at eliciting their degree of altruism.

We asked all subjects in a session how much of 10 euros they would share with an anonymous, randomly chosen partner, knowing that, once the choice is made, the computer will assign them at random to the group of dictators or to the group of bene…ciaries. We convert the monetary payo¤ into an index of altruism (IA) by dividing the amount by 10. Thus, the maximum degree of altruism is 1, and maximum sel…shness is 0.

In a control treatment with 56 subjects (giving "without pledge" treatment), we analyzed the decision of a similar pair, with the only di¤erence that they are not o¤ered the possibility to pledge. They learn their income and must chose their contribution to the joint donation.

At the end of the experimental session, subjects learned their gain from the two parts. The sessions lasted for 20 minutes on average; participants earned 9.15 euros on average. The transfers to charities totaled 560 euros.

Empirical results

We provide descriptive statistics for the full sample of 142 subjects in Appendix 2. In line with a standard result in the experimental research on giving, on average individuals donate approximately 50% of their endowment.

Our report focuses on the with-pledge treatment with 86 subjects. The average donation was of 3.81 euros; subjects who received 5 euros donated on average 2.46 euros, those who received 10 euros donated on average 5.04 euros.

On average, those who received 5 euros pledged 2.71 euros, and those who received 10 euros pledged 5.63 euros, which suggests a positive relationship between the pledge and the income of the participant (in line with the Proposition 1).

Figure 1 shows the average donation, depending on the income of the agent (5 or 10 euros) and on the income of the partner (5 or 10 euros). For constant income of the partner, "rich" subjects give more than "poor" subjects (the lines have a positive slope). Furthermore, the income of the partner has its own e¤ect on the donation. A "poor" subject paired with a "rich" subject will give on average more than if paired with another poor subject (1.73 vs. 3.17 euros ; p<0.01). 13 A "rich" subject paired with a "poor" subject will give less than if paired with a "rich" partner, albeit in our small sample this di¤erence is not signi…cant (4.69 vs. 5.35 euros; p=0.49). This can be seen as suggestive evidence for the in ‡uence of the announcement on the giving behavior of the players (in line with Proposition 2).

The numerical example presented at the end of the previous section describes a situation where agents will overestimate their donation (in a homogenous income pair). Across the 86 subjects, the average donation was of 3.81 euros, while the average pledge totalled 4.24 euros, a 10% di¤erence (p<0.01), which suggests that they might have a tendency to make excessive pledges. Furthermore, actual donations depend not only on the income of the subject, but also on the income of the partner. In Table 1 we display the di¤erence between the donation and the pledge, depending on the income of the subject and the income of the partner in the pair. Because on average "poor" subjects paired with "rich" subjects will end up by giving more than they have promised (by 0.12 euros), the 10% spread as indicated before is hiding stronger pledge excesses in the other groups. On average, subjects pledged approximately 60% of their income. In line with Proposition 1, the endowment is a signi…cant factor in explaining the pledge; however, the endowment explains only 25% of the variance of the pledge. This di¤erence from the theoretical model is not surprising, unlike before, in the lab individuals are heterogeneous with respect to many personal characteristics, while the theoretical model was built on the assumption of identical agents. In particular, individuals are heterogenous with respect to their degree of altruism, and this may be re ‡ected in more or less generous pledges and donations.

Because the altruism measure was elicited after the execution of the main donation task, the subject's choices in the dictator game could be impacted by his experience in the …rst task, and in particular by his endowment; subjects who received the high amount could be more generous in the second stage, compared to subjects who received the low amount. Data analysis shows that this is not the case. In the full sample of 142 subjects, at 0.29, the mean IA for subjects who received an endowment of 5 euros is not di¤erent from the mean IA of subjects who received 10 euros, at 0.32 (p=0.52) (Appendix Table 9). It turns out that "pledge only" models (models 4 to 6) have a better explanatory power than models 1 to 3 (which are strict replica of the theoretical model). In the "pledge only" models the altruism index looses some of its signi…cance, which suggests that individuals are aware of their degree of altruism, and take into account this information when making their pledge. While the pledge is related to the income, the former is a better predictor of the giving behavior. This departure from the theoretical model can be justi…ed by the heterogeneity of the subjects in the lab.

Model These empirical results corroborate our Proposition 2. First, the subject's donation is positively and signi…cantly related to the pledge of the subject, and the pledged amount is related to his endowment. Second, the donation is positively and signi…cantly related to the pledge of the partner. This positive e¤ect suggests that, in our sample, the taste for conformity is strong enough ( > 0:5).

We can check for the consistency of the beliefs in this experiment, using the information provided by the subject's guess of the partner's donation. Remember that, in the last stage of the giving game, after having received the pledge of the other, subjects not only had to decide on the gift, but were also asked to make an incentivized guess about the gift of the partner.

Overall, subjects made rather correct guesses about the donations of their partner, as shown in the Appendix Table 11.

More insights can be obtained from a regression model of the "Guessed gift of the partner".

We report the results of the OLS regression models in Table 4. The key independent variable is the pledge of the partner (which is known by the subject at the time of the guess). The results point out that in this experiment individuals' beliefs are strongly consistent: individuals use almost the same linear model to forecast the gift of the partner (guessed gift by partner=0.77*pledge partner + 0.2*own pledge) as the model that explains their own gifts (Models 4 to 6 in Table 3: gift = 0.87*pledge + 0.2*pledge partner). The slight di¤erence in coe¢ cients (0.77 vs. 0.87) can be explained by the altruism measure, a known characteristic in the …rst model (the individual knows his degree of altruism) and unknown (and non-signi…cant) in the guessed gift equation.

Finally we remark that, at 3.81 euros, the average donation in the with-pledge treatment (86 subjects) is not di¤erent from the average donation in the without-pledge treatment (56 subjects) at 3.69 euros (p=0.79). 15 This is not surprising, since in heterogenous income pairs the positive impact of a high pledge on the low income agent is canceled by the negative impact of the low pledge on the high income agent. Also recall that in this experiment pledging and giving occur almost simultaneously, thus neutralizing other important e¤ects, such as the commitment e¤ect or the reputation / social image e¤ect.

Conclusion

In this paper, we study the generosity inducement e¤ect of pledges on altruistic donors, using data collected on a speci…c "pledge and give" experiment, backed by a simple communication game.

The model shows that an intuitively appealing linear pledge function can be an equilibrium of the game. The lab experiment corroborates this result, as the pledge appears to be strongly related to the endowment, with subjects pledging on average approximately 60% of their endowment.

For altruistic agents with a strong taste for conformity, the equilibrium donation is increasing in both their income and the income of the partner. Their own pledge has a positive e¤ects on the donation of the partner. The lab experiment corroborates this result too,In particular, the subject will increase his donation by 0.20 euros if the partner increases his pledge by 1 euro, an estimation obtained while controlling for the altruism of the donor.

In pairs with an uneven (5,10) endowment distribution, agents with a low endowment will end up giving more, and those with a higher endowment will give less, as suggested by the "social conformism" principle incorporated in the theoretical model. Since "rich" and "poor" persons are equally distributed in our sample, donations are no larger in the with-pledge treatment than in the without-pledge treatment.

In the game, the care for the charity prompts agents to make excessive pledges, although the lying cost sets a cap on it. This form of altruistic deception is also observed in the lab data (with a 10% positive gap between the pledge and the donation, on average).

This analysis comes with its own limitations. The game is symmetric, agents di¤ering only in their endowment. A more realistic model would allow for heterogeneity in agents'preferences, which would be better aligned with the situation in the lab. The lab experiment has elicited altruism; ideally we should have also measured the preference for conformity, another important parameter of the model. If implemented, a treatment where the pledge is private information to the donor could allow us to control directely for the strategic e¤ect of pledges. 16 Despite its limitations, this paper sheds light on the social in ‡uence motive of pledging in charitable giving. The implications of our simultaneous game can be extended to a sequential setting; because pledges in ‡uence the decisions of the others, whenever the organization of the fundraising campaign allows it, it makes sense to let the wealthiest pledge …rst.

believes that the partner does the same (a j = r j ); the linear pledge is optimal and consistent with the equilibrium of the game.

If both players use the linear pledge function in the equilibrium, observing the pledge a j allows agent i to infer the income of his partner. Estimating r j by the ratio a j = and replacing a i with r i ; the optimal gift g i can be written as a function of one players's own income and of the pledge of his partner (itself related to the income of the latter). For player i; the expression of the optimal gift can be written as:

g i = 2r i (1 + + k) (1 + 2k ) a j 1 + 2k (1 2 ) 4 (1 + + k) 2 (1 2 ) 2 ; (14)
or, in a compact form, as:

g i = 1 r i + 2 a j ; (15)
an expression in which t 1 = 2(1+ +k)(1+2k ) 4(1+ +k) 2 (1 2 ) 2 and 2 =

(

1 +2k)(1 2 ) 4(1+ +k) 2 (1 2 ) 2 .
The symmetric expression de…nes g j :

A.1.2 The optimal pledge (…rst stage)

We now move back to the …rst stage of the game and determine the optimal pledge of player i. At this stage, his/her utility depends on the expected donations (ĝ i and ĝj ) at the last stage, which both depend on his own pledge, and the yet unknown income and pledge of the partner, as shown in Eq. ( 14).

The …rst order condition for utility maximization (Eq. 1) can be written d Vi =da i = 0:

dg j da i [(r i ĝi ) + 2 (ĝ i ĝj )]+ dg i da i (r i ĝi ) dg i da i (ĝ i + ĝj ) 2 (ĝ i ĝj ) dg i da i 2k(a i ĝi ) 1 dg i da i = 0; (16)
leading to an optimal pledge:

a i = ĝi + dgj dai [(r i ĝi ) + 2 (ĝ i ĝj )] + dgi dai [(r i ĝi ) 2 (ĝ i ĝj ) (ĝ i + ĝj )] 2k 1 dgi dai ; (17)
where, according to Equations ( 12) and ( 13)

: dgi dai = 4k(1+ +k) 4(1+ +k) 2 (1 2 ) 2 and dgj dai = 2k(1 2 ) 4(1+ +k) 2 (1 2 ) 2 :
According to Eq. ( 15), the last stage optimal gift g i is a linear function of the income of player i and of the pledge of his partner. If player j also uses the linear pledge function a j = r j ; the values ĝi and ĝj as expected by player i depend on the expectation by player i of player's j income contingent on the pledge of i; formally:

For agent i :

8 > > < > > : ĝi = 1 r i + 2 âj = 1 r i + 2 rj ĝj = 1 rj + 2 a i : (18)
At the …rst stage, player i does not know r j . Since all he/she knows is that the partner will receive an endowment drawn from the same unknown distribution, his/her best estimation of the income of the other must be his/her own income, or rj = r i :

Therefore, the expected value of the partner's gift is:

ĝj = 1 rj + 2 a i = 1 r i + 2 r i = r i ; (19)
with:

= 1 + 2 = 1 + 2k 3 + 2k : (20)
In a symmetric way, player i expects to give:

ĝi = 1 r i + 2 rj = r i : (21)
At stage one, both players expect their partners to make gifts identical to their own gifts, ĝi = ĝj :

Therefore, the optimal pledge (Eq. 17) can be written in a simpler form as:

a i = ĝi + dgj dai (r i ĝi ) + dgi dai [(r i ĝi ) (ĝ i + ĝj )] 2k 1 dgi dai = r i " + dgj dai (1 ) + dgi dai (1 3 ) 2k(1 dgi dai ) # : (22)
The linear pledge function can be an equilibrium of the game if a …xed point condition is ful…lled.

A.1.3 The …xed point condition

The …xed point condition for a linear pledge a i = r i to be an equilibrium is:

= + dgj dai (1 ) + dgi dai (1 3 ) 2k(1 dgi dai ) 20
According to Eq. ( 20), is de…ned as:

= 1+2k 3+2k .
Replacing by its explicit form, the …xed point condition becomes:

2k(1 dg i da i ) = 2 k(1 dg i da i ) + dg j da i (1 ) + dg i da i (1 3 ) 2k = 2 k + dg j da i (1 ) 2k = (1 + 2k ) (3 + 2k) 2k + dg j da i 1 (1 + 2k ) (3 + 2k) = 1 3 + dg j da i (1 + k k ) 3k (A.23)
We replace now dgj dai by its explicit form, as resulting from the expression of the optimal donation (Eq. 14):

dg j da i = 1 + 2k (1 2 ) 4 (1 + + k) 2 (1 2 ) 2 :
The …xed point condition becomes:

= 1 3 1 + 2k (1 2 ) 4 (1 + + k) 2 (1 2 ) 2 (1 + k k ) 3k 1 3 = (2 1) 1+k + k + 2k 2 (1 ) 3k (1 + 2k + 4 ) (3 + 2k) (A.

24)

Proposition 4 A solution exists under the su¢ cient, not necessary condition > 0:5: Finaly, given Eq. ( 15)

Proof. Let L( ) = 1 3 ; and R( ) = (2 1)[ 1+k +k+2k 2 (1 )] 3k(1+2k+4 )(3+2k) : Then, lim !0 R( ) = +1; lim !1 R( ) =
g i = 1 r i + 2 a j = 1 r i + 2 r j (A.25) = q 1 r i + q 2 r j (A.26) an expression in which q 1 = 2(1+ +k)(1+2k ) 4(1+ +k) 2 (1 2 ) 2 and q 2 = (1+2 k)(1 2 ) 4(1+ +k) 2 (1 2 ) 2 .

Good morning

We thank you for participating to this experiment that should last about 15 minutes. It is important to pay attention to these instructions because your compensation will depend on your decisions. This experiment comprises two parts. Payo¤s are denominated in euros. You will be paid in cash for both parts at the end of the experiment. In addition, you will receive a 2 euros …xed participation amount.

Your decisions are strictly anonymous. The other participants will not be informed about your decisions or about your identity.

From now on, please do not communicate in other way than it is indicated in the instructions.

Stay focus and turn o¤ cell phones and personal computers, otherwise you will be asked to leave the lab. You will be matched at random with another participant in a pair. None of you will learn the name of the other. Each of you will receive an endowment drawn at random by the computer from the same statistical distribution.

Your pair will be allowed to make a donation to one of the charities present on the list. The computer will match at random the charity with a pair of donors.

Each pair will make a gift to only one charity and a charity will not receive more than the gift of one pair.

In a …rst stage, the computer will ask each participant in the pair to indicate his/her gift promise.

Figure 1 :

 1 Figure 1: Mean donation by subject's income and the partner's income

14 A

 14 second regression model analyzes the determining factors of the donation. Models 1 to 3 analyze donations depending on the individual's endowment and the pledge of the partner; models 4 to 6 analyze donations depending on the individual's own pledge and the pledge of the partner.

  1 and dR( ) d < 0. The two functions, L( ) and R( ); should cross only once, for an equilibrium value, with > 1=3: The linear pledge function, a = r ; as assumed at the onset of the problem, is an equilibrium of the pledge game.

Table 1 :

 1 Actual donation -pledge, by income and partner income (s.d.)Regression models allow us to bring additional insights.A …rst regression model analyses the relationship between the pledge (at time t) and various covariates, including the endowment of the subject (5 or 10 euros), the altruism index, gender, age, how well the subject is informed about the charities on the list (Stated knowledge about charities) and whether the subject belongs to the economics admission track (Econ adm. track).

			Income, partner
			5	10	Total
	Own	5	-0.68 (1.41) 0.14 (1.00	-0.26 (1.27)
	income 10	-0.60 (0.86) -0.58 (2.11) -0.59 (1.63)
		Total -0.63 (1.15) -0.24 (1.71) -0.43 (1.47)
			Model 1	Model 2	Model 3
	Endowment		0.583*** (0.07) 0.590*** (0.09) 0.586*** (0.10)
	Altruism index		-	3.626** (0.91)	3.723*** (0.97)
	Gender (Fe=1)		-	-0.194 (0.72)	-0.260 (0.84)
	Age		-	0.073 (0.06)	0.068 (0.04)
	Stated knowledge charities -	-	0.130 (0.43)
	Econ track		-	-	-0.238 (0.16)
	Constant		-0.194 (0.42)	-2.891* (1.21)	-3.113 (2.22)
	Nb. observations		86	86	86
	R2		0.256	0.340	0.343

Note: *** Signi…cant at 1%. ** Signi…cant at 5%. OLS, Errors clustered by session.

Table 2 :

 2 The Pledge Regression Model

Table 3 :

 3 The Donation Regression Model1 4 A regression model of the altruism index over endowments and treatments (factor variable) shows no statistically signi…cant e¤ect of the treatment and endowment (Appendix Table10).

Table 4 :

 4 The Guess of Partner's Donation Regression Model

Table 6 :

 6 Table6presents the average donation in the main and control treatment. Mean donation per treatement and endowment (s.d.)Table7presents OLS estimates of the full sample donation equation. Model 1 includes only the endowment and a dummy for the pledge treatment, model 2 includes other covariates.

		Pair donation, with pledge Pair donation, without pledge
	Nb. of subjects	86		56
	Age (years)	22.31 (0.26)		21.52 (0.21)
	Female (=1)	0.61 (0.61)		0.63 (0.65)
	Eco. adm. track	0.33 (0.05)		0.37 (0.06)
	Altruism index (0-1)	0.31 (0.02)		0.29 (0.02)
	Stated knowledge of charities (1-5) 3.50 (0.10)		3.59 (0.12)
	Endowment (5 or 10).	7.61 (0.27)		7.50 (0.31)
	Table 5: Descriptive statistics
	A.2 Appendix 2. Descriptive statistics
	A.2.1 Full sample
	Treatment		5 euros	10 euros	Total
	Donation, with pledge treatment (N=86)	2.46 (1.78) 5.04 (3.21) 3.81 (2.92)
	Donation, without pledge treatment (N=56) 2.78 (1.86) 4.59 (2.91) 3.69 (2.59)
	Total		2.59 (1.81) 4.86 (3.09) 3.76 (2.78)
			Model 1 Model 2
	Endowment		0.455*** 0.431***
	With-pledge treatment (=1) 0.074	-0.028
	Altruism index		-	3.809***
	Gender (Fe=1)		-	0.330
	Age		-	0.049
	Stated knowledge	-	0.227
	Econ. adm. track	-	-0.543
	Constant		0.276	-2.550
	Nb. observations	142	142
	R2		0.168	0.275

The symmetric expression de…nes g i : Note: *** Signi…cant at 1%. Errors clustered by session.

Table 7 :

 7 Donation equationA.2.2 Robustness checks -Altruism Index (AI)The altruism index, mesured by the task 2 -Dictator game, does not depend on the task 1 treatement or endowment of the subject (in task 1).

	Regression of IA by endowment and treatement.

Table 9 :

 9 Altruism index by endowmentA.3 Other tablesActual donation and guessed donation Two tailed t-tests con…rm that the di¤erences between the mean actual and guessed donations are not signi…cant regardless of the case studied.

Table 10 :

 10 Do you have any question? If so, please rise your hand and wait for the Administrator. Altruism with respect to treatement and endowment

	Slide 2. First part

Table 11 :

 11 Actual and guessed partner's donation (s.d.) This …rst task will allow you to make a donation to one major charity in France. The list of the target organizations is:

	La Croix Rouge Française	Secours Catholique
	Médecins sans Frontières	Secours Populaire
	Médecins du Monde	La Fondation Abbé Pierre
	AFM-Téléthon	Action contre la faim
	Sidaction	Apprentis d'Auteuil
	Handicap International	Comité Français pour l'Unicef
	La Ligue Nationale contre le Cancer Greenpeace
	Les Restos du Coeur

His action arrived at a time when the US Administration was dragging feet to pay its agreed contribution to the organization.

See David Callahan, This Huge Gift Made History, 18 Sept 2017, Inside Philanthropy. www.insidephilanthropy.com/home /2017/9/18/this-huge-gift-made-philanthropic-history-whats-been-the-impact

[START_REF] Koessler | Setting new behavioural standards: Sustainabilty pledges and how conformity impacts their outreach[END_REF] andBarett and Dannenberg (2016) claim that the preference for conformity of the agents explains why.pledges can support cooperation in public good experiments.

Other assumptions, leading to a similar guessing rule, such as di¤use priors, could be considered. Information structures that involve more sophisticated prior updating mechanisms would greatly increase the complexity of the problem that would prevent us from obtaining an analytical solution.

The litterature on lying and deception is booming. See the surveys by Rosembaum et al. (2014),[START_REF] Jacobsen | Why do we lie? A practical guide to the dishonesty literature[END_REF],[START_REF] Abeler | Preferences for truth-telling[END_REF] and the meta-analysis by[START_REF] Gerlach | The truth about lies: A meta-analysis on dishonest behavior[END_REF].

See for instance[START_REF] Van Huyck | Commitment versus discretion in the peasant-dictator game[END_REF] for a negotiation game with commitement, and[START_REF] Kartik | Strategic communication with lying costs[END_REF] for a quadratic lying cost in a sender-receiver game.

With two players, both use the linear pledge, and believe that the other is using the linear pledge.

3 In this text, p-values correspond to two-tailed t-tests.

5 See also the Table7in the Appendix. It presents a regression model for the whole sample, with giving as the dependent variable. The treatment dummy is not signi…cant.

6 In this treatment pledges should perfectly match donations in identical income pairs.

Acknowledgement 3 This research bene…ted from the grant ANR-11-LBX-0023 of the LABEX MME-DII. Authors would like to thank Delphine Dubart for the implementation of the experiment at the ESSEC Experimental Lab. They would like to thank two anonymous referees, the Associate Editor of the journal, Ann-Kathrin Koessler, Oana Peia, participants to the ESSEC Behavioral Research Seminar on January 24, 2018, Cergy, participants to the PSE Workshop in Economics and Psychology, March 12, 2019, Paris, participants to the Annual Conference of the French Association of Experimental Economics, June 21-22, 2019, Toulouse, for their suggestions and remarks on an early version of this paper.

A Appendix (Online)

A.1 Appendix 1. Solving the formal model A.1.1 The optimal gift (last stage)

In this resolution, we use the indices i and j of the generic z: At the last stage all variables are known.

The net utility of player i (and j) is given by :

2 k(a j g j ) 2 (A.9)

The FOC for maximizing utility are:

At the Nash equilibrium we have:

All things equal, for > 0:5 a higher pledge a i entails a higher gift g j at the later stage. The opposite relationship holds for < 0:5:

In the following, we look for a speci…c intuitive solution in which the pledge a i is a linear function of the individual's income r i , a i = r i ; with as a positive constant. In the last subsection, we will prove that, when player i chooses his pledge according to this function and After …lling in this information, the computer will simultaneously convey this information to the other participant.

At the second stage, you will have to decide how much you want to give. At that stage, you know: your endowment, your gift promise, and the gift promise of your partner.

We commit on transferring the amounts donated to the charities shortly after the experiment.

At the end of the experiment you will receive in cash the di¤erence between the endowment received at the beginning of the experiment and your gift.

Do you have any question? If so, please rise your hand and wait for the Administrator.

Slide 3.

Following the random draw, your endowment is 5 (or 10) euros.

Please


MD, PhD Cédric Annweiler
email: cedric.annweiler@chu-angers.fr

MD Bérangère Hanotte

MD Claire Grandin De L'eprevier

PhD Jean-Marc Sabatier

MD Ludovic Lafaie

MD Thomas Célarier

Vitamin D and survival in COVID-19 patients: A quasi-experimental study

Keywords: 2, Figure count: 2, Reference COVID-19, SARS-CoV-2, vitamin D, therapeutics, quasi-experimental study, older adults HIGHLIGHTS

Vitamin D may be a central biological determinant of COVID-19 outcomes. The objective of this quasi-experimental study was to determine whether bolus vitamin D3 supplementation taken during or just before COVID-19 was effective in improving survival among frail elderly nursing-home residents with COVID-19. Sixty-six residents with COVID-19 from a French nursing-home were included in this quasi-experimental study. The "Intervention group" was defined as those having received bolus vitamin D3 supplementation during COVID-19 or in the preceding month, and the "Comparator group" corresponded to all other participants. The primary and secondary outcomes were COVID-19 mortality and Ordinal Scale for Clinical Improvement (OSCI) score in acute phase, respectively. Age, gender, number of drugs daily taken, functional abilities, albuminemia, use of corticosteroids and/or hydroxychloroquine and/or antibiotics (i.e., azithromycin or rovamycin), and hospitalization for COVID-19 were used as potential confounders. The Intervention (n=57; mean±SD, 87.7±9.3years; 79%women) and Comparator (n=9; mean,87.4±7.2years; 67%women) groups were comparable at baseline, as were the COVID-19 severity and the use of dedicated COVID-19 drugs. The mean follow-up time was 36±17days. 82.5% of participants in the Intervention group survived COVID-19, compared to only 44.4% in the Comparator group (P=0.023). The full-adjusted hazard ratio for mortality according to vitamin D3 supplementation was HR=0.11[95%CI:0.03;0.48],P=0.003. Kaplan-Meier distributions showed that Intervention group had longer survival time than Comparator group (log-rank P=0.002). Finally, vitamin D3 supplementation was inversely associated with OSCI score for COVID-19 (β=-3.84[95%CI:-6.07;-1.62],P=0.001). In conclusion, bolus vitamin D3 supplementation during or just before COVID-19 was associated in frail elderly with less severe COVID-19 and better survival rate. 3

INTRODUCTION

Since December 2019, the COVID-19 caused by SARS-CoV-2 is spreading worldwide from China, affecting millions of people and leaving thousands of dead, mostly in older adults.

With the lack of effective therapy, chemoprevention and vaccination [START_REF] Ahn | Current Status of Epidemiology, Diagnosis, Therapeutics, and Vaccines for Novel Coronavirus Disease 2019 (COVID-19)[END_REF], focusing on the immediate repurposing of existing drugs gives hope of curbing the pandemic. Importantly, a most recent genomics-guided tracing of the SARS-CoV-2 targets in human cells identified vitamin D among the three top-scoring molecules manifesting potential infection mitigation patterns through their effects on gene expression [START_REF] Glinsky | Quercetin, and Estradiol manifest properties of candidate medicinal agents for mitigation of the severity of pandemic COVID-19 defined by genomics-guided tracing of SARS-CoV-2 targets in human cells[END_REF]. In particular, by activating or repressing several genes in the promoter region of which it binds to the vitamin D response element [START_REF] Hossein-Nezhad | Vitamin D for health: a global perspective[END_REF], vitamin D may theoretically prevent or improve COVID-19 adverse outcomes by regulating (i) the renin-angiotensin system (RAS), (ii) the innate and adaptive cellular immunity, and (iii) the physical barriers [START_REF] Grant | Evidence That Vitamin D Supplementation Could Reduce Risk of Influenza and COVID-19 Infections and Deaths[END_REF]. Consistently, epidemiology shows that hypovitaminosis D is more common from October to March at northern latitudes above 20 degrees [START_REF] Hossein-Nezhad | Vitamin D for health: a global perspective[END_REF], which corresponds precisely to the latitudes with the highest lethality rates of COVID-19 during the first months of winter 2020 [START_REF] Ahn | Current Status of Epidemiology, Diagnosis, Therapeutics, and Vaccines for Novel Coronavirus Disease 2019 (COVID-19)[END_REF]. In line with this, significant inverse associations were found in [START_REF] Dijkman | Replication-Dependent Downregulation of Cellular Angiotensin-Converting Enzyme 2 Protein Expression by Human Coronavirus NL63[END_REF] European countries between serum 25-hydroxyvitamin D (25(OH)D) concentration and the number of COVID-19 cases, as well as with COVID-19 mortality [START_REF] Ilie | The role of vitamin D in the prevention of coronavirus disease 2019 infection and mortality[END_REF]. This suggests that increasing serum 25(OH)D concentration may improve the prognosis of COVID-19.

However, no randomized controlled trial (RCT) has tested the effect of vitamin D supplements on COVID-19 outcomes yet. We had the opportunity to examine the association between the use of vitamin D3 supplements and COVID-19 mortality in a sample of frail elderly nursing-home residents infected with SARS-CoV-2. The main objective of this quasiexperimental study was to determine whether bolus vitamin D3 supplementation taken during or just before COVID-19 was effective in improving survival among frail elderly COVID-19 patients living in nursing-home. The secondary objective was to determine whether this intervention was effective in limiting the clinical severity of the infection.

MATERIAL AND METHODS

Study population

The study consisted in a quasi-experimental study in a middle-sized nursing-home in Rhône, South-East of France, the residents of which were largely affected by COVID-19 in March-April 2020 (N=96, including n=66 with COVID-19). Data were retrospectively collected from the residents' records of the nursing-home.

The nursing-home is dedicated to residents with physical disabilities, major neurocognitive and psychiatric disorders. The facility includes 56 single rooms and 21 double rooms, along with communal dining and activity areas. There are no closed units. All residents were allowed to move around the building until 21 March 2020, when social distancing and other preventive measures were implemented. Residents were isolated in their rooms with no communal meals or group activities. No visitors, including families, were allowed in the nursing-home since 10 March 2020. Enhanced hygiene measures were implemented, including cleaning and disinfection of frequently touched surfaces, permanent face masks, and additional hand hygiene stations for staff members.

The inclusion criteria for the present analysis were as follows: 1) residents with clinically obvious or diagnosed COVID-19 with RT-PCR in March-April 2020; 2) data available on the treatments received, including vitamin D supplementation, since the diagnosis of COVID-19 and during the previous month at least; 3) data available on the vital status and COVID-19 evolution as of May 15, 2020;4) no objection from the resident and/or relatives to the use of anonymized clinical and biological data for research purposes. Sixty-six residents had COVID-19 during the study period. They all met the other inclusion criteria and were included in the present analysis.

Intervention: bolus vitamin D3 supplementation during or just before COVID-19

All residents in the nursing-home receive chronic vitamin D supplementation with regular maintenance boluses (single oral dose of 80,000 IU vitamin D3 every 2-3 months), without systematically performing serum control test as recommended in French nursing-homes [START_REF] Rolland | Vitamin D supplementation in older adults: searching for specific guidelines in nursing-homes[END_REF] due to the very high prevalence of hypovitaminosis D reaching 90-100% in this population [START_REF] Annweiler | Vitamin D in the elderly: 5 points to remember[END_REF]. Here, the "Intervention group" was defined as all COVID-19 residents who received an oral bolus of 80,000 IU vitamin D3 either in the week following the suspicion or diagnosis of COVID-19, or during the previous month. The "Comparator group" corresponded to all other COVID-19 residents who did not receive any recent vitamin D supplementation. None received D2 or intramuscular supplements. All medications were dispended and supervised by a nurse.

Primary outcome: COVID-19 mortality

The primary outcome was mortality of COVID-19 residents during follow-up. Follow-up started from the day of COVID-19 diagnosis for each patient, and continued until May 15, 2020, or until death if applicable.

Secondary outcome: OSCI score for COVID-19 in acute phase

The secondary outcome was the score on the World Health Organization's Ordinal Scale for Clinical Improvement (OSCI) for COVID-19 [START_REF]Coronavirus disease (COVID-2019) R&D[END_REF]. The score was calculated by the geriatrician of the nursing-home during the most severe acute phase of COVID-19 for each patient. The OSCI distinguishes between several levels of COVID-19 clinical severity according to the outcomes and dedicated treatments required, with a score ranging from 0 (benign) to 8 (death). A score of 4 corresponds to the introduction of oxygen (nasal oxygen catheter or oral nasal mask), and a score of 6 to intubation and invasive ventilation [START_REF]Coronavirus disease (COVID-2019) R&D[END_REF].

Covariables

Age, gender, number of drugs daily taken, functional abilities, nutritional status, COVID-19 treatment with corticosteroids and/or hydroxychloroquine and/or dedicated antibiotics, and hospitalization for COVID-19 were used as potential confounders. The number of drugs usually taken per day was recovered from prescriptions in the nursing-home and served as a measure of the burden of disease, as previously reported [START_REF] Decker | Number of drug classes taken per day may be used to assess morbidity burden in older inpatients: a pilot cross-sectional study[END_REF]. The use of corticosteroids and/or hydroxychloroquine and/or dedicated antibiotics (i.e., azithromycin or rovamycin) were noted from prescriptions in the nursing-home or during hospitalization, as appropriate. Functional abilities prior to COVID-19 were measured from 1 to 6 (best) with the Iso-Resources Groups (GIR) [START_REF] Vetel | AGGIR. Practical use. Geriatric Autonomy Group Resources Needs[END_REF]. Finally, the prognosis related to the nutritional status prior to COVID-19 was evaluated using the last measure of serum albumin concentration during the past semester, as appropriate [START_REF] Cabrerizo | Serum albumin and health in older people: Review and meta analysis[END_REF].

Statistical analysis

Participants' characteristics were summarized using means and standard deviations (SD) or frequencies and percentages, as appropriate. As the number of observations was higher than 40, comparisons were not affected by the shape of the error distribution and no transform was applied [START_REF] Rochon | To test or not to test: Preliminary assessment of normality when comparing two independent samples[END_REF]. First, comparisons between participants separated into Intervention and Comparator groups were performed using Mann-Whitney U test or the Chi-square test or Fisher test, as appropriate; and then according to mortality. Secondly, a full-adjusted Cox regression was used to examine the associations of mortality (dependent variable) with bolus vitamin D3 supplements and covariables (independent variables). The model produces a survival function that provides the probability of death at a given time for the characteristics supplied for the independent variables. Third, the elapsed time to death was studied by survival curves computed according to Kaplan-Meier method and compared by log-rank test.

Finally, univariate and multiple linear regressions were used to examine the association of bolus vitamin D3 supplementation (independent variable) with OSCI score (dependent variable), while adjusting for potential confounders. P-values<0.05 were considered significant. All statistics were performed using SPSS (v23.0, IBM Corporation, Chicago, IL) and SAS® version 9.4 software (Sas Institute Inc).

Standard Protocol Approvals, Registrations, and Patient Consents

The study was conducted in accordance with the ethical standards set forth in the Helsinki Declaration (1983). No participant or relatives objected to the use of anonymized clinical and biological data for research purposes. The study was approved by the Ethical Committee of Angers University Hospital (2020/67). The study protocol was also declared to the National Commission for Information Technology and civil Liberties (CNIL).

RESULTS

Sixty-six participants (mean±SD age 87.7±9.0years, range 63-103years; 77.3% women) were infected with SARS-CoV-2 and included in this quasi-experimental study. The mean followup was 36±17days. Fifty-one people survived COVID-19, while 15 died. The two groups were comparable at baseline with no significant difference regarding the age (P=0.699), gender (P=0.731), the mean number of drugs usually taken per day (P=0.053), the GIR score (P=0.209) and the serum albumin concentration (P=0.263) (Table 1). Regarding care dedicated to COVID-19, only the proportion of patients who received a bolus of vitamin D3 during or just before COVID-19 differed between deceased participants and survivors, with a higher prevalence in survivors (respectively 92.2% versus 66.7%, P=0.023). In contrast, there was no between-group difference in the proportion of patients treated with corticosteroids, hydroxychloroquine or dedicated antibiotics, or hospitalized for COVID-19.

Table 1 indicates the characteristics of participants separated into Intervention (n=57) and Comparator (n=9) groups. Their baseline characteristics (age, gender, albuminemia) did not differ between groups, with the exception of the number of drugs usually taken (which involved the use of vitamin D supplements) and the disability score (Table 1). Similarly, the proportion of participants at each OSCI severity level as well as the use of dedicated COVID- Finally, the linear regression model illustrated in Table 2 showed that bolus vitamin D3 supplementation during or just before COVID-19 was inversely associated with the OSCI score for COVID-19 in acute phase. Similar results were found before (β=-2.96 [95%CI: -4.79;-1.12], P=0.002) and after adjusting the analyses for potential confounders (β=-3.84 [95%CI: -6.07;-1.62], P=0.001).

DISCUSSION

The main finding of this nursing-home-based quasi-experimental study is that, irrespective of all measured potential confounders, bolus vitamin D3 supplementation during or just before COVID-19 was associated with less severe COVID-19 and better survival rate in frail elderly. No other treatment showed protective effect. This novel finding provides a scientific basis for vitamin D replacement trials attempting to improve COVID-19 prognosis.

To our knowledge we provide here the first quasi-experimental data examining the effect of vitamin D supplementation on the survival rate of COVID-19 patients. To date, only rare observational data, all of which are consistent, are available on the link between vitamin D and COVID-19. The first reports in COVID-19 patients indicated that hypovitaminosis D is highly prevalent in this population, reaching 85% [START_REF] James | Vitamin D Level of Mild and Severe Elderly Cases of COVID-19: A Preliminary Report[END_REF], and that serum 25(OH)D concentrations are lower in COVID-19 patients compared to controls [START_REF] D'avolio | 25-Hydroxyvitamin D Concentrations Are Lower in Patients with Positive PCR for SARS-CoV-2[END_REF]. Similarly, significant inverse correlations were found in 20 European countries between the mean serum 25(OH)D concentrations and the number of COVID-19 cases/1M, as well as with mortality/1M [START_REF] Ilie | The role of vitamin D in the prevention of coronavirus disease 2019 infection and mortality[END_REF]. The severity of hypovitaminosis D appears to relate to the prognosis of COVID-19 since the mortality rate was multiplied by 7.6 among people with hypovitaminosis D <75nmol/L, and by 10.1 among those with hypovitaminosis D <50nmol/L (P<0.001) [START_REF] Raharusun | Patterns of COVID-19 Mortality and Vitamin D: An Indonesian Study[END_REF].

Similarly, another observational study in 212 COVID-19 cases showed that, for each standard deviation increase in serum 25(OH)D, the probability of mild rather than severe COVID-19 was multiplied by 7.9 (P<0.001), while the probability of mild rather than critical COVID-19 was multiplied by 19.6 (P<0.001) [START_REF] Alipio | Vitamin D supplementation could possibly improve clinical outcomes of patients infected with Coronavirus-2019 (COVID-2019[END_REF]. These results suggest that increasing serum 25(OH)D concentration may improve the prognosis of COVID-19. Interventional studies dedicated to COVID-19 are yet warranted for investigating the role of vitamin D supplementation on COVID-19 outcomes. Interestingly, previous meta-analyses found that high-dose prophylactic vitamin D supplementation was able to reduce the risk of respiratory tract infections [START_REF] Martineau | Vitamin D supplementation to prevent acute respiratory tract infections: systematic review and meta-analysis of individual participant data[END_REF].

Based on this observation, we and others are conducting an RCT designed to test the effect of high-dose versus standard-dose vitamin D3 on 14-day mortality in COVID-19 older patients (https://clinicaltrials.gov/ct2/show/NCT04344041). While waiting for the recruitment of this RCT to be completed, the findings of the present quasi-experimental study strongly suggest a benefit of bolus vitamin D3 supplementation on COVID-19 outcomes and survival.

How vitamin D supplementation may improve COVID-19 outcomes and survival is not fully elucidated. Three mechanisms are possible: regulation of (i) the RAS, (ii) the innate and adaptive cellular immunity, and (iii) the physical barriers [START_REF] Grant | Evidence That Vitamin D Supplementation Could Reduce Risk of Influenza and COVID-19 Infections and Deaths[END_REF]. First, vitamin D reduces pulmonary permeability in animal models of acute respiratory distress syndrome (ARDS) by modulating the activity of RAS and the expression of the angiotensin-2 converting enzyme (ACE2) [START_REF] Kong | VDR Attenuates Acute Lung Injury by Blocking Ang-2-Tie-2 Pathway and Renin-Angiotensin System[END_REF]. This action is crucial since SARS-CoV-2 reportedly uses ACE2 as a receptor to infect host cells [START_REF] Hoffmann | The Novel Coronavirus 2019 (2019-nCoV) Uses the SARS-Coronavirus Receptor ACE2 and the Cellular Protease TMPRSS2 for Entry into Target Cells[END_REF] and downregulates ACE2 expression [START_REF] Dijkman | Replication-Dependent Downregulation of Cellular Angiotensin-Converting Enzyme 2 Protein Expression by Human Coronavirus NL63[END_REF]. ACE2 is expressed in many organs, including the endothelium and the pulmonary alveolar epithelial cells, where it has protective effects against inflammation [START_REF] Annweiler | Counter-regulatory 'Renin-Angiotensin' System-based Candidate Drugs to Treat COVID-19 Diseases in SARS-CoV-2-infected patients[END_REF]. During COVID-19, downregulation of ACE2 results in an inflammatory chain reaction, the cytokine storm, complicated by ARDS [START_REF] Ji | TWIRLS, an Automated Topic-Wise Inference Method Based on Massive Literature, Suggests a Possible Mechanism via ACE2 for the Pathological Changes in the Human Host after Coronavirus Infection[END_REF]. In contrast, a study in rats with chemically-induced ARDS showed that the administration of vitamin D increased the levels of ACE2 mRNA and proteins [START_REF] Yang | Effect of Vitamin D on ACE2 and Vitamin D receptor expression in rats with LPS-induced acute lung injury[END_REF]. Rats supplemented with vitamin D had milder ARDS symptoms and moderate lung damage compared to controls.

Second, many studies have described the antiviral effects of vitamin D, which works either by induction of antimicrobial peptides with direct antiviral activity against enveloped and nonenveloped viruses, or by immunomodulatory and anti-inflammatory effects [START_REF] Fabbri | Editorial -Vitamin D status: a key modulator of innate immunity and natural defense from acute viral respiratory infections[END_REF]. These are potentially important during COVID-19 to limit the cytokine storm. Vitamin D can prevent ARDS [START_REF] Dancer | Vitamin D Deficiency Contributes Directly to the Acute Respiratory Distress Syndrome (ARDS)[END_REF] by reducing the production of pro-inflammatory Th1 cytokines, such as TNFα and interferon γ [START_REF] Fabbri | Editorial -Vitamin D status: a key modulator of innate immunity and natural defense from acute viral respiratory infections[END_REF]. It also increases the expression of anti-inflammatory cytokines by macrophages [START_REF] Fabbri | Editorial -Vitamin D status: a key modulator of innate immunity and natural defense from acute viral respiratory infections[END_REF]. Third, vitamin D stabilizes physical barriers [START_REF] Grant | Evidence That Vitamin D Supplementation Could Reduce Risk of Influenza and COVID-19 Infections and Deaths[END_REF]. These barriers are made up of closely linked cells to prevent outside agents (such as viruses) from reaching tissues susceptible to viral infection. Although viruses alter the integrity of the cell junction, vitamin D contributes to the maintenance of functional tight junctions via E-cadherin [START_REF] Grant | Evidence That Vitamin D Supplementation Could Reduce Risk of Influenza and COVID-19 Infections and Deaths[END_REF]. All these antiviral effects could potentiate each other and explain our results.

We also found that none of the other dedicated drugs used in this quasi-experimental study was associated with better survival rate in COVID-19 patients. The interest of these drugs in COVID-19 is still debated, whether for corticosteroids [START_REF] Ye | Efficacy and safety of corticosteroids in COVID-19 based on evidence for COVID-19, other coronavirus infections, influenza, community-acquired pneumonia and acute respiratory distress syndrome: a systematic review and meta-analysis[END_REF], hydroxychloroquine [START_REF] Ferner | Chloroquine and hydroxychloroquine in covid-19[END_REF] or azithromycin [START_REF] Gbinigie | Should azithromycin be used to treat COVID-19? A rapid review[END_REF]. However, it should be noted that these drugs were given here as part of patient care in the most severe clinical situations, which could have biased and masked their effectiveness (if any).

The strengths of the present study include i) the originality of the research question on an emerging infection for which there is no scientifically validated treatment, ii) the detailed description of the participants' characteristics allowing the use of multivariate models to measure adjusted associations, and iii) the standardized collection of data from a single research center.

Regardless, a number of limitations also existed. First, the study cohort was restricted to a limited number of nursing-home residents who might be unrepresentative of all older adults. Second, although we were able to control for the important characteristics that could modify the association, residual potential confounders might still be present such as the serum concentration of 25(OH)D at baseline -a low level classically ensuring the effectiveness of the supplementation [START_REF] Heaney | Guidelines for optimizing design and analysis of clinical studies of nutrient effects[END_REF]. As this analysis was not initially planned, no concerted efforts were made to systematically measure the serum 25(OH)D concentration before and after supplementation. Third, the quasi-experimental design of our study is less robust than an RCT. Participants in the Comparator group did not receive vitamin D placebo, and there was no randomization. It should yet be noted that the characteristics of the two groups did not differ at baseline, which allows interpreting the survival difference as linked to the vitamin D3 supplementation.

CONCLUSIONS

In conclusion, we were able to report among frail elderly residents that bolus vitamin D3 supplementation taken during or just before COVID-19 was associated with less severe COVID-19 and better survival rate. No other treatment showed protective effect.

Vitamin D3 supplementation may represent an effective, accessible and well-tolerated treatment for COVID-19, the incidence of which increases dramatically and for which there are currently no validated treatments. Further prospective, preferentially interventional, studies are needed to confirm whether supplementing older adults with bolus vitamin D3 during or just before the infection could improve, or prevent, COVID-19. and TC.

-Obtained funding: Not applicable.

-Statistical expertise: CA.

-Administrative, technical, or material support: CA and TC.

-Study supervision: CA.

  19 drugs did not differ between Intervention and Comparator groups. At the end of the follow-up, 82.5% of patients from the Intervention group survived COVID-19, compared to only 44.4% in the Comparator group (P=0.023).

Figure 1

 1 Figure1shows a statistically significant and clinically relevant protective effect against

Figure 1 .

 1 Figure 1. Hazard ratio for mortality in frail elderly COVID-19 patients according to the

Figure 2 .

 2 Figure 2. Kaplan-Meier estimates of the cumulative probability of participants' survival

Table 1 . Characteristics and comparison of COVID-19 participants (n=66) according to the study arm All COVID-19 participants (n=66)

 1 : coefficient of regression corresponding to a change in OSCI score for COVID-19 in acute phase (/8); CI: confidence interval; COVID-19: Coronavirus Disease 2019; GIR: Iso Resource Groups; OSCI: Ordinal Scale for Clinical Improvement of the World Health Organization; *: measured in preceding 6 months.

	Study arm
	Comparator group *	Intervention group *	P-value
	(n=9)	(n=57)

β

ACKNOWLEDGMENTS

The authors wish to thank their collaborators:

-Angeline MEY, MD, from the Department of Clinical Gerontology, University Hospital of Saint-Etienne, Saint-Etienne, France; -Pauline GUILLEMIN, MD, from the Geriatric Hospital of Saint Symphorien sur Coise, Saint Symporien sur Coise, France; -Jennifer GAUTIER, MS, from the Research Center on Autonomy and Longevity, University Hospital, Angers, France.

There was no compensation for their contribution.

CONFLICT OF INTEREST STATEMENT

The authors report no conflict of interest with this paper.

SPONSOR'S ROLE

No sources of funding were used to assist in this study.

AUTHORS CONTRIBUTION

-CA has full access to all of the data in the study, takes responsibility for the data, the analyses and interpretation and has the right to publish any and all data, separate and apart from the attitudes of the sponsors. All authors have read and approved the manuscript.

-Study concept and design: CA, BH, CGE, JMS, LL and TC.

-Acquisition of data: BH and CGE.

-Analysis and interpretation of data: CA and TC.