François Cochard
email: francois.cochard@univ-fcomte.fr

Julie Le Gallo
email: julie.le-gallo@agrosupdijon.fr

Nikolaos Georgantzis
email: nick.geogantzis@gmail.com

Jean-Christian Tisserand
email: jean-christian.tisserand@bsb-education.com

Social preferences across different populations: Meta-analyses on the ultimatum and dictator games

Keywords: ultimatum game, dictator game, meta-analysis, social preferences JEL Classification: C13, C78, D03, D64

We perform meta-regressions on a single database containing 96 observations of simple ultimatum games and 144 observations of simple dictator games to disentangle the fairness hypothesis based on the degree of economic development of a country. According to the fairness hypothesis, offers in the two games should not differ if they were motivated by a subject's fairness concerns. Using the difference across countries between offers in ultimatum and dictator games, we address the effect of being exposed to the market mechanism on pure fairness concerns and other-regarding, expectations-driven fairness. Our results show in particular that the lower the level of economic development in a country, the less likely the rejection of the fairness hypothesis.

Introduction

What is fair and what is not? People's perceived norms of fairness and overall expectations depend of their experience in their cultural, social and economic environments and are thus very likely to vary across countries. For example, Corneo (2001) argue that people from a country in which intense redistribution takes place exhibit greater concerns for fairness than individuals in countries with less tradition in redistribution. In fact, according to Alesina and Giuliano (2011), such "home grown" values are carried over and persist among immigrants in the destination country. Related to our study, Alesina and Angeletos (2005) point that in countries in which people trust the market mechanism as a fair device for the allocation of resources, unequal outcomes are more likely to be tolerated as the result of differences in the citizens' efforts. Therefore, exposure to a more developed market may lead agents to accept more unequal outcomes. In this paper, we investigate the relationship between market exposure and pure fairness concerns and other-regarding, expectations-driven fairness. We are the first, to our knowledge, to use experimental data to investigate this issue.

Fairness is an essential component of the decisions taken by the subjects who participate in economic experiments. Among many related experimental paradigms, the dictator game (DG) and the ultimatum game (UG) have sparked much discussion regarding fairness. In a DG, when splitting a pie between one's self and another person, giving anything to another player may be the result of a preference for more fair outcomes (see e.g. [START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF]. Similarly, in an UG (Güth et al., 1982), the proposer may also have fairness concerns and raise offers above the minimum predicted by the subgame perfect equilibrium. According to Forsythe et al. (1994), under the "fairness hypothesis", distributions of offers in both the DG and UG should be the same. Alternatively, if the distribution of offers differs between the two games, then fairness alone is not enough to explain the subjects' choices in the UG.

Of course, one can fairly expect higher offers in an UG than in the DG because the first player in an UG must also account for the second player's Minimum Acceptable Offer (MAO), apart from any other-regarding motivation leading to a fairer split of the pie. Therefore, the receivers' veto power and the beliefs of proposers regarding their counterparts' MAO create an extra motivation for higher than minimum offers in the UG than in the DG. The share of this extra belief-and risk-related motivation in proposers' offers in an UG can be measured by the difference between donations in a DG and proposals in an UG. If the two coincide the "fairness hypothesis"), non-minimal proposals in an UG are fully motivated by fairness concerns, whereas if they significantly differ from each other, then the fairness motivation in UG offers can be rejected. One could envisage within-subject designs, measuring the difference between UG offers and DG donations, which would capture heterogeneity across individuals as far as the underlying behavioral motivations are concerned. Alternatively, one could look at differences across countries to identify socio-economic factors which could determine the size of the difference. Addressing the second type of question requires distinguishing between socioeconomic determinants of the other-regarding component in dictators' behavior, on one hand and, on the other hand, determinants of proposers' behavior, which we interpret under some assumptions as the (expected) MAO of UG receivers. 1 In this article, we investigate whether the "fairness hypothesis" is supported for experimental data collected in different countries.

Our article is motivated by the fact that experimental economics has provided very interesting but contradictory evidence to the debate. [START_REF] Henrich | In search of homo economicus: behavioral experiments in 15 small-scale societies[END_REF]Henrich et al. ( , 2005) ) investigate how the social and economic environment as well as cultural differences can shape the subject's behavior in experimental games. Their studies focus on small-scale societies' inhabitants rather than university students. Using various proxies for market integration in different locations such as individuals' participation in wage labor, reliance on cash cropping, competence in national language, or the amount of land an individual devotes to cash cropping, their UG and DG results show that exposure to markets is positively correlated to higher offers from proposers. 2However, a recent meta-study by [START_REF] Engel | Dictator games: a meta study[END_REF] provides results that may appear somehow different for the DG. He summarizes the evidence about the past 25 years of dictator experiments and addresses the issue of the relation between DG offers and the level of development of a country. Using an ordered categorical variable to distinguish between more or less developed countries, Engel finds a strongly significant negative correlation between DG offers and the level of development of a country. He concludes that "the more a society is primal, the more dictators are willing to share". As development shows high correlation with market integration, this may suggest a contradiction with previous results.

To investigate this issue, we implement a meta-analysis on a large number of UG and DG experiments to test whether the fairness hypothesis depends on the level of economic development of a society. For this purpose, we gathered a total of 96 standard UG and 144 standard DG. Apart from the obvious advantage of statistical power, the meta-analysis method is particularly crucial for investigating such a research question. Indeed, very few cross-country experimental studies comparing offers in both games are available in the literature. The only ones we are aware of are Henrich (2000), Ensminger (2004) and [START_REF] Henrich | Markets, religion, community size, and the evolution of fairness and punishment[END_REF], and only for small-scale societies. The meta-analyses by [START_REF] Engel | Dictator games: a meta study[END_REF] and [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] focus respectively on the DG and on the UG, they do not compare the two.

The first attempt to investigate cultural differences in bargaining behavior was undertaken by Roth et al. (1991). Their paper reports data from UGs that were collected in four different countries: Israel, Japan, the United States, and Yugoslavia. The data show that the observed bargaining outcomes are significantly different from the perfectequilibrium predictions in every country. Yet, offers made by proposers vary substantially across countries: the highest offers are observed in the United States and Yugoslavia while the lowest offers are made in Israel. Using data from 75 UGs, [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] conduct a meta-analysis in which they investigate the forces that shape the amount offered by the proposer in the UG. They found no significant differences in proposers' offers across different countries. However, it is useful to bear in mind that the authors clustered the countries of their database by continents, using a dummy variable for each continent.

The large number of existing experiments on DG and UG allows us to study the existence of significant country differences in the divergence between DG donations and UG offers. Our results are the first evidence that such differences have their origin in the effect of economic development on the preference for fairness.

Specifically, the more developed a country is, the lower are the donations in the DG. Interestingly, no effect of economic development is found on UG offers. Therefore, the gap between DG donations and offers in the UG and, thus, the rejection probability of the fairness hypothesis increases as we move from less to more developed countries. This implies that a country's development does not affect the expectation of UG proposers on their counterparts' MAO but, rather, the extent to which UG proposers and dictators in the DG have a preference for more fair outcomes.

Our meta-analysis has two major implications. First, the socioeconomic environment and, in particular, the economic development of a country influences negatively the share of fairness among the motivations of more generous actions. Hence, the fairness hypothesis will not be rejected for the less developed countries, contrary to what has been found in all past studies generally based on western countries' subject pools. Second, there is a large value-added potential in the combined meta-analysis of data obtained from different games, as a source of information on the underlying motivations of observed behavior.

The remaining part of the article is organized as follows. Section 2 offers a very simple model clearly defining the fairness hypothesis. Section 3 presents the data and design of the meta-analysis. We describe the procedure used to select the studies, the explanatory variables of the meta-regression, and the estimation methods. Section 4 is devoted to the comparative meta-analysis of DG and UG offers. Section 5 concludes.

The fairness hypothesis

According to Forsythe et al. (1994), under the fairness hypothesis, distributions of offers in the DG and in the UG should not be different. Alternatively, if the distribution of offers differs between the two games, then fairness alone is not enough to explain the subjects' choices in the UG. In other words, the veto power conferred on the respondent in the UG creates a strategic environment that may cause the proposer to offer a greater amount, in order to avoid rejection by the respondent. In fact, this only occurs if the proposer believes that the amount he would be willing to give in the absence of veto power would not satisfy the receiver's MAO. In this situation, the proposer will consider the recipient's preferences and increase his offer to meet their MAO and avoid rejection. This extra gift can be described as a "strategic gift".

Fairness is not an absolute but a relative concept that depends on the norms that apply in a given culture or in a given geographical area. As stated by [START_REF] Chaudhuri | Experiments in economics: playing fair with money[END_REF], "notions of fairness may vary across cultures in that offers that are considered unfair and routinely turned down in one society are readily accepted in others." Giving 30% of your endowment can appear greedy if recipients expect you to give 50% but can also appear decidedly generous if they expect you to give 10%. For simplification purposes, it is reasonable to consider that fairness is a matter of norms and expectations where giving the norm is fair while giving under the norm is not.

According to this definition of fairness, measuring and comparing proposers' fairness would require estimating the difference between the actual norm of fairness in a given country or in a given culture and the share proposers give when nothing forces them to (the DG). But what defines the norm? The average "maximal acceptable offer" (hereafter MAO) of recipients in a population does. More precisely, the average MAO can be interpreted as the recipient's aversion to disadvantageous inequality. Supposing that proposers are good at guessing the threshold below which their offer will be refused, average UG offers just reflect the norm of tolerance to disadvantageous inequality. [START_REF] Tisserand | Le jeu de l'ultimatum, état de l'art et méta-analyse de 30 années de recherches expérimentales[END_REF] performed meta-analysis of the UG and focused on the respondants' MAO. The results show that, on average, respondents' MAO that can be found in papers using the strategic method coincides with the average amount offered by proposers. Under the as-sumption that proposers' beliefs about recipients' MAO are accurate, our study aims at investigating whether proposers' fairness depends on the level of economic development of countries.

We formalize this interpretation for more clarity. Let w denote the proposer's initial endowment and y ult ∈ [0, w] (respect. y dic ∈ [0, w]) denote his offer in the ultimatum (respect. dictator) game. The value x ∈ [0, w] denotes the recipient's MAO. The distribution of x in the population may vary with the country where the UG is played. Let θ denote the proposer's belief about x. We need to formulate two assumptions:

Assumption 1: Proposers have a correct belief about the distribution of the recipients' MAOs: θ = x, so that they correctly infer the recipients' average MAO, i.e. E( θ) = E(x).

Assumption 2: Proposers are risk-neutral.

Under these assumptions, the proposers' payoff maximizing strategy in the UG is clearly to offer y ult = E(x). A straightforward definition follows:

Definition: In the DG, under Assumptions 1 and 2, a fair (respect. unfair) offer y dic is characterized by y dic = y ult (respect. y dic < y ult ).

Of course if proposers are risk-or ambiguity-averse, they may add a safety margin in addition to the mean MAO in the UG. In this case, our reasoning still holds if that safety margin is constant across all countries, i.e. supposing that risk aversion does not substantially differ across different countries. Similarly, if proposers' MAO estimates are inaccurate, our result will still hold provided that the estimation biases do not vary across countries.

Hence, under our assumptions, if we consider that the recipients' MAO reflects the norm of fairness in a given culture or in a given geographical area, our study allows us to investigate whether proposers' fairness depends on a country's level of economic development.

Data and design

Design of meta-analysis

For this comparative meta-analysis of the UG and the DG, we first constructed two separate databases that were merged later. We used Econlit and Google Scholar with four different combinations of keywords: "Dictator game", "Dictator experiment", "Ultimatum game", and "Ultimatum experiment". The search for "dictator game" and "dictator experiment" papers in Econlit yielded respectively 368 and 110 results while we obtained 396 and 105 results for "ultimatum game" and "ultimatum experiment".

On Google scholar, results are much more numerous but may also become less relevant over subsequent pages. 3 We found a large number of articles in common in the two databases which still provided a fair amount of studies with which to perform metaanalyses. The search was completed in February 2014.

Our main purpose is to establish a comparative study of the DG and the UG. It is thus important that the average offers of both games are computed on the basis of very similar criteria and are not affected by protocol differences which is why we focused our analysis on standard UG and DG. Restraining our observations to these games allows us to ensure that our analysis is not confounded by unobserved heterogeneity associated to protocol designs departing largely from the standard design. 4For the DG we chose to retain only standard protocols referring to the DG under its original form initially proposed by Forsythe et al. (1994), with two anonymous subjects and a random entitlement. Restricting the selected studies to such DG led us to exclude a large number of studies, for example those with the following characteristics:

-Studies in which the available set of actions is too restricted, constraining subjects to choose either the selfish outcome or the equal share. We thus chose to keep the studies in which the number of feasible actions for dictators was at least eight.

-Studies in which no money or fake money is at stake. As assumed by economic theory, subjects are sensitive to monetary incentives. We have no reason to believe that a dictator would make the same decision when earnings are hypothetical. Moreover, we also excluded studies in which dictators are endowed with less than $4. We made this choice to ensure that monetary incentives were non-negligible.

-Studies in which dictators are asked to give their money to a charity association. In this case subjects' behavior may obviously be altered compared to a standard protocol.

-Studies in which subjects "earned" the dictator position. In this case, since subjects deserve the dictator position, they may behave more selfishly than usual.

-Studies in which the subjects' anonymity is not totally guaranteed. Such a protocol can improve subjects' generosity in case of observability of their choice.

-Studies which involve any form of competition. Such protocols modify subjects' incentives depending on the purpose of the competition.

-Studies in which subjects played any other game prior to the DG. This restriction is set to avoid contagion effect across treatments.

-Studies implying more than two subjects or computerized subjects.

-Framed studies in which players do not have neutral denominations. For exemple protocols in which the recipient is called the "partner" can increase dictators' generosity toward the recipient. Similarly, framed studies in which players are called "sellers" and "buyers" can influence players' decisions and are thus excluded.

These restriction criteria led us to rule out many studies and, for many others to select only one observation, the control treatment. As we seek to estimate the average proportion of the endowment offered in a standard DG, we set these restrictions to ensure that our estimate is not altered by uncontrolled protocol differences.

For the ultimatum games we also chose to retain only standard protocols, referring to the UG under its original form initially proposed by Güth et al. (1982) with two anonymous subjects and a random entitlement. We thus, for example, excluded studies having the following characteristics:

-Repeated games in which subjects are matched with the same subject for all the periods (partner protocols). We retained only stranger protocols to avoid reputation effects.

-Repeated games in which players' mean offer is revealed at the end of each period. This information can influence players' decisions.

-Studies which involve any form of competition.

-Studies in which the information differs between the proposer and the responder. In these studies, there is uncertainty about the endowment or the offer made by the proposer.

-Studies in which decisions are taken in groups and studies involving more than two subjects or computerized subjects.

-Studies in which the set of actions available to proposers is not at least eight.

-Studies in which there is no real money at stake. Studies in which proposers are not endowed with at least $4 (or equivalent).

-Studies in which proposers earned the proposer position.

-Studies in which anonymity is not totally guaranteed.

-Studies in which subjects played another game prior to the UG.

-Studies in which the Nash perfect sub-game equilibrium is revealed to players before they make a decision. This information can obviously influence players' decisions.

-Framed studies in which players are called seller/buyer in the experiment. Again, we chose to set these restrictions to ensure an accurate estimate of the average proportion of the endowment offered by a proposer in the standard UG, excluding protocol differences.

Although these criteria may seem restrictive, they are necessary for the DG and the UG to be comparable.5 Finally, the DG database contains a total of 144 observations collected from 96 articles. The articles were published between 1994 and 2013, 2008 being the median date of the sample. Regarding the geographical distribution of the studies, observations were collected from 30 different countries (see the list of included countries in Table 8 in the Appendix). Our UG database contains a total of 96 observations collected from 42 articles and one book. The articles were published between 1983 and 2011, 2001 being the median date of the sample. The selected UGs were collected from a set of 29 different countries.

Variables

For each selected article, ultimatum or dictator, we recorded three categories of information. First, the essential information are:

-The average offer of the study, as it is our dependent variable of interest, expressed as a proportion of the initial proposer's endowment.

-The standard error of the average offer, which allows us to define the appropriate weights of the studies. Some papers did not report information about the dispersion of offers or the distribution of offers. In these cases, we sent emails to the authors whose papers did not allow us to record the standard deviations of offers.

The second category of variables are the explanatory variables of interest, related to the game (ultimatum or dictator) and to the degree of economic development of the country where the experiment was run. Since there is no standard method to measure how developed a country is, we used three different proxies, each measuring economic development level by very different approaches. We choose to rely on heterogeneous measures of economic development as a robustness check of our results. Table 8 in the Appendix shows the value of these variables for each country involved in the analysis.

-Macroeconomic development

Our first proxy relies exclusively on macroeconomic indicators. It considers the GDP per capita, the HDI (Human Development Indicator) index, the poverty rate of the country where the experiment was run, and the year it was run. This information was available on the World Bank database (www.worldbank.org). As including them simultaneously in the meta-regressions would lead to serious multicolinearity problems, we built a synthetic variable based on a principal component analysis (PCA) of these three variables. From this analysis, we only kept the first axis since it explained more than 80% of the total variation.

-Ease of doing business The second proxy refers to the regulatory environment of business. We use the "ease of doing business" indicator of the World Bank website (www.doingbusiness.org/rankings). The ease of doing business index ranks economies from 1 to 189 with first place referring to the most business-friendly economy. A high ranking (a low numerical rank) means that the regulatory environment is conducive to business operation. The index averages the country's percentile rankings on 10 topics covered in the World Bank's Doing Business. 6In contrast to the other two proxies, highly developed countries are thus among the lowest numerical values. Hence, for comparison purposes, we report the results obtained when the scale of this variable is reversed.

-Bank account penetration The third proxy is a measure of financial inclusion. We use the "account penetration" rate of the global Findex database (datatopics.worldbank.org/ financialinclusion) to assess countries' financial inclusion. This rate is the percentage of inhabitants (aged 15+) who possess a bank account with a financial institution.

There is no consensus about the effect of the degree of economic development of a country on subjects' choices in both games. On the one hand, the results of Henrich (2000), [START_REF] Henrich | Markets, religion, community size, and the evolution of fairness and punishment[END_REF]Ensminger (2004) show that exposure to markets is positively correlated to higher offers from proposers in both the UG and DG. On the other hand, Engel (2011) finds a strongly significant negative correlation between DG offers and the level of development of countries. As regards the UG, [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] find no evidence that countries influence subjects' choices. However, it is useful to bear in mind that the authors clustered the countries of their database by continent.

The third category of variables are control variables: -Amount of money to share (proposer's endowment). We systematically converted this amount into dollars in Purchasing Power Parity (PPP) at the date we wrote the article. Neither in the UG (Hoffman et al., 1996;Slonim and Roth, 1998;Cameron, 1999) nor in the DG (Carpenter et al., 2005;Cherry et al., 2002;Forsythe et al., 1994) has the literature shown evidence of any significant effect of the amount of money at stake on the subject's choices. We nevertheless include this variable for better control.

-Whether or not subjects are students in economics. 7 In the UG, Carter and Irons (1991) show that economics students behave in a more selfish way than other subjects. In the DG, this variable has not been studied to our knowledge.

-Whether or not the experiment has been run in a laboratory. Note that since the proportion of experiments that are run in a laboratory is greater in developed countries than in undeveloped countries, this variable is highly correlated with our development variables. Moreover, since all except one study in labs involve students, we did not introduce an additional dummy variable when the subjects are students or not.

-Whether or not the game has been run with a double blind procedure. Players may be more selfish when they are ensured anonymity toward experimenters. In his meta-study, [START_REF] Engel | Dictator games: a meta study[END_REF] finds no significant effect for the double blind procedure in the DG.

-Whether or not the game is repeated. Since proposers in repeated UGs have the opportunity to play the game multiple times, it is possible that the mean of the mean offers across all periods in a repeated UG could differ from the mean of one-shot offers in non-repeated UGs. Studies by [START_REF] Roth | Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term[END_REF] and [START_REF] Tisserand | Le jeu de l'ultimatum, état de l'art et méta-analyse de 30 années de recherches expérimentales[END_REF] suggest that proposers' offers remain constant over time. [START_REF] Cooper | The dynamics of responder behavior in ultimatum games: a meta-study[END_REF] find that proposers adjust their behavior according to responders' choices in order to maximize their profit.

-We also coded two dummy variables to control for the two possible strategy methods. These variables are equal to 1 when the strategy method is used, and 0 when it is not. As these protocols may be more likely to be used in field experiments for organizational purposes, their effect could be correlated with economic development. There are two possible uses of the strategy method. The first strategy method concerns the responder in the UG. The responder must announce his minimum acceptable offer before receiving the proposer's offer. The second strategy method, in both the UG and DG, consists of making decisions for the two possible roles before players' roles are randomly drawn.

Table 1 provides an overview of the main descriptive statistics of our two databases. a) In US dollars at the time the paper was published.

Meta-regression and the fairness hypothesis

In this section, we aim at exploiting statistical power of the meta-analysis to investigate the fairness hypothesis across different countries. In particular, we test the fairness hypothesis for different degrees of economic development, measured by three different proxies.

Our analysis is based on the collection of two separate datasets that have been merged. The DG database contains a total of 144 observations, while the UG database contains a total of 96 observations. Since meta-studies by [START_REF] Engel | Dictator games: a meta study[END_REF] and [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] provided an exhaustive analysis of both the DG and UG, we only provide here the results. Regarding the amount offered in both games at the aggregate level, our results are very similar to those of [START_REF] Engel | Dictator games: a meta study[END_REF] for the DG and [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] for the UG. Using a simple FAT-PET-MRA model for each game, it appears that estimated mean offers are 30.6% and 42.58% respectively of the total amount to share and the estimated 95% confidence intervals for the average value of the offer are respectively of [29.05%; 32.17%] and [40.88%; 44.28%].

Subsequently, we perform a meta-regression on the merged sample to investigate the fairness hypothesis across different types of populations.8 Our baseline model is specified as follows:

γis = γ 0 + γ 1 D is + x is β 1 + x is D is β 2 + z 1is β 3 + z 2is D is β 4 + is (1)
where γis represents the s th mean offer sampled from study i; γ 0 is the intercept; D is is a dummy variable aimed at distinguishing between the DG and the UG, this dummy is equal to 1 if the s th in study i is an UG and 0 otherwise.

x is is the vector with the variables of interest (macroeconomic development, account penetration, ease of doing business). β 1 is then the corresponding vector of parameters to be estimated that capture the specific impact of these variables on the estimated offer in the DG. β 2 is the vector of parameters to be estimated that should be interpreted as the difference in impact of these variables between the UG and the DG. We consider three specifications, depending on the content of x is : the three variables of interest are included separately.

z 1is is the vector with the control variables relevant for the DG: Amount at stake, Economist, Laboratory, Double.blind, Repeated and Strategic2. β 3 is the corresponding vector and indicates the specific impact on the estimated offer of these variables for the DG.

z 2is is the vector with the control variables relevant for the UG: Amount at stake, Economist, Laboratory, Repeated, Strategic1 and Strategic2. β 4 is the corresponding vector to be estimated. For these variables common to the DG, the corresponding coefficients must be interpreted as the difference in impact of these variables between the UG and the DG. For the variable Strategic1, which is specific to the UG, the coefficient should be interpreted as the specific impact of this variable on the estimated offer in the UG.

Finally, is is the error term with is ∼ iid(0, σ 2 is ). In order to control for potential publication bias, [START_REF] Stanley | Beyond publication bias[END_REF] proposes to include the estimated standard error as an additional moderator variable. We, therefore, also consider the following specification:

γis = γ 0 + γ 1 D is + x is β 1 + x i D is β 2 + z 1is β 3 + z 2is D is β 4 + φ 1 se(γ is ) + φ 2 D is se(γ is ) + is (2)
where se(γ) is the estimated standard error of the mean offer. Considering that this publication bias might be different for the UG and the DG, we also include an interaction variable between se(γ) and D is . Hence, φ 1 controls for publication bias in the DG and φ 2 controls whether there is a difference in publication bias between the UG and the DG.

Several methods can be used to estimate these models. Obviously, a fixed effect estimator is not relevant for our purpose, since it assumes that all studies share the same real variable of interest. Because of unobserved protocol differences, it is impossible to reliably apply this estimation method. This is further confirmed by the fact that the Qtest of heterogeneity applied to equations ( 1) and ( 2) strongly rejects the null hypothesis of between-study homogeneity for all three configurations of x is , as shown in the first column of Table 2. Conversely, the random effects estimator allows the real variables of interest to vary from one study to the other but this method may be sensitive in case of publication bias. [START_REF] Stanley | Beyond publication bias[END_REF]Doucouliagos (2015, 2016) propose to estimate equations ( 1) and ( 2) using an unrestricted least squares (WLS) model, which consists in estimating equations ( 1) or (2) using weighted least squares with 1/se 2 (γ is ) as the weights. We performed Breusch-Pagan (BP) tests against (i) all variables (ii) a clustering variable related to each study to check for within-study dependence and (iii) se(γ) to check whether the WLS method is necessary.

The results are reported in the last three columns of Table 2. They are consistent across all specifications. The null hypothesis of homoscedasticity is strongly rejected when the BP test is computed with all variables. It is also rejected when the BP test is computed with the study variable. However, the null hypothesis of homoscedasticity cannot be rejected when the BP test is computed with se(γ). Hence, we estimated equations ( 1) and ( 2) using both random effects and OLS with White robust-standard errors. The results with standard errors clustered at the study level are similar and available from the authors upon request.

Dependent variable: Mean offer

Macroeconomic development

Account penetration

Ease of doing business First, with respect to the control variables, we have computed marginal effects. In the case of the DG, they simply correspond to β 3 . For the UG, they correspond to β 3 + β 4 . Our results show that the amount of money at stake does not significantly influence the choice of the proposer in the UG or the DG. In other words, the remuneration of subjects is unable to explain the observed average offers. These results are consistent with studies by Cameron (1999), Hoffman et al. (1996) and Slonim and Roth (1998) for the amount of money at stake. Regarding the environment of the experiment, in all specifications, the dummy variable Repeated does not significantly affect the amount offered in the UG or in the DG. The two types of strategic protocols we considered do not significantly affect players' offers in the UG. In the DG, the strategy method shows a small negative and significant effect in two of our 6 models. Economists tend to give significantly less than their non-economist counterparts in both games. Finally, the results show that offers made in laboratory experiments differ significantly from offers made in field experiments. DGs that are run in a laboratory rather than in the field lead to significantly lower offers (for instance, -5.10% in specification (1)) while UGs that are run in a laboratory lead to significantly greater offers (for instance, -5.10% + 9.20% = 4.1% in specification (1)).

OLS-White

Second, the level of economic development of countries in which experiments were run as measured by the different proxies significantly influences subjects' offers in all specifications for the DG and the UG. While dictators' offers tend to fall as the level of economic development rises (β 1 is significant and negative), offers in the UG stay stable with the level of economic development (β 1 + β 2 is never significant). These results are robust across the two estimation methods and the three different proxies considered.

Hence, conclusions are similar for the three specifications : the gap between UG and DG offers increases as the level of economic development increases which suggests that the fairness hypothesis is more likely to be rejected in highly developed countries.

To illustrate the differences between the UG and the DG, we made in-sample predictions for the DG and UG offers for the minimum and maximum values of the three variables of interest. Computing these offers involved defining a value for each control variable of each regression. For each control variable, we chose to set the value according to the number of observations available. In other words, except for Lab, each dummy control variable was set to its most common value. The continuous control variable amount at stake is set to its average value. The estimated offers displayed in tables 4 and 5 thus correspond to the following configuration: non-economist subject, non-strategic protocol, non-double blind, average amount at stake. In each case, we also performed tests of mean difference between UG and DG offers for both the most and the least developed country. These F-tests were based on the estimation results in Table 3 for the OLS-White specification9 . Specifically, we tested the following null hypothesis: 1), where z2 has its variables set as explained above. If H 0 cannot be rejected for a given value of x is , then there is no significant difference between the UG and the DG offers for this value of x is . We performed these tests for min(x is ) and max(x is ) for each variable of interest. The results of the in-sample predictions and the F-tests are displayed in Table 4 for predicted offers in labs (Lab = 1) and in Table 5 for predicted offers not in labs (Lab = 0). Notes: *** significant at 1%, ** significant at 5%, * significant at 10% 5% prediction confidence intervals in parentheses. Notes: *** significant at 1%, ** significant at 5%, * significant at 10% 5% prediction confidence intervals in parentheses.

H 0 : γ 1 + x is β 2 + z2 β 4 = 0 in equation (
In the lab the results are homogeneous across all models. While the fairness hypothesis cannot be rejected at the 5% error threshold for the least developed country in all specifications, the same does not hold true for the most developed country.

First, regarding the highest values of the three proxies of economic development, we find that in specification (1), the average UG offer is estimated at 43.79% of the amount at stake while it amounts to 20.98% in the DG. This 22.81% gap between UG and DG offers allows rejecting the fairness hypothesis for the most developed country without any doubt. Specifications (2) and (3) provide very similar results with respectively 20.00% and 20.33% difference between UG and DG offers.

Turning to the lowest values of economic development proxies, results are also homogeneous across all models. In specification (1), the average UG offer is estimated at 43.30% of the endowment while it amounts to 35.51% in the DG. The estimated difference between UG and DG offers proves to be much tighter (7.79% difference) for the least developed country10 . Specifications (2) and (3) show similar results with respectively 5.50% and 6.07% difference between UG and DG offers.

In the field, the results are also consistent across all models. In all specifications, as shown by the F-tests, the difference between UG and DG offers is not significant at conventional significance levels for the least developed country while it is highly significant for the most developed country.

The estimated differences between UG and DG offers are less important in field experiments, in both highly and least developed countries. In the least developed country, the DG offer is estimated to be higher than the UG offer in all specifications. In the most developed country, dictators keep an extra 13.64% of the pie compared to the UG in specification (1)11 .

These results show that the more developed countries are more likely to reject the fairness hypothesis of Forsythe et al. (1994). This is even more true in field experiments where DG offers are estimated to be higher than UG offers. According to the definition of fairness we proposed in section 2, we conclude that people from less developed countries tend to be fair whereas people from more developed countries tend to be unfair12 .

Conclusion

In this study, we report results from a meta-analysis of the UG and the DG. Specifically, we have studied offer differences across the two games and interpreted them as departures from the fairness hypothesis driven by differences in exposure to market mechanism. The differences observed have been studied, looking for country differences.

Our results show that the degree of economic development of a country influences DG but not UG offers. In particular, we find that people from more developed countries tend to give less in the DG. Thus, we provide support to the findings of [START_REF] Engel | Dictator games: a meta study[END_REF] regarding the DG but differ from the results of [START_REF] Henrich | In search of homo economicus: behavioral experiments in 15 small-scale societies[END_REF] and Henrich et al. (2005), who find offers to be positively correlated with market integration in both games.

The interesting consequence of our results is that the gap between UG and DG offers widens as the level of economic development increases, implying that the fairness hypothesis is rejected for the highest developed countries but not for lowest developed ones.

The conclusions that can be drawn about fairness from these results depend on how fairness is defined. As first suggested by [START_REF] Henrich | In search of homo economicus: behavioral experiments in 15 small-scale societies[END_REF], notions of fairness vary across cultures and countries, and situations that constitute the norm in some societies might be perceived as unfair in others. In our study, we defined fairness as a relative, rather than absolute concept, liable to vary across different countries or cultures. In particular, for simplification purposes we define fairness as a matter of norms and expectations where giving the norm (or above) is fair while giving less than the norm is not. Following this definition, we then seek to compare fairness across different populations by comparing the difference between the actual norm of fairness in a given country or in a given culture and the share proposers give when nothing forces them to (the DG). We assume that, on average, UG proposers have an accurate idea of recipients' MAO and take the mean UG offers as a measure of fairness norms in different locations. Under this hypothesis and considering our simplified definition of fairness, we show that the less developed a country, the fairer its inhabitants. Our results suggest that this effect is due rather to differences in what proposers are willing to give in the absence of fear of rejection than to variation in fairness norms across different countries.

As interesting as the result may be, it relies on rather bold assumptions: we assume that UG offers are representative of fairness norms in a given country or population. Indeed, it is reasonable to suppose that UG offers are not simple predictions of recipients' MAO but also include a safety margin in addition to the proposer's guess as to the recipient's MAO. In particular, different people from different countries or societies might have different levels of risk aversion, implying that the UG offer may not be a suitable proxy for fairness norms in a given society. Supposing that the aforementioned hypotheses do not hold, our results can still be interpreted in the narrow sense that the fairness hypothesis of Forsythe et al. (1994) is rejected in highly developed countries while it is not for the least developed county of our database.

Overall, our results do not seem to be in line with Montesquieu's famous thesis of "doux commerce", that the involvement in market interactions tends to pacify relations with others. Neither do they support the virtuous side of markets regarding fairness defended by [START_REF] Paganelli | Commercial relations: From adam smith to field experiments[END_REF]. In a more developed country, market integration supposedly makes subjects more sensitive to fairness. The subject has nevertheless long been debated. Authors such as Marx insist on the negative impact of commercial interactions on the moral foundations of society.13 Moreover, in a complex society, where life is regulated and protected almost exclusively by large and anonymous institutions (the constitution, laws, social security, big companies, etc.), close relationships with others may become less frequent and less necessary. In more developed societies, individuals know that social security will provide them with some protection in the event of severe illness or if their home is destroyed by fire. So, what is the purpose of maintaining good relations with others? Things are completely different in traditional (small) societies, where life-changing events require the help of relatives, friends, but also other acquaintances.

  ** significant at 1%, ** significant at 5%, * significant at 10% OLS-White means OLS with White robust standard errors in brackets.

Table 1 : Descriptive statistics of the ultimatum and dictator game studies

 1

			Ultimatum game			Dictator game
		Mean	Median	Min	Max.	Mean	Median	Min.	Max.
	Mean offer	41.50%	42.40%	25%	56.80%	25.31% 25%	6.30% 47%
	Number of subjects	82	40	14	320	66	35.5	12	426
	Publication date	2001	2001	1983 2011	2006	2008	1994	2013
	GDP per capita (a)	23096	28459	349	87998	32592	38175	350	93352
	HDI	0.779	0.880	0.4	0.93	0.8756	0.951	0.467	0.961
	Poverty	13.08%	8.50%	1%	35%	17.86% 13.20%	1.16% 63.70%
	Account penetration	80.04%	94%	7%	99%	82.15% 94%	7%	100%
	Ease of doing business (reversed) 147.1	173	19	184	152.6	181	19	188
	Amount at stake (PPP) (a)	41.21	11.56	2	700	19.25	10	1	110
	Economist subjects	18.75%				10.42%
	Laboratory	72.92%				75%
	Double-blind	-				49.31%
	Repeated	18.75 %				2.77%
	Strategy method 1	14.58%				-
	Strategy method 2	3.12%				7.63%

Table 2 :

 2 Specification tests.

			Q-test	BP test	BP test BP test
			of heterogeneity All variables	Study	se(γ)
	Macroeconomic development	eq (1)	2389.847***	72.974***	10.979***	0.381
		eq (2)	2367.498***	76.487***	12.162***	0.320
	Account penetration	eq (1)	2389.847***	80.236***	12.187***	0.276
		eq (2)	2383.591***	83.901***	13.078***	0.314
	Ease of doing business (reversed) eq (1)	2738.345***	78.258***	9.938***	0.889
		eq (2)	2302.408***	80.779***	10.840***	0.884
	Notes: *** significant at 1%, ** significant at 5%, * significant at 10%

Table 3

 3

	provides the estimation results for equation (1).

Table 3 : Meta-regression for the pooled sample: Random effects and OLS estimation results

 3

Table 4 :

 4 Estimated offers of non-economist subjects for an average amount at stake, non strategic settings, non-double blind, in the lab

		Minimum macro. development	Maximum macro. development
	Ultimatum	43.30%	43.79%
		(38.86% -48.74%)	(39.98% -47.59%)
	Dictator	35.51%	20.98%
		(31.08% -39.94%)	(18.50% -23.45%)
	F-test of difference between	3.57*	337.38***
	ultimatum and dictator offer
		Minimum account penetration	Maximum account penetration
	Ultimatum	41.87%	43.98%
		(36.49% -47.25%)	(41.72% -46.24%)
	Dictator	36.27%	23.98%
		(31.89% -40.65%)	(22.10% -25.87%)
	F-test of difference between	2.927*	316.56***
	ultimatum and dictator offer
		Ease of doing business, last rank Ease of doing business, first rank
	Ultimatum	42.73%	43.79%
		(37.72% -47.73%)	(41.58% -45.99%)
	Dictator	36.66%	23.46%
		(32.16% -41.15%)	(21.53% -25.40%)
	F-test of difference between	0.4215	62.734***
	ultimatum and dictator offer

Table 5 :

 5 Estimated offers of non-economist subjects for an average amount at stake, non strategic settings, non-double blind, in the field

		Minimum macro. development	Maximum macro. development
	Ultimatum	39.28%	39.75%
		(34.71% -48.83%)	(34.05% -45.46%)
	Dictator	40.65%	26.11%
		(37.06% -44.24%)	(21.84% -30.38%)
	F-test of difference between	0.247	16.41***
	ultimatum and dictator offer
		Minimum account penetration	Maximum account penetration
	Ultimatum	38.17%	40.28%
		(33.39% -42.95%)	(36.50% -44.07%)
	Dictator	42.50%	30.21%
		(38.60% -46.41%)	(27.11% -33.32%)
	F-test of difference between	1.741	17.433***
	ultimatum and dictator offer
		Ease of doing business, last rank Ease of doing business, first rank
	Ultimatum	38.93%	39.99%
		(34.82% -43.04%)	(35.95% -44.03%)
	Dictator	41.24%	28.04%
		(37.66% -44.82%)	(24.48% -31.60%)
	F-test of difference between	0.685	18.866***
	ultimatum and dictator offer

This is further detailed in section

The term market integration here refers to the simplicity for a country or for a community to have access to common markets.

On Google Scholar, we obtained 70,800 results for the DG and

42,700 for the UG. Among these numerous results, approximately the first 30 pages, with 10 results per page, displayed articles related to our study.

A large number of protocols are unique or have only been performed a few times. Even if it might technically be possible to include a dummy variable to control for the effect of one particular study, the statistical power of such a "control" would be too weak to draw any conclusions.

Given the purpose of our meta-analysis, we did not systematically supplement our list of papers by those from the meta-analyses of[START_REF] Engel | Dictator games: a meta study[END_REF] and[START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF]. We found it more important to avoid biases between the searches for UG and DG papers than to find as many papers as possible.

This ranking takes account of the ease of doing the following actions: resolving insolvency, enforcing contracts, trading across borders, protecting minority investors, obtaining credit, registering property, getting electricity, paying taxes, dealing with construction permits, and starting a business.

This does not include business students. This variable equals 1 only if 100% of the subjects in the study are students of economics.

The meta-analysis focuses on the value of the variable of interest whereas the meta-regression focuses on the variables that influence this variable. For further details, see[START_REF] Stanley | Meta-regression analysis in economics and business[END_REF].

The results for the random effects specification are similar and are available upon request from the authors.

This difference is mildly statistically significant at the 10% significance level in the two first specifications, whereas it is not in the third specification.

10.07% in specification (2), 11.95% in specification (3).

To ensure the robustness of these results we have also investigated some alternative specifications. First, as there are only four studies with a repeated protocol in DGs and only three studies with a Strategic2 protocol in UGs, we removed these studies from the sample and reestimated the model without the variables Repeated and Strategic2*Ultimatum. On this specification, we then performed the same in-sample predictions as before. Our findings are robust to this change. The results are available upon request from the authors. Second, we estimated equation (2) with the standard error and its interaction with the UG dummy as additional moderator variables. The corresponding coefficients are either not significant, or only significant at 10%. The in-sample predictions based on this specification are reported in Table6 and Table 7in Appendix A for experiments run in labs and in the field respectively, setting the value of the standard error to 0. They show that our main result still holds.

For further details on this debate, see[START_REF] Hirschman | Rival interpretations of market society: Civilizing, destructive, or feeble[END_REF] and the summary for example inEnsminger (2004).

Appendix A

Notes: *** significant at 1%, ** significant at 5%, * significant at 10% 5% prediction confidence intervals in parentheses. Notes: *** significant at 1%, ** significant at 5%, * significant at 10% 5% prediction confidence intervals in parentheses.

Appendix B


Margueritat

M Shepilov

A Anan'ev

L Maksimov

A Onushchenko

J Margueritat

Anomalously low Rayleigh scattering by CdS/CdSe doped glasses

Keywords: Glass, CdS/CdSe nanocrystals, Light scattering, Landau-Placzek ratio, Interference effects, Computer simulation

Anomalously low Rayleigh scattering by CdS/CdSe doped glasses

Introduction

Silicate glasses with CdS-CdSe nanocrystals (quantum dots) are being extensively studied during decades by a lot of structural sensitive methods such as absorbance and luminescence spectroscopy, transmission electron microscopy (TEM), small-angle X-ray scattering (SAXS), and the others [START_REF] Champagnon | Size of Cd(S,Se) quantum dots in glasses: Correlation between measurements by high-resolution transmission electron microscopy, small-angle x-ray scattering, and low-frequency inelastic Raman scattering[END_REF][START_REF] Champagnon | CdSxSe1-x Nanocrystallites: New materials for non linear optics[END_REF][START_REF] Azhniuk | Evidence for formation of Se molecular clusters during precipitation of CdSe 1-x S x nanoparticles in glass[END_REF][START_REF] Irmer | Size analysis of nanocrystals in semiconductor doped silicate glasses with anomalous small-angle X ray and Raman scattering[END_REF][START_REF] Ivanda | Low-wave-number Raman scattering from CdS x Se 1-x quantum dots embedded in a glass matrix[END_REF].

Recently it was found that Rayleigh and Mandelshtam-Brilloun scattering (RMBS) spectroscopy can be effectively used for separating the contributions of nano-scaled fluctuation inhomogeneites of a glass host and semiconductor nanocrystals (NCs) into Rayleigh scattering intensity that opened a new path for estimating NCs number density in glasses [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF].

In the present work, we apply RMBS spectroscopy to study of color glasses (optical cut-off filters) with CdS x Se 1-x NCs and obtain an unexpected result: Rayleigh scattering intensity of some samples was practically as low as that for initial glasses without NCs. This result is discussed in terms of interparticle interference effects in scattering. To illustrate a possibility of interparticle interference in the case of small volume fraction of NCs, simulation of light scattering for a model system is carried out.

Experimental

Glass samples

Cut-off filter glasses of Russian production have been chosen for studying. Compositions of ZhS (yellow), OS (orange) and KS (red) glasses included 64SiO 2 , 3B 2 O 3 , 13ZnO, 20(Na 2 O+K 2 O) (wt%) while the concentration of dopants (Cd, S and Se) did not exceed 1wt %. As-melted glasses were colorless but their heat-treatment in the range 530-620 C induced their coloring. This effect is caused by precipitation of NCs and depends on both S/Se concentration ratio and heat-treatment conditions.

Glass samples of 151510 mm 3 dimensions with all polished faces were prepared for RMBS measurements. The same samples were studied by low frequency Raman scattering (LFRS) spectroscopy. Absorbance spectra were measured using plane-parallel plates with thickness of 0.10.25 mm.

Measurement procedures 2.2.1. RMBS spectroscopy

RMBS spectra of the glass samples were taken by piezoelectric scanned Fabry-Perot interferometer. Light scattering was excited by He-Ne laser (λ = 633 nm, power  35 mW) and measured at the scattering angle θ = 90. To enhance contrast, the spectra were processed with Bayesian deconvolution technique. Spectral shifts of inelastic longitudinal (Mandel'shtam-Brillouin) and transverse components of light scattering spectra, ∆ν l , and Δν t , and Landau-Placzek ratio R L-P = I R /(2I MB ), where I R and I MB are the intensities of elastic (Rayleigh) and Mandel'shtam-Brillouin components, were measured within 0.5%, 1% and (15)%, respectively. Using I MB as an inner reference, I R may be estimated as a function of glass composition and external impact.

Details of the measurement procedure can be found elsewhere [START_REF] Maksimov | Rayleigh and Brillouin scattering spectroscopy for oprical material science[END_REF][START_REF] Anan'ev | Origin of Rayleigh scattering and anomaly of elastic properties in vitreous and molten GeO 2[END_REF].

Optical absorption

Absorbance spectra of the glass samples were measured in 0.2-0.7 μm spectral range by spectrophotometer Shimadzu UV 3600 using a round diaphragm of 2 mm diameter. High optical homogeneity of the samples was evidenced by coincidence of results of measurements carried out at different areas of a sample surface.

LFRS spectroscopy

The experiments were performed by means of LabRam HR equipped with super notches ensuring the resolution down to 7 cm -1 . The spectra were obtained at λ = 532 nm with an x50 objective with a NA of 0.35, at full power (100%) using a confocal pinhole of 100µm. Spectra taken at different scattering points of the samples were reproducible.

Additional data

Densities ρ, refractive indices n and glass transition temperatures T g of the glasses under study were taken from [START_REF]Colored optical glass, Specifications[END_REF] 3. Results

RMBS

Typical RMBS spectra of as-melted and heat-treated glass ZhS 10 (I) are shown in Fig. 1 where two doublets are caused by modulation of light wave by longitudinal (L) and transverse (T) hypersonic waves. The Rayleigh components (R) are surrounded by the ellipse, and their ordinates should be multiplied by 1000.

The obtained RMBS spectroscopy data [spectral sifts of longitudinal (Mandel'shtam-Brillouin) and transverse components, Δν l and Δν t , and Landau-Placzek ratio R L-P ] and some reference data of the glasses are summarized in Table 1.

Table 1

RMBS and reference data for some color filter glasses. melt "frozen-in" at its cooling [START_REF] Schroeder | Light scattering of glass[END_REF]. In the case of multicomponent silicate optical glasses, Rayleigh scattering intensity is mainly originated from "frozen-in" concentration fluctuations [START_REF] Anan'ev | Characterization of optical glasses by Rayleigh and Mandel'shtam-Brillouin scattering and high temperature acoustic methods, Part 1, Flint glasses[END_REF].

Heat-treatment does not affect the spectral shifts, the density, and the refractive index, but leads to coloring the glasses from yellow to red (Table 1) which is caused by precipitation of CdS x Se 1-x NCs (see below). Even if the volume fraction of NCs is very small ( 10 4 ), their precipitation should lead to a significant increase in the intensity of Rayleigh scattering whereas the intensity of Mandel'shtam-Brillouin scattering remains unchanged [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF]. Hence, the precipitation of NCs should cause significant increase in Landau-Placzek ratio. For example, Landau-Placzek ratio of glasses with CdS or CdSe NCs of 1.3 nm radius and volume fraction of 0.001 was estimated by equations of [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF] as R L-P > 1000.

For some heat-treated glasses presented in Table 1, this conclusion is confirmed by experimental data (e.g., for color glasses #5, #7, and #8, L-P 1000 R  ). However, for most heattreated color glasses in Table 1, Landau-Placzek ratio is significantly smaller, L-P 300 R ˆ, and, for two of them (#10: ZhS16(II) and #11: ZhS17(I)), this ratio is close to that of the as-melted colorless glass.

Thus, the data of Table 1 show that Rayleigh scattering of a system of NCs may be suppressed.

To understand a degree of such suppression, we studied manifestation of NCs in optical absorption and LFRS. Using these methods, we estimated size and volume fraction of NCs for a number of the glasses, calculated Landau-Placzek ratio associated with Rayleigh scattering by NCs and compared the results of calculation with experimental data.

Optical absorption

Examples of absorption spectra of the glasses are illustrated in Fig. 2. The absorption of the asmelted glass #1 (ZhS10(I)) is negligible at wavelengths  > 350 nm. Heat-treatment shifts absorption edge to longer wavelengths, so that intensive absorption bands appear in visible spectral range. These bands in 250-450 nm spectral range can be explained by precipitation of CdS x Se 1-x NCs in a glass host (curves 2-5) while the absence of the bands in the spectrum of as-melted Zh10(I) glass evidences the absence of any crystal inclusions (curve 1) [START_REF] Borrelli | Quantum confinement effects of semiconducting microcrystallites in glass[END_REF].

Fig. 2.

Absorption spectra of the glasses: 1 -ZhS10(I), as-melted; 2 -ZhS16(I); 3 -ZhS17(I); 4 -ZhS18(I); 5 -OS13(II); 6experimental silicate glass with CdSe NCs of 1.5 nm radius [START_REF] Golubkov | The kinetics of the formation of CdSe nanocrystals in sodium-zinc-silica glass[END_REF]. µ = log 10 (τ)/d, where τ is internal transmittance, d is sample thickness in cm. The arrows show the wavelengths of the maxima of the lowest (in energy) absorption bands for CdSe doped glass (curve 6) and for silicate glass with CdS NCs of 1.4 nm radius [START_REF] Ekimov | Size quantization of the electron energy spectrum in a microscopic semiconductor crystal[END_REF].

Obviously, only CdSe NCs with a radius less than 1.5 nm could give absorption spectra similar to those presented by curves 2-4 in Figure 2. However, the glasses contain sulfur as well and its content is higher than the content of selenium. Therefore one can expect that radii of CdS x Se 1-x NCs in these samples should be larger than 1.4 nm.

As stated above, it should result in very large values of R L-P for the volume fraction of the crystalline phase appropriate for such glasses. To compare quantitatively experimental values of R L- P ratios with those obtained within a simple model of independent Rayleigh scatters [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF] one need getting values of NCs size and their volume fraction primarily. These data will be given below.

Absorption spectra of glasses of the same code but taken from various batches are shown in than 6 cm 1 that corresponds to optical density 3 for the usual filter thicknesses of 5 mm. Thus, being similar functionally samples of different batches may differ in parameters such as composition, volume fraction and/or size of NCs precipitated due to heat-treatment.

Fig. 3.

Absorption spectra for glass samples of the same code taken from different batches.

These parameters though to a different extent may affect Rayleigh scattering intensity.

However, there is no obvious correlation between the spectra and the measured values of R L-P for them (given in Fig. 3). Actually, absorption coefficients being proportional to the volume fraction of CdS x Se 1-x NCs is larger for samples of the batch (I) whereas R L-P is larger for ZhS16(I) glass only. Moreover, the R L-P value for OS13(I) is more than three times lower than that of OS13(II) and it looks especially strange because the second derivative of their absorption spectra is quite similar evidencing the same composition and size of the precipitated NCs.

As to ZhS17 and ZhS18 glasses, some spectral shift to shorter wavelengths observed for samples of batch (II) can be due to a change of NCs composition but not due to a decrease of their size. In any case, extremely small values of R L-P for these glasses unambiguously cannot be explained with a simple model of independent Rayleigh scatters. The results give an additional hint for the existence of another factor governing Rayleigh scattering intensity strongly.

LFRS

Having in mind the qualitative estimations of NC dimensions in Section 3.2 one can use the data of LFRS for determination of CdS x Se 1-x NC size in the commercial filter glasses under study.

Appropriateness and reliability of this technique applied to glasses with NCs, namely, CdS x Se 1-x was confirmed earlier by results of a lot of studies [START_REF] Champagnon | Size of Cd(S,Se) quantum dots in glasses: Correlation between measurements by high-resolution transmission electron microscopy, small-angle x-ray scattering, and low-frequency inelastic Raman scattering[END_REF][START_REF] Irmer | Size analysis of nanocrystals in semiconductor doped silicate glasses with anomalous small-angle X ray and Raman scattering[END_REF][START_REF] Duval | Vibration eigenmodes and size of microcrystallites in glass: Observation by very-low-frequency Raman scattering[END_REF][START_REF] Champagnon | Nanocrystallites vibration modes of CdS x Se 1-x semiconductors in glasses: Size determination by Raman scattering[END_REF].

Using formalism of [START_REF] Duval | Vibration eigenmodes and size of microcrystallites in glass: Observation by very-low-frequency Raman scattering[END_REF], radius a of NCs can be estimated as LFRS NC 0.9 v /(2 )

l ac   (1)
where  LFRS is the frequency of LFRS peak, v NCl is the longitudinal sound velocity in precipitated NCs and c is the light velocity in vacuum.

Normalized spectra of the ratio of scattered light intensity in polarized Raman spectra (VV) to the intensity in unpolarized ones (VH) for a selected set of samples are presented in Fig. 4. v NCl = 3700 m/s, (CdSe).

(2b)

The data obtained for samples whose absorption spectra are given in Fig. 3 are summarized in Table 2.

Table 2

The frequencies of LFRS peaks, radii, and volume fractions of NCs. To estimate volume fraction of nanocrystals, the absorption coefficient of glasses in 250-300 nm spectral range can be used under the assumption that deviations due to quantum confinement effects can be practically neglected [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF]. For this estimation, extinction coefficients of CdS and CdSe NCs were calculated using the optical constants of bulk crystals for a number of wavelengths [START_REF] Adachi | Optical Constants of Crystalline and Amorphous Semiconductors, SpringerScience + Business Media[END_REF] and the expression for the efficiency of extinction of small particles (Eq. (5.7) in [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]). The calculations showed that the contribution of scattering in extinction (Eq. (5.8) in [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]) can be neglected in comparison with that of absorption in all variants presented below.

Glass code

In the calculations, the radii of NCs were taken from Table 2 and

Discussion

For a two-phase system consisting of NCs in a glassy matrix, Landau-Placzek ratio [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF] is

h NC L-P L-P L-P , R R R  (3)
where h L-P R and NC L-P R are Landau-Placzek ratios of the glass host and contribution of NCs into L-P R , respectively.

In the case of independent Rayleigh scattering of light by identical spherical NCs distributed randomly in a glassy host with a real refractive index n, the contribution of Ncs into the Landau-Placzek ratio L-P R can be written as [START_REF] Shepilov | Landau-Placzek ratio of an inorganic glass with PbS quantum dots[END_REF]: indices m in Eq. ( 3) are taken from [START_REF] Adachi | Optical Constants of Crystalline and Amorphous Semiconductors, SpringerScience + Business Media[END_REF]: . Using I MB as an inner reference it seems easy to estimate the coefficient of Rayleigh scattering suppression, I R (exp)/I R (cal).

   
        (4)
n p = 2.

Table 3

Concentration of nanocrystals, N 0 , and coefficient of Rayleigh scattering suppression I R (exp)/I R (cal).

For CdS nanocrystals

Glass Code N 0, particle/cm Explanation for the effect of suppression of the Rayleigh scattering in glasses may be given on the basis of the approach where interference of light scattered by different elements of structure of nanostructured glass is considered [START_REF] Andreev | Scattering of visible light by glasses undergoing phase separation and homogenization[END_REF]. In the case of NCs (or phase-separated particles) distributed in a glass matrix, two approaches are used for theoretical description of interference.

In the first approach, interference of light scattered by particle (NC) and its diffusion zone is considered [START_REF] Goldstein | Theory of scattering for diffusion-controlled phase separation[END_REF]. The approach is applicable to early stage of particle growth when diffusion zones do not overlap. At certain conditions, such interference leads to suppression of dipole component in scattering, so that the intensity of scattering becomes significantly less than the intensity of the Rayleigh scattering by single particle. For randomly located zones (which are centered at randomly located particles), the theory of transformations predicts [START_REF] Christian | The Theory of Transformation in Metals and Alloys, Part 1[END_REF] that diffusion zones may be treated as isolated if their volume fraction is small, v z < 0.1. Under this condition, particles grow independently and their volume fraction v f is less than one tenth of volume fraction of particles in the end of growth stage, v max .

Let us estimate the maximum volume fraction of NCs in the glasses under consideration on the basis of batch composition. The glasses contain about 0.5 wt.% of CdS. The simple estimation

shows that this content corresponds to volume fraction v max  0.0027. The letter value may be considered as the upper limit for precipitation of CdS phase because of evaporating sulfur during melting process and equilibrium solubility of CdS in the glass matrix. Thus, one can conclude that the volume fractions v f presented in Table 2 are close to maximum volume v max , the condition v f < 0.1v max is not satisfied and the first approach is inapplicable.

In the second approach, interference of radiation scattered by different particles (NCs) is considered (interparticle interference) [START_REF] Guinier | Small-Angle Scattering of X-Rays[END_REF]. This approach was applied to small-angle X-ray (e.g., [START_REF] Wright | Nucleation and crystallization of cordierite-TiO 2 glass ceramic: Part 1. Small angle neutron scattering measurements and simulations[END_REF][START_REF] Alekseeva | Morphology of Silicate Glasses with Lead Sulfide Nanocrystals[END_REF]) and light (e.g., [START_REF] Andreev | Scattering of visible light by glasses undergoing phase separation and homogenization[END_REF][START_REF] Shepilov | On the problem of theoretical description of anomalous light scattering by phase separated glasses[END_REF]) scattering in glasses. In this approach, it is assumed that the particles are distributed in homogeneous matrix. Thus, the approach is applicable to the stage of transformation when the active diffusional growth of particles from supersaturated solution is completed, the volume fraction of particles is close to maximum, and the concentration of precipitating component in the matrix is close to equilibrium. Basing on the estimate of volume fraction of NCs in the glasses under investigation, we consider that their structures are close to the structures accepted in the approach of interparticle interference. Below the interference effects will be analyzed in terms of this approach.

Interparticle interference was often observed in glass-ceramics with appreciable volume fraction of NCs, v f  0.10.4 (for example, see [START_REF] Wright | Nucleation and crystallization of cordierite-TiO 2 glass ceramic: Part 1. Small angle neutron scattering measurements and simulations[END_REF][START_REF] Shepilov | Anomalously low light scattering in the Na 2 O-Nb 2 O 5 -SiO 2 glass-ceramics[END_REF][START_REF] Shepilov | Effect of low NiO doping on anomalous light scattering in zinc aluminosilicate glass-ceramics[END_REF]). The color glasses under consideration are classified as glass-ceramics [START_REF] Deubener | Updated definition of glass-ceramics[END_REF]. However, the volume fraction of NCs in these glass-ceramics is very small, v f  0.001 (Table 2), and it is usually assumed that such sparsely distributed NCs scatter the light independently without interference effects [START_REF] Mishchenko | Scattering, Absorption, and Emission of Light by Small Particles[END_REF]. Nevertheless, the interparticle interference was observed in SAXS experiments for glasses containing small volume fraction v f  0.005 of NCs [START_REF] Alekseeva | Morphology of Silicate Glasses with Lead Sulfide Nanocrystals[END_REF].

Because the notion of interparticle interference for sparsely distributed scatterers (NCs) contradicts the usually accepted opinion [START_REF] Mishchenko | Scattering, Absorption, and Emission of Light by Small Particles[END_REF], it would be interesting to give any theoretical explanation for interparticle interference effects in a system with small volume fraction of scatterers. To give such an explanation, we use the model of phase transformation in a system with simultaneous nucleation of diffusional growth of NCs [START_REF] Shepilov | On some aspects of the calculation of phase separation kinetics in metastable solutions[END_REF][START_REF] Shepilov | Calculation of kinetics of metastable liquid-liquid phase separation for the model with simultaneous nucleation of particles[END_REF]. It should be noted that the conception of diffusion-limited growth of particles was used for explanation of interference effects in light scattering [Erreur ! Source du renvoi introuvable.], however, the authors of this work modeled two-dimensional case for significant area fraction 0.25 of two-dimensional particles.

We consider the model [START_REF] Shepilov | On some aspects of the calculation of phase separation kinetics in metastable solutions[END_REF][START_REF] Shepilov | Calculation of kinetics of metastable liquid-liquid phase separation for the model with simultaneous nucleation of particles[END_REF] where nuclei of a new phase are distributed in a glass matrix randomly with a mean number density n 0 and begin the diffusion-limited growth simultaneously.

The model was proposed for description of transformation kinetics in a system with small supersaturation, in which the volume fraction of particles at the late stage of transformation is much less than unity. At the first stage (the growth stage), the particles (NCs) grow due to diffusion of precipitating component from supersaturated matrix and, as shown in [START_REF] Shepilov | On some aspects of the calculation of phase separation kinetics in metastable solutions[END_REF], the possibility of Ostwald ripening can be neglected at this stage. Here we simulated the structure and light scattering of the system at time when the growth from supersaturated solution is completed, but effect of Ostwald ripening is not yet significant. As shown in [START_REF] Shepilov | On some aspects of the calculation of phase separation kinetics in metastable solutions[END_REF][START_REF] Shepilov | Calculation of kinetics of metastable liquid-liquid phase separation for the model with simultaneous nucleation of particles[END_REF], the distribution of volumes of particles at this time may be approximated by distribution of volumes of cells in the spatial Poisson-Voronoi tessellation [START_REF] Chiu | Stochastic Geometry and its Applications[END_REF]. Volume fraction of particles at this time is close to the maximum and remains practically constant during of the Ostwald ripening stage.

It should be noted that the results obtained in the model are in agreement with experimental data on metastable liquid-liquid phase separation in glasses [START_REF] Shepilov | Calculation of kinetics of metastable liquid-liquid phase separation for the model with simultaneous nucleation of particles[END_REF]. The similar models were developed for the case of continuous nucleation [START_REF] Shepilov | A model for calculation of isothermal kinetics of the nucleation-and-growth type phase separation in the course of one-step heat treatment[END_REF][START_REF] Bruna | On the validity of Avrami formalism in primary crystallization[END_REF].

As argued above, the volume fraction of NCs in the glasses under consideration is close to maximum. Hence, the stage of phase transformation in these glasses relates to the end of growth stage or to the Ostwald ripening stage. Our model gives the quantitative results only for the end of growth stage. Possibly, some conclusion may be qualitatively applied to the beginning of the Ostwald ripening stage.

Using the model, one can simulate the random location of particles (NCs) and calculate the volumes of cells of the spatial Poisson-Voronoi tessellation, that is, the diameters of particles at the end of growth stage if the particle volume fraction is given. Particles in the simulated system are polydisperse (e.g., see Fig. 5 in [START_REF] Shepilov | Calculation of kinetics of metastable liquid-liquid phase separation for the model with simultaneous nucleation of particles[END_REF] and Fig. 6 below). If particles in our polydisperse system are sufficiently small and conditions of the Rayleigh scattering [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF][START_REF] Shepilov | Light scattering in optical material containing polydisperse spherical nanoparticles[END_REF] are satisfied, the intensity of scattered light may be calculated by the equations derived earlier (Eqs. ( 16)-( 19) in [START_REF] Shepilov | Light scattering in optical material containing polydisperse spherical nanoparticles[END_REF]). For respectively. The probability density () fd of the diameter distribution was estimated using the kernel method with the Gaussian kernel [START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF] and the window width (smoothing parameter) 0.006 h ; at which the mean integrated square error is minimum (Fig. 6). The statistical error of () fd is less than 0.005 and cannot be displayed in the scale of Fig. 6. The density () fd is almost symmetric with respect to d  1. It is practically equal to zero for d < 1.4 and d >1.6. The density of distribution over 12 size groups is also shown in Fig. 6.  ˆ, so that the probability of close spacing of particles is less than for random system where ( ) 1 g   . Second, the pair correlation function in Fig. 7a shows maximum that means existence of preferred spacing between particles. The similar behavior takes place for the other simulated partial pair correlation functions. It should be noted that in monodisperse system these correlations in particle arrangement lead to significant interference effects in scattering [START_REF] Wright | Nucleation and crystallization of cordierite-TiO 2 glass ceramic: Part 1. Small angle neutron scattering measurements and simulations[END_REF]. 18) in [START_REF] Shepilov | Light scattering in optical material containing polydisperse spherical nanoparticles[END_REF]) for a number of values of dimensionless scattering vector magnitude

1/3 4/3 1/3 0f 4 sin( / 2) / ( ) (4 ) sin( / 2) (3 ) / n n n v a          (6)
where  is the scattering angle.

The obtained values of the partial structure factors and Eq. ( 16) of Ref. [START_REF] Shepilov | Light scattering in optical material containing polydisperse spherical nanoparticles[END_REF] were used to evaluate the ratio indep sc sc / II of scattering intensity for the simulated system, sc I , to intensity indep sc I of independent scattering by particles which have the same diameter distribution as in the simulated system (see Fig. 6). The calculated ratio indep sc sc / II is presented in Fig. 8 as a function of dimensionless scattering vector magnitude  for a number of series of computer experiments.  ˆ (Fig. 8). This means that scattering should be low if the wavelength of light is large enough. This conclusion has a simple qualitative explanation.

Indeed, in our model, the volume of a particle is proportional to the volume of its cell, i.e., the concentration of precipitated substance averaged over the cell is the same for all cells, and if the wavelength is much greater than the mean size of the cells (the interparticle distance), the light "sees" the material as almost homogeneous. In this case, the interference effects determine the scattering properties of the system.

Let us estimate values of dimensionless scattering vector magnitude  (Eq. ( 6)) for the experiments described above. Because volume fraction of particles is small (Table 2), the refractive index of the glass matrix is equal to that of the material, . For such values, our model predicts the suppression of the scattering intensity in comparison with the case of independent scatterers by a factor of 30 and more. Such a situation is realized in glasses ZhS17(I) and ZhS18(I).

However, in some other glasses the suppression is less. There are various reasons for different suppression of scattering. For example, different glasses may be characterized by somewhat different stages of phase transformation; in particular, different times of the Ostwald ripening stage may be realized. To discuss the situation in more detail, one needs more information on the thermal history and structure of glasses.

Conclusions

Main parameters of CdS x Se 1-x nanocrystals in glasses (radius, volume fraction, number density) can be determined by study of light scattering (Raman, Rayleigh and Mandelshtam-Brillouin) and absorption.

For the first time, it was observed that precipitation of CdS x Se 1-x nanocrystals in silicate glasses induced by their heat-treatment does not cause an appreciable increase in Rayleigh scattering intensity.

Anomalously low Rayleigh scattering of some glasses with CdS x Se 1-x nanocrystals has been explained in terms of interference of light scattered by different nanocrystals (interparticle interference). To illustrate that interparticle interference effects may be significant in in the case of small volume fraction of nanocrystals, computer simulation of structure and light scattering of a system of nanocrystals has been carried out using the model proposed earlier. In the model, the simultaneous nucleation and diffusion-limited growth of nanocrystals are considered. The structure of such a system was simulated at the stage when the diffusional growth from supersaturated solution is completed, but effect of Ostwald ripening is not yet significant. It was shown that in a long-wavelengths range the Rayleigh scattering by the system is suppressed due to interparticle interference effects even if the volume fraction of nanocrystals is small.

Fig. 1 .

 1 Fig. 1. RMBS spectra of ZhS 10 (I) glass samples after deconvolution (256 iterations): as-melted (dots) and heat-treated (line). Scattering angle is 90, VV+VH polarization. Longitudinal (Mandelshtam-Brillouin), transverse and Rayleigh components are labeled by L, T and R, respectively.

  I), as-melted, R L-P = 44 2 ZhS 10 (I), heat-treated, R L-P

Fig. 3 .

 3 Fig.3. As one can see, they differ noticeably in the whole but coincide well only at µ values less

  ZhS 16 (I), R L-P = 245 ZhS 16 (II), R L-P = 48 ZhS 18 (I), R L-P = 58 ZhS 18 (II), R L-P = 116 (c)

Fig. 4 .

 4 Fig. 4. Normalized LFRS spectra (VV/VH) for the glasses: 1 -ZhS10(I), as-melted; 2 -ZhS10(I?); 3 -ZhS16(I); 4 -ZhS17(I); 5 -ZhS18(I); 6 -OS13(II).

Fig. 4

 4 Fig.4illustrates the absence of LFRS peak in the spectrum of as-melted glass and its appearance in the spectra after heart-treatments. Both intensity and position of the peak vary from one glass sample to another evidencing the change of NCs size. Lack of information about the composition and crystal modification of ternary compounds CdS x Se 1-x enforces to use the average values of longitudinal sound velocity in cubic and wurzite modifications[START_REF] Adachi | Properties of group-IV, III-V and II-VI semiconductors[END_REF][START_REF] Adachi | Properties of semiconductor alloys: group-IV, III-V and II-VI[END_REF]: v NCl = 4300 m/s, (CdS) (2a)

  their volume fractions were varied to obtain satisfactory agreement between the calculated and experimental values of extinction (absorption) coefficient of a given glass in the spectral range 250-300 nm. The result of such fitting the volume fraction for the glass ZhS18(I) is shown in Fig. 5a. The estimated values of volume fractions are given in Table2. To illustrate a degree of applicability of this procedure to volume fraction determination, we used it for the glasses containing CdSe NCs, for which volume fraction was measured independently by SAXS. The comparison of volume fraction values obtained by two methods (Fig.5b) shows a satisfactory agreement of the results.

Fig. 5 .

 5 Fig. 5. (a): Absorption spectrum of the glass ZhS18(I) (line) and the values of absorption coefficient calculated for NCs of CdS (solid circles) and CdSe (crosses). (b): Volume fraction values found from absorption (v fA ) vs those determined by SAXS (v fS ) (solid circles) for glasses containing CdSe NCs and their linear fit by relation v fA  0.85v fS (solid line).

I

  is the intensity of Rayleigh scattering by NCs; N 0 and v f are number density and volume fraction of NCs; p 12 and v l are Pockels coefficient and the hypersonic velocity of glass; and T is the ambient temperature (T = 300 K). Relative refraction index of NCs, m = n NC /n, is expressed in terms of their complex refractive index n NC and the refractive index of glassy host n.Hypersonic velocity v l can be determined from the spectral sift Δν l of the longitudinal (Mandel'shtam-Brillouin) component by[START_REF] Schroeder | Light scattering of glass[END_REF] where ν is the frequency of exciting light,  is the scattering angle. The obtained value of v l is 5550 ± 40 m/s. The values of refractive indices of NCs at  = 633 nm for evaluation of relative refractive

  calculation, particles are divided into size groups, and pair correlation functions for particles of different size groups (partial pair correlation functions) should be obtained in the course of simulation. Then partial structure factors and intensities of scattered light can be calculated for a given scattering vector. These calculations have been carried out in several series of computer experiments. Dimensionless representation of quantities was used. The volumes of simulated cubic samples were different in different series of computer experiments. To eliminate surface effects, particles belonging to surface layer of certain thickness did not used in calculation of particle radius distribution and partial pair correlation functions. Let us present some results of the simulation. First of all, the distribution of dimensionless diameters / d D D    was simulated where D and D  are particle diameter and its mean value,

Fig. 6 .

 6 Fig. 6. The probability density () fd of the distribution of relative diameters / d D D    ( D  is the mean diameter) (smooth curve) and the probability density corresponding to dividing particles into size groups l (l  1, 2, …, 12) (stepwise line).

Fig. 7 .

 7 Fig. 7. Examples of simulated partial pair correlation functions ' () ll g  (the results for two series of

l = 6 ,Fig. 8 . 1  1 

 6811 Fig. 8. The ratio of scattering intensities of for the simulated system, sc I , and the system of independent polydisperse scatterers, indep sc I , as a function of dimensionless scattering vector

  47 (CdS) and 2.85 + i 0.32 (CdSe). Finally, p 12 =  R was found from comparison of RMBS spectrum of colorless as melted glass (See Table1) and RMBS spectrum of commercial silica glass with p 12 =  0.28 in

	0.22 taken for calculations of NC L-P
	accordance with [21]. Results of calculating NC L-P R are summarized in Table 3. Taking into account
	that the calculated values NC L-P R	are significantly greater than h L-P R , it is possible to adopt
	(	NC L-P calc ) RR  L-P calc ) (

Table 1 )

 1 . The scattering angle 

	n 	1.525	(

Table 2 .

 2 As a result, the estimation gives the values

		;	0.6 1.0 

Acknowledgements

M.S. expresses his gratitude to the Russian Foundation for Basic Research (19-03-00855) for partial support of this work.