Aurélie Méjean
email: aurelie.mejean@cnrs.fr

Antonin Pottier

Stéphane Zuber

Marc Fleurbaey

Opposite ethical views converge under the threat of catastrophic climate change -preprint

Keywords: Climate change, Catastrophic risk, Equity, Population, Climate-economy model JEL Classification: D63, Q01, Q54, Q56, Q5

Climate change is seen by economists as an issue of intertemporal consumption trade-off: consume all you want today and face climate damages in the future, or sacrifice consumption today to implement costly climate policies that will bring future benefits through avoided climate damages. If one assumes enduring technological progress, a controversial conclusion ensues: to reduce intergenerational inequalities, we should postpone climate policies and let future, richer generations pay. Growing evidence however suggests that the trade-off is more complex: abrupt, extreme, irreversible changes to the climate may cause discontinuities to socio-economic systems, possibly leading to a sharp decline of human population and consumption per capita. The most relevant trade-off would then be between present consumption and the size of future generations. In this paper we show that when accounting even for a very small risk of catastrophic climate change, it is optimal to pursue stringent climate policies to postpone the catastrophe. Our results conform with the well-known conclusion that tight carbon budgets are preferred when aversion towards inequalities between generations is low. However, by contrast with previous studies, we show that stringent policies are also optimal when inequality aversion is high. This is because a higher inequality aversion makes the scenario of a small and relatively poor population (obtained when mitigation is low) especially unattractive. The size of the optimal carbon budget decreases with the social preference for large populations, but this parameter plays almost no role at extreme levels of inequality aversion. Our result demonstrates that views from opposite sides of the ethical spectrum in terms of inequality aversion converge in terms of climate policy recommendations, warranting immediate climate action. We identify ethically-robust climate policies as new spaces of compromise between contrasted ethical stances may emerge to set the ambition of climate policies.

Introduction

The risk of abrupt and irreversible changes to the climate is one of the five reasons for concern identified by the IPCC [START_REF] Mccarthy | Climate Change 2001: Impacts, Adaptation, and Vulnerability. Contribution of the Working Group II to the Third Assessment Report of the Intergovernmental Panel on Climate Change[END_REF][START_REF] Smith | Human Health: Impacts, Adaptation, and Co-Benefits[END_REF]. Extreme climate events, climate tipping points [START_REF] Lenton | Tipping elements in the Earth's climate system[END_REF] -such as the shutoff of the Atlantic thermohaline circulation, the collapse of the West Antarctic ice sheet or the dieback of the Amazon rainforest -may have direct impacts like a catastrophic loss of biodiversity [START_REF] Christopher | The projected timing of abrupt ecological disruption from climate change[END_REF]. Climate change may more generally favour the spread of vector-borne diseases [START_REF] Rocklov | Climate change: an challenge for vector-borne disease prevention and control[END_REF]. Extreme climate events, such as droughts, may have indirect impacts too, for instance through increased migration and conflicts [START_REF] Reuveny | Climate change-induced migration and violent conflict[END_REF][START_REF] Hsiang | Quantifying the Influence of Climate on Human Conflict[END_REF], triggering a 'cycle of conflict and climate disaster', as stated by the UN security council [START_REF] Un | Fragile countries risk being 'stuck in a cycle of conflict and climate disaster[END_REF]. Without appropriate policy responses, chain reactions could very well follow, possibly leading to general warfare, which would challenge the physical, political and social infrastructures of global society and possibly lead to its near collapse. Taking catastrophes into account has been a daunting task of economics [START_REF] Dowlatabadi | Climate Change Thresholds and Guardrails for Emissions[END_REF][START_REF] Richard | Is the Uncertainty about Climate Change too Large for Expected Cost-Benefit Analysis?[END_REF][START_REF] Azar | Catastrophic Events and Stochastic Cost-benefit Analysis of Climate Change[END_REF]. The discipline has traditionally mainly considered climate change as an issue of intertemporal consumption tradeoff [START_REF] William | The economics of Global Warming[END_REF][START_REF] William D Nordhaus | Managing the global commons: the economics of climate change[END_REF]]. In reality, the possible cascade of tipping points may represent an existential threat to civilization [START_REF] Lenton | Climate tipping points -too risky to bet against[END_REF] and may introduce an irreversible regime shift in the sense that post-catastrophe welfare is independent from pre-catastrophe actions [Cropper, 1976, Clarke and[START_REF] Clarke | Consumption/pollution tradeoffs in an environment vulnerable to pollution-related catastrophic collapse[END_REF], and could be very small. The possibility that social welfare may drop to very low levels can be interpreted as a reduction of the size of human population combined with a reduction of per capita consumption. The trade-off would then be not only between present and future consumption, but between present consumption and catastrophic risk reduction [Weitzman, 2009]. This trade-off has been little studied in the literature, with the exception of [START_REF] Bommier | Models-as-usual for unusual risks? On the value of catastrophic climate change[END_REF] and [START_REF] Martin | Averting Catastrophes: The Strange Economics of Scylla and Charybdis[END_REF].

Climate policy, and catastrophic risks in particular, raise very difficult ethical questions pertaining to how policies should be evaluated [START_REF] Kolstad | Social, Economic, and Ethical Concepts and Methods[END_REF]. A tool to incorporate ethical views is the use of social welfare functions that may represent distributive judgements (on how goods or consumption should be distributed among individuals) and views about the value of total population (this is the population ethics problem). The issue of evaluating policies with varying population size [START_REF] Broome | Climate matters: ethics in a warming world[END_REF] has been largely ignored in the literature, with the exception of [START_REF] Adler | Prioritarianism and Climate Change[END_REF], [START_REF] Scovronick | Impact of population growth and population ethics on climate change mitigation policy[END_REF] and [START_REF] Méjean | Catastrophic climate change, population ethics and intergenerational equity[END_REF]. This paper contributes to fill this gap by examining the issue of population ethics when there is a risk that population size may be affected by climate change. A key parameter to represent distributive judgements is the parameter of inequality aversion (η), also named elasticity of marginal utility of consumption in the economics literature: the larger the η, the more we want to equalize the distribution of consumption among individuals (see Methods). A key parameter we introduce to represent the value of total population is the social preference for large populations (β): the larger the β, the more we are willing to pay to increase population size (see Methods). Population ethics raises several paradoxes and impossibility results so that no consensus has emerged about the value of population. There are also debates about the correct value of inequality aversion, in particular in relation to social discounting [START_REF] Kolstad | Social, Economic, and Ethical Concepts and Methods[END_REF][START_REF] Dasgupta | Discounting climate change[END_REF]. Given such ethical disagreements, can we find policy recommendations that are ethically robust, or can we at least find limited consensus when ethical views differ?

The paper is structured as follows. Section 2 presents the framework used to evaluate climate policies. The numerical analysis presented in section 3 shows that the preferred climate policy depends on the relative impact of ethical parameters on welfare. Section 4 concludes.

Methods

This paper builds on [START_REF] Méjean | Catastrophic climate change, population ethics and intergenerational equity[END_REF], which examines the terms of the trade-off between present consumption and the existence of future generations when there is a risk of extinction due to climate change. Here, we extend that work by providing an extensive numerical analysis of the influence of ethical parameters on the optimal climate policy when accounting for the risk of catastrophic climate change. We evaluate the social welfare associated with 250 climate policies, corresponding to 250 carbon budgets. We follow two steps. Policies are first specified in terms of a savings rate, which is set at a constant value, 25.8%, for all scenarios [START_REF] Golosov | Optimal Taxes on Fossil Fuel in General Equilibrium[END_REF]. This is consistent with the observed world average gross savings rate [The World Bank, 2017, [START_REF] Dennig | Inequality, climate impacts on the future poor, and carbon prices[END_REF]. Policies are also specified in terms of carbon budgets, spanning 400-5800 GtCO 2 from 2015 to 2250 across scenarios. The 400 GtCO 2 budget is consistent with the objective of limiting warming to 1.5 • C, while the 5800 GtCO 2 budget corresponds to warming above 6 • C. We define the climate change related catastrophe in very simple economic terms as a drop in human population size (90% compared to the case without catastrophic climate change in the central scenario) combined with a drop in consumption per capita (after the catastrophe, consumption per capita is assumed to be constant, and is set at 1000 USD per capita per year in the central scenario). Each policy is first translated into a social outcome x using a climate-economy model linking the risk of catastrophe (or hazard rate) to temperature change. Social outcomes are defined in terms of consumption per capita c and hazard rate p over time. The catastrophe is defined as a drop in terms of human population size and consumption per capita. We assume that the catastrophic risk depends linearly on temperature, and on temperature only. We introduce parameter b as the marginal hazard rate (per annum per • C), which is the additional risk of a catastrophe occurring due to an increase in temperature of 1 • C above a given temperature threshold (see below). Due to the catastrophic risk, the cumulative population over the whole horizon is unknown ex-ante. The generation size stream n depends on the policy, although it is common to all social outcomes before the catastrophe occurs. Generation t is of size n t and enjoys consumption per capita c t . Consumption is assumed to be equally distributed among individuals of the same generation. The cumulative size of the human population depends on the policy, as the generation size drops at date H.

Date H depends on temperature via the hazard rate. After the catastrophe, generation size is n (constant), and consumption per capita is ĉ (constant). Social outcomes, i.e., the streams of consumption and catastrophic risk over time, are then translated into social welfare using a social welfare function [START_REF] Kolstad | Social, Economic, and Ethical Concepts and Methods[END_REF]. This function embodies ethical views in terms of inequality aversion and social preference for large populations. The chosen function treats all generations in a symmetric way, giving no a priori preference to the present [START_REF] Ramsey | A mathematical theory of saving[END_REF][START_REF] Stern | The economics of climate change: the Stern review[END_REF]. Policies are then ranked according to social welfare. Below, we introduce in turn the social welfare function, the climate-economy model, the catastrophic risk, and its calibration.

Social welfare function

First consider sure social outcomes (i.e., without risk) with a bounded time horizon H (meaning that extinction occurs then for sure). We follow most of the existing literature and write welfare W as a generalized utilitarian form, which can be written as the product of average utility and a population weight N β H .

W H = N β H H t=0 n t N H c 1-η t 1 -η - c 1-η 1 -η (1)
where N H = H t=0 n t is the total (cumulated) population that comes into existence in a given social outcome, and

c 1-η t 1-η -c 1-η
1-η is the utility enjoyed by an individual of generation t. There is no pure time discounting, i.e., we treat generations in a symmetric way. The social welfare function embodies the preferences of a social planner through two ethical parameters.

Parameter η is the intergenerational inequality aversion coefficient. It determines the marginal utility of individual consumption. A more inequality averse social welfare function means that the social planner is willing to sacrifice more to equalize consumption levels across generations. Indeed, increasing consumption by $1 brings a social value of c -η . A person with consumption 2c has a relative priority for consumption increase of 2 -η compared to a person with consumption c. Or equivalently, taking $1 from the person with consumption 2c while giving only $2 -η to the person with consumption c would leave social welfare unchanged. This means that the larger the η, the larger the loss we accept when making transfers to equalize income across individuals. This is the so-called leaky-bucket experiment [START_REF] Arthur | Equality and Efficiency: The Big Tradeoff[END_REF]. We test a wide range for this parameter, from 1 to 4, similar to the range of inequality aversion values reviewed in the latest IPCC assessment report [START_REF] Kolstad | Social, Economic, and Ethical Concepts and Methods[END_REF].

Parameter β is the social preference for large populations. For a population with equal consumption, a crucial parameter is the elasticity of substitution between population and consumption, that is how much average consumption must increase to compensate for a drop in population size. The larger the β, the more we are willing to sacrifice in terms of consumption to increase population size. The social welfare function embeds well-known views of utilitarianism when population size varies. Total (or classical) utilitarianism values the total sum of utilities (β = 1).

Average utilitarianism values average utility (β = 0). We vary β between 0 and 1, spanning cases between total and average utilitarian views.

The social welfare function also includes a consumption per capita threshold, c. In the case of total utilitarianism (β = 1), it is the critical level of consumption, i.e., the level of consumption such that, if enjoyed by an additional individual, total welfare is left unchanged when that individual is added to the population. When β < 1 however, the critical level is not constant and depends on population size and average utility. Here, the consumption per capita threshold is calibrated on an extreme poverty consumption level (1000 USD per capita per year). Now consider a risky social prospect. At each time step, three possible events may occur (conditional on humanity not being extinct and on the world not having switched to a catastrophic state): (i) humanity goes extinct, with probability p ex , which is unrelated to climate change ;

(ii) the catastrophe occurs and there is a switch to an exogenous path of consumption per capita and population (ĉ, n), with probability p c (t) (the hazard rate at date t); (iii) the world continues as before, with probability 1-(p ex +p c ). The probability of not becoming extinct and not having switched to a catastrophic state at the beginning of period H is:

P ≥H = H-1 t=0 (1 -(p ex + p c ))
. After the catastrophe, the world exists for τ additional periods with probability

p ex •(1-p ex ) τ -1 .
For each time horizon H, we write the contribution to expected total welfare of the event that humanity may go extinct, W ex,H , and the contribution to expected total welfare of the event that the catastrophe may occur, W c,H . Expected total welfare, W , is then written as the sum of those for all possible time horizons:

W ex,H = P ≥H • p ex • N β-1 H H t=0 n t • u(c t ) (2) W c,H = P ≥H • p c (H) ∞ τ =1 p ex (1 -p ex ) τ -1 (N H + τ n) β-1 H t=0 n t • u(c t ) + τ n • u(ĉ) (3) W = ∞ H=0 W ex,H + W c,H (4)
The above welfare function is used to evaluate the social outcomes computed by the climateeconomy model.

Climate-economy model

The trophe as an abrupt event that occurs over the course of a decade. This is a harsh simplification, as the processes involved would be gradual, with feedbacks between institutions, consumption, mortality and fertility, and would probably take decades to unfold. We consider that after the catastrophe occurs, the course of events is inevitably set and cannot be acted upon [Cropper, 1976, Clarke and[START_REF] Clarke | Consumption/pollution tradeoffs in an environment vulnerable to pollution-related catastrophic collapse[END_REF]]. Whether such a catastrophic chain of events would actually lead to a sharp decline of human population size and consumption per capita is controversial and may be considered as highly unlikely. Given the large uncertainty surrounding the probability of such events, we attempt at covering a wide range of possible values for that parameter (as described below).

Catastrophic risk

We assume that the hazard rate p t at date t depends on the temperature increase T only, and not on wealth or adaptive capacity. We therefore disregard any factor, social or natural, that may mitigate the effect of climate change on the risk of a catastrophe occurring. More precisely, we model the hazard rate p c (t) as a linear function of temperature increase T , above a temperature threshold T 0 . The hazard rate at date t, p c (t), is truly p c (T ), a function of temperature increase T at date t. We assume that this function is valid in a range of temperature increase spanning from 1 • C to 10 • C, which includes the values of temperature increase presented in the analysis.

Given the very nature of the risk considered, there is no available data about its realization from which the parameters of the hazard rate function could be inferred. The calibration can thus only be illustrative and tentative. The calibration does not rely on the frequentist approach to

Parameter ranges

The numerical analysis presented below will explore the following parameter ranges.

parameter description range

X

The role of inequality aversion and the preference for large populations

In this section we present the numerical analysis on how the preferred climate policy depends on the relative impact of ethical parameters on welfare. We compare the cases with and without accounting for a risk of catastrophic climate change, and explore wide parameter ranges, as summarised in Table 1. Each figure shows the preferred carbon budget (i.e., the carbon budget that gives the highest welfare in the range of carbon budgets considered) for each pair of ethical parameters (η, β).

The case without a risk of catastrophe

When the risk of catastrophe related to climate change is not accounted for (marginal hazard rate b = 0), large carbon budgets are optimal if the social planner is averse to inequalities across generations (high η). This is a standard result in the economics literature 1 : as future generations are assumed to be richer thanks to technological progress, a high aversion towards inequalities across generations means postponing climate policy (i.e., choosing larger carbon budgets) to spare the present, poorer generation [START_REF] Dasgupta | Discounting climate change[END_REF]. This is illustrated in Figure 2, where carbon budgets leading to warming of at least 3 • C are preferred for an inequality aversion η ≥ 1.7 (darker colors represent cases where a larger carbon budget results in higher welfare, and is thus the preferred option).

Accounting for the risk of catastrophic climate change

This result changes dramatically when accounting for a risk of catastrophe which depends on temperature increase. In the central scenario presented in Figure 3, the post-catastrophe per capita consumption (ĉ) is set at 1000 USD per year, and the post-catastrophe population (n) is set at 1 billion individuals. In that case, even with a very small risk (e.g., b = 10 -7 per annum per • C), it is optimal to achieve tigher carbon budgets than in the case without catastrophic risk. This result holds for all combinations of inequality aversion (η) and preference for large populations (β), although to various degrees. For instance in the case (η = 2.5, β = 0.5), with

1 However, this relies on analysis that ignores intragenerational inequalities and the fact that the most vulnerable populations will be severely hit by climate change. The role of inequality aversion is substantially altered when this is taken into account [START_REF] Budolfson | The comparative importance for optimal climate policy of discounting, inequalities and catastrophes[END_REF] a very small catastrophic risk (b = 10 -7 per annum per • C), the optimal carbon budget is 40% lower than in the case without risk (from 4600 GtCO 2 without risk, to 2800 GtCO 2 with risk).

With a stronger preference for large populations (larger β), it is optimal to implement a tighter carbon budget. This is shown on Figure 3, where for a given inequality aversion (say η = 3), increasing β from 0 to 1 (i.e., giving a stronger preference to large populations by switching from the average utilitarian to the total utilitarian case) reduces the optimal carbon budget by almost 70% from 5000 GtCO 2 (corresponding to a stabilization above 3 • C) to less than 1600 GtCO 2 , thus bringing the temperature increase close to the 2 • C objective. This result is intuitive, as a tighter carbon budget delays the catastrophe due to climate change, which leads to a larger cumulative number of individuals across generations.

The impact of inequality aversion (η) on the optimal carbon budget is very different than in the case without catastrophic risk. When accounting for a risk of catastrophe due to climate change, the standard result that high inequality aversion leads to postponing climate policy to spare the present, poorer generation does not hold. Instead, the results show that when accounting for an endogenous risk of climate catastrophe, both cases of high and low inequality aversion command tight carbon budgets (Figure 3). This is because with a low inequality aversion, the benefit of preserving a large and on average richer population makes mitigation attractive, whereas a high inequality aversion makes the scenario of a short-lived and relatively poor population particularly unattractive. For instance, with a strong preference for large populations (β = 1), both low inequality aversion (η < 1.5) and high inequality aversion (η > 3) command carbon budgets which may limit global warming to 2 • C (Figure 3).

Influence of the level of risk

This pattern remains as we consider higher risks of catastrophe due to climate change (from 10 -7 to 10 -5 per annum per • C). However, in those cases, tight carbon budgets are optimal for a wider range of ethical views: the results show a larger number of combinations of inequality aversion (η) and preference for large populations (β) for which tight carbon budgets are warranted (Figure 4). For even higher risks of catastrophic climate change (b set at 10 -4 and 10 -3 per annum per • C), the most stringent carbon budget (400 GtCO 2 ) is optimal for all social preferences on population size and inequality aversion across generations. Such a carbon budget is compatible with limiting global warming to 1.5 • C above the pre-industrial level.

The case of extinction

Allowing for the possibility of extinction assuming n = 0) does not qualitatively change the influence of the ethical parameters on the preferred policy, as the shape of the results remains the same (Figure 5). In particular, the case of total utilitarianism (β = 1) gives stricly identical results in the case of extinction compared to the case of a drop in population size and consumption per capita (labelled 'existential catastrophe' in the figure). This is due to the fact that in the central case presented here, post-catastrophe consumption (ĉ) is set to be equal to the threshold per capita consumption (c). Therefore u(ĉ) = 0, which means that adding or removing individuals who consume ĉ does not change aggregate welfare in the case of total utilitarianism.

The result is different for lower values of β. In the case of average utilitarianism (β = 0), it appears that the possibility of extinction leads to prefer larger carbon budgets, compared with the case of a catastrophe causing a drop in population size and consumption per capita (darker points on the β = 0 line on Figure 5). In other words, assuming that more people survive the catastrophe warrants more ambitious climate policies in the case of average utilitarianism. This

The influence of post-catastrophe consumption and population size

A lower higher level of post-catastrophe consumption does not qualitatively change the influence of the social preference for large populations and of inequality aversion on the optimal climate policy (Figure 6). For a given set of ethical parameters (η, β), the optimal carbon budget is smaller for a lower consumption level after the catastrophe (ĉ), which conforms with intuition. This is also the case for total utilitarianism: adding individuals with a level of consumption below c reduces aggregate welfare, hence the preference more stringent climate policies to postpone the catastrophe. Conversely, the optimal carbon budget is larger for a higher consumption level after the catastrophe (ĉ > c). As in the central case shown on Figure 4, for a given pair (η, β), the optimal climate policy is more stringent as the catastrophic risk increases (Figure 7 shows the case ĉ = 500 USD per capita). The same qualitative result is obtained in the case ĉ = 2000 USD per capita.

Finally, maintaining population size at the level reached before the catastrophe (i.e., 10 billion individuals) also gives similar qualitative results (Figures 8 and9). A larger post-catastrophe population size warrants more stringent climate policy for a given (η, β), for the opposite reasons as in the case of extinction (section 3.4).

Conclusion

In this paper, we show that the possibility that climate change may induce a sharp decline in human population size and per capita consumption, as unlikely as it may be, changes the terms of the trade-off to solve the climate puzzle. A catastrophic risk as small as 10 -7 per annum per • C above 1 • C commands much tighter carbon budgets than in the case where that risk is not accounted for. The size of the optimal carbon budget decreases with the social preference for large populations, but this parameter plays almost no role at extreme levels of inequality aversion. Further, the results show an unexpected ethical consensus on the ambition of climate change: when accounting for an endogenous risk of climate catastrophe, both cases of high and low inequality aversion command tight carbon budgets. With a low inequality aversion, the benefit of preserving a large and on average richer population makes mitigation attractive, while with a high inequality aversion, the scenario of short-lived and relatively poor population particularly unattractive. We also show that a catastrophic risk of 10 -4 per annum per • C above 1 • C commands strong, immediate action to limit global temperature increase to 1.5 • C for the whole range of ethical values considered. This result is robust to the level of post-catastrophe consumption per capita and population size explored in the sensitivity analysis. Accounting for the risk of catastrophe due to climate change reveals new spaces of compromise between opposing ethical stances to set the ambition of climate policies. Coalitions could therefore arise between opposite sides of the ethical spectrum in terms of inequality aversion.

A Economic equations of the RESPONSE model

The main equation of economic module of the climate-economy model RESPONSE [START_REF] Dumas | Comprehensive Description of RESPONSE[END_REF] are presented below.

Production function

The economy produces a unique good Y t from capital K t and labour L t . The production function is a traditional Cobb-Douglas. Population L t is an exogenous variable (before the catastrophe or extinction occurs). Total factor productivity A t is exogenous.

Y t = A t • K α t • L 1-α t (A-1)

Emissions

Emissions E t are endogenous. They are a by-product of production. Their level depends on output Y t , on the carbon intensity of production σ t , and on abatement a t . The carbon intensity of production is expected to decline over time through exogenous technical change.

E t = σ t • (1 -a t ) • Y t (A-2)
The climate module is described through a three reservoirs linear carbon-cycle model, similar to that used in the DICE model.

Non-catastrophic damages

Damages D depend on the atmospheric temperature increase T t . Damages are expressed as a share of output Y t . Damages are assumed to be a quadratic function of temperature change.

D(T t ) = κ • T 2 t (A-3)

Abatement costs

Abatement costs C a depend on the abatement level at t a t , and possibly on past abatement

a t-

Capital accumulation

We consider a continuum of identical households, who derive instantaneous utility from consumption of a composite good. C t is the aggregate consumption, and capital K t evolves endogenously according to the following equation:

K t+1 = (1 -δ) • K t + Y t • [1 -C a (

Figure 1 :

 1 Figure1: Catastrophic risk (hazard rate p c ) as a function of temperature increase T. p c (T ) is the hazard rate as a function of temperature increase T , p 0 is the minimum hazard rate, set at 10 -3 per annum, T is the temperature increase compared to pre-industrial levels ( • C), T 0 is the temperature increase above which the hazard rate starts and rising with temperature, set at 1 • C, b is the marginal hazard rate per • C above T 0 .

Figure 2 :

 2 Figure 2: Optimal climate budget as a function of the social preference for large populations (β) and the inequality aversion coefficient (η) for an exogenous hazard rate (b = 0)

Figure 3 :Figure 4 :

 34 Figure 3: Optimal climate budget as function of the social preference for large populations (β) and the inequality aversion coefficient (η) for a hazard rate that depends on temperature (b = 10 -7 per annum per • C, ĉ = 1000 USD per capita, n = 1 billion)

  is due to the fact that the catastrophe leads to a world with many future poor. As ĉ = c, the consumption of the surviving individuals is too low to contribute to aggregate welfare, while the fact that they are many (1 billion) reduces average welfare. Therefore, for low values of β, a low level of post-catastrophe consumption is more threatening than extinction from a welfare point of view, and warrants more abatement for a given set of ethical parameters (η, β).

( a )Figure 5 :

 a5 Figure 5: Optimal climate budget as a function of the social preference for large populations (β) and the inequality aversion coefficient (η) in the case of extinction (n = 0) vs. the case of a drop in consumption and population size (central case: b = 10 -7 per • C per annum, ĉ = 1000 USD per capita, n = 1 billion)

(

  a) ĉ = 500 USD per capita (b) ĉ = 1000 USD per capita (c) ĉ = 2000 USD per capita

Figure 6 :

 6 Figure 6: Optimal climate budget as a function of the social preference for large populations (β) and the inequality aversion coefficient (η) for b = 10 -7 per annum per • C and n = 1 billion individuals, in the central case (ĉ = 1000 USD per capita) vs. a case with lower post-catastrophe consumption (ĉ = 500 USD per capita) vs. a case with higher post-catastrophe consumption (ĉ = 2000 USD per capita)

(a) b = 10 - 3 Figure 7 :

 1037 Figure 7: Optimal climate budget as a function of the social preference for large populations (β) and the inequality aversion coefficient (η) for a range of values of the marginal hazard rate (b), ĉ = 500 USD per capita, n = 1 billion individuals

Figure 8 :Figure 9 :

 89 Figure 8: Optimal climate budget as a function of the social preference for large populations (β) and the inequality aversion coefficient (η) for b = 10 -7 per annum per • C and ĉ = 500 USD per capita, in the central case (n = 1 billion) vs. a case with higher post-catastrophe population (n = 10 billion)

  climate economy model RESPONSE[START_REF] Ambrosi | Optimal Control Models and Elicitation of Attitudes towards Climate Damages[END_REF][START_REF] Dumas | Comprehensive Description of RESPONSE[END_REF][START_REF] Pottier | The Comparative Impact of Integrated Assessment Models' Structures on Optimal Mitigation Policies[END_REF] belongs to the tradition of cost-benefit analysis integrated assessment models[START_REF] William D Nordhaus | Managing the global commons: the economics of climate change[END_REF][START_REF] Hope | Policy analysis of the greenhouse effect -An application of the PAGE model[END_REF][START_REF] Richard | On the optimal control of carbon dioxide emissions: an application of FUND[END_REF], combining a simple representation of the economy and a climate module. The economic module is a Solow-like growth model with capital accumulation and exogenous population. It includes climate mitigation costs and a climate damage function that is chosen to be equivalent to that of the DICE model. The main equations of the economic module are presented in the appendix (section A). The climate module describes the evolution of global temperature increase and radiative forcing as a function of emissions. The temperature increase feedbacks on the economy through the damage function. We model the climate catas-

Table 1 :

 1 Parameter ranges

		CO 2 budget	[ 400 , 5800 ] GtCO 2
	η	inequality aversion parameter	[ 1.0 , 4.0 ]
	β	population parameter	[ 0 , 1 ]
	b	marginal hazard rate	[ 0 , 10 -3 ] per annum per • C
	n	post-catastrophe population	[ 0 , 10 ] billion
	ĉ	post-catastrophe per capita consumption [ 500 , 1000 ] USD per year

  1 , if inertia is accounted for. Abatement costs are expressed as a share of output Y t . Note that abatement a t is expressed as a fraction of emissions.C a (a t , a t-1 ) = P T t • a t • ζ + (BK -ζ) •

	a ν t ν	+ ξ 2 • (a t -a t-1 ) 2	(A-4)

Table A :

 A a t , a t-1 ) -D(T t )] -C t (A-5)The main variables and parameters of the model are summarised in table A. Selected variables and parameters

	name description
	t	time
	A t	exogenous productivity factor
	C t	aggregate consumption
	α	share of capital in revenues
	δ	depreciation rate of capital
	D	damages (share of output Y t )
	C a	abatement costs (share of output Y t )
	T t	atmospheric temperature increase
	P T t exogenous technical progress (abatement technologies)
	BK backstop price
	ζ	marginal cost of abatement when abatement is null
	a ν t ν	absolute level of abatement
	ξ	inertia parameter
	κ	damage parameter
	ν	power coefficient of the abatement cost function


Justine Saint-Aubert
email: justine.saint-aubert@inria.fr

Ferran Argelaguet
email: ferran.argelaguet@inria.fr

Anatole L Écuyer
email: anatole.lecuyer@inria.fr

Tangible Avatar : Enhancing Presence and Embodiment During Seated Virtual Experiences with a Prop-Based Controller

Keywords: Prop, Controller, Tangible, Avatar, Virtual environments, Presence, Embodiment

We investigate using a prop to control human-like avatars in virtual environments while remaining seated. We believe that manipulating a tangible interface, capable of rendering physical sensations and reproducing the movements of an avatar, could lead to a greater virtual experience (presence) and strengthen the relationship between users and the avatar (embodiment) compared to other established controllers. We present a controller based on an instrumented artist doll that users can manipulate to move the avatar in virtual environments. We evaluated the influence of such a controller on the sense of presence and the sense of embodiment in 3 perspectives (thirdperson perspective on a screen, immersive third-person perspective, and immersive first-person perspective in a head-mounted display). We compared the controller with gamepad controllers to control the movements of an avatar in a kick-in-a-ball game as illustration. The results showed that the prop-based controller can increase the sense of presence and fun in all three perspectives. It also enhances the sense of embodiment in the immersive perspectives. It could therefore enhance the user experience in various simulations involving human-like avatars.

INTRODUCTION

Users are increasingly represented by human-like avatars in virtual environments that they can control with diverse controllers. These controllers have different levels of mapping between user inputs and avatar outputs, ranging from "arbitrary (unrelated to the function being performed) to natural (related to the function being performed)" [START_REF] Steuer | Defining virtual reality: Dimensions determining telepresence[END_REF].

For instance, controllers based on body movements (e.g., Nintendo Wiimote, walk-in-place controllers [START_REF] Boletsis | The new era of virtual reality locomotion: A systematic literature review of techniques and a proposed typology[END_REF][START_REF] Freiwald | Walking by cycling: A novel in-place locomotion user interface for seated virtual reality experiences[END_REF]) induce a natural mapping between user and avatar movements while joysticks, buttons, or hand gestures (e.g. [START_REF] Buttussi | Locomotion in place in virtual reality: A comparative evaluation of joystick, teleport, and leaning[END_REF][START_REF] Lee | The control of avatar motion using hand gesture[END_REF]) lead to high abstraction. Controllers that map natural body actions tend to enhance the users' experience in virtual environments, previous works suggesting that they increase the sense of presence (e.g [START_REF] Seibert | Control mapping in virtual reality: effects on spatial presence and controller naturalness[END_REF][START_REF] Skalski | Mapping the road to fun: Natural video game controllers, presence, and game enjoyment[END_REF][START_REF] Tamborini | The role of presence in the experience of electronic games. Playing video games: Motives, responses[END_REF][START_REF] Terziman | Shake-your-head: Revisiting walking-in-place for desktop virtual reality[END_REF][START_REF] Usoh | Walking¿ walking-in-place¿ flying, in virtual environments[END_REF]) related to the feeling of "being in the virtual world" [START_REF] Schuemie | Research on presence in virtual reality: A survey[END_REF], and the sense of embodiment [START_REF] Fribourg | Avatar and sense of embodiment: Studying the relative preference between appearance, control and point of view[END_REF] related to the feeling of "being inside, controlling and having a virtual body" [START_REF] Kilteni | The sense of embodiment in virtual reality[END_REF]. However, controllers with natural mapping tend to generate fatigue [START_REF] Terziman | Shake-your-head: Revisiting walking-in-place for desktop virtual reality[END_REF], so they cannot be used for long or by disabled users (e.g., people undergoing rehabilitation, older people). They are also not suitable for desktop use.

In this paper, we investigate using a prop-based controller consisting of a small-scale model of the avatars that can be manipulated directly by hand (Figure 1). Such a controller could be considered a trade-off between arbitrary and natural controllers: it allows one to We propose to use a tangible prop to control a humanlike avatar during seated virtual experiences. By manipulating the prop, users perform kinematics similar to those of the avatar and experience physical interactions with it. We show that such controllers can enhance the sense of presence and embodiment compared to traditional gamepad controllers.

control various movements in miniature so it does not generate fatigue while having kinematics similar to the avatars. It also conveys physical interactions that could enhance the user's virtual experience [START_REF] Ishii | Tangible bits: towards seamless interfaces between people, bits and atoms[END_REF][START_REF] Meehan | Physiological measures of presence in stressful virtual environments[END_REF]. With immersive Virtual Reality, the prop can even be co-located, i.e., share the same physical location, with an avatar in a third-person perspective for a potentially improved user experience.

Prop-based controllers have been used in the past to control the off-line pose or motion capture of virtual characters (e.g., [START_REF] Esposito | Of mice and monkeys: a specialized input device for virtual body animation[END_REF][START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Johnson | Sympathetic interfaces: using a plush toy to direct synthetic characters[END_REF]); as users perform more efficiently, these tasks with prop-based controllers than with a computer mouse [START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Yoshizaki | An actuated physical puppet as an input device for controlling a digital manikin[END_REF]. However, the interest in using prop-based controllers for real-time control of avatars in virtual environments has not been studied. In particular, the influence on the sense of presence and the sense of embodiment, important factors in users' virtual experiences, has not been explored.

This paper aims to fill this gap by evaluating the influence of a prop-based controller on users' virtual experiences (presence, embodiment, and fun). A prop has been designed and implemented to control an avatar from 3 different perspectives-a classic thirdperson perspective on a desktop screen. Using a head-mounted display (HMD), we propose co-locating the prop with a third-person avatar to transmit co-located haptic sensations (immersive thirdperson perspective). The controller was also implemented with an avatar observed from a first-person perspective in an HMD (immersive first-person perspective). In the three perspectives, the prop was compared with arbitrary mapping controllers (buttons or Dpad) in a scenario of kicking in a ball. The results show that the prop enhances the virtual experience of users from all perspectives.

RELATED WORK

In the following, we discuss existing prop-based controllers (also known as dolls) and their influences on virtual experiences.

Existing prop-based controllers

Different prop-based controllers have been proposed in the past to control virtual characters. For instance, rigid elements (e.g., objects [START_REF] Held | 3d puppetry: a kinect-based interface for 3d animation[END_REF] mobile phone [START_REF] Ye | Aranimator: in-situ character animation in mobile ar with user-defined motion gestures[END_REF], Vive controller [START_REF] Garcia | Spatial motion doodles: Sketching animation in vr using hand gestures and laban motion analysis[END_REF]) can be manipulated by hands and tracked to control the movements of virtual objects. However, with these interfaces, users can only control global motions, not local ones (e.g., arm or leg).

Other interfaces have been proposed to control local motions (e.g., [START_REF] Barnes | Video puppetry: a performative interface for cutout animation[END_REF][START_REF] Esposito | Of mice and monkeys: a specialized input device for virtual body animation[END_REF][START_REF] Feng | Motion capture data retrieval using an artist's doll[END_REF][START_REF] Hiroki | A puppet interface for the development of an intuitive computer animation system[END_REF][START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Johnson | Sympathetic interfaces: using a plush toy to direct synthetic characters[END_REF][START_REF] Knep | Dinosaur input device[END_REF][START_REF] Numaguchi | A puppet interface for retrieval of motion capture data[END_REF][START_REF] Raffle | Topobo: a constructive assembly system with kinetic memory[END_REF][START_REF] Slyper | Mirror puppeteering: Animating toy robots in front of a webcam[END_REF]). For instance, Knep et al. [START_REF] Knep | Dinosaur input device[END_REF] designed instrumented dolls representing dinosaurs to perform motion capture for movies. Esposito et al. [START_REF] Esposito | Of mice and monkeys: a specialized input device for virtual body animation[END_REF] proposed an interface called "the monkey" representing a virtual human model, assuming that pose capture would be easier with the 3D device than with a 2D mouse input. A commercial controller lighter and smaller than the monkey and therefore easier to manipulate was suggested by celsys QUMARION. To represent other types of characters and overcome the fixed topology of classical props, researchers also proposed to use reconfigurable systems, in which users assemble components to achieve different shapes (e.g., [START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Raffle | Topobo: a constructive assembly system with kinetic memory[END_REF]). Going a step further, Yoshizaki et al. [START_REF] Yoshizaki | An actuated physical puppet as an input device for controlling a digital manikin[END_REF] proposed an actuated interface to simulate specific poses automatically.

However, all these interfaces are have to be built from scratch. Other works then suggest tracking the movements of existing artist dolls using vision and markers to perform motion capture [START_REF] Feng | Motion capture data retrieval using an artist's doll[END_REF][START_REF] Hiroki | A puppet interface for the development of an intuitive computer animation system[END_REF]. Our prop-based controller draws on these works, consisting of an artist's doll tracked with accelerometers but provides built-in tracking that is not sensitive to occlusion that is likely to happen during the manipulation of the interface.

Influence on virtual experiences

The benefits of using a prop-based controller over arbitrary mapping controllers have been investigated in some studies [START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Johnson | Sympathetic interfaces: using a plush toy to direct synthetic characters[END_REF][START_REF] Yoshizaki | An actuated physical puppet as an input device for controlling a digital manikin[END_REF]. Jacobson et al. [START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF] compared mouse and prop-based controllers for pose capture. They asked participants to perform a 3D targeting task with both controllers, and results show that they moved fewer parts with the prop than with the mouse for the same result. Yoshizaki et al. [START_REF] Yoshizaki | An actuated physical puppet as an input device for controlling a digital manikin[END_REF] also compared prop and mouse interfaces in a 3D targeting task and found that participants were significantly faster with the prop than with the mouse and that the prop was easiest to learn, easiest to use, and least fatiguing. Prop-based controllers are then helpful for pose or motion capture.

In the context of real-time control of virtual characters, Johnson et al. [START_REF] Johnson | Sympathetic interfaces: using a plush toy to direct synthetic characters[END_REF] asked some users to test a chicken plush interface mapped with a virtual chicken on a screen and asked for feedback on the usability of the controller. Users gave positive feedback, even if they did not make a comparison with other input systems. This work suggests that prop-based controller dolls can be used with real-time simulations. In this paper, we are interested in the influence of prop-based controllers on the sense of presence and the sense of embodiment, which are important factors during virtual experiences. The prop has a kinematic similar to one of the avatars, which may increase the user experience. Active control is more positively associated with presence than arbitrary mapping control [START_REF] Seibert | Control mapping in virtual reality: effects on spatial presence and controller naturalness[END_REF][START_REF] Skalski | Mapping the road to fun: Natural video game controllers, presence, and game enjoyment[END_REF][START_REF] Slater | A virtual presence counter[END_REF][START_REF] Tamborini | The role of presence in the experience of electronic games. Playing video games: Motives, responses[END_REF][START_REF] Terziman | Shake-your-head: Revisiting walking-in-place for desktop virtual reality[END_REF][START_REF] Usoh | Walking¿ walking-in-place¿ flying, in virtual environments[END_REF]. Props also convey haptic feedback that is considered an important factor for the sense of presence [START_REF] Meehan | Physiological measures of presence in stressful virtual environments[END_REF][START_REF] Slater | Presence in immersive virtual environments[END_REF]. Getting haptic feedback from virtual objects also seems to positively impact the embodiment of users (e.g., [START_REF] Fröhner | Can wearable haptic devices foster the embodiment of virtual limbs[END_REF][START_REF] Richard | Studying the role of haptic feedback on virtual embodiment in a drawing task[END_REF]).

Controlling avatars with a prop-based controller could then enhance users' virtual experiences. As avatars are more likely to be present in virtual environments than other virtual objects, such a result would be interesting for future applications. But it has to be demonstrated, and this in different perspectives. In the case of an immersive first-person perspective, for instance, the avatar and prop have different scales, which can negatively affect virtual experiences.

PROP-BASED CONTROLLER TO CONTROL AVATARS

Hardware

The prop was designed to provide an accessible device that could be easily replicated. It consisted in a 20 cm tall commercial artist's doll that we instrumented (see Figure 2). Rotations of the right forearm and right leg were inferred using 2 IMU (Inertial Measurement Unit) stations (Bosh BNO055, 100Hz), providing absolute rotations along the three axes (360°). BNO055 sensors fused data from an accelerometer, compass, and gyroscope to accurately estimate the angle. We limited the measurements to the right arm and right leg since these body parts are involved in most actions right-handed people perform. The detection process is simple enough to extend to other prop parts if necessary (e.g., torso, head, etc.). Capacitive sensors were added on the right leg and arm to detect the user grip (copper tape and 1 MΩ resistor). The sensors communicate with the computer through 2 Arduino Micro boards at 38400 Bauds. The Arduino boards were placed in a base so that the interface was only connected to the computer by two USB wires.

The controller rested on a table thanks to a base made of 3D printing material and the doll was attached to it by a pillar (10 cm, Ø2mm ). The pillar was grounded such that no rotation was allowed between the doll and the base, however, the base was free to move on the table. Global motions (3 rotations and 3 translations of the overall system) were inferred using a Vive tracker attached to the base, detected by 2 HTC Vive 2.0 stations. Users moved the prop's body part with 2 hands to control local movements only. The interface (including the base and tracker) weighed 380 g and could be easily moved.

Mapping with avatar motions

The virtual environment was implemented with Unity 3D on a computer equipped with an Intel(R) (Xeon(R) CPU E5-1603 V4, (2.80GHz)) processor and an NVIDIA (GeForce GTX 1080) graphic card. The simulation ran at frequencies above 90 HZ. In the simulations, users controlled human-like avatars downloaded from Adobe Mixamo website 1 .

The avatar's movements could have been mapped directly to the movements of the prop's body parts. However, Johnson et al. [START_REF] Johnson | Sympathetic interfaces: using a plush toy to direct synthetic characters[END_REF] noticed that some users complained about this mapping because the virtual character's movements seemed unnatural. Glauser et al. [START_REF] Glauser | Rig animation with a tangible and modular input device[END_REF] presented a mapping method based on inverse kinematics methods to render more natural movements. Here we propose an alternative mapping, suggesting to drive animation clips of the avatars (prerecorded actions used in virtual environments to animate the avatar). The method is similar to usual controllers, for example, pressing a button triggers an animation. However, the prop-based controller allows continuous control of the movements, i.e at each frame the system maps the physical and virtual body motion to guarantee smooth movement. The speed can also be controlled intuitively compared to button inputs.

Hybrid Animation

The animation clips were managed with the Mecanim animation system of Unity. The time of the animation clip was modified based on the information about the rotations of the prop's body parts. The mapping was calculated to achieve maximum synchronization between the avatar's and the prop's movements. In the example in Figure 3, when the prop's leg rotation was maximum, the motion parameter was set to the animation time when the leg was higher and vice versa for the low position. We used a linear relationship to achieve continuous motion between these positions. The same method can be applied to various animations (punch, pick, etc.).

We noticed in a pilot experiment that users sometimes made additional movements that were not simulated in the animation. For example, some users tend to put the prop's leg back much higher than in the animation when kicking. We then implemented a hybrid mapping, extending the animation mapping with direct mapping to simulate these extra movements (Figure 3).

Avatar perspectives

The prop-based controller was implemented with 3 avatar perspectives (see Figure 4). In the third-person perspective ([3PP]), the virtual scene was displayed on a desktop screen in front of the user.

In the immersive third-person perspective ([3PPi]), the avatar is observed from a third-person perspective in an HMD. The avatar and the prop were co-located using an HMD (HTC Vive pro eyes2 ), with the prop and the avatar having the same size and location. The colocation gave users the impression that the avatar was truly tangible. The pose of the HMD was calibrated for each user using SteamVR software. The prop's pose was calibrated using a Leap Motion3 attached to the HMD and capable of detecting the user's right hand (see Figure 5). Users were asked to place their index finger on the prop's head, and the avatar's head was placed in the same location.

In the immersive first-person perspective ([1PPi]), users saw the virtual scene through the eyes of the avatar thanks to a HMD. Users then saw a full-size avatar while the prop was small, which may explain why it has never been paired. We hypothesized that users would be able to overcome this difference. The virtual camera was placed at virtual eye level, and the avatar's head orientation was updated to follow the user's head orientation.

Extra features were implemented to increase the connection between users, the prop, and the avatars. In 3PP, the avatar was trans- parent and became opaque after a "magic" outline effect appeared for 3s when users grabbed the prop, detected by the capacitive sensors. In 1PPi, the same features were implemented, except for the outline effect. The avatar was also placed in front of a virtual mirror at the beginning of the simulation to help with appropriation. During the simulations, the avatar's shadow was displayed in front of the avatar.

Leap motion

USER STUDY

We conducted a user study to evaluate the influence of the propbased controller on the users' virtual experience. We chose a kick in a ball play as an illustration. The ethics committee of Inria Rennes approved the protocol.

Overview and hypotheses

Seated participants controlled an avatar shooting balloons in the virtual environment (see Figure 6). They used either the prop-based controller or buttons to control the movements. Buttons were chosen because they are popular controllers with arbitrary mapping. Since the prop allows for movements related to those of the avatar and conveys physical interaction, we expected that the prop would increase the sense of being in the virtual scene, the relationship with the avatar, and the fun. Therefore, we made several hypotheses:

• (H1) The sense of presence is higher with the prop-based controller than with the button controller. • (H2) The sense of embodiment is higher with the prop-based controller than with the button controller. • (H3) The fun would be higher with the prop-based controller than with the button controller.

To test several configurations, participants performed the prop/buttons comparisons in the three perspectives: third-person [3PP], immersive third-person [3PPi], and immersive first-person perspectives [1PPi].

Details on the experimental set up

During the experiment, participants were represented by a male (visible in Figure 6) or female avatar (visible in Figure 5). They shot at 25 virtual balloons placed in front of them. To keep the task from being too repetitive, they had to kick slowly if the ball was yellow and faster if the ball was orange, the orange balloons going farther. Participants controlled the avatar's action by pulling with the prop's right leg. They did not control the global movement to assess the prop system's interest only. From a third-person perspective, we chose the buttons on a PS4 game controller as the input, as this is a popular input with the screen display. Participants pressed the cross button to control slow kicks and the circle button to control fast kicks. In both immersive third-person and immersive firstperson perspectives, the D-pad of the Vive controller was used as the button. Participants clicked in the center to control slow kicks and up to control fast kicks. In the immersive first-person perspective, they observed the avatar in a virtual mirror for one minute before beginning.

Measures and Participants

Presence, embodiment, and fun were measured using questionnaires. Presence was self-reported by the IPQ questionnaire (final scores from 0 to 100) [START_REF] Schubert | The experience of presence: Factor analytic insights[END_REF]. The embodiment was self-assessed by the Peck et al. questionnaire (final scores from 0 to 6) [START_REF] Peck | Avatar embodiment. a standardized questionnaire[END_REF]. These two questionnaires refer to the users' experience in the virtual environment (e.g., "I felt that I could control the virtual body as if it was my own body") to ensure that they assess their presence and embodiment, not the one of the prop. The fun was measured on a 5-point Likert scale, with participants answering the question: "I had fun while using the interface.". All of these data were measured after each controller/perspective configuration.

A sample of 36 volunteers (mean age = 24.65 years; SD = 3.13 years; 25 men) participated in the experiment. They were students from our research center at Inria-Rennes. In total, only two participants reported being left-handed. They were all naive about the purpose of the experiment and had normal or correct to-normal vision. They were not compensated for their participation.

The experiment had one between factor (Perspectives) that had three levels ([3PP],[3PPi], and [1PPi]) and one within factor (Controllers) with two levels (Prop and Button). The controllers' presentations were counter-balanced across participants. After each simulation, participants filled in questionnaires. Overall, the experiment lasted 25 minutes.

Results

The results of the user study are shown in Figure 7. The assumption of normality and homogeneity of variance were tested with Shapiro-Wilk and Bartlett tests. If these hypotheses were satisfied, Student's t-tests for paired samples were performed, otherwise Wilcoxon signed-rank tests were performed. In both cases, Holm-Bonferroni correction was applied. For the presence scores, the normality assumption was met in all perspectives. The participants' sense of presence was significantly higher (+ 10%) with the prop (m = 46.3, std = 14.7) than with the button interface in 3PP (m = 36.9, std = 15.2, p = 0.002, d = 0.6). It was significantly higher (+ 13%) with prop (m = 65.1, std = 10.7) than with the button interface in 3PPi (m = 52.5, std = 13.3, p = 0.001, d = 0.92). It was also significantly higher (+11 %) with the prop (m = 73.1, std = 10.3) than with the button interface in 1PPi (m = 65.2, std = 10.8, p = 0.028, d = 0.71).

**

For the embodiment scores, the normality assumption was met in 1PPi but not in 3PP and 3PPi. The embodiment was significantly higher (+ 7%) with the prop (m = 1.9, std = 0.6) than with the button interface in 3PP (m = 1.4, std = 0.9, p = 0.003, r = 0.8). It was significantly higher (+ 13%) with the prop (m = 2.6, std = 0.6) than with the button interface in 3PPi (m = 1.7, std = 1.3, p = 0.001, r = 0.78). It was also significantly higher (+11 %) with the prop (m = 3.6, std = 1.4) than with the button interface in 1PPi (m = 2.8, std = 1.4, p = 0.003, d = 0.5).

The normality assumption was not met in all the perspectives for fun. Participants had significantly more fun (+ 48%) with the prop (m = 4.8, std = 0.9) than with the button interface in 3PP (m = 2.4, std = 1.2, p = 0.006, r = 0.86). It was also significantly higher (+ 45%) with the prop (m = 4.75, std = 0.5) than with the button interface in 3PPi (m = 2.5, std = 1.0, p = 0.002, r = 0.89). Finally, It was significantly higher (+28%) with the prop (m = 4.7, std = 0.4 ) than with the button interface in 1PPi (m = 3.3, std = 1.1, p = 0.007, r = 0.77).

DISCUSSION

Influence of the prop on users' experiences

Previous studies demonstrated that users' performances were increased when controlling the pose of a virtual character with propbased controllers when compared to arbitrary controllers [START_REF] Jacobson | Tangible and modular input device for character articulation[END_REF][START_REF] Yoshizaki | An actuated physical puppet as an input device for controlling a digital manikin[END_REF]. Our study extends these results, showing that prop-based controllers can increase the users' virtual experiences.

The study first aimed to check if the sense of presence of seated users could be improved by using the prop-based controller. The results show that the prop-based controller induces a higher sense of presence than arbitrary mapping controllers, supporting (H1). The effect was comparable across the perspectives. Overall, the presence was enhanced by 10% on average, a significant amount considering that participants remained seated. These results are consistent with previous work that demonstrate a higher sense of presence when introducing a physical object in virtual environments [START_REF] Meehan | Physiological measures of presence in stressful virtual environments[END_REF][START_REF] Slater | Presence in immersive virtual environments[END_REF]. We show here that a higher sense of presence can be induced by a tangible object representing the avatar that is the entity the more likely to be present in virtual environments. Previous work also suggests that active control is more positively associated with presence than arbitrary mapping control [START_REF] Seibert | Control mapping in virtual reality: effects on spatial presence and controller naturalness[END_REF][START_REF] Skalski | Mapping the road to fun: Natural video game controllers, presence, and game enjoyment[END_REF][START_REF] Slater | A virtual presence counter[END_REF][START_REF] Tamborini | The role of presence in the experience of electronic games. Playing video games: Motives, responses[END_REF][START_REF] Terziman | Shake-your-head: Revisiting walking-in-place for desktop virtual reality[END_REF][START_REF] Usoh | Walking¿ walking-in-place¿ flying, in virtual environments[END_REF]. The prop-based controller does not strictly induces active control, but our results suggest that it still induces a better sense of presence than arbitrary mapping controllers, supporting the use of controllers based on miniaturized motions (e.g., hand gestures [START_REF] Kim | Finger walking in place (fwip): A traveling technique in virtual environments[END_REF], smartphone-based motion control [START_REF] Ye | Aranimator: in-situ character animation in mobile ar with user-defined motion gestures[END_REF]).

Our results also indicate that the sense of embodiment increases with the prop-based controller, supporting (H2). The effects are stronger in immersive perspectives [3PPi] and [1PPi], with an increase of more than 0.7 points, which is significant considering that participants remained seated. While a higher sense of embodiment was demonstrated with haptic feedback displayed on specific objects (e.g [START_REF] Fröhner | Can wearable haptic devices foster the embodiment of virtual limbs[END_REF][START_REF] Richard | Studying the role of haptic feedback on virtual embodiment in a drawing task[END_REF]), we show that the haptic feedback can be displayed on the avatar itself. Besides haptic feedback, the kinematics similar to the avatar could have helped the process, but further research should be led to investigate its influence. The embodiment enhancement is particularly interesting in the case of the immersive first-person perspective [1PPi] because this configuration is widely used to maximize the embodiment. Users also seem able to link the miniature prop-based controller with a life-scale avatar, suggesting that scale changes are supported between real and virtual environments. The sense of embodiment is higher in the first-person perspective [1PPi] than in [3PP] and [3PPi] perspectives for the button controller, con-sistent with previous work that demonstrated a greater embodiment in this perspective [START_REF] Fribourg | Avatar and sense of embodiment: Studying the relative preference between appearance, control and point of view[END_REF][START_REF] Gorisse | First-and third-person perspectives in immersive virtual environments: presence and performance analysis of embodied users[END_REF].

The perceived fun was also investigated. The results show that it was significantly higher in all perspectives, supporting (H3). It suggests that the prop-based controller could be used to motivate users to use virtual environments, which could be interesting for some applications (e.g., video games, rehabilitation, etc).

Future Directions of Research and Use cases

This study shows that a prop-based controller can increase users' virtual experiences. As it limits fatigue while keeping interesting virtual experiences, it could interest all seated users. It could also benefit users with disabilities and older people, so it would be interesting to replicate our study with these populations.

Seated users used the interface but could be used by standing users by attaching it with a harness [START_REF] Numaguchi | A puppet interface for retrieval of motion capture data[END_REF]. Natural mapping controllers are more likely to be used in this situation. Still, we think the propbased controller could be a good way to limit user fatigue while preserving the user experience. We may make comparisons with natural mapping controllers in future work.

Extending the results to other actions could also be interesting in the future. Many local actions can be explored with the prop's current design, and the IMU concept can also be extended to other body parts to perform even more actions. Actions related to global movements like jumping or flying are also possible, as they can be detected with the Vive Tracker.

Some use cases are presented in Figure 8. Our first example involves locomotion with a virtual visit to a museum. Users moved the prop's leg to control forward motion. Moving one leg with each hand makes it difficult to control the rotation, and since the leg movements are antagonistic, controlling one leg is enough. The rotation of the base was used to manage the rotation of the avatar directly. The second and third examples show a rugby game, with users kicking a ball between rugby posts, and a sword game, where fruits were thrown at a ninja who had to cut them with a sword. These scenarios illustrate the value of controlling a continuous action. The impact forces can be computed from the prop's leg velocity in the rugby game. In the sword game, the prop allows good timing to interact with the dynamic objects.

CONCLUSION

In this paper, we explored the use of a prop-based controller for realtime control of human-like avatars. We proposed a prop-based controller based on an artist's doll and accelerometers. We implemented it third-person perspective on a screen, immersive third-person perspective, and immersive first-person perspective in a head-mounted display. Compared to arbitrary mapping controllers, we show that the prop increases the sense of presence and fun in all perspectives, and the sense of embodiment in immersive perspectives. It could then be used to improve the virtual experiences of seated users in many applications.

Figure 1 :

 1 Figure 1:We propose to use a tangible prop to control a humanlike avatar during seated virtual experiences. By manipulating the prop, users perform kinematics similar to those of the avatar and experience physical interactions with it. We show that such controllers can enhance the sense of presence and embodiment compared to traditional gamepad controllers.

Figure 2 :

 2 Figure 2: Overview of the prop-based controller including IMU stations, capacitive sensors and a Vive tracker.

Figure 3 :

 3 Figure 3: With the animation mapping alone, the avatar does not kick backward like the prop's leg because it is not part of the animation. With hybrid mapping, the avatar follows the prop's movement.

Figure 4 :

 4 Figure 4: The different perspectives : (Top) Desktop third-person perspective [3PP]. (Bottom Left) Immersive third-person perspective [3PPi]. (Bottom Right) Immersive first-person perspective [1PPi].

Figure 5 :

 5 Figure 5: Calibration to co-locate the prop and the avatar.

Figure 6 :

 6 Figure 6: (Left) Screenshots of the avatar shooting in balloons in the user study. (Middle) The scene from the third-person perspective and (Right) from the immersive first-person perspective [1PPi].

Figure 7 :

 7 Figure 7: Presence, embodiment and fun scores in the different perspectives. Higher is better for all these measures.

https://www.mixamo.com

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