The usual framework of probability discounting is, more or less explicitly, this of a representation of decision under risk where the set Λ of probability distributions is typically defined over {0, x}, x being the possible gain, of probability p, of a 2-issues lottery L belonging to Λ. After Rachlin et al. (1991) , a common feature of the probability discounting contributions is that L is related to the valuation of a decision through a waiting time l, which can be interpreted as “odds against” in repeated gambles⁵. Though rather intuitive, this interpretation could be given a firmer basis than the usual one, which draws on the comparison with a gambler betting on a horse race, in terms of repeated Bernoulli trials. It is well known that the expected value of the random variable representing the number of trials before winning (the winning trial included) is 1/p. If we take the interval between two trials as the unit of time, the expected value of the physical delay l before the winning trial is therefore given by:

$l=\frac{1}{p}-1=\frac{1-p}{p}$ (1)

Such representation of the link between probability and physical delay has been currently admitted, at least since Rachlin et al. (1991) , as the initial moment of a procedure leading to transform probabilities. Insofar as we remain in the framework of 2-issues lotteries, and as the counterpart of the transformation of the probability p of success is a parallel and consistent transformation of the probability $1-p$ of failure, the immediate link in (1) between probability and delay is not contentious. But regarding the more general case of n-issues lotteries is less simple. Assume these lotteries L are the laws of probability of discrete random variables X with finite support:

$L=\left(x_1,\cdots ,x_i,\cdots ,x_n;p_1,\cdots ,p_i,\cdots ,p_n\right)$ (2)

in which the outcomes $x_i$ are ranked in increasing order, $x_1<\cdots <x_i<\cdots <x_n$, and ${\displaystyle {\sum }_{i=1}^n{p}_i}=1$.

Let G be the decumulative distribution function of the random variable X whose probability law is given by the lottery L: $G\left(x_i\right)=\text{Pr}\left(X\ge x_i\right)$. It is obvious that $G\left(x_1\right)=1$ and $G\left(x_n\right)=p_n$. Consider now not the isolated probability $p_i$ of obtaining $x_i$, but the probability of obtaining at least $x_i$, that is, $G\left(x_i\right)$. We can derive from $G\left(x_i\right)$ a Bernoulli trial whose issues are either sucess, with an outcome between $x_i$ and $x_n$ both included, or failure, with an outcome between $x_1$ and $x_{i-1}$ also included. $G\left(x_i\right)$ is therefore the probability of success, and $F\left(x_i\right)=1-G\left(x_i\right)$ the probability of failure ( $F\left(x_i\right)$ standing for the usual cumulative distribution function). The expected number of Bernoulli trials to obtain one success (that is, getting at least $x_i$) is $1/G\left(x_i\right)$. And going on transposing probability into a physical delay before winning in a repeated gamble like in (1), the average delay $l_i$ for success, that is for obtaining at least $x_i$ is given by:

$l_i=\frac{1-G\left(x_i\right)}{G\left(x_i\right)}\left(i=1,\cdots ,n\right)$ (3)

As far as p is an objective probability, l can be viewed as “physical waiting time” ( Takahashi et al., 2012 : p. 13.). Drawing on the reintroduction of Fechnerian-like perspectives in psychophysics ( Dehaene, 2003 ), Takahashi and his co-authors assume that in a 2-issues lottery, the subjectively perceived waiting time τ is a logarithmic function of the physical waiting time

$\tau =a\ln \left(1+bl\right)$ (4)

with $a,b>0$⁶. The same principles hold in the general case of a n-issues lottery: the subjectively perceived waiting time ${\tau }_i$ before winning at least $x_i$ is logarithmically related to the physical waiting time:

${\tau }_i=a\ln \left(1+b{l}_i\right)\left(i=1,\cdots ,n\right)$ (5)

Replacing the physical delay by decumulated probability (i.e., the probability of winning at least a certain outcome) like in (3), the probability of winning at least $x_i$ is related as follows to the perceived delay before winning at least $x_i$:

${\tau }_i=a\ln \left(1+b\frac{1-G\left(x_i\right)}{G\left(x_i\right)}\right)\left(i=1,\cdots ,n\right)$ (6)

The third step provides a separate treatment of time discounting. In the case of a 2-issues lottery explored by standard literature on probability discounting, things are rather simple. The basic idea is this of an exponential discounting whose argument is the perceived delay τ, instead of the physical delay l:

$\mu =\text{exp}\left(-r\tau \right)$ (7)

where μ and r stand respectively for the discounting factor and the discount rate⁷ for an outcome x, whose expected perceived delay before winning it, is τ. From (4) and (7) we therefore have:

$\mu ={\left(1+bl\right)}^{-ra}$ (8)

which amounts to a generalized hyperbolic discounting factor⁸. Exponential discounting, relatively to perceived time, has therefore generated hyperbolic discounting, relatively to physical time.

However, such determination of the discounting factor would be seriously flawed if extended as such to n-issues lotteries: if, drawing on (5), $\text{exp}\left(-r{\tau }_i\right)={\left(1+b{l}_i\right)}^{-ra}$ can rightly be viewed as a discounting factor, it depends on the expected time (perceived or physical) before winning at least $x_i$—not before winning exactly $x_i$. The discounting factor associated to the outcome $x_i$ is therefore the difference between two discounting factors: the one related to the expected time before winning at least $x_i$ and the one related to the expected time before obtaining strictly more than $x_i$, that is at least $x_{i+1}$. So that, assuming that $l_{n+1}\to +\infty$:

${\mu }_i={\left(1+b{l}_i\right)}^{-ra}-{\left(1+b{l}_{i+1}\right)}^{-ra}\left(i=1,\cdots ,n\right)$ (9)

After the work of ( Cajueiro, 2006 ), the expression of the discounting factor has been currently rewritten, through a change in the parameters, as a q-exponential discounting based on Tsallis’ statistics. This change leads to set a pair of alternative parameters, k and q defined as $k=rab$ and $q=1-1/ra$. Extending this redefinition to the expression of ${\mu }_i$, (9) can be rewritten as⁹:

${\mu }_i=\psi \left(l_i\right)-\psi \left(l_{i+1}\right)$ (10)

where $\psi \left(l_i\right)={\left(1+k\left(1-q\right)l_i\right)}^{-\frac{1}{1-q}}\left(i=1,\cdots ,n\right)$.

Or, using Cajueiro’s notation for q-exponential discounting:

${\mu }_i={\text{exp}}_q\left(k{l}_i\right)-{\text{exp}}_q\left(k{l}_{i+1}\right)\left(i=1,\cdots ,n\right)$ (11)

The discounting factor ${\mu }_i$ can therefore be equivalently expressed as the difference between the values $\psi \left(l_i\right)$ and $\psi \left(l_{i+1}\right)$ of two generalized hyperbolic discountings (10) or, equivalently, between two q-exponential discountings (11). Cajueiro’s presentation introducing in 2006 q-exponential discounting can be found in the literature as early as the following year¹⁰. It will be considered that, because of the definition of a, b and r in (4) and (5), the parameters k and q are, by construction, such that $k\ge 0$ and $-\infty <q<1$. The possibility that q is negative does not appear in the article by Cajueiro (2006) , nor in that of Takahashi (2007b) . However, when he resumes q-discounting during the same year or the following year but in an intertemporal choice framework, Takahashi (2007a , 2008) explicitly considers the possibility that q is less than 0¹¹. The interpretation of the parameters k and q will be discussed in Section 3.

The recourse to an explicit representation of the desirability of the reward is lacking in the works on probability discounting cited above. The emphasis placed on the transposition of probabilities into delays, as well as the binary structure of lotteries, justified a minimum treatment allowing to ignore it. It was sufficient to work with a simple function $V\left(x,t\right)$ whose two arguments, the outcome x and the delay t before winning had each one only two possible values: $x=0$ in case of failure or $x=\bar{x}$ in case of success; $t=0$ for an immediate (because certain) gain, $t=l$ for a delayed reward (because its probability p is such that $l=\left(1-p\right)/p$). Assuming that $V\left(0,t\right)=0$, the immediate or certain value of the reward $\bar{x}$ writes $V\left(\bar{x},0\right)$, and its delayed or with probability p, value is $V\left(\bar{x},l\right)$. This is enough to get

$V\left(x,l\right)=\mu V\left(x,0\right)$ (12)

which is all we need to focus on the specification and the discussion of the discounting factor μ. But such simplicity must be abandoned when moving on to the more general case of n-issues lotteries which require comparisons between the desirability of the various possible outcomes when they are immediate or certain. This desirability can be represented by an increasing utility function u of x, calibrated so that $u\left(0\right)=0$, and defined up to a positive linear transformation. So that the utility of a lottery $U\left(L\right)$ can be given, like for utility in time, as the sum of the undiscounted utilities of each possible outcome $u\left(x_i\right)$ weighted by its discounting factor ${\mu }_i$ defined as in (10):

$U\left(L\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\mu }_{i}u\left(x_i\right)$ (13)

Now, because of the probability discounting perspective, ${\mu }_i$ in (13) can be understood either as a discounting factor whose expression is given by ${\mu }_i=\psi \left(l_i\right)-\psi \left(l_{i+1}\right)$ in (10), or as probability decision weights. Relying on (3) and (10) we get an alternative expression of ${\mu }_i$, as the decision weight for obtaining an outcome $x_i$. ${\mu }_i$ is the difference between the transformed probability $G\left(x_i\right)$ of winning at least $x_i$ and the transformed probability of winning strictly more than $x_i$, $G\left(x_{i+1}\right)$:¹²

${\mu }_i=\phi \left(G\left(x_i\right)\right)-\phi \left(G\left(x_{i+1}\right)\right)$ (14)

where $\phi \left(G\left(x_i\right)\right)={\left(1+k\left(1-q\right)\frac{1-G\left(x_i\right)}{G\left(x_i\right)}\right)}^{-\frac{1}{1-q}}\left(i=1,\cdots ,n\right)$.

It can be shown that the probability weighting function φ is an increasing transformation of $\left[0,1\right]$ into itself with the following properties:

$\begin{array}{l}\phi \left(0\right)=0\\ \phi \left(1\right)=1\\ {\phi }^{\prime }>0\end{array}$ (15)

As a result, what was first perceived as discounting factors, the ${\mu }_i$’s, now appear as transposed probabilities whose sum is obviously equal to 1.

The combination of a utility function u with decision weights ${\mu }_i$ (such that ${\displaystyle {\sum }_{i=1}^n{\mu }_i}=1$) determined by a probability weighting function φ, given by (13) and (14), amounts to what is currently known as “rank-dependent utility”¹³:

$\begin{array}{l}U\left(L\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\mu }_{i}u\left(x_i\right)\\ =\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left(\phi \left(G\left(x_i\right)\right)-\phi \left(G\left(x_{i+1}\right)\right)\right)u\left(x_i\right)\\ =\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left({\left(1+k\left(1-q\right)\frac{1-G\left(x_i\right)}{G\left(x_i\right)}\right)}^{-\frac{1}{1-q}}-{\left(1+k\left(1-q\right)\frac{1-G\left(x_{i+1}\right)}{G\left(x_{i+1}\right)}\right)}^{-\frac{1}{1-q}}\right)u\left(x_i\right)\\ =\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left({\exp }_q\left(k\frac{1-G\left(x_i\right)}{G\left(x_i\right)}\right)-{\exp }_q\left(k\frac{1-G\left(x_{i+1}\right)}{G\left(x_{i+1}\right)}\right)\right)u\left(x_i\right)\end{array}$ (16)

It is well-known that when rank-dependent utility prevails, the acknowledged drawbacks of a direct transformation of each single probability, like this of the probability of success in a 2-issues lottery, (the sum of the decision weights might be different from zero and violation of first degree stochastic dominance might occur) do not hold anymore (see, for instance, ( Abdellaoui, 2009 )). The probability weighting function φ possesses the expected properties (see (15)) of decision weights in rank-dependent utility, but its shape is more specific, since it is generated by the whole process of probability discounting¹⁴. Some consequences of the properties of the probability weighting function are discussed in the following section.

A common way to approach time-rationality and risk-rationality is to agree that they rest, respectively, on the fulfillment of axiomatic properties regarding the underlying preferences: stationarity for decision in time, and independence for decision in risk¹⁵. Stationarity and independence enter crucially in the axiomatic basis which make, respectively, preferences in time represented by discounted (exponential) utility, and preferences over random variables (lotteries) represented by expected utility. Both are, in their respective domain, a condition for avoiding preference reversal: stationarity guarantees time-consistency, i.e. the constancy of preferences between two gains at different dates, whether close or remote, provided they are separated by the same interval of time; independence preserves our order of preference between two lotteries, whatever the proportions in which they are combined with a third lottery.

Since the decision weights ${\mu }_i$ can be viewed equivalently as discounting weights ( ${\mu }_i=\psi \left(l_i\right)-\psi \left(l_{i+1}\right)$; see (10) or the reformulation of n. 13 supra) or as probability weights ( ${\mu }_i=\phi \left(G\left(x_i\right)\right)-\phi \left(G\left(x_{i+1}\right)\right)$; see (14)), a peculiarity of probability discounting is that the issue of rationality is raised simultaneously in relation to time and in relation to risk.

Now, on the one hand, time-rationality is obtained only when q is tending to 1, which yields exponential discounting (and therefore, stationarity and time-consistency) because the ratio between ψ in (10) and its first derivative is a constant equal to −k, so that ${\mu }_i=\text{exp}\left(-k{l}_i\right)-\text{exp}\left(-k{l}_{i+1}\right)$. On the other hand, risk-rationality is a special case of simple hyperbolic discounting like in Herrnstein (1981) or Mazur (1987) , obtained with $q=0$ in ψ (10). In this case, ${\mu }_i={\left(1+k{l}_i\right)}^{-1}-{\left(1+k{l}_{i+1}\right)}^{-1}$: it occurs with the additional condition that $k=1$, which makes that φ in (14) is such that $\phi \left(G\left(x_i\right)\right)=G\left(x_i\right)$, whatever $x_i$. As a result, when $q=0$ and $k=1$, ${\mu }_i=p_i$, so that probability discounting has generated expected utility (and hence, independence).

This sheds light on the relationship between time-rationality and risk-rationality generated by the transposition of a decision in risk into a decision in time. When moving from the first to the second, we loose time-rationality if the parameters are such that they preserve risk-rationality. Conversely, if we reach time-rationality, we have to give up risk-rationality. Such a conclusion might seem disturbing, but it should not be overestimated. The simple fact that ${\mu }_i$ can be understood at the same time as a discount factor and as a probability weight, referring respectively to a specific case of generalized hyperbolic discounting (10) and of a probability weighting function in rank-dependent utility (14), means that probability discounting should satisfy the criteria of rationality, obviously weaker, which characterize each of these two approaches: the Thomsen condition of separability ( Fishburn & Rubinstein, 1982 : pp. 686-687) for time-rationality¹⁶, and comonotonic tradeoff consistency ( Wakker, 1994 : p. 13) for risk-rationality¹⁷. Taking seriously the idea on which probability discounting is based, namely that deciding in risk might be viewed as a way of deciding in time, entails that something has to be abandoned in our requirements in terms of rationality: either one of the two types of rationality (in time or in risk), when the parameters k and q are given appropriate values or, in the general case, the strong versions of risk-rationality and time-rationality, in favour of the weaker versions consistent with rank-dependent utility and generalized hyperbolic discounting.

Let us start with the properties of the probability weighting function φ defined as in (14). We know that this function is increasing, since its first derivative is positive on $\left[0,1\right]$:

${\phi }^{\prime }\left(p\right)=k{p}^{-2}{\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-\frac{2-q}{1-q}}>0$ (17)

Its second derivative is

${\phi }^{″}\left(p\right)=-k{p}^{-4}{\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-\frac{2-q}{1-q}}\left(2p-k\left(2-q\right){\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-1}\right)$ (18)

The part played by the parameters k and q is crucial. According to their values, ${\phi }^{″}$ is either positive, or negative, or of alternate signs, so that φ is either convex, or concave, or inverse S-shaped (firstly concave, then convex), or S-shaped (firstly convex then concave).

${\phi }^{″}$ can be rewritten: ${\phi }^{″}\left(p\right)=A\left(p\right)\times B\left(p\right)$

where $A\left(p\right)=-k{p}^{-4}{\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-\frac{2-q}{1-q}}$ and $B\left(p\right)=\left(2p-k\left(2-q\right){\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-1}\right)$

$A\left(p\right)$ is always negative. Hence, the sign of ${\phi }^{″}\left(p\right)$ depends on the sign of $B\left(p\right)$, which writes also:

$B\left(p\right)=\frac{2p\left(1-k\left(1-q\right)\right)-kq}{p+k\left(1-q\right)\left(1-p\right)}$ (19)

Let us analyse the sign of $B\left(p\right)$ with respect to the values of q.

a) If $q=0$ then replacing q by 0 in $B\left(p\right)$ leads to $B\left(p\right)=\frac{2p\left(1-k\right)}{p+k\left(1-p\right)}$.

Since the denominator is always positive, then the sign of $B\left(p\right)$ depends on the sign of its numerator. As a consequence, $B\left(p\right)$ is positive if and only if $k\le 1$. Hence when $q=0$, φ is concave if and only if $k\le 1$, and φ is convex otherwise.

b) If $q\in \left]0,1\right[$ then according to equation (19), two cases can occur: the case where $k<\frac{1}{1-q}$ and the case where $k\ge \frac{1}{1-q}$.

• If $k\ge \frac{1}{1-q}$ then $B\left(p\right)$ is negative whatever $p\in \left[0,1\right]$. This leads to ${\phi }^{″}\left(p\right)\ge 0$ ( $\phi \left(p\right)$ convex) on the interval $\left[0,1\right]$.

• If $k<\frac{1}{1-q}$ then $B\left(p\right)$ is negative (see the numerator of $B\left(p\right)$) on the interval $\left[0,p_0\right]$ and is positive on the interval $\left[p_0,+\infty \right]$, where $p_0=\frac{kq}{2\left(1-k\left(1-q\right)\right)}$. However p (a probability) cannot go beyond 1. This means that $p_0$ is either less than 1 or higher than 1. $p_0$ is less than 1 if and only if $k<\frac{1}{1-\frac{q}{2}}$. Hence:

- when $q\in \left]0,1\right[$, if $k<\frac{1}{1-q}$ and $k<\frac{1}{1-\frac{q}{2}}$ then $B\left(p\right)$ is negative on the

interval $\left[0,p_0\right]$ and positive on the interval $\left[p_0,1\right]$; that is, ${\phi }^{″}\left(p\right)\ge 0$ ( $\phi \left(p\right)$ convex) on the interval $\left[0,p_0\right]$ and ${\phi }^{″}\left(p\right)\le 0$ ( $\phi \left(p\right)$ concave) on the interval $\left[p_0,1\right]$;

- when $q\in \left]0,1\right[$, if $k<\frac{1}{1-q}$ and $k\ge \frac{1}{1-\frac{q}{2}}$ then $B\left(p\right)$ is negative on the

interval $\left[0,1\right]$; that is, ${\phi }^{″}\left(p\right)\ge 0$ ( $\phi \left(p\right)$ convex) on the interval $\left[0,1\right]$.

c. If $q<0$ then according to equation (19), two cases can occur: the case where $k\le \frac{1}{1-q}$ and the case where $k>\frac{1}{1-q}$.

• If $k\ge \frac{1}{1-q}$ then $B\left(p\right)$ is positive whatever $p\in \left[0,1\right]$. This leads to ${\phi }^{″}\left(p\right)\le 0$ ( $\phi \left(p\right)$ concave) on the interval $\left[0,1\right]$.

• If $k>\frac{1}{1-q}$ then $B\left(p\right)$ is positive (see the numerator of $B\left(p\right)$) on the interval $\left[0,p_0\right]$ and is negative on the interval $\left[p_0,+\infty \right]$, where $p_0=\frac{kq}{2\left(1-k\left(1-q\right)\right)}$. However p (a probability) cannot go beyond 1. This means that $p_0$ is either less than 1 or higher than 1. $p_0$ is less than 1 if and only if $k>\frac{1}{1-\frac{q}{2}}$. Hence:

- when $q<0$, if $k>\frac{1}{1-q}$ and $k>\frac{1}{1-\frac{q}{2}}$ then $B\left(p\right)$ is positive on the

interval $\left[0,p_0\right]$ and negative on the interval $\left[p_0,1\right]$; that is, ${\phi }^{″}\left(p\right)\le 0$ ( $\phi \left(p\right)$ concave) on the interval $\left[0,p_0\right]$ and ${\phi }^{″}\left(p\right)\ge 0$ ( $\phi \left(p\right)$ convex) on the interval $\left[p_0,1\right]$;

- when $q<0$, if $k>\frac{1}{1-q}$ and $k\le \frac{1}{1-\frac{q}{2}}$ then $B\left(p\right)$ is positive on the interval $\left[0,1\right]$; that is, ${\phi }^{″}\left(p\right)\le 0$ ( $\phi \left(p\right)$ concave) on the interval $\left[0,1\right]$.

What are the implications of the above results on the shape and properties of the graph of φ?

Since $\phi \left(0\right)=0$ and $\phi \left(1\right)=1$, then it is obvious that $\phi \left(p\right)\le p,\forall p$ (respectively: $\phi \left(p\right)\ge p,\forall p$) when φ is (fully) convex (resp., concave) on the interval $\left[0,1\right]$.

However when φ is firstly convex then concave (S-shaped; case with $0<q<1$ and $k<\frac{1}{1-\frac{q}{2}}$) or when it is firstly concave then convex (inverse S-shaped; case with $q<0$ and $k\ge \frac{1}{1-\frac{q}{2}}$), it is not straightforward to conclude whether its

graph crosses the first bisector, or whether it does not cross it, because it lies entirely above or under this bisector. The difference between the two situations (φ crossing the bisector, or not crossing it) amounts to the existence (in the first situation) or to the non-existence (in the second) of $p^{\text{*}}\in \left]0,1\right[$ such that

$\phi \left(p^{\text{*}}\right)=p^{\text{*}}$ (20)

Remind (see (14)) that $\phi \left(p\right)={\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-\frac{1}{1-q}}$. Hence equation (20) writes:

${\left(1+k\left(1-q\right)\frac{1-p}{p}\right)}^{-\frac{1}{1-q}}=p$ (21)

Since $\phi \left(p\right)$ is a positive and monotonic function on its domain of definition, (21) writes: $1+k\left(1-q\right)\frac{1-p}{p}=p^{\left(q-1\right)}$. That is,

$\frac{\left(1-k\left(1-q\right)\right)p+k\left(1-q\right)-p^q}{p}=0$

As a consequence, we want to know if the below equation (22) admits a root belonging to the interval $\left]0,1\right[$, in which case it crosses the first bisector (otherwise, it does not):

$-p^q+\left(1-k\left(1-q\right)\right)p+k\left(1-q\right)=0$ (22)

Denote $\eta \left(p\right)=-p^q+\left(1-k\left(1-q\right)\right)p+k\left(1-q\right)$.

• Let us take the case of φ S-shaped, with $0<q<1$ and $k<\frac{1}{1-\frac{q}{2}}$. We can see that $\eta \left(0\right)=k\left(1-q\right)>0$, $\eta \left(1\right)=0$, and ${\eta }^{\prime }\left(p\right)=-q{(p)}^{q-1}+\left(1-k\left(1-q\right)\right)$.

So that ${\eta }^{\prime }\left(p\right)\ge 0$ if and only if $p\ge {\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}$.

- If $k<1$ then ${\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}<1$. As a consequence $\eta$ will decrease on the interval $\left[0,{\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}\right]$ and will increase on the interval $\left[{\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}},1\right]$. However since $\eta \left(0\right)>0$, and $\eta \left(1\right)=0$ then it is necessarily the case that there exist $p^{\text{*}}<{\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}$ such that $\eta \left(p^{\text{*}}\right)=0$. This proves that when $0<q<1$, $k<\frac{1}{1-\frac{q}{2}}$ and $k<1$, there exists $p^{\text{*}}$ such that $\phi \left(p^{\text{*}}\right)=p^{\text{*}}$. This means that φ is S-shaped and crosses the bisector at $p^{\text{*}}<{\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}$.

- If $k\ge 1$ then ${\left(\frac{q}{1-k\left(1-q\right)}\right)}^{\frac{1}{1-q}}\ge 1$, and $\eta$ decreases on the interval $\left[0,1\right]$. This means that when $0<q<1$, $k<\frac{1}{1-\frac{q}{2}}$ and $k\ge 1$, then φ is S-shaped and fully under the bisector.

• Likewise if we take the case of φ inverse S-shaped, with $q<0$ and $k\ge \frac{1}{1-\frac{q}{2}}$.

- if $k>1$, there exists $p^{\text{*}}$ such that $\phi \left(p^{\text{*}}\right)=p^{\text{*}}$ - i.e. φ crosses the bisector;

- if $k\le 1$, φ is fully above the bisector.

We can therefore write, in the following propositions, the first line after the headers, from which the rest of the tables proceeds (see comments below, in Section 3.2.3).

Proposition 1 The table below indicates the link between attitude toward probabilities, attitude toward outcomes and attitude toward risk when the discounting parameter q lies on the interval $\left]0,1\right[$ ( Table 1 ).

Remarks: • $G_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(x\right)}{u^{\prime }\left(y\right)}$; • $g\ge 1$ is such that $u^{\prime }\left(x\right)\le g\frac{u\left(x\right)-u\left(y\right)}{x-y}$, for $x>y$, and $\phi \left(p\right)\le p^g$.

Proposition 2 The table below indicates the link between attitude toward probabilities, attitude toward outcomes and attitude toward risk when the discounting parameter $q=0$ ( Table 2 ).

Remarks: • $T_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(y\right)}{u^{\prime }\left(x\right)}$; • $h\ge 1$ is such that $u^{\prime }\left(y\right)\le h\frac{u\left(x\right)-u\left(y\right)}{x-y}$, for $x>y$, and $\phi \left(p\right)\le 1-{\left(1-p\right)}^h$; • $G_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(x\right)}{u^{\prime }\left(y\right)}$; • $g\ge 1$ is such that $u^{\prime }\left(x\right)\le g\frac{u\left(x\right)-u\left(y\right)}{x-y}$, for $x>y$, and $\phi \left(p\right)\le p^g$.

Proposition 3 The table below indicates the link between attitude toward probabilities, attitude toward outcomes and attitude toward risk when the discounting parameter q is strictly negative ( Table 3 ).

Remarks: • $T_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(y\right)}{u^{\prime }\left(x\right)}$; • $h\ge 1$ is such that $u^{\prime }\left(y\right)\le h\frac{u\left(x\right)-u\left(y\right)}{x-y}$, for $x>y$, and $\phi \left(p\right)\le 1-{\left(1-p\right)}^h$.

a) On the attitudes toward probabilities

Drawing on (18), the properties of the probability weighting function in relation to the parameters q and k are listed in the first lines of the tables in Propositions 1, 2 and 3.

The block of lines which follows immediately the header in each table deals with the attitude toward probabilities embodied in the probability weighting function. Generally speaking, it amounts to pessimism or optimism, which can be approached from two different points of view, each one is linked to a way of considering the generic term in the expression of the rank-dependent utility of a lottery: either a utility multiplied by a difference between transformed probabilities, or a transformed probability multiplied by a difference between utilities.

The first point of view ( Yaari, 1987 ; Chateauneuf & Cohen, 1994 ) contrasts strong pessimism with strong optimism (which meets ( Wakker, 1994 )’s distinction between “probabilistic risk aversion” and “probabilistic risk seeking”), associated respectively to the convexity and to the concavity of φ. The weight of a typical element $u\left(x_i\right)$ in (16) is given by a transformed probability ${\mu }_i=\phi \left(G\left(x_i\right)\right)-\phi \left(G\left(x_{i+1}\right)\right)$. In particular, a finite variation of G in the neighborhood of 1 or of 0, corresponding to the lowest or to the highest outcomes, indicates its decisional weight ${\mu }_i$ at the endpoints of the domain of definition of φ by the corresponding variation in ordinate. The convexity ( Figure 3 or Figure 5 ) (resp., the concavity ( Figure 4 or Figure 6 )) of φ therefore amounts to strong pessimism (resp., strong optimism), insofar as the probability of the lowest outcomes is overweighted (resp., underweighted), whereas the probability of the highest outcomes is underweighted (resp., over- weighted). Strong pessimism (resp., strong optimism) can be interpreted as in- creasing (resp., decreasing) sensitivity to probability changes when moving from the low probabilities of getting at least the higher outcomes to the high probabilities of getting at least the lower outcomes. This makes easier the interpretation of the intermediate situations of an inverse S-shaped (first concave, then convex; see Figure 7 and Figure 8 ) or S-shaped (first convex, then concave; see Figure 1 and Figure 2 ) probability weighting function (see the seminal paper of ( Gonzalez & Wu, 1999 ). In the case of an inverse S-shaped function ( Figure 7 and Figure 8 ), the probabilities of the lowest and of the highest outcomes are overweighted relatively to those of the medium outcomes (in the neighbourhood of the inflexion point $p_0$) which are underweighted. This boils down to strong optimism toward medium to high outcomes (the concave part of φ), and strong pessimism toward low to medium outcomes (its convex part). Commonly used in cumulative prospect theory (see ( Tversky & Kahneman, 1992 )), the inverse S-shaped probability weighting function is interpreted in terms of cognitive ability after Wakker (2010: pp. 203 sqq) who called it “likelihood insensitivity”, in the sense that people fail to distinguish sufficiently variations of probabilities for medium, usual outcomes, but are overly sensitive when these changes concern best ranked and worst ranked unusual outcomes. Obviously, a symetrical interpretation can be given to the less common S-shaped probability weighting function ( Figure 1 and Figure 2 ), which can be viewed as an expression of what might be called “unlikelihood insensitivity”.

The second point of view makes a distinction between what is usually refered to as weak pessimism and weak optimism ( Cohen, 1995 ). At the difference of strong pessimism and strong optimism, weak pessimism and weak optimism are implicitly based on the interpretation of $\phi \left(G\left(x_i\right)\right)$ as the transformed probability which we associate to a minimum additional utility $u\left(x_i\right)-u\left(x_{i-1}\right)$ (see supra n. 13). In an expected utility framework, we know that $\phi \left(G\left(x_i\right)\right)=G\left(x_i\right)$ for each i. So that pessimism can be seen as doing worse than expected utility, and optimism as doing better than it. Weak pessimism therefore occurs (resp., weak optimism) when $\phi \left(G\left(x_i\right)\right)\le G\left(x_i\right)$ (resp., $\phi \left(G\left(x_i\right)\right)\ge G\left(x_i\right)$), the probabilities of additional utilities being underweighted (resp., overweighted). It is obvious that strong pessimism implies weak pessimism (and strong optimism implies weak optimism), whereas the reverse is not true. The previous issue of the convexity or concavity of the probability weighting function is here replaced by the question of knowing whether φ lies below the bisector (weak pessimism) (see Figure 2 , Figure 3 , Figure 5 ) or above it (weak optimism) (see Figure 4 , Figure 6 , Figure 7 ). Consequently, S-shaped or inverse S-shaped probability weighting functions are now significant only when they cross the bisector. When φ is inverse S-shaped crossing the bisector ( Figure 8 ), weak optimism prevails locally for relatively high outcomes (with probabilities of winning at least this outcome belonging to the interval between 0 and the abciss $p^{\text{*}}$ of the point of intersection of φ and the bisector) because the corresponding part of φ lies above the bisector; and weak pessimism prevails locally for relatively low outcomes (with probabilities of winning at least this outcome belonging to the interval between $p^{\text{*}}$ and 1) because the corresponding part of φ lies below the bisector. Of course, an S-shaped φ crossing the bisector ( Figure 1 ) is interpreted in a symetrical way.

b) On the attitudes toward risk

Following ( Rothschild & Stiglitz, 1970 )’s seminal paper, we are used to distinguish weak and strong risk-aversion (resp., risk-seeking; risk-neutrality being equivalent to risk-aversion and risk-seeking). Both provide answers to different questions. A decision-maker is said to be weakly risk-averse (resp., weakly risk-seeker), if he or she prefers a lottery L to its expected value $E_L\left(x\right)$ (resp., the expected value $E_L\left(x\right)$ of a lottery L to this lottery). By contrast, a decision-maker is strongly risk-averse (resp., strongly risk-seeker) when, given a pair of lotteries L₁ and L₂ with equal means such that L₁ is stochastically dominating L₂ at degree 2¹⁸, L₁ (resp., L₂) is preferred to L₂ (resp., L₁). Weak risk attitude is the result of a comparison between a risky distribution and a certain outcome, whereas strong risk attitude denotes a comparison between two risky distributions. An intermediary concept was introduced by Quiggin (1992) in relation to what was to become known as rank-dependent utility: monotone risk-aversion (resp., monotone risk-seeking) denotes a situation where a decision-maker prefers L₁ to a lottery L₂ (resp., L₂ to L₁) when L₂ is a monotone increase in risk of L₁¹⁹. Strong, monotone and weak risk attitudes are equivalent in standard expected utility, when the decision weights are equal to the corresponding probabilities, since they all depend on the concavity (risk-aversion) or the convexity (risk-seeking) of the utility function, which incorporates the whole relevant information on the attitude toward risk. Such is the case when $q=0$ and $k=1$, so that the decision weights ${\mu }_i$ are equal to the corresponding probabilities $p_i$. Because his or her behaviour boils down to expected utility when $k=1$, a simple hyperbolic probability discounter ( $q=0$) who is weakly risk-averse (weakly risk-seeker) is also strongly risk-averse (strongly risk-seeker) and monotonely risk-averse (monotonely risk seeker).

But in all other cases, when the utility of a lottery is given by (16), the properties of the utility function u alone are not sufficient to determine the attitude toward risk: it now depends on the properties of both the utility function u and the probability weighting function φ. Let us therefore turn to the properties of the utility function. Assume, for sake of simplicity, that it is bi-differentiable, and either concave or convex. The concavity and the convexity of u are currently interpreted as, respectively, a decreasing sensitivity and an increasing sensitivity to outcomes. In a probability discounting framework like the one of (16), the risk attitude carried on by the utility function can be either reinforced or thwarted by the attitude toward probabilities carried on by the probability weighting function. We rely explicitly on some results concerning rank-dependent utility and adapted to q-discounting in order to account for the effects on risk attitude of the interaction between the sensitivity to outcomes (u) and the attitude toward probability (φ).

The first result is from Quiggin (1992) and Cohen (1995) . It shows that strong risk aversion implies monotone risk aversion which implies weak risk aversion and, in the same way, that strong risk seeking implies monotone risk seeking which implies weak risk seeking. The second result, from Hong, Karni, & Safra (1987) states on the one hand, that decreasing sensitivity and strong pessimism is equivalent to strong risk aversion, on the other it states that increasing sensitivity and strong optimism is equivalent to strong risk seeking. The third result is due to Chateauneuf & Cohen (1994) . It highlights the link between weak attitude toward risk and weak attitude toward probability, in the sense that weak risk aversion implies weak pessimism and weak risk seeking implies weak optimism. The fourth result is also from Chateauneuf & Cohen (1994) . It aims at finding the extent of weak pessimism (resp., weak optimism), which can overcome increasing sensitivity (resp., decreasing sensitivity) so that weak risk aversion (resp., weak risk seeking) is made possible. It states that whatever $x,y$, with $x>y$, whatever $p\in \left[0,1\right]$, if

there exists $g\ge 1$ such that $u^{\prime }\left(x\right)\le g\frac{u\left(x\right)-u\left(y\right)}{x-y}$ and $\phi \left(p\right)\le p^g$, then weak risk aversion is satisfied. Likewise, whatever $x,y$, with $x>y$, whatever $p\in \left[0,1\right]$, if there exists $h\ge 1$ such that $u^{\prime }\left(y\right)\le h\frac{u\left(x\right)-u\left(y\right)}{x-y}$ and $\phi \left(p\right)\ge 1-{\left(1-p\right)}^h$,

then weak risk seeking is satisfied. The fifth result is from Quiggin (1982, 1992 ); see also ( Chateauneuf & Cohen, 1994 )). It says that when u is concave (resp., convex), monotone risk aversion, weak risk aversion and weak pessimism are equivalent (respectively, monotone risk seeking, weak risk seeking and weak optimism are equivalent). Finally the last result that we use is due to Chateauneuf, Cohen, & Meilijson (2005) . It improves Chateauneuf & Cohen (1994) by relying on indexes of pessimism or optimism on the one hand, and on indexes of non-concavity or non convexity of the utility function on the other hand. This result states that monotone risk aversion is equivalent to $G_u\le P_{\phi }$, and monotone risk seeking is equivalent to

$T_u\le O_{\phi }$, where $G_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(x\right)}{u^{\prime }\left(y\right)}$ is an index of non-concavity ( $G_u\ge 1$ and is equal to 1 when u is concave), $T_u=\underset{y<x}{\text{sup}}\frac{u^{\prime }\left(y\right)}{u^{\prime }\left(x\right)}$ is an index of non-convexity ( $T_u\ge 1$ and is equal to 1 when u is convex), $P_{\phi }=\underset{0<p<1}{\text{inf}}\frac{\frac{1-\phi \left(p\right)}{\phi \left(p\right)}}{\frac{1-p}{p}}\ge 1$ is an index of pessimism, and $O_{\phi }=\underset{0<p<1}{\text{inf}}\frac{\frac{\phi \left(p\right)}{1-\phi \left(p\right)}}{\frac{p}{1-p}}$ an index of optimism. The result of Chateauneuf, Cohen, & Meilijson (2005) therefore expresses situations where pessimism (resp. optimism)

compensates the convexity (resp. concavity) of the utility function. It can be shown that when q-discounting occurs, $P_{\phi }=k$ and $O_{\phi }=1/k$, which are both obtained when p tends to 1. So that the result of Chateauneuf, Cohen, & Meilijson (2005) can be reformulated as:

$\{\begin{array}{l}\text{Monotone Risk Aversion}\iff G_u\le k\hfill \\ \text{Monotone Risk Seeking}\iff T_u\le 1/k\hfill \end{array}$

Drawing on the above results from the literature, it it has become possible to determine, in Propositions 1, 2 and 3, the various types of attitudes toward risk generated by the combination between an attitude toward probabilities expressed by the properties of φ, and an attitude toward output which comes from the properties of u.

It is commonsense to claim that all this depends on the action of the two parameters, k and q. In the case of decision in time, their respective function seems rather clear (see, for instance, ( Takahashi, 2007a : pp. 639-640) and ( Munoz Torrecillas et al., 2018 : pp. 191-192)). k is usually perceived as a parameter of “impulsivity”, which we can understand as “impatience”, since it increases the discounting weight of physical waiting time. And q is a parameter of (time-) consistency, since when it moves away from 1, it also makes exponential discounting more and more distant. Regarding decision in risk, q separates situations of non-optimism (in which global risk-seeking of any type is impossible) when it is greater than 0 and smaller than 1 (Proposition 1) from situations of non-pessimism (in which global risk-aversion, also of any type, is impossible) when it is less than 0 (Proposition 3). Rather than a parameter of “risk-aversion”, as Takahashi et al. (2013: p. 877) first called it, k plays a crucial part as a sophisticated parameter of pessimism: it constitutes the upper-bound for the index of non-concavity $G_u$ in order to obtain monotone risk-aversion; or it represents, through 1/k, the upper-bound for the index of non-convexity $T_u$ to produce monotone risk-seeking. This shows that appropriate values of k can compensate either the concavity or the convexity of the utility function to produce either monotone risk-seeking in the first case, or monotone risk-aversion in the second case. And if k is either too large or too small for this, it remains possible to have at least sufficient conditions to obtain weak risk-aversion or weak-risk seeking ( Chateauneuf & Cohen, 1994 ). When it is smaller than 1 (when $0<q<1$) or greater than 1 (when $q<0$), k generates S-shaped or inverse S-shaped probability weighting functions φ which cross the bisector, so that none of the basic global attitudes toward risk can exist. In all other cases, at least weak optimism or weak pessimism occurs, so that the necessary condition for any conception of risk aversion or risk seeking is satisfied ( Chateauneuf & Cohen, 1994 ). At last, the relation between both parameters, k and q, allows determining the range of their relative values for which strong risk attitudes are possible: if q

lies between 0 and 1, $k\ge \frac{1}{1-\frac{q}{2}}$ generates strong pessimism, thus determining strong risk-aversion with u concave; symmetrically, if q is less than 0, $k\le \frac{1}{1-\frac{q}{2}}$

generates strong optimimism, and strong risk-seeking with u convex ( Hong, Karni, & Safra, 1987 ).