# Ergodicity transformation for additive-ruin wealth dynamic

Carlos Rodríguez Raposo, Pablo Coello Pulido

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Carlos Rodríguez Raposo, Pablo Coello Pulido. Ergodicity transformation for additive-ruin wealth dynamic. 2021. hal-03198073v3

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# Ergodicity transformation for additive-ruin wealth dynamic

Carlos Rodríguez Raposo and Pablo Coello Pulido*

OpSeeker Tech SL

October 6, 2021

## Abstract

Ergodicity economics provides a framework to derive utility functions that assure growth optimality without the need for psychological considerations. It takes into account the wealth dynamic in which the agent evolves. In this article we define a modified additive wealth dynamic. It behaves as a pure additive dynamic for positive values of wealth. However, for negative values, individuals are penalized by having to borrow money at a certain interest rate. We propose two different versions of this dynamic: one in which unpaid interests make the principal grow if the agent cannot afford them at a given time, and another in which principal is frozen when the agent cannot pay the whole interest. We use the word ruin for the negative levels of wealth in which the agent cannot pay back the debt. We propose these two dynamics to better describe individuals that earn periodic wages and that do not invest them, but who cannot take debt for free. Unlike in the pure additive case, we show that ergodicity economics predicts risk averse behavior for agents in the additive-ruin dynamics. This result gives an explanation to why agents might refuse positive expected value lotteries for some levels of wealth and not for others by only having to make mechanistic hypothesis about the wealth dynamic.

## I. INTRODUCTION

As explained by O. Peters and A. Adamou in [1], ergodicity economics, EE, is a recent branch of economic theory that takes into account the difference between time averages and expectation values. EE postulates that agents behave seeking growth optimality. According to Peters and Adamou, individual agents are therefore concerned with time average growth rate of wealth.

An observable is said to be ergodic if its time average equals its expectation value, as explained by Peters in [2]. EE shows that, in general, wealth changes are not ergodic. This is proven by Peters and Gell-Mann for discrete multiplicative dynamics in [3] and by Peters and Klein for wealth following a geometric brownian motion in [4].

In [1], Peters and Adamou argue that in expected utility theory, EUT, utility functions can be interpreted as a way of restoring empirical evidence to the theoretically irrelevant expectation values in order to accomodate to behavioral data. In the same paper they introduce a method to derive transformations of wealth, which if used as utility functions assure growth optimality. We will refer to this method as the EE framework, and to these transformations

of wealth as ergodicity transformations. The observables obtained by applying these ergodicity transformations to changes of wealth are ergodic. This implies equality between time average and expectation value.

In EE a wealth dynamic is a stochastic process which depends on the economic circumstances of the agent but not on its preferences. And since the ergodicity transformation can be interpreted as the utility function in EUT, EE framework provides a method to derive utility functions which depend only on wealth dynamics and not on subjective hypothesis made about the agent.

In additive dynamics the ergodicity transformation is linear. A linear utility function implies risk neutrality. Peters and Adamou argue in [1] that "an agent with labour income and no savings would be well modelled as experiencing additive changes in wealth, while for an agent with significant invested capital a multiplicative dynamic would be more realistic.". This would suggest that agents that have no capital invested and are completely dependent on their labour income are likely to show risk neutral behaviors.

In this paper we make a new contribution to the set of dynamics already studied in ergodicity economics by proposing a modified additive wealth dynamic. We argue that a pure additive dynamic might not be representative

<footnote>*Emails: carlosr@opseeker.com, pablo@opseeker.com</footnote>
for individuals who would need to pay interest to cover for debt. As a matter of fact, when in debt, the interest owed will lead to a slower wealth growth. For a certain level of debt, the interest owed per period will be as big as the regular income of the individual. We will call this point, the perpetuity point, since the only kind of loan the individual can accept is an interest-only loan, which is itself, a perpetuity. We propose two variations of the dynamic to study what happens for bigger levels of debt, i.e. agent cannot pay back whole owed interest.

In the first variation of the dynamic the creditor accepts as the best deal the whole income of the ruined individual, without increasing the owed principal. In the second variation of the dynamic the creditor accepts as the best deal the whole income of the ruined individual, however, the principal gets increased in the amount of the unpaid interest.

We call ruin the negative levels of wealth beyond the perpetuity point. In this region, the agent, even using the whole income, cannot pay any of the principal back after each period.

After applying EE to our modelization of the first variation of the dynamic we find that the associated ergodicity transformation is a linear and exponential function, or linex, already studied in economic theory [5]. For the second variation of the dynamic we obtain a logarithmic function in a case that can be solved analytically.

Both results show a certain degree of risk aversion which, in contrast to pure additive dynamics, seems to be more aligned with the empirical evidence of risk aversion in controlled laboratory settings [6].

We will proceed in the following order. We first introduce the additive-ruin dynamic and argue why it might be relevant for agents with labour income and no invested savings. Second, we analyze the EE framework exposed in [1]. Third, we introduce our models for the additive-ruin wealth dynamic and derive the associated utility functions. Fourth, we show the results of a simulation in which agents be-

have according to one of the previous utility functions. We end this paper with a brief discussion of the results.

## II. ADDITIVE-RUIN WEALTH DYNAMIC

As an attempt to better describe the wealth dynamic of agents with labour income and no invested savings we introduce the additive-ruin wealth dynamic. The intuition behind this proposal is that a dynamic that aims to reflect the reality of these agents must take into account the fact that they cannot borrow money for free. The owed interests will burden the subsequent evolution of wealth.

The proposed dynamic looks like an additive dynamic for positive values of wealth. However, it differs for negative levels, in which it is assumed that the agent needs to borrow money at a certain interest rate to cover for the debt.

We can define this dynamic using the following values:

* The drift coefficient of the positive wealth region: $p$. It can be interpreted as the recurring income the agent receives.

* The interest rate $r$ that the agent would need to pay if wealth is negative to cover the debt

Using these two values we can define an important point in the negative wealth region: the perpetuity point. This is the point in which the individual is able to pay back the full interest but no principal. An agent that reaches this level of debt would not be able to recover in a finite amount of time since the principal would never be paid back. This point is defined by $-p/r$. For bigger debts, the agent is not able to pay even for interest. We call this state ruin.

Denoting wealth by $x$, the general outline of this dynamic would therefore present three regions:

* Positive wealth $x > 0$: additive dynamics with growth factor $p$

* Transition region $-p/r < x \le 0$: wealth increases each period less than $p$ due to
the reimbursement of interest $rx$

* Ruin region $x \le -\frac{p}{r}$: agent is not able to pay back any of the principal, or even the full interest

In this paper we present two different versions of this dynamic. Both versions have in common the previously described additive-ruin behaviour but differ from each other in terms of how the ruin region is treated.

We call the first version of the dynamic: Dynamic I. In this dynamic, when the individual cannot pay back the totality of the owed interest per period, the creditor accepts as the best deal the whole income of the individual, without increasing the owed principal.

It could be argued that Dynamic I is not realistic because no one would forgive owed interest for free. However, this could be understood as a bankruptcy deal, in which creditors reach an agreement for the existing assets.

<figure><img src="image_3.png" /><figcaption>**Figure 1:** Wealth drift for Dynamic I. In the green region agent experiences an additive dynamic. In the yellow region a loan is paid back at interest rate $r$. In the red region the agent cannot afford interest and creditor accepts $p$ as best deal with the principal frozen. Source: elaborated by the authors.</figcaption></figure>

In the second version of the dynamic, Dynamic II, when the individual cannot pay back the totality of the owed interest per period, the creditor accepts as the best deal the whole income. The principal gets increased in the amount of the unpaid interest.

<figure><img src="image_4.png" /><figcaption>**Figure 2:** Wealth drift for Dynamic II. In the green region agent experiences an additive dynamic. In the yellow region a loan is paid back at interest rate $r$. In the red region the agent cannot afford interest and creditor accepts $p$ as best deal, with debt increasing due to unpaid interest. Source: elaborated by the authors.</figcaption></figure>

### III. METHODOLOGY

Peters and Adamou show in [1] that for expected utility theory to be equivalent to optimisation over time, utility must follow a Lévy process. In the case that only continuous paths are considered, it is shown that utility $u$ must follow a Brownian motion with drift:

$$du = a_u dt + b_u dW \quad (1)$$

Where $dW$ represents the infinitesimal increment of the Wiener process and $a_u, b_u$ are constants. Note that we try to keep the same notation as in [1].

Only dynamics that can be expressed as an autonomous Itô process are considered:

$$dx = a_x(x)dt + b_x(x)dW \quad (2)$$

Note that, unlike $a_u$ and $b_u$, $a_x(x)$ and $b_x(x)$ are in general functions of wealth $x$. The question is: what needs to satisfy a dynamic defined by $a_x(x)$, $b_x(x)$ so that it can be mapped to a utility process which is a Brownian motion with drift (i.e $a_u$ and $b_u$ are constants)?

To answer this question, Peters and Adamou, apply Itô's lemma to (2) with $u = u(x)$. For this to be possible $u(x)$ needs to be twice differentiable.
$$du = \left\{ a_x(x) \frac{du(x)}{dx} + \frac{1}{2} b_x(x)^2 \frac{d^2 u(x)}{dx^2} \right\} dt + b_x(x) \frac{du(x)}{dx} dW \quad (3)$$

Note that we use a total derivative notation and not partial because the utility function depends only directly on wealth and not explicitly on time.

We can identify the coefficients of $dt$ and $dW$ in (3) with the constant coefficients of (1):

$$a_u = a_x(x) \frac{du(x)}{dx} + \frac{1}{2} b_x(x)^2 \frac{d^2 u(x)}{dx^2} \quad (4)$$

$$b_u = b_x(x) \frac{du(x)}{dx} \quad (5)$$

Differentiating (5):

$$0 = \frac{db_x(x)}{dx} \frac{du(x)}{dx} + b_x(x) \frac{d^2 u(x)}{dx^2} \quad (6)$$

Solving for $\frac{d^2 u}{dx^2}$:

$$\frac{d^2 u(x)}{dx^2} = - \frac{1}{b_x(x)} \frac{db_x(x)}{dx} \frac{du(x)}{dx} \quad (7)$$

Substituting this result in (4) using (5):

$$a_x(x) = \frac{a_u}{b_u} b_x(x) + \frac{1}{2} b_x(x) \frac{db_x(x)}{dx} \quad (8)$$

This is a consistency condition that $a_x(x)$ and $b_x(x)$ need to satisfy in order for $u(x)$ to be a Brownian motion with drift.

If a given dynamic $a_x(x), b_x(x)$ satisfies (8), then, the utility function can be calculated with (5):

$$\frac{du(x)}{dx} = \frac{b_u}{b_x(x)} \quad (9)$$

$$u(x) = C + \int \frac{b_u}{b_x(x)} dx \quad (10)$$

Summarizing: if a wealth dynamic $dx$ is an autonomous Itô process defined by $a_x(x), b_x(x)$ which satisfies condition (8), then it is possible to obtain the associated ergodicity transformation using (10).

## IV. EE SOLUTIONS FOR ADDITIVE-RUIN WEALTH DYNAMICS

As already discussed in section II, we propose two dynamics that differ from each other in the ruin region. In Dynamic I, the lender accepts $p$ as best deal, and principal of the debt is frozen. In Dynamic II, the lender accepts $p$ as best deal, and the unpaid interest increases principal.

To integrate EE equations for Dynamic I we use a logistic function. We apply the EE framework and obtain the associated ergodicity transformation which is a linear and exponential function, or linex, already studied in economic theory [5]. This function implies the presence of risk aversion in agents. Furthermore it predicts one switch preferences: a decision maker might refuse a positive expected value lottery for a given level of wealth, but accept it after a certain value of wealth has been exceeded.

For Dynamic II we use an inverted ramp exponential function. We can only integrate analytically for a trivial case, however, numerical integration can be used for other cases. The obtained ergodicity transformation is a logarithmic function which implies risk aversion as in the previous case and has a vertical asymptote at the perpetuity point.

Note that in both cases, ruin is a state in which is hard to escape from. As a matter of fact, with the given parameters, the only way to escape would be thanks to volatility. Some models in ruin theory take into account the time spent with negative reserves to validate the ruin condition. One example is the parisian ruin model [7]. In our models, ruin is sticky, and a time condition can be easily studied.
i. Solution for Dynamic I

Sigmoid functions are good candidates to model this dynamic since they can be considered a smooth representation of curve shown in Figure 1. For the purpose of this paper we decide to use the logistic function. The three states can be easily modeled: ruin corresponds with big negative values, in which volatility tends to zero. Additive state corresponds with big positive values, in which volatility tends to a constant value. Between them, there is a smooth transition. The fact that the width and position of the step can be parametrized and that this function can be easily differentiated makes the logistic function a suitable candidate. Other sigmoids could as well be used. We apply the same process in Appendix I to an algebraic sigmoid function reaching similar results.

We would like that either $a_x(x)$ and $b_x(x)$ have logistic geometries. A logistic function can be defined as:

$$h_{x_r, x_a}(x) = \frac{1}{1 + e^{-\frac{x-x_r}{x_a}}} \quad (11)$$

In this equation $x_r$ defines the center of the smooth step and $x_a$ defines its size. Without worrying about the scales, we would like our logistic function to match in the best way possible the geometry of the curve shown in Figure 1, which can be defined by pieces as:

$$\begin{cases} I) x \le p/r : y = 0 \\ II) p/r < x \le 0 : y = p + rx \\ III) 0 < x : y = p \end{cases} \quad (12)$$

The piece II corresponds to the transition. Its center is located at $x = -p/2r$. We therefore set the center of the smooth transition to match this point $x_r = -p/2r$. We can as well conclude that, since $x_a$ is related to the width of the smooth transition, it would be wise to think that it needs to be proportional to the width of piece II, which is $p/r$, and therefore set $x_a = \theta p/r$. We set $\theta = 1/4$ because in this

case, the smooth transition is tangent to piece II exactly at $x_r$.

We can now say that we would like $a_x(x)$ and $b_x(x)$ to have a geometry proportional to:

$$h_{p,r}(x) = \frac{1}{1 + e^{-2(\frac{2r}{p}x+1)}} \quad (13)$$

We see that actually the geometry depends only on the ratio $r/p$. By defining the parameter $\gamma = \frac{2r}{p}$ we can rewrite equation 13 as:

$$h_{\gamma}(x) = \frac{1}{1 + e^{-2(\gamma x + 1)}} \quad (14)$$

We can therefore define the volatility function $b_x(x)$ as:

$$b_{x;\beta,\gamma}(x) = \beta h_{\gamma}(x) \quad (15)$$

Where $\beta$ is a constant which defines the intensity of the volatility.

Using the consistency condition (8) we can derive the associated drift term:

$$a_x(x) = \frac{a_u}{b_u} \beta h_{\gamma}(x) (1 + \varepsilon(x)) \quad (16)$$

Where $\varepsilon(x)$ is:

$$\varepsilon(x) = \frac{b_u}{a_u} \gamma \beta \frac{e^{-2(\gamma x + 1)}}{(1 + e^{-2(\gamma x + 1)})^2} \quad (17)$$

$\varepsilon(x)$ can be interpreted as a term that, if small enough, can make $a_x(x)$ have the same logistic geometry as $h_\gamma(x)$. This would imply that it is consistent with our Dynamic I drift term.

It is easy to prove that the term $\varepsilon(x)$ has only one maximum at $x = -\frac{1}{\gamma} = -\frac{p}{2r}$ which is the center of the smooth transition. And that when $x \to \pm\infty$ then: $\varepsilon(x) \to 0$.

Therefore, $\varepsilon(x)$ is bounded by $\varepsilon(-\frac{1}{\gamma})$:

$$\varepsilon_{max} = \frac{b_u}{a_u} \frac{\gamma \beta}{4} \quad (18)$$

By taking the limit when $x$ tends to infinity in 16 we obtain:
$$ \lim_{x \to +\infty} a_x(x) = \frac{a_u}{b_u} \beta \qquad (19) $$

In order to match the dynamic, the drift term in this limit needs to be equal to $p$, and therefore we conclude:

$$ \frac{a_u}{b_u} = \frac{p}{\beta} \qquad (20) $$

In order to make $\varepsilon_{max} \ll 1$, and using equations 18 and 20, we conclude that if the following condition is met, $a_x(x)$ and $b_x(x)$ have logistic geometries:

$$ \beta \ll p \sqrt{\frac{2}{r}} \qquad (21) $$

Note that this imposes a limit on how much volatility this model can handle regarding $p$ and $r$.

The closed form of the dynamic is:

$$ dx = h_{\gamma}(x) \{p(1 + \varepsilon(x))dt + \beta dW\} \qquad (22) $$

In Figure 3 we observe several paths issued from integrating the dynamic defined in (22). Agents beginning with positive wealth see their wealth evolve in a linear manner (additive dynamics). However, those affected by ruin ($x_0 < 0$) are either frozen or take more time to begin the additive growth.

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Dynamic I drift term compared to logistic drift term. Source: elaborated by the authors.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3:** Sample paths for dynamic defined in (22) for different values of initial wealth, in terms of $p/r$. Note that $x = -p/r$ is the perpetuity point. Note how for big negative values of initial wealth, wealth is frozen, and for less extreme negative values, additive dynamics takes time to appear. The simulations were made with $p = 2$, $r = 0.03$ and $\beta = 1.5$. Source: elaborated by the authors.</figcaption></figure>

We have shown that $b_x(x)$ and $a_x(x)$ are consistent according to (8). We can therefore use (10) to derive the utility function.

$$ u(x) = \frac{b_u}{\beta} \left[ x - \frac{1}{2\gamma} e^{-2(\gamma x+1)} + C \right] \qquad (23) $$

Setting arbitrarily the integration constant $C$ so that $u(0) = 0$, we can obtain a dimensionless utility function dividing by $\frac{b_u}{\beta\gamma'}$ in terms of a dimensionless wealth $\tilde{x} = \gamma x$:

$$ \tilde{u}(\tilde{x}) = \tilde{x} + \frac{1}{2e^2} (1 - e^{-2\tilde{x}}) \qquad (24) $$
<figure><img src="image_7.png" /><figcaption>**Figure 5:** Dimensionless utility function obtained in (24) compared to linear utility function (green). Source: elaborated by the authors.</figcaption></figure>

This utility function is known in the literature as linex function [5] [8]. It has the one-switch property. This means that an agent might switch preferences for a risky gamble depending on the level of wealth at most once. In particular, an agent might refuse a risky gamble even if expected value is positive for low levels of wealth, but accept it after a certain wealth level in which risk aversion is smaller.

Moreover, due to the shape of the linex function, given a certain level of wealth, the same agent could reject a bet of a certain magnitude that could potentially bring him closer to ruin but accept another bet of a smaller magnitude that would not bring him as close.

Note that for big values of wealth, this utility function behaves as a linear one, retrieving the result of pure additive dynamics.

We highlight the fact, that in contrast to the solution that we obtain for Dynamic II in the next section, this utility function does not present a vertical asymptote. This is because the logistic function used to model volatility and drift equals zero only when $x$ tends to minus infinity. And therefore, utility does not reach minus infinity for any finite amount of debt.

Risk aversion can be studied using the Pratt measure [9], defined as: $A(x) = -\frac{u''}{u'}$.

$$A(\tilde{x}, \gamma) = \frac{2\gamma}{1 + e^{2(\tilde{x}+1)}} \quad (25)$$

Note that $A(\tilde{x})$ is a logistic function with the transition centered in $\tilde{x} = -1$ what implies $x = -\frac{1}{\gamma} = -\frac{p}{2r}$. This means that the center of the transition of the risk aversion function coincides with the center of the drift term in our dynamic.

On top of that, we see that the risk aversion is proportional to $\gamma$. This implies that the harder it is to pay back the debt, the higher the risk aversion is. Higher interest rates, or lower income would imply higher risk aversion.

<figure><img src="image_8.png" /><figcaption>**Figure 6:** Absolute risk aversion for the case $\gamma = 0.5$. Note that $\tilde{x} = -2$ corresponds with $x = -p/r$ which is the perpetuity point and beginning of the ruin region. Source: elaborated by the authors.</figcaption></figure>

We can check, as predicted by Bell in [5], that the utility function that we have derived verifies the condition of decreasing risk aversion: $u''u'' < u'u'''$ [9]. In [10], Bell refers to this fact as "reasonable". With EE we conclude that for an agent in Dynamic I, this behavior maximises the average growth rate of wealth.

## ii. Solution for Dynamic II

We obtain the solution for this dynamic in an equivalent way. We define our volatility function to match the geometry of the model. We then define the consistent drift function and verify that it satisfies the required geometry. We end up by integrating in the same manner as before to obtain the utility function.
To model this dynamic we use the following smooth ramp function for the volatility function:

$$b_x(x) = \frac{\beta_0 x}{1 + e^{kx}} + \beta_1 \quad (26)$$

This ramp function can match the geometry of the curve shown in Figure 2. This curve can be defined by pieces as:

$$\begin{cases} I) x \le 0: y = p + rx \\ II) 0 < x: y = p \end{cases} \quad (27)$$

<figure><img src="image_9.png" /><figcaption>**Figure 7:** Drift function for the modelization of Dynamic II: the red area corresponds to the ruin state, the yellow area to the transition state and the green area to the additive region. Source: elaborated by the authors.</figcaption></figure>

Using the consistency equation 4 we obtain:

$$a_x(x) = b_x(x) \frac{a_u}{b_u} (1 + \varepsilon(x)) \quad (28)$$

$$\varepsilon(x) = \frac{b_u \beta_0}{2a_u} \left( \frac{1}{1 + e^{kx}} - \frac{kx e^{kx}}{(1 + e^{kx})^2} \right) \quad (29)$$

It can be shown that $\varepsilon(x)$ is bounded and tends to zero for big positive values of $x$ and to a positive constant for big negative values. This means that the drift function defined in equation 28 will still have a smooth ramp function geometry, and thus, it can represent the additive-ruin Dynamic II.

$$\lim_{x \to +\infty} a_x(x) = \frac{a_u}{b_u} \beta_1 = p \quad (30)$$

Parameter $k$ can be set to a value according to the precision desired to match the transition between pieces I and II. The bigger, the sharper the transition is.

On top of these conditions, one more condition can be imposed so that drift is zero at the perpetuity point:

$$a_x(-p/r) = 0 \quad (31)$$

Analizing equation 28 it can be concluded, that equation 31 is equivalent to:

$$b_x(-p/r) = 0 \quad (32)$$

And therefore:

$$\frac{\beta_1}{\beta_0} = \frac{\frac{p}{r}}{1 + e^{-k \frac{p}{r}}} \quad (33)$$

Utility functions can be multiplied and divided by constants. By knowing $\frac{\beta_0}{\beta_1}$ in terms of $p$ and $r$ as defined in 33, the utility function can be obtained by integrating:

$$u(x) = \int \frac{dx}{\frac{\beta_0 x}{\beta_1} + 1} \quad (34)$$

We have only found an analytic solution for the case in which $\beta_0 = \beta_1 = k = 1$. After dividing and rearranging constants we get:

$$u(x) = \ln \left| \frac{1 + x + e^x}{2} \right| \quad (35)$$

Note that this utility function has a vertical asymptote at the $x$ which verifies $1 + x + e^x = 0$. This is the point in which volatility equals zero, which is the perpetuity point. This means that in the modelization of Dynamic II, utility decreases much faster than in the modelization of Dynamic I. This is not necessarily suprising
since in Dynamic II the wealth of the agent decreases once she is not able to pay back interest, whereas in Dynamic I the wealth of the agent remains constant.

It can be shown that along the same asymptote, as expected, risk aversion grows to infinity.

Note as well that for big values of $x$ this utility behaves as a linear utility, retrieving the result of pure additive dynamics.

<figure><img src="image_10.png" /><figcaption>**Figure 8:** Utility function for Dynamic II with $\beta_0 = \beta_1 = k = 1$ setting integration constant so that $u(0) = 0$. Source: elaborated by the authors.</figcaption></figure>

For other values of $\beta_0, \beta_1$, or what is equivalent, for other values of $p$ and $r$, utility functions can be found applying numerical methods to integrate equation 34.

## V. SIMULATION

One way to make visible the superiority of the utility function derived via the ergodicity framework regarding the growth rate of wealth is simulation. We ran a simulation for two sets of individuals evolving in our modelization of the Dynamic I presented in this paper.

For this simulation we define two sets of individuals. The first set contains 1000 individuals who make decisions using a linear utility function, like the one predicted by pure additive dynamics. Hereinafter for the sake of brevity we will call these individuals linear agents. The second set contains the same number of individuals, but that make decisions using the linex

<table><caption>**Table 1:** Magnitude of the bets proposed for each scenario. Values in wealth units. Source: elaborated by the authors.</caption><thead><tr><th rowspan="2">Scenario</th><th>Win</th><th>Lose</th></tr><tr><th>(50% probability)</th><th>(50% probability)</th></tr></thead><tbody><tr><td>Soft</td><td>45</td><td>40</td></tr><tr><td>Moderate</td><td>65</td><td>60</td></tr><tr><td>Severe</td><td>120</td><td>100</td></tr></tbody></table>

function predicted by EE for Dynamic I. Hereinafter, linex agents.

The simulation consisted on the same experience for each individual: one initial bet followed by 10 repetitions made up of 12 periods according to the additive-ruin dynamic, plus one potential bet at the end of each repetition. Each individual will be proposed a maximum of 11 bets. Bets are only proposed if the individual's current wealth is positive or zero.

We ran the simulation for three different scenarios. The three scenarios have the same probability of winning or losing the bet (50% - 50%) and differ from each other in the size of the proposed bets. For each scenario, the bet is always the same and has positive expected value. Their magnitude can be seen in the Table 1

Linear agents would accept the bet whenever it is proposed to them, while linex agents may or may not accept each bet depending on the potential of the bet to bring their wealth closer to the ruin region. The difference in the results between the two sets will be more pronounced the larger the magnitude of the proposed bets (holding all other parameters constant).

We use the same parameters as in Figure 3: $p = 2, \beta = 1.5$ and $r = 0.03$. These parameters position the perpetuity point at -66.66 units of wealth. On the other hand, the threshold at which linex agents start to take bets depends on the magnitude of the bet and is located in a different position for each scenario: 63.54 units of wealth in the severe scenario, 45.80 units of wealth in the moderate scenario and 23.51 units of wealth in the soft scenario. All individuals start the simulation with zero wealth, starting
subsequent repetitions with their wealth result-
ing from the previous repetitions.

The decision making comes with the individ-
uals having to decide if they take the bet or
not. The condition of *wealth* ≥ 0 was added to
give more realism to the game. It is equivalent
to saying that individuals must have paid off
their previous debts before they can take on a
new bet.

The results for the three scenarios can be seen
in graphical form in Figure 9 and in a more
detailed fashion in Table 2. In the soft scenario,
with relatively small bets, the distribution of
the final wealth of both groups of individu-
als is fairly similar. None of the individuals
in this scenario has gone into the ruin region.
This happens due to the concurrence of two
reasons: individuals cannot take new bets if
their wealth is negative and the magnitude of
the loss they can incur by losing a bet is not
enough to position them below the perpetuity
point. However, it is possible to see a difference
in the number of individuals who, at the end
of the simulation, are in the transition zone
(with negative wealth but above the perpetuity
point), with 15 linex agents compared to 63
linear agents.

Regarding the moderate scenario, there are not
ruined individuals either for any of the utility
functions. This is due to the same reason pre-
sented in the soft scenario: the magnitude of
the loss due to a bet is not sufficient to position
an individual's wealth below the perpetuity
point. However, in this scenario it is possible to
see a more pronounced difference in the num-
ber of individuals that end the simulation in
the transition zone, being 41 linex agents as
opposed to 290 linear agents.

Finally, the severe is the most aggressive sce-
nario. It allows for the possibility of getting
ruined by losing a bet that brings a given in-
dividual's wealth below the perpetuity point.
In this case, the differences between the two
groups of individuals are remarkable with 588
linear agents compared to 0 linex agents get-
ting ruined. In this case it is possible to see a

higher number of linex agents in the transition
region with 74 compared to 31 for the linear
agents. Finally, the highest number of linex
agents is located in the additive region with
926 individuals as opposed to 381 for the linear
utility function.

It is also possible to see notable differences in
the behaviour of the two groups of individuals
by studying the number of proposed, accepted
and rejected bets. The aggregate numbers for
each set of individuals and scenario are shown
in Table 2.

For each set of 1000 individuals who behave
according to a particular utility function the
maximum number of bets that can be offered
to them is 11000 (an initial one plus one at the
end of each repetition for each individual).

Each bet may be accepted or rejected at the
time it is proposed. Whether it is accepted or
rejected is determined by the utility function
describing the behaviour of the individual to
whom the gamble is proposed as well as the
wealth available to him at that time.

Linear agents will accept any positive expected
value bet that is proposed to them due to their
risk neutrality. However, we can see how the
number of bets proposed to this type of indi-
viduals decreases as the severity of the scenario
increases. This is due to the increasing number
of linear agents whose wealth becomes nega-
tive at any point.

Linex agents would accept or reject a pro-
posed bet depending on how close losing this
bet might bring their wealth to the perpetuity
point. This behaviour translates into a higher
number of proposed bets since the number
of individuals following this utility function
whose wealth becomes negative is lower. In
none of the three scenarios the number of pro-
posed bets goes below 10,000.

The difference in behaviour for linex agents
between the three scenarios could be observed
in the number of bets accepted or rejected.
Ranging from 8620 bets accepted against 2358
bets rejected in the soft scenario, compared to
<figure><img src="image_11.png" /><figcaption>(a) Soft scenario</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Moderate scenario</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) Severe scenario</figcaption></figure>

**Figure 9:** Kernel density estimation of the final wealth results from the simulation of the three scenarios. The plot was made using the gaussian kernel. The black line represents the 0-wealth threshold, the red line represents the perpetuity point and the blue line represents the wealth threshold at which individuals who behave according to a linex utility begin to accept bets. Source: elaborated by the authors.
<table><caption>**Table 2:** Results of the different scenarios of the simulation. The total number of individuals per u(x) and simulation is 1000. The total number of bets per u(x) and simulation is 11000. Source: elaborated by the authors.</caption><thead><tr><th rowspan="2">Scenario</th><th rowspan="2">u(x)</th><th colspan="3">Region (number of individuals)</th><th colspan="3">Bets (number of bets)</th></tr><tr><th>Ruin</th><th>Transition</th><th>Additive</th><th>Proposed bets</th><th>Accepted bets</th><th>Rejected bets</th></tr></thead><tbody><tr><td rowspan="2">soft</td><td>linex</td><td>0</td><td>15</td><td>985</td><td>10978</td><td>8620</td><td>2358</td></tr><tr><td>linear</td><td>0</td><td>63</td><td>937</td><td>9254</td><td>9254</td><td>0</td></tr><tr><td rowspan="2">moderate</td><td>linex</td><td>0</td><td>41</td><td>959</td><td>10994</td><td>6911</td><td>4083</td></tr><tr><td>linear</td><td>0</td><td>290</td><td>210</td><td>7042</td><td>7042</td><td>0</td></tr><tr><td rowspan="2">severe</td><td>linex</td><td>0</td><td>74</td><td>926</td><td>10277</td><td>5172</td><td>5105</td></tr><tr><td>linear</td><td>588</td><td>31</td><td>381</td><td>5292</td><td>5292</td><td>0</td></tr></tbody></table>

5172 bets accepted against 5105 bets rejected in the severe scenario. It can be seen from these values that the higher magnitude of the bets included in the severe scenario forces linex agents to be more cautious in accepting bets.

It was discussed how empirical evidence suggests that individuals show risk aversion to some degree as explained in [6]. This simulation makes visible how this risk aversion can be understood as a means of optimizing growth rate of wealth by avoiding ruin.

## VI. DISCUSSION

In this article we have derived the utility functions predicted by ergodicity economics corresponding to two dynamics that we called additive-ruin wealth dynamics. The difference between the two lied in how debt is treated once the individual cannot pay back the totality of the owed interest.

Dynamic I, consists in an agreement between the agent and the creditor, in which the creditor will accept the full income of the agent without increasing the debt principal. We have modelled this dynamic using the logistic function. The obtained utility is a linex function. It implies that an agent, evolving in this dynamic, optimises the time average growth rate of wealth, by being more risk averse the closer to the ruin state.

Dynamic II consists in adding the unpaid interest to the owed principal. We have modelled this dynamic using an inverted smooth ramp

function. The obtained utility is logarithmic. Utility decreases much faster for negative values than in the modelization of Dynamic I. It contains a vertical asymptote at the perpetuity point.

These results give an explanation to why people might refuse lotteries of the kind: $110 with 50% chance or -$100 with 50%. Note that before EE, the majority of the explanations given to this problem were circular. A behavior is observed, and a utility function is constructed accordingly. However, the utility function is used as well as the justification for the behavior. There are exceptions like the one of Kelly [11]. With EE we start from a principle, agents optimise the time average growth rate of wealth. Then, applying the equations to a certain wealth dynamic, an associated utility function is found. It is noted that hypothesis need to be made about the dynamic in which the agent evolves in. However, there is no circularity in the resolution.

This paper provides EE with another example of dynamic in which risk aversion is present. For a lottery such as the one defined in the previous paragraph ergodicity is broken when ruled players cannot continue playing the game. Expected value of one round will not capture what happens in the long run. Decision makers need to avoid ruin in order to assure long term growth. This is the reason why risk aversion is advisable in the vicinity of ruin. Yechiam argues in [12] that agents overweight big losses but not small ones. This could be explained with an increasing risk aversion in the vicinity
of ruin as in our results. This conclusion under
EE needs no subjective or psychological consid-
erations, only mechanistic assumptions about
the wealth dynamic.
REFERENCES

[1] Ole Peters and Alexander Adamou. The time interpretation of expected utility theory. 1 2018.

[2] Ole Peters. The ergodicity problem in economics. *Nature Physics*, 15:1216–1221, 12 2019.

[3] O. Peters and M. Gell-Mann. Evaluating gambles using dynamics. *Chaos: An International Journal of Nonlinear Science*, 26(2):023103, Feb 2016.

[4] O. Peters and W. Klein. Ergodicity breaking in geometric brownian motion. *Physical Review Letters*, 110, 3 2013.

[5] David E. Bell. One-switch utility functions and a measure of risk. *Management Science*, 34(12):1416–1424, 1988.

[6] Glenn W. Harrison and E. Elisabet Rutström. Risk aversion in the laboratory. *Risk aversion in experiments*, 2008.

[7] Angelos Dassios and Shanle Wu. Parisian ruin with exponential claims, 2008.

[8] Michel M. Denuit, Louis Eeckhoudt, and Harris Schlesinger. When ross meets bell: The linex utility function. *Journal of Mathematical Economics*, 49(2):177–182, 2013.

[9] John W. Pratt. Risk aversion in the small and in the large. *Econometrica*, 32(1/2):122–136, 1964.

[10] D. Bell and P. Fishburn. Strong one-switch utility. *Manag. Sci.*, 47:601–604, 2001.

[11] J. L. Kelly. A new interpretation of information rate. *The Bell System Technical Journal*, 35(4):917–926, 1956.

[12] Eldad Yechiam. Acceptable losses: the debatable origins of loss aversion. *Psychological Research*, 83, 10 2019.
A. APPENDIX I: ALGEBRAIC SIGMOID FUNCTION AS AN ALTERNATIVE TO THE LOGISTIC FUNCTION FOR DYNAMIC I

As discussed previously, the choice of the lo-
gistic function to represent the additive-ruin
Dynamic I is not unique. Other sigmoid func-
tions can model the three states of ruin, transi-
tion and additive wealth. As an example, we
show the derivation of the utility function for
an algebraic sigmoid volatility function:

$$
b_x(\tilde{x}) = \beta \frac{\tilde{x} + \sqrt{1 + \tilde{x}^2}}{2\sqrt{1 + \tilde{x}^2}} \quad (36)
$$

Where:

$$
\tilde{x} = \frac{x}{x_a} \tag{37}
$$

$\tilde{x}$ is a dimensionless measure of wealth, with $x_a$ an arbitrary wealth amount used to remove the dimension. Note that for simplicity matters we have not added any extra parameters to control the transition center and width. However, it can be easily seen that the transition is centered at 0. As with the logistic function, we check that using the consistency condition (8) we obtain a realistic drift term (i.e. featuring the three states as well):

$$
a_x(\tilde{x}) = b_x(\tilde{x}) \frac{a_u}{b_u} \{1 + \varepsilon(\tilde{x})\} \quad (38)
$$

$$
\epsilon(\tilde{x}) = \frac{\frac{b_u \beta}{4x_a a_u}}{(1 + \tilde{x}^2)^{\frac{3}{2}}} \quad (39)
$$

It is the case since $\epsilon(\tilde{x})$ is bounded by a maxi-
mum located at $\tilde{x} = 0$. It tends to zero when
wealth tends to plus and minus infinity. We
can therefore calculate the utility function:

$$
u(\tilde{x}) = \int \frac{b_u x_a}{b_x(\tilde{x})} d\tilde{x} \quad (40)
$$

$$
u(\tilde{x}) = \frac{2b_u x_a}{\beta} \left( \tilde{x} + \frac{\tilde{x}^3}{3} - \frac{1}{3} (1 + \tilde{x}^2)^{\frac{3}{2}} \right) + C \quad (41)
$$

We can arbitrarily choose the integration con-
stant so that:

$$
u(0) = 0 \tag{42}
$$

<figure><img src="image_14.png" /><figcaption>**Figure 10:** Dimensionless utility function (blue) obtained in 43 compared to linear utility function (green). Source: elaborated by the authors.</figcaption></figure>

Additionnaly, we can divide by $\frac{2b_u x_a}{\beta}$ and ob-
tain the following dimensionless utility func-
tion:

$$
\tilde{u}(x) = \frac{1}{3} \left\{ 1 + 3\tilde{x} + \tilde{x}^3 - (1 + \tilde{x}^2)^{\frac{3}{2}} \right\} \quad (43)
$$

An inspection of this utility function shows
that for big values of wealth, utility behaves
as a linear function. This is the same result
already obtained for the logistic function. For
wealth values below the ruin threshold utility
decreases much faster (order of $\tilde{x}^3$). This be-
havior is present as well in the logistic case.
However, in that case, utility decreased at an
exponential rate for negative values.

Calculating the dimensionless Pratt measure
for absolute risk aversion we find:

$$
\tilde{A}(\tilde{x}) = -x_a \frac{\tilde{u}''}{\tilde{u}'} = \frac{\frac{\tilde{x}^2}{\sqrt{1+\tilde{x}^2}} + \sqrt{1+\tilde{x}^2} - 2\tilde{x}}{1+\tilde{x}^2 - \tilde{x}\sqrt{1+\tilde{x}^2}} \quad (44)
$$

<figure><img src="image_15.png" /><figcaption>**Figure 11:** Dimensionless absolute risk aversion for the algebraic volatility case. Source: elaborated by the authors.</figcaption></figure>
Note that this risk aversion function is similar to the one obtained with the logistic function in that it captures an increment in risk aversion when close to ruin. However, it is different because for big negative values of wealth risk aversion tends to zero.