# A Thought Experiment on Sustainable Management of the Earth System

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## Abstract

**:**

## 1. Introduction

## 2. A Thought Experiment

Assume there is a well-defined infinite sequence of generations of humanity, the current one being numbered 0, future ones 1, 2, …, and past ones −1, −2, … . At each point in time, one generation is “in charge” and can make choices that influence the “state of the world”. The possible states of the world can be classified into just four possible overall states, abbreviated L, T, P, and S, and we assume that this overall state changes only slowly, from generation to generation, due to the inherent dynamics of the world and humanity’s choices. We assume the overall state in generation k + 1, denoted X(k + 1), only depends on the following three things: (i) on the immediately preceding state, i.e., that in generation k, denoted X(k); (ii) in some states on the aggregate behavior of generation k, denoted U(k) and called generation k’s “choice”; and (iii) in some states also on chance; all this in a way that is the same for each generation (i.e., does not explicitly depend on the generation number k). Being in state X(k) implies a certain overall welfare level for generation k, denoted W(k). We assume the possible choices and their consequences depend on the state X(k) as follows:

Up until generation 0 and including it, all generations have been in state X(k) = L, where welfare is “high”, denoted W(k) = 1. When in state X(k) = L, generation k has two choices, A (which is considered the “default” choice that all generations before 0 have made) and B.

- ○
If generation k chooses option A, the next state is either L or T, depending somewhat on chance. It will be again X(k + 1) = L with some (typically large) probability η > 0, which is a time-independent constant, and will be X(k + 1) = T with probability π = 1 − η > 0.- ○
If they choose option B, the next state will be X(k + 1) = P for sure. In state X(k) = T, welfare is low, denoted W(k) = 0, and the state will never change again, X(k’) = T for all k’ > k; In state X(k) = P, welfare is also “low”, W(k) = 0, but the next state will be X(k + 1) = S for sure; and Finally, in state X(k) = S, welfare is again high, W(k) = 1, and the state will never change again, X(k’) = S for all k’ > k.We assume all this is known to generation 0 and all later generations.

On an island very far away from any land lives a small tribe whose main food resource are the fruits of a single ancient big tree despite which only grass grows on the island. Although the tree is so strong that it would never die from natural causes, every year there is a rainy season with strong storms, and someday one such storm might kill and blow away the tree. In fact, until just one generation ago, there was a second such tree that was blown away during a storm. If the same happens to the remaining tree, the tribe would have to live on grass forever, having no other food resource. Every generation so far has passed down the knowledge of a rich but unpopulated land across the large sea that can be safely reached if they build a large and strong boat from the tree’s trunk. Still, the tribe is so small and the journey would be so hard that they would have to send all their people to be sure the journey succeeds. Also, the passage would take so much time that a whole generation would have to live aboard and hope to catch the odd fish for food, causing deep suffering, and would not be able to see the new land with their own eyes, only knowing their descendants would live there happily and safely for all generations to come. No generation has ever set off on this journey.

_{A}

_{S}

^{ρ}).

_{A}but growing due to emissions E; Y is the gross world economic product, growing at a basic rate β slowed by climate-related damages; θ is the sensitivity of this slowing to A; S is the global knowledge stock for producing renewable energy, decaying at rate τ

_{S}, but growing due to learning-by-doing in proportion to produced renewable energy R; energy efficiency ε stays constant so that total energy use, U, is proportional to Y; energy is supplied by either fossils, F, or renewables, R, in proportions depending on relative price G; σ is the break-even level of S at which fossils and renewables cost the same; ρ is a learning curve exponent; and, finally, emissions are proportional to fossil combustion with combustion efficiency ϕ.

## 3. Analyses Using Rationality-Based Frameworks

#### 3.1. Optimal Control Framework with Different Intergenerational Welfare Functions

#### 3.1.1. Terminology

- “XcLT” = (L, …, L, T, T, …), with c > 0 times L and then T forever (so that c is the time of “collapse”);
- “XkLPS” = (L, …, L, P, S, S, …), with k > 0 times L, then once P, then S forever; and
- “all-L” = (L, L, …), which is possible, but has a probability of zero.

- “rc10” = (1, …, 1, 0, 0, …) with c > 0 ones and then zeros forever;
- “rk101” = (1, …, 1, 0, 1, 1, …) with k > 0 ones, then one zero, then ones forever; and
- “all-1” = (1, 1, …), which is possible, but has a probability of zero.

- infinite sequences (p(0), p(1), …) with all p(t) > 0,
- finite sequences (p(0), p(1), …, p(k − 1), 0) with p(t) > 0 for all t < k.

- “all-A” = (1, 1, …),
- “directly-B” = (0),

- “Bk” = (1, 1, …, 1, 0) with k + 1 ones, where the case k = 0 is “directly-B” and k → ∞ is “all-A”,

- “Ax” = (x, x, x, …) with 0 < x < 1, where the case x → 0 is “directly-B” and x → 1 is “all-A”.

- P(XcLT|p) = P(rc10|p) = p(0) η p(1) η … p(c − 2) η p(c − 1) π
- P(XkLPS|p) = P(rk101|p) = p(0) η p(1) η … p(k − 2) η (1 − p(k − 1))
- P(all-L|p) = P(all-1|p) = 0

#### 3.1.2. Aggregation of Welfare over Time

^{c})/(1 − δ). Thus, with exponential discounting, “rB” > “rc10” iff 1 − δ + δ² > 1 − δ

^{c}or, equivalently, δ

^{c−1}+ δ > 1, i.e., the policy “directly-B” is preferable iff δ is large enough or c is small enough. Since 1/δ can be interpreted as a kind of (fuzzy) evaluation time horizon, this means that “directly-B” will be preferable iff the time horizon is large enough to “see” the expected ultimate transition to state T at time c under the alternative extreme policy “all-A”. At what δ exactly the switch occurs depends on how we take into account the uncertainty about the collapse time c, i.e., how we get from preferences over RSs to preferences over RSLs, which will be discussed later. A variant of the above evaluation v due to Chichilnisky [21] adds to v(r) some multiple of the long-term limit, lim

_{t→}

_{∞}r(t), which is 1 for “rB” and 0 for all “rc10”, thus making “directly-B” preferable also for smaller δ, depending on the weight given to this limit.

^{k+1}+ δ

^{k+2})/(1 − δ), which grows strictly with growing k. Thus, if the collapse time c was known, the best policy among the “Bk” would be the one with k = c − 1, i.e., initiating the transition at the last possible moment right before the collapse, which is evaluated as (1 − δ

^{c}+ δ

^{c+1})/(1 − δ) > (1 − δ

^{c})/(1 − δ), hence, it would be preferred to “all-A”. However, c is, of course, not known, but a random variable, so we need to come back to this question when discussing uncertainty below.

#### 3.1.3. Social Preferences over Uncertain Prospects: Expected Probability of Regret

^{c−1}π to RS “rc10”, and other policies result in RSLs with more complicated probability distributions. e.g., with exponential discounting, rB > rc10 iff δ

^{c−1}+ δ > 1, hence, “gB” > “gA” iff the sum of η

^{c−1}π over all c with δ

^{c−1}+ δ > 1 is larger than ½. If c(δ) is the largest such c, which can be any value between 1 (for δ → 0) and infinity (for δ → 1), that sum is 1 − η

^{c(δ)}, which can be any value between π (for δ → 0) and 1 (for δ → 1). Similarly, with rectangular discounting, “rB” > “rc10” iff H > c + 1, hence, “gB” > “gA” iff 1 − η

^{H−1}> ½. In both cases, if η < ½, “directly-B” is preferred to “all-A”, while for η > ½, it depends on δ or H, respectively. In contrast, under hyperbolic discounting, “directly-B” is always preferred to “all-A”.

^{c})/(1 − δ). If c > k, we get an evaluation of (1 − δ

^{k+1}+ δ

^{k+2})/(1 − δ), which is larger than (1 − δ

^{c})/(1 − δ) iff δ

^{c−k−1}+ δ > 1. Thus, RSL(Bk) > gA iff the sum of η

^{c−1}π over all c > k with δ

^{c−k−1}+ δ > 1 is larger than ½. Since the largest such c is c(δ) + k, that sum is η

^{k}(1 − η

^{c(δ)}), so whenever “Bk” is preferred to “all-A”, then so is “directly-B”. Let us also compare “Bk” to “directly-B”. In all cases, “directly-B” gets (1 − δ + δ

^{2})/(1 − δ), while “Bk” gets the larger (1 − δ

^{k+1}+ δ

^{k+2})/(1 − δ) if c > k, but only (1 − δ

^{c})/(1 − δ) if c ≤ k. The latter is < (1 − δ + δ

^{2})/(1 − δ) iff c ≤ c(δ). Thus, “directly-B” is strictly preferred to “Bk” iff 1 − η

^{min(c(δ),k)}> ½, i.e., iff both c(δ) and k are larger than log(½)/log(η), which is at least fulfilled when η < ½. Conversely, “Bk” is strictly preferred to “directly-B” iff either c(δ) or k is smaller than log(½)/log(η). In particular, if social preferences were based on the expected probability of regret, delaying the choice for B by at least one generation would be strictly preferred to choosing B directly whenever η > ½, while at the same time, delaying it forever would be considered strictly worse at least if the time horizon is long enough. Basing decisions on this maxim would, thus, lead to time-inconsistent choices: in every generation, it would seem optimal to delay the choice B by the same positive number of generations, but not forever, so no generation would actually make that choice.

^{½}. A similar result holds for the geometric distribution with parameter δ. Thus, while the probability of regret idea can lead to time-inconsistent choices, the formally similar veil of ignorance idea may not be able to differentiate enough between choices. Another problematic property of our veil of ignorance-based preferences is that they can lead to preference cycles. e.g., assume H = 3 and compare the RSs r = (0, 1, 2), r’ = (2, 0, 1), and r’’ = (1, 2, 0). Then it would occur that r > r’ > r’’ > r, so there would be no optimal choice among the three.

#### 3.1.4. Evaluation of Uncertain Prospects: Prospect Theory and Expected Utility Theory

_{r}w(P(r|g)) f(v(r)).

_{r}P(r|g) v(r) = E

_{g}v(r), the expected evaluation of the RSs resulting from RSL g. If combined with a v(r) based on exponential discounting, this gives the following evaluations of our polar policies:

_{all-A}v(rc10) = ∑

_{c>0}η

^{c−1}π(1 − δ

^{c})/(1 − δ) = 1/(1 − δη).

_{crit}(η) with δ

_{crit}(0) = 0 and δ

_{crit}(1) = 1. The result for rectangular discounting is similar, while for hyperbolic discounting “directly-B” is always preferred to “all-A”, and all of this as expected from the considerations above.

^{1−a}with 0 < a < 1 (isoelastic case) or f(v) = −exp(−av) with a > 0 (constant absolute risk aversion) (welfare economists might be confused a little by our discussion of risk aversion since they are typically applying the concept in the context of consumption, income or wealth of individuals at certain points in time, in which context one can account for risk aversion already in the specification of individual consumers’ utility function, e.g., by making utility a concave function of individual consumption, income, or wealth. Here we are, however, interested in a different aspect of risk aversion, where we want to compare uncertain streams of societal welfare rather than uncertain consumption bundles of individuals. Thus, even if our assessment of the welfare of each specific generation in each specific realization of the uncertainty about the collapse time c already accounts for risk aversion in individual consumers in that generation, we still need to incorporate the possible additional risk aversion in the “ethical social planner”). This basically leads to a preference for small variance in v. One can see numerically that in both cases increasing the degree of risk aversion, a, lowers δ

_{crit}(η), not significantly so in the isoelastic case but significantly in the constant absolute risk aversion case, hence, risk aversion favors “directly-B”. In particular, the constant absolute risk aversion case with a → ∞ is equivalent to a “worst-case” analysis that always favors “directly-B”. Conversely, one can encode risk-seeking by using f(v) = v

^{1+a}with a > 0.

^{k}(1 − δ

^{k+1}+ δ

^{k+2}) + ∑

_{c=1…k}η

^{c−1}π(1 − δ

^{c}),

^{1+a}, however, we have:

^{k}(1 − δ

^{k+1}+ δ

^{k+2})

^{1+a}+ ∑

_{c=1…k}η

^{c−1}π(1 − δ

^{c})

^{1+a},

^{1−b}with 0 ≤ b < 1, then increasing the degree of optimism b, one can move δ

_{crit}(η) arbitrarily close towards 1, which is not surprising. We will however not discuss this form of probability reweighting further but will use a different way of representing “caution” below. Since that form is motivated by its formal similarity to a certain form of inequality aversion, we will discuss the latter first now before returning to risk attitudes.

#### 3.1.5. Inequality Aversion: A Gini-Sen Intergenerational Welfare Function

_{2}(r) = (∑

_{t=0…H−1}∑

_{t’=0…H−1}min[r(t), r(t’)])/H².

_{a}(r) that get more and more inequality averse as a is increased from 1 (no inequality aversion, “utilitarian” case) to infinity (complete inequality aversion), where the limit for a → ∞ is the egalitarian welfare function:

_{1}(r) = [r(0) + … + r(H − 1)]/H

_{a}(r) = (∑

_{t1=0…H−1}… ∑

_{ta=0…H−1}min[r(t

_{1}), …, r(t

_{a})])/H

^{a}

_{∞}(r) = min[r(0), …, r(H − 1)]

_{2}(r)/V

_{1}(r) is the Gini index of inequality and the formula V

_{2}(r) = V

_{1}(r) (1 − I) is often used as the definition of the Gini-Sen welfare function.

_{a}(rc10) = min(c/H, 1)

^{a}, while “rk101” gets V

_{a}(rk101) = [(H − 1)/H]

^{a}if k < H and V

_{a}(rk101) = 1 if k ≥ H. Together with expected utility theory for evaluating the risk about c, this makes:

_{a}(all-A) = η

^{H}+ ∑

_{c=1…H}η

^{c−1}π(c/H)

^{a}

_{a}(directly-B) = [(H − 1)/H]

^{a}. Numerical evaluation shows that even for large H, “all-A” may still be preferred due to the possibility that collapse will not happen before H and all generations will have the same welfare, but this is only the case for extremely large values of a. If we use exponential instead of rectangular discounting and compare the policies “directly-B”, “Bk”, and “all-A”, we may again get a time-inconsistent recommendation to choose B after a finite number of generations. e.g., Figure 2a shows V(Bk) vs. k for the case η = 0.985, δ = 0.9, a = 2, where the optimal delay would appear to be five generations. If we restrict our optimization to the time-consistent policies “Ax”, the optimal x in that case would be ≈0.83, i.e., each generation would choose A with about 83% probability and B with about 17% probability, as shown in Figure 2b. Still, note that the absolute evaluations vary only slightly in this example.

#### 3.1.6. Caution: Gini-Sen Applied to Alternative Realizations

_{1}, …, t

_{a}at random, we draw a ≥ 1 many realizations r

_{1}, …, r

_{a}of an RSL g at random and use the expected minimum of all the RS-evaluations V(r

_{i}) as a “cautious” evaluation of the RSL g?

_{a}(g) = ∑

_{r1}… ∑

_{ra}g(r

_{1}) × … × g(r

_{a}) × min[v(r

_{1}), …, v(r

_{a})].

_{a}(g) = ∫

_{x≥0}P

_{g}(v(r) ≥ x)

^{a}dx,

_{g}(v(r) ≥ x) is the probability that v(r) ≥ x if r is a realization of g. In that form, a can be any real number ≥ 1 and it turns out that the evaluation is a special case of cumulative prospect theory [23], with the cumulative probability weighting function w(p) = p

^{a}. Focusing on “all-A” vs. “directly-B” again, we get V

_{a}(all-A) = (1 − η

^{aH})/(1 − η

^{a})H and V

_{a}(directly-B) = (H − 1)/H, hence, “all-A” is preferred iff (1 − η

^{aH})/(H − 1) > 1 − η

^{a}, i.e., iff H and a are small enough and η is small enough. In particular, regardless of H and η, for a → ∞ we always get a preference for “directly-B” as in the constant absolute risk aversion. This is because with the Gini-Sen-inspired specification of caution, the degree of risk aversion effectively acts as an exponent to the survival probability η, i.e., increasing risk aversion has the same effect as increasing collapse probability, which is an intuitively appealing property.

#### 3.1.7. Fairness as Inequality Aversion on Uncertain Prospects

_{a}(g) = (∑

_{t1=0…H−1}… ∑

_{ta=0…H−1}min[V(g, t

_{1}), …, V(g, t

_{a})])/H

^{a},

_{a}(g) is the expected minimum of how two randomly drawn generations within the time horizon evaluate their uncertain rewards under g. Using exponential discounting instead, the formula becomes:

_{a}(g) = (1 − δ)

^{a}∑

_{t1=0…H−}

_{1}… ∑

_{ta=0…H−}

_{1}δ

^{t1+...+ta}min[V(g, t

_{1}), …, V(g, t

_{a})].

_{a}(g), the result looks similar to Figure 2b, i.e., the optimal time-consistent policy is again non-deterministic. A full optimization of V

_{a}(g) over the space of all possible probabilistic policies shows that the overall optimal policy regarding V

_{a}(g) is not much different from the time-consistent one, it prescribes choosing A with probabilities between 79% and 100% in different generations for the setting of Figure 2.

#### 3.1.8. Combining Inequality and Risk Aversion with Fairness

^{1}, …, g

^{4}listed in Table 1 in a way that makes V(g

^{1}) > V(g

^{2}) because the latter is more risky, V(g

^{2}) > V(g

^{3}) because the latter has more inequality, and V(g

^{3}) > V(g

^{4}) because the latter is less fair. Then we can achieve this by applying the Gini-Sen technique several times to define welfare functions V

^{0}… V

^{6}that represent more and more of our aspects as follows:

- Simple averaging: V
^{0}(g) = E_{r}E_{t}r(t) where E_{r}f(r) is the expectation of f(r) w.r.t. the lottery g and E_{t}f(t) is the expectation of f(t) w.r.t. some chosen discounting weights; - Gini-Sen welfare of degree a = 3: V
^{1}(g) = E_{r}E_{t1}E_{t2}E_{t3}min{r(t_{1}), r(t_{2}), r(t_{3})}; - Overall risk-averse welfare: V
^{2}(g) = E_{r1}E_{r2}min{E_{t}r_{1}(t), E_{t}r_{2}(t)}; - Fairness-seeking welfare of degree a = 3: V
^{3}(g) = E_{t1}E_{t2}E_{t3}min{E_{r}r(t_{1}), E_{r}r(t_{2}), E_{r}r(t_{3})}; - Inequality- and overall risk-averse welf.: V
^{4}(g) = E_{r1}E_{r2}min{v^{4}(r_{1}), v^{4}(r_{2})} with v^{4}(r) = E_{t1}E_{t2}E_{t3}min{r(t_{1}), r(t_{2}), r(t_{3})}; - Inequality and overall risk index: I
^{4}(g) = 1 − V^{4}(g)/V^{0}(g); - Generational risk averse and fair welfare: V
^{5}(g) = E_{t1}E_{t2}E_{t3}min{V^{5}(g, t_{1}), V^{5}(g, t_{2}), V^{5}(g, t_{3})} with V^{5}(g, t) = E_{r1}E_{r2}min{r_{1}(t), r_{2}(t)}; - Generational risk and fairness index: I
^{5}(g) = 1 − V^{5}(g)/V^{0}(g); and - All effects combined: V
^{6}(g) = V^{4}(g)V^{5}(g)/V^{0}(g) = V^{0}(g)[1 − I^{4}(g)][1 − I^{5}(g)]

^{1}… g

^{4}can be seen in Table 1. We chose a higher degree of inequality-aversion (a = 3) than the degree of risk-aversion (a = 2) so that V

^{6}(g

^{2}) > V

^{6}(g

^{3}) as desired. Applied to our thought experiment, V

^{6}can result in properly probabilistic and time-inconsistent policy recommendations, as shown in Figure 3 for two example choices of η and discounting schemes. An alternative way of combining inequality and risk aversion into one welfare function would be to use the concept of recursive utility [25], which is, however, beyond the scope of this article.

#### 3.2. Game-Theoretical Framework

_{t’}and the set of generations t > t’ by G

_{>t’}and focus on generation t’ = 0 at first. Let us assume that V = V

^{4}, V

^{5}, or V

^{6}with exponential discounting encodes their social preferences over RSLs. Given G

_{0′}s beliefs about G

_{>0′}s behavior, q(0, t) for all t > 0, we then need to find that x in [0, 1] which maximizes V(RSL(p

_{x,q})), where p

_{x,q}is the resulting policy p

_{x,q}= (x, q(0, 1), q(0, 2), …). If G

_{0}believes G

_{1}will choose B for sure (i.e., q = (0, …) = “directly-B”) and chooses strategy x, the resulting RSL(p

_{x,q}) produces the reward sequence r

_{1}= (1, 0, 0, …) with probability xπ, r

_{2}= (1, 0, 1, 1, …) with probability 1 − x, and r

_{3}= (1, 1, 0, 1, 1, …) with probability xη. Hence:

^{4}(RSL(p

_{x,q})) =

x² [(1 − (1 − δ)δ²)³ η² − (1 − δ)³η² + 2(1 − δ)³η − 2(1 − (1 − δ)δ)³η − (1 − δ) + (1 − (1 − δ)δ)³]

+ 2x (−(1 − δ)³η + (1 − (1 − δ)δ)³η + (1 − δ)³ − (1 − (1 − δ)δ)³] + (1 − (1 − δ)δ)³.

^{4}is maximal for either x = 0, where it is (1 − δ + δ²)³, or for x = 1, where it is (δ³ − δ² + 1)³η² + (δ − 1)³(η² − 1), which is always smaller, so w.r.t. V

^{4}, x = 0 (choosing B for sure) is optimal under the above beliefs. For V

^{5}, we have V

^{5}(RSL(p

_{x,q}), t) = 1 for t = 0, (xη)² for t = 1, (1 − x)² for t = 2, and (1 − xπ)² for t > 2. If x < 1/(1 + η), we have (xη)² < (1 − x)² < (1 − xπ)² < 1, while for x > 1/(1 + η), we have (1 − x)² < (xη)² < (1 − xπ)² < 1. For x ≤ 1/(1 + η), V

^{5}(RSL(p

_{x,q})) is again quadratic in x with a positive x² coefficient with value 1 + (1 − δ)³ − (1 − δ²)³ at x = 0 and, again, a smaller value at x = 1/(1 + η). Additionally, for x ≥ 1/(1 + η), V

^{5}(RSL(p

_{x,q})) is quadratic in x with positive x² coefficient and a value of:

^{6}, as shown in Figure 4, blue line, for the case η = 0.95 and δ = 0.805, where G

_{0}will choose A if they believe G

_{1}will choose B, resulting in an evaluation V

^{6}≈ 0.43. The orange line in the same plot shows V

^{6}(RSL(p

_{x,q})) for the case in which G

_{0}believes that G

_{1}will choose A and G

_{2}will choose B if they are still in L, which corresponds to the beliefs q = (1, 0, …). Interestingly, in that case, it is optimal for G

_{0}to choose A, resulting in an evaluation V

^{6}≈ 0.42. Since the dynamics and rewards do not explicitly depend on time, the same logic applies to all later generations, i.e., for that setting of η and δ and any t ≥ 0, it is optimal for G

_{t}to choose A when they believe G

_{t+1}will choose B and optimal to choose B when they believe G

_{t+1}will choose A and G

_{t+2}will choose B.

^{6}and believe that all generations G

_{t}with even t will choose A and all generations G

_{t}with odd t will choose B. Then it is optimal for all generations to do just that. In other words, these assumed common beliefs form a strategic equilibrium (more precisely, a subgame-perfect Nash equilibrium) for that setting of η and δ. However, under the very same set of parameters and preferences, the alternative common belief that all even generations will choose B and all odd ones A also forms such an equilibrium. Another equilibrium consists of believing that all generations choose A with probability ≈83.7% which all generations evaluate as only V

^{6}≈ 0.40, which is less than in the other two equilibria. The existence of more than one strategic equilibrium is usually taken as an indication that the actual behavior is very difficult to predict even when assuming complete rationality. In our case, this means G

_{0}cannot plausibly defend any particular belief about G

_{>0}′s policy on the grounds of G

_{>0′}s rationality since G

_{>0}might follow at least either of the three identified equilibria (or still others). In other words, for many values of η and δ a game-theoretic analysis based on subgame-perfect Nash equilibrium might not help G

_{0}in deciding between A and B. A common way around this is to consider “stronger” forms of equilibrium to reduce the number of plausible beliefs, but this complex approach is beyond the scope of this article. An alternative and actually older approach [26] is to use a different basic equilibrium concept than Nash equilibrium, not assuming players have beliefs about other players policies encoded as subjective probabilities, but rather assuming players apply a worst-case analysis. In that analysis, each player would maximize the minimum evaluation that could result from any policy of the others. For choosing B, this evaluation is simply v(1, 0, 1, 1, …), while, for choosing A, the evaluation can become quite complex. Instead of following this line here, we will use a similar idea when discussing the concept of responsibility in the next section, where we will discuss other criteria than rationality and social preferences.

## 4. Solutions Based on Other Ethical Principles and Sustainability Paradigms

#### 4.1. Responsibility

_{0}is the absolute difference between the probability of low welfare in generation 1 when choosing A rather than B, which equals η. In other words, G

_{0}would have a degree of η responsibility to choose A in order to avoid the EUO that G

_{1}gets low welfare. If they choose B instead, they will have a degree of factual backward-looking responsibility for G

_{1}′s low welfare equaling again η since this is the amount by which they could have reduced the probability of the EUO. If they behave “responsibly” by choosing A, G

_{1}′s welfare might also be low (with probability π), but G

_{0}would still not have backward-looking responsibility since they could not have reduced that probability.

_{2}rather than G

_{1}, the assessment of G

_{0}′s responsibility must consider the possible actions of G

_{1}in addition to those of G

_{0}. If G

_{0}chooses B, the probability of the EUO is zero, while if they choose A, it depends on G

_{1}′s choice. If G

_{1}would choose B, the EUO has probability 1 so that G

_{0}′s choice would make a difference of 1, while if G

_{1}would choose A, the EUO has probability 1 − η² < 1 and G

_{0}′s choice would make a difference of only 1 − η² < 1. In both cases, however, they have considerable forward-looking responsibility to choose B since by that they can reduce the probability of the EUO significantly. If choosing B, no backward-looking respectively accrues. If G

_{0}and G

_{1}both choose A and the collapse occurs at time 2, G

_{1}has no factual responsibility since they could not have reduced that probability, but G

_{0}has factual responsibility of degree 1 − η². If G

_{0}chooses A and G

_{1}B, G

_{1}′s factual responsibility is η as seen above, but G

_{0}′s is even larger, since in view of G

_{1}′s actual choice, G

_{0}could have reduced the probability from one to zero by choosing B instead. Thus, G

_{0}has factual responsibility of 1. It might seem counterintuitive at first that the sum of the factual responsibilities of the two agents regarding that single outcome would be larger than 100%, but our theory is actually explicitly designed to produce this result in order to show that responsibility cannot simply be divided. Otherwise, each individual in a large group of bystanders at a fight in public could claim to have almost no responsibility to intervene (diffusion of responsibility). Finally, if both G

_{0}and G

_{1}choose B and no collapse happens, G

_{0}still has counterfactual responsibility since the collapse could have happened and G

_{0}could have reduced that probability by 1 − η². This distinction between factual and counterfactual responsibility would also allow a discussion of Nagel’s concept of moral luck in consequences [27] and responses to it, such as [28] but we will not go there here.

_{3}, it becomes more complicated. By choosing B, G

_{0}can avoid the EUO for sure, but when choosing A they might hope G

_{1}will choose B and the EUO will be avoided for sure as well, in which case they might claim to have a rather low responsibility to choose B which amounts only to π, the probability that G

_{1}will have no chance of choosing B due to immediate collapse. Common sense, however, shows that while wishful thinking regarding the actions of others might affect one’s own psychological assessment of responsibilities, it cannot be the basis for an ethical observer’s assessment of responsibility. Otherwise, even in a group of just two bystanders, neither one would be ethically obliged to intervene since both could hope the other does. Here we even take the opposite view and argue that G

_{0}′s degree of forward-looking responsibility should equal the largest possible amount by which they might be able to reduce the probability of the EUO, maximized over all possible behaviors of the other agents. This means that rather than being optimistic about G

_{1}′s action, they need to be pessimistic about both G

_{1}′s and G

_{2}′s behavior. The worst that can happen regarding the welfare of G

_{3}when G

_{0}chooses A is that G

_{1}would choose A and G

_{2}, B. In that case, the EUO has probability 1, so G

_{0}would still be fully responsible (degree 1) to choose B in order to avoid the EUO.

_{0}responsible for G

_{1}′s suffering.

#### 4.2. Safe Operating Space for Humanity

#### 4.3. Sustainability Paradigms à la Schellnhuber

#### 4.3.1. Standardization

#### 4.3.2. Optimization

#### 4.3.3. Pessimization

#### 4.3.4. Equitization

#### 4.3.5. Stabilization

## 5. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Formal version of the thought experiment. A generation in the good state “L” can choose path “B”, surely leading to the good state “S” via the bad state “P” within two generations, or path “A”, probably keeping them in “L”, but possibly leading to the bad state “T”.

**Figure 2.**Inequality-averse evaluation of deterministic but delayed policies (

**a**) and time-consistent but probabilistic policies (

**b**) for the case η = 0.985, δ = 0.9 and a = 2.

**Figure 3.**(

**Left**) Evaluation V

^{6}for the case of η = 0.68, rectangular discounting with very short horizon 3 and choosing A for sure in generation 1, by probability of chosing A in generation 0, showing an optimal probability of approximately 82%. (

**Right**) Optimal policy for the first 20 generations according to V

^{6}for the case of η = 0.97 and exponential discounting with δ = 0.9.

**Figure 4.**Evaluation V

^{6}for the case of η = 0.95 and δ = 0.805 depending on the first generation’s probability of choosing A (horizontal axis), for the case where they expect the next generations to choose B (blue line) or to choose A and then B (orange line).

**Table 1.**Comparison of the effects of inequality aversion, overall and generational risk aversion, and fairness on the evaluation of four simple reward sequence lotteries (RSLs). All effects are implemented in the Gini-Sen style (see main text for details), inequality aversion with a larger degree of a = 3, risk aversion and fairness with a lower degree of a = 2, which is reflected in the preference for the coin toss between the “no-inequality” reward sequences (0, 0) and (1, 1) over the coin toss between the “equal average” reward sequences (0, 1) and (1, 0).

RSL | V^{0}: No Effects | V^{1}: Only Inequality Aversion | V^{2}: Only Overall Risk Aversion | V^{3}: Only Fair-ness | V^{4}: Inequality and Overall Risk Aversion | V^{5}: Generational Risk Aversion and Fairness | V^{6}: All Effects Combined |
---|---|---|---|---|---|---|---|

g^{1}: (0.5, 0.5) for sure | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

g^{2}: coin toss between (0, 0) and (1, 1) | 0.5 | 0.5 | 0.25 | 0.5 | 0.25 | 0.25 | 0.125 |

g^{3}: coin toss between (0, 1) and (1, 0) | 0.5 | 0.25 | 0.5 | 0.5 | 0.125 | 0.25 | 0.0625 |

g^{4}: (0, 1) for sure | 0.5 | 0.25 | 0.5 | 0.25 | 0.125 | 0.125 | 0.03125 |

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Heitzig, J.; Barfuss, W.; Donges, J.F.
A Thought Experiment on Sustainable Management of the Earth System. *Sustainability* **2018**, *10*, 1947.
https://doi.org/10.3390/su10061947

**AMA Style**

Heitzig J, Barfuss W, Donges JF.
A Thought Experiment on Sustainable Management of the Earth System. *Sustainability*. 2018; 10(6):1947.
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**Chicago/Turabian Style**

Heitzig, Jobst, Wolfram Barfuss, and Jonathan F. Donges.
2018. "A Thought Experiment on Sustainable Management of the Earth System" *Sustainability* 10, no. 6: 1947.
https://doi.org/10.3390/su10061947