Numerous economic scenarios involve correlated risks or uncertainties. An insurance company, who insures two houses against fire, apparently faces a positive correlation among the events that a claim will be made if they lie in the same neighborhood. But what if the houses lie in different countries or if the company offers an insurance against flood damages? At second glance, even those liability cases can be correlated by an omitted effect, e.g., climate change. Similarly, a policy maker, needing to decide how to mitigate and adapt to climate change, faces correlated uncertainties, since the likelihoods of climate-change related catastrophes are positively correlated. A climate scientist may have a good understanding of individual aspects of the earth climate (e.g., temperature, glaciers, oceans), but may face uncertainty regarding the correlation of these aspects. How should the insurer price the insurance cover? How should the policy maker decide how much to invest in climate mitigation and adaptation? How should the climate scientist estimate the overall expected climate damage in light of the lack of probabilistic information regarding correlations among the individual aspects? In this paper, we propose a theoretical framework to tackle such choice problems. We assume that the set of possible states of the world Ω follow a product structure, i.e., Ω " Ω 1 ˆ¨¨¨ˆΩ n . The subspaces Ω i contain the set of states that are relevant to an aspect i P t1, . . . , nu of the choice problem. In our benchmark case, the decision maker (DM) tries to arrive at a belief on the whole state space from probabilistic information about each subspace Ω i in a statistically consistent way. We demonstrate that this consistency requirement leaves ample space for correlations among the subspaces. We identify this space, denoted by P, derive its dimension as well as its extreme points, and show that it is the core of a non-convex, but exact capacity. This clarifies the scope of correlations among subspaces in full generality, paving the way to the analysis of what can be called "aversion towards correlation uncertainty". That is, we allow for uncertainty regarding the correlations between the subspaces and aversion towards this uncertainty. Furthermore, we provide an axiomatic analysis of independence among subspaces. That is, we answer axiomatically the question of when groups of subspaces are independent of each other, a case that is a crucial base case for numerous economic applications, e.g., the insurance sector. Our model and results aim to aid concrete decision-making in scenarios where the potential outcomes of given actions depend on multiple subspaces, such as the scenarios sketched above. For concreteness, we demonstrate the usefulness of our framework and the results in this paper through the following three economically relevant applications. At the end of each example, we refer to the sections relevant to the respective topics and phenomena discussed in the example. Example 1 (climate uncertainties). The question of how to react to a chang-ing climate is a key challenge to society. The Intergovernmental Panel on Climate Change (IPCC) aims to scientifically clarify the uncertainties that society faces on different global warming pathways and evaluates possible mitigation and adaptation as well as climate engineering options. The "classic" mitigation strategy is to reduce carbon emissions or to remove it from the atmosphere. More recently, in light of the severity of the problem, alternative climate engineering options such as solar radiation modification (SRM) have received much attention, in order to artificially increase the amount of sunlight that is being reflected back into space. The most-studied idea is to increase the albedo, i.e., the fraction of sunlight reflected back into space, through stratospheric aerosol injection (SAI) (Masson-Delmotte et al., 2022): aerosols are dispersed into the atmosphere to reflect solar radiation to get a cooling effect, an imitation of volcanic eruptions. Other examples for SRM are marine cloud brightening, i.e. spraying sea water into the air to increase cloud reflection, and surface-based albedo modification. Relative to mitigation, SRM strategies are estimated to be low cost Bala et al. (2023). They do however introduce new uncertainties into the choice problem regarding the impact on the environment, ecosystems and climate as well as ethics and social acceptability Harrison et al. (2021). 1 These "new" uncertainties are potentially correlated with the "old" uncertainties that were relevant already before the inclusion of SRM. Lack of scientific understanding of this correlation, due to the relative novelty of SRM, leads to what we refer to as correlation uncertainty. Its presence makes finding the optimal response to a changing climate even more challenging. Our framework provides a theoretical model in which decision problems involving correlation uncertainty can be represented and analyzed, as demonstrated more concretely in what follows. At present-day there is overwhelming evidence that human-caused greenhouse gas emissions cause global warming, threatening severe disruptions of the environment, ecosystems, economies, and societies. Mitigation strategies aim to reduce the amount of greenhouse gases in the atmosphere. Their effectiveness crucially depend on the parameter climate sensitivity, a key measure in climate science, cf. Weitzman (2009). It measures how much the temperature of earth increases when the concentration of atmospheric greenhouse gas doubles compared to the level of the pre-industrial age. Estimating it requires understanding of several feedback effects due to interactions of atmospheric temperature with clouds, ice-albedo, and water vapor. Let the state space Ω 1 represent the uncertainty regarding climate sensitivity, i.e., the relevant uncertainty for the evaluation of mitigation strategies. The consideration of climate engineering as (part of the) strategy against climate change introduces new uncertainties that were not relevant before this inclusion. 2 We model these "new" uncertainties introduced through climate engineering via the state space Ω 2 . The overall state space, modeling the uncertainty relevant to mitigation as well as climate engineering, is thus Ω " Ω 1 ˆΩ2 . The subspaces Ω 1 and Ω 2 cannot be treated as being independent of each other. For instance, climate sensitivity interacts with the ozone layer (Gómez-Leal et al., 2019).
Thus SAI, which also impacts the ozone layer, may trigger feedback effects that were not present before its implementation. Certain SRM methods influence precipitation patterns as well as the albedo (Masson-Delmotte et al., 2021), which in turn are connected with climate sensitivity. 3 Thus, the interaction of atmospheric SRM methods with the climate system of our planet demonstrate that the relevant states of the subspaces Ω 1 and Ω 2 are potentially correlated. Knowledge regarding such potential correlations, and the lack thereof, is crucial for deciding on the optimal response against climate change, as demonstrated in the following stylized example. Consider a policy maker, searching for the most effective and efficient response to climate change. It faces the challenging question of how much to invest in CO2 reduction (mitigation) and how much into SRI (climate engineering), being aware that the uncertainties relevant for these strategies are correlated through feedback effects. The policy maker considers three strategies: "Business as usual", in which no measures against climate change are undertaken, "mitigation" in which atmospheric CO2 is reduced by a certain amount, and "climate engineering", in which the SAI method is used to artificially cool the temperature. Consider Ω 1 " tHcs, Lcsu, where Hcs{Lcs stands for high/low climate sensitivity and Ω 2 " tHa, Lau, where Ha{La stands for high/low robustness of the atmosphere when interacting with aerosols. The overall state space is thus given by Ω " tpHcs, Haq, pHcs, Laq, pLcs, Haq, pLcs, Laqu. The policy maker has probabilistic information p 1 P ∆pΩ 1 q and p 2 P ∆pΩ 2 q regarding the states in Ω 1 and Ω 2 , but lacks information about the correlation of these two subspaces, i.e., it faces uncertainty as to how exactly climate sensitivity is correlated with the robustness of the atmosphere after aerosol injection. 4 This correlation uncertainty is reflected by a prior set C Ď PpΩ; p 1 , p 2 q, where PpΩ; p 1 , p 2 q is the set of probability distributions over Ω which have marginals p 1 and p 2 . Note that the independent product p 1 b p 2 , reflecting no correlation among Ω 1 and Ω 2 , is an element of PpΩ; p 1 , p 2 q, but there are many more. The costless "business as usual" option leads to further global warming, resultacts with the ozone layer and how the dispersed aerosols manipulate clouds Storelvmo et al. (2013). Surface-based albedo modification aims to alter the albedo of the earth's surface in order to reflect more sunlight. Large-scale application could cause changes in precipitation patterns Masson-Delmotte et al. (2021). There are also societal issues. The successful use of SRM may reduce the ambition for climate mitigation, a moral hazard problem Hansen et al. (2025). Furthermore, the rapid effect of SRM would quickly fade out if not sustained Masson-Delmotte et al. (2021). 3 For instance, a higher temperature reduces the ice area which decreases the albedo. 4 The interpretation of this assumption is that science has provided a good understanding of the climate sensitivity as well as the robustness of the atmosphere. However, the feedback effects among these systems are considerably more complex and lacks scientific rigor, leading to uncertainties regarding the correlation. ing in an economic damage of ´D ! 0 if climate sensitivity is high, i.e., Hcs is true. It leads to a normalized damage 0 if Lcs is true. In the "mitigation" strategy, the amount c ą 0 is invested to absorb the negative climate change effects to some extent when Hcs occurs, resulting in damage dpcq, where d : R Ñ R is decreasing with dp0q " D.5 For the third strategy, "climate engineering", Ω 2 becomes relevant. It costs c 1 ą 0 and solves the climate change problem when pHcs, Laq occurs. However, due to feedback effects there are devastating ecological consequences when Hcs and Ha jointly occur, resulting in the additional loss x ą 0. The policy maker evaluates each strategy at the worst expected payout over the prior set C, which reflects aversion towards the uncertainty regarding the correlation of Ω 1 and Ω 2 . Note that both the business as usual and the mitigation strategy are independent of the states in Ω 2 . They are thus unaffected by any correlation among the subspaces Ω 1 and Ω 2 . Uncertainty regarding the correlation thus plays no role.6 However, the evaluation of climate engineering does depend crucially on the set C. The expected utilities are given as

V pBusiness as usualq " ´p1 pHcsqD

V pMitigationq " ´c ´p1 pHcsqdpcq

V pClimate engineeringq " ´c1 ´x min pPC ppHcs, Haq. Note that, whenever x `c1 ă D, the climate engineering strategy can be seen as a hedge against climate change, so you would expect aversion towards unknown probabilities to make this option attractive. However, it is clear from the above formula for V pClimate engineeringq that uncertainty regarding the correlation has the opposite effect: more uncertainty leads to climate engineering being less attractive. Indeed, consider a larger prior set C 1 Ą C, reflecting strictly more uncertainty regarding the correlation compared to C. Note that min pPC 1 ppHcs, Haq ď min pPC ppHcs, Haq, implying that in general a larger prior set makes the "safer" option climate engineering weakly less attractive relative to the other strategies. At first sight this may be puzzling, however it becomes clear once recalling that the other two strategies are unaffected by an increase in correlation uncertainty. In this simple example, the lack of knowledge regarding the correlations of the states in Ω 1 and Ω 2 , i.e., the presence of correlation uncertainty, leads to a reluctance in implementing "new" climate engineering strategies. It models the difficulty to commit to strategies which introduce new uncertainties: if correlations between "new" and "old" states are unknown, aversion towards this uncertainty will downgrade their appeal.
The larger this uncertainty regarding correlations is, the less attractive these strategies become. Our results in Section 3 lead the way to modeling aversion towards correlation uncertainty by pinning down the maximal extent of correlation uncertainty PpΩ; p 1 , . . . , p n q over the state space Ω in full generality, hereby clarifying the scope of possible correlations among the subspaces. Section 4.1 clarifies axiomatically when probabilistic information on the subspaces is consistently aggregated, see Proposition 3. Theorem 2 axiomatizes the important benchmark case where behavior is consistent with the subspaces being completely independent, i.e., the probabilistic assessment of the uncertainties on Ω is represented by the independent product p 1 b . . . b p n . In Section 4.4, we provide a decision-theoretic characterization of correlation aversion. We introduce a logical comparative notion of "more correlation averse", allowing tracking down the above described behavioral phenomena. Example 2 (insurance). This example demonstrates that, in an insurance market, asymmetry in beliefs regarding the correlation of events can lead to undesired consequences such as unfair prices and market failure, even if the marginal beliefs regarding the payoff-relevant events coincide for all agents. Consider a company that offers house insurance (the "insurer") as well as a house owner who considers buying such an insurance for her house near the coast (the "insuree"). For simplicity, we assume that both are risk-neutral. The insuree faces the risk of the house being destroyed through fire as well as being destroyed due to a storm tide flooding. The payoff relevant states for these two events of damage are modeled as follows. Let Ω 1 " pBpurnq, N poqBpurnqq model the uncertainty associated with the house being burnt down, Ω 2 :" tF ploodq, N poqF ploodqu models the uncertainty associated with the house being destroyed by flooding. The overall state space is Ω " Ω 1 ˆΩ2 . If either B or F happens, the house is completely destroyed. The house is worth v ą 0. Without any insurance, the payoff to the insuree is ´v in case either F or B happens and 0 if neither happens. The insurer offers full insurance against fire at cost c ą 0. If the state pB, F q is realized, i.e. both events of damage occur, the insurer pays fraction x P p0, 1q of the damage.7

The insuree faces the problem of whether to buy insurance or not. The payoffs for both cases are illustrated in Table 2 for the insurer and in Table 3 the insuree. We assume that both agents have probabilistic beliefs over Ω. The insurer has belief p P ∆pΩq, the insuree has belief p P ∆pΩq. Let p i " p Ωi and pi " p Ωi be the marginals on Ω i , i P t1, 2u. We assume that these marginals coincide,

Insurance against B i.e. p 1 " p1 and p 2 " p2 . This reflects that agents agree on the individual likelihoods of the states in Ω 1 and Ω 2 . In particular, insurer and insuree agree on the likelihood of F as well as B. However, we do allow for different beliefs regarding the correlation of these events, i.e., p ‰ p is not ruled out. Let V p and V p reflect the expected returns of the insurer and insuree, respectively, given their beliefs p and p. For the insurer we have V p pNo insuranceq " 0 V p pInsurance against Bq " c ´vpxp 1 pBq `p1 ´xqppB, N F qq. For the insuree we have V ppNo insuranceq " ´vp1 ´ppN B, N F qq V ppInsurance against Bq " ´c ´vpp1 ´xqp 2 pF q `xppN B, F qq. The minimum price at which the insurer is willing to offer insurance is c " vpxp 1 pBq `p1 ´xqppB, N F qq. The maximum amount that the insuree is willing to pay for the insurance is c " vpxp 1 pBq `p1 ´xqppB, N F qq. This shows that when p " p, these reservation prices coincide. Thus agreeing on the correlation necessitates the existence of a unique price c ˚" vpxp 1 pBq `p1 ´xqppB, N F qq. Note that if these beliefs are in addition correct, the insurer makes zero expected profit, i.e., the contract is fair. We now allow for p ‰ p. We assume that p, the belief of the insurer, correctly reflects the correlation structure of the relevant events. 8 For any price c in the interval rvpxp 1 pBq `p1 ´xqppB, N F qq, vpxp 1 pBq `p1 ´xqppB, N F qqs, both the insurer and the insuree are willing to sign the contract. Thus, recalling p 1 pBq " p1 pBq, such a price exists whenever ppB, N F q ď ppB, N F q, which in turn is equivalent to ppB, F q ě ppB, F q. This happens whenever the insuree underestimates the correlation of F and B. He assigns a too low probability to the "good state" pB, N F q in which full compensation for the damage due to the event B is paid. The insurer can thus charge the price c " vpxp 1 pBq`ppB, N F qq, resulting in a positive expected profit of vp1´xqpppB, N F q´ppB, N F qq. Assume now that the interval rvpxp 1 pBq`p1´xqppB, N F qq, vpxp 1 pBq`p1´xqppB, N F qqs is empty, i.e., ppB, F q ă ppB, F q. The insuree overestimates the correlation of B and F . This incorrect belief thus leads to a market failure as no contract will be signed.9 Note that a particularly interesting scenario is where the insuree ignores the possible correlation of B and F altogether. That is, his prior p coincides with the independent product p1 b p2 . According to the above analysis, a positive correlation of B and F , i.e.
ppB, F q ą p 1 pBqp 2 pF q, leads to an overpricing (and thus positive profit) of the insurance, whereas a negative correlation, i.e. ppB, F q ă p 1 pBqp 2 pF q, leads to market failure. Such neglect is not unrealistic in this context. Indeed, whereas it can be convincingly argued that there is no direct causality between Ω 1 and Ω 2 , there may still be correlation due to correlation to the third, payoff-irrelevant, factor climate sensitivity.10 Specifying the uncertainty regarding climate sensitivity by the state space Ω 3 , the model becomes Ω " Ω 1 ˆΩ2 ˆΩ3 . The IPCC report clarifies (up to a certain extent) the uncertainties regarding Ω 3 as well as its correlations to the other subspaces Ω 1 and Ω 2 . This allows the insurer to model the correlation among B and F through the causal relationships with Ω 3 and thus to determine the correct value of ppB, F q. In Section 4.5, we provide a decision-theoretic characterization of "revealed correlations". That is, we clarify what kind of behavior reveals that the decision maker treats certain events as positively or negatively correlated. We suggest a comparative notion of revealed positive correlation, allowing comparative statics, as touched upon in this example. Applying this comparative notion to comparisons with the "no correlation" benchmark, axiomatized in Theorem 2, directly leads to an absolute notion of revealed correlations. Example 3 (finance). The returns of an asset on the stock market depend on different economic indicators, like recession, inflation, market entry of a competitor or technological advances, but also on psychological effects, like market sentiment or market bubbles. Understanding how states that influence an asset's return are correlated is thus crucial for constructing robust and well-performing portfolios. However, experimental data shows that individuals tend to ignore correlation of relevant states and treat them as independent when making financial decisions, even if they are aware of their dependencies, a phenomenon widely known as correlation neglect, cf. (Kallir and Sonsino, 2009;Eyster and Weizsacker, 2010). Disregarding well-known causalities between states can thus lead to a wrong assessment of expected returns and thus to unfavorable decisions when buying financial assets in order to make profits or diversify portfolios. Consider the following stylized example, in which a risk-neutral decision maker thinks about investing in gold on the stock market if this yields positive expected returns. For her assessment, she takes into account three different factors influencing the stock market value of gold in the future: inflation, economic uncertainties and the discovery of new gold deposits. For different scenarios, the decision maker faces the returns depicted in Table 4 which reflect that a high inflation and high economic uncertainty raises the market price of gold, while the discovery of new large gold deposits decreases its price. Table 4: Returns from the gold stock market as a function f , depending on inflation (high/low, H infl /L infl ), economic uncertainty (high/low, H unc /L unc ), and the discovery of new gold deposits (G/N G). Assume the decision maker expects to face a high inflation with probability 1 3 , a high economic uncertainty with probability 1 2 and estimates the discovery of a new gold deposit to be made with probability 1 4 . We consider two cases in which the decision maker believes in different correlation structures among the state spaces: Firstly, she may neglect all correlation among the considered state spaces and treats them as happening mutually independently from one another. Her expected payoff from buying the asset can be computed to be 0. Consequently, she is indifferent between not investing and investing and might thus refrain from buying gold. 11 Secondly, she believes that inflation and economic uncertainty are positively correlated, i.e., the probability for both states being high (resp. low) is larger than the value assigned under independence considerations, which is 1 3 ¨1 2 " 1 6 (resp. 2 3 ¨1 2 " 1 3 ). At the same time, she might say that inflation and economic uncertainty as economic indicators are independent of the geological discovery of new gold deposits. 12 As a result, the decision maker averages the returns of gold over the possibility of finding new deposits, eventually facing the returns and probabilities in Table 5. A positive correlation between inflation and economic uncertainty is represented by the condition a ą 1 6 . Under any such positive correlation, the expected return of buying gold is strictly positive, making it a profitable investment. 13 av. returns

Returns of gold (averaged)

prob. Table 5: Returns of gold, when averaged over the prior probability of discovering new deposits, and the corresponding probabilities over inflation and economic uncertainty that reflect their possible correlations, depending on the parameter a P r0, 1 3 s.

The above example can be modeled by a product state space Ω " Ω 1 ˆΩ2 ˆΩ3 with #Ω i " 2, i P t1, 2, 3u and given marginal probabilities p i on Ω i . Mutual independence of the economic indicators (inflation and economic uncertainty) and the discovery of new gold mines formally requires ppω 1 , ω 2 , ω 3 q " p Ω1ˆΩ2 pω 1 , ω 2 q ¨p3 pω 3 q for a joint probability p in consideration, while independence of all subspaces entails ppω 1 , ω 2 , ω 3 q " p 1 pω 1 q ¨p2 pω 2 q ¨p3 pω 3 q. We provide a full description of all independence structures and correlation uncertainty in this 2 ˆ2 ˆ2 case in Example 4. More generally, Section 3.2 mathematically describes the set of probability distributions over Ω that reflect independence of collections of disjoint subspaces. In Section 4.3 we axiomatically describe the properties that preferences over (collections of) assets must have in order to exhibit these kinds of relationships. The notion of positive correlation is made precise in Section 4.5. Related literature.
Closest related to us are Grabisch et al. (2023), who also characterize when exactly the probability vector of a subjective expected utility representation on the product of two measurable state space factors into the independent product of its marginals. Most notably, the authors' necessary condition in Proposition 4 for their axiom of stochastic independence A7 is most similar to our Axiom 2 of subspace independence, with the exception that we restrict the conditioning to constant outcomes instead of more general acts. 12 The decision maker thus ignores that economic crises can foster the exploration for and exploitation of new mining sites. 13 Under low risk-aversion, buying the asset in presence of positive correlation is still profitable. Precisely, consider the CRRA Bernoulli utility function upxq " p c 6 q 1´ρ and assume positive correlation, i.e., a ą 1 6 . Buying the asset is profitable if and only if the risk aversion parameter ρ does not exceed 1 `log 2 p 6a 1`6a q ą 0. Our results elevate these insights to an arbitrary number of state spaces and allow for partial independence among subspaces. On a state space without a Cartesian product structure, Monet and Vergopoulos (2022) provide an axiom of subjective independence that can replace the classical sure-thing principle and event-wise monotonicity to arrive at a subjective expected utility representation within a Savage framework. Swapping the order of horse races and lotteries by use a Cartesian product structure, Monet and Vergopoulos (2024) axiomatically characterize Expected Choquet Utility and identify it as the Choquet Expected Utility with respect to a modified capacity that represents the reversed evaluation timing of uncertainty. Epstein and Halevy (2019) introduce a three-fold distinction between risk, Knightian uncertainty and ambiguous correlation by providing an Ellsberg-type thought experiment, that they have also tested in a lab experiment. Participants prefer ambiguous bets to bets on the correlation on ambiguous urns. Levy and Razin (2022) study rationalizable forecasts based on given a finite set of observations and a joint information structure. Their main result characterizes rationalizable predictions in terms of the "naïve Bayesian approach", which plays a similar role as the independent product p ind of the given marginals in our decomposition of correlation. Ignoring correlation can be seen as a behavioral trait of individuals, known also as correlation neglect, playing a crucial role in belief formation Enke and Zimmermann (2019), portfolio choice in finance (Kallir and Sonsino, 2009;Eyster and Weizsacker, 2010) and in school choice Rees-Jones et al. (2020). Zhang ( 2021) provides an axiomatic characterization of correlation neglect and narrow-bracketing for preferences on lotteries. A game theoretic equilibrium notion that encompasses ignorance of correlation is known as cursed equilibrium, cf. Eyster and Rabin (2005). Hereby, an agent occasionally forgets that the opponent's action depends on the their private type, while still correctly identifying the opponent's distribution of actions. In a recent work, Preker (2024) develops two notions of cursed equilibrium under Knightian uncertainty. The author makes the surprising finding that uncertainty increases the frequency of trade in a simple trading game if agents are cursed, while uncertainty has no effect if agents are not cursed. The remainder of the article is organized as follows. Section 2 briefly introduces the formal setup and notations. In Section 3, we classify the set of probabilities consistent with given marginals as a convex polytope with and without partial independence assumptions on the subspaces and identify its extreme points. Section 4 develops an axiomatic approach to correlation uncertainty, pinning down axioms that identify the previously classified subsets of (partial) independence. Finally, we sketch an extension to our model in Section 5 that allows the decision maker to face ambiguity already on the subspace level. Let Ω be a finite state space that is the product of subspaces Ω 1 , . . . , Ω n such that Ω " Ω 1 ˆ¨¨¨ˆΩ n . 14 A state is thus of the form ω " pω 1 , . . . , ω n q, where ω i P Ω i . A set of states is called an event and denoted by E Ď Ω, likewise E i Ď Ω i for subspaces. Ω ´i :" Ω 1 ˆ¨¨¨ˆΩ i´1 ˆΩi`1 ˆ¨¨¨ˆΩ n is the Cartesian product of all subspaces but Ω i . For ω i P Ω i we define rω i s :" tω i u ˆΩ´i Ď Ω, i.e., rω i s coincides with the set of states that agree with ω i in the ith index. For E i Ď Ω i we analogously define rE i s :" E i ˆΩ´i Ď Ω, i.e., rE i s coincides with the set of states that agree with an element of E i in the ith coordinate. For ω ´i P Ω ´i we define rω ´is :" tω ´iu ˆΩi Ď Ω, and rE ´is :" E ´i ˆΩi Ď Ω for any E ´i Ď Ω ´i. We denote by ∆pΩq and ∆pΩ i q the set of probability distributions on Ω and Ω i , respectively. When dealing with probability measures, we use the operator b to construct independent products. For instance, if p i P ∆pΩ i q, then p 1 b . . . b p n is the unique probability measure, the weight of which on the atom ω " pω i q n i"1 is given by ś n i"1 p i pω i q. Formalizing preferences in Section 4, we fix a set X of consequences and consider acts, i.e., maps f : Ω Ñ X. Let F " tf : Ω Ñ Xu denote the set of acts. An act f P F is called independent of Ω ´i if f is constant on rω i s for all ω i P Ω i .
For two acts f, g P F and event E Ď Ω, f E g denotes the act that results in f pωq on E and gpωq on E c : " ΩzE. Let F i " tf i : Ω i Ñ Xu denote the set of acts on Ω i . Preferences over all acts are modeled by a binary relation ě on F while preferences over F i are modeled by ě i . An event E Ď Ω is called null, if f E g " g for all f, g P F , otherwise we say that E is non-null. Having precise probabilistic assessments of all isolated properties, given by the subspaces, does not uniquely determine a single probability of the whole state space. For instance, statistical evidence on climate sensitivity as well as the albedo of the earth does not uniquely pin down their intricate relation towards one another. An insurer may not want to neglect the dependence of the risk of an insured house burning down or being destroyed by a flood, as these events are correlated through the effects of climate change. It is hence important to understand the different ways of aggregating the given probabilistic information, which is given by a set P of probability measures with identical marginals, but different correlation structures between them. In this section, we first characterize this set as a convex polytope with its extreme points, by a system of linear equations and as the core of an exact capacity υ. Second, we introduce 14 The finiteness assumption guarantees set P of all probabilities with the same marginals to be a convex polytope with finitely many extreme points and admit a simple notion of dimension. A suitable generalization to measurable spaces still allows to decompose a probability into the independent product of its marginals plus a measurable function capturing the correlation structure. The definitions and results in Section 4 can immediately be extended to finitely many measurable spaces when considering acts with finite range. the notion of independence among arbitrary (sets of) subspaces and analyze how this restricts the set P.

Recall Ω " Ω 1 ˆ. . . ˆΩn and fix corresponding, fully supported probability measures p 1 , . . . , p n in the following. For any probability distribution p P ∆pΩq, i P t1, . . . , nu, we define the marginal p Ωi P ∆pΩ i q of p on Ω i by setting p Ωi pω 0 i q :" pprω 0 i sq "

for any ω 0 i P Ω i . The key object of interest in this section is the set

of all probability distributions with marginals p i , also known as multi-marginal couplings, where we drop the state space or the marginals from the notation if they are clear from the context. The set is non-empty, as it contains the independent product p ind :" p 1 b. . .bp n , which is given by p ind pωq " ś n i"1 p i pω i q on the atoms ω P Ω. As we are going to see, the set P contains many more elements, each of which describe a different possible correlation between the marginals. We often treat the marginals p i as implicitly and thus suppress them from the notation. To ease future notation, let n i :" #Ω i , N :" #Ω " ś i n i and embed Ω Ď R N . Discarding zero-elements from our considerations, we may assume that each p i has full support on Ω i . In the following, we characterize the set P of all probability measures on Ω the marginals of which coincide with the resp. p i , as defined in expression (2). Note that we do not impose any further restrictions on possible correlations. In that sense, P " PpΩ; p 1 , . . . , p n q describes a setting in which any compatible correlation is possible. We tackle this characterization from two angles: First, through a set of linear equations, pinning down P as a convex polytope and its dimensionality. Second, as the core of an exact capacity, providing a description of P's extreme points. A linear equation system & the dimension of P By the defining property over the marginals, any p P P Ď R N (indexed by the values ppωq for ω P Ω) is a solution to the finite system of linear equations p Ωi pω 0 i q " ÿ ω´iPΩ´i ppω 0 i , ω ´iq " p i pω 0 i q @i P t1, . . . , nu @ω 0

with ř N k"1 n k equations and N variables. Vice versa, any vector p P R N that solves (3) and has non-negative values is an element of P. Since we already have the particular solution p ind " p 1 b . . . b p n to the inhomogeneous system (3), it suffices to consider the homogeneous system ÿ ω´iPΩ´i qpω 0 i , ω ´iq " 0 @i P t1, . . . , nu @ω 0

the solutions of which we denote by q its solution set Q " QpΩq, which is a sub vector space of R N . 15 Note that the difference between two solutions to (3) is a solution of (4) and p ind `q solves (3) for any q P Q and vice versa. In particular, any p P P can be decomposed into p " p ind `q (5)

for some q P Q. Thereby, p ind captures the information of p about its marginals while q describes its correlation as a shift of mass away from the independent product that leaves the marginals unchanged. We can thus also construct all elements of P by considering p ind `q for all q such that the sum remains a probability.
Consequently,

As a first observation, we notice that P is the intersection of finitely many halfspaces defined by the above system of linear equations and the requirement that its elements are all in R N ě0 . Consequently, it is a convex polytope and has thus a nice geometric structure. Lemma 1. The following properties hold. (i) P is a compact and convex subset of R N . In particular, it is a convex polytope with a finite set of extreme points. (ii) The independent product p ind is an interior point of P.

Proof. All proofs are delegated to the appendix. As a convex polytope, P is the convex hull of its finite set of "extreme points". We give an easy description of these by means of a capacity later and now focus on the "size" of correlation admitted by P: Since the independent product p ind is an interior point that supports the solution set of (3) (an affine space), we can move in any direction q P Q. As a result, the "dimension" of P can be measured by the vector space dimension of Q over the reals and, notably, does not depend on the concrete values of the marginals p i . dim

Knowing Q thus tells us what how many and what kind of correlations are possible (independent of the precise values of the marginals p i ). In order to answer these questions, we give a precise formula for its dimension. 15 Indeed, let ι : t1, . . . , N u Ñ Ω and κ : t1, . . . , ř N k"1 n k u Ñ Ů i Ω i be bijections of Ω resp. of the elements in all the Ω i . Then Q is the kernel of the matrix M " pm ij q P R p ř k n k qˆN the elements of which are m ij " 1 if ιpjq P rκpiqs and 0 otherwise. Applying the homomorphism theorem, the dimension of Q is given by dim Q " N ´rankpM q. Proposition 1. The dimension of P is independent of the given marginals p i and equal to dimpPq "

The intuition behind the dimension is as follows. There are ś n i"1 n i variables, one for each element of any Ω i . For each i P t1, . . . , nu, the marginal conditions (3) or (4) reduce the variables by n i ´1 many degrees of freedom, one for each but one element ω i , since p i is a probability on Ω i . Finally, the condition that all variables sum up to 1 reduces the dimension by one more degree of freedom. The extreme points of P given by maximally zero elements and mutual information As a convex polytope, P is equal to the convex hull of its so-called extreme points. These are particularly useful, not only as a simple geometric description of the set P, but also since worst-case expected utility preferences (maxmin expected utility) are evaluated at them. In the following, we give a two-fold characterization the extreme points of P: Firstly, as those probabilities that assign mass zero to as many states as possible. Secondly, as the local maximizers of mutual information, a measure of correlation from information theory. A probability vector p P P Ĺ R N is called an extreme point of P, if for all 0 ‰ x P R N with p `x P P we have p ´x R P. Equivalently, extreme points can not be written as the convex combination of two distinct elements of P. Let ExtpPq denote the set of extreme points of P. Since P is convex, it is the convex hull of its extreme points, i.e., convpExtpPqq " P. We are going to identify the extreme points of P as exactly those probability vectors with "as many zeros as possible", defined as follows. Definition 1. A probability vector p P P is called maximally zero, if for any p 1 P P with ppωq " 0 ñ p 1 pωq for all ω we have p " p 1 . A maximally zero probability vector p assigns probability 0 to as many states ω P Ω as possible in the sense that there no other probability p 1 P P that vanishes in the same states as p. One can equivalently think of probabilities having minimal support. The intuition behind maximally zero probability vectors being extreme points is that an element in the interior resp. on a face of P can be pushed to the boundary resp. frontier with another face to create another zero state. The extreme points of P can also be linked to the concept of mutual information or, equivalently, entropy. To this end, recall that any p P P can be written as p " p ind `q where p ind " p 1 b. . .bp n is the independent product of p's marginals and q P Q is a mass shift of p ind that does not change the marginal. Note that q thus characterizes the type and degree of correlation, i.e., the component that makes p different from p ind . As a quantitative measure for correlation of a probability distribution, we employ the notion of mutual information. Definition 2. For any probability measure p P ∆pΩq, the mutual information Ippq of p is defined as the Kullback-Leibler divergence (relative entropy) between the joint distribution p and p ind " p Ω1 and defined by

where we adopt the common convention that 0 ¨logp0q " 0 justified by continuity arguments. Using entropy, one can also write Ippq " ř n i"1 Hpp i q ´Hppq where Hpp i q " ř ωiPΩi p i pω i q logpp i pω i q and Hppq " ř ωPΩ ppωq logpppωq.
Mutual information is non-negative and strictly convex with its unique minimum at p " p ind with value 0, cf. Cover and Thomas (2006). The base of the logarithm is uncritical, but typically 2, so that the unit of mutual information is the bit. Mutual information describes how much information the knowledge of the realization on a (set of) subspace(s) reveals about the others. Based on the principle of maximum information preservation (Infomax), Linsker (1988), maximizing mutual information has become an exceptionally important technique for feature selection, clustering, dimensionality reduction and representation learning in machine leaning, (Vergara and Estévez, 2014;Belghazi et al., 2018;Hjelm et al., 2018;Kim et al., 2022). The following theorem ties the concept of mutual information from information theory and maximally zero probability vectors to the geometric property of extreme points of correlation uncertainty. Theorem 1. The following are equivalent for a probability p P P.

(i) p is an extreme point of P.

(ii) p is maximally zero. (iii) p is a local maximizer of mutual information. P as the core of an exact capacity A decision maker, who only has objective or subjective probabilities p i for each factor Ω i can not with certainty form a probabilistic assessment of the state space Ω. She faces uncertainty about how to aggregate the given information, i.e., the marginals' correlation. The set P quantifies her uncertainty as a set of possible probabilities. If the decision maker is wary of picking wrong distribution, she might deal with uncertainty in a cautious manner, as she fears to always end up in the worst-case scenario. Pondering about the likelihood of an event E Ď Ω, this (extreme) form of uncertainty aversion is represented by the functional υpEq :" min pPP ppEq. (10)

Note that υpHq " 0, υpΩq " 1 and that υ is set-monotonic, i.e., υpAq ď υpBq whenever A Ď B Ď Ω. A measure of probability with this property is called a capacity and generalizes additive probability measures. We have the following first observations. Lemma 2. The function υ : 2 Ω Ñ r0, 1s defines an exact capacity, i.e., corepυq :" tp P ∆pΩq | ppωq ě νpωq @ω P Ωu " P. Furthermore, for every rE i s Ď E we have υpEq " p i pE i q `υpEzrE i sq. Since υ is exact, the correlation set P can be recovered from the values of υ. In particular, knowing either P or υ is equivalent. On Ω i -independent events, υ is additive. Furthermore, the value assigned to projections of Ω i -independent events on Ω i are evaluated at the corresponding marginal p i . The capacity υ also provides insight into the extreme points of P. In line with Definition 1, we call an event E Ď Ω maximally zero, if υpEq " 0, but υpE 1 q ą 0 for all E Ĺ E 1 . Evidently, an event E is maximally zero, if and only if E " tω P Ω | ppωq " 0u for a maximally zero p P P. Consequently, υ characterizes the extreme points of P, interpreted as a convex polytope in R N , cf. Theorem 1. We conclude our discussion of the capacity υ by some considerations about ambiguity-averse preferences on correlation. Firstly, utilities can be suitable integrated with respect to a capacity, leading to the Choquet expected utility. Secondly, a decision maker might suspect to always and up on the loosing end when considering correlation, given by the maxmin expected utility over the set P. If υ was a convex capacity, these notions coincide, cf. Schmeidler (1989). However, υ is not convex, showing that uncertainty about correlation allows for a larger variety of ambiguity preferences. Remark 1. The capacity υ is not convex. To see why, consider the state space Ω " Ω 1 ˆΩ2 with Ω i " tω 1 i , ω 2 i u, i P t1, 2u. Take the uniform probability distributions p i P ∆pΩ i q, specified by p i pω 1 i q " 1 2 , i P t1, 2u. Let E " tpω 1 1 , ω 1 2 q, pω 1 1 , ω 2 2 qu and F " tpω 1 1 , ω 1 2 q, pω 2 1 , ω 1 2 qu. We have υpEYF q`υpEXF q " 1 2 `0 ă 1 2 `1 2 " υpEq `υpF q, thus υ is not convex. Non-convexity gives the decision maker access to a larger variety of rules when it comes to decision making. In particular, the prominent Choquet and maxmin expected utility do not coincide, as is illustrated in Remark 2. Remark 2. There are multiple ways how a decision maker might deal with the uncertainty about correlation, encoded by P, resp. the capacity υ. Two prominent decision theoretic decision rules that can reflect "aversion towards correlation uncertainty" are the Choquet expected utility (CEU), where the utility of an act is evaluated with respect to its Choquet integral against υ, and the Maxmin expected utility (MEU), which is the minimum over all expected utilities of the act indexed by p P P. Let the outcome space be the reals and the Bernoulli utility function on constants u " id the identity map. By Remark 1, the capacity is not convex and thus, CEU and MEU preferences do not coincide.
Indeed, consider again Ω i " tω 1 i , ω 2 i u, i P t1, 2u with uniform priors p i and write f " pf 1 , f 2 , f 3 , f 4 q for an act f P F with f

Considering the acts f " p4, 3, 2, 1q and g " p5, 3, 2, 0q, we find f " MEU g, but f ą CEU g.

Having studied the correlation set PpΩq of the whole state space, we now allow agents to believe that certain (sets of) subspaces are independent from one another. We characterize how this restriction splits the set PpΩq of full correlation, compatible with the resp. marginals, into factors of correlation sets of smaller subspaces. To this end, we first extend our notation to allow for sets of indices. For any nonempty I Ď t1, . . . , nu we define Ω I :" ˆiPI Ω i with generic elements ω I , subsets E I , complement Ω ´I " Ω t1,...,nuzI with generic elements ω ´I and events E ´I and the natural embeddings rω I s :" tω I u ˆΩ´I and rE I s :" E i ˆΩ´I Ď Ω. If p P ∆pΩq is a probability distribution on Ω, its marginal on Ω I is given by p ΩI pω I q :" pprω I sq. To motivate and illustrate our idea of independence, consider the state space Ω " Ω 1 ˆΩ2 ˆΩ3 and fix the indices 1 and 2. Drawing on classical probability theory, under an element p P P, the subspaces Ω 1 and Ω 2 should be regarded as independent, if the probability of the intersection of two events is the product of the individual probabilities, holding all other subspaces constant. As the intersection of two events E 1 Ď Ω 1 , E 2 Ď Ω 2 is given by their Cartesian product within Ω, i.e., rE 1 s X rE 2 s " rE 1 ˆE2 s, this can be formalized as

In order to have a flexible approach that allows to distinguish families of subspaces with respect to independence, we more generally define independence on what we call a collection of subspaces. Definition 3 (Collection (of subspaces)). A collection (of subspaces) I is a family of at least two, non-empty, pairwise disjoint subsets I Ď t1, . . . , nu. In the above example for instance, we may have reason to believe that under p P P, events on the subspace Ω 1 are independent from those on its complementary subspaces Ω 2 , Ω 3 . We model this formally by requiring for all events

The probability p thus captures the idea that events on the subspace defined jointly by Ω 2 and Ω 3 occur independently of events on the subspace Ω 1 . We now give a general definition of independence on a collection that is build analogously to the concept of mutual independence in probability theory. Definition 4 (Independence on a collection). A probability p P PpΩq is called independent on the collection I of pair-wise disjoint subsets of t1, . . . , , nu, if for all tE I u IPI , with H ‰ E I Ď Ω I resp.,

Let furthermore P I " P I pΩq denote the set of all p P PpΩq that are independent on I. Note that P I is a subset of P. It differs from P since is must not contain the elements of P where Ω I and Ω J are dependent for I, J P I. With regards to the notion of independence in probability theory, note that r Ś IPI E I s " Ş IPI rE I s, so that indeed, the probability of an intersection factors out as a product with respect to p. The following Lemma shows that it suffices to consider the independence property element-wise. Lemma 3. A probability p P PpΩq is independent on a collection I if and only if for all pω I q IPI P Ś

Having defined independence on collections of pair-wise disjoint subsets of the set t1, . . . , nu of indices, one may wonder whether this property decomposes to independence on singleton subsets. For instance, one may ask if independence of t1u and t2, 3u is equivalent to independence of t1u paired with both, {2} and t3u, individually. As with classical independence, this is not the case, cf. Example 4. However, independence gets inherited on sub collections, i.e., independence of t1u and t2, 3u implies independence of t1u with both, t2u and {3}. Hereby, a sub collection I 1 of I, is a collection with the property that for each I 1 P I 1 , there is I P I 1 with I 1 Ď I.

Lemma 4. Let I 1 be a sub collection of I. Then P I Ď P I 1 , i.e., independence on I implies independence on I 1 . If I is a partition, i.e., 9 Ť IPI I " t1, . . . , nu, the independent probability measures on I can be decomposed into the independent ones of the correlation sets PpΩ I q of the respective family of subspaces. In particular, the dimension of P I pΩq can be directly computed by Proposition 1. Proposition 2. Let the collection I be a partition. Then

i.e., the set of probabilities independent on Ω is the independent product of the correlation sets on the subspace collections Ω I . In particular,

Example 4. Consider Ω " Ω 1 ˆΩ2 ˆΩ3 with Ω i " tω j i u j"1,2 and given probabilities p i P ∆pΩ i q. By slight abuse of notation, we write p i " p i pω 1 i q. From Proposition 1, we know dimpPq " 4.
For the free parameters a, b, c, d ě 0, Figure 1 depicts any probability p P PpΩq, as long as it defines an element within the simplex, i.e., all terms are positive. First, consider the case where Ω 1 and Ω 2 are considered independent, i.e., we are looking for all p P PpΩq with pprpω 1 , ω 2 qsq " p 1 pω 1 q ¨p2 pω 2 q for any ω 1 P Ω 1 , ω 2 P Ω 2 , cf. Lemma 3. Indeed, each of these conditions is equivalent to a `d " p 1 ¨p2 . Effectively, P t1u,t2u has thus dimension 4 ´1 " 3. In contrast, P t1u,t2,3u " tp 1 u b PpΩ 2 ˆΩ3 q has dimension 1, according to Propositions 1 and 2. Explicitly, P t1u,t2,3u is given by the equations b " 1´p1 p1 a, c " p 1 p 3 ´a, d " p 1 p 2 ´a and thus parametrized by the single variable a. Interestingly, considering P t1u,t2u X P t1u,t3u , i.e, considering Ω 1 independent of Ω 2 and Ω 3 separately, only yields the conditions d " p 1 p 2 ´a and c " p 1 p 3 ´a from before, but misses the one restricting b. Consequently, P t1u,t2u X P t1u,t3u is of dimension 2. In this section, we consider a decision maker (DM) with preferences over a state space with a product structure. The object of study is thus a binary relation ě over the set of acts F " tf : Ω Ñ Xu, where Ω " Ω 1 ˆ¨¨¨ˆΩ n . In addition, the DM has preferences ě i over F i " tf i : Ω i Ñ Xu, i P t1, . . . , nu. We consider the framework of Anscombe and Aumann (1963). To this end, let the set of consequences X be a mixture-space over some set of prizes. We assume that for all i P t1, . . . , nu, ě i has a subjective expected utility (SEU) representation on Ω i , Savage (1954). That is, there exists an affine utility function u i : X Ñ R and a probability distribution p i P ∆pΩ i q such that ě i is represented by the functional V i : F i Ñ R, where

The interpretation is that the DM faces no uncertainty regarding the likelihood of events within the subspaces Ω 1 , . . . , Ω n . Further, we assume that ě has a maxmin expected utility representation on Ω (Gilboa and Schmeidler, 1989). That is, there exists an affine utility function u : X Ñ R and a non-empty, convex and compact set of probability distribution C Ď ∆pΩq such that ě is represented by the functional V : F Ñ R, where

The interpretation is that, while the DM has probabilistic beliefs about the subspaces Ω 1 , . . . , Ω n , she may not be able to precisely determine the overall probability in Ω. The obvious question now is how ě is connected with ě i , i P t1, . . . , nu. In general, the p i P ∆pΩ i q must not be the marginals of the elements of C Ď ∆pΩq. Our first result, developed in Section 4.1, is that this being the case is equivalent to a logical consistency axiom connecting ě and ě i . Put differently, we axiomatically characterize the case C Ď PpΩ; p 1 , . . . , p n q. In Section 4.2, we characterize the important benchmark case C " tp ind u through our axiom Subspace Independence 2. In the preceding three subsections we provide behavioral foundations of the phenomena discussed in the three motivating examples. In Section 4.3, we clarify axiomatically, under the assumption that ě is SEU, when a collection of subspaces is regarded as being independent, hereby building a bridge to the finance Example 3. In Section 4.4, we provide a definition of comparative correlation aversion, hereby providing a preference-based justification for the reluctance to implement climate engineering options against climate change as it introduces new, possibly correlated uncertainties into the choice problem (Example 1). In Section 4.5, we clarify what it means behaviorally that one decision maker perceives a collection of events as being more correlated, compared to the perception of another decision maker. Building on this, we also provide an absolute notion for revealed positive/negative correlation. Recall that F i is the set of acts on Ω i . Each f i P F i has a well-defined embedding rf i s P F , where rf i spω 1 , . . . , ω n q :" f i pω i q for all ω " pω 1 , . . . , ω n q P Ω. From the DMs perspective, f i and rf i s are simply the same act. We can thus identify the set F i as a subset of F . That being established, a desirable rationality condition is to demand consistency between ě and ě i when the ranking of such acts is considered. This consistency condition is formalized as our first axiom. Axiom 1 (Subspace Consistency). For all i P t1, . . . , nu and all acts f i , g i P F i , f i ě i g i ðñ rf i s ě rg i s.

Remark 3. In our model, we take on the view of starting off with exogenously given preferences on Ω and the respective Ω i , requiring compatibility by means of Subspace Consistency. However, one can also start from a preference ľ on Ω and restrict it to preferences ľ i on Ω i by defining f i ľ i g i : ðñ rf i s ľ rg i s, thereby guaranteeing Subspace Consistency. If these restrictions turn out to be SEU preferences, the setting is the same as the one presented.
The following proposition shows that, in addition to implying consistent risk preferences across (sub-)spaces, Subspace Consistency has a concrete implication for the relationship between the probability distributions p i P ∆pΩ i q and C Ď ∆pΩq. Proposition 3. The following statements are equivalent. (i) Subspace Consistency holds. (ii) For all i P t1, . . . , nu, (a) There exist a i ą 0 and b i P R such that u i " a i u `bi , (b) For all p P C, p Ωi " p i , i.e. C Ď PpΩ; p 1 , . . . , p n q.

Subspace Consistency is maintained throughout the rest of the article. Proposition 3 shows that we can assume without loss that u i " u for all i P t1, . . . , nu, as the affine utilities of MEU and SEU functionals are unique precisely up to positive affine transformations. Further, it implies that all elements of C have the same marginals, namely the p i P ∆pΩ i q that represent ě i . Note that Subspace Consistency does allow for correlations among the subspaces, i.e., C may contain elements that do not coincide with the independent product p ind " p 1 b . . .b p n . Put differently, C Ď P. The prior set C can thus be interpreted as capturing the DM's uncertainty about correlation among the subspaces. Having a pessimistic assessment of correlation, she expects to face the worst-case scenario for any choice she makes. A crucial benchmark case for economic applications is the special case C " tp ind u, i.e., the scenario where there is no uncertainty regarding the correlation and in addition subspaces are regarded as independent by the DM. As demonstrated in Theorem 2 below, the following axiom classifies this case. Axiom 2 (Subspace Independence). For all f i , g

An interpretation of this axiom is in order. Consider two acts f i , g i P F i , that is, the resulting consequences of these acts are independent of what happens on the subspaces Ω j , j ‰ i. The axiom states that the preference among these acts is inherited when we "condition" on some rE ´is " E ´i ˆΩi . For a concrete illustration, consider the case Ω " Ω 1 ˆΩ2 with Ω i " tω 1 i , ω 2 i u, i P t1, 2u, and the two acts f and g illustrated in Table 6. Note that f, g P F 1 since they are independent of Ω 2 . The acts f rω 1 2 s x and g rω 1 2 s x coincide with f and g, respectively, on the two states resulting in ω 1 2 and result in the consequence x otherwise. Assuming that rω 1 2 s is non-null, Subspace Independence requires that f ě g if and only if f rω 1 2 s x ě g rω 1 2 s x. Table 6: The acts f and g are independent of Ω 2 . Subspace Independence thus implies that f ě g ðñ f rω 1 2 s x ě g rω 1 2 s x.

Theorem 2. The following are equivalent:

(i) ľ satisfies Subspace Independence, 2. (ii) C " tp ind u. Note that Subspace Independence is, somewhat surprisingly, strong enough to force ě to being SEU, indeed with the specific independent product p ind as its corresponding probability. That is in particular, SEU on the subspaces plus Subspace Independence is strong enough to force an MEU preference on Ω to be SEU itself. Of course, simply assuming the standard Independence axiom of Anscombe and Aumann (1963), instead of Subspace Independence, merely gives us a singleton prior set C " tpu, p P PpΩ; p 1 , . . . , p n q. Subspace Independence is thus stronger than the Independence axiom, given our context (state space having a product structure) and assumptions (SEU on the subspaces and Subspace Consistency). In the motivating Example 3 we have discussed the behavioral phenomenon of correlation neglect, meaning that individuals treat well-established correlations as independent when making financial decisions. Our framework allows a sophisticated approach to model arbitrary degrees and structures of independence the decision maker may consider. Consider the case where ě is SEU, i.e., it is represented by a singleton prior set C " tpu, p P PpΩ; p 1 , . . . , p n q. We turn to the case where p does not necessarily coincide with p ind , but instead reflects independence among collections (of subspaces), a considerably weaker property. We show, in full generality, that this property can be classified axiomatically. Recall that Ω I " ˆiPI Ω i and Ω ´I " ˆiRI Ω i for I Ď t1, . . . , nu. We define F I to be the acts in F that are independent of Ω ´I , i.e., f P F I is constant on rω I s for all ω I P Ω I . Let I be a collection of subspaces, i.e., a family of at least two non-empty, pairwise disjoint subsets of t1, . . . , nu and let P I again be the set of all probabilities that exhibit mutual independence on I. The following axiom is the generalization of Axiom 2 that allows for mutual independence on the collection I of subspaces. Axiom 3 (I-Independence). For all I 0 P I, f I0 , g I0 P F I0 , E J0 Ď Ω J0 , s.t. rE J0 s is non-null and x P X we have

where J 0 :" ď IPIztI0u I. Note that when I coincides with the partition tt1u, . . . , tnuu of singletons, Axiom 3 coincides with Axiom 2.
To illustrate the axiom, consider the state space Ω " Ω 1 ˆΩ2 ˆΩ3 and the case I " tt1, 2u, t3uu. Assume for concreteness that Ω i " tω 1 i , ω 2 i u, i P t1, 2, 3u, and that there are no null states. To save space we denote by pi, j, kq the state pω i 1 , ω j 2 , ω k 3 q. Consider the two acts f and g illustrated in Table 7. Note that these acts are independent of Ω 3 and are thus elements of F t1,2u . tt1, 2u, t3uu-Independence states that f ě g is equivalent to f rω 1 3 s x ě g rω 1 3 s x.

p1, 1, 1q p1, 1, 2q p1, 2, 1q p1, 2, 2q p2, 1, 1q p2, 1, 2q p2, 2, 1q p2, 2, 2q

Table 7: The acts f and g are independent of Ω 3 . Thus pt1, 2u, t3uq-Independence requires that f ě g ðñ f rω 1 3 s x ě g rω 1 3 s x. The following theorem shows that I-Independence is equivalent to p being an element of P I . That is, the axiom characterizes behavior that treats the collection of subspaces in I as being independent. More concretely, considering the above example, tt1, 2u, t3uu-Independence is necessary and sufficient for p reflecting mutual independence of events in Ω 1 ˆΩ2 and events in Ω 3 , i.e., for all E t1,2u Ď Ω 1 ˆΩ2 and E 3 Ď Ω 3 , ppE t1,2u ˆE3 q " p Ω1ˆΩ2 pE t1,2u q ¨p Ω3 pE 3 q. (23)

Theorem 3 (I-Independence characterization). The following are equivalent:

(i) ě satisfies I-Independence. (ii) p P P I . Note that I need not be a partition. However, if I " tI 1 , . . . , I m u is a partition, then Theorem 3 reveals that I-Independence is equivalent to p " p ΩI 1 b . . . b p ΩI m , factoring into the independent product of m probability measures, one on each Ω Ij . Written differently, P I " Â m j"1 PpΩ Ij q, where, say PpΩ I q, is the full correlation set of the state space Ω I " ˆiPI Ω i and its dimension is computed in Proposition 2. In Motivating Example 1, we suggested that reluctance in choosing strategies against climate change which make "new" uncertainties relevant (e.g. Climate engineering) can be explained by aversion towards uncertainty regarding the correlation of subspaces. We demonstrated that "more" correlation uncertainty leads to less willingness to implement such strategies when compared to established strategies that are independent of correlation "uncertainties" (e.g. Mitigation). We now provide a behavioral foundation of this concept by constructing a comparative notion for correlation aversion. As before, we consider preference relations that are SEU on Ω i , i P t1, . . . , nu, and MEU on Ω. That is, recalling the maintained Subspace Consistency 1 and its implication through Proposition 3, the only uncertainty regarding probabilities is about the correlation of subspaces. Our notion of comparative correlation aversion builds on the following idea. Let g be an act which does not depend on the correlation among subspaces, i.e., regardless of what the precise correlation is, the expectation of g is always the same. If a DM prefers some act f to g, then a less correlation averse DM should do so as well. This concept is inspired by the celebrated idea of Yaari (1969) for defining comparative risk aversion. The candidates for "correlationally neutral" acts g are quickly found, namely the acts in F i , i P t1, . . . , nu. Indeed, recall that an act f i P F i is independent of Ω j for all j ‰ i. This implies in particular, that the evaluation of such an act does not depend on the correlation among the subspaces in any way. These acts can thus be called "correlation neutral". An inclination for such acts thus reveals an aversion towards correlation of subspaces, which explains the following definition. Note that the strategy "Mitigation" in Motivating Example 1 conforms to such a correlationally neutral act, whereas "Climate engineering" does not. Definition 5 (Comparative correlation aversion). ě reflects more correlation aversion than ě 1 if for all f P F , i P t1, . . . , nu, f i P F i ,

The following proposition shows that this behavioral comparative notion is characterized by a straightforward subset relation of the respective prior sets representing the preferences. Proposition 4. Let ě and ě 1 be preference relations obeying Subspace Consistency 1 with prior sets C and C 1 and utility functions u, u 1 , respectively. The following are equivalent:

(i) ě is more correlation averse than ě 1 . (ii) C Ě C 1 and u, u 1 are positive affine transformations of one another. Proposition 4 implies in particular that ě being more correlation than ě 1 forces the prior sets C and C 1 to be subsets of the same correlation set PpΩ; p 1 , . . . , p n q. Further, note that for general MEU preferences, that is without Subspace Consistency 1 working in the background, C Ě C 1 is equivalent to the weaker notion of "ě is more ambiguity averse than ě 1 " as suggested by Ghirardato and Marinacci (2002), 16 see their Theorem 17. The inclusion of Subspace Consistency implies that our notion of "more correlation averse than" is equivalent to their "more ambiguous averse than". This is because acts that are elements of some F i , i P t1, . . . , nu, are independent of Ω ´i and therefore "unambiguous" since the only ambiguity that Subspace Consistency allows is regarding the correlation of subspaces.
In Motivating Example 2 we considered two agents, the "insurer" and the "insuree", who had different assessments of the correlations of states in the subspaces. In this subsection, we provide a behavioral foundation for what it means for one preference relation to reflect more correlation among states or events than another preference relation. Throughout we consider SEU preferences only. To clarify our approach, consider some ω 1 P Ω 1 and ω 2 P Ω 2 . What does it mean for these two states of the subspaces Ω 1 and Ω 2 to be positively correlated? Intuitively, this is the case when the probability that they occur together is larger than the product of their marginal likelihoods, i.e., pprtω 1 u ˆtω 2 usq ą p 1 pω 1 q¨p 2 pω 2 q. In this section, we pin down this concept of "positively correlated events" in behavioral terms. First, we provide a comparative notion for positive correlation for collections of events. After defining correlation neutrality in the canonical way, we achieve also an absolute notion for positive correlation among events. To further motivate the theory that follows, consider the following example. Consider the case Ω " Ω 1 ˆΩ2 with Ω i " tω 1 i , ω 2 i u, i P t1, 2u. Consider the two acts in Table 8. Now consider a preference relation ě with marginals p 1 P ∆pΩ 1 q and p 2 P ∆pΩ 2 q. Assume that ş upf qdp ind " upzq, i.e., the evaluation of f through the independent product p ind " p 1 b p 2 results in the utility of z. What does it mean for ě to reflect positive or negative correlation among the states ω 1 1 and ω 1 2 ? Well, if x ą y, then a positive correlation among these states implies that the good consequence x gets more probabilistic weight compared to the weight that p ind assigns to it. This is revealed through f ą z. Equivalently, if x ă y, then a positive correlation of these states is revealed by f ă z. This demonstrates that correlation among events can be observed when compared to the correlation neutral counterpart characterized through p ind . Table 8: The preference among f and z reveals the perception of how ω 1 1 and ω 1 2 are correlated. In the following we pin this down behaviorally. First we propose a comparative notion for revealed correlation. Definition 6 (Comparative revealed correlation). Let I be a collection. Let ě and ě 1 be two preference relations with the same marginal distributions. Consider the family of events pE I q IPI , E I Ď Ω I . We say that ě reveals more positive (negative) correlation than ě 1 on pE I q IPI if for all x, y, z P X with x ą y, z ě pďqx r ą

The interpretation of Definition 6 is that if ě prefers a constant act to a binary act that results in the good consequence if all considered events occur, then a second preference relation reflecting less correlation among these events also prefers the constant act. Put differently, if a DM rejects a bet on a family of certain events happening simultaneously, then another DM who believes in less correlation among these events should also reject this bet. Proposition 5. Let I be a collection. Let ě and ě 1 be two SEU preference relations, represented by probabilities p and p 1 which have the same marginal distributions, i.e., p, p 1 P PpΩ; p 1 , . . . , p n q. Consider the family of events pE I q IPI , E I Ď Ω I . The following are equivalent:

(i) ě reflects more positive (negative) correlation among events pE I q IPI . (ii) ppr Ś IPI E I sq ě pďqp 1 pr Ś IPI E I sq. Proposition 4 allows us to define an absolute notion for positive (negative) correlations. The idea is to define a preference relation to reflect positive (negative) correlation among events if it reflects more positive (negative) correlation than its correlation neutral counterpart. That is, we need to consider Definition 6 with ě 1 being correlation neutral. This correlation neutral counterpart is easily found: for ě with marginals p 1 , . . . , p n , the correlation neutral counterpart is the well-defined preference relation characterized by p ind " p 1 b . . . b p n . We refer to it as ě cn , where "cn" stands for "correlation neutral". Definition 7 (Absolute revealed correlation). Let I be a collection. Let ě and ě 1 be two preference relations with the same marginal distributions. Consider the family of events pE I q IPI , E I Ď Ω I . We say that ě reveals more positive (negative) correlation than ě 1 on pE I q IPI if for all x, y, z P X with x ą y, z ě pďqx r Ś IPI EI s y ùñ z ě cn pď cn qx r Ś IPI EI s y. The following corollary is a direct consequence of Proposition 4. Corollary 1. Let I be a collection. Let ě and ě 1 be two preference relations with the same marginal distributions. Consider the family of events pE I q IPI , E I Ď Ω I . The following are equivalent:

In this paper, we assume sufficient statistical knowledge of the associated subspaces when considered in isolation, represented by subjective expected utility preferences on each subspace. This alone already induces a natural notion of uncertainty, or ambiguity, a decision maker faces about the correlation of the given marginals. An interesting extension would be to allow ambiguity attitudes already on the subspace level, capturing the idea that only incomplete probabilistic information is available on each subspace in isolation.
Economic applications of such a scenario are manifold. For instance, in our motivating example one could allow for ambiguity regarding the consequences of solar radiation modification (SRM) due to the lack of scientific understanding of the relevant uncertainties. This ambiguity regarding the likelihood of states within one subspace has consequences for the overall assessment of the choice problem, in particular if there is also uncertainty regarding the correlation of the subspaces. An extension of our axiomatic analysis to such cases promise to be a fruitful contribution to the literature on choice under ambiguity. , ω 1 ´i0 q pω 1 i0 , ω ´i0 q pω 1 i0 , ω 1 ´i0 q q ω´i 0 pxq:

A.1 Proofs of Section 3

Proof of Lemma 1. As the intersection of an affine space and the simplex ∆pΩq, P is compact and convex and thus a convex polytope. Since all p i have full support, so does p ind . There thus exists ε ą 0 such that for all q P Q within ε distance (by the Euclidean norm) we have p ind `q P P. Hence, p ind is an interior point of P.

Proof of Proposition 1. We prove the formula by induction over each n i . To start with, note that if n i " 1 for all i " 1, . . . , n, there is only one probability measure on the singleton set Ω and thus dimpPq " 0, which agrees with the formula given. By means of induction, it suffices to show that, for any i 0 P t1, . . . , nu, adding one element ω ni 0 `1 i0

to Ω i0 , forming Ωi0 with cardinality n i0 `1 increases the dimension by the amount ´1 `śj‰i0 n j from the stated formula. To this end, consider for any ω ´i0 P Ω ´i0 that is not equal to ω 1 ´i0 :" pω 1 j q j‰i0 the following mass shift, denoted by q ω´i 0 pxq, of a mass x P R depicted in Figure 2. It is straightforward to see that q ω´i 0 pxq does not change marginals and is thus an element of Q -the kernel of the homogeneous linear equation system of Ω. First, observe that there are ´1 `śj‰i0 n j of these mass shifts, one for each ω ´i0 P Ω ´i0 ztω 1 ´i0 u. Now consider any q P Q. The mentioned mass shifts can be used to extract all q-mass from the elements pω ni 0 `1 i0

, ω ´i0 q and bestow them upon pω ni 0 `1 i0

, ω 1 ´i0 q, resulting in q 1 P Q. By the marginal property, we then also have q 1 ppω ni 0 `1 i0

, ω 1 ´i0 qq " 0, thus indeed q 1 P Q, the kernel of the homogeneous problem for Ω. Thus, adding ω to Ω increases the dimension by at most ´1 `śj‰i0 n j . Indeed, one cannot dispose of any of the mentioned mass shifts, as there is no way to absorb the mass of pω ni 0 `1 i0 , ´i0 q. So indeed, the dimension grows exactly by the mentioned amount and the added mass shifts can be added to a basis of Q to form a basis of Q. Alternative proof of Proposition 1. Let dpω, ω 1 q :" #ti P t1, . . . , nu | ω i ‰ ω 1 i u denote the Hamming distance. In the following, fix any ω P Ω and realize that # tω P Ω| dpω, ωq ě 2u "#Ω ´# tω P Ω| dpω, ωq ď 1u

´n ÿ

i"1 pn i ´1q. For any ω with dpω, ωq ě 2, we can pick any two indices i, j for which ω, ω disagree and consider ω 1 , ω : which result from ω by replacing the coordinate of i with ω i , resp. j with ω j . Note that we can shift any mass x P R from ω towards, say, ω 1 , by shifting the same mass from ω to ω : . This operation is a vector in Q that allows us to eject (all) mass from ω by changing masses on elements in Ω with a smaller Hamming distance than dpω, ωq. Collect any such operation (for a normalized mass of x " 1) for each element ω with dpω, ω ě 2 and call its set B. We claim that B is a basis of Q: First, observe that B generates Q: If q P Q is any element, we can iteratively eject all mass from ωs with dpω, ωq ě 2 by applying appropriately scaled operations from B, starting with the elements of maximal distance. We argue that the resulting q P Q is indeed the constant 0. Take any ω with dpω, ωq " 1 and write ω " pω i , ω ´iq. By the defining marginal property and since qpω 1 q " 0 if dpω 1 , ωq ě 2, 0 " qprω i sq " ÿ ω´iPΩ´i qpω i , ω ´iq " qpω i , ω ´iq " qpωq. Eventually, also 0 " qprω i sq " ř ω´i qpω i , ω ´iq " qpωq, thus, B generates P. Second, B is a basis, i.e., we cannot do without any of its elements. If we do not add the operation for a ω with dpω, ωq ě 2, there is no way to shift mass away from it (we can assume that no other operation with a higher distance element involves ω as an intermediate step). Proof of Lemma 2. It is obvious that υpHq " 0, υpΩq " 1 and that υpEq ď υpE 1 q whenever E Ď E 1 .
Recall that corepυq " tq P ∆pΩq | qpEq ě υpEq for all Eu. By definition of υ, we have P Ď corepυq. Consider now any q P corepυq and fix any i P I. Note that qprω i sq ě υprω i sq " p i pω i q by definition. Consequently, qprω i sq " q Ωi pω i q " p i pω i which means q P P. Letting F :" EzrE i s, the second statement follows from υpEq " υprE i s 9 YF q " min pPP pprE i sq `ppF q " p i pE i q `υpF q. Proof of Theorem 1. (i) ñ (ii): If p is not maximally zero, there is p 1 P P, p ‰ p 1 , which vanishes in all ω in which p vanishes (and possibly more). In other words p ´1p0q Ď p 1´1 p0q. Note that x :" p 1 ´p P Q and that xpωq " 0 whenever ppωq " 0. We can thus scale down x by a scaling factor κ ą 0 such that p `κx, p ´κx P P (note that κx P Q). Consequently, p is not an extreme point. (ii) ñ (iii): Let p be maximally zero. We show that Ip.q is increasing locally if moving towards p from any direction. To this end, fix p 1 P P with p ‰ p 1 and consider the convex combination λp `p1 ´λqp 1 for λ P r0, 1s. We have

Ipλp `p1 ´λqp 1 q " ÿ ωPΩ pppωq,p 1 pωqq‰p0,0q pppωq ´p1 pωqq ¨"log ˆλppωq `p1 ´λqp 1 pωq p ind pωq

where the summation runs over all states ω which do not get assigned probability zero by both, p and p 1 . We claim that lim λÑ1 B Bλ Ipλp`p1´λqp 1 q " `8, showing that Ip.q strictly increases locally when moving towards p. To see this, we make a case distinction w.r.t. ω for each summand. If ppωq ‰ 0, the respective summand in (26) has a finite limit for λ Ñ 1. If ppωq " 0 (and p 1 pωq ‰ 0), the limit of the corresponding summand in (26) is lim λÑ1 ´p1 pωq ¨"log ˆp1 ´λqp 1 pωq p ind pωq

Since p is maximally zero, there actually is at least one ω with ppωq " 0 ‰ p 1 pωq, thus proving the claim. (iii) ñ (i): Let p be no extreme point. Then there is 0 ‰ x P R N such that p `x, p ´x P P, in particular x P Q. We are going to show that in arbitrarily close environment to p there is a probability p 1 P P with Ipp 1 q ą Ippq, proving that p is not a local maximum of Ip.q. To this end, let ε ą 0 be arbitrary. Defining y :" x x 2 ¨ε 2 , we see that p 1 :" p `y and p 2 :" p ´y are both in P and ε 2 -close to p (in the Euclidean norm . 2 ). Wlog, assume Ipp 1 q ě Ipp 2 q and set p 1 :" p 1 . Since p " 1 2 p 1 `1 2 p 2 and Ip.q is strictly convex, we find Ippq ă 1 2 Ipp 1 q `1 2 Ipp 2 q ď Ipp 1 q, concluding the proof. Proof of Lemma 3. It suffices to show that if ( 16) holds for all pω I q IPI P Ś IPI Ω I , then the property holds for a product of events Ś IPI E I , i.e., (15). For a better overview write I " tI 1 , . . . , I m u. Proof of Lemma 4. For ease of notation, let I 1 " tI 1 1 , . . . , I 1 , m 1 u, I " ti 1 , . . . , I m u with m 1 ď m and I 1 k Ď I K for all k " 1, . . . , m 1 . Note that k P t1, . . . , m 1 u, we have rE I 1 k s " rE I k s, where E I k :" E I 1 k ˆΩI k zI 1 k Ď Ω I k . Consequently, for p P P I p

pp rE I k s lo omo on

"

only if upzq " V pzq ě V pf q " ppr Ś IPI E I sq. Since V 1 pzq " upzq " V pzq and V 1 pf q " p 1 pr Ś IPI E I sq, the equivalence follows. IPI,ωI PEI rω I1 ˆ. . . ˆωIm s ˙(29) " ÿ IPI,ωI PEI p prω I1 ˆ. . . ˆωIm sq (30) " ÿ IPI,ωI PEI p prω I1 sq ¨. . . ¨p prω Im sq (31) "pprE I1 sq ¨. . . ¨pprE Im sq " ź IPI pprE I sq. (32)

Harrison et al. (2021), p.2: "[Some regard SRM] as a responsible technological option in the face of a climate emergency, while others view it as a hubristic attempt to play God, posing disastrous consequences for the planet and humanity." It is not yet fully understood how SRM techniques effect the ecosystems and how it impacts regional climates. For instance, there are significant uncertainties as to how SAI inter-

Of course, mitigation nests business as usual. We list both for clarity. More precisely, each element of PpΩ; p 1 , p 2 q, and thus C, results in the same expectation. The rationale behind this is that two events occurred that would both have completely destroyed the house. So if the house is only insured against one of these events, the contract specifies the fraction of the damage being paid.
Another interpretation is that the insurance pays the full price v in case the insured event happens first, which has probability x. This assumption is not unrealistic, since insurances tend to have more accurate information compared to insurees. This is especially problematic when we allow the insuree to be risk-averse. Under this assumption, correct beliefs allow for insurance prices which make both the insurer and the insuree strictly better off. An overestimation of the correlation of B and F leads however to a market failure as no contract will be signed. From the IPCC reports we know that both, the likelihood of B and F , are correlated with climate sensitivity, since a higher temperature makes extreme weather events, such as fires and storm tides, more likely. Thus, B and F are positively correlated. Under any degree of risk-aversion, the decision maker strictly prefers not to buy gold if considering the three states to be independent. For instance, taking constant relative risk aversion (CRRA) upxq " p c 6 q 1´ρ as a Bernoulli utility function, the expected utility from not buying the asset is larger than the one from buying the asset if and only if ρ ą 0, i.e., for risk aversion. This is the case whenever, for all f P F and x P X, f ě x implies f ě 1 x.