Can you infer causality from correlation in this example of dictator game? I've just had en exam where we were presented with two variables. In a dictator game where a dictator is given 100 USD, and can choose how much to send or keep for himself, there was a positive correlation between age and how much money the participants decided to keep.  
My thinking is that you can't infer causality from this because you can't infer causation from correlation. My classmate thinks that you can because if you, for example, split the participants up into three separate groups, you can see how they differ in how much they keep and how much they share, and therefore conclude that age causes them to keep more. Who is correct and why?
 A: I can postulate several causalities from your data.


*

*The age is measured and then the amount of money kept. Older participants prefer to keep more money (maybe they are smarter or less idealistic, but that's not the point).

*The amount of money kept is measured and then the age. People who keep more money spend more time time counting it and are therefore older when the age is measured.

*Sick people keep more money because they need money for (possibly life-saving) medication or treatment. The actual correlation is between sickness and money kept, but this variable is "hidden" and we therefore jump to the wrong conclusion, because age and likelihood of sickness correlates in the demographic group of persons chosen for experiment.
(Omitting 143 theories; I need to keep this reasonably short)


*The experimenter spoke in an old, obscure dialect which the young people did not understand and therefore mistakenly chose the wrong option.


Conclusion: you are correct, but your classmate might claim to be 147 times correcter.
Another famous correlation is between low IQ and hours of TV watched daily. Does watching TV make one dumb, or do dumb people watch more TV? It could even be both.
A: Inferring causation from correlation in general is problematic because there may be a number of other reasons for the correlation. For example, spurious correlations due to confounders, selection bias (e.g., only choosing participants with an income below a certain threshold), or the causal effect may simply go the other direction (e.g., a thermometer is correlated with temperature but certainly does not cause it). In each of these cases, your classmate's procedure might find a causal effect where there is none.
However, if the participants were randomly selected, we could rule out confounders and selection bias. In that case, either age must cause money kept or money kept must cause age. The latter would imply that forcing someone to keep a certain amount of money would somehow change their age. So we can safely assume that age causes money kept.
Note that the causal effect could be "direct" or "indirect". People of different age will have received a different education, have a different amount of wealth, etc., and for these reasons might choose to keep a different amount of the $100. Causal effects via these mediators are still causal effects but are indirect.
A: The relationship between correlation and causation has stumped philosophers and statisticians alike for centuries. Finally, over the last twenty years or so computer scientists claim to have sorted it all out. This does not seem to be widely known. Fortunately Judea Pearl, a prime mover in this field, has recently published a book explaining this work for a popular audience: The Book of Why.
https://www.amazon.com/Book-Why-Science-Cause-Effect/dp/046509760X
https://bigthink.com/errors-we-live-by/judea-pearls-the-book-of-why-brings-news-of-a-new-science-of-causes
Spoiler alert: You can infer causation from correlation in some circumstances if you know what you are doing. You need to make some causal assumptions to start with (a causal model, ideally based on science). And you need the tools to do counterfactual reasoning (The do-algebra). Sorry I can't distill this down to a few lines (I'm still reading the book myself), but I think the answer to your question is in there.
A: Correlation is a mathematical concept; causality is a philosophical idea.
On the other hand, spurious correlation is a mostly technical (you won't find it in measure-theoretical probability textbooks) concept that can be defined in a way that's mostly actionable. 
This idea is closely related to the idea of falsificationism in science -- where the goal is never to prove things, only to disprove them. 
Statistics is to mathematics as medicine is to biology. You're asked to make your best judgement with the support of a wealth of technical knowledge, but this knowledge is never enough to cover the whole world. So if you're going to make judgements as a statistician and present them to others, you need to follow certain standards of quality are met; i.e. that you're giving sound advice, giving them their money's worth. This also means taking the asymmetry of risks into consideration -- in medical testing, the cost of giving a false negative result (which may prevent people from getting early treatment) may be higher than the cost of giving a false positive (which causes distress). 
In practice these standards will vary from field to field -- sometimes it's triple-blind RCTs, sometimes it's instrumental variables and other techniques to control for reverse causation and hidden common causes, sometimes it's Granger causality -- that something in the past consistently correlates with something else in the presence, but not in the reverse direction. It might even be rigorous regularization and cross-validation.
A: Causal claim for age would be inappropriate in this case
The problem with claiming causality in your exam question design can be boiled down to one simple fact: aging was not a treatment, age was not manipulated at all. The main reason to do controlled studies is precisely because, due to the manipulation and control over the variables of interest, you can say that the change in one variable causes the change in the outcome (under extremely specific experimental conditions and with a boat-load of other assumptions like random assignment and that the experimenter didn't screw up something in the execution details, which I casually gloss over here).
But that's not what the exam design describes - it simply has two groups of participants, with one specific fact that differs them known (their age); but you have no way of knowing any of the other ways the group differs. Due to the lack of control, you cannot know whether it was the difference in age that caused the change in outcome, or if it is because the reason 40-year olds join a study is because they need the money while 20-year olds were students who were participating for class credit and so had different motivations - or any one of a thousand other possible natural differences in your groups.
Now, the technical terminology for these sorts of things varies by field. Common terms for things like participant age and gender are "participant attribute", "extraneous variable", "attribute independent variable", etc. Ultimately you end up with something that is not a "true experiment" or a "true controlled experiment", because the thing you want to make a claim about - like age - wasn't really in your control to change, so the most you can hope for without far more advanced methods (like causal inference, additional conditions, longitudinal data, etc.) is to claim there is a correlation.
This also happens to be one of the reasons why experiments in social science, and understanding hard-to-control attributes of people, is so tricky in practice - people differ in lots of ways, and when you can't change the things you want to learn about, you tend to need more complex experimental and inferential techniques or a different strategy entirely. 
How could you change the design to make a causal claim?
Imagine a hypothetical scenario like this: Group A and B are both made up of participants who are 20 years old. 
You have Group A play the dictatorship game as usual. 
For Group B, you take out a Magical Aging Ray of Science (or perhaps by having a Ghost treat them with horrifying visage), which you have carefully tuned to aging all the participants in Group B so that they are now 40 years old, but otherwise leaving them unchanged, and then have them play the dictator game just as Group A did. 
For extra rigor you could get a Group C of naturally-aged 40-year olds to confirm the synthetic aging is comparable to natural aging, but lets keep things simple and say we know that artificial aging is just like the real thing based on "prior work".
Now, if Group B keeps more money than Group A, you can claim that the experiment indicates that aging causes people to keep more of the money. Of course there are still approximately a thousand reasons why your claim could turn out to be wrong, but your experiment at least has a valid causal interpretation. 
A: Generally you can't jump from correlation to causation. For example, there's a well-known social science phenomenon about social status/class, and propensity to spend/save. For many many years it was believed that this showed causation. Last year more intensive research showed it wasn't. 
Classic "correlation isn't causation" - in this case, the confounding factor was that growing up in poverty teaches people to use money differently, and spend if there is a surplus, because it may not be there tomorrow even if saved for various reasons.
In your example, suppose the older people all lived through a war, which the younger people didn't. The link might be that people who grew up in social chaos, with real risk of harm and loss of life, learn to prioritise saving resources for themselves and against need, more than those who grow up in happier circumstances where the state, employers, or health insurers will take care of it, and survival isn't an issue that shaped their outlook. Then you would get the same apparent link - older people (including those closer to their generation) keep more, but it would only apparently be linked to age. In reality the causative element is the social situation one spent formative years in, and what habits that taught - not age per se.
A: No. There is a one-way logical relationship between causality and correlation.
Consider correlation a property you calculate on some data, e.g. the most common (linear) correlation as defined by Pearson. For this particular definition of correlation you can create random data points that will have a correlation of zero or of one without having any kind of causality between them, just by having certain (a)symmetries.
For any definition of correlation you can create a prescription that will show both behaviours: high values of correlation with no mathematical relation in between and low values of correlation, even if there is a fixed expression.
Yes, the relation from "unrelated, but highly correlated" is weaker than "no correlation despite being related". But the only indicator (!) you have if correlation is present is that you have to look harder for an explanation for it.
A: There are a few reasons why this conclusion doesn't make sense.


*

*It's not a prespecified hypothesis.

*There is no control group.

*Age is not a modifiable risk factor... depending on what question you're trying to ask.


A suggested improvement to the design is the following cross-over type study.
Same setting: random despots of any age who rule lands. 
Design: Select matched pairs of young and old dictators. Give them money pot, inspect proportion-difference withheld (old - young = $p_1$). Steal the money back so the country and the ruler have basically the same assets as before. Depose them from their respective thrones and install them in the other's land. Reperform the pot-giving, inspect proportion-difference withheld (old - young = $p_2$).
A: That must be one heavy cat!  Clearly he must be responsible for crushing the awning.
I found this one on LinkedIn.  Just because you saw some things does not mean that one caused the other.  We are free to assume and to entertain different hypotheses, but correlation does not imply causation.

A: In general you should not assume that correlation implies causality - even in cases where it seems that is the only possible reason.
Consider that there are other things that correlate with age - generational aspects of culture for example.  Perhaps these three groups will remain the same even as they all age, but the next generation will buck the trend?
All that being said, you are probably right that younger people are more likely to keep a larger amount, but just be aware there are other possibilities.
A: Causality and correlation are different categories of things. That is why correlation alone is not sufficient to infer causality.
For example, causality is directional, while correlation is not. When infering causality, you need to establish what is cause and what is effect.
There are other things that might interfere with your inference. Hidden or third variables and all the questions of statistics (sample selection, sample size, etc.)
But assuming that your statistics are properly done, correlation can provide clues about causality. Typically, if you find a correlation, it means that there is some kind of causality somewhere and you should start looking for it.
You can absolutely start with a hypothesis derived from your correlation. But a hypothesis is not a causality, it is merely a possibility of a causality. You then need to test it. If your hypothesis resists sufficient falsification attempts, you may be on to something.
For example, in your age-causes-greed hypothesis, one alternative hypothesis would be that it is not age, but length of being a dictator. So you would look for old, but recently-empowered dictators as a control group, and young-but-dictator-since-childhood as a second one and check the results there.
A: 
My thinking is that you can't infer causality from this because you
can't infer causation from correlation.

You cannot infer causation from correlation alone. For achieve causal conclusion you need causal assumptions.
So your question should be: in the example causal assumptions are declared? If yes what they are and what statistical results they imply in the data?
So:

I've just had en exam where we were presented with two variables. In a
dictator game where a dictator is given 100 USD, and can choose how
much to send or keep for himself, there was a positive correlation
between age and how much money the participants decided to keep.

No clear causal assumptions are given, so no clear causal conclusion can be achieved. Maybe we can glimpse some causal assumptions here: "In a dictator game where a dictator is given ..." because this can look like a sort of experiment. However this connection is too crude and, among others, control group are absent. In other words causal assumptions are absent or, at best, unclear.
Therefore

My classmate thinks that you can because if you, for example, split
the participants up into three separate groups, you can see how they
differ in how much they keep and how much they share, and therefore
conclude that age causes them to keep more. Who is correct and why?

if some measures  differ or not among groups tell us not much about causal relations. Your classmate is wrong.
A: A number of readers of this post think that Pearl has authored the definitive opinion on causality. However entertaining this may be, it is flawed. That post is a highly rated answer, and in that light the downvotes for this post were mistakenly awarded. A better approach to this arises from mechanics, i.e., from physics as epitomized by Newton's laws, e.g., "For every action, there is an equal and opposite reaction," in that we refer to a cause as an 'action' and an effect as a 'reaction'. Note the atemporality correctly implied by the phrase 'equal and opposite;' that is, effect is not subordinate to cause, they are equal, simultaneous and opposite.
Why atemorality? To begin with "Statistics means never having to say you’re certain”. Statistics can be used to screen for probable causes, but more is needed for a conviction regarding causality, both legally, and in experiments. Taken alone, statistical arguments do not reduce to cause and effect because the Sir Arthur Conan Doyle criterion, "When you have eliminated the impossible, whatever remains, however improbable, must be the truth" does not adjudicate between multiple improbabilities, and neither does statistics. Furthermore, naive attempts at defining causality based upon untested assumptions lead to outright rediculous statements. For example, in the Merriam Webster Dictionary one reads with dismay that Causality is "the relation between a cause and its effect or between regularly correlated events or phenomena." True enough, language is fluid and people use words without worrying about whether those words are used self-consistently, and if the reader is one of those, then my concern about defining causality in a self-consistent fashion is irrelevant, and subjects like Resolving the black hole causality paradox cannot be understood, because the definition of causality used is strict, unambiguous, and uninterpretable using sloppy definitions of causality.
In that light, we turn to physics to investigate causality, and if we do not, we will never sort out just how confusing causality is. Regarding atemporality, there are those who claim, without proof, that cause must precede effect because that "seems" reasonable. Is it? "The arrow of time" is ambiguous at the quantum level, and effects can precede causes, e.g., see The arrow of causality and quantum gravity. If a cause is persistent in time, an effect may have temporal duration, but the effect may still precede the cause in temporal sequence at the quantum level, the durations can be in negative time, and the clock can run backwards. It takes a while staring at Feynman diagrams to understand why this is the case, e.g., "Feynman used Ernst Stueckelberg's interpretation of the positron as if it were an electron moving backward in time.[3] Thus, antiparticles are represented as moving backward along the time axis in Feynman diagrams."
The OP is correct from a physical sciences point of view. In simplest form, the possibility of a physical time-independent view of causality is at the basis of the deductive-nomological (D-N) view of scientific explanation, considering an event to be explained if it can be subsumed under a scientific law. In the D-N view, a physical state is considered to be explained if, applying the (deterministic) law, it can be derived from given initial conditions. (Such initial conditions could include the momenta and distance from each other of binary stars at any given moment.) Such 'explanation by determinism' is sometimes referred to as causal determinism.
Getting a bit more complete about this, one would include Hempel's inductive-statistical model to form a scientific explanation, which link offers a more complete discussion of causality.
As for the problem at hand, age can be related to experience, but the relationship is not simple, moreover, brain function at different ages is different (time demarcation dilates with age). Experience as a modifier of behaviour is quite variable, and just because a cohort in a certain territorial and temporal sense may have similar historical experiences does not imply that any behaviour resulting from those experiences can be extrapolated to other cohorts without fear of contradiction. With respect to a controlled trial, the commonality of experiences is an uncontrolled variable that introduces an unknown and unexplored amount of spurious correlation into any binary comparison such that any difference found should not be thought of as revealing a probably causal linkage. Moreover, a probable cause, when found, would only constitute a suspicion and not something one can state with conviction; it is at best a working hypothesis not a best conclusion. Convictions concerning causality should only be drawn from a body of evidence that is inclusive enough for those convictions to be without reasonable doubt. That is not the case for the question above for which there is not enough information to claim any causal relationship beyond a coincidental context from cohort grouping. One can, indeed, formulate so many hypotheses, for example, that the evolution of generosity with age is modified by cultural/historical epoch experience, that no firm conclusions can be drawn from the problem as stated.
