Ethical egoism is the consequentialist theory that morality is based on everyone promoting their own individual value that is, the right act is the act that maximises the agent's value. By contrast, in classicial act utilitarianism, the right act is the act that maximises everyone's (aggregative) value and in desire-based accounts such as desire utilitarianism (or desirism for short) the right act is the act that a person with good desires would perfom.
The issue underlying ethical egoism is what is meant by maximising one's own value? Such values are usualy summed up in the terms "self-interest" or "selfish" motives. However the first term (self interest) has a variety of related meanings, only one of which is the latter term (selfish). This inderminancy weakens any descriptive evidence that could be used as support for an egoist normative theory. As a descriptive theory if ethical egoism is to be shown to be pragmatically likely true it must be an emprical argument, it must be the result of provisional, fallible and defeasible claims. What this theory needs to show is that a narrow construal of self interest is empirically correct to support the normative case.
Why is this? Well self interest as a broad conception includes other regarding desires as ends of the agent and, as a narrow conception, this excludes such other regarding desires as ends of the agent, the agents only ends are all self regarding desires. Only the latter narrow conception can be equated with selfish motives. Both self regarding and other regarding desires could be motivating reasons, however egoism asserts that there are only final selfish motivations, only self regarding desires as ends. That is narrow self interest requires that other regarding desires as ends do not exist, that they are all reducible to self regarding desires,that is all other regarding desires are instrumental, only means to such self regarding desires as ends.
Why should this be an issue? Well if a broad conception of self interest is correct, that there are other regarding desires as ends as well as self regarding desires as ends, then this cannot be used as evidence in support of a normative egoism since any agent is maximising something more than their own (selfish) value and it is quite unclear how to resolve this, unless one is happy to allow for an ad hoc metaphysical basis for normative egoism. That is the egoist cannot assume that everyone only has (ultimate or final) selfish motives as support in formulating their normative theory.
So if the egoist assumes only a broad self interest then this is not substantive evidence that can be used in support of a normative egoist argument, since agents therefore have other motives apart from selfish ones which a normative egoist theory cannot account for. The egoist must argue for a narrow self interest which is certainly substantive support for a normative egoism, but can empirical support for this be demonstrated? If not the default remains broad self interest and this narrow construal is likely false and ethical egoism is likely false or at least empirically unsupported (note there are other additional objections to egoism not covered in this post).
Why is the broad conception the default? This is due to shared assumptions, parsimony and error likelihood and minimisation in this comparative analysis. Both the broad and narrow conceptions of self interest have in common a desire-based (or equivalent, such as preference-based) framework, this is not in dispute and must be accepted as empirically sound for such comparative analysis. However the egoist has to make two types of additional empirical claims to show that other regarding desires as ends do not exist. This entails that the egoist position is both less parsimonious and more surprising in a neutral adbuctive and positive sense. However failing to demonstrate these claims implies that the default of a broad self interest remains likely empirically correct.
First what they need to do to empirically demonstrate this is to provide what would count as disconfirming evidence for this claim and show that this disconfirming evidence does is absent. It is not sufficient, although of course it is certainly necessary (as without this there is no argument at all), to show that other regarding desires can be means to self regarding ends. This, alone, is not an empirical demonstration, it could just as easily be considered a post hoc rationalisation to claim how any action, however apparently driven or claimed to be driven by other regarding desires is a self regarding desire, this still has to be shown to be the case in fact.
Secondly, the egoist has to show that this was in fact the true motivating reason behind the action, that is to also show that any claim for an other regarding desire as ends was due to self-deception, confusion, lying or similar.
Suffice it to say that no-one has empirically demonstrated support for these claims, only provided the aforementioned insufficient post hoc rationalisations.
Noting the above the egoist cannot trade over the indeterminacy of self-interest, noting evidence for a broad self interest (which no-one denies) and then equivocating to this being an evidence of a narrow self interest. To do so would be leg hopping.
To sum up showing a broad self interest exists is trivially true and vacuous, whereas asserting narrow self interest exists is a substantive claim but is most likely false. There is no empirical support here to date for ethical egoism.