Correct Option: D

Let’s first get some background knowledge into what has been described in the passage. Von Nuemann and Morgenstern were amongst the most famous mathematicians and economists; they together formed the mathematical field of ‘game theory’ and how it could be applied to economics. The given paragraph talks about the theory they developed: they formulated a theory which one can use to make a choice among the different available options. The paragraph mentions a term, ‘expected utility’ which is basically a way to balance risk versus reward with the help of a mathematical function. Utility can be described as a measure of one’s preferences amongst the options or choices available. We just have to grasp the context of passage in order to pick the right options. So, they are basically talking about a seemingly complex and thorough mathematical model which is aimed at helping someone make a better choice.
Option ( d ) is correct, as apparently is an adverb which means ‘as far as one can see or knows’. This adverb is trying to indicate that the rest of the sentence followed by this word is an information which one can see or know after analyzing this decision framework. So, this word makes proper sense. Option (a) is incorrect, because regrettably means unfortunately or unluckily, so this is used when telling something which is not good or not desired, but as we can see, the phrase which follows it is only adding more information about what has been talked about previously in the passage. This is being done dispassionately. It does not tell anything that is unfortunate or undesired.
Option (b) is wrong, because obviously is used for something that is very ‘visibly clear and unmistakably true’. But for someone who is just reading about this new theory or system for the first time or someone who does not know much about it, what this sentence goes onto say cannot be deemed ‘obvious’. It will not be appropriate to use that word.