If you are under instructions not to use "for example" in a paper, I can't imagine that using e.g. (short for "exempli gratia," which in Latin—according to Merriam-Webster's Eleventh Collegiate Dictionary—means "for example") will take care of the problem—unless your adviser is easily bamboozled by abbreviations or really loves Latin.
Like Drew in the comments above, I am mystified by the prohibition against using "for example" to introduce an example. Perhaps, as a matter of style, the adviser feels that you should simply give the example without announcing that it is one? Even so, a blanket prohibition against using the phrase strikes me as being odd and excessive.
Still, if you feel obliged to abide by the prohibition, and yet you don't like to launch into an example without hinting to the reader that that's what it is, you might try working around the adviser's rule with a one-two punch of the following type:
The state of New York is a case in point [or "is an illustrative case" or "demonstrates how this works in practice" or "has adopted one such approach"]. It uses Auctions to assign...
Arbitrarily disallowing the use of a common and helpful phrase like "for example" seems to me to be counterproductive and exceedingly idiosyncratic, bordering on capricious. But you can work around the prohibition in ways other than simply replacing "for example" with e.g., without having to go through too many contortions. On the bright side, at least your adviser didn't tell you that you shouldn't use examples in academic work.
Update (May 25, 2018)
Having discussed how to deal with the problem that your advisor opposes the use of "for example" in constructions such as "The state of New York, for example, uses Auctions to assign..." let me turn to relevant counsel from the MLA Style Manual, a major style guide for academic writing in the United States. Here is what the MLA Style Manual, second edition (1998) says generally about the use of abbreviations in scholarly research:
Abbreviations are used regularly in the list of works cited and in tables but rarely in the text of a research paper (except within parentheses). In choosing abbreviations, keep your audience in mind. While economy of space is important, clarity is more so. Spell out a term if the abbreviation may puzzle your readers.
The upshot of this guidance is that using an abbreviation such as e.g. is fine in a chart, a bibliography, or a footnote or end note, but is rarely appropriate in main text (as your sentence seems to be). It follows that simply replacing "for example" with "e.g." in main text would probably be a step backward in satisfying MLA style.
Meanwhile, the MLA Style Manual says nothing about avoiding the use of "for example." It does, however, use that phrase itself. For example, section 3.2 ("Language and Style") contains this paragraph:
Careful writers do not use language that implies unsubstantiated or irrelevant generalizations about such personal qualities as age, economic class, ethnicity, political or religious belief, race, sex, or sexual orientation. Many journals and book publishers have strong editorial policies concerning the avoidance of such language. Since 1981, for example, PMLA's statement of editorial policy has urged "its contributors to be sensitive to the social implications of language and to seek wording free of discriminatory overtones."
I see no support in the MLA Style Manual for the proposition that use of "for example" is inappropriate in academic work. It follows that your problem isn't with universal views of proper scholarly style, but with your adviser's idiosyncratic preferences—and gauging how to satisfy those preferences returns us to the practical advice I offered in the first part of my answer.