hinter historians.

Neater:

Since the aim of this book is to ...

... demonstrate to historians ...

Willst du die Historiker präsentieren oder doch eher den Historikernetwas zeigen?

Ansonsten: Wenn ich Deinen Satz richtig verstanden habe, gehört "with this volume" hinter "approaches". Möglich wäre auch noch nach "since" oder (mit Kommata abgegrenzt) nach "intend".

Edit: Schließe mich #2 an, eine andere Formulierung wäre schöner.
Edit 2: Hinter "historians" würde ich das nur setzen, wenn der Teil mit der usefulness vorgezogen wird, also "...I intend to demonstrate the usefulness of ... approaches to historians" (was ich allerdings auch schöner finde als Deine variante)

If you want to retain the original phrasing, I'd rearrange the sentence a bit:

Since I intend to demonstrate the usefulness of economic or game theoretic approaches to historians in this volume, [...]

That said, all the other options given above are just as valid.

Zu 3, Edit 2: I would not put "with this volume" after "historians" either. Also, I strongly prefer leaving "historians" in its current position.

One more thing: game theory approaches, not game theoretic approaches.

If you really want to keep the rest of the sentence as it is, I would add it parenthetically after intend:

"Since I intend, with this volume, to demonstrate to historians the usefulness of economic or game theory approaches, I ..."

The problem is that the sentence is overloaded.
Since it is obvious that it is the author's intention, the passive would probably be more appropriate:

"Since this volume is intended to demonstrate to historians the usefulness of economic or game theory approaches, I ..."

Another problem is that a clause with "since" may make an important intention of the author appear incidental. Unless you have discussed this intention previously, it might be preferable to use a main clause and express the causality or purpose differently. "This volume . . .; so/therefore/for this reason . . ."

Use of "because" rather than "since" would be another way of making the reason less incidental (but bear in mind that a few readers, having misunderstood some advice, may think that one should not begin a sentence with because).

It depends a bit on the context.

I think #2 was being kind with "neater" about adding "to" to "historians". Or did "neater" just refer to "Since the aim of this book is to ..."? Anyway, the text in #0 would not be grammatically correct without "to", IMO, unless it is rephrased.