Same is one of the most common words in the English language. It has been long used as an adjective, a pronoun or quasi-determiner, or even as a noun proper. But being roped into use as an adverb is now fairly rare in today’s literature, and is almost only ever used to mark rustic or uneducated speech. You probably should not do that.
I’m afraid that the referenced learner’s dictionary has compressed matters down so far as to leave you with a misrepresentation. It is not possible for a multiword phrase to “be” a part of speech like a noun, verb, adjective, or adverb. When we write:
The man walked the dog.
Then your subject and your object, respectively “the man” and “the dog”, are not themselves nouns the way man and dog alone are. Rather, they are noun phrases, which have their own anatomies and anomalies. Here you know you have a noun phrase because of the definite article the preceding both nouns. You also know you have a noun phrase because only noun phrases can serve as subjects and objects.
Now let’s expand our sentence:
The man walked the dog and the cat at the same time.
We’ve created a compound direct object by connecting the dog and the cat using a coördinating conjunction, and then we’ve added a prepositional phrase at the very end. That “at the same time bit there at the end is neither a noun or a verb nor an adjective or an adverb. It’s a prepositional phrase composed of the preposition the and the noun phrase the same time. In that noun phrase, the is a definite article, same is an adjective, and time is a noun. The entire prepositional phrase is an adverbial modifier of the predicate. But it isn’t an adverb, and it contains no adverbs within it, either.
Prepositional phrases are a type of syntactic constituent that can be used in the same way that you can use a modifying adjective or adverb.
- I’ll see you then.
- I’ll see you on the last Sunday in April.
Notice how there I’m using the preposition on followed by a noun phrase. It is a property of noun phrases in English that they can sometimes be used as modifying phrases despite having no connecting preposition. For example, here we can delete the preposition with no change in meaning:
- I’ll see you the last Sunday in April.
That’s what’s going on with your example:
An uncommitted player reacts to different alliance types the same.
That’s not an adverb: it’s a noun phrase being used as an adverbial modifier of the predicate.
Here’s something that looks the same as your example, but isn’t. When someone says:
Person 1: I’d like a fresh beer.
Person 2: I’d like the same.
That one isn’t a modifier, though. That’s a substantive use because it’s the object. The word same itself is probably best thought of as a sort of pronoun that originated as a nominalized adjective meaning same thing. We make pronouns like this a lot in English, like when we talk about the other meaning a person. It is rather rare but possible to use same in this way without the article, or with actual nominal inflection into the plural. Here’s one of each:
- 1926 in H. W. Fowler Mod. Eng. Usage 512/1
Sir,—Having in mind the approaching General Election, it appears to me that the result of same is likely to be as much a farce as the last.
- 1977 Trans. Philol. Soc. 1975 9
Certain configurations in languages typically result from the principled (‘lawful’) divergence over time of original sames.
That first example from Fowler still shows up from time to time in legal language, but it’s not very well thought of. The second example is a highly specialized use you’ll rarely come across.
Just as we can omit the defined article from “the same” in select substantive uses, we can also do so when using it as an an adverbial modifying phrase. Doing so is more common than the strange-sounding examples I just provided, but it remains little done in literature, with most modern examples considered to be representative of dialect, illiterate, or jocular speech.
When you use the noun phrase “the same” as an adverbial modifying phrase, it mean “in the same manner”. Here are some examples of that from the OED over the past century and a half or so:
- 1857 ‘S. Sondnokkur’ Ryde fro Ratchda to Manchistur (ed. 2) iv. 9
Aw kuddunt elp wundurin..wether it wur to put iz grund coffi in, saym uz wi dun o whoam.
- 1857 ‘S. Sondnokkur’ Ryde fro Ratchda to Manchistur (ed. 2) vi. 14
Thir wur o rattlin saym uz uv o lot a peawur looms.
- 1861 ‘G. Eliot’ Silas Marner xviii. 325
You'll never think the same of me again.
- 1884 ‘M. Twain’ Adventures Huckleberry Finn ii. 24
Strange niggers would..look him all over, same as if he was a wonder.
- 1930 W. Faulkner As I lay Dying 4
She ought to taken those cakes when she same as gave you her word.
- 1933 M. Lowry Ultramarine i. 16
He knows bloody well same as myself it doesn't pay to shout and be unkind to youngsters.
- 1957 L. P. Hartley Hireling viii. 65
But I shouldn't be able to serve them personally, same as I do now.
- 1975 Listener 6 Feb. 174/1
There was no work... They were all bad years, because, same as I say, there was nothing.
Those examples of same without a definite article really are places where it’s an adverb. They’re also considered ‘sloppy’ speech, not something educated speakers say, deliberately used that way to convey that point. The same thing happens in the phrase “same like” where it means ‘just like’:
- 1959 A. Christie Cat among Pigeons ix. 107
‘See no evil, hear no evil, think no evil. Same like the monkeys,’ observed Sergeant Percy Bond.
- 1968 ‘L. Egan’ Serious Investigation vi. 78
But same like the gent in Holy Writ, Beware the anger of a patient man.
- 1973 G. Mitchell Murder of Busy Lizzie xv. 185
Ain't going to be no share-out. Same like the boy with the apple-core, if you happen to know that story.
- 1980 I. Murdoch Nuns & Soldiers vii. 382
I have rich friends, same like you.
Another rare adverbial use is the little-seen compound same-ways:
- 1887 Sir W. Thomson in Nature 3 Oct. 546/2
Every A is at the centre of an equal and similar, and same-ways oriented, tetrahedron of O's.
You’re safest using same with the if you’re going to use it adverbially:
- I always cook it the same as my grandma made it.
But be again warned that this is now rare in literary use. A more carefully worded version of that might be:
- I always cook it the same way (that) my grandma made it.
That said, I can find little fault in examples where “the same” is used to mean the opposite of differently:
- We never treat our children differently.
- We strive to treat all our children the same.
The OED still thinks that flavor doesn’t show up much in literature any longer.