I question your question. Even Christians believe in punishment, which is all that is required in the case of defection.
A pedestrian and more fully expressed version of the Golden Rule from mathematics would be like the following: Do unto others as you have them do unto you. But if they do harm, punish in like kind.
These are really key theorems in game theory, that the maximally optimal behavior is to "cooperate" with others whenever possible, but if others "defect" or harm you, optimality is obtained by retaliating in like kind.
Since you are interested in this kind of thing, I will point out that these assumes rough parity in the nature of the agents involved. If one posits strong disparities in the fundamental capabilities (not natures) of the agents (e.g. man versus animals), then the optimality of interaction looks quite a bit different. Generally speaking, humans tend towards optimal behaviors on average. The reason is simple economics; optimal behavior is powerful multiplier of utility. Grossly sub-optimal behaviors are terribly expensive and not sustainable, particularly in the face of competition from groups that behave in more optimal ways.
(One can make some get into some really interest theological discussions from game theory due to some of the interesting consequences of asymmetric agents e.g. God versus man. Frankly though, I don't think this forum is up for it; people have a hard time dealing with asymmetric game theory.)
To clarify something which was badly written, the narrow subset of game theory normally applied to human interaction assumes rough parity of intelligence in the general abstract sense. You can have any odd mixture of good people and bad people, but the game theory will always work out the same if all the parties involved have roughly equivalent intelligence. Having lots of "bad" people or lots of "good" people in a population does not change the outcomes of the basic theory, which is also why cultural indoctrination and behaviors tend to have only a mild impact on game theoretic outcomes around the world at large. The aggregate nature of the behaviors of a population don't change the basic game theory that one applies to determine the optimality and outcomes of said behaviors.
Where things become very interesting is when you leave the narrow case of parity with respect to intelligence of the agents in a system. Our justice system is premised on this parity, but one can show that if there is sufficient disparity (e.g. if you have a brilliant criminal and stupid police/prosecutors or vice versa) then the likely game theoretic outcome will NOT be "justice" in the sense you would expect if there was parity. Parity of intelligence is a useful fiction, but sometimes you can see where it strains the system as expectations meet theoretical reality. Game theory as applied to strong asymmetries of intelligence represents a real conundrum, particularly as it applies to social and cultural institutions.
My study of asymmetric systems has given me a much better idea of just how fragile and unreconcilable many of our institutions are in a larger sense, and changed my views on some issues. It would take a relatively small asymmetry to affect radical changes in the basic nature of the systems, and with the progress of technology, such asymmetries are feasible within our lifetimes. Our social and cultural institutions as we know them literally cannot exist in an environment where such asymmetries exist; they are not stable constructs outside narrow game theoretic premises that just happen to be the rule for now as practical matter.
I believe that game theoretics can tell us a great deal about why people behave as they do and also give us much insight into history and the likely outcome of current events. It would be very useful to discuss this, but because of its wide applicability - I’d like to see it on a separate thread where "atheism, science, and the Golden Rule" is not the main focus. You said:
But Christianity is unique! Here is the Golden Rule as you asserted it from Scripture, i.e. Matthew 7:12 (emphasis mine)
And the second [is] like unto it, Thou shalt love thy neighbour as thyself. On these two commandments hang all the law and the prophets.
Going back to the Sermon on the Mount, Christ explicitly describes this fulfillment of the law in Matthew 5:17-48 (excerpted):
For I say unto you, That except your righteousness shall exceed [the righteousness] of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven...
Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth: But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
And if any man will sue thee at the law, and take away thy coat, let him have [thy] cloke also. And whosoever shall compel thee to go a mile, go with him twain. Give to him that asketh thee, and from him that would borrow of thee turn not thou away.
Ye have heard that it hath been said, Thou shalt love thy neighbour, and hate thine enemy. But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; …
Be ye therefore perfect, even as your Father which is in heaven is perfect.
Enter ye in at the strait gate: for wide [is] the gate, and broad [is] the way, that leadeth to destruction, and many there be which go in thereat: Because strait [is] the gate, and narrow [is] the way, which leadeth unto life, and few there be that find it.