Game Theory: Your enemy won't allow you to be a friend of his/her enemy

There is a famous proverb, "The enemy of your enemy is your friend". But, it is questionable to regard that all individuals unconditionally behave by means of what this proverb suggests. In order to lead you to this situation, you and the enemy of your enemy has to trust each other very deeply (E.g. You guys are related, already known to each other, and/or guaranteed to gain the mutual benefit and interest together already).

Let's imagine that there are three individuals with the same power and no emotional attachment on the others (All of them are completely anonymous to each other). These individuals are homogeneous in terms of their characteristics. These three individuals are named, A, B, and C accordingly.

When A is in a conflict with both B and C (I.e. B is the enemy of C's enemy A, and also C is the enemy of B's enemy A), A loses in this situation if A accepts this situation and does nothing. Because A has to fight with both B and C at the same time, and B's and C's characteristics is identical to A's, A has a big disadvantage to be alive unless there is an exogenous advantage like a geographic advantage.

The philosophy of "The enemy of your enemy is your friend" looks also irrelevant to B and C to defeat A. Even though B and C do not make an alliance (Becoming a friend of each other), A's loss is inevitable under this situation.

Therefore, A will have to intervene into the relationship between B and C in order to survive in this situation. Then, what Friedrich Nietzsche sounds more striking as being most likely. He said "The best weapon against an enemy is another enemy".

Under the situation that both B and C do not cooperate together, A will advice either B or C, or both, to antagonise the other. When B and C are already cooperating together to plot to defeat A, then A will attempt to break up the coalition of B and C until either B or C betrays the other.

So, let's take a look of this following graph:

First of all, it is the least likely option for A not to intervene into B and C because it is a suicidal choice for A as explained previously. Then, B and C suffer from a dilemma to choose whether they corporate together to defeat A or not.

Secondly, B and C can cooperate together, and so reject A's attempt of dissolving their alliance. However, when B or C takes an offer from A and betray his/her ally, this betrayer can not only survive in this battle but also gain A's reward for the betrayal. Then, there is always a suspicion that B or C will be convinced to A's attempt. If B and C are stranger for each other, then B and C are always possible to betray the other. Therefore, B is suspicious about C, and C is suspicious about C.

If B trusts C, and C betrays B, then C gains A's reward, and B is defeated. If C trusts B, and B betrays C, then B gains A's reward, and C is defeated. If both B and C trust each other, and ensure both of them do reject A's offer, they corporate. Nonetheless, if both B and C are anonymous (Neither related to each other nor sharing the mutual benefit and interest guaranteed), trusting their partner involves a high risk of being defeated. Then, both of them eventually accept A's offer plotting them to antagonise each other. This choice is cruel but the safest option for B and C to survive.

All in all, the most likely outcome in this game is that all A, B, and C are fighting altogether. B will betray C, and C will betray C in order to secure their "safest" option to survive. But, it does not mean that both B and C will be friendly with A. Both B and C will eventually realise that the cost of accepting A's reward for betraying their former ally is high. So, they start hating A for making them antagonising each other. Furthermore, B and C do not remain enough power to defeat both A, their old enemy, and their new enemy, their former ally because all of them have to fight with two enemies. Unless there is a shock to provide any of them with an exogenously gained advantage, all of these three individuals keep fighting together until all these three of them cease away by exhaustion.


Hence, "The enemy of your enemy is your friend" is wrong because your enemy won't allow you to be an enemy of his/her enemy, and so "The best weapon against an enemy is another enemy" (Friedrich Nietzsche) tends to be correct (realistic?!).