Consider a war game where each player starts with a building and maybe a bomber. A bomber can destroy a building but cannot attack other bombers.
Once a player loses his building he loses the game. The final score of a player is based on the order he loses the game; the first to lose will have $0$ point and the $n$th to lose will have $n-1$ points .etc; the winner counts as the last person to lose the game)
For example, if we have three players $A, B, C$ and $A$ loses first, $B$ loses next and $C$ wins, the scores of them will be $A=0, B=1, C=2$.
If two players loses simultaneously they got the same average point. For example, if $A$ and $B$ loses at the same time and $C$ wins, $A=0.5, B=0.5, C=2$. If the three players all lose at the same time they all get one point.
In our scenario we have three players $A, B, C$ playing the above game. The buildings of the players are placed in an equilateral triangle fashion and only player $A$ and $B$ each have a bomber. $C$ has no bomber so he can only sits and wait for the fate. All bombers have the same speed and starts at the building location of their respective players.
Intuitively, player $A$ and $B$ clearly have advantage over player $C$ because they have what $C$ have (the building) and they also got bombers which $C$ do not have.
Now consider the Nash equilibrium for player $A$:
(1) If $B$ goes for bombing $C$, then $A$ should go for bombing $B$ which he will guarantee a win and get $2$ points rather than goes for $C$ which he will end up with $0.5$ point.
(2) If $B$ goes for bombing $A$, then $A$ should also go for bombing $B$ which will give $A$ a $0.5$ point; if $A$ choose to bomb $C$ he will end up with $0$ point.
Similarly the Nash Equilibrium for player $B$ is also that he should go for bombing $A$.
This seems to me the only Nash equilibrium of the game. Ironically under this equilibrium situation player $A$ and $B$ will kill each other and $C$ will end up with winning the game.
This does not really make any sense to me intuitively because $C$ is clearly inferior to $A$ and $B$. Is there anything wrong with my logic or is there any explanation that can make sense of this intuitively?