Game without Nash eqm

Take a 2 person and 2 choice game.

Rule of the game:

A coin is tossed.

For Person A, he can call head (H) or tail (T).
If Person B calls the same, ie HH or TT, then A gets $1.
If A calls H and B calls T, then A loses $1.

The reverse is true for B.
In other words, if B calls H and A calls H, then B loses $1.
But if B calls T and A calls H, then B gets $1.

The game is played infinite number of times.
Interestingly, the following result is observed...

When A calls H and B calls H, A gains $1 and A continues to call H.
But B loses $1 and will switch to T.

Once B switches his choice, A realizes that he is losing money, so he will switch to T too.

They can keep playing and continue switching but neither party will gain or lose!


Well~ I thought this is an amazing game. No winner or loser and no Nash equilibrium!