Which Nash equilibrium is this?

Question

There are two stores, A and B with the following demand functions,

The open circles indicate an open interval, the filled circle is a closed interval. Both firms wants to maximize profit, and profit is just demand times price. The best response function for both stores are plot in the right figure, where the blue line is the best response for store A, and the red line is the best response for store B.

The dashed line indicate that the best response is such that the price difference is "zero". It is not really zero, for store A it is '0 - epsilon' and for store B it is '0 + epsilon'. When both stores choose the same price, both stores has demand 15. However, store A can increase its demand to 20 by only decreasing its price to '0 - epsilon'. Therefore the dashed lines do not really intersect and it is nog a pure NE.

Is this also a kind of a Nash equilibrium? If so, which one is this? Ik can't find it on google.

Thanks in advance.

Answer 1

I found that it is an Epsilon equilibrium. An Epsilon equilibrium is a super set of Nash equilibra. It means that a player will only change there reaction if this increase there profit by more than epsilon, where epsilon is a given positive number.

In my case, a store can increase it revenue by decreasing its price slightly. However, if the value of epsilon is large enought both stores are satisfied with the current situations.

See: http://en.wikipedia.org/wiki/Epsilon-equilibrium

Answer 2

This is indeed not a NE in pure strategies because, as you point out, one player has an incentive to deviate when you fix the strategy of her opponent. This is not a NE in mixed strategies either because the BR functions do not intersect.

However, this looks very much like a trembling hands perfect equilibrium with a trembling of $\epsilon$.