# Nash Equilibrium and Dominant Pure Strategy when payoffs identical

What happens if the payoff is the same across strategies? I have some thoughts but I am not sure.

$$\begin{array}{c|cc} &C&D\\ \hline A& 1,1& 1,1\\ B& 0,1&1,-1 \end{array}$$

Dominant strategy By definition, a player's strategy X is dominant over his other strategies if strategy X is the best strategy regardless of what the other player chooses as his/her strategy.

For player 1 (Row), is ‘A’ a dominant strategy for player 1? I'm not so sure here. Yes, A is the best response if player 1 plays C, but both A and B are the best responses to player 1 when he plays D. In this case, should strategy A be considered a dominant strategy for player 1?

Same goes for player 2, if player 1 plays A, then player 2 can respond with either C or D. If player 1 plays B, then player 2 respond with C. Can i say C is a dominant strategy for player 2 then?

Nash Equilibirum Nash Equilibrium is defined such that no other player can do better by unilatarally change his strategy.

Is (A,C) Nash Equilibrium? By definition its Nash Equilibirum since player 1 will not change from A to B since payoff of 0 is not better than 1. player 2 will not change from C to D, since payoff of 1 and not better than 1. (Is this thinking correct?)

Is (A,D) Nash Equilibrium? By definition its Nash Equilibrium since player 1 will not change from A to B since payoff of 1 is not better than 1. player 2 will not change from C to D, since payoff of 1 and not better than 1. (Is this thinking correct?)

Thanks.

• Instead of your 'feeling' could you please edit the question by adding the definitions of these concepts so we can see that you did your research? Commented Apr 28, 2017 at 23:19
• Hi, sorry about that. I believe i wasn't too obvious about that. I do understand the concept but i encountered a situation (as above) where the definition seems to be on the grey area. I have edited the question accordingly.
– Jax
Commented Apr 29, 2017 at 12:21
• I think your edit is better, you are more likely get an answer this way. In the meantine allow me refine your notion of "dominant strategy" a little bit by directing you to this definition. Commented Apr 29, 2017 at 12:43