<p>First, make sure you have a definition of strategy clear. Strategy describes the actions a player chooses conditional on information etc. In your case it's a one-shot game so perhaps the distinction between an action and a strategy is not that important.</p>

<p>In game theory there is the notion of a "weakly" or "strictly" dominant (or equivalently "weakly" or "strictly" dominated) action. Nash equilibrium does not allow for any strictly dominated action to be chosen. It can be, however that a weakly dominated action is chosen. Hence you cannot cross out a whole row or column because an action is weakly dominated.</p>

<p>Your thinking regarding to NEs is correct. In both AC and AD no player can be strictly better off by changing his/her action conditional on opponent's action being held constant.</p>

<p>Finally, game theory done properly is rigorous enough (at least at undergraduate and graduate level), that there is never grey area. The definitions you gave are not stated mathematically and not 100% correct either, hence they seem a bit vague. </p>

CC BY-SA 3.0
2017-04-29T17:03:31.423Z
Hi ElChorro, thanks for the clarification.
Jax
CC BY-SA 3.0
2017-04-29T14:59:13.460Z
2017-04-29T14:59:13.460Z
ElChorro
<p>What happens if the payoff is the same across strategies? I have some thoughts but I am not sure.</p>

<p>$$
\begin{array}{c|cc}
 &amp;C&amp;D\\
\hline
A&amp; 1,1&amp; 1,1\\
B&amp; 0,1&amp;1,-1
\end{array}
$$</p>

<p><strong>Dominant strategy</strong>
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.</p>

<p>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?</p>

<p>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?</p>

<p><strong>Nash Equilibirum</strong>
Nash Equilibrium is defined such that no other player can do better by unilatarally change his strategy. </p>

<p>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?)</p>

<p>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?)</p>

<p>Thanks.</p>

CC BY-SA 3.0
2017-04-28T23:19:36.317Z
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?
Giskard
CC BY-SA 3.0
2017-04-29T12:21:19.523Z
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
CC BY-SA 3.0
2017-04-29T12:43:01.153Z
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](http://www.gametheory.net/dictionary/WeaklyDominantStrategy.html).
Giskard
<game-theory>
Nash Equilibrium and Dominant Pure Strategy when payoffs identical
Jax
# Nash Equilibrium and Dominant Pure Strategy when payoffs identical

(Asked by: username_0 on 2017-04-28)

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.

**Tags:** game-theory
> **Comments:**
> **username_1**: 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?  
> **username_0**: 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.  
> **username_1**: 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](http://www.gametheory.net/dictionary/WeaklyDominantStrategy.html).  

---

## 1 answers

### Answer by username_2 on 2017-04-29. Score: 1

First, make sure you have a definition of strategy clear. Strategy describes the actions a player chooses conditional on information etc. In your case it's a one-shot game so perhaps the distinction between an action and a strategy is not that important.

In game theory there is the notion of a "weakly" or "strictly" dominant (or equivalently "weakly" or "strictly" dominated) action. Nash equilibrium does not allow for any strictly dominated action to be chosen. It can be, however that a weakly dominated action is chosen. Hence you cannot cross out a whole row or column because an action is weakly dominated.

Your thinking regarding to NEs is correct. In both AC and AD no player can be strictly better off by changing his/her action conditional on opponent's action being held constant.

Finally, game theory done properly is rigorous enough (at least at undergraduate and graduate level), that there is never grey area. The definitions you gave are not stated mathematically and not 100% correct either, hence they seem a bit vague.

> **Comments:**
> **username_0**: Hi ElChorro, thanks for the clarification.  

---

