Prove the assertion on the game theory.

Question

if a dominant strategy for player1 is added to finite normal form game then the payoff to player1 at any equlibrium of the new game must be at least as great as any nash equlibrium payoff of player1 in the orginal game.

This statement is true or not? Please show me its proof. Thank you.

Answer 1

I think this is false.

Consider a two-player game where player $1$ has only one action $L$ and player $2$ has two actions, $A$ and $B$. Let payoffs be as follows:

$\begin{array}{lcl} & L \\ A& (1^*,1)\\ B & (5^*,5^*) \end{array} $

Now, add the following dominant strategy $R$ for player $1$

$\begin{array}{lccc} & L & R\\ A& (1,1) & (2^*,500^*)\\ B & (5,5^*) & (6^*,0)\\ \end{array} $

Player $1$ is worse off in the only Nash equilibrium of the new game.