In Martin Osbourne and Ariel Rubenstien's book A Course In Game Theory (page 20) the authors say that a 2 person game is symmetric the A1 = A2 (both players have access to the same action set)
and
(a1,a2) is weakly preferred by player 1 to (b1,b2) IFF (a2,a1) is weakly preferred by player 2 to (b2,b1) and then we are asked to show that the NE is (a1*,a1*)
What does (a1,a2) mean? Player 1 plays a and player 2 plays a also? If so what does (a1,a1) mean?
thanks
Yes you are correct, $(a_1,a_2)$ is what we call a strategy profile: in this profile, player 1 plays $a_1$ while player 2 plays $a_2$. Remember the definition of a strategic game in Osborne: players, strategy sets for each one of the players and payoff functions. The payoff functions are maps from the space of strategy profiles (also referred as outcomes) to the real numbers. For the two player case, $u_i:A_1\times A_2\rightarrow \mathbb{R}$ for $i=1,2$. What Osborne is trying to say in a more general way is that for a symmetric game: $$u_1(a_1,a_2)=u_2(a_2,a_1).$$ Also $(a_1,a_1)$ gives a payoff in the diagonal (if you write the game in matrix form): it is the profile where both players choose $a_1$.