I am doing a beginners course in game theory, and I have been repeatedly coming across situations of the following type:
Consider the following game:
So one would say that strategy $R$ is strictly dominated by strategy $C$, because for every choice of $R$, a better payoff can be obtained by choosing strategy $C$. So then you might do something like remove strategy $R$ completely in a process of iterative removal.
But this is only considering the absolute payoff of $R$ relative to the absolute payoff of strategy C. Surely it makes much more sense to consider the payoff of $R$ and and $C$ relative to the payoff of $U$, $M$, and $D$.
I know that if I was an agent in this game and another agent played $U$, I would definitely rather play $R$ than $C$, because then I would not be losing. Whereas if you play the 'dominant' strategy, $C$, you would be losing, but you would have more points, but having more points is irrelevant, because you would be losing rather than drawing...
Why is it that a strategies payoff is always analysed in absolute terms rather than relative terms?
Maybe I am not far enough into the course yet.
When we write down payoffs in game theory, we specify them so that the numbers listed capture everything that a player cares about in the game. If there are other concerns over outcomes that aren't captured in the payoffs, then the payoffs need to be modified so that they do.
Going to your example, if you are thinking of a game where players get points (according to the table you've given), and whomever has the most points wins, then the correct payoffs are (assuming a payoff of $1$ for a win, $0$ for a draw, and $-1$ for a loss):
\begin{array}{|c|c|c|c|} \hline 1 / 2& L & C & R \\ \hline U & 1,-1 & 1,-1 & 0,0\\ \hline M & 0,0 & 0,0 & 1,-1\\ \hline D & -1,1 & 1,-1 &-1,1\\ \hline \end{array}
Clearly, in this formulation of the game, $C$ does not dominate $R$, because it gives a worse payoff than $R$ when player $1$ plays $U$.