I am having trouble converting a signalling game with an extensive form representation to the normal form matrix.
E.g. this one from wikipedia:
I know each player has four strategies: P1: $\{AA, AB, BA, BB\}$ and P2: $\{XX, XY, YX, YY\}$ so we'll have a 4 by 4 table but then I am stuck.
For example, if player 1 plays $AA$, we are in the top part of the game, now if player 2 plays $X$ we get 2 possible payoffs, how do I discern which payoff is related to strategy $XX$ and which is $XY$ ?
Best Answer
The first step in the game is the central node labeled "N". It encodes that Nature (or the referee, or Lady Luck) makes a random choice (with probabilities $q$ and $1-q$) of whether we're in the left or right part of the game. Player 1 knows the outcome of this step, but player 2 doesn't (indicated by the dotted lines in the diagram).
If player 1's strategy is AA, then the second part of player 2's strategy (namely what P2 plays if P1 plays B) will never matter, and the outcome is the same in both cases.
However, since there is randomness involved, the payoff matrix should contain expected payoffs.
So in the case AA, XX (or AA, XY), the payoff is $1$ to each player with probability $q$ and $0$ to each player with probability $1-q$. The expected payoff for these strategies is therefore $q$.