Consider a simple static, incomplete-information entry game.
Two players, $i=1,2$. where player $1$ is new entrant, and $2$ is incumbent.
Player 2 can be two types: rational or belligerent, with common prior $(p,1-p)$ over her types.
Action set: $A_1=\{Enter, Out\}$, $A_2=\{Fight, Accomodate\}$
When we turn this extensive form game into normal form, we compute the expected payoff for each pair of strategies $(x,zy)$ where $x$ refers to what player 1 chooses at the information set, and $zy$ refers to what player 2 chooses in case of each type.
My question:
When we make this incomplete information game into a complete-imperfect one, we introduce Nature and let her draw the type for player 2. However, she knows her type when she is called upon to move, so why do we calculate an expected payoff for player 2 when converting the game into an normal-form? Is this because when we collapse the extensive form into the normal-form, player 2's pure strategy set takes into account each type that she can be (rational or belligerent), thus she is also forced to calculate an expected pay off? I mean, for any pair of pure strategy set, you have to deal with two terminal nodes and their payoff, so it makes sense in this perspective, you need some sort of expected payoff calculation. But it seems odd that player 2 who supposedly knows her type also has to form expectation on her payoff.
My guess is the way strategy is defined in incomplete information is that it prescribes each type of a player what she should do if this is the type Nature draws. So, as long as the cardinality of type space is greater 1, any pair of pure strategy will force each player to face more than 1 terminal nodes (eg. $(Enter, (Fight, Fight)$).
It just sounds odd that all along, Harsanyi allows a player's type to be unknown only to other players, but in the end, no matter what, whether you have various types or not, the payoffs must be computed in expectation.