For the clairvoyance game, example 11.1 on pages 111 to 112, the ex-showdown payoff entry when Y the nuts and X folds is 0, but when Y bluffs and X folds the value is +P. This convention is continued throughout the analysis. I don't understand why we count the pot size only if it is won through a bluff. It seems to me the value should be +P when Y has the nuts, bets, and is folded to as much as when Y bluffs and is folded to. What am I missing here?

Ex-showdown helps you focus on the pure value of a strategy. Plus, it makes the math easier. Like with energy in thermodynamics classes, calculating the value ex-showdown sets the scale so that external terms can be conveniently ignored.

I still don't understand why the pot size is included in the ex-showdown value if it is won via a non-bluff bet and not if it is won via a bluff.

When the second player has the nuts, the pot is already his.
When he bluffs successfully he has taken the pot away from the first player.

I see your point, but to discount the value of the pot from the ex-showdown value when player has the nuts seems needlessly complicated and not very useful. The treatment would be simpler and more intuitive if it were more uniform.

One problem with this formulation would arise in game with ties. For example, consider a half-street two-person game where the pot is P, where Y, the first player to bet can bet one unit or check, and where X can call or fold. Suppose each player is independently dealt a uniform random integer between 1 and 10. At showdown, the highest number gets the pot; tie chops.

Now, what is the ex-showdown value of Y's hand if he has the 10? There's no clear answer, because the concept of discounting pot size from ex-showdown value in certain special cases is inherently obscure.

The beauty of [0,1] games are there are no ties.

Zero.

Hands have showdown value. Strategies have ex-showdown value. If you play in some way that alters the outcome compared to a showdown, then your strategy has a non-zero ex-showdown value. Since the book focuses on strategies, ex-showdown is a more natural way of looking at things. Plus, it makes the math easier, since terms that would cancel out in the final result anyway don't appear in the equations in the first place.

Nothing has changed. Y wins 1 unit whenever X calls with the losing hand, else wins 0 units as no money exchange hands on ties or folds by X.

This is the clairvoyant game with Y as the clairvoyant.
When it's Y's turn to act he already knows who holds
the best hand. With the loser Y is assured of zero
by checking. Y only bets if his EV is positive by
betting. Ex-showdown is Y's value from betting.
If the ex-showdown matrix shows that Y will receive
a -EV by betting, he will always check.
The ex-showdown is the extra EV from betting when Y
has the best hand.
In a manner of speaking the treatment is uniform. It's about
what can be made from betting.