Reading: McAllister paper on "Consipracy Theory"?

What are the big points in this paper?

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Big Idea:  
-look at the set of nodes "conspirators" that make the value of a node change
-allow incomplete trees
-How many conspirators?

      root (max)
      /\
     /  \        suppose x's value = 0
min /    \       Then, root value = 0.
   /\    /\
  /  \  /  \
 /    \ X   1
-1    0			Now, Change a leaf value of X

			Q. What's the
			best/worst change?

			A. +infinity, -infinity
			Q. What effect does
				this have on the root?


			Change x's value to -infinity --> change root
			to -1

			 Change x's value to +infinity --> change root
			to +1


*	If it is likely that one leaf can change but unlikely that two leaves
can change, tehn the value of the root in this tree is in [-1, +1]

*	In a large, incomplete tree, might have to change a large number of
different leaves before the root value would change.
	(i.e.  the SBE would have to be wrong many times.  probability?)

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Definition:
-If C is a set of leaves, s.t. by changing the values of the nodes in C, one
can change the value of a node "n" to the value v  , we call C, the set of
conspirators for assigning v to "n".
			(conspirators n v) ==> C

NB: the more conspirators (higher conspiracy numbers), are needed to
achieve a value, the less likelh that value will be achieved.

Questions:
*			How do we find critical leaf nodes?
*			How do we compute bounds on the root value range?
*			Use conspiracy numbers (|C|) to control search depth



Meaning of Conspiracy Theory

1. the goal of alpha-beta search is to construct strategies for each player that
together establish a value for the root.
	-Maximum strategy establishes a lower bound
	-Minimum strategy establishes an upper bound

2. Conspiracy numbers can be used to measure the "safety" of these two
strategies.
	-Maximum strategy is safe to the extent that there exist "
options (alternative moves) to the max moves selected in the strategy.
		-if one of the lines of play turns out not to work
(i.e., not to achieve the desired root value then a different live of
play can be chosen, provided that options exist to the selected moves
in the strategy.
	-so...conspiracy numbers measure the "density" of the solution
space...  The number of paths to winning.

3.  A strategy is usafe if there are options to the selected moves,
i.e. the selected moves are forced.

	-Forced lines of play should be explored more deeply, beyond search
horizon. (Shanonn and Turing, 1950, 1953)

This leads to the implementation of "Quiescence Search"

Definition: The quiescense value of a position is the value computed form
searching the stand-put or capture tree, SPOC, from that position.
	-(at each postition, the player to move can either elect not to move
"stand pat" or can elect a capture move) 

Fact:  The SPOC tree can be seear ched quickly in practice, especeially when
the static values fall outside the alpha-beta window.

Fact:  Almost all competitive programs use some form of quiescence values at
the search leaves.

Can interpret this using conspiracy theory!

Definition:  The conspiracy depth of a strategy is the least number of leaf
nodes where value must be changed in order to defeat the strategy.

-"depth" in a maximum strategy is called maximum depth.

1) The maximum depth (in a max strategy) only increases at max moves---This is
where  the maximum player has options supporting the safety of the strategy.

2) If a maximum move in a max strategy is forced (i.e. there are
options to that move), then max depth does not increase across that
move and the search must be carried further to achieve a desired
safety (depth measurement).

3) If a min move happens to be forced in a max strategy, this fact does not
influence max depth.  Max depth only measures the options of the max player!

1), 2) and 3): intution hairy

Observation: Capture moves not to increase conspiracy depth!
	-this observation underlies both quiescence search and capture
extensions--extensions granted at capture moves.

Fact:  Generating an extension at each capture generates too large a tree!  eg: 
Capture exchanges!  P/P; R/R; Q/Q; R/R.

Question: How can we use conspiracy theory to auromatically search deeper under
a restricated class of capture moves? (research)


Summary:

* Conspiracy numbers provide a game--independent theoretical framework
for guiding non-uniform growth in min-max game trees

*	 an extension of alpha-beta

*	Idea: grow search trees for which one has confidence in the root min-max
value.

*	Conspiracty theory measures confidence by measuring the number of leaf
values that would have to be changed to bring about a given change in the root
value.

*	Crisp, clean theorems.