Hi Ahmed and Bryan:

I really like the unification we had on Thursday:
integrating absolute and relative (or as I prefer to call them:
global and individual).

Below is my analysis of the situation.

-- Karl

Two dimensions:
extensional / intensional
absolute / relative

unify:
use idea of default solution for secret solution.
absolute: same default solution for game, part of game configuration.
relative: carefully selected secret solution.

better words:
global solution = default solution
individual solution = secret solution

challenge = (predicate, price)

One formula:
P = price = plays the role of approximation factor
BQ = quality achieved by buyer on problem = buyer quality
SQ = quality achieved by seller = seller quality
  (global or individual solution quality) 
PF = profit factor, determined in game configuration

income is after having paid price p and if you win.

win = if BQ >= P*SQ (Bryan did not like >=; prefers > Why?)
For global, we usually have even: BQ >> SQ.
For global case always win because can always take default solution
which is known from configuration file.

For global game: 

income = P + (BQ - P*SQ)*PF

issue:
SQ may be 0 for global (absolute) case:
P + BQ*PF: looks good

It should be much easier to beat global solution
than individual solution.

Why does it make sense to use price as approximation factor?
Because seller has advantage over buyer: seller can hide secret
carefully so that it is difficult to just approximate it.
The seller might not have an algorithm to retrieve the secret
if it would not be known already to the seller.

Let's consider the CSP version of the game to check whether the new
version of the game with a default solution can still be analyzed using the old techniques.
Let's assume a set of relations RR that are false under the
assignment all 0. For a global challenge (RR, price) with default assignment all 0,
we still need to solve the exactly same min max problem where
the worst-case are the symmetric formulas.

If RR contains relations that are satisfied under all 0, we need to subtract
an expression from the formula. But if the default solution is simple,
like all 0 or all 1, it is easy to find the expression.
The min max problem needs to solve this modified formulas and the
symmetric formulas are still the worst case.


contest: tournament | group.
tournament : full | swiss.