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{\large\bf A power algebra of games of choice}

Ingrid Rewitzky

{\small \em Department of Mathematics and Applied Mathematics,
University of Cape Town}
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A very simple game of choice is one which starts in a fixed position and
whose outcome is determined by a single pair of moves: the player selects
a set of possible outcomes, then the adversary selects an element of this
set. A game is lost by the participant who is faced with the choice from
an empty set: if the family of sets of outcomes from the starting position
is empty then the player loses, if the set chosen by the player is empty
then the adversary loses. Such games may be viewed as forming a power
algebra of the power set Boolean algebra of outcomes.

Allowing a set of possible starting positions yields a more interesting
game which may be viewed as a binary multirelation from positions to
sets of outcomes. The family of these games has a rich lattice-theoretic
structure inherited from the power algebra of games of choice.

With this algebraic view of simple games of choice we invoke (a 
generalisation of) the techniques of J\'{o}nsson/Tarski duality to 
calculate, given a game and a set of outcomes as winning positions 
for the player of the game, the initial positions from which the 
player will win. 
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