66 research outputs found
Rigorous computer analysis of the Chow-Robbins game
Flip a coin repeatedly, and stop whenever you want. Your payoff is the
proportion of heads, and you wish to maximize this payoff in expectation. This
so-called Chow-Robbins game is amenable to computer analysis, but while
simple-minded number crunching can show that it is best to continue in a given
position, establishing rigorously that stopping is optimal seems at first sight
to require "backward induction from infinity". We establish a simple upper
bound on the expected payoff in a given position, allowing efficient and
rigorous computer analysis of positions early in the game. In particular we
confirm that with 5 heads and 3 tails, stopping is optimal.Comment: 10 page
Partially ordered secretaries
The elements of a finite nonempty partially ordered set are exposed at
independent uniform times in to a selector who, at any given time, can
see the structure of the induced partial order on the exposed elements. The
selector's task is to choose online a maximal element. This generalizes the
classical linear order secretary problem, for which it is known that the
selector can succeed with probability and that this is best possible. We
describe a strategy for the general problem that achieves success probability
for an arbitrary partial order.Comment: 5 page
From heaps of matches to the limits of computability
We study so-called invariant games played with a fixed number of heaps of
matches. A game is described by a finite list of integer vectors
of length specifying the legal moves. A move consists in changing the
current game-state by adding one of the vectors in , provided all
elements of the resulting vector are nonnegative. For instance, in a two-heap
game, the vector would mean adding one match to the first heap and
removing two matches from the second heap. If , such a
move would be permitted provided there are at least two matches in the second
heap. Two players take turns, and a player unable to make a move loses. We show
that these games embrace computational universality, and that therefore a
number of basic questions about them are algorithmically undecidable. In
particular, we prove that there is no algorithm that takes two games
and (with the same number of heaps) as input, and
determines whether or not they are equivalent in the sense that every
starting-position which is a first player win in one of the games is a first
player win in the other.Comment: 13 pages, 7 figure
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