89,306 research outputs found
Endless love: On the termination of a playground number game
A simple and popular childhood game, `LOVES' or the `Love Calculator',
involves an iterated rule applied to a string of digits and gives rise to
surprisingly rich behaviour. Traditionally, players' names are used to set the
initial conditions for an instance of the game: its behaviour for an exhaustive
set of pairings of popular UK childrens' names, and for more general initial
conditions, is examined. Convergence to a fixed outcome (the desired result) is
not guaranteed, even for some plausible first name pairings. No pairs of top-50
common first names exhibit non-convergence, suggesting that it is rare in the
playground; however, including surnames makes non-convergence more likely due
to higher letter counts (for example, `Reese Witherspoon LOVES Calvin Harris').
Different game keywords (including from different languages) are also
considered. An estimate for non-convergence propensity is derived: if the sum
of digits in a string of length obeys , convergence
is less likely. Pairs of top UK names with pairs of `O's and several `L's (for
example, Chloe and Joseph, or Brooke and Scarlett) often attain high scores.
When considering individual names playing with a range of partners, those with
no `LOVES' letters score lowest, and names with intermediate (not simply the
highest) letter counts often perform best, with Connor and Evie averaging the
highest scores when played with other UK top names.Comment: 12 pages, 9 figure
On the convergence of iterative voting: how restrictive should restricted dynamics be?
We study convergence properties of iterative voting procedures. Such procedures are defined by a voting rule and a (restricted) iterative process, where at each step one agent can modify his vote towards a better outcome for himself. It is already known that if the iteration dynamics (the manner in which voters are allowed to modify their votes) are unrestricted, then the voting process may not converge. For most common voting rules this may be observed even under the best response dynamics limitation. It is therefore important to investigate whether and which natural restrictions on the dynamics of iterative voting procedures can guarantee convergence. To this end, we provide two general conditions on the dynamics based on iterative myopic improvements, each of which is sufficient for convergence. We then identify several classes of voting rules (including Positional Scoring Rules, Maximin, Copeland and Bucklin), along with their corresponding iterative processes, for which at least one of these conditions hold
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