### Voting Model Java Applet, by
Jeffrey S. Rosenthal,
July 2015
(instructions below)

This applet models a vote by allowing each of "numplayers" different
players to either stay out (red square; payoff=0), or play at some position
in the interval [0,1] (green circle; payoff = 1 if they win outright, or 1/k if
they are one of "k" players tied for the most votes, or -1 otherwise).
Voters are assumed to be uniformly distributed on the interval [0,1],
and to always vote for whichever player is closest to their position.
Players' voting ranges and payoff scores are shown on the right.

This applet is inspired by a conjecture
made years ago by Martin J. Osborne.

To use this applet, use your mouse to "click" on the yellow REFRESH
box to get a fresh random position,
or on the white "+" and "-"
spots to increase or decrease the corresponding parameter. (For
example, increase the speed to make the optimisations run faster.)

You can also click on a yellow circle to optimise the actions of that
player and all subsequent players (hence, click next to player #1 to
do a full optimisation). The optimisation is done via a Monte Carlo
simulation, which repeatedly tweaks the positions of that player and
higher-numbered players, rejecting tweaks which lower the score.

Or, click the ORIG/CONTIN box to toggle between the original model,
and a continuous version with payoff for entering players of:

[range(i)^power / (sum_j range(j)^power)] - penalty

In a pinch, after "clicking" on the applet, you
can type the key "p" to pause/unpause an optimisation,
or "x" to abort an optimisation in progress.

You can also view or download the
source code
and class file
and html file.

If you have trouble running the applet,
see these notes.

Applet by Jeffrey S. Rosenthal
(jeff@math.toronto.edu).

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