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.