# [EM] Phragme’n method for a participatory budget election

Ross Hyman rahyman at sbcglobal.net
Sun Feb 11 08:21:33 PST 2018

```Phragme’n method for a participatory budget election.

In several wards in Chicago, Alderman allow their constituents to vote their spending priorities, on a number of items, each with a different cost, in an election.  Sometimes this is done with approval ballots.  Phragme’n’s method is a natural way to make this process honor proportional representation since Phragme’n's method produces a proportional ordered list, which is precisely what the Alerman needs to set spending priorities. The nth spending item is the item whose election minimizes the Phragme’n load (a measure of amount of money that a ballot has successfully voted for) of the ballot with the greatest Phragme’n load.

The method:
Spending items are elected sequentially.  The first elected spending item is spending item 1.  The next is spending item 2, etc.  The first item on the priority list is the one that minimizes
s_1 = C_1/V_1

and the nth item on the spending priority list is the one that minimizes

s_n = (C_n + sum_m V_m,n * s_m)/V_n

C_n is the monetary cost of the nth spending item. (for electing people to a legislature set each C_n to 1.  The only change from the normal Phragme'n method is the setting of C_n to the cost of the item).

sum_m is a sum over the elected spending items 1 to n-1.

V_m,n = is the number of approval ballots that approve spending items m and  n, and do not approve any of the spending items between m and n.

V_n is the number of approval ballots that approve spending item n.
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