So, by my count the equations seem to go something like this…

O is your set of outstanding orders.
W is your set of possible states.

Each order, o, specifies a maximum quantity q(o), a price-per-unit p(o) and a proportion of each state wanted (A(o,w) for each state w in W).

At the end of the day, x(o) (amount satisfied) is found for each order, and the order submitter receives A(o,w)*x(o) shares in state w, at a cost of p(o)*x(o).

We add some “phantom” orders to round things off:

The “creation” order will create or destroy shares, so assigning it the identifier C, with A(c,w) = 1 for all w, and p(o) = 1. x(c) is unconstrained.

Some “leftovers” orders, L_w, will account for leftover shares for each w (say Alice buys share 1 for 50c, B buys share 2 for 50c, the share 3 onwards are leftover). In this case A(L_w, w’) = 1 if w=w’ and 0 otherwise, p(L_w) = 0, and x(L_w) \gteq 0.

Then it’s just a question of balancing everything. So, set O’ to all the orders (O + {C} + {L_w…}), then…

(a) All the money goes somewhere:

Sum {o \in O’} x(o) * C(o) = 0

(b) All the shares are accounted for:

Forall {w \in W}
Sum {o \in O’} x(o) * A(o,w) = 0

For completeness there’s also:

Forall {o in O} 0 \lteq x(o) \lteq q(o)
Forall {o in O} x(o)*p(o) \lteq BALANCE(o)

Sum {w \in W} p(L_w) represents an increment of profit for the marketplace.

x(C)*p(C) represents how much (less) the marketplace is holding in trust until the market can be settled as a result of these trades.

I’m not following quite how that all works though — shouldn’t this be just as hard as the (multiple?) knapsack problem, ie Hard?

But aren’t the inequalities defining a continuous solution space that includes the origin, and thus, in some sense Easy? It’s been a long time since I did inequalities though so maybe I’m just confused.

O is your set of outstanding orders.
W is your set of possible states.

Each order, o, specifies a maximum quantity q(o), a price-per-unit p(o) and a proportion of each state wanted (A(o,w) for each state w in W).

At the end of the day, x(o) (amount satisfied) is found for each order, and the order submitter receives A(o,w)*x(o) shares in state w, at a cost of p(o)*x(o).

We add some “phantom” orders to round things off:

The “creation” order will create or destroy shares, so assigning it the identifier C, with A(c,w) = 1 for all w, and p(o) = 1. x(c) is unconstrained.

Some “leftovers” orders, L_w, will account for leftover shares for each w (say Alice buys share 1 for 50c, B buys share 2 for 50c, the share 3 onwards are leftover). In this case A(L_w, w’) = 1 if w=w’ and 0 otherwise, p(L_w) = 0, and x(L_w) >= 0.

Then it’s just a question of balancing everything. So, set O’ to all the orders (O + {C} + {L_w…}), then…

(a) All the money goes somewhere:

Sum {o \in O’} x(o) * C(o) = 0

(b) All the shares are accounted for:

Forall {w \in W}
Sum {o \in O’} x(o) * A(o,w) = 0

For completeness there’s also:

Forall {o in O} 0 long time since I did inequalities though so maybe I’m just confused.

I believe the constraint corresponding to outcome w is something like:

Sum_O q_O*x_O(1_{w\in O} – p_O) <= 0

where O is an order; we sum over all O
p_O & q_O are the price & quantity of order O
x_O is the fraction of order O accepted
1_{w\in O} equal 1 if the outcome is part of order O, and equals 0 otherwise

there is one such contraint for every outcome w

But this approach doesn’t strike me as an alternative to the phantom bid method, instead it seems like a more general/more powerful way to generate the phantom bids.

Also, it seems to me that there is nothing inherent in bundling that makes it incompatible with, say, the existing phantom bid approach of Newsfuture. If Newsfuture wanted to enable bundling, they could define the necessary relationships (i.e., candidates 1…3 are in party A, candidates 4…9 are in party B) and then accept bids/asks for either a candidate to win or a party to win.

But using the LP formulation seems inherently more flexible and likely to be more efficient at finding matches among a diverse set of bids.

Why not tell readers exactly what that constraint is?