I am trying to understand how to find the equilibrium price in a double auction.
From wikipedia:
Natural ordering
- Order the buyers in decreasing order of their bid: b1≥b2≥...≥bn.
- Order the sellers in increasing order of their bid: s1≤s2≤...≤sn. Let k be the largest index such that bk≥sk (the "breakeven index"). Every price in the range [max(sk,bk+1),min(bk,sk+1)] is an equilibrium price, since both demand and supply are k. It is easier to see this by considering the range of equilibrium prices in each of the 4 possible cases (note that by definition of k, bk+1 < sk+1)
What does it mean for the "supply and demand to be at k"?
Saying something like "the demand is 23" doesn't make any sense to me.
You should have seen this when a demand curve was defined. Saying the demand at $50$ is $23$ means if you price the object at $50$ you will sell $23$ of them. Another point on the demand curve could be that the demand at $40$ is $29$, which would say that there are six people who will buy at $40$ but will not buy at $50$. The demand curve is just the number that will be bought as a function of the price.
In your example, let $n=10$ and let there be one bid each of $10,20,30\ldots 100$. The demand at $55$ is $5$ because the five highest bidders, $b_1$ through $b_5$, are willing to buy at $55$ while the other five are not willing to buy.
Now make up a set of sales offers. I would avoid having any at an exact multiple of $10$. It doesn't have to be $10$ of them. Follow the process described. Note how it finds a price range where you have the same number of buyers and sellers that will accept the price.