# Auction theory - Selling Multiple Items via Social Networks

I am trying to understand this paper.

The one thing that I cannot wrap my head around is the payment policy for the Generalized Information Diffusion Mechanism. It is explained on page 72 with examples on page 73.

I understand it as follows:

So for any node $$i \in N$$, the payment is: $$\mathcal{SW}_{-D_{i}} - (\mathcal{SW}_{-C_{i}^{\mathcal{K}}}-v_{i}^{'})$$

$$D_{i}$$ is the union of $$i$$ and its critical children,

$$\mathcal{SW}_{-D_{i}}$$ sums over all nodes except the nodes in $$D_{i}$$.

$$\pi_{j}(\theta^{'})$$ is 1 if $$j$$ receives an item, else 0.

Thus, $$\mathcal{SW}_{-D_{i}}$$ is the sum of the bids of the nodes that receive an item and are not $$i$$ and not a critical child of $$i$$.

So for node D (figure 5) I get: $$\mathcal{SW}_{-D_{D}} = 19 + 17 + 20 = 56$$

$$C_{i}^{\mathcal{K}}$$ is the set of the top $$\mathcal{K}$$ ranked critical children of $$i$$ according to their bid. When the set is smaller that 5, we just take all critical children.

For node D: $$\mathcal{SW}_{-C_{i}^{\mathcal{K}}} = 19 + 17 + 20 + 14 = 70$$ (all nodes that receive an item that are not D's critical children).

Finally, $$v_{i}^{'}$$ is just node $$i$$'s reported valuation (bid).

For node D: $$v_{D}^{'} = 14$$.

Thus when I calculate D's payment: $$\mathcal{SW}_{-D_{D}} - (\mathcal{SW}_{-C_{D}^{\mathcal{K}}} - v_{D}^{'}) = 56 - (70 - 14) = 0$$.

Of course this is wrong because it always comes out as 0 and in reality D has to pay 10.

the calculation of $$\mathcal{SW}_{-D_i}$$ and $$\mathcal{SW}_{-C_i^\mathcal{K}}$$ is incorrect. There are still $$\mathcal{K}$$ items to be allocated in both allocations, so for node $$D$$, the first is 20+19+17+11+10 and the second is 20+19+17+14+11. So D's payment is 10.