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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.

Has anyone read this paper already and can help me?

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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.

p.s. there is a typo in one of the constraints for the two allocations, which is updated in the paper.

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