I am working with Generalized Second Price Auction with $n$ players and $n$ slots to allocate.
GSP auctions is defined with a utility function : $u_{i}(b) = \alpha_{i}(v_{i} - p_{i})$ where $p_{i}=b_{i+1}$ and $\alpha_{i}$ the click-through rate for slot $i$.
A Nash Equilibrium is a profile of bids $b_{1} \geq b_{2} \geq, ... , \geq b_{n}$ such that, for any player $j$, for $k < j$: $\alpha_{j}(v_{j}-b_{j+1}) \geq \alpha_{k}(v_{j}-b_{k})$ and, for $k \geq j$, $\alpha_{j}(v_{j}-b_{j+1}) \geq \alpha_{k}(v_{j}-b_{k+1})$
We assume that an allocation is reasonable if for each pair $i$, $j$ of slots : $\frac{\alpha_{j}}{\alpha_{i}} + \frac{v_{i}}{v_{j}} \geq 1$
Now my problem is to show that when we have a Nash Equilibrium, the allocation is reasonable (the formula is right).
Thanks.