A Simple Auction Game
The problem is equivalent to online knapsack problem. Let $n$ represents the number of items. Let the knapsack has capacity $B$. Every item $i$, has value $v_i=1$ and weight $w_i=p_i$.

The offline optimal solution: 


Sort the items in increasing order of their weights ($p_{\sigma(1)}\leq p_{\sigma(2)}\leq \cdots \leq p_{\sigma(n)}$ for some permutation $\sigma$).
Let $S\gets\emptyset$.
For $i\gets1$ to $n$: add $\sigma(i)$ to $S$ unless $\sum_{i\in S}p_i&gt;B$.
For each $i\in S$, bid $b_i=p_i$ and for each $i\notin S$, bid $b_i=0$.


Claim: No online algorithm can achieve any non-trivial competitive ratio [1].

If we assume that $p_i=o(B)$ and $L\leq p_i\leq U$, then there is a competitive ratio of $\ln(U/L)+1$ [2].

For the regret, I don't know if there is some algorithm that achieves low regret (under some assumptions).