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You have a budget $B$ and a satisfaction level $L$. An indivisible product is available for sell and you are interested in buying. The product is divided into $n$ parts. Part $i$ of the product has positive value $v_i$.

On day $i$, you get to know $v_i$ and you have to decide how much price $p_i\geq0$ you would pay for this part of the product.

After $j$ days, given the values $v_1, v_2, \ldots, v_j$ and the prices you paid $p_1, p_2, \ldots, p_j$ for each value, you know that you can reach a satisfaction level of $(1+v_1p_1)(1+v_2p_2)\cdots(1+v_jp_j)$.

Your objective is to buy the product after $n$ days while you reach the satisfaction level of $L$ and pay no more than $B$.

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  • $\begingroup$ "indivisible product... divided into $n$ parts"?! $\endgroup$ – Aryeh Oct 29 '19 at 12:05
  • $\begingroup$ @Aryeh you can see it as $n$ products. $\endgroup$ – zdm Oct 29 '19 at 14:35
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    $\begingroup$ @zdm A very, very rough similar-ish problem is the secretary problem. $\endgroup$ – orlp Oct 30 '19 at 0:01
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    $\begingroup$ For n=2 there is a strategy. Once you see v1, compute the minimum value of v2, say v2*, such that the problem (v1, v2*) is feasible. Then choose p1 and p2 to be the values that work for (v1, v2*). This suffices because the actual v2 will be at least v2*. @orlp, can you prove your claim that any online strategy will fail on at least one input? I'm not so sure. $\endgroup$ – Neal Young Oct 30 '19 at 17:15
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    $\begingroup$ Note, also, for intuition, that for the modified the problem where we replace each $1+v_ip_i$ by $\exp(v_i p_i)$, there is an online strategy: set $p_i=0$ for all $i$ except the first where $v_i\ge \ln(L)/B$, and set that one to $B$. (If there is no such $i$, then the instance is not feasible, because $$\prod_i \exp(v_ip_i) = \exp(\sum_i v_i p_i) < \exp(\ln(L)/B \sum_i p_i) \le \exp(\ln(L)/B \cdot B) = L.)$$ $\endgroup$ – Neal Young Oct 30 '19 at 17:31
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@orlp's intuition is correct.

lemma. No online algorithm solves the problem in the worst case.

Proof. Consider the following instance:

$$f(p) = (1+p_1)(1+p_2)(1+p_3) \ge 8 \text{ with } B=3.$$

This instance has just one solution (one $p$ that satisfies it), namely $p=(1,1,1)$. So, given $B=3$ and $L=8$ and just the first factor $1+p_1$, the algorithm has to choose $p_1 = 1$.

But then the adversary instead continues by defining the following instance:

$$g(p) = (1+p_1)(1+ 1.49 p_2)(1+0\cdot p_3) \ge 8 \text{ with } B=3.$$

This instance is feasible (e.g. for $p_1=p_2=3/2$ we have $g(p)\approx 8.1$). However, the algorithm committed to $p_1 = 1$, so the residual problem it is faced with is now $2(1+1.49 p_2) \ge 8$ with $p_2 \le 2$. Since $2(1+1.49\cdot 2) < 2\cdot 4 = 8$, this residual problem is not feasible.

So the online algorithm cannot guarantee a solution, even if it knows in advance that it will face just one of the two instances above. $~~~\Box$

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