SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Evangelia Gergatsouli
evagers@cs.wisc.edu
& University of Athens
PANDORA'S BOX is a central problem in decision making under uncertainty that can model various real life scenarios. In this problem we are given  $n$  boxes, each with a fixed opening cost, and an unknown value drawn from a known distribution, only revealed if we pay the opening cost. Our goal is to find a strategy for opening boxes to minimize the sum of the value selected and the opening cost paid.
In this work we revisit PANDORA's BOX when the value distributions are correlated, first studied in Chawla et al. [2020]. We show that the optimal algorithm for the independent case, given by Weitzman's rule, directly works for the correlated case. In fact, our algorithm results in significantly improved approximation guarantees compared to the previous work, while also being substantially simpler. We also show how to implement the rule given only sample access to the correlated distribution of values. Specifically, we find that a number of samples that is polynomial in the number of boxes is sufficient for the algorithm to work.
In various minimization problems where uncertainty exists in the input, we are allowed to obtain information to remove this uncertainty by paying an extra price. Our goal is to sequentially decide which piece of information to acquire next, in order to minimize the sum of the search cost and the value of the option we chose.
This family of problems is naturally modeled by PANDORA'S BOX, first formulated by Weitzman [1979] in an economics setting, with multiple application in consumer search, housing markets and job search (see [McCall and McCall, 2007] for more applications). In this problem where we are given  $n$  boxes, each containing a value drawn from a known distribution and each having a fixed known opening cost. We can only see the exact value realized in a box if we open it and pay the opening cost. Our goal is to minimize the sum of the value we select and the opening costs of the boxes we opened.
In the original work of Weitzman, an optimal solution was proposed when the distributions on the values of the boxes were independent [Weitzman, 1979]. This algorithm was based on calculating a reservation value  $(\sigma)$  for each box, and then choosing the box with the lowest reservation value to open at every step. Independence, however, is an unrealistic assumption in real life; in a housing market neighboring houses' price are affected the same way, or in a job search setting, candidates might share qualifications that affect them similarly. Wanting to tackle a more realistic setting, Chawla et al. [2020] first studied the problem where the distributions are correlated, and designed an algorithm giving a constant approximation guarantee. This algorithm is quite involved, it requires solving an LP to convert the PANDORA's BOX instance to a MIN SUM SET COVER one, and then solving this instance to obtain an ordering of opening the boxes. Finally, it reduces the problem of deciding when to stop to an online algorithm question corresponding to SKI-RENTAL.
In this work we revisit PANDORA's BOX with correlations, and provide simpler, learnable algorithms with better approximation guarantees, that directly generalize Weitzman's reservation values. More specifically, our results are the following.
- Generalizing: we first show how the original reservation values given by Weitzman [1979] can be generalized to work in correlated distributions, thus allowing us to use a version of their initial greedy algorithm.
- Better approximation: we give two different variants of our main algorithm, that each uses different updates on the distribution  $\mathcal{D}$  after every step.
1. Variant 1: partial updates. We condition on the algorithm not having stopped yet.
2. Variant 2: full updates. We condition on the exact value  $v$  revealed in the box opened.
Both variants improve the approximation given by Chawla et al. [2020] from 9.22 to 4.428 for Variant 1 and to 5.828 for Variant 2. It is worth noting that our result for Variant 1 is almost tight, since the best possible approximation factor we can obtain is 4, implied by Feige [1998]. We include more details on the lower bound in Section A.4 of the Appendix.
- Simplicity: our algorithms are greedy and only rely on the generalized version of the reservation value, while the algorithms in previous work rely on solving a linear program, and reducing first to MIN SUM SET COVER then to SKI-RENTAL, making them not straightforward to implement. A 9.22 approximation was also given in Gergatsouli and Tzamos [2022], which followed the same approach but bypassed the need to reduce to MIN SUM SET COVER by directly rounding the linear program via randomized rounding.
- Learnability: we show how given sample access to the correlated distribution  $\mathcal{D}$  we are able to still maintain the approximation guarantees. Specifically, for Variant 1 only  $\mathrm{poly}(n,1 / \varepsilon ,\log (1 / \delta))$  samples are enough to obtain  $4.428 + \varepsilon$  approximation with probability at least  $1 - \delta$ . Variant 2 is however impossible to learn.
Our analysis is enabled by drawing similarities from PANDORA's BOX to MIN SUM SET COVER, which corresponds to the special case of when the values inside the boxes are 0 or  $\infty$ . For MIN SUM SET COVER a simple greedy algorithm was shown to achieve the optimal 4-approximation [Feige et al., 2002]. Surprisingly, Weitzman's algorithm can be seen as a direct generalization of that algorithm. Our analysis follows the histogram method introduced in Feige et al. [2002], for bounding the approximation ratio. However, we significantly generalize it to handle values in the boxes and work with tree-histograms required to handle the case with full-updates.
Since Weitzman's initial work [Weitzman, 1979] on PANDORA's BOX there has been a renewed interest in studying this problem in various settings. Specifically Doval [2018], Beyhaghi and Kleinberg [2019], Beyhaghi and Cai [2023a], Fu et al. [2023] study PANDORA's BOX when we can select a box without paying for it (non-obligatory inspection), in Boodaghians et al. [2020] there are tree or line constraints on the order in which the boxes can be opened. In Chawla et al. [2020, 2021] the distributions on the values inside the boxes are correlated and the goal is to minimize the search and value cost, while finally in Bechtel et al. [2022] the task of searching over boxes is delegated by an agent to a principal, while the agent makes the final choice. The recent work of Chawla et al. [2020] is the first one that explores the correlated distributions variant and gives the first approximation guarantees. The recent survey by Beyhaghi and Cai [2023b] summarizes the recent work on Pandora's Box and its variants.
This problem can be seen as being part of the "price of information" literature [Charikar et al., 2000, Gupta and Kumar, 2001, Chen et al., 2015b,a], where we can remove part of the uncertainty of the problem at hand by paying a price. In this line of work, more recent papers study the structure of approximately optimal rules for combinatorial problems [Goel et al., 2006, Gupta and Nagarajan, 2013, Adamczyk et al., 2016, Gupta et al., 2016, 2017, Singla, 2018, Gupta et al., 2019].
For the special case of MIN SUM SET COVER, since the original work of Feige et al. [2002], there has been many follow-ups and generalizations where every set has a requirement of how many elements
contained in it we need to choose [Azar et al., 2009, Bansal et al., 2010, Azar and Gamzu, 2011, Skutella and Williamson, 2011, Im et al., 2014].
Note also that multiple results on problems related to Pandora's box have been published in ML-related conferences, as this is a problem that encompasses both algorithmic and learning aspects (e.g. Esfandiari et al. [2019], Gergatsouli and Tzamos [2022], Bhaskara et al. [2020], Cesa-Bianchi et al. [2021], Guo et al. [2021]).
In PANDORA's BOX  $(\mathcal{PB})$  we are given a set of  $n$  boxes  $\mathcal{B}$ , each with a known opening cost  $c_{b} \in \mathbb{R}^{+}$  and a distribution  $\mathcal{D}$  over a vector of unknown values  $\boldsymbol{v} = (v_{1}, \ldots, v_{n}) \in \mathbb{R}_{+}^{n}$  inside the boxes. Each box  $b \in \mathcal{B}$ , once it is opened, reveals the value  $v_{b}$ . The algorithm can open boxes sequentially, by paying the opening cost each time, and observe the value instantiated inside the box. The goal of the algorithm is to choose a box of small value, while spending as little cost as possible "opening" boxes. Formally, denoting by  $\mathcal{O} \subseteq \mathcal{B}$  the set of opened boxes, we want to minimize
$$
\mathbb {E} _ {v \sim \mathcal {D}} \left[ \min  _ {b \in \mathcal {O}} v _ {b} + \sum_ {b \in \mathcal {O}} c _ {b} \right].
$$
A strategy for PANDORA's BOX is an algorithm that in every step decides which is the next box to open and when to stop. We measure the performance of our algorithm using the competitive (or approximation) ratio; a strategy  $\mathcal{A}$  is  $\alpha$ -approximation if  $\mathbb{E}[\mathcal{A}] \leq \alpha \mathrm{OPT}$ , where  $\mathrm{OPT}$  is the optimal online algorithm<sup>1</sup>
A strategy can pick any open box to select at any time. To model this, we assume without loss of generality that after a box is opened the opening cost becomes 0, allowing us to select the value without opening it again. In its full generality, a strategy can make decisions based on every box opened and value seen so far. We call this the Fully-Adaptive (FA) strategy.
Different Benchmarks. As it was initially observed in Chawla et al. [2020], optimizing over the class of fully-adaptive strategies is intractable, therefore we consider the simpler benchmark of partially-adaptive (PA) strategies. In this case, the algorithm has to fix the opening order of the boxes, while the stopping rule can arbitrarily depend on the values revealed.
2.1 Weitzman's Algorithm
When the distributions of values in the boxes are independent, Weitzman [1979] described a greedy algorithm that is also the optimal strategy. In this algorithm, we first calculate an index for every box  $b$ , called reservation value  $\sigma_{b}$ , defined as the value that satisfies the following equation
$$
\mathbb {E} _ {\boldsymbol {v} \sim \mathcal {D}} \left[ \left(\sigma_ {b} - v _ {b}\right) ^ {+} \right] = c _ {b}, \tag {1}
$$
where  $(a - b)^{+} = \max(0, a - b)$ . Then, the boxes are ordered by increasing  $\sigma_{b}$  and opened until the minimum value revealed is less than the next box in the order. Observe that this is a partially-adaptive strategy.
3 Competing with the Partially-Adaptive
We begin by showing how Weitzman's algorithm can be extended to correlated distributions. Our algorithm calculates a reservation value  $\sigma$  for every box at each step, and opens the box  $b \in \mathcal{B}$  with the minimum  $\sigma_{b}$ . We stop if the value is less than the reservation value calculated, and proceed in making this box free; we can re-open this for no cost, to obtain the value just realized at any later point. The formal statement is shown in Algorithm 1.
We give two different variants based on the type of update we do after every step on the distribution  $\mathcal{D}$ . In the case of partial updates, we only condition on  $V_{b} > \sigma_{b}$ , which is equivalent to the algorithm
The optimal online has the exact same information as our algorithm  $\mathcal{A}$  but has infinite computation time to solve the problem.
not having stopped. On the other hand, for full updates we condition on the exact value that was instantiated in the box opened. Theorem 3.1 gives the approximation guarantees for both versions of this algorithm.
Algorithm 1: Weitzman's algorithm, for correlated  $\mathcal{D}$
Input: Boxes with costs  $c_{i}\in \mathbb{R}$  distribution over scenarios  $\mathcal{D}$    
1 An unknown vector of values  $v\sim \mathcal{D}$  is drawn   
2 repeat   
3 Calculate  $\sigma_{b}$  for each box  $b\in \mathcal{B}$  by solving:  $\mathbb{E}_{\boldsymbol {v}\sim \mathcal{D}}[(\sigma_b - v_b)^+ ] = c_b.$    
4 Open box  $b = \mathrm{argmin}_{b\in \mathcal{B}}\sigma_{b}$    
5 Stop if the value the observed  $V_{b} = v_{b}\leq \sigma_{b}$    
6  $c_{b}\gets 0 / /$  Box is always open now or can be reopened   
7 Update the prior distribution - Variant 1:  $\mathcal{D}\gets \mathcal{D}|_{V_b > \sigma_b}$  (partial updates) - Variant 2:  $\mathcal{D}\gets \mathcal{D}|_{V_b = v_b}$  (full updates)   
8 until termination;
Theorem 3.1. Algorithm 1 is a 4.428-approximation for Variant 1 and 5.828-approximation for Variant 2 of PANDORA'S BOX against the partially-adaptive optimal.
Proof. We separately show the two components of this theorem in Theorems 3.2 and 3.3.
Observe that for independent distributions this algorithm is exactly the same as Weitzman's [Weitzman, 1979], since the product prior  $\mathcal{D}$  remains the same, regardless of the values realized. Therefore, the calculation of the reservation values does not change in every round, and suffices to calculate them only once at the beginning.
Scenarios To proceed with the analysis of Theorem 3.1, we assume that  $\mathcal{D}$  is supported on a collection of  $m$  vectors,  $(\pmb{v}^s)_{s\in S}$ , which we call scenarios, and sometimes abuse notation to say that a scenario is sampled from the distribution  $\mathcal{D}$ . We assume that all scenarios have equal probability. The general case with unequal probabilities follows by creating more copies of the higher probability scenarios until the distribution is uniform.
A scenario is covered when the algorithm decides to stop and choose a value from the opened boxes. For a specific scenario  $s \in S$  we denote by  $c(s)$  the total opening cost paid by an algorithm before this scenario is covered and by  $v(s)$  the value chosen for this scenario.
Reservation Values To analyze Theorem 3.1, we introduce a new way of defining the reservation values of the boxes that is equivalent to (1). For a box  $b$ , we have that
$$
\sigma_ {b} = \min  _ {A \subseteq S} \frac {c _ {b} + \sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ] v _ {b} ^ {s}}{\sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ]}
$$
The equivalence to (1), follows since  $\sigma_{b}$  is defined as the root of the expression
$$
\begin{array}{l} \mathbb {E} _ {s \sim \mathcal {D}} \left[ \left(\sigma_ {b} - v _ {b} ^ {s}\right) ^ {+} \right] - c _ {b} = \sum_ {s \in \mathcal {S}} \mathbf {P r} _ {\mathcal {D}} [ s ] \left(\sigma_ {b} - v _ {b} ^ {s}\right) ^ {+} - c _ {b} \\ = \max _ {A \subseteq S} \sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ] (\sigma_ {b} - v _ {b} ^ {s}) - c _ {b}. \\ \end{array}
$$
If we divide the above expression by any positive number, the result will not be affected since we require the root of the equation;  $\sigma_{b}$  being the root is equivalent to  $\sigma_{b}$  being the root of the numerator.
Thus, dividing by  $\sum_{s\in A}\mathbf{Pr}_D[s]$  we get that  $\sigma_{b}$  is also the root of
$$
\max  _ {A \subseteq S} \frac {\sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ] \left(\sigma_ {b} - v _ {b} ^ {s}\right) - c _ {b}}{\sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ]} = \sigma_ {b} - \min  _ {A \subseteq S} \frac {c _ {b} + \sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ] v _ {b} ^ {s}}{\sum_ {s \in A} \mathbf {P r} _ {\mathcal {D}} [ s ]}. \tag {2}
$$
This, gives our formula for computing  $\sigma_{b}$ , which we can further simplify using our assumption that all scenarios have equal probability. In this case,  $\mathbf{Pr}_{\mathcal{D}}[s] = 1 / |S|$  which implies that
$$
\sigma_ {b} = \min  _ {A \subseteq S} \frac {c _ {b} | \mathcal {S} | + \sum_ {s \in A} v _ {b} ^ {s}}{| A |}. \tag {3}
$$
3.1 Conditioning on  $V_{b} > \sigma_{b}$
We start by describing the simpler variant of our algorithm where after opening each box we update the distribution by conditioning on the event  $V_{b} > \sigma_{b}$ . This algorithm is partially adaptive, since the order for each scenario does not depend on the actual value that is realized every time. At every step the algorithm will either stop or continue opening boxes conditioned on the event "We have not stopped yet" which does not differentiate among the surviving scenarios.
Theorem 3.2. Algorithm 1 is a 4.428-approximation for PANDORA's BOX against the partially-adaptive optimal, when conditioning on  $V_{b} > \sigma_{b}$ .
In this section we show a simpler proof for Theorem 3.2 that gives a  $3 + 2\sqrt{2} \approx 5.828$ -approximation. The full proof for the 4.428-approximation is given in section A.2 of the Appendix. Using the equivalent definition of the reservation value (Equation (3)) we can rewrite Algorithm 1 as follows.
Algorithm 2: Weitzman's rule for Partial Updates
Input: Boxes with costs  $c_{i}\in \mathbb{R}$  , set of scenarios S.   
1  $t\gets 0$    
2  $R_0\gets S$  the set of scenarios still uncovered   
3 while  $R_{t}\neq \emptyset$  do   
4 Let  $\sigma_t\gets \min_{b\in \mathcal{B},A\subseteq R_t}\frac{c_b|R_t| + \sum_{s\in A}v_b^s}{|A|}$    
5 Let  $b_{t}$  and  $A_{t}$  be the box and the set of scenarios that achieve the minimum   
6 Open box  $b_{t}$  and pay  $c_{bt}$    
7 Stop and choose the value  $v_{b_t}$  at box  $b_{t}$  if it is less than  $\sigma_t$  (see also Fact 3.2.1)   
8 Set  $c_{b_t}\gets 0$    
9  $R_{t}\leftarrow R_{t}\backslash A_{t}$    
10  $t\gets t + 1$    
11 end
Structure of the solution. An important property to note is that by the equivalent definition of the reservation value (3) the set of scenarios that stop at each step are the ones that give a value at most  $\sigma$  for the box opened, as we formally state in the following fact.
Fact 3.2.1. The value at box  $b_{t}$  is less than  $\sigma_{t}$  if and only if  $s \in A_{t}$ .
In equation (8) the set  $A_{t}$  that maximizes the expression contains all the scenarios with value at most  $\sigma_{b}$  for the box  $b$ . Therefore, the set  $A_{t}$  are exactly the scenarios covered at each step  $t$  of the algorithm, and can be removed from consideration.
Before showing our result, observe that this algorithm is partially adaptive; the order of the boxes does not depend on the scenario realized. This holds since we only condition on "not having stopped" (i.e.  $\mathcal{D}_{V_b > \sigma_b}$ ) and therefore each scenario either stops or uses the same updated prior as all other surviving scenarios to calculate the next reservation values. If we were to draw our solution, it would look like a line, (see also Figure 2 in Appendix A.2), which as we observe in Section 3.2 differs from Variant 2.
Moving on to show the proof, we first start by giving a bound on the cost of the algorithm. The cost can be broken down into opening cost plus the value obtained. Since at any time  $t$ , all remaining
scenarios  $R_{t}$  pay the opening cost  $c_{b_t}$ , we have that the total opening cost is  $\sum_{t}c_{b_t}|R_t|$ . Moreover, the chosen value is given as  $\sum_{t}\sum_{s\in A_{t}}v_{b_{t}}^{s}$ . Overall, we have that
$$
\mathrm {A L G} = \sum_ {t} \left(c _ {b _ {t}} | R _ {t} | + \sum_ {s \in A _ {t}} v _ {b _ {t}} ^ {s}\right) = \sum_ {t} | A _ {t} | \frac {c _ {b _ {t}} | R _ {t} | + \sum_ {s \in A _ {t}} v _ {b _ {t}} ^ {s}}{| A _ {t} |} = \sum_ {t} | A _ {t} | \sigma_ {t}.
$$
Defining  $\sigma_{s}$  to be the reservation value of scenario  $s$  at the time it is covered, i.e. when  $s\in A_t$ , we get  $\mathrm{ALG} = \sum_{s\in S}\sigma_s^2$ . We follow a histogram analysis similar to the proof of Theorem 4 in Feige et al. [2004] for MIN SUM SET COVER and construct the following histograms.
- The  $\mathrm{OPT}_o$  histogram: put the scenarios on the x-axis on increasing opening cost order  $c_s^{\mathrm{OPT}}$  according to OPT, the height of each scenario is the opening cost it paid.
- The  $\mathrm{OPT}_v$  histogram: put the scenarios on the x-axis on increasing covering value order  $v_s^{\mathrm{OPT}}$  according to OPT, the height of each scenario is the value with which it was covered.
- The ALG histogram: put scenarios on the x-axis in the order the algorithm covers them. The height of each scenario is  $\sigma_{s}$ . Observe that the area of the ALG histogram is exactly the cost of the algorithm.
Proof of Theorem 3.2. Initially, observe that the algorithm will eventually stop; every time we open a box we cover at least one scenario (since line 3 is cannot be  $\infty$  while scenarios are left uncovered).
To show the approximation factor, we scale the histograms as follows;  $\mathrm{OPT}_o$  scale horizontally by  $1 / \alpha_{o}$  and vertically by  $1 / (\beta \cdot \gamma)$ , and  $\mathrm{OPT}_v$  scale by  $1 / \alpha_v$  horizontally, for some constants  $\alpha_o,\alpha_v,\gamma ,\beta \in (0,1)$  to be determined later. We align the ALG histogram with  $\mathrm{OPT}_v$  and  $\mathrm{OPT}_o$  so that all of them have the same right-hand side. Observe that the optimal opening cost is the area below the histogram  $\mathrm{OPT}_o$  and has increased by  $\beta \cdot \gamma \cdot \alpha_o$ , and similarly the area below  $\mathrm{OPT}_v$  has increased by  $\alpha_v$  as a result of the scaling.
To conclude the proof it suffices to show that any point in the ALG histogram is inside the sum of the rescaled  $\mathrm{OPT}_v$  and  $\mathrm{OPT}_o$  histograms. Consider any point  $p$  in the ALG histogram, and let  $s$  be its corresponding scenario and  $t$  be the time this scenario is covered. We have that the height of the ALG histogram is
$$
\sigma_ {s} = \frac {c _ {b _ {t}} \left| R _ {t} \right| + \sum_ {s \in A _ {t}} v _ {b _ {t}} ^ {s}}{\left| A _ {t} \right|} \leq \frac {c _ {b} \left| R _ {t} \right| + \sum_ {s \in A} v _ {b} ^ {s}}{\left| A \right|} \tag {4}
$$
where the last inequality holds for all  $A \subseteq R_t$  and any  $b \in \mathcal{B}$ .
Denote by  $c^*$  the opening cost such that  $\gamma |R_t|$  of the scenarios in  $R_t$  have opening cost less than  $c^*$ , and by  $R_{\mathrm{low}} = \{s \in R_t : c_s^{\mathrm{OPT}} \leq c^*\}$  the set of these scenarios. Similarly denote by  $v^*$  the value of scenarios in  $R_{\mathrm{low}}$  such that  $\beta |R_{\mathrm{low}}|$  of the scenarios have value less than  $v^*$  and by  $L = \{s \in R_{\mathrm{low}} : v_s^{\mathrm{OPT}} \leq v^*\}$  these scenarios. This split is shown in Figure 1, and the constants  $\beta, \gamma \in (0,1)$  will be determined at the end of the proof.
Figure 1: Split of scenarios in  $R_{t}$ .
Let  $B_{L}$  be the set of boxes that the optimal solution uses to cover the scenarios in  $L$ . Let  $L_{b} \subseteq L \subseteq R_{t}$  be the subset of scenarios in  $L$  that choose the value at box  $b$  in OPT. Using inequality (4) with
2Throughout this proof we omit the normalization term  $1 / |S|$  both on the algorithms cost and on the optimal cost, without loss of generality, since our guarantee is multiplicative.
${}^{3}$  Scaling horizontally means that we duplicate every scenario and scaling vertically we just multiply the height at every point by the scale factor.
$b \in B_L$  and  $A = L_b$ , we obtain  $\sigma_s|L_b| \leq c_b|R_t| + \sum_{s \in L_b} v_s^{\mathrm{OPT}}$ , and by summing up the inequalities for all  $b \in B_L$  we get
$$
\sigma_ {s} \leq \frac {\left| R _ {t} \right| \sum_ {b \in B _ {L}} c _ {b} + \sum_ {s \in L} v _ {s} ^ {\mathrm {O P T}}}{| L |} \leq \frac {\left| R _ {t} \right| c ^ {*} + \sum_ {s \in L} v _ {s} ^ {\mathrm {O P T}}}{| L |} \leq \frac {c ^ {*}}{\beta \cdot \gamma} + \frac {\sum_ {s \in L} v _ {s} ^ {\mathrm {O P T}}}{| L |} \tag {5}
$$
where for the second inequality we used that the cost for covering the scenarios in  $L$  is at most  $c^*$  by construction, and in the last inequality that  $|L| = |R_t| / (\beta \cdot \gamma)$ . We consider each term above separately, to show that the point  $p$  is within the histograms.
Bounding the opening cost. By the construction of  $c^*$ , the point in the  $\mathrm{OPT}_o$  histogram that has cost at least  $c^*$  is at distance at least  $(1 - \gamma)|R_t|$  from the right hand side. This means that in the rescaled histogram, the point that has cost at least  $c^* / (\beta \cdot \gamma)$  is at distance at least  $(1 - \gamma)|R_t| / \alpha_o$  from the right hand side.