## 2 Results
### Calculating the Upper Bound  
Authors in [19] used \(EBA\) to give an upper bound for the bi-directional preemptive conversion problem with constant price interval and unknown \(k\). The upper bound is based on the idea that there is a sequence, on which the maximum ratio that can be achieved within a period is achieved within every period (see Theorem 2). **Theorem 2**. : _For the competitive ratio of an optimal online algorithm for the bi-directional preemptive conversion problem with given \(M\), \(m\), and \(T\) holds \(c\leq\left(\dfrac{2}{\left(\frac{m}{M}\right)^{\frac{1}{2}}+1}\right)^{T-1}\)._  
Proof of Theorem 2.: The upper bound for the competitive ratio follows from \(EBA\). From Equation (16) follows the highest possible ratio between OPT and \(EBA\), within one period. There are \(T-1\) periods in a sequence with \(T\) prices. Thus, the ratio between OPT and ON can raise \(T-1\) times. Therefore, an optimal algorithm has to achieve a competitive ratio smaller than or equal to the right side of Equation (16), with the power of \(T-1\). Nevertheless, the upper bound is still not tight and can be improved. To achieve this, the competitive ratio of \(EBA\) for \(T=3\) can be determined. First the definition of a symmetric algorithm is introduced and it is shown that \(EBA\) is a symmetric algorithm. Therefore, \(b_{t}\) is defined as function of the free variable \(q_{t}\), i.e., \(b_{t}(q_{t})\). **Definition 1**. : _An online algorithm is called symmetric if it holds \(b_{t}(q_{t})=1-b_{t}\left(\frac{M\cdot m}{q_{t}}\right)\forall t=1,\ldots,T\)._  
The authors in [19] showed that for a symmetric algorithm that holds that on each sequence \(\mathbf{q}=q_{1},q_{2}\ldots,q_{T}\), the ratio between OPT and ON is the same as on sequence \(\mathbf{q}^{\prime}=\frac{M\cdot m}{q_{1}},\frac{M\cdot m}{q_{2}},\ldots, \frac{M\cdot m}{q_{T}}\). Thus, it is only necessary to consider a part of the possible sequences to determine the worst-case-sequences, e.g., it is sufficient to consider only sequences that start with a raising price. **Lemma 3**. : \(EBA\) _is a symmetric online algorithm._  
Proof. : An algorithm is symmetric if it holds \(b_{t}(q_{t})=1-b_{t}\left(\frac{M\cdot m}{q_{t}}\right)\). Thus, for \(EBA\) must hold  
\[\frac{1-\frac{q_{t}}{M}}{2-\frac{q_{t}}{M}-\frac{m}{q_{t}}}=1-\frac{1-\frac{ \frac{q_{t}}{M}}{M}}{2-\frac{\frac{M\cdot m}{q_{t}}}{M}-\frac{m}{\frac{M\cdot m }{q_{t}}}}. \tag{19}\]  
Rearranging (19) leads to  
\[\frac{1-\frac{q_{t}}{M}}{2-\frac{q_{t}}{M}-\frac{m}{q_{t}}}=\frac{1-\frac{q_{t }}{M}}{2-\frac{q_{t}}{M}-\frac{m}{q_{t}}}. \tag{20}\]  
Thus, it follows that \(EBA\) is a symmetric online algorithm. **Theorem 3**. : _For the competitive ratio of an optimal online algorithm for the bi-directional preemptive conversion problem with given \(M\), \(m\), and \(T=3\) holds_  
\[c\leq\frac{(M\cdot m-2\cdot M\cdot q_{wc1}+q_{wc1}^{2})\cdot(M\cdot m-2\cdot M \cdot q_{wc2}+q_{wc2}^{2})}{q_{wc1}\cdot(m-M)\cdot(M\cdot(m-q_{wc1}-q_{wc2})+ q_{wc1}\cdot q_{wc2})} \tag{21}\]  
_with_  
\[q_{wc1}=\frac{(f+g)^{\frac{2}{3}}-(f+g)^{\frac{1}{3}}\cdot m\cdot M-m^{2} \cdot M^{2}+m^{3}\cdot M}{(m-2\cdot M)\cdot(f+g)^{\frac{1}{3}}}\]  
_and_  
\[q_{wc2}=\frac{m^{2}\cdot M-m\cdot q_{wc1}\cdot(2\cdot M+q_{wc1})+2\cdot M \cdot q_{wc1}^{2}}{m\cdot(M-2\cdot q_{wc1})+q_{wc1}^{2}}\]  
_with_  
\[f=-2\cdot m^{4}\cdot M^{2}+4\cdot m^{3}\cdot M^{3}-2\cdot m^{2}\cdot M^{4}\]\[g=(m^{9}\cdot(-M)^{3}+7\cdot m^{8}\cdot M^{4}-19\cdot m^{7}\cdot M^{5}+25\cdot m^{6 }\cdot M^{6}-16\cdot m^{5}\cdot M^{7}+4\cdot m^{4}\cdot M^{8})^{\frac{1}{2}}.\]  
**Proof of Theorem 3. ** Using \(EBA\), there are four possible types of worst-case-sequences for \(T=3\). These differ by their price development:  
1. the price raises twice,
2. the price first raises and then falls,
3. the price first falls and then raises,
4. the price falls twice. As \(EBA\) balances the errors that can occur at the last period, type 1 and 2, and also type 3 and 4 are balanced. As \(EBA\) is symmetric, the errors of type 1 and 4 are also balanced.