## 2 Results
### Comparison of the Competitive Ratios  
One question is how \(EBA\) performs compared to the algorithms for the bi-directional preemptive CP with a constant price interval which exist in literature. The authors in [3,4] do not provide a formula for the competitive ratio which can be rearranged to \(c\). This makes it hard to proof if one of the algorithms is better than the others. Therefore, an experimental analysis is used to compare the algorithms. Furthermore, since [3,4] use the number of runs \(k\) in the formula for their competitive ratios, new competitive ratios for these algorithms have to be determined which do not depend on \(k\). In fact, only lower bounds for the competitive ratio will be determined. Therefore, the performance ratio for these algorithms is calculated on sequences where the algorithms do not convert.  
Theorem 5: _For the competitive ratio of \(bTH^{\prime}\) for the bi-directional preemptive conversion problem with a constant price interval and given \(M,m,\) and \(T\) holds \(c(bTH^{\prime})\geq c(uTH^{\prime})^{\left\lfloor\frac{T}{2}\right\rfloor}\)._  
Proof of Theorem 5.: Only sequences on which \(bTH^{\prime}\) does not trade are considered. The highest ratio between OPT and ON results on the sequence \(\mathbf{q_{1}}=c(uTH^{\prime})\cdot m,m,c(uTH^{\prime})\cdot m,m,\ldots,c(u TH^{\prime})\cdot m,m\) for even \(T\geq 2\). It holds that:  
\[OPT(\mathbf{q_{1}})=\left(\frac{c(uTH^{\prime})\cdot m}{m}\right)^{\frac{T}{2} },ON(\mathbf{q_{1}})=1,\frac{OPT(\mathbf{q_{1}})}{ON(\mathbf{q_{1}})}=c(uTH^{ \prime})^{\frac{T}{2}} \tag{31}\]  
In case of odd \(T\), OPT and ON have the same wealth as in case of \(T-1\). It holds\[c(bTH^{\prime})\geq c(uTH^{\prime})\lfloor^{\frac{T}{2}}\rfloor. \tag{32}\]  
**Theorem 6**.: _For the competitive ratio of bTH for the bi-directional preemptive conversion problem with a constant price interval and given \(M,m,\) and \(T\) holds \(c(bTH)\geq c(uTH)\lfloor^{\frac{T}{2}}\rfloor.\)_  
Proof of Theorem 6.: The proof is analogous to the proof of Theorem 5. It holds that  
\[c(bTH)\geq c(uTH)\lfloor^{\frac{T}{2}}\rfloor. \tag{33}\]  
Since it holds that \(c(uTH)<c(uTH^{\prime})\) (see [4]), it also holds that \(c(uTH)\lfloor^{\frac{T}{2}}\rfloor<c(uTH^{\prime})\lfloor^{\frac{T}{2}}\rfloor.\) Therefore, \(EBA\) is only compared with \(bTH.\) For the experimental analysis, the following parameters are used: \(M\in\{2,10,100,1000\},T\in\{2,5,7,13\}\) and \(m=1.\) Table 1 shows the values for the upper bound of \(EBA\) and the lower bound of \(bTH.\)  
From Table 1, it follows that both algorithms achieve the same competitive ratio for \(T=2\). This is not surprising. \(bTH\) is optimal for a sequence consisting of one run. A sequence with \(T=2\) points in time consists of one run. Thus, \(bTH\) is also optimal in the single-period model. The upper bounds for the competitive ratio of \(EBA\) are lower than the lower bounds of the competitive ratio of \(bTH\) for \(T\in\{7,13\}.\) Only for \(T=5\) and small values of \(M\) the results of \(bTH\) are lower than the results of \(EBA.\) In these cases, it is not known which algorithm is better. The results indicate that \(EBA\) is getting better compared to \(bTH\) with increasing \(T\) and with an increasing ratio \(\frac{M}{m}\). This is not surprising. \(EBA\) is designed with the objective to perform well in the case of given \(T\) and unknown \(k\). In contrast, \(bTH\) is designed with the objective to perform well in the case of given \(k\). Nevertheless, these results show that there is a lack in the literature considering the bi-directional preemptive CP with a constant price interval, and given \(T\) and unknown \(k\).