We introduce the compound interest rate into the continuous version of the online leasing problem and discuss the generalized model by competitive analysis. On the one hand, the optimal deterministic strategy and its competitive ratio are obtained; on the other hand, a nearly optimal randomized strategy is constructed and a lower bound for the randomized competitive ratios is proved by Yao's principle. With the help of numerical examples, the theoretical results show that the interest rate puts off the purchase date and diminishes the uncertainty involved in the decision making.

In this paper, we consider the continuous version of the online leasing problem in a market with compound interest rate. In this problem, an online player needs some equipment (e.g., a computer, a car) for an initially unknown amount of time. To use the equipment, he may choose from the following two options: to lease it for a leasing fee

Online leasing problem is one of the fundamental problems in financial decision making and has been widely concerned using the theory of competitive analysis. In the context of competitive analysis, an online algorithm, which has no knowledge of the future events, is measured by the ratio of its performance to the performance of an optimal offline algorithm, which has full knowledge of the future events. Competitive analysis was explicitly formulated by Sleator and Tarjan [

Now, we illustrate the theory of competitive analysis using the cost minimization problem. Let

From the definitions above, we have that

The online leasing problem has been investigated by many authors [

The rest of this paper is arranged as follows. In Section

In this section, we introduce the compound interest rate in the market into the traditional online leasing problem of continuous version and discuss its optimal deterministic strategy and optimal competitive ratio.

Let

The net present value of leasing for

From

Suppose that

When

Based on competitive analysis, we can obtain the optimal deterministic strategy and its competitive ratio.

For the online leasing problem in a market with compound interest rate

We prove the theorem by worst-case analysis. The worst case for the problem is that the offline player ends the game immediately when the online player buys the equipment. That is, the input

Notice that

In this section, we aim to construct a randomized competitive strategy, which is illustrated to be nearly optimal later. Let

On the time interval

When

Differentiating (

When

Now we have constructed a randomized strategy

The effect of the interest rate

In this section, we pursuit the lower bound for the randomized competitive ratios by Yao's principle (see, e.g., [

For the leasing problem in a market with compound interest rate

Construct a probability distribution over inputs as follows:

It naturally holds that the lower bound for the competitive ratios is less than the upper bound obtained in Section

As the interest rate tends to 0, the lower and upper bounds for the randomized competitive ratios tend to the same limit; that is,

In this section, we provide some numerical examples for exploiting the effect of the interest rate

First, we consider the effect of the interest rate on deterministic and randomized strategies. In Table

Values of

0 | 10.0 | 2.0 | 1.582 | 1.582 |

0.01 | 10.5 | 1.9 | 1.535 | 1.503 |

0.02 | 11.2 | 1.8 | 1.481 | 1.426 |

0.03 | 11.9 | 1.7 | 1.438 | 1.352 |

0.04 | 12.8 | 1.6 | 1.387 | 1.281 |

0.05 | 13.9 | 1.5 | 1.333 | 1.214 |

Next, we consider the effect of the interest rate on competitive ratios. In Table

In this paper, we have generalized the continuous version of the online leasing problem by introducing the compound interest rate. On the one hand, we obtained the optimal deterministic strategy and its competitive ratio; on the other hand, we constructed a nearly optimal randomized strategy and obtained a lower bound for the randomized competitive ratios. Moreover, thanks to numerical examples, we realized that the market interest rate postpones the purchase date and diminishes the uncertainty involved in the decision making. However, we fail to obtain the optimal randomized competitive ratio but its upper and lower bounds. We conjecture that the randomized competitive strategy we have obtained is exactly optimal, which will be verified in our future research.

This work was supported by the National Natural Science Foundation of China (nos. 70825005 and 70801027), China Postdoctoral Science Foundation (no. 20110490090), and GDUPS (2010).