Algorithms
Goal
- Make the ratio of the time spent on projects and entertainment fluctuated around your set PER.
- Make the interest value reflects your preference accurately.
- Keep the user as motivated as possible.
Reward
After harvesting a tomato, you are required to self-evaluate yourself between 0% to 100% according to your performance in this pomodoro period. If you are not sure how to conduct the evaluation, here are some beneficial questions that might enlight you.
- Is the outcome up to your expectation when you start this pomodoro?
- Are you fully immersed into this task?
- What about your performance compared to the last time you did similar tasks?
The coins you get from this pomodoro are calculated as
coins = (10 * period_h * (1 + (self_eval - 0.7) * 0.5 + epsilon * 0.1)) / PER;
where period_h represents how long this pomodoro is in unit of hours,
self_eval represents your self-evaluation score, epsilon is a random variable
of a truncated normal distribution between -2 and 2.
Price
One coin corresponds to one hour of entertainment on average. But we have multiple algorithms to adjust the price with the following thoughts.
- The price is based on how much time this event costs. (Standard unit price)
- If an entertainment is more appealing to you, it has a higher price. (Interest)
- If an entertainment is recommended, it has a higher price. (Recommendation)
You can switch to a different strategy but sometimes additional information are required in another strategy. So you have to fill in some information collection forms.
Fixed
The edition of the price is all in your hand. We will not change it or give any recommendations.
Zen Master
This strategy is suitable for those freelances who have no holiday or, in other words, have holidays everyday -- what day it is today is not in their mind.
If the recent frequency of your entertainment is higher than before, it would be assigned a higher score. This strategy encourages you to have a stable entertainment schedule.
price = clamp(avg_interval / last_time_till_now, 0.6, 1.5)
Gradient Descent
The initial price is set by the user. This strategy would reevaluate the price of entertainments based on your interest and their value.
Suppose we have \(m\) types of entertainments with prices \(p_1, p_2, \cdots, p_m\) , durations \(h_1, h_2, \cdots, h_m\), and normal frequencies \(f_1, f_2, \cdots, f_m\). The normal frequency represents how often you buy this entertainment in common cases. For instance, it can be \(1/1, 1/7, 1/30, 1/365\), representing daily, weekly, monthly and yearly respectively. The real frequency is different from normal frequency, it is \(s_i f_i\) where \(s_i\) is a ratio like \(0, 1.2, 1.5\). It is recommended that the ratio is less than \(2\) so that our algorithm could work the best.
We connect \(s_i\) and \(p_i\) by the following formula:
\[s_ i = A(h_i/p_i + \mu_i - \sigma_i)\]
where the activation function \(A(x) = \text{max}\{ 0, \text{log}(1+x) \}\) and \(\mu_i\) represents the interest value \(\sigma_i\) represents the recommendation value. Using this relation and historical data, we can calculated the \(\mu_i\). This interest value can be used to predict the change on \(s_ i\) in the next period when the price varies.
\[s'_ i = A(h_i/p'_ i + \mu_i - \sigma_i)\]
where \(s'_ i\) are predicted ratios in the next period based on historical data and \(p'_ i\) are the price in the next period.
We are going to optimize the following cost using gradient descent with a small learning rate:
\[ \left(\sum_i s'_ i f_i p'_ i - \text{Income}\right)^2 + \left(\sum_i s'_ i f_i h_i-\text{WorkingHour / PER}\right)^2 \]