target function approximation

excluding use of floats, and division, only +,-,* are allowed.

target function emulation

target function

target fuction T: T = L * \phi(\sigma) = L * (1- (1 - f)^{\sigma})
\sigma is relative stake.
f is tuning parameter, or the probability of winning have all the stake
L is field length

`\phi(\sigma)` approximation

\phi(\sigma) = 1 - (1-f)^{\sigma}
= 1 - e^{\sigma ln(1-f)}
= 1 - (1 + \sum_{n=1}^{\infty}\frac{(\sigma ln (1-f))^n}{n!})
\sigma = \frac{s}{\Sigma}
s is stake, and \Sigma is total stake.

target T n term approximation

k = L ln (1-f)^1
k^{'n} = L ln (1-f)^n
T = -[k\sigma + \frac{k^{''}}{2!} \sigma^2 + \dots +\frac{ k^{'n}}{n!}\sigma^n]
= -[\frac{k}{\Sigma}s + \frac{k^{''}}{\Sigma^2 2!} s^2 + \dots +\frac{k^{'n}}{\Sigma^n n!} s^n]

comparison of original target to approximation

consequences

hard coded tunning.
public reward function.

conclusion

as the derivative of deltas graph shows, starting for term 2, the derivatives is ~ 0, and it's the optimal number of terms in approximation accuracy that has the least number of terms.

README.md