*I will write a couple of mathematics posts about deriving an optimal strategy for maximising your voting value since I could not find a solid post with proofs. This post was sparked by a question of @iamstan.* 

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![businessmen-1050341_640.jpg](https://cdn.steemitimages.com/DQmSRwfi9hzkfjNX9ptcspcXx2mex4rX6GaZPRQf3KsKKid/businessmen-1050341_640.jpg)
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On steemit you use voting power (VP) to vote. Luckily it is an infinte resource since it regenerates over time.  The amount you can reward and the speed at which VP is replenished follows specific formulas. You can find an overview of them over [here](https://www.steem.center/index.php?title=Rewards:Formulas). In this post I will derive a formula for the VP you have  left after a single vote. From this we will then derive an easy formula for the amount of VP left after n-votes at equally spaced times.  Your vote value is equal to VP x Constant where the constant depends on a number of account specific properties.  So using these formulas we can derive the value of casting n-votes. In a later post we can then optimise the total value of the votes cast in relation to voting value. 

The intermediate calculations require some basic calculus, but you can also skip these if you are yet uninitiated to the calculus cult.


## Assumptions and notations

I will just be looking at votes cast at 100% weight. Also I will be assuming that all votes are cast at equally spaced times.  

<center>
v<sub>n </sub>  is  VP rate at the n-th vote and &Delta;t  is the time interval in minutes between two votes
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VP is generated at fixed amount per time interval. I will just be working with minutes. So denote by

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R  the VP generated per minute
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## v<sub>1 </sub> and  v<sub>2 </sub> : VP at the first two votes

So the first vote you cast is equal to your starting VP so v<sub>1 </sub> . So that's easy. After you voted   your VP will be reduced by a value determined by your previous vote which is  c<sub> 1</sub>(v<sub>1 </sub> + c<sub> 0</sub>) where  c<sub>0 </sub>,  c<sub>1 </sub> are two positive constants.  In the time between your first vote and your second vote you regenerate  VP equal to   R  &Delta;t  . Observe that the time between votes could be so big that our VP has already been fully replenished. So the VP directly after you cast your second vote is equal to

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![wwwwww.png](https://cdn.steemitimages.com/DQmQ9tMtVnGiMaMbxPhNthgrxs7bFiwvmbp63fE3o9epnEh/wwwwww.png)
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## v<sub>n +1</sub> : VP at the n+1-th vote

The cost of a vote on your VP is determined by your previous VP. Making it a recursive relation. After you cast your  n+1-th vote  your VP will be reduced by a value determined by your previous vote which is  c<sub> 1</sub>(v<sub>n </sub> + c<sub> 0</sub>) where  c<sub>0 </sub>,  c<sub>1 </sub> are the two same positive constants as mentioned earlier.  In the time between your n-th vote and your (n+1)-th vote you regenerate  VP equal to   R  &Delta;t  . Then the equation we end up with is

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![wwwwwwwwwwwwwwwwwwwwwwwww.png](https://cdn.steemitimages.com/DQmU6geuFeLE6YPATtaZmumAcrDHfRKnmPaj8vXbEae1amC/wwwwwwwwwwwwwwwwwwwwwwwww.png)
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Now assume that f<sub>k </sub>   ≤ 1 for all  k ≤ n+1. This means that there is no time interval where our VP is 1. Note that under this assumption the VP rate can equal 1 at the exact time we vote. Then, with some calculus, we obtain this cool formula for the VP after n+1-votes:
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![simplevn.png](https://cdn.steemitimages.com/DQmZN91gNssvTzVVvdQxvkHrQaaxwwLH2fE5BBaiMcjfNk3/simplevn.png)
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## v<sub>tot </sub> : the sum of VP
Recall that the value of a vote is equal to VP x constant. So the total value of  N-votes is determined by 

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![vtot.png](https://cdn.steemitimages.com/DQmb8yqJzFiY8VnKT79ZrFNeLnYfrAHgdrdVXHfDciN8wgz/vtot.png)
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times a constant. Let us assume that  f<sub>k </sub>   ≤ 1 for all  k ≤ N  .  Then, yet again with the magic of  calculus we can show that
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![wwww.png](https://cdn.steemitimages.com/DQmVLWG3i1mR8Ug4obwzgxJSgzsPp8n3EqkbGxcFbNCmDPi/wwww.png) 
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That is all for now, all the opimisation stuff postponed to the next post.

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**Sources:** [Rewards: formulas from steemit](https://www.steem.center/index.php?title=Rewards:Formulas) . Top image from [Pixabay - Geralt - CC0 creative commons](https://pixabay.com/en/businessmen-silhouettes-man-economy-1050341/).  All equations made using [quicklatex](http://quicklatex.com/) it is free.  :D

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<h2> Merchandise :D </h2><br />
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<p><img src="https://steemitimages.com/0x0/https://cdn.steemitimages.com/DQmdwbunfcePxAncstQN21fDefS97B3ofRrAABmi29zD26B/g7592-1-9.png" alt="2018-06-07 (2).png" /><br />
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<center><img src="https://steemitimages.com/0x0/https://cdn.steemitimages.com/DQmYSEjNhZDkm4FQPsnPgBdLszb82txzj4fNtKh2Bhb5uiU/banner.png" /></center><center>Image courtesy: <a href="https://steemit.com/@mathowl">Me ;) </a></center>

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