EXAMPLE 5
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start with 4 people:
  A has 10k and qualifies for a 50% bonus on all stakes
  B has 12k and qualifies for a 25% bonus on all stakes
  C has 15k and doesn't get any bonuses
  D has 10k and qualifies for a 20% bonus on all stakes
note that 150% of 10k = 125% of 12k = 100% of 15k, so A, B, C can all get the same number of shares

day 1:

all of them open stakes on day 1; the price is initially 1 hex per share. A,B,C get 15k shares each, D gets 12k shares. total 57k shares
let's say the pool pays a return of 10% to 0% bonus stakes on the first day. to do that we need to calculate the effective amounts staked. the 10k guy with the 50% bonus is effectively staking 15k for example. so we have 57k hex effectively staked, and 57k shares. the payout for will be 5700 hex:

'effective stake' (col 4) is the amount staked (col 2) with the bonus (col 3) added on
'stake value' (col 7) is hex staked (col 2) plus daily profit (col 6) - it is the amount the staker actually gets if they end their stake
'effective value' (col 8) is effective stake (col 4) plus daily profit (col 6) - it is the amount their stake would be worth if their bonus had multiplied their coins as well as their shares
'next price' (col 9) is effective value (col 8) divided by number of shares (col 5) - note that you can use any of the 4 stakes to determine tomorrow's price - they all give the same number (and it's 1.1 because the daily return was 10%), and of course dividing the sum of (col 8) by the sum of (col 5) also gives the same price

                            (2)+(3)                   (2)+(6)     (4)+(6) (8)/(5)
                          effective             daily   stake   effective    next
   who   staked   bonus       stake   shares   profit   value       value   price
   ---1  ------2  -----3  ---------4  ------5  ------6  -----7  ---------8  -----9
     A    10000     50%       15000    15000     1500   11500       16500     1.1
     B    12000     25%       15000    15000     1500   13500       16500     1.1
     C    15000      0%       15000    15000     1500   16500       16500     1.1
     D    10000     20%       12000    12000     1200   11200       13200     1.1
   ---   ------   -----   ---------   ------   ------   -----   ---------   -----
                              57000    57000     5700   52700       62700     1.1

day 2:

let's say none of them pull out, and that today the pool pays 10% again (to 0% bonus stakes). C has no bonus, and is worth 16.5k. 10% means he should get 1650. he's getting 15k/57k of the profit, so the whole pool needs to get 1650*57/15 = 6270 (which is 10% of the effective value (col 8))

                            (2)+(3)                  (2)+(6s)    (4)+(6s) (8)/(5)
                          effective             daily   stake   effective    next
   who   staked   bonus       stake   shares   profit   value       value   price
   ---1  ------2  -----3  ---------4  ------5  ------6  -----7  ---------8  -----9
     A    10000     50%       15000    15000     1650   13150       18150     1.21
     B    12000     25%       15000    15000     1650   15150       18150     1.21
     C    15000      0%       15000    15000     1650   18150       18150     1.21
     D    10000     20%       12000    12000     1320   12520       14520     1.21
   ---   ------   -----   ---------   ------   ------   -----   ---------   -----
                              57000    57000     6270   58970       68970

note that the pool has paid out a compounded 10% both days, and the price per share has also increased 10% per day 1 -> 1.1 -> 1.21
the price per share is the same as the return made by a 0% bonus staker since the start. C's 15k has turned into 15*1.21 = 18.15k

let's check this by having a day with 40% returns. everyone stays staked again:

day 3:

pool pays 40% of effective value: 0.4 * 68970 = 27588

                            (2)+(3)                  (2)+(6s)    (4)+(6s) (8)/(5)
                          effective             daily   stake   effective    next               total   percentage
   who   staked   bonus       stake   shares   profit   value       value   price              profit       profit
   ---1  ------2  -----3  ---------4  ------5  ------6  -----7  ---------8  -----9
     A    10000     50%       15000    15000     7260   20410       25410   1.694               10410        104.1  : 50% more than C's profit [69.4 * 1.5 = 104.1]
     B    12000     25%       15000    15000     7260   22410       25410   1.694               10410         86.75 : 25% more than C's profit
     C    15000      0%       15000    15000     7260   25410       25410   1.694               10410         69.4  :    40% on 10% on 10% (1.1*1.1*1.4 = 1.694)
     D    10000     20%       12000    12000     5808   18328       20328   1.694                8328         83.28 : 20% more than C's profit
   ---   ------   -----   ---------   ------   ------   -----   ---------   -----
                              57000    57000    27588   86558       96558   1.694

now the next price is 1.694, which is 1.1 * 1.1 * 1.4, as expected after three days returning 10%, 10%, 40%

so by now we're happy that the new share price multiplier each day is simply the return multiplier. let's check what happens when a staker with a bonus tries to externally compound

day 4:

if C ends stake he gets his stake value (col 7) [it's his initial 15k plus the profits from days 1 (1500), 2 (1650), and 3 (7260): 15e3+1500+1650+7260 = 25410]
   at a price of 1.694 that gets him 25410/1.694 = 15000 shares - the same as he had already

if B ends stake. he gets his stake value (col 7) [it's his initial 12k plus the profits from days 1 (1500), 2 (1650), and 3 (7260): 12e3+1500+1650+7260 = 22410]
   at a price of 1.694, and assuming he's still getting a 25% bonus that gets him 22410*1.25/1.694 = 16536 shares - that's more than he had before. it looks like he can beat the system by externally compounding, but that's because the example was dumb. nobody gets a 25% bonus and can end stake without penalty after 3 days. it does make me worry however that the system maystill be gameable.

  I can get 20% bonus for each extra year I stake.
  If I stake for 12 sets of 30 days I get a bonus of 1.6438% for each stake   1 + 0.20 *  30/365 = 1.016
  If instead I stake for the whole 371 days I get 20.3287% for the stake      1 + 0.20 * 371/365 = 1.203 (12 30 day stakes take 371 days due to having to skip a day between stakes)

Compounding 12 30 day stakes gives a return that is (1 + 0.20 *  30/365) ** 12 = 1.216 times better than having no bonus
A single 371 day stake gives a return that is       (1 + 0.20 * 371/365) **  1 = 1.203 times better than having no bonus

The 12 sets of monthly stakes pays slightly better...

The problem is that you're trying to offset an exponential bonus (from manually restaking) with a linearly growing bonus (longer pays better). Same old story: exponential beats linear!