How to determine lead ranking score with changeable variables An auctioneer holds an open reverse auction between two bidding parties. Prior to the auction starting, each party fills out a survey which is scored by the auctioneer, who scores it out of a possible 10 percentage points. Price bidding then commences and the bid price makes up the remaining 90 percentage points. Auction rank is determined by highest auction score using this calculation:


*

*let x = the current bidders lowest bid amount (lower is better)

*let y = the overall lowest current bid amount

*let z = the current bidders survey score


score = $(90 * (100 * (1 - (x - y) / y)) / 100) + z$
Party A scores 5 on the survey and bids £85, giving a rounded score of 95.00 and a ranking of 1(st). Party A score = $(90 * (100 * (1 - (85 - 85) / 85)) / 100) + 5 = 95$
Party B scores 8 on the survey and bids £90, giving a rounded score of 92.71 and a ranking of 2(nd). Party B score = $(90 * (100 * (1 - (90 - 85) / 85)) / 100) + 8 = 92.70588235294118$
Question: For valid values of x, y and z, what is the calculation that I can use to calculate the highest possible money amount that the 2nd place supplier can bid to move them into first ranking position?
Edit: what happens when Party B bids a new leading low amount of £80?
Party B still scores 8 on the survey and now bids £80 (which becomes the new value of $y$), giving a rounded event score of 98 and moves into 1st ranked position. Party B score = $(90 * (100 * (1 - (80 - 80) / 80)) / 100) + 8 = 98$
Party A still scores 5 on the survey and bid remains £85, giving a rounded event score of 89.38 and a new ranking of 2nd. Party A score = $(90 * (100 * (1 - (85 - 80) / 80)) / 100) + 5 = 89.375$
In the example above, I used a target bid price of £80. I would like to understand how to calculate the target bid price so that the difference in score between A and B is only 0.01.
(I can't figure this out because by changing the lead score you change the other score as well. I have made a number of attempts, none satisfactory, I either overshoot or undershoot the target price. Apologies for poor terminology or lack of formatting, mathematics is not my forte.)
 A: Let $x_1$ and $x_2$ be the bids, $s_1$ and $s_2$ the corresponding survey scores and $m=\min(x_1,x_2)$. The bidders score respectively (I simplified a little your expression):
$$
s_1+90\left(2-{x_1\over m}\right)
\quad\hbox{and}\quad
s_2+90\left(2-{x_2\over m}\right)
$$
and these have the same value if
$$
s_1+90\left(2-{x_1\over m}\right)
=
s_2+90\left(2-{x_2\over m}\right)
$$
that is if:
$$
90{x_2-x_1\over m}=s_2-s_1.
$$
Suppose, without loss of generality, that $s_2\ge s_1$: the above equation then tells us that $x_2\ge x_1$ at the equilibrium point. We can thus substitute $m=x_1$ into that equation, to get the equilibrium equation:

$$ {x_2\over x_1}=1+{s_2-s_1\over 90}. $$

In your first example, $x_1=85$, $s_1=5$, $x_2=90$, $s_2=8$. Equilibrium is reached when 
$$
{x_2\over x_1}=1+{3\over90}={31\over30}.
$$ 
With $x_1=85$ the second bidder can thus make a winning offer at $x_2<85\cdot{31\over30}=87.83$.
In your second example, $x_1=85$, $s_1=5$, $x_2=80$, $s_2=8$. Equilibrium equation is the same as above, thus with $x_2=80$ the first bidder should bid less then $80\cdot{30\over31}=77.42$ to get a better score.
