Lot split ratios: ten lots read 1.6 bidders per lot if they compete and 0.5 if they share

Type the tender's volume, each bidder's capacity and the number of lots. Five bidders on ten lots of 10 read 1.6 per lot competing and 0.5 sharing; five lots read 1.4 and 0.8. More lots raised one ratio and lowered the other. No split is recommended.

Live result

Ten lots of 10 read 1.60 bidders per lot if the five compete and 0.50 if they share.

Lot size10
Bidders able to bid5 biddersA · B · C · D · E
Bidding capacity in lots16 lots
Optimistic ratio1.60
Pessimistic ratio0.50

Five lots read 1.40 and 0.80: adding lots raised the competing reading and lowered the sharing one.

Givens

Three proposed splits

The lots are equal; volume and capacities use one unit.

Tender volume: Tender volume100

From 1 to 1,000,000 in the unit you use.

Each capacity is your estimate, from 0.01 to 1,000,000 in the tender's unit.

Bidder 1
Name 1A
Capacity A60
Bidder 2
Name 2B
Capacity B40
Bidder 3
Name 3C
Capacity C30
Bidder 4
Name 4D
Capacity D20
Bidder 5
Name 5E
Capacity E15

Number of equal lots: Number of equal lots10

A whole number from 1 to 20.

Two readings of the same split

Optimistic ratioPessimistic ratio

Each step reads an equal split; the two flags compare the typed count with the first count giving the highest sharing reading.

This page shows what a mechanism does under stated rules and says nothing about any public procurement regulation. It scores your split and recommends none. Capacities are your estimates. The competing and sharing readings bracket what these ratios can say without forecasting a tender result; both matter.

Every lot count from 1 to 20

Lot count	Lot size	Bidders able to bid	Bidding capacity in lots	Optimistic ratio	Pessimistic ratio
1	100	0	0	0.00	0.00
2	50	1	1	0.50	0.50
3	33.33	2	2	0.67	0.67
4	25	3	4	1.00	0.75
5	20	4	7	1.40	0.80
6	16.67	4	7	1.17	0.67
7	14.29	5	10	1.43	0.71
8	12.5	5	11	1.38	0.63
9	11.11	5	12	1.33	0.56
10typed	10	5	16	1.60	0.50
11	9.09	5	16	1.45	0.45
12	8.33	5	17	1.42	0.42
13	7.69	5	18	1.38	0.38
14	7.14	5	21	1.50	0.36
15	6.67	5	24	1.60	0.33
16	6.25	5	24	1.50	0.31
17	5.88	5	26	1.53	0.29
18	5.56	5	27	1.50	0.28
19	5.26	5	28	1.47	0.26
20	5	5	33	1.65	0.25

Here is the maths

Count bidders in two ways

Dividing the volume makes each lot smaller. Capacity determines who can bid and how many lots each bidder can cover. Competing credits that capacity; sharing counts each bidder once.

Divide the tender into equal lots:

s=\frac{V}{L}

where

s

is the lot size and

V

the tender volume and

L

the lot count.

Count whole lots within each capacity, capped at the tender's lot count:

t_i=\min\!\left(L,\left\lfloor\frac{K_i}{s}\right\rfloor\right)

where

t_i

is the number that one bidder can cover and

i

identifies that bidder and

K_i

is its capacity.

Count each bidder who can cover a lot once:

N=\sum_i\mathbf{1}_{\{t_i\ge1\}}

where

N

is the number of distinct bidders able to bid; the indicator contributes one when the condition holds.

Add all their whole lot capacities:

n=\sum_i t_i

where

n

is bidding capacity counted in lots.

The competing reading credits every lot of capacity:

OR=\frac{n}{L}

where

OR

is the optimistic ratio.

The sharing reading credits each eligible bidder once:

PR=\frac{N}{L}

where

PR

is the pessimistic ratio.

A worked example

Smaller lots admit one more bidder here, but that gain does not keep pace with the extra lots; the two readings move apart.

The tender volume is 100 and the capacities, in row order, are 60, 40, 30, 20 and 15; at ten lots the size is:
$s=\frac{100}{10}=10$
Divide each capacity by that size and retain whole lots:
$(t_1,t_2,t_3,t_4,t_5)=(6,4,3,2,1)$
Every bidder can cover a lot, and their lot capacities add to:
$N=5\qquad n=6+4+3+2+1=16$ $\begin{gathered}N=5 \\ n=6+4+3+2+1=16\end{gathered}$
The two readings at ten lots are:
$OR=\frac{16}{10}=1.60\qquad PR=\frac{5}{10}=0.50$ $\begin{gathered}OR=\frac{16}{10}=1.60 \\ PR=\frac{5}{10}=0.50\end{gathered}$
At five lots the size is:
$s=\frac{100}{5}=20$
The same capacities now cover:
$(t_1,t_2,t_3,t_4,t_5)=(3,2,1,1,0)$
The last bidder can no longer cover a lot:
$N=4\qquad n=3+2+1+1+0=7$ $\begin{gathered}N=4 \\ n=3+2+1+1+0=7\end{gathered}$
The two readings at five lots are:
$OR=\frac{7}{5}=1.40\qquad PR=\frac{4}{5}=0.80$ $\begin{gathered}OR=\frac{7}{5}=1.40 \\ PR=\frac{4}{5}=0.80\end{gathered}$

What each control can move

The lot count changes size, eligibility and both denominators; the volume changes size and which capacities reach a whole lot.

Each capacity can change how many lots its bidder covers and whether that bidder is counted at all; adding or removing a bidder can change both ratios.

Names identify the bidders read back in the finding; they carry no weight in either ratio.

The sweep holds volume and capacities fixed across every count from 1 to 20; its comparison takes the smallest count tied for the highest pessimistic ratio.

Capacities exactly at a whole lot boundary count that whole lot, allowing a tiny tolerance for computer arithmetic.

What this does not model

Unequal lots.

Bidders whose capacity is not measured in the tender's unit.

Package bids or collusion itself.

Source basis

Nicola Dimitri, Gustavo Piga and Giancarlo Spagnolo, Handbook of Procurement, supplies both ratios: capacity under strong competition and distinct bidders under suspected sharing, including the risk when lots match strong incumbents. Paul Klemperer, Auctions: Theory and Practice, describes spectrum sales in which matching licences to incumbents yielded little revenue and an extra differently sized licence attracted more bidders.

Scale: Linear axes; whole counts from 1 to 20; ratios to 2 decimals and lot sizes to at most 2
Basis: Equal lots, typed volume and estimated capacities in one unit
Model version: 1.0.0

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These ratios describe your stated split and capacities. They predict neither participation nor conduct, and recommend no lot count, bidder or award.

Givens

Two readings of the same split

Every lot count from 1 to 20

Here is the maths

Count bidders in two ways

A worked example

What each control can move

What this does not model

Source basis

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