Project 1 A FAIR TREATMENT FOR ALL GROWERS AT THE AUCTION Corvaisier, Aude

Joseph Fourier University, B.P. 53 38041 Grenoble cedex 9, France. E-mail: [email protected]

Moreau-Gaudry, Alexandre

TIMC-UJF-IMAG, CHU Grenoble, Faculte de Medecine, Institut Albert Bonniot, 38706 LA TRONCHE CEDEX, France. E-mail: [email protected]

Kelle, Olavi

Department of Mathematics, University of Joensuu, PL 111, Yliopistonkatu 7, SF-80101 Joensuu, Finland. E-mail: [email protected].

Reis, Anja

Department of Mathematics, University of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, Germany. E-mail: [email protected]

Takkula, Tuomo

Department of Computing Science, Chalmers University of Technology, Eklandagatan 86, S-41296 Goteborg, Sweden. E-mail: [email protected]

Instructor: Ivo Adan

Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

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Abstract

This paper deals with the question, whether the current procedure owers are auctioned at a major Dutch auctioning centre can be considered as fair or not from the growers view. Some growers have stated, that this is not the case. We rst introduce a notion of \fairness", which is meant to approximate the real feeling of a fair or unfair treatment. Then we compare the current procedure used at the auctioning centre with other procedures which we developed and present their eect on our approximation of fairness in a simulation environment. We conclude that the auctioning centre should revise their auctioning system, since it shows a signi cant bias in the growers' chances to sell their products. We show how this drawback can be overcome with.

1 Introduction Flowers Auction is an international trading center with auctioning centers in Naaldwijk and Bleiswijk, the Netherlands. In these auctioning centers a wide range of high quality owers and plants are traded. The heart of the auction are the bidding halls with computer controlled auctioning clocks. The clock runs from the highest price to lower prices. The rst buyer to press button (who therefore pays the highest price) becomes the owner of that lot (or a part of it). Those who pressed too late can try next round. The outcoming price does not depend on the quality of the owers only, but also on the moment the owers are auctioned. For a given type of owers, the price increases a little in the rst rounds, then stays at a certain level and eventually decreases. If supply exceeds demand, then the price produced in the last rounds may be very low or even drop below a minimum price (in which case the remaining

owers are destroyed). Therefore the order in which the lots are auctioned is important. A sample price behaviour is shown in Figure 1. The order in which the growers products arrive at the auction is nearly xed, since the transportation is done by professional agencies, whose lorries use nearly the same route every day to collect the owers. This order determines the order in which the owers are placed into the dedicated areas in the cool store (the dierent types of owers are auctioned one at a time).1 A dedicated area consists of several lines with at most 23 trolleys per line. Lines may not completely be lled, because a growers lot (which may consist of several trolleys) is always in exactly one line. A consequence of this scheme is, that the lot of one grower is always in the front, whereas that of another grower is always at the back of the area. Since the auction center is aware of the eect of the order in which the lots appear to the buyers, the following procedure was installed in order to guarantee fair chances for all growers: once a year lists of rst lines are computed for each 1

In order to protect small producers large and small lots are also traded separately

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1. Fair Treatment at the Auction

19

cool store

21

price

20

00 11 01 11 00 00 11 11 00 11 01 00 00 11 11 00

Start price

Minimum price

first line

9

32

31 19 24 27 30 18 23 26 29 17 22 25 28

36 35 4 34 3 33 2

16 8 7 12 15 6 11 14 5 10 13

1

11 00

clock sold

11 11 00 00 00 11 11 00

rounds

Buyers

cutoff

Figure 2: Order in the auction.

Figure 1: Price behaviour.

day and for each possible amount of lines and kept disclosed in a safe. Every morning the appropriate sheet is taken from the safe and handed out to the workers involved with the trolley movement. Subsequently the lines from the dedicated area are auctioned cyclically starting with the rst line associated with the numbers of lines in the current auction stated on the sheet. A sample of this procedure is shown in Figure 2. The number on the trolleys show the order in which they appear to the auction. For areas consisting of one line only the starting lot is selected and the remaining lots follow cyclically. β1

β2

α1

α2

Figure 3: Sample unfair coolhouse setting. Although the choice of the rst line is made randomly, some of the growers complained that this selection mechanism is not fair. One of the arguments used is that the rst line always follows the last, except when the rst line is selected. The rst aim of this investigation is to nd out whether or not the growers are right. This paper is organized as follows. The next section gives a brief description

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The Ninth ECMI Modelling Week

of the existing auctioning procedure and an answer to the question posed above. Then we explain the requirements on new rules and we introduce the model we use to evaluate dierent rules. We de ne dierent measures of \fairness" and describe our simulation environment. The third section summarises the results of our investigations and the last section draws some conclusions.

2 Problem modelling A rst answer to the question Is the current rule fair ? We give an answer

to this question by a small example. Consider the case that for some type and quality combination of a ower there are only two lines lled in their dedicated areas in the cool house. Assume, that the positions of the lots in the dedicated area is exactly the same each day and call the lots in the front of the lines 1; 2 , and the lots at the back of a line 1 ; 2 | see Figure 3. Suppose, that the cuto takes place when 75% of the lots have been sold. Then we have the following distribution of cuto-hits among the growers, when we apply according to the current rule: The owner of a -lot will never be concerned by a cuto whereas the owner of a -lot suers a cuto on the average half of the time. This can hardly be considered as fair. After this statement we have to develop a rule which is in some sense more \fair" than the current rule. One of the main diculties we face is, that we have to describe the growers feelings and their subjective views of things. Since we have no direct analytical approach to these matters, we try the following: rst we develop a numerical approximation of what \fairness" could be. Then we use this measure to build a simulation of the auctioning process over a given time. In addition, we compute the eect of changes in this process, i.e other line selection schemes. This enables us to compare the resulting \fairness" values for the various line selection schemes.

2.1 Some de nitions

Let G = f1; : : : ; jGjg denote throughout the paper an (index-)set of growers. Without loss of generality we can restrict ourselves to the treatment of lots rather than trolleys. Since the number of lots equals jGj, we can conveniently use lotsubscripts i 2 G to de ne the lot of a certain grower i. For the sake of simplicity, we assume that jGj is arbitrary (positive), but xed in our experiments. The sequence of lots appearing to the auction in the order determined by some rule is called chain. The position of a lot within the chain is called position and applying the note from the last paragraph, denoted by pos(g); g 2 G. The cool house is partitioned into lines which we number as usual from left to right by 1; 2; : : : ;. This set is denoted by L and we assume that L only contains those lines, which are not empty at the beginning of the auction.

1. Fair Treatment at the Auction

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The number of auctions that we consider usually equals the memory period, that is the number of days in the past, which we assume, that growers remember when we determine their happiness. There should be no ambiguity in the following when we use these values. They are used most of the time synonymously and are both denoted by N . In our experiments we always performed a startup period to extinguish memory periods, where nothing happened.

2.2 Problems and requirements we have to cope with

It is easy to invent totally new selling systems which can be considered as quite fair, but which nevertheless will neither be accepted by the growers nor by the auctioning center. For example, one could just use the existing procedure and collect the income achieved over an auction and redistribute it among the growers according to their lotsize and the quality of their owers. But even if the person judging on the quality is highly competent and appreciated, some growers will still for example doubt the quality produced by other growers. So this \communistic" approach will not work. We have collected a list of requirements which have to be ful lled by any new rule system. One should not believe that this list is complete since as a rule of thumb, for every new rule, there is at least one person which complains... At least this list may help to understand what the problem consists of.

 Complexity

The rule installed must not have a signi cant greater complexity than the existing rule. For example, the idea of choosing the rst lot randomly instead of the rst line (and let the others follow cyclically) will be rejected as too complicated and as impracticable, since there is usually not enough \parking" and \maneuvering space" for the part of the \ rst" line, that has to wait until the other lines are sold.

 Price and position in the auction

Growers will usually not complain about achieved prices in certain auctions since these are unpredictable anyway. But they pay a lot of attention to the position of their lot in the chain. If someone feels that he or she is always behind a certain other grower, thus having always a smaller income than this competitor, this grower feels treated unfair, even if you can prove that the same grower is always in front of a certain other grower. We have already shown in our small example, that a rule showing this behaviour may in fact cause problems with respect to equal chances.

 Predictability

The growers know that the last of them in the chain have smaller chances to sell their lots than the rst. Thus, if they were informed about their position in the chain in advance, they would determine the size of their

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The Ninth ECMI Modelling Week lot according to that. It is obvious that major growers could abuse their production potential to create a monopoly for certain days while smaller growers can not. Therefore the rule must be unpredictable.

 Limited growers memory

As all human beings the growers pay more attention to occasions that happened recently compared to occasions that happened a long time ago. More speci cally, a rule guaranteeing equal chances in the very long run but allowing a signi cant bias over a short period will no be considered as sucient.

2.3 The simulation environment

We developed a simulation model where we follow the ow of decisions and of information during the auction of an arbitrary kind of ower over a given number of auction days. We compute the eect of various line selection methods under reasonable parameter settings like the number of growers, the variance of their lot sizes, the demand and so on. Subsequently we brie y describe the major building blocks of our simulator | see also Figure 4. Details about the selection rules and the approximation of fairness and unfairness are given later.

 Create Input Data In this rst step, we determine for a given N and jGj the lotsize of each grower. For this we use some reasonable individual mean and standard deviation to compute the daily lotsizes.

 Fill Cool House

The cool house is lled sequentially respecting the constraints given in the introduction, that is: the maximum number of trolleys is 23 and the lot of a grower will not be stored in two dierent lines. We assume, that the number of trolleys in a single lot is always smaller than 23, since a violation of this constraint is very unlikely anyway. Consequently, the number of lines in an auction is not xed but determined by the growers' lotsizes.

 Create Chain This module applies one of the line-selecting rules and

computes the order in which the lots appear to the auction. We investigated the eect of dierent rules for selecting the rst line and studied their eect on global fairness. These rules are Random Choice, the Monte Carlo Method, the Monte Carlo Method with Contrast Enhancement, the Minimum Variance Method and the Permutation Method. These rules will be discussed in more detail in Subsection 2.4.

 Evaluate Happiness, Happiness functions

It is obvious that after one single auction some growers are certainly happier than others. We use two dierent functions to express this fact by some approximate numerical values. Of course, one can imagine several alternatives. See Section 2.5.

1. Fair Treatment at the Auction

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Input: |G|, |N|, lotsizes

Input data for a large number of days is produced and auctionwise Create Input Data feeded into the simulator loop

Output: daywise auction data

Fill Cool House

Output: filled cool house Accumulated happiness

Accumulated unfairness

Rules

Create Chain

Happiness Function

Evaluate Happiness

Output: effect on growers’ happiness

No

enough days ? Yes

Unfairness Function

Output: order of trolleys in the auction

to next day

Evaluate Unfairness

No to next iteration

enough iterations? Yes

Results

Figure 4: The simulation data and execution ow.

 Evaluate Unfairness, Unfairness Functions

The individual fairness determined in the last building block, is used to determine the \unfairness" in the current auction. This value is computed in this block using one of the unfairness-functions and stored. Some of the line-selection rules access this data to determine the next \ rst" line. See Section 2.6.

 Results If all iterations (auctions) are done, then the data accumulated is reported in the last block.

2.4 Rules

 Random choice This rule models the rule currently used and the reason why some growers started to complain: the rst line is chosen randomly

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The Ninth ECMI Modelling Week among a uniform distribution such that each line has the same probability to be selected. The way this rule works is explained in the introductory Section.  Monte Carlo method The aim of this method is to take into account the past to improve the probability of a recently unlucky grower to be in the rst position in the chain. Let hij be the happiness of grower i in auction j , computed by one of the happiness functions from Section 2.5, and de ne N X 1 h := h i

ij

N j =1

This scheme is shown in Table 1. Growers Happiness : Auction 1 Happiness : Auction 2 ... Happiness : Auction N mean(N)

1

::: ::: :::

h11 h12

...

...

jGj

hjGj1 hjGj2

...

h1N : : : hjGjN h1 : : : hjGj

Table 1: Accumulated-Happiness(N). We know the state of the cool-house at the N +1-th auction and therefore the positions of the growers in the dierent lines.2 We then use these positions to adjust the probability PN +1 (l) of the l-th line to be chosen rst in the current day N + 1 . Let ?l = fj j Lot j 2 Line lg be the set of lots in the l-th line. We calculate the mean h (l) of the mean happiness of the growers in the l-th line: Let 1 X h (l) := j?l j k2? hk h (l) represents a sort of mean \happiness" of the growers in the l-th line. But what we want is the \unhappiness" of the l-th line: the more \unhappy" the l-th line is, the greater we want to have their chance to be chosen as a rst-line. So, by inversion and normalisation, we obtain for each line l the probability l

1

PN +1 (l) =  PjLj 1 h(l) k=1 h (k) 2

Of course, this data must be collected somehow.

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Another possible approach for calculating PN +1(l) can be the following:

?h (l) + maxj h (j )   k=1 (?h(k ) + maxj h(j ))

PN +1 (l) = PjLj

Obviously, with the convention that when all h( j ) are equal, then also the probabilities PN +1(l) are equal.  Monte Carlo method with contrast enhancement In this rule, the idea is to modify the relative weights of the probabilities of the lines to be chosen the rst, to favour the most "unhappy" line even more. For instance, if we have two lines with the probabilities PN +1 (1) = 0:51 and PN +1 (2) = 0:49 respectively, we transform these probabilities to obtain for example PN +1(1) = 0:80 and PN +1(2) = 0:20. A way to do this is the use of a power (  1): P  (l) PN +1 (l) PjLjN +1 i=1 PN +1 (l) We want to "force" the fairness but we want to stay unpredictable, too. In fact, we have to nd a fair balance between these two points of view.  Minimum variance method Another way to achieve fairness is to de ne some unfairness measure (see Section 2.7) and to select the line that minimizes it. To do that, we calculate all possible accumulated happinesses (Table 1) after the current auction. Note, that because the current auction is here technically considered as past, we actually use information about the (N ? 1) last auctions, while in Monte-Carlo method we have used N . Let hli be the happiness of grower i, when line l would be chosen. There are jLj possible choices for the rst line for current, N -th auction. For all selections we have dierent happiness distributions among growers, compare Table 2. Growers line 1 chosen line 2 chosen ... line jLj chosen

G1 : : : GjGj h11 : : : h1jGj h21 : : : h2jGj

...

...

...

hj1Lj : : : hjjLGjj

Table 2: Happinesses for selection. Considering the lines in Table 2 as dierent choices for N -th auction in Table 1, we calculate accumulated happinesses hl . We use the variance of

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The Ninth ECMI Modelling Week accumulated happinesses to model unfairness. (The general meaning of the auction unfairness will be discussed in Section 2.6.) According to some unfairness function F (in Section 2.6 several unfairness functions will be speci ed) we will choose line arg min fF j line l chosen g l2L So the rule is to select the line which yields minimal unfairness. That rule is impractical in reality because it's predictability, but is important in simulation analysis. The rule can be also modi ed such that it is less predictable. Namely, we can add to the selection unfairness some Gaussian noise G (with mean 0 and standard deviation ) and choose line arg min fF + G(0; ) j line l chosen g l2L Again, we have to nd some trade-o between predictability ( = 0) and randomness ( much greater than standard deviation of F (Rk ) corresponds to total randomness).  Permutation method In order to cancel the eect caused by the growers' nite memory (see above) one can use a \random" rule with the property, that the frequency of an arbitrary grower on an arbitrary position is theoretically equal for all growers. This can be done choosing the start line from sequences of random permutations of f1; 2; : : : ; jLjg. This gives a zero unfairness, if jLj and the cuto position are constant over N auctions and if jLj is a divider of N . Unfortunately, in this case this rule is quite predictable. But, since these assumptions are pretty academical, predictability might not be a problem in practice.

2.5 Happiness functions

First, we have to motivate out notion of \happiness functions". Of course, one could choose the happiness proportional to some kind of estimated auction prices. But this way of computation is arguable by itself and hides the more important eect of an auction: Even if the price dierence between the highest achieved price and the lowest price may be big, the loss of the growers hit by a cuto is much more dramatic since these lose all of their investments. The second of our happiness functions describes this fact only, while the rst one just weights the position within the chain.

 Happiness function 1

Happiness function h1 : G ?! IN is de ned by h1 (g ) := jGj ? pos(g )

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Figure 5 shows a graph of h1. Happiness

11 00 00 11 Number of growers - k

11 00 00 11

Happiness

11 00 00 11

11 00 00 11

11 00 00 11

1

11 00 00 11

11 00 00 11

11 00 00 11

11 00 00 11

kth grower

0011 0011 1100 0011 1100 1100 1100 1100 0011 1100 0011 1100

0011 0011 1100 1100 1100 0011

0

11 00 00 11

Growers before the cutoff

Growers

Figure 5: Happiness function h1.

Growers

Growers after the cutoff

Figure 6: Happiness function h2 .

 Happiness function 2 Happiness function h2 : G ?! f0; 1g is de ned by h2 (g ) :=

(

1 if pos(g) < cuto 0 else

Figure 6 shows a graph of h2.

2.6 Unfairness functions

We would say that a series of auctions is \fair", if the happiness of all growers is the same after every auction, and since that is not possible, the variance of the mean happiness among all growers is minimal. To be more precise, we de ne the following unfairness functions.

 Unfairness function 1 and 2

The rst unfairness functions about which we think naturally, are to compare the dierent means hj of the happiness function values after N days. For instance, we can use: or

F1 := max jhi ? hj j i;j



hi F2 := j1 ? max j i;j hj

So, the greater F1 or F2 are, the more unfair is the auction.

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The Ninth ECMI Modelling Week  Unfairness function 3

We can now use a more evoluate function. First, we compute the mean of the means hi : jGj X 1 E := h

jGj i=1

i

Then, we estimate the variance which gives: F3 :=  2 =

jGj

X

i=1

(hi ? E )2

The greater F3 is, the more unfair were the N last auctions. With these functions we can judge, whether a rule is \fair" or not: it is fair, if the unfairness value computed with respect to this rule is minimal under all rules.

3 Results The model described above was implemented and tested in a MatLab environment [2]. From the various experiments we made we selected the following to illustrate the results of our investigations. In the following gures a dashed line denotes the current rule, a dashed-dotted line the Monte-Carlo method with contrast enhancement, a dotted line the Permutation method and a incessant line the Minimum variance method. The numerical values on the unfairness axis should be mostly ignored, since we had to apply normalization in order to be able to compare the dierent methods properly. Thus, only the relations between the dierent methods are important. Figure 7 shows a sample unfairness graph over 50 auctions with 10 growers, a mean lotsize of 10 and a lotsize variance of 3. The Enhanced Monte-Carlo method used the parameter  = 10. One of the most arguable parameter settings is the length of the memory period we suppose. Figure 8a shows the unfairness depending on the memory period under standard settings: 10 growers, 50 experiments, 40 auctions per experiment, mean lot size 9, standard deviation of lot sizes 3. The relations between the the methods remain pretty the same with varying memory periods. Only the permutation method takes advantage of longer memory periods compared to the other methods, since this method levels happiness with a horizon of length jLj. If the memory period is chosen greater than the number of growers, then the change depending on the memory period is quite small for all methods. Figure 8b shows an interesting eect. Using 60 experiments with 30 auctions per experiment, 10 growers, a mean lot size of 9 and a memory period length of 10, we see, that unfairness decreases, as the variance of the lot sizes increases. This happens, because increasing variance has a mixing eect on the ordering in

1. Fair Treatment at the Auction

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Random

0.045 Monte-Carlo with Contrast Enhancement

0.04

Permutation Method

0.035

unfairness

0.03

Minimum Variance Method

0.025 0.02 0.015 0.01 0.005 0 0

10

20

30

40

50

auctions

Figure 7: 50 auctions, standard settings. the cool house and thus a similar eect on the positions in the chain. The permutation method though relies on more or less constant cool house con gurations and performs therefore worse with greater lot size variance. Generally, increasing lot size variance improves fairness. But this dependency is rather sensible to other parameters.

4 Conclusion There were very many parameters to choose: the number of growers participating some auction series, the length of the memory period etc. To simulate change of growers lot sizes, the sizes were created randomly with given mean value and standard deviation. In addition, the Monte-Carlo method and the Minimum Variance methods had one parameter to deal with the tradeo between fairness and unpredictability. Also, three dierent unfairness measures were tested. One thing to choose was also the number of auctions per experiment to simulate. The results of these experiments were the mean numbers of fairness within the auction series for dierent rules (but same "coolhouse data" { growers lot sizes for every auction { otherwise the comparison would have been meaningless). To get reliable information about mean unfairness dependencies, also experiments with same parameter settings were repeated and again the results averaged. Also many parameter settings were tested. All results assured our basic assumption: there exist better line selection

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The Ninth ECMI Modelling Week 0.08 0.07

unfairness

0.06 0.05 0.04 0.03 0.02 0.01 0

5

10 15 memory period

20

25

0.022 0.02

unfairness

0.018 0.016 0.014 0.012 0.01 0.008 0

0.5

1

1.5 2 2.5 standard deviation of lot size

3

3.5

4

Figure 8: Memory period length and variance of lot sizes. schemes than just random choice. The dierence in fairness between random choice and best possible, minimum variance rule, is always remarkable and obvious not only in average value terms, but even just in practically every experiment. Also the three other rules, i.e. the Monte-Carlo Rules and the Permutation Method perform signi cantly better than the current random rule. All these results remain true for dierent happiness and unfairness functions. Functions h1 and F3 (used for all gures) were a little more stable, i.e. the variance of 'unfairness' was smaller. The implementation requirements are quite dierent for the various rules. The (probably predictable) Permutation Method can be installed with roughly the same eort (and the same cost) as the Random Choice Rule, whereas the Minimum Variance and the Monte-Carlo Methods require a great deal of data collection (cut-o times, lot positions for each lot) and information management. It is not at

1. Fair Treatment at the Auction

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all obvious how this three-fold tradeo between implementation eort, predictability and fairness can be dealt with, but we believe, that each of our methods is worth considering when the current system is replaced { and replacement seems inevitable.

References [1] Ivo Adan, private communication. [2] Matlab, The MathWorks, Inc. 24 Prime Park Way Natick, MA 01760, FAX: 508-653-2997.