[Type here]

Reverse Auction Simulation Analysis

Reference No: LM0105

Issued by: ADAS

Date: 31 August 2015

Annex 3

Contents

1 The Simulation Environment ..................................................................................................... 1

1.1 Farmers: Fundamentals of Decision-Making .............................................................................1

1.2 Foregone Income: 𝑐 ...................................................................................................................2

1.3 Transaction Costs: 𝑡𝑐 .................................................................................................................3

1.4 Risk Aversion: 𝛾 .........................................................................................................................3

1.5 Probabilities of Success: 𝑃𝑟𝑤𝑖𝑛|𝑏 .............................................................................................6

1.6 Strategic Bidding Logic ........................................................................................................... 10

1.7 Information Feedback: Update Logic ..................................................................................... 13

1.8 Guide Prices: Update Logic ..................................................................................................... 14

1.9 Multiple activities in a budget-constrained auction ............................................................... 15

1.10 References .............................................................................................................................. 16

Tables

Table 1: Certainty equivalents to risky future income flows from agriculture .............................................3

Table 2: Auction outcomes under different assumptions regarding farmer risk aversion ............................5

Table 3: Auction outcomes under different assumptions regarding prior bias ............................................9

Table 4: Auction outcomes under different assumptions regarding bidding sophistication ..................... 12

Figures

Figure 1: Bidding behaviour under different assumptions regarding risk aversion ......................................4

Figure 2: Proportion of farmers participating in the auction under different assumptions regarding risk

aversion .........................................................................................................................................................5

Figure 3: Bidding behaviour under different assumptions regarding prior bias ...........................................8

Figure 4: Proportion of farmers participating in the auction under different assumptions regarding prior

bias ................................................................................................................................................................9

Figure 5: Bidding behaviour under different assumptions regarding the sophistication of bidding ......... 11

Figure 6: Proportion of farmers participating in the auction under different assumptions regarding the

sophistication of bidding ............................................................................................................................ 12

[Type here]

© ADAS 2014 1

1 The Simulation Environment

In this annex we provided a detailed description of the mechanics of the auction simulator and

investigate the sensitivity of the research findings recorded in the main report to some key assumptions.

The analyses reported here use the simulation environment to explore a budget-constrained scheme

where the government is looking to contract farmers to undertake a single agri-environment activity.

As described in the main report, the set of farms eligible for the scheme is taken to be a random sample

of a 1,000 farms drawn from the farm business survey. We assume that the agri-environment activity

requires each farm to take 1 ha out of production such that the costs of participation in the scheme

amount to each farm’s gross margin per ha. Each farm has a quality score that is drawn at random from

a uniform distribution centred on 100 and with a range of 100. The simulation code allows us to allocate

those quality scores to farms at random, or in such a way that cost and quality show positive or negative

correlation. The budget for the scheme is set at just over £153,500 an amount that allows for around a

third of the eligible farms to be offered a contract. In each case we imagine the scheme is run each year

for five years with a new (though identical) set of farmers participating each year.

1.1 Farmers: Fundamentals of Decision-Making

From the outset, we assume that farmers are rational decision makers; that is to say, each farmer,

within the bounds of the information provided to them and the amount of effort they are prepared to

put into figuring out how to bid, attempts to make choices which maximise their returns. To that end,

we posit a utility function, u(c), which gives the utility that a farmer gets in the event that they earn an

income of c. Moreover, we assume that farmers exhibit risk aversion and make the standard assumption

that the farmer’s utility function can be approximated by the Constant Relative Risk Aversion (CRRA)

specification:

u(c) =c1−γ

1 − γ

where γ is the coefficient of relative risk aversion. As we shall explore shortly, the mathematical

properties of this utility function are such that higher values of γ equate to higher levels of risk aversion.

In bidding in an auction, a farmer’s actual earnings are uncertain though farmers can use the

information at their disposal to estimate the probability of different possible earnings outcomes.

Accordingly, we assume farmers evaluate uncertain outcomes by calculating expected utility;

E[u(c)] = ∑ Pr(a) u(ca) = ∑ Pr(a) ∙ca

1−γ

1 − γa∈Aa∈A

where 𝐴, is the set of possible earnings outcomes, Pr(a) is the probability of earnings outcome a

occurring and ca is income in state a.

In an auction a farmer’s key decision relates to the payment they request in the event that they are

successful. We shall label that bid amount, 𝑏. Of course, famers want to earn as much as possible which

encourages them to enter a high 𝑏. At the same time, the probability of winning falls the higher the level

of 𝑏, since asking for a higher payment makes a bid less attractive to the government relative to other

farmers’ bids. In formulating a bid a farmer must weigh up these counter-acting factors. More formally,

we can write the farmer’s bidding problem as;

© ADAS 2014 2

Maxb : Pr(win|b) ∙ (b1−γ

1 − γ) + (1 − Pr(win|b)) ∙ (

c1−γ

1 − γ)

where in this case c is specifically the income they have to forego if their bid of 𝑏 is accepted and they

join the agri-environment scheme.

A final complicating factor comes in the form of transaction costs. In our analyses we assume that

bidding in the auction is itself a costly undertaking. Indeed, the process of entering a bid imposes costs

of 𝑡𝑐 on a farmer, costs that they can avoid if they decide not to participate in the auction. If a farmer

believes that their chances of winning are rather slim, then it could well be a good strategy to avoid

incurring the transaction costs by not entering a bid in the auction. Accordingly, in our simulator the

decision problem faced by each farmer amounts to solving the following problem:

𝑀𝑎𝑥𝑏,𝜃: 𝜃 [Pr(𝑤𝑖𝑛|𝑏) ∙ ((𝑏 − 𝑡𝑐)1−𝛾

1 − 𝛾) + (1 − Pr(𝑤𝑖𝑛|𝑏) ∙ (

(𝑐 − 𝑡𝑐)1−𝛾

1 − 𝛾)] + (1 − 𝜃) [

𝑐1−𝛾

1 − 𝛾] (A.1)

where 𝜃 is a binary decision variable taking the value 1 if the farmer decides to bid and 0 otherwise.

Notice that the introduction of transaction costs has two main effects on the decision problem. First it

means that the minimum payment required by a farmer to make it worthwhile bidding in the auction

must at least cover their foregone income and the transaction cost; 𝑏 > 𝑐 + 𝑡𝑐. Second, it may lead to

some farmers deciding not to participate in the auction.

In operationalising the decision problem A.1 for the purposes of construction the auction simulator, a

number of key issues need to be addressed. First with regards to the data; what are appropriate levels

of foregone income, 𝑐, and transaction costs, 𝑡𝑐? Second, what is an appropriate level of risk aversion,

𝛾? And finally, how do farmers assess their chances of winning, 𝑃𝑟(𝑤𝑖𝑛|𝑏)?

1.2 Foregone Income: 𝑐

As explained in the main report, in the simulator we make the assumption that the activities to be

undertaken in the agri-environment scheme require a farmer to take 1 hectare of land out of

production. The Steering Group provided us with data from the Farm Business Survey that recorded the

distribution of farm gross margins (FGM) per hectare for 2013. Having adjusted the data to remove the

influence of the very high returns realised in horticulture, a sample of 1,000 FGMs per hectare were

drawn at random from the resultant distribution. Each of those FGMs per hectare was taken to be the

foregone income of a farm in our simulation.

Of course, when participating in an agri-environment scheme, fluctuations in yields and market

conditions mean that farmers are uncertain as to the actual levels of income they will forego by

participating in the scheme. Accordingly, we used data published by DEFRA (DEFRA, 2014) to calculate

the average annual percentage change in FGMs across the period 2011 to 2014 and for each farm

constructed two further possible foregone income quantities that were this percentage higher and

lower than the foregone income derived from the 2013 data.

In contrast, income from an agri-environment scheme is fixed and certain over the length of that

contract. In order to compare the uncertain income flow from agriculture to the certain flow from an

agri-environment contract, we assume that farmers process the variable agricultural income into a

single ‘certainty equivalent’ amount. A certainty equivalent is a certain income amount that a farmer

regards as giving the same utility as the set of uncertain incomes that might arise from agriculture. In

our case the certainty equivalent cost, 𝑐, is implicitly defined as;

© ADAS 2014 3

𝑐1−𝛾

1 − γ=

1

3. (

𝑐𝑙𝑜𝑤1−𝛾

1 − γ) +

1

3. (

𝑐𝑚𝑒𝑑1−𝛾

1 − γ) +

1

3. (

𝑐ℎ𝑖𝑔ℎ1−𝛾

1 − γ) (A.2)

where 𝑐𝑙𝑜𝑤 is the low estimate, 𝑐𝑙𝑜𝑤 the medium estimate and 𝑐ℎ𝑖𝑔ℎ the medium estimate of possible

agricultural income. Notice that for wont of a better assumption, in calculating the certainty equivalent

income we attribute an equal probability to each possible income.

Table 1 uses equation A.2 to calculate some illustrative certainty equivalent figures for different levels of

risk aversion. The uncertain agricultural incomes of £300, £500 and £700 are not untypical of those

derived from the FBS data. Notice that if 𝛾 takes a value of zero, then a farmer shows no aversion to risk

and the certainty equivalent is simply their expected income (in our case the average of the three

possible incomes). Increasing levels of 𝛾, imply increasing risk aversion; farmers regard the uncertain

income with increasing distaste preferring to accept a lower certain income 𝑐 as equivalent to that risky

prospect.

Table 1: Certainty equivalents to risky future income flows from agriculture

Risk

Aversion (𝜸)

Uncertain Income: Certainty

Equivalent

Income (𝒄) Low (𝒄𝒍𝒐𝒘) Medium (𝒄𝒎𝒆𝒅) High (𝒄𝒉𝒊𝒈𝒉)

0 £300 £500 £700 £500.00

0.5 £300 £500 £700 £486.03

3 £300 £500 £700 £418.22

6 £300 £500 £700 £367.18

In the simulator, farmers’ foregone incomes (that is, the cost of winning a contract) are taken to be

certainty equivalents calculated using equation A.2. Notice that the higher the level of risk aversion

assumed for farmers, the lower the value of this foregone income and hence the lower the perceived

cost associated with taking on an agri-environment contract; higher risk aversion implies farmers

increasingly preferring the relative certainty of income from an agri-environment contract.

1.3 Transaction Costs: 𝑡𝑐

There is some evidence that transaction costs in participating in agri-environment schemes are not

insubstantial. In the simulator we rely on the work of Mettepenningen et al. (2009) who estimate such

costs to around £20 per ha. The main report provides a detailed investigation of the impacts on auction

outcomes from assuming a lower and higher transaction cost.

1.4 Risk Aversion: 𝛾

As well as determining how farmers calculate the costs of taking on an agri-environment contract,

equation A.1 makes clear that risk aversion is fundamental to shaping farmers’ bidding behaviour. To

explore that impact further, consider Figure 1 which summarises the bids received in auctions run over 5

consecutive years in which the VFM of the winning bids in each year’s auction are communicated to

farmers. In these simulations, farmers face transactions costs of £20.

© ADAS 2014 4

The left panel simulates bidding behaviour when farmers have low risk aversion (𝛾 = 0.5), the middle

panel moderate risk aversion (𝛾 = 3) and the right panel high risk aversion (𝛾 = 6). To understand

Figure 1, note that it plots out information for each farmer, organising those farmers from left to right in

order of declining quality-for-cost. Remember quality-for-cost is calculated by dividing a farmer’s quality

score by their foregone income. Farmers to the left of the plot are those with the highest quality-for-

cost and are those who have the potential to offer the highest VFM bids. Moving to the right, the

quality-for-cost of each successive farmer declines, such that those farmers are progressively less well-

positioned to enter high VFM bids.

Six lines are visible on the figure. The black line represents farmers’ costs. As a matter of fact, this line is

a ‘smooth’; that is to say it gives the average foregone income for farmers around each level of quality-

for-cost. In this simulation, foregone income and quality are not correlated such that we observe that

these averaged costs tend to rise from left to right: those capable of offering the best VFM are also

those that tend to have the lowest costs. Notice also that this locally-averaged cost curve is not always

increasing due to the fact that at some particular quality-for-cost we may have a simulated farmer with

an unusually high (low) cost but an equally unusually high (low) quality.

The five progressively paler blue lines are also smooths. They illustrate the bidding behaviour of farmers

for each of the five years of the auction. Recall that the budget will fund around a thrid of the 1,000

simulated farmers. Since those with low quality-for-cost have very little chance of being successful in the

auction, the plot is truncated at the farmer with the 500th lowest quality-for-cost.

Figure 1: Bidding behaviour under different assumptions regarding risk aversion

Notice from Figure 1 that as we move from low to moderate to high risk aversion the cost line

progressively falls, a result that follows from each farmer’s costs being calculated as certainty

equivalents to their variable returns from agriculture (see section 1.2): that is to say, the minimum

amount that the farmer would accept for certain (perhaps as an agri-environment contract) rather than

face the prospect of an uncertain return from agricultural output. The greater a farmer’s risk aversion,

the greater their distaste for uncertain outcomes and hence the lower their certainty equivalent cost.

The general pattern of bidding behaviour over successive years of the auction (as shown by the blue

lines) is very similar, though bids are noticeably lower the greater the level of risk aversion assumed for

farmers. That observation is confirmed by the data from the numerical summary of these three

simulations presented in Table 2. Observe, for example, that in the first year’s auction the average

payment made to a winning participant is £484 when risk aversion is low, £458 when risk aversion is at

its medium level and £439 when risk aversion is high. It appears that the lower perceived opportunity

costs that result from higher risk aversion, induce relatively lower bids from farmers. In particular, since

other farmers with high risk aversion are prepared to accept relatively lower payments to participate in

the agri-environment scheme, the auction is more competitive forcing farmers to enter lower bids.

© ADAS 2014 5

Table 2: Auction outcomes under different assumptions regarding farmer risk aversion

Risk Aversion

Year of

Scheme

Low ( = 0.5)

Moderate ( = 3)

High ( = 6)

Part1 Avg Bid2 Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

One 39.3% £483.71 0.224

40.1% £457.92 0.235

40.2% £439.05 0.243

Two 44.5% £503.99 0.232

44.5% £476.14 0.243

45.9% £454.22 0.255

Three 40.5% £500.56 0.231

40.5% £478.66 0.242

39.5% £464.53 0.250

Four 41.0% £504.98 0.230

42.0% £480.97 0.240

41.3% £464.32 0.251

Five 40.4% £507.04 0.229

41.0% £482.89 0.240

39.9% £470.50 0.250

Notes: 1 Participation rate in percent;

2 Average of winning bids;

3 Cost efficiency calculated as quality points per £

of govt. expenditure on contracts

Notice also from Table 2 that levels of overall participation in each year’s auction remain quite similar no

matter what the level of risk aversion. A more detailed exposition of the participation data is provided

by Figure 2 which plots out the proportion of farmers at each point on the quality-for-cost distribution

that choose to enter a bid in the auction. From those diagrams, it is apparent that participation rates in

the first year (dark blue line) trace out a similar pattern for each assumption regarding risk aversion. For

all three simulations, over subsequent auctions (progressively lighter blue lines) participation tends to

increase for farmers with high quality-for-cost as they learn from feedback that they are well-placed to

enter a successful bid in the auction, but fall for low quality-for-cost farmers who learn that their

circumstances make them relatively less competitive. In all cases, the pivot between these two

behaviours is around the 333rd highest quality-for-cost farmer; that is to say, around the number of

farmers that stand to be successful in the auction given the budget.

Figure 2: Proportion of farmers participating in the auction under different assumptions regarding risk aversion

There are subtle variations in the patterns of participation across the simulations. For example, it

appears that participation amongst the group of farmers spanning the 200th to the 333rd positions on the

quality-for-cost distribution is generally lower at higher levels of risk aversion in later year’s auctions.

This group represent what we shall term the ‘competitive fringe’: farmers who have relatively good

quality-for-cost that and from whom we might hope to elicit reasonably competitive bids.

© ADAS 2014 6

Accordingly, the impact of risk aversion on the overall cost-efficiency of the auction appears to comprise

two opposing effects. On the one hand, greater risk aversion leads to lower bids from those that

participate in the auction (see Figure 1). On the other, greater risk aversion discourages participation

from those in the competitive fringe who have the capacity to offer cost-efficient bids (see Figure 2). The

aggregate of those two effects can be deduced from the data on overall cost-efficiency shown in Table

2. Notably, for each year’s auction we observe that the higher the level of assume risk aversion, the

greater the cost-efficiency of the simulated auction: clearly the bidding effect outweighs the

participation effect.

In deciding upon a level of risk aversion to include in the simulation, we reviewed the evidence from the

literature. Many analyses support a value for the (constant relative) risk aversion coefficient, 𝛾, in the

range 2 to 4 (e.g. Palsson, 1996; Meyer and Meyer, 2005; , Chiappori and Paiella, 2011). Some have

argued for a much higher value of up to 30 (e.g. Janecek, 2002). In the specific context of farming a

similarly broad range of estimates can be found. For example, Bar-Shira et al. (1997) estimate a

coefficient of 0.6 for farmers in Israel, Chavas and Holt (1996) suggest a coefficient between 1.41 and

6.81 for US farmers a result similar to the value range estimated by Abdulkadri and Langemeier (2000)

of 2.85 to 6.33. At the top end Binswanger et al. (1981) report a risk aversion coefficient in the range 10

to 30 for rural Indian farmers.

In the rest of the analysis we make the relatively conservative assumption that farmers have a risk

aversion parameter with a value of 3. That value is consistent with the literature, results in attitudes to

risk that seem plausible to us1 and does not overly favour an auction mechanism in comparison to a

fixed price scheme by inducing unrealistically conservative bidding.

1.5 Probabilities of Success: 𝑃𝑟(𝑤𝑖𝑛|𝑏)

The final element of the decision problem in Equation A.1 that must be specified in order to implement

a simulation of the auction concerns how farmers calculate their probability of being successful in the

auction at any particular bid.

The approach we take in our simulator is based upon the idea of farmers having some prior knowledge

on the costs and qualities of a selection of other farmers. We motivate this as being either information

that a farmer has on the costs and qualities of real farms (perhaps those of other local farmers or of

business associates) or as a farmer’s efforts to estimate the levels of costs and qualities of other farms

that might participate in the auction.

In our simulator we assume that out of the 1,000 farmers in the population each farmer has information

on 25 other farms. Those 25 observations of costs and qualities constitute a farmer’s prior information

set. We introduce a small element of error in each farmer’s observation of these other farms

characteristics. In particular, for each quality or cost value we add a random element drawn from a

normal distribution with a standard deviation of 5% of that value. Accordingly, approximately 96% of the

time a farmer’s estimate of the cost and quality of a farm in their prior information set is within 10% of

the true value.

Crucially, in our simulator we assume that each farmer takes their prior information set to provide a

representative sample of the full set of 1,000 farms that might participate in the auction. As a result,

1 For example, the analysis in Table 1 suggests that a farmer with risk aversion parameter 3, would see a certain

amount of £418 as being equivalent to the risky prospect of receiving either £700, £500 or £300 with equal probability. That feels plausible compared to the £367 certainty equivalent calculated for a risk aversion parameter of 6 which feels overly cautious.

© ADAS 2014 7

from a farmer’s point of view it is reasonable to conjecture how the 25 farms in their prior information

set might bid and then use that distribution to work out how likely they are to be successful when

entering a bid of a certain level in the auction.

In order to make that calculation, farmers need two other pieces of information. First they need to know

the size of the budget being allocated through the auction. We assume that this information is made

public by the government. Second, they need to know how many farmers in total are eligible to

participate in the auction. Here we assume that each farmer makes an unbiased guess as to what that

number is. In particular, since our simulations assume that there are 1,000 eligible farmers, each

farmer’s guess at that number is taken as a random draw from a uniform distribution on the range 750

to 1,250.

Bringing, those various elements together, we can now describe the logic that is used by the simulator

to represent the thought process that farmer’s go through in calculating the probability that they will be

successful in the auction when bidding at a certain level.

For each farmer (i = 1 to 1000):

o Estimate Bids of Prior Information Set: Estimate how each farmer in their prior information

set of 25 farmers might bid in the auction (we shall explain how this is done in the following

section on Bidding Logic) and from that calculate the VFM of the bids of each of those other

farmers

o Rescale Budget: Given a farmer’s estimate of the number of farmers eligible to bid in the

scheme, let us call that �̂�, rescale the full budget by a factor of 25 �̂�⁄ in order to arrive at

budget that provides the same competitive pressures on a population of 25 farmers as the

full budget does on a population of �̂� farmers.

o Bootstrap a Distribution for the Marginal VFM in the Auction: For each bootstrap sample (j =

1 to 100):

Draw a random sample of size 25 with replacement from the Prior information set

Order those 25 observations from best to worst VFM

Working down that list, deem bids to have been accepted up until the point at

which the rescaled budget is exhausted.

Record the VFM offered by the last bid to be accepted as one estimate of the

marginal VFM

o Loop back to repeat for next bootstrap sample (j = j + 1)

o Evaluate Possible Bids: For each unique VFM in the bootstrapped distribution of 100

marginal VFMs, calculate the payment the farmer would have to request for their bid to

offer that VFM. These constitute the set of possibly maximal bids from which the farmer

could choose. In particular, it would never be worth entering a bid that resulted in a VFM

between two points on the bootstrapped VFM distribution because the farmer could always

do better by increasing their payment request until their VFM was just (fractionally above)

that of the next lowest VFM in the distribution.

o Calculate the probability of winning at each possible bid: The probability of winning for each

possible bid level is evaluated empirically from the bootstrap distribution of marginal VFMs

as the proportion of times a VFM of that level or better appears in the distribution.

© ADAS 2014 8

o Calculate Optimal Bid: Using Equation A.1, use those probabilities to establish which of the

possible bids (or not bidding at all) maximises the farmer’s expected utility

Loop back and repeat next farmer (i = i + 1)

This bootstrap method for selecting bids for each farmer has the advantage of being intuitive, relatively

quick to compute and a plausible approximation to the sort of thought processes a farmer might pursue

in working out how to bid.

Clearly, using this logic farmers’ bidding behaviour is shaped by the prior information they hold on the

costs and qualities of other farms. If a farmer’s prior information is based on an exact representative

sample of all farmers then it will provide them with an unbiased assessment of the cost and quality

distribution. Accordingly, in working out how to bid farmers’ decisions are driven by information which

gives a fair representation of the level of competitiveness they will face in the actual auction.

Alternatively, and possibly more likely, it might be the case that farmers prior information sets are not

representative but instead are made up of other farms that have costs and qualities relatively more like

their own. Allowing for such bias reflects our expectation that farmers will know relatively more about

the costs and qualities of farms in their local region which are likely to be more like their own.

To create biased prior information sets we use the following procedure. First, for each farmer we draw a

random a sample of farms from the full population that is larger than the 25 required to make a prior

information set. Next, we compare the farmer’s cost and quality to those of each farm in the

oversample and select those 25 farms which have costs and qualities closest to the farmer (where

closest is measured using a standardised Euclidean measure). To create more bias we simply increase

the size of the oversample.

Figure 3 illustrates bidding behaviour under three different assumptions regarding the degree of bias in

prior information; no bias, moderate bias (oversample=100) and high bias (oversample= 200).

Participation details for the same three simulation set-ups are provided in Figure 4.

Figure 3: Bidding behaviour under different assumptions regarding prior bias

Notice that bids, especially in the earlier auctions, are generally lower at increasing levels of bias. To

explain that observation, consider a low cost farmer whose biased prior information leads them to

believe that other farms resemble their own more than is the case in reality. Clearly, under that

misapprehension, such a farmer would assume that bidding environment is highly competitive and

hence would tend to enter a relatively competitive bid themselves. Accordingly, the simulation suggests

that increasing prior bias will tend to lead to lower bids especially amongst relatively competitive

farmers. Of course, in subsequent years of the auction, feedback on the VFM of successful bidders alerts

farmers to the inaccuracies of their prior information which they subsequently update to reflect the

reality they observe in the feedback. Accordingly, when prior information is biased we observe bids

© ADAS 2014 9

progressively increasing over years of the auction. In contrast, when prior information is not biased,

there is relatively little change in bidding patterns over successive years of the auction.

The patterns of participation shown in Figure 4 reflect the same story. When priors are unbiased,

farmers’ participation decisions remain relatively unchanged through successive rounds of auctions. In

contrast, as prior bias increases we observe that in the first year’s auction high quality-for-cost farmers,

believing other farms to be more like their own, overestimate the level of competition. Participation

rates amongst this group are significantly lower than in the no bias case. Likewise, low quality-for-cost

farmers underestimate the level of competition and hence participation rates amongst that group are

somewhat higher than in the no bias case. Over successive years’ auctions farmers revise their prior

information set as a result of feedback such that participation rates in the biased cases tend towards

those in the no bias case.

Figure 4: Proportion of farmers participating in the auction under different assumptions regarding prior bias

Table 3 summarises the three simulations. Notice how average bids are lower the higher the level of

bias which results in significantly higher levels of cost efficiency in the simulations with high prior bias

compared to low prior bias.

Table 3: Auction outcomes under different assumptions regarding prior bias

Prior Bias

Year of

Scheme

None

Moderate

High

Part1 Avg Bid2 Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

One 43.8% 514.53 0.229

40.1% 457.92 0.235

39.7% 451.09 0.236

Two 41.1% 500.91 0.234

44.5% 476.14 0.243

46.9% 463.05 0.250

Three 40.4% 500.58 0.235

40.5% 478.66 0.242

40.9% 471.25 0.246

Four 40.7% 502.11 0.235

42.0% 480.97 0.240

41.8% 470.36 0.243

Five 40.4% 499.99 0.235

41.0% 482.89 0.240

41.9% 475.83 0.241

Notes: 1 Participation rate in percent;

2 Average of winning bids;

3 Cost efficiency calculated as quality points per £

of govt. expenditure on contracts

© ADAS 2014 10

Since, we suspect that farmers will know relatively more about the costs and qualities of farms in their

local region which are likely to be more like their own, in the simulations recorded in the main report we

make the assumption that farmers have the moderate level of bias in their prior information.

1.6 Strategic Bidding Logic

The process by which a farmer determines how to bid (described in the last section) is part of a

potentially more complex strategic logic that the simulator assumes a farmer follows in deciding upon

their final bid. Indeed, the simulator assumes that there are progressively more sophisticated levels of

strategic logic that farmers can use in deciding upon how much to ask for as a payment in the auction.

The first level of logic is extremely basic and assumes that farmers only consider their own

circumstances. They decide upon a percentage mark-up that they feel represents a suitable recompense

for joining the agri-environment scheme and enter a bid amounting to their costs plus this mark-up. To

arrive at that amount in our simulator we randomly allocate each farmer a mark-up on the range 5% to

25%. Indeed, using this logic farmers ignore their prior information set altogether.

Farmers employing the second level of sophistication go one step further and consider how their

chances of winning might be affected by other farmers’ bids. In particular, using their assumed level of

suitable mark-up, a farmer calculates the bids that they anticipate the other farmers in their prior

information set might enter in the auction if those other farmers only used the first level of bidding

sophistication. Looking at the distribution of those bids and considering the amount of money in the

scheme budget, a farmer using the second level of sophistication chooses a bid that maximises their

expected earnings according to the logic described in the last section.

Of course, a farmer might go one step further and employ a third level of bidding sophistication. In this

case, they would assume that each farmer in their prior information set decided on a bid using the

second level of bidding sophistication. Reasoning out those other bids, a farmer would again arrive at a

bid distribution and formulate their own bid as a best response.

A farmer using a fourth level of bidding sophistication would repeat that same cycle of logic, formulating

their bid as a best response to the bids of other farmers assumed to be using three levels of bidding

sophistication. And so on through increasing rounds of sophistication.

The full strategic logic used in the simulator is as follows:

For each farmer (i = 1 to 1000):

o Calculate naive bid: Calculate own bid as percentage mark-up above costs

o Calculate naïve bids for Prior Information Set: Use same percentage mark-up to estimate

bids for 25 farms in prior information set

o For each round of strategic logic (j = 1 to Num logic rounds)

Bootstrap a Distribution for the Marginal VFM in the Auction: Using the bootstrap

logic described in previous section generate a distribution for the marginal VFM

based on the current supposed bids of farmers in the prior information set

Calculate strategic bid: Using the bidding logic described in the previous section

calculate an optimal bid given the assumed distribution of marginal VFM.

If j < Num logic rounds

Calculate strategic bids for Prior Information Set: For each farm in the prior

information set calculate an optimal bid given the assumed distribution of

© ADAS 2014 11

marginal VFM. These become the current supposed bids of the prior

information set

o Loop back for next round of strategic logic (j = j + 1)

o Final bid: Farmers final bid is determined after all rounds of strategic reasoning have

finished

Loop back to repeat for next farmer (i = i+ 1)

Figure 5, plots bidding behaviour for three different simulations making different assumptions regarding

the number of rounds of strategic logic farmers pursue in deciding upon a bid. Notice that when farmers

do not employ strategic logic, their bids are just a straight percentage above costs. In bidding farmers

pay no attention to the possible bidding behaviour of other farmers nor do they consider whether some

other level of bidding might result in higher returns. Notice from Figure 5, that when farmers bid naively

the information on VFM provided between rounds is irrelevant to their decision-making. Accordingly,

the bids in each successive year of the auction are identical.

Figure 5: Bidding behaviour under different assumptions regarding the sophistication of bidding

Of course, assuming that farmers bid with such naivety is unrealistic. The second and third panels of

Figure 5 show how bidding behaviour and auction outcomes change as farmers adopt ever more

sophisticated strategic logic in formulating their bids. The key outcome is a significant rotation in the

bidding curve; more sophisticated reasoning leads high quality-for-cost farmers to the conclusion that

they can shade up their bids and low quality-for-cost farmers to the realisation that their only chance of

success is to bid more competitively. Indeed, the more rounds of strategic logic that farmers employ,

the more competitive they assume they will have to be in their bidding.

Figure 6 presents participation data for the same three simulations. Of course when farmers employ no

strategic logic, they are assumed to simply enter a bid in the auction at a percentage mark-up above

costs without considering whether they have any chance of being successful in the auction. As a result,

under this assumption all farmers participate in the auction, and this is true for each year of the scheme.

The more rounds of strategic logic that farmers employ, however, the more sophisticated they assume

others will be in their bidding and so the more competitive they assume the auction will be. As a result

we observe generally increasing numbers of farmers at increasing high levels of quality-for-cost

choosing not to bid in the auction.

Those bidding and participation patterns are reflected in the summary statistics reported in Table 4. The

outcome of the auction under the assumption of naïve bidding is summarised in the first column of the

table. Since bidding a percentage over costs results in low bids from low-cost farmers the auction

delivers relativley high cost efficiency, and that efficiency does not decline over successive years.

© ADAS 2014 12

Figure 6: Proportion of farmers participating in the auction under different assumptions regarding the sophistication of bidding

When farmers employ one round of strategic logic, the strategic shading-up of bids by those with high

quality-for-cost results in an increase in the average level of bids in the auction. Those increases in bids

precipitates a reduction in the cost-efficiency of the auction. When farmers employ more rounds of

strategic logic the more competitive they assume they will have to be in their bidding. Accordingly, as

shown in Table 4 the cost efficiency of the auction increases the greater the level of sophistication we

assume farmers employ in formulating their bids.

Table 4: Auction outcomes under different assumptions regarding bidding sophistication

Rounds of Strategic Logic

Year of

Scheme

None

One

Four

Part1 Avg Bid2 Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

Part1 Avg

Bid2

Cost

Eff.3

One 100% £446.79 0.262

33.6% £454.30 0.231

27.7% £427.18 0.246

Two 100% £446.79 0.262

44.1% £517.68 0.234

26.1% £450.15 0.256

Three 100% £446.79 0.262

32.3% £502.57 0.234

23.5% £431.80 0.263

Four 100% £446.79 0.262

29.8% £504.36 0.234

20.5% £427.07 0.271

Five 100% £446.79 0.262

29.2% £502.80 0.235

17.4% £412.18 0.282

Notes: 1 Participation rate in percent;

2 Average of winning bids;

3 Cost efficiency calculated as quality points per £

of govt. expenditure on contracts

In reality we suspect that some farmers bid naively while others employ sophisticated bidding

strategies. To reflect that heterogeneity, we assign a level of bidding sophistication to each farmer

where the number of rounds of Nash logic they employ is a random draw from a Poisson distribution.

That procedure results in 14% of farmers bidding naïvely, 53% using one or two rounds of Nash logic and

33% three rounds or more. For the purposes of comparison, the auction outcome from that mixed

population is shown in the final column of Table 4.

© ADAS 2014 13

1.7 Information Feedback: Update Logic

In a scheme where funding is allocated across multiple years, there is substantial scope for learning via

informational updating over time. In reality there are many ways in which an agent might choose to use

the information revealed to them regarding bidding in previous year’s auctions. For example, some

individuals may simply choose to ignore such information. Another individual might compare their own

priors and predictions about expected auction outcomes with the actual outcomes observed from a

previous year. Observing differences between the two would alert them to the fact that their prior

information set is somehow biased and might precipitate them updating that prior information set to

tally with their observations of real behaviour.

In our main report we describe the outcome of a series of simulations in which the government provides

feedback on the bids that were successful in previous years’ auctions. That feedback comes in one of

three forms; information on the VFM of winning bids, information on the environmental quality scores

of winning bids or information on the payment requests of winning bids. Here we discuss the technical

details of the logic employed by the simulator to describe the way in which farmers process that

information.

Consider first a scheme in which the government releases details of the payments asked for in the bids

made by successful farmers in an auction. That information is useful to future auction participants since

it allows them to identify the levels of bid that have previously proved successful. For example, if those

payments are markedly lower than a farmer had expected, then that will alert the farmer to the fact

that there are likely to be more farms capable of offering low bids than suggested by their prior

information set. As a result, we assume that farmers update their prior information sets to reflect the

new information. In doing so we assume that farmers follow the following logic

Prior expectations of winning bids: First, farmers process their prior information set in the

manner described above so as to derive their expectations on the payment requests that will be

made in the auction. In processing the feedback on actual payment request from the previous

auction, farmers first look to see if the observed payment requests look anything like their

expected payment requests.

Equivalent winning payment requests from feedback: By observing the actual number of

winners in the previous auction, farmers can work out the number representing the equivalent

proportion from a population the size of their prior information set (i.e. 25). The farmer wishes

to make that number of new observations for their prior information set based on the feedback

data. To do that they bin the observed feedback data on payments into that number of

categories and create observations with payment requests calculated as the average payment

level in each bin. Those new observations of payment requests have approximately the same

distribution as the original feedback on payments from the previous auction.

Create new observations: That set of payments is approximately the set of winning payment

requests the farmer would have hoped to have derived from their prior information.

Accordingly, the update progresses by replacing the set of winning observations from the prior

information set with a new set of observations based on the observations of winning payment

requests derived from the auction feedback. To create those observations the farmer needs to

move from requested payments to underlying farm costs. In the simulator that is achieved by

rescaling payment requests by the average level of mark-up that a farmer estimated was

claimed by winning bidders in their prior information set. Finally, qualities are assigned to these

new observations as a random selection of qualities from the prior information set.

Update Prior Information Set: Replace old ‘winning’ observations from the prior information set

with the new ones based on the feedback data.

© ADAS 2014 14

The logic describing responses to feedback on the environmental quality scores of winning bids is very

similar to that just described for feedback on payment requests. Clearly the qualities of the new

observations are derived straight from the feedback information. This time, however, it is costs and not

the qualities of the new observations that are generated as a random selection of costs from the prior

information set. In fact, farmers have a little more information to go on in estimating these costs. Given

their estimate of the mark-ups used by winning bidders, they can move from these costs to predicted

payment requests and check whether the sum of those payment requests is compatible with the

budget. If not then the costs can be rescaled to ensure that they are.

Finally, the logic describing responses to feedback on the VFM of winning bids in the previous auction

follows a similar pattern. First new winning observations for the prior information set are constructed

that have VFMs with a distribution equivalent to that in the feedback. A selection of qualities are chosen

at random from the prior information set and assigned to these observations. An estimate of the

payment request for each new observation can then be calculated by dividing each observation’s VFM

by its quality score. Those payments are rescaled to ensure they are compatible with the budget

constraint and finally converted to underlying costs by removing the mark-up a farmer estimated would

be claimed by winning bidders in their prior information set.

1.8 Guide Prices: Update Logic

Another form of information that can be provided to farmers in the auction is that of a guide price. We

provide a thorough analysis of the impact of guide prices on bidding behaviour in the main report, and

here focus our discussion on the technical details of the logic used in the simulator to model farmers’

response to that information.

As with feedback on previous winning bids, we assume that farmers respond to decision-relevant

information by updating their prior information sets. In particular, we assume that farmers interpret this

guide price as the government’s estimate as to the level of the marginal bid; that is to say, the sort of

bid that will just achieve funding. In the simulator, farmers first calculate the bids they expect to see in

the auction based on their prior information set. Taking the mean quality of the bids they expect to win

from their prior information set, they convert the guide price to a VFM and take this to be the

government’s expectation of what the marginal VFM in the auction will look like.

Next, farmers compare that government signal as to the likely marginal VFM to the one they themselves

estimate from their prior information set. The ratio of those two quantities indicates to the farmer

whether the government believes the auction will be more or less competitive than the farmer had

assumed based on their prior information set. Putting equal weight on the government signal and their

own prior information, farmers adjust the costs and qualities of all their prior observations in such a way

as to halve the difference between the government signal of the marginal VFM and the one they

calculate from their prior information set.

When quality and cost are uncorrelated we observe generally higher bidding than is observed in either

of the correlated cases. The lack of correlation leads to farmers’ prior information sets showing greater

dispersion of cost and quality scores. In those circumstances, low cost providers have a greater insight

into their comparative advantage and thus tend to bid relatively higher even in the initial years of the

scheme. We continue our analysis assuming no cost-quality correlation.

© ADAS 2014 15

1.9 Multiple activities in a budget-constrained auction

Yet more complex logic is required to simulate a scheme where the government is looking to contract

farmers to undertake two different agri-environment activities. Our particular interest is in situations

where the government would like to encourage farmers to undertake both activities on their farm since

the combination of activities delivers additional environmental benefits. In order to encourage the

uptake of both activities we consider a scheme in which the government confers a bonus of 25pts on to

the quality score of a farmer bidding for both activities. Clearly the bonus increases the VFM of such bids

and hence increases a farmer’s chance of being successful in the auction.

To fix ideas, consider the case where each farmer has two 1 ha fields. Under agricultural production, the

fields generate different incomes; incomes that are foregone if the field is entered into an agri-

environment scheme. Under the simulated agri-environment scheme, each farmers’ first field is eligible

for activity A, while their second field is eligible for activity B. For simplicity, we assume that the

environmental benefit conveyed by activities A and B are the same but differs across farmers. This

provides differences in quality-for-cost across the two fields but allows us to focus on the role of

differences in foregone income; as such, the extension to heterogeneous qualities is trivial.

There are a number of ways in which an auction for the provision of two or more activities can be set

up, we examine two potential options which we refer to as i) an All or Nothing format and ii) a Pick and

Mix format. In an All-or-Nothing format, farmers enter a single bid, which describes whether they are

offering to do one, the other, or both activities. The All or Nothing format forces bidders to choose

between one of three potential bid options: bid for activity A; bid for activity B; or bid for activities A and

B together.

The decision making process of farmers is slightly different when there are multiple activities to choose

from and adjusts in response to the format of the auction. When the auction has an All or Nothing

format, each farmer must decide whether to bid for any activities, and if so, whether to bid for one

activity or several (activity A only, activity B only or A and B in our simulations).

As in a single activity auction, farmers observe their own foregone incomes and quality scores, however

now they observe these for two eligible fields. With an All or Nothing auction format, the farmer faces a

two-part decision: i) decide which activities to bid for and ii) decide the value of their bid.

In our simulation code, each farmer chooses the activities and bid to maximise their expected utility,

based on their probability of success in the auction and the quality score that they receive if they are

successful. The probability of success in the auction is determined by the quality score – which itself

varies by activity choice - and the bids of other farmers.

The bidding logic in the All or Nothing format is as follows, the farmer:

i) Calculates his potential value for money for each activity and chooses the activity with the

highest – this provides the greatest probability of success for a given level of shading

ii) Estimates, from their prior information set, the bids made by other farmers and therefore the

distribution of VFM

iii) Evaluates his best response bid for the activity chosen in i), where the best response is defined

as the bid value that maximises expected utility conditional on the anticipated distribution of

VFM.

Finally, the farmer repeats steps ii) and iii) through successive rounds of strategic logic, taking

into account the best responses of competing farmers.

© ADAS 2014 16

With a Pick and Mix auction format, each farmer instead chooses their bid value for each activity. The

bidding logic in the code is similar to the All or Nothing format:

i) The farmer calculates his potential value for money for each activity (A only, B only, and A and

B). This replicates the procedure that will be undertaken with the bids received in the auction.

ii) If activities A and B together have the highest potential VFM,

o The farmer submits bids that even out the VFM for activity A and activity B such that the

individual bids are ranked equally but offer a lower VFM than their combined option of

A and B.

Otherwise,

o The farmer differentiates his bids.

iii) Using a sample of other farmers, each farmer estimates the competing bids and therefore the

distribution of VFM.

iv) The farmer evaluates his best response bids given the anticipated distribution of VFM. Where

the best response is defined as the bid value that maximises expected utility conditional on the

anticipated distribution of VFM.

Finally, the farmer repeats steps iii) and iv) through successive rounds of Nash logic, taking into

account the best responses of competing farmers.

1.10 References

Abdulkadri, A. O., and Langemeier, M. R., (2000). “Using farm consumption data to estimate the

intertemporal elasticity of substitution and relative risk aversion coefficients”, Agricultural Finance

Review, 60(1), pp. 61 – 70.

Azar, S. A., (2006). “Measuring relative risk aversion”, Applied Financial Economics Letters, 2, pp. 341–

345

Bar-Shira, Z., Just, R. E., and Zilberman, D. (1997). “Estimation of farmers' risk attitude: an econometric

approach”, Agricultural Economics, 17(2), 211-222.

Binswanger, H. (1981). “Attitudes toward risk: Theoretical implications of an experiment in rural India”,

The Economic Journal, 9, pp. 867-890.

Chavas, J. P., and Holt, M. T. (1996). “Economic behavior under uncertainty: A joint analysis of risk

preferences and technology”, The Review of Economics and Statistics, 78(2), pp. 329-335.

Chiappori, P. A., and Paiella, M. (2011). “Relative risk aversion is constant: Evidence from panel

data”. Journal of the European Economic Association, 9(6), pp.1021-1052.

Defra (2014). Farm Business Income by type of farm in England, 2013/14, FBS Statistics Notice.

Janecek, K., (2002). “What is a realistic aversion to risk for real-world individual investors?” Working

Paper.

Mettepenningen, E., A. Verspecht and G. Van Huylenbroeck (2009). “Measuring private transaction costs

of European agri-environmental schemes,” Journal of Environmental Planning and Management,

52(5), pp. 649-667.

© ADAS 2014 17

Meyer, D. J., and Meyer, J. (2005). “Relative risk aversion: What do we know?”, Journal of Risk and

Uncertainty, 31(3), pp. 243-262.