A Spatial Model of Competitive Bidding for Govt Grant

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Journal of Theoretical Politics

DOI: 10.1177/09516298070840392008; 20; 47Journal of Theoretical Politics

Hugh Ward and Peter JohnEfficiency Gains Are Limited

A Spatial Model of Competitive Bidding for Government Grants: Why

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A SPATIAL MODEL OF COMPETITIVE BIDDING

FOR GOVERNMENT GRANTS: WHY EFFICIENCYGAINS ARE LIMITED

Hugh Ward and Peter John

ABSTRACT

With a view to improving public sector efficiency many governments now

make public sector bodies competitively bid for funding. We model the bid-

ding process as a game of spatial competition. Using Monte Carlo simula-

tions we show that in efficient equilibria many bidding groups will not be

under competitive pressure. The model suggests that this is because their ideal

projects are inherently valuable for the funding agency and other groups can-

not match this without departing so far from their ideal that they would

rather not be successful. We test the hypothesis that competition will be lim-

ited largely to groups whose preferred projects are of medium quality on data

from the UK Single Regeneration Budget. Using resubmitted bids to track

the impact of competition, we find evidence consistent with this hypothesis.

KEY WORDS . competition . bidding . public sector . grants

In this article we develop a model of competitive bidding for public funding of

projects. In such bidding tournaments (Lane, 2001) or challenge programmes

(Foley, 1999) an agency funds the best set of projects from a larger pool of bids,

constrained by a fixed budget. Our aim is to understand the degree to which

competition pushes bidders to improve the quality of their projects. This ques-

tion is significant because bidding tournaments have become commonplace at

national and at international levels. For instance, the Single Regeneration Bud-

get (SRB) was introduced by the Conservative government in the UK in 1994 to

generate competition between projects for urban regeneration, partly following

precedents set by competitive US Federal Department of Housing and Urban

Development programmes. Another example is bidding under the Montreal Pro-

tocol to the Multilateral Fund through the World Bank to fund projects to reduce

the production and use of ozone-depleting substances in developing and transi-

tion economies.

Bidding tournaments are an important component of the New Public Manage-

ment reforms, introduced in many countries in an attempt to increase efficiency

by introducing market or market-like forces into the public sector (Walsh, 1994;

Journal of Theoretical Politics 20(1): 4766 Copyright 2008 Sage PublicationsDOI: 10.1177/0951629807084039 Los Angeles, London, New Delhi and Singaporehttp://jtp.sagepub.com

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Boyne, 1998; Robinson, 2000; Lane, 2001). For instance, since the early 1980s,

central government in the UK has compelled ministries and local authorities to

put out to competitive tender more and more functions that were once carried

out in-house (Boyne, 1998: 10310). In the 1990s, central government devel-oped the Private Finance Initiative (PFI) and the similar Public Private Partner-

ships (PPP), whereby private consortia provide capital funding for public

projects. In New Zealand in the 1990s such measures were worked up into the

method of contract budgeting, where the executive decides on what outputs

it requires and quasi-independent government agencies, the private sector, or

publicprivate partnerships competitively tender for contracts (Robinson, 2000).

There have been extensive measures to encourage competition for school

places, such as the Danish voucher system (Anderson and Serritzlew, 2007).

On common sense grounds there may be significant efficiency gains whencompetitive bidding is introduced, because the monopoly of bureaucracies is

broken (cf. Boyne, 1998: 1067) and because the executive as the sole or main

purchaser can drive hard bargains (cf. Robinson, 2000: 86), although the transac-

tion costs of forging contractual relationships may be high (Foley, 1999; Lane,

2001: 35). But there is an absence of theoretical models from which hypotheses

can be derived about bidding tournaments. First thoughts might be to treat them

as some sort of auction and to try to borrow or adapt a model from the vast

literature (Milgrom, 2004; Menezes and Monteiro, 2005). An instance of such

adaptation is to model lobbying as an all-pay auction in which the prize goes tothe lobby making the biggest bid, with all other bids forfeit (e.g. Baye, Kovenock

and De Vries, 1993; cf. Grossman and Helpman, 2002). Also standard auction

theory can be applied to bidding to carry out a public sector contract (Rothkopf

and Harstad, 1994). However, bidding tournaments are not like the auctions dealt

with by economic theory. While groups that fail to get funding do waste their

efforts in bid preparation and lobbying, the analogy with the all-pay auction

breaks down because competition may force those who do get funding to pay an

additional price: carrying through a project which is not what they exactly wanted

to do. As far as we know auction theory does not deal with situations where the

number of prizes depends on a budget constraint and the bids made, and is endo-

genous to the game-equilibrium, not determined in advance.

Our contribution is to develop the first model specifically tailored to under-

standing bidding tournaments. Our claim is that gains from competition are lim-

ited, though not necessarily negligible, because competition does not impinge

much on bidders whose preferences happen to be similar to those of the funding

agency. We use Monte Carlo simulations to show that in equilibria derived from

formal results many bidders are not under competitive pressure.

We know little empirically about efficiency gains from bidding tournaments.

While some find significant gains from competitive tendering for public con-tracts, there are questions about the degree to which results generalize, whether

quality of output has been maintained when production costs are cut, and

48 JOURNAL OF THEORETICAL POLITICS 20(1)



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whether cost-cuts are maintained once contractors have established a quasi-

monopoly position (Walsh, 1994: 22249; Boyne, 1998: 14752 and 16267;

Lane, 2001: 423). Apparently large reductions in unit costs of the order of

20% (Boyne, 1998: 13546, Domberger and Jensen, 1997) divert attention fromtransaction and other social costs, such as unemployment, and poverty among

low-paid workers (Lane, 2001: 423). PFIs and PPPs have generated claims for

efficiency savings, but evidence for actual gains is limited (IPPR, 2001). Com-

petitive bidding to carry out particular functions within the public sector may be

seen as part of the phenomenon of quasi-markets (Bartlett and Le Grand, 1993),

where similar mixed results have been found (Anderson and Serritzlew, 2007).

The evidence discussed so far may tell us little about bidding tournaments,

because they do not involve tendering for a single well-defined contract. Doubts

have been expressed about efficiency gains from bidding tournaments (Foley,1999; Taylor, Turok, and Hastings, 2001; John, Ward and Dowding, 2004; John

and Ward, 2005), but the number of studies is small. We provide additional evi-

dence by re-analysing data from the SRB programme.

1. The Model

In this section we state the assumptions of our model.

1) If a group makes a bid, it must propose a single project to the funding agency at

the same time as all other bids are placed.

2) Bid characteristics are reduced to a single bid score, s0, that reflects how far the

bid is away from the funding agencys ideal point, and the level of funding

demanded, y.

3) For any bid (s, y) is known with certainty by the funding agency.

A portfolio of bids is some subset of a set of bids. A portfolio is feasible if

the total funding demanded is less than or equal to the agency budget, G > 0. No

bid can be added to a maximal portfolio without breaking the funding limit. Theagency tries to find the maximal portfolio of bids which has the minimum

summed score, a version of the 01 knapsack problems (Martello and Toth,

1990).1 Algorithms giving exact solutions are computationally complex (Mar-

tello and Toth, 1990: 1377) and analytically difficult in our context. Approxi-

mate solutions for n> 2, where n is the number of bids, generally rely on

calculating the value ratio x0= s0=y0. (Here low values of x0 mean the bid is ofbetter value.) One method is called the greedy algorithm. Suppose for the

moment that no two bids have the same x-score. Order the bids according to

1. So called because they are analogous to filling a knapsack up to a weight limit so that the subset

of all the items taken has the highest possible usefulness on the hike (Dantzig, 1957).

WARD & JOHN: BIDDING FOR GRANTS 49



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their value x0a, x0b; . . . ; x

0n1, x

0n, where x

0a is the bid with the best value. Then for

the agency:

4) i) Start with x0a; if y0a G, funds this bid; else do not fund it; ii) if the amount of

funding left after stage 1 is G1; repeat stage 1 with x0b replacing x0a and G1 repla-

cing G; iii) repeat the second stage until there are no more bids to consider or

there is no more remaining funding.

Taking bids in value-for-money order until no more can be taken without

breaching the budget constraint is relatively efficient if the value of the marginal

item considered but left out is small compared to the total value of those chosen

(Martello and Toth, 1990: 28), which ought to apply where funds demanded are

small compared to the overall budget.Set the baseline payoff for a group that makes no bid at 0. There are bid pay-

offs enjoyed if a bid is made and succeeds in getting funding and process pay-

offs from the act of making a bid, whether or not the bid succeeds. Process

payoffs derive from: satisfying local electors that you are trying to bring home

the pork; the process of bid construction, itself, helping build community rela-

tions; useful experience for future bidding (Ward, 1997). Specifically:

5) For group i the process payoff associated with any bid made is the same, and

greater than 0.

Groups know what project they ideally want to carry out, its characteristics

and the funding they would like to deliver those characteristics. Not only can

they get too little funding for their ideal project, but they can also get too much.

For one thing they ultimately have to account for the efficient use of public

money. Their ideal bid has a certain score on the agencys metric. Our assump-

tion about group preferences is that moving in any given direction away from

their ideal point is s/y space they get a lower payoff, if their bid is funded. It is

easier to work with groups induced preferences in x/y space. Points in s/y spacecorrespond one-to-one with points in x/y space, so a group has a derived ideal

point in x/y space, say (xi, yi for the i-th group.2 Group is bid payoff if their

bid succeeds and obtains funding is given by an ordinal utility function ui(x, y).

Set the notional bid payoff from a bid that receives no funding at zero. Then:

6) ui(x, y) is a continuous, strictly decreasing, function of the distance between (x,

y) and (xi; yi) on the Euclidean metric, such that uixi, yi) > 0.

2. Consider (s0, y0) and (s00, y00) in s/y space mapping into (s0=y0, y0) and (s00=y00, y00) in x/y space.They are only the same point ify0= y00, hence ifs0= s00.




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Thus groups indifference curves are circles centred on their ideal points in

x/y space.3 For bids close enough to their ideal point groups would prefer to get

funded rather than not; but for bids too far away from what they ideally want to

do they would rather not succeed as the price of delivering on the bid is toohigh. The set of bids such that group i is just indifferent between the bid suc-

ceeding and failing have a bid payoff of zero and are all located on some circle

of radius ri centred on (xi, yi. Assume:

7) For any two groups i and j, ri= rj= r; r> 0;8) ui(x, 0) < 0 for any x

0;.

9) For any group i, ui(0, y0) < 0

which implies r < xi for all i.

10) The utility functions for bid payoffs and process payoffs of each bidding group

are common knowledge among the groups and the funding agency.

First to prevent ambiguity assume that:

11) For any two groups i, j xi 6 xj so that groups can be completely ranked inascending order of the ideal points on the x-dimension and correspondingly num-

bered 1, 2, . . . n.

Then:

12) If two or more groups put in bids with the same x-value, the funding agency

considers them in the same order that the groups are numbered, starting with the

bid from the group with the lowest number.

Thus faced with bids of equal value, the agency first funds the bid coming

from the group that ideally in the absence of competition would like to put

in the better value bid. Given competition improves bids, if anything, an agency

breaking ties this way allocates funds first to the group that stands to gain most,as the funds go to the group whose ideal bid is closer to its actual bid.

We refer to assumption 12 together with assumption 4 as sequential funding.

Then:

13) Sequential funding is common knowledge among bidding groups.

We consider pure-strategy equilibria in a one-round bidding game.4

3. This can be shown to be consistent with the fundamental assumption about group preferences ins/y space that things must get worse for the group along any line starting at its ideal point.

4. Repeat bidding could lead to collusion to get higher payoffs, for instance by taking it in turns to

put in serious bids (cf. Pesendorfer, 2000).




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2. Limits on Competition

Say a group g is under competitive pressure in equilibrium @ if its bid obtains

funding and x0g G y1 y2. So the critical group identified in Theorem 1 is group 3; andTheorem 1 asserts that group 1 cannot be under pressure in an efficient equili-

brium, because x1 < x, where x= x3 r. One equilibrium is {(x1, y1, (x*,

y2, (x*, G y1}. Here group 1 proposes its ideal project but group 2 comesunder competitive pressure from group 3. In this equilibrium group 3 does not

get funded: although it proposes a bid with the same value to the agency as

group 2, under sequential funding group 2s bid is funded first, because it is

nearer what it ideally wants to do than 3s bid. It is still worth group 3 making a

token bid that is not funded because it gets a positive process payoff from

doing so. (Because of the positive process payoff, it is a general feature of equi-

libria that all groups bid.) Group 2 cannot successfully make a bid to the right of

x* and nearer its ideal bid: if it does so, 3s bid will be funded and there will be

no funds left over. In equilibrium if a group comes under competitive pressure itis because some other group locates at the same value of x and bids for an

amount that would leave the group under pressure worse off if it moved to the




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right. What if group 3 located somewhere to the right in the interval (x*, x2],say at x#, and still asked for G y1? Holding group 1s position fixed, group 2

would now also locate at x and ask for y2. But this cannot be an equilibrium: if

3 relocated to the left of x, it would get the funding. One can imagine a bid-

ding war between groups 2 and 3: starting from any position to the right of x*,

group 3 would sequentially relocate putting 2 under more and more competitive

pressure until the configuration under discussion is reached. Notice that if group

1s bid is (x1, y1 and group 2s bid is (x*, y2, it would not pay group 3 tolocate even further to the left than x*: although its bid might succeed in getting

funding if it did not ask for too much, the bid payoff would be negative, because

the project is outside the circle of radius r around 3s ideal point that contains

projects with bid payoffs greater than or equal to zero.

Relative to bidding groups preferences, the equilibrium under discussion so

far is an efficient one: in no alternative equilibrium does 3 get funded; and in no

alternative equilibrium can 1 and 2 more closely approach their ideal bids. How-

ever, there are infinitely many other equilibria; and in some of them even group

1 comes under competitive pressure. For instance it is not difficult to see that

there is an equilibrium at {(x1 , y1, (x1 , y2, (x1 , G)}, as long as x1 > x2 r. In this equilibrium group 3 also makes a token bid and gets no fund-

ing. While groups 1 and 2 both get funded under sequential funding, 3s bidlocks them into carrying out projects that are less than ideal. However, this equi-

librium is no better for group 3 and is worse for groups 1 and 2 than the efficient

y

G

i

y3

y2y1

G y1 y2

xx1 x2 x3

x

Figure 1. An Illustration of Theorem 1




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equilibrium discussed in the last paragraph; so the fact that group 1 is under

competitive pressure does not contradict Theorem 1. The proof of Theorem 1

essentially just comprises two steps: i) if a group like 1 to which the theorem

applies is under competitive pressure, it is because a group like 3 is pushing it tothe left of its ideal bid; ii) but if a group like 1 is locked into a position to the

left of its ideal in some equilibrium, continuity guarantees we can always con-

struct another equilibrium where it is only 0 to the left of its ideal point, 0 < ,

and this equilibrium leaves no group worse off and group 1, at least, better off.

A group enjoys a rent from putting in a successful bid if its bid payoff is

strictly positive. Theorem 1 identifies a set of bidders that not only continue to

enjoy rents but whose rents are maximal. Thus bidding tournaments do not

achieve a socially efficient allocation of resources even if the funding agency

represents the social interest and bidders do not engage in competition that isinefficient relative to their own interests.

Although efficient equilibria relative to group interests might plausibly be

a focal point for a common conjecture, it is relevant to ask whether something

approaching efficiency could be attained in actual bidding. As suggested

when discussing the efficient equilibrium above, such equilibria can be reached

through a tattonement process whereby groups sequentially adjust their bids

under incomplete information about others preferences. If groups initially bid

at-or-near their ideal point, adjustment would gradually drive groups towards

the funding agencys conceptions of higher value, because it would not pay tooffer more than a minute increment in value above the other group. In this

circumstance it is plausible that the equilibrium arrived at would be efficient.

Another motivation for only considering efficient equibria is that the funding

agencys hand might tremble in an inefficient equilibrium, funding token bid-

der 2 instead of 1 by mistake. For instance it might mistake what the two bid-

ders ideal projects were. In this case 2 would be cursed with the need to carry

out a project that was worse than not getting funding at all. To avoid even a tiny

possibility of such an outcome 2 would not make bids having a strictly negative

bid payoff; so equilibria could be expected to be efficient.

Let C be the set of groups such that for each member g xg < xc r. The fol-lowing corollary is an immediate consequence of Theorem 1:

COROLLARY 1: No group j such that j C and yj r> Gygg C isunder competitive pressure in Pareto efficient equilibrium @, so long as G > y1.

This is because there is not enough left over from G to give them a positive bid

payoff once groups in C have had their bids funded, so they make token bids.Consider a group, j, to which Corollary 1 applies. It is possible that js token

bid would not correspond to its ideal point. However we can invoke twoarguments that suggest that it will do so. First, as already suggested, the possibi-

lity of trembles by the funding agency make it implausible that token bids




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associated with strictly negative bid payoffs would be observed, which means

the token bids of such groups would not be too far from their ideal points. Sec-

ond, as already discussed, equilibrium may be reached through sequential

adjustment starting with groups ideal bids. If so j will be clear to such a groupfrom early on that it cannot complete, for initial bids disclose which groups are

potentially competitive. Why would j bother to change the value of its bid when

it knows that this is futile?

Given the above, we would not expect competition to impact either on the set

of groups with ideal projects that the funding agency likes to which Theorem 1

applies or to groups to which corollary 1 applies: both can be expected to tender

their ideal projects. It is among middle-range groups to which neither Theorem 1

nor corollary 1 applies that competition will push them to put in high value bids.

Hence:

Hypothesis: Competition will do most to improve the value of bids among

groups with ideal projects that are mid-range in value.

3. Monte Carlo Simulations

If Theorem 1 applied only to a small subset of groups given empirically reason-

able values of parameters, it would do little to alter our thinking about whetherbidding tournaments are a significant innovation.

However, we show that if the ideal funding of each group is of a lower order

of magnitude than the total budget and groups cannot be pushed too far away

from their ideal project without leaving them with something they would rather

not carry out, Theorem 1 will apply to a considerable proportion of groups.

The basis of our simulations are x-values of ideal points, assumed to be inde-

pendent draws from a distribution. The y-values of each ideal point are assumed

to not be conditional on x-values and are also independent of each other. Bids

are drawn from probability distribution over the x/y space. We repeat the experi-

ment 1000 times with 30 groups. For each draw the proportion of groups to

which Theorem 1 applies can be calculated. The experiment is repeated 1000

times and, according to the basic Monte Carlo principle, the average proportions

are estimates of the expected proportions given the assumed distributions.

We report an estimate of average proportion of groups to which Theorem 1

applies and an estimate of average proportion of groups to which Corollary 1

applies. What matters in Theorem 1 is not G per se but the proportion of ideal

bids that can be funded, and this parameter must be varied. A distribution of

ideal points over the bidding space must be assumed. The one chosen approxi-

mates a uniform distribution over the unit square. (The exact assumption will bestated shortly.) Thus on average groups ideal funding level would be around

0.5 units. What proportion of its ideal budget for its ideal project could a group




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lose without being forced into something it would rather not carry out? Twenty

percent seems a liberal estimate. Given an average ideal budget of around 0.5,

this implies a value of r of around 0.1. The conclusions are somewhat sensitive

to the assumption about r, with lower values of r corresponding to less competi-tion. Results are also presented for r= 0.05. The assumption in the model is that

ideal points cannot be within distance r from the axes, measured both vertically

and horizontally. Thus the actual distribution used in the simulation was uni-

form over the square such that the x and y values of ideal points both belonged

to the interval (r, 1]. For r= 0.1 the average ideal funding level would then be

0.55 and the average proportion of ideal bids that could be fully funded is G/

(0.55 * n), or around 1.8 * G/n. Results are shown in Table 1.

As a larger proportion of ideal bids can be funded, the simulations suggest

that Theorem 1 applies to a greater and greater proportion of groups. The sumof columns 2 and 3 can be thought of as a lower limit on the proportion of

groups not subject to competition or making token bids. (Theorem 1 does not

say other groups will be under competitive pressure.) This total is soon domi-

nated by groups covered by Theorem 1. When r= 0.05, the lower limit is close

to the proportion whose ideal bids could be expected to be funded. When

r= 0.1, it is about 0.15 less.

In summary the simulations suggest that if about a half of ideal bids could be

funded, the proportion of groups not subject to competition is surprisingly high:

at least a third of groups will get funding for their ideal project and at most asixth will have to improve the quality of their proposals to get funding.

4. Empirical Evidence on competition in the SRB Process

In this section we attempt to test our hypothesis about the limited impact

of competition using a unique dataset on a series of bidding rounds for English

urban policy funds, the Single Regeneration Budget (SRB), which contains

detailed information on the bidders and the bid documents for both successful

and unsuccessful bidders. Although there has always been an element of compe-

tition for urban funds (Ball, 1995), there was a break in UK urban policy in the

early 1990s when the Conservatives moved decisively away from allocation

according to measures of spending need towards competitive bidding by local

partnerships, usually led by local authorities (Oatley, 1998; Foley, 1999; Stewart,

1994; Taylor, Turok, and Hastings, 2001). In 1994 the government unified

existing urban regeneration funds into the SRB. The overall budget was around

1.4bn in 1994/5. The administering government Offices for the Regions each

had a separate budget. The programme was continued under Labour, with a fourth

round in 1997/8 (DETR, 1997).The SRB bidding process provided opportunities for sequential adjustment

under incomplete information. Because most bids were led by officials from




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different local-government units who met quite regularly, including at pre-brief-

ings on annual rounds of the SRB, groups had considerable knowledge of what

others wanted to do. These meetings also gave officials from Government

Offices of the Regions information about groups ideal projects. In the first

annual round groups put in outline bids, then some were invited to put in modi-

fied bids. Although this was subsequently dropped, there was still considerable

interaction between groups and officials, whereby bids were further refined.Knowing the quality of bids currently in the frame, officials had incentives to

pass on this information, to provoke bid improvement. Their role may have been

Table 1. Monte Carlo Simulations

(a) Bids are drawn from the uniform distribution on the set such that x , y [0.1 , 1].

Repetitions of the experiment 1000. Number of groups, n= 30, r= 0.1.

G (expected proportion

of ideal bids that

can be funded

1:8G=30)

Estimate of average

proportion of

groups to which

Theorem 1 applies

Estimate of average

proportion of groups

to which Corollary 1

applies

Sum of column 2

and column 3

1 (0.06) 0.011 0.013 0.024

3 (0.18) 0.041 0.010 0.051

5 (0.3) 0.141 0.010 0.151

7 (0.42) 0.267 0.008 0.275

9 (0.54) 0.388 0.005 0.393

11 (0.66) 0.510 0.003 0.513

13 (0.78) 0.631 0.002 0.63315 (0.9) 0.749 0.001 0.750

17 (1.02) 0.831 0.000 0.831

(b) Bids are drawn from the uniform distribution on the set such that

x , y [0.05, 1]. Repetitions of the experiment 1000.

Number of groups, n= 30, r= 0.05.

G (expected proportion

of ideal bids that

can be funded

1:9G=30)

Estimate of average

proportion of groups to which

Theorem 1 applies

Estimate of average

proportion of groups

to which

Corollary 1 applies

Sum of column 2

and column 3

1 (0.06) 0.006 0.060 0.066

3 (0.19) 0.099 0.055 0.154

5 (0.32) 0.218 0.046 0.264

7 (0.44) 0.343 0.035 0.378

9 (0.57) 0.473 0.029 0.502

11 (0.70) 0.602 0.019 0.621

13 (0.82) 0.725 0.011 0.736

15 (0.95) 0.838 0.003 0.841

17 (1.08) 0.899 0.001 0.900




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analogous to that of the Walrasian auctioneer, spreading information about the

current configuration of bids until equilibrium was reached.

We coded the bid documents on various indicators of bid quality for almost

all the bids submitted over four rounds from 1994/5 to 1997/8. We developed acomposite index of bid quality, index, derived by standardizing and adding indi-

vidual indicators. We could not directly observe what groups ideal bids were,

nor did we have drafts of bid documents which would have enabled us directly

to observe competition at work within a given round. We showed that round-by-

round improvements in index largely occurred at or below the median, not at the

high-quality end of the distribution (John, Ward and Dowding, 2004; John and

Ward, 2005). Assuming it took more than one round for competition to have its

full effects, this is consistent with the idea that competition has less effect

among bidders with high-quality ideal projects. Here we adopt a differentempirical strategy.

First, we now focus on bid value, not bid quality as in our previous work.

Our measure of this is value= index/srbtot, where srbtot is the total SRB fund-

ing asked for. (Note bigger scores on value suggest higher value bids.) Theory

suggests that under sequential funding bids will not get funded unless value is

greater than some cut-off point. The amount of funding a bid obtained from the

SRB process is total. If value is greater than the cut-off, total should be an

increasing function of how much grant was requested, srbotot. The overall

observed success rate of bids was around 54%. So we constructed a variableh46value that took on the value 0 if the value index was less than the 46th per-

centile point of the distribution by value and was equal to srbtot otherwise.

Because many bids were not successful and total is censored below at zero, we

used Tobit Regression. We controlled for the size of the regional budget in the

round, rbudget, and the number of bids in the region/round, nobids. As shown in

Table 2, h46value powerfully predicts the amount of funding bids received,

which suggests that value is a reasonable proxy for what the funding agency

was looking for.

Second, we considered the impact of competition on bids that were resub-

mitted. The advantage of this is that for resubmitted bids we had a measure of

how the value of that particular bid changed between rounds, whereas our pre-

vious method did not track the effects of competition on particular bids. We

identified 43 bids that were resubmitted in subsequent rounds. This coding was

based on the criteria that the aims of the bid and the main members of the bid-

ding group were the same, identification being aided by comparing bid titles. 5

5. We exclude a bid submitted in the first and second rounds, entitled Safe in Tesside a very

large, high-leverage outlier in models for the value of resubmitted bids. The very large increase infunding asked for between the first and second rounds, from 63,000 to 1,743,000, suggests that a

different and far more ambitious law-and-order-related project was resubmitted. If a dummy variable

for this bid is included in the model, results are substantively the same as those reported.




14/21

Although we cannot directly test our hypothesis we can test the related claim

that resubmitted bids will show the greatest improvements in value if they were

mid-range on value the first time they were submitted: i) if the bid was of high

value at time t, it could be expected to be competitive at time t+ 1 without much

improvement, conditional on a similar competitive environment in the next

round; ii) if it was mid-range in value at t, it could be expected to face the stron-gest competition at time t+ 1, requiring a bigger improvement to be competi-

tive; iii) finally if it was low value at time t, the likelihood is it would not be

competitive at time t+ 1, so it would not be worth putting much extra effort in.

Notice that our expectations about resubmitted bids follow if groups carry

expectations over to the next round that parallel our hypothesis; so although our

test is indirect, it seems appropriate, given the fact that we could not observe the

process of competition within a round.

Valuenext is the value of the resubmitted version of a bid, measured only for

bids subject to the selection filter of being resubmitted. We hypothesise an

inverted U-shaped relationship between this variable and value, which we tested

for by including value and its square, valuesq as explanatory variables. To con-

trol for the competitive environment in the round the bid was resubmitted in

which we included the average value of bids at t + 1, avaluenext, the budget of

the region the bid was submitted to at t+ 1, rbudgetnext, and the number of bids

in the region concerned at t+ 1, nobidsnext. To allow for the selection effect

due to the fact only some bids were resubmitted we estimated a Heckman selec-

tion model. We examined a number of possible influences on whether a bid

was re-submitted, the most powerful of which proved to be shortfall, the differ-

ence between the amount of SRB funding that was asked for at time t, srbtot,and the amount the bid obtained, total. We tested for regional effects, based on

interviews that suggested that some regions were more helpful than others when

Table 2. Tobit Regression for Total Funding Obtained

Total

h46value .633(30.69)

Nobids 47.6

(5.87)

Rbudget .00785

(2.82)

Constant 1048

(3.20)

obs. 1194

Note. Absolute value of t statistics in parentheses

significant at 10%;

significant at 5%;

significant at 1%log likelihood=6703.92

prob>2 = 686.64




15/21

bidding groups wanted to resubmit; and we found that the dummy variable for

the Eastern Region, east, was significant in selection models. In Table 3 the first

column contains the selection component of the Heckman model and the second

the explanatory component. In column 2, valuesq has a highly significant nega-

tive coefficient. When we added value to the model, its coefficient was nowhere

near significant. As the mean ofvalue is .0163, the evidence is for a significant

inverted U-shaped relationship between value and valuenext centred at zero,

near the mean ofvalue. That is, bids of mid-range value at time t had the highest

value when resubmitted at time t+ 1. The competition variables behaved as

expected, except that nobidsnext was insignificant and was dropped. The valueof rho is 0.91, and the chi-square test indicates rho is significantly different

from 0, justifying the idea that it was important to allow for selection to avoid

biased estimates.

5. Conclusion

This article has developed a model to understand how an agency allocates a

budget to a group of bidders and how these bidders strategies evolve. The main

engine of our approach is that, given a list of bids is presented to the fundingagency before it makes a choice, the agency uses a relatively efficient algo-

rithm to pack its knapsack. Given this assumption, the theoretical results and

Table 3. Heckman Selection Model for the Value of Resubmitted Bids

(1) (1)

Selection Equation Valuenext

Valuesq 5.14

(5.36)

avaluenext .244

(2.48)

rbudgetnext 3.26e-08

(2.73)

shortfall .0000312

(3.47)

East .438

(2.07)

Constant 1.94 .0315

(24.19) (3.80)

obs. 1191 uncensored obs. 43

Note: z statistics in parenthesessignificant at 10%; significant at 5%; significant at 1%

log likelihood= 25.05

rho=0.91

prob > 2 = .023




16/21

simulations suggest that competition will be confined to groups whose ideal pro-

jects are neither too near what the funding agency wants nor too far away. Com-

petition is at the margin, when much of the budget has already been allocated to

groups with good ideal projects. Our simulations suggest that relatively highproportions of groups will get funding for their ideal projects.

We test our model on a unique dataset of urban policy bids, where the results

suggest that the SRB process did little to induce groups already inclined to pro-

vide high value to do any better. This is in line with theoretical expectations and

supports the view that there are limits on gains to be made from introducing

competition for grants, especially as transaction costs are often high and it can-

not be guaranteed that the funding agency will act in the public interest. This is

not to say that competition will fail to induce greater efficiency among groups

inclined to produce lower-quality projects. These findings have implicationsfor those wishing to develop specific models of public sector competition, for

students of public administration seeking to understand the operation of such

competitive processes, and for policy-makers themselves who wish to introduce

competitive bidding schemes.

Appendix

LEMMA 1: if@ is a pure strategy equilibrium, x0i xi for each group i whose bid

is funded.

Suppose to the contrary that for some group x0i > xi. As funding is sequential

for x00i [xi, x0i), i would be able to make a bid which would receive at least as

much funding as its bid under @, given other groups strategies. As x00i is nearer x0i

given it receives the same funding or more, it is strictly better than x0i, which

contradicts the assumption that @ is an equilibrium.Hypothetically allocate funding sequentially to the ideal bids of groups

according to their index numbers until for the c-th group it is not possible tofund that groups ideal bid. C+ is the set of groups with x-ideal points xc.

C is the set of groups whose with x-ideal points xc.

LEMMA 2: if xg < xc r, g cannot be under competitive pressure in any pure-

strategy equilibrium @ in which no member ofC+ puts in a bid with a lowerx-value than xc r.

At most all members of C/g put in bids with lower x-values than xg in the

equilibrium. Suppose they all did so. Given the x-values of their bids, as they

could also get their ideal funding level under sequential, if their bids are part ofequilibrium @ each must ask for its ideal amount. But this means that gs bidmust be its ideal point: as no member of C+ locates to gs left and all members




17/21

of C-/g bid for their ideal level of funding, gs ideal bid will be funded under

sequential funding. If only some members of C-/g put in bids with lower x-

values than xg, the argument still holds as gs ideal bid will be funded before the

funding constraint bites under sequential funding.

LEMMA 3: let f be the lowest-indexed group under competitive pressure in a

pure-strategy equilibrium @. Let fs equilibrium bid be (x0f, y0f). Then x

0f < xf,

so long as G > y1.

Suppose this were not the case. Then as f is under competitive pressure and x 0f xf in equilibrium (Lemma 1), it must be the case that x

0f= xf and y

0f < yf.

Suppose f= 1. Under sequential funding 1s bid is the first to be considered,

even if other groups locate at x1, and 1 will get the amount it asks for so long as

this is feasible. By assumption G > y1. If 1 asks for y01 < y1, @ cannot be an

equilibrium, then. So either x01 < x1, or 1 is not the first group under competitivepressure. Suppose the latter is the case. F is the set of groups with bids less

than x0f in @. This set is non-empty as 1 is not the first group under competitivepressure. In @ each group in Fmakes its ideal bid and gets funding, as none isunder competitive pressure. Now suppose that x0f= xf, and f is only under com-

petitive pressure because y0f < yf. Again this cannot be the case. Under sequen-tial funding the first bid considered for funding among the bids at xf will be f s,

even if more than one bid is made at this x-value. It must be the case that Ff C- because xf < xc and for each member of F, say g, xg < xf. Thereforethe amount already allocated to members of F plus yf must be less than or

equal to G, since G is large enough to fund all the ideal bids of members of C.Hence if x0f= xf and y

0f < yf, fs bid cannot be a best response and @ is not an

equilibrium: f could ask for yf and get it under sequential funding.

THEOREM 1: @ is not among the subset of pure strategy equilibria that is Paretoefficient if there exists some group g such that xg < xc r which is under compe-

titive pressure, so long as G > y1.

If f is under competitive pressure in @ the set of groups, F, such that their bidshave the same x-value as fs must be non-empty, or else f could put in a bid for

the same amount but a higher x-value, nearer its ideal point, get the bid funded

and obtain a higher bid payoff (Lemma 2). Divide F into subsets F1;F2 con-taining groups putting in token bids and groups putting in funded bids with ideal

points at higher x-values than x0f. (Recall that groups never put in funded bids

with higher x-values than their ideal point in equilibrium, by Lemma 1.) Let the

next bid(s) to the right of x0f in @ be located at x0f+. Now consider an alternativeset of strategies @0 which differs from those in @ only in that f and the membersof F1 and F2 put in bids for the same amount as under @ but this time at x0f+ ,




18/21

> 0;x0f+ < x0

f+. For small-enough , we show that @0 is an equilibrium if@

is an equilibrium.

F is the set of groups with bids less than x 0f in @. By construction they bid

their ideal points in @ and @0 and these bids get funded under sequential funding,so they cannot strictly increase their payoffs by switching strategy in @0.

F+ is the set of groups with bids with x-values greater than x0f in @. We needonly consider whether members of F+would get a higher payoff for some bid

on the interval [x0f, x0

f+ ] rather than in their assumed position in @0: given the

strategies of the other groups, for any position outside this interval their payoff,

contingent on the amount of funds asked for, is exactly the same as in @, becausefunding is sequential and the rank order of bids on the x-dimension is the same;

and groups in F+ cannot strictly increase their payoff by moving to any such

position in equilibrium @.First consider any group j in the set F+ C+. By Lemma 3 x0f < xf, xf < xc

r, and xj xc; so can be chosen such that x0

f+ < xj r. Thus for smallenough the most that j can make by relocating in the interval [x0f, x

0f+ ] is the

payoff associated with a token bid, for any bid there that obtained funding would

have a negative bid payoff. Group js payoff in @0 is the same as their payoff in@: as x0f+ < x

0f+ and funding is sequential, the same amount is divided among

the members of F+ in @0 as in @; and as their bids are the same, so must theirpayoffs be the same. As @ is assumed to be an equilibrium, no group can get less

than its payoff from a token bid with zero bid payoff, because they can guaran-tee this amount by making a token bid. Thus no j in the set F+ C+ can strictly

increase its payoff by relocating in the interval [xf0 , xf0 + ], so long as is small

enough, because it would receive a maximum of its bid payoff, which is less

than or equal to its current payoff.

Next consider any group in the set F+C: The payoff of each such group,say j, is the same in both @ and @0. By assumption group j is making the samebid in each case. As x0f+ < x

0f+ and x

0j x

0f+ and each group is asking for the

same amount in both @ and @0, the amount that has yet to be allocated when jsbid is considered under sequential funding is the same in both cases; so js pay-

off is the same. We show that the funding j asks for in @ must be yj the levelthat corresponds to its ideal bid. By definition of the set C,

yjGj C

that is, each member of C can be funded up to its ideal level of bidding, so long

as no other bid is funded first under sequential bidding made by groups outside

C and no member of the set C asks for more than its ideal level of funding.Any group in the set F+ C+ that makes a bid with an x-value lower than

xc r, that might be considered before js bid, must be making a token bid that

gets no funding, otherwise its bid payoff would be negative. Conditional on thex-value of its bid, the level of funding that maximizes a groups payoff is always

that corresponding to its ideal bid. Thus no member of C that is considered for




19/21

funding before j asks for more than this amount. Thus when it is the turn of js

bid to be considered j can feasibly ask for yj, and this maximizes its payoff con-

ditional on the x-value of its bid. By making a bid in the interval [x 0f, x0

f+ ]

rather than its assumed bid in @0, j might feasibly have access to more fundingthan the amount it already gets, yj, because its bid might be considered earlier

under sequential funding as it would be at or to the left of those of f, and groups

in F1 and F2. But it would not be optimal to ask for more than yj. Any such bid

would be for yj, otherwise j could increase its payoff conditional on the x-value

of the bid, and it would be funded for this amount under sequential funding.

As @ is an equilibrium, x0j xj (Lemma 1); so by moving into the interval[x0f, x

0f+ ] j would be moving further away from its ideal point parallel to the

x-axis and would get the same level of funding. But this means its payoff would

be lower if it made this move. So if @ is an equilibrium, no group in the setmember of the set F+ C would have an incentive to switch strategy intothe interval [x0f, x

0f+ ] in @

0.

To show that @0 is an equilibrium it remains to prove that neither f, nor anygroups in F1 and F2 have any incentive to change strategy. We only need con-

sider whether it pays any such group to change its strategy under @0 by locatingin the interval [x0f, x

0f+ since the payoffs of such groups are the same in @

and in @0 for any other move they can make and, as @ is an equilibrium, it cannotstrictly increase groups payoffs to relocate outside this interval as their payoffs

in @0

are greater than or equal to those in @. Group j in F1 makes a token bid. Ifin equilibrium @ it cannot move further to the left to obtain more funding, bybeing considered earlier under sequential funding, and remain with a radius r of

its ideal point, for small enough this will still hold in @0. The feasible level offunding nearest to yj available to j in @ if it moves just to the left of x

0f is the

same that is obtainable in @0 on the interval [x0f, x0

f+ ; and if x0

f is too far from

js ideal measured along the x-axis for the bid to be made with a positive bid

payoff just to the left of x0f, the same will be true for bids on the interval

for small enough , since x0f+ will still be too far from js ideal point along

the x-axis.

Now consider a group j in f F2. Suppose, first, that in equilibrium @ j getsless than yj units of funding. The next bid below those located at x

0f, if any, is

that of group (f 1) located at its ideal point. If j is located anywhere on the

interval (xf1, x0

f0), or on the interval [0, x0

f0) if f = 1, the maximum feasible

amount of funding it could obtain is the same under sequential funding. Suppose

this was yj + > y0

j. Then @ could not be an equilibrium: just to the left of x0

f j

could put in a bid for an amount nearer its optimal level of funding that would

be nearer its ideal point. Thus on the interval (xf1, x0

f) the most j can obtain in

funding must be y0j. Now consider @0. So long as x0f+ < x

0f+ , the most j can

obtain on the interval [xf00 , xf0 + in @0 is the same as the most it can obtain onthe interval it (xf1, x

0f0

) in @, which is to say y0j as @ is an equilibrium. But thismeans that it will not pay for j to relocate on the interval [x 0f, x

0f+ in @

0, so




20/21

long as is small enough: by moving in this way it can only obtain the same

amount of funding as from its strategy in @0, but it moves further away from itsideal point measured along the x-dimension, so long as xj > x

0f+ . Group f is

chosen to be the group with the lowest index number that is under competitivepressure. By Lemma 3 x0f < xf and xf xj, or else j would be the group with the

lowest index number under competitive pressure; so we can chose small

enough to ensure that x0f+ < xj.

Finally suppose that in @ j gets yj units of funding. Then in @0 j also gets yj

units of funding. On the interval [x00f , x0

f+ j might feasibly be able to obtain

more than this, but it would have no incentive to do so. As x 0f < xf (Lemma 3)

and xf xj, for small enough , x0

f+ < xj. To move into the interval [x00

f ,

x0f+ would be to move further away from its ideal point relative to the x-

dimension and there would be no advantage in terms of extra funding. Hence j

cannot strictly increase its payoff by making such a move.

We have shown that by choosing a small enough value of it is possible to

ensure that @0 is an equilibrium if@ is an equilibrium. Moreover @0 is a Paretoimprovement on @. Apart from f and any groups in F2, the payoff of all groupsare the same in each case. Group f and any groups in F2 get the same level of

funding but, for small enough , x0f+ is nearer to their ideal point than x0

f mea-

sured along the x-dimension, as x0f < xf < xj for any j F2 (Lemma 3). There-

fore it cannot be the case that any group g such that xg < xc r is under

competitive pressure and @ is a Pareto optimal equilibrium, for under theseassumptions it is possible to construct an alternative equilibrium no worse for

any group and strictly better for at least one group.

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A Spatial Model of Competitive Bidding for Govt Grant

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