The Scope of Some Scoring Rules Applying

  • Andrei IvanovEmail author
Living reference work entry



For the award of public procurement contracts, contracting authorities use one of the two different approaches, namely, the lowest price or the most economically advantageous tender. The latter is applied when price and quality matter. In this case, scoring rule, which transfers the price and quality of bidder’s offer into the points, is needed.


In the modern public procurement, contracts can be awarded according to results of bids comparing on one or more criteria. In the second case, the contracting authority (or, other words, public buyer) has to assign scores to suppliers’ bids on each criterion. In the international practice, there are a lot of assignment rules, which have been applied for a long time without serious trouble.

However, the experience of the Russian Federation has revealed that in the institutional environment, which is characterized by corruption, low competition in the tenders, and regulatory policy, aimed at the restriction of public buyer’s discretion power, the applying of some famous rules cannot be recommended at all (linear scoring rule without threshold (Dini et al. 2006, p. 305)) or can be recommended with some restrictions (highest bid–lowest bid scoring rule (Dini et al. 2006, p. 309)). Thus, the accumulated experience can give to researchers and regulatory bodies some valuable information about their applicability in institutional conditions that are typical for developing countries and, in some extent, for developed ones too.

The Three Scoring Rules

Let us consider a contracting authority seeking to procure an indivisible differentiated good by means an open tender. We suppose that there are m + 1 awarding criteria (m of them for the quality assessment, including the delivery terms, volume and duration of the warranty, costs of operations and, may be, utilization, and so on, and price), and any supplier’s bid has the form: B = (q 1, q 2, …, q m , p).

The contracting authority attributes to supplier’s bid B the total score:
$$ U(B)=\sum_{i=1}^m{w}_{qi}{Q}_i+{w}_pP, $$
where w qi and w p are the weights of awarding criteria defined by the contracting authority under some restrictions established by the regulator and \( \sum_{i=1}^m{w}_{qi}+{w}_p=1; \) and Q i and P are the scores of the same scale, assigned to the values of criteria according to the corresponding scoring rule.
Thus, Formula 1 demands the specific scoring rule, which transfers the price and quality of bidder’s offer into the points. To explain what scoring rule is and which rules exist, we assume that there are two awarding criteria (quality and price) and total score of each supplier’s bid B = (q, p) is:
$$ U(B)={w}_qQ+{w}_pP,\ {w}_q+{w}_p=1. $$
Suppose that selecting stage of the tender have passed N (N > 1) suppliers with bids (q 1, p 1), …, (q N , p N ). We designate
$$ {q}_{\mathrm{min}}=\underset{1\le i\le N}{\min }{q}_i,\ {q}_{\mathrm{max}}=\underset{1\le i\le N}{\max }{q}_i,{p}_{\mathrm{min}}=\underset{1\le i\le N}{\min }{p}_i,\ {p}_{\mathrm{max}}=\underset{1\le i\le N}{\max }{p}_i. $$

There are many ways how the contracting authority can assign scores Q and P to the supplier’s offer on quality q and price p (Dini et al. 2006, pp. 304–310). We assume that quality is an increasing criterion: for any scoring rule, the higher value of quality gives the higher score, and price, in turn, is an decreasing criterion: for any scoring rule, the higher value of price bid gives the lower score.

Below we will consider following three scoring rules: the highest bid–lowest bid scoring rule, the linear scoring rule, and the highest/lowest bid scoring rule (Ibid). These rules have been applied in the Russian Federation; and the accumulated experience can give to researchers and regulatory bodies some valuable information about their applicability in corresponding institutional conditions.

Highest Bid–Lowest Bid Scoring Rule

The basic formula of the highest bid–lowest bid scoring rule is
$$ {Q}_i=\frac{q_i-{q}_{\mathrm{min}}}{q_{\mathrm{max}}-{q}_{\mathrm{min}}}\times L,\ {P}_i=\frac{p_{\mathrm{max}}-{p}_i}{p_{\mathrm{max}}-{p}_{\mathrm{min}}}\times L,\ L\ge 1,i=1,\dots, N, $$
where integer L is the maximum number of points available to the bidder for his quality/price offer.
Correspondingly, the total scores of suppliers’ bids are
$$ U\left({q}_i,{p}_i\right)={w}_q{Q}_i+{w}_p{P}_i,i=1,\dots, N. $$
According to Formula 3, the rule (Rule 1.1) gives maximum score (L points) to the best bid and minimum one (0 points) to the worst bid and scores all other bids proportionally their distance from the worst bid (Fig. 1a, b, L = 1).
Fig. 1

(a) Rule 1.1 for the increasing criterion. (b) Rule 1.1 for the decreasing criterion

Sometimes, the regulator prefers to use the range of points [1, L] (L > 1) and prescribes to apply the following variation of Formula 3 (Rule 1.2):
$$ {Q}_i=1+\frac{q_i-{q}_{\mathrm{min}}}{q_{\mathrm{max}}-{q}_{\mathrm{min}}}\left(L-1\right),\ {P}_i=1+\frac{p_{\mathrm{max}}-{p}_i}{p_{\mathrm{max}}-{p}_{\mathrm{min}}}\times \left(L-1\right),\ L>1. $$
Figure 2a, b illustrates Rule 1.2 for L = 10.
Fig. 2

(a) Rule 1.2 for the increasing criterion. (b) Rule 1.2 for the decreasing criterion

The Linear Scoring Rule

The linear scoring rule (Dini et al. 2006, p. 305) is usually applied to the price criterion and described by the following expression:
$$ {P}_i=\left\{\begin{array}{l}0,\ {p}_i>{P}_r,\hfill \\ {}\frac{P_r-{p}_i}{P_r-{P}_{thr}}\times L,\ {P}_{thr}\le {p}_i\le {P}_r,\hfill \\ {}L,\ 0\le {p}_i<{P}_{thr},\hfill \end{array}\right. $$
where P r is the reserve price and P thr is the price threshold.
Thus, the bidding at the reserve price and above does not give to the supplier any points; the bidding at the price threshold and below gives to him L points (Fig. 3). To define the price threshold, the buyer has to bear some additional costs, and, if he is going to avoid it, it is possible to apply the linear scoring rule without threshold (P thr = 0):
$$ {P}_i=\frac{P_r-{p}_i}{P_r}\times L,\ 0\le {p}_i\le {P}_r. $$
Fig. 3

Linear scoring rule with and without threshold

The most clear interpretation of Formula 6 can be done, if L = 100. In this case, the supplier’s price score just equal to his price discount: 0% discount (p i = P r ) gives 0 points, 10% discount (p i = 0.9P r ) gives 10 points, 50% discount (p i = 0.5P r ) gives 50 points, and the only zero price bid (100% discount) gives to the supplier 100 points. Figure 3 illustrates the linear scoring rules with and without threshold (the thick dotted and solid lines, correspondingly, L = 100) and the main difference between them: to obtain the same score P, the second rule demands significantly more discount.

The Highest/Lowest Bid Scoring Rule

The highest/lowest bid scoring rule (Dini et al. 2006, p. 309) assigns to values of quality and price criteria following scores:
$$ {Q}_i=\frac{q_i}{q_{\mathrm{max}}}\times L,{P}_i=\frac{p_{\mathrm{min}}}{p_i}\times L. $$

Similar to Rule 1.1, this rule assigns L points to the best bid, but in contrary to it, the points of the worst bid can be close to L as well to 0, depending on the difference between best and worst offers. The highest/lowest bid scoring rule generates the incentives to aggressive bidding: the highest (lowest) bid on quality (price) is awarded maximum point, on the one hand, and decreases the competitors score, on the other hand.

The Public Procurement Institutional Environment in the Developing Countries: The Russian Case

The functions of a state involve the provision of various services to the population in areas such as health, education, utilities, etc. In this case, any society, first and foremost, democratic, in varying degrees, exerts pressure on the state, demanding higher quality, range, and quantity of such services and/or reducing the cost of their provision.

Around since the early 1980s of the last century, countries, where such pressure was in a greater degree, began to implement a variety of tools, improving the efficiency and quality of services, including decentralization, performance auditing, contracting out, and so on. By the mid-1990s, this kind of policy was formed under the name of New Public Management (hereafter – NPM).

On the one hand, NPM tools have focused on the introduction of competition in the provision of public services and, on the other hand, gave rise to various forms of collaboration between the public and the private sector in the creation of infrastructure and the provision of services on its basis to the population. For this kind of collaboration gradually, beginning from the Great Britain, stuck title public-private partnership (hereafter – PPP).

In the Russian Federation, like other countries with emerging economy, market reforms have created such political and economic conditions, in which the tools of NPM had become in demand. In particular, great efforts have been directed at the restoration of institutions, which were not used in the time of the Soviet Union (1917–1991): public procurement and public-private partnership.

In the Russian Federation, the introduction of NPM tools occurred in the context of intense economic reforms and was objectively accompanied by growth of corruption (Huntington 1968, p. 59). The development of public procurement has created a new area for corrupt behavior and formed specific public procurement corruption – practice of kickbacks: public buyer sets up inflated price of purchasing goods (works, services), then restricts competition in the tender, after that pays to the preselected winner the inflated price, and, finally, shares with the latter his surplus.

In the modern information-transparent world, the corruption in the public procurement inevitable reflects in the mass media in the form of the numerous corruption cases and creates the demand on the strong anti-corruption policy. It is a common knowledge that there are three pillars of anti-corruption policy: restricting the discretionary power of public buyers, preventing them from rent-seeking behavior, and strengthening the institutions (Aidt 2003, p. F633). However, it is not a simple task to fit the balance between the combating corruption and efficiency of public procurement. Due to the lack of experience in the public procurement, in the corresponding legislation, created by 2006, Russian regulators applied a complex of extremely strong tools for restriction of the public buyer’s discretionary power:
  • Very low price thresholds had been established, extending the area for corrupt behavior and creating the demand on the additional monitoring efforts.

  • The applying of the qualification criteria had been restricted, giving to all firms an easy access to public procurement market.

  • The second-price (English) auction was selected as the main procurement method; thus, it was impossible to award contract by the criterion of the most economically advantageous offer for all homogenous goods and even for some differentiated goods.

The second and third tools created a risk of awarding contract to the mala fide suppliers, which cannot or are not going to satisfy public buyer’s demand. That is why, the bona fide contracting authorities have to restrict competition in the tenders similar to mala fide ones, which do the same for bribes.

Thus, the competition in the Russian tenders, which are divided to the auctions (the price is a single criterion) and contests (there are, as minimum, two criteria: price and something else), is essentially lower than, for example, in European Union (Fig. 4a (an average number bids for tender)-4b (an average number and median bids for tender)).
Fig. 4

(a) The competition in tenders in RF (Source: Federal State Statistics Service). (b) The competition in tenders in EU (Source: Public procurement in Europe, 2011)

Thus, most of Russian contests, which demand the applying of scoring rules, have not more than two bidders.

The Applying of Some Scoring Rules in the Russian Institutional Environment: Several Lessons for Developing and Developed Countries

In this section, we will consider a contracting authority who is seeking to procure indivisible differentiated goods in the institutional environment, which is characterized by corruption, low competition in the tenders, and regulatory policy, aimed at the restriction of buyer’s discretion power.

Highest Bid–Lowest Bid Scoring Rule

In the Russian Federation, the highest bid–lowest bid scoring rule has been applied in the public procurement and public-private partnership tenders. The legal acts, which have introduced the rule, are summarized in the table below (Table 1).
Table 1

The Russian legal basics of the highest bid–lowest bid scoring rule


The legal act





The letter of Ministry of Economic Development (for public procurement)




Federal Law #115 “On Concession Agreements” (art. 32-5) (for public-private partnership)




Federal Law #224 “On Public-Private Partnerships …” (art. 28-5) (for public-private partnership)



For the public-private partnership tenders, the regulator prescribes to apply the rule in the form (3) with L = 1 (Fig. 1a, b):

$$ {Q}_i=\frac{q_i-{q}_{\mathrm{min}}}{q_{\mathrm{max}}-{q}_{\mathrm{min}}},\ {P}_i=\frac{p_{\mathrm{max}}-{p}_i}{p_{\mathrm{max}}-{p}_{\mathrm{min}}}, $$
and for the public procurement tenders – in the form (4) with L = 10 (Fig. 2a, b):
$$ {Q}_i=1+\frac{q_i-{q}_{\mathrm{min}}}{q_{\mathrm{max}}-{q}_{\mathrm{min}}}\left(10-1\right)=1+\frac{q_i-{q}_{\mathrm{min}}}{q_{\mathrm{max}}-{q}_{\mathrm{min}}}\times 9,{P}_i=1+\frac{p_{\mathrm{max}}-{p}_i}{p_{\mathrm{max}}-{p}_{\mathrm{min}}}\times 9 $$

All properties, discussed below, are the same for the rules, defined by Formulas 8 and 9, and further, for definiteness reasons, we treat the highest bid–lowest bid scoring rule in the form (8). Below, we will examine the properties of the rule in the institutional environment similar to the Russian Federation one.

We begin with consideration of a public buyer, who for assessment of bids sets up two criteria, quality and price, and their weights w q and w p , correspondingly. Russian regulatory policy makes the buyers to give the higher weight to the price criterion, and we will follow this restriction: w p > w q .

Let us assume that at the tender two bids have been submitted: B 1 = (q 1, p 1) and B 2 = (q 2, p 2). We exclude from the consideration the case when the one bid dominates the other one different from it or, other words, has the higher (or the same) quality and lower (or the same) price and suppose for definiteness reasons that q 1 > q 2 and p 1 > p 2.

Given assumptions, the highest bid–lowest bid scoring rule assigns to suppliers’ quality and price bids following scores (Formula 8, Fig. 1a, b): Q 1 = 1, Q 2 = 0, P 1 = 0, P 2 = 1. Thus, the total scores of bids will be:
$$ U\left({B}^1\right)={w}_q\times 1+{w}_p\times 0={w}_q,U\left({B}^2\right)={w}_q\times 0+{w}_p\times 1={w}_p. $$

Thereby, the supplier, who submitted the bid with the lower price, will be awarded the contract independently of “quality” of his quality suggestion, and, respectively, there are no incentives for him to improve the latter, if the supplier knows ex ante (i.e., before the submitting the bid) that he is a price (or quality, if w p < w q ) “leader” (this proposition demands some additional assumptions; however, the exact consideration of them is beyond the scope of this article and can be found in Ivanov (2016)).

Let us move to the consideration of the general case. Suppose that there are m + 1 awarding criteria: m of them for the quality assessment and price and public buyer attributes to each supplier’s bid B = (q 1, q 2, …, q m , p) the total score:
$$ U(B)=\sum_{i=1}^m{w}_{qi}{Q}_i+{w}_pP,\ \sum_{i=1}^m{w}_{qi}+{w}_p=1. $$
We will assume that there is a criterion with its weight more than 0.5, for example, w p > 0.5, two suppliers have submitted their bids, and the price bid of the first supplier (we will call him the Winner and mark his bid components by the index W) is less than the second one (the Looser, index L). Given assumption, we have
$$ {\displaystyle \begin{array}{l}U\left(\mathrm{Winner}\right)=\sum \limits_{i=1}^m{w}_{qi}{Q}_{iW}+{w}_p{P}_W\hfill \\ {}=\sum \limits_{i=1}^m{w}_{qi}{Q}_{iW}+{w}_p\times 1>0.5,U\left(\mathrm{Looser}\right)=\sum \limits_{i=1}^m{w}_{qi}{Q}_{iL}\\ {}+{w}_p{P}_L=\sum \limits_{i=1}^m{w}_{qi}{Q}_{iL}+{w}_p\times 0\le \sum \limits_{i=1}^m{w}_{qi}<0.5.\hfill \end{array}} $$

Thus, the Winner will be awarded the contract independently of “quality” of his quality suggestion, and, respectively, he has no incentives to improve the latter (with the stated above remarks). This property is not good for public procurement contracts and unacceptable for the long-run public-private partnership agreements.

However, when and why does the supplier can ex ante know that he has the single rival and he is a price (or quality) “leader” toward to him? Some of such cases are described below:
  1. 1.

    The market structure is the differentiated duopoly.

  2. 2.

    There is a corruption, and public buyer, who has received the single bid by the end of submitting time, asks the other supplier of lower-quality goods submit the bid at lower price.

  3. 3.

    There is collusion, and the second supplier is a dummy bidder.


It is clear that in the case of two awarding criteria of different weights, the applying of the highest bid–lowest bid scoring rule is very risky, because one of them will be more than 0.5.

Thus, if the regulator prescribes the applying of discussed rule, she must demand from the public buyer to designate in the tender documentation:
  • Three awarding criteria as minimum.

  • Weights of the criteria less than 0.5.

  • The minimum number of suppliers’ bids for the tender to be performed (following the (UNCITRAL Model Law, 2011, 53-j)).

  • That if two bidders have passed the selection stage, the other scoring rule has to be applied.

There is a special recommendation for the case of PPP. As it mentioned above, the laws on concession and PPP (# 115-FZ “On Concession Agreements” and # 224-FZ “On Public-Private Partnership, …”) directly prescribe to apply the highest bid–lowest bid scoring rule in the form (8). Because there is a very long way to correct the law, it would be better to give some discretion power to change the scoring rules to the executive power (in RF, Ministry of Economic Development).

We finish with the consideration of the case, in which the corrupt public buyer has preselected tender’s winner – his favorite. The public buyer has tried to restrict the competition in the tender, receives the bid from the favorite, and is going to award him the contract when, unexpectedly, the other supplier submits the bid. We assume that the latter has the advantage in one of the quality criteria toward to the favorite.

Since envelops (or electronic data) are still sealed (are closed), the corrupt buyer does not know what the total score of second bidder will be but knows that the undesirable supplier has the advantage in the value of specific criterion (for definiteness, quality one) toward to the favorite (q W < q L , we have reserved the index W for the favorite and L for the undesirable supplier). Thus, the contribution of this criterion into the total score will be w q  × 1 for newcomer and w q  × 0 = 0 for the favorite (where w q is the weight of the criterion).

Being afraid of the corrupt transfer losing, the public buyer asks the third supplier (or asks favorite to find such supplier), we call him the Dummy one, to submit a bid with q dummy < q W < q L (and values of other criteria, which do not change the corresponding scores of the favorite; that is possible since favorite’s bid is known to the buyer). After that, the contribution of the criterion into the total score will be still w q  × 1 for the newcomer and, according to (9), \( {w}_q\times \frac{q_W-{q}_{\mathrm{dummy}}}{q_L-{q}_{\mathrm{dummy}}} \) for the favorite.

Thus, in the case of applying highest bid–lowest bid scoring rule, the corrupt buyer can use his discretion power to increase the total score of bid of the supplier selected by him for the organizing of the corrupt transfer.

There is a simple way to combat this type of the public buyer’s mala fide behavior: the suppliers’ bids must be submitted to the third party. In the Russian Federation, the role of such third party can play (but doesn’t play now) the electronic trade platforms which had been established some years ago for providing of electronic auctions.

Linear Scoring Rule Without Threshold

In 2009, the highest bid–lowest bid scoring rule, which had some undesirable properties (subsection “Highest Bid–Lowest Bid Scoring Rule”), had been substituted for public procurement tenders with the linear scoring rule (Table 2).
Table 2

The Russian legal basics of the linear scoring rule


The legal act





The Government Decree #722



For the assessment of price proposals of suppliers, the linear scoring rule without threshold was selected. The regulator definitely knew that in the international practice, this rule is applied with the threshold, since in the assessment of delivering time she left the choice of rule, with or without threshold, at the discretion of the public buyer. However, there were, as minimum, two reasons against setting up the price threshold. Firstly, as it was stressed above, to define the price threshold, the buyer has to bear some additional costs. Secondly, the bidding below the price threshold does not give to the supplier additional points, and he has no incentives to do so. It doesn’t fit the modern Russian public procurement policy, which, from its beginning, was sharply aimed at the budget cuts, as it was reflected, in particular, in the title of Presidential Decree # 826, which was the basis of the public procurement system of the Russian Federation in the period 1997–2006: “On urgent arrangements to prevent corruption and budget cuts in the organization of purchasing goods for public needs.”

Let us consider the linear scoring rule without threshold, which was introduced for the price bids assessment by the Government Decree #722:
$$ {P}_i=\frac{P_0-{p}_i}{P_0}\times 100,\ 0\le {p}_i\le {P}_0, $$
where the reserve price P r (Formula 6) was substituted with the initial price of contract P 0, which, according to the Russian legislation, must be pointed out in the solicitation to the tender and cannot be exceeded in the suppliers’ price bids.

Formula 10 assigns price score to the supplier’s price bid which is just equal to the price discount, and we can reasonably assume that the examined rule must generate two different strategies of suppliers: to bid aggressively on price or to bid aggressively on quality (and do not give any price discounts).

The first strategy does not fit well with the antidumping mechanism of the regulation; the second one demands the mechanism of rejecting of abnormal quality bids. The latter tool is not easy to implement: mala fide public buyers usually can justify that the selected awarding criterion, which gives any necessary advantage to preselected supplier, is objective and does not restrict the competition.

Let us consider how the introduction of linear scoring rule without threshold had influenced the effectiveness of the tender procedure. Given the fact that the rule was introduced in late 2009, we have compared the average price reduction at the contests, which took place in the Russian Federation in 2008 and 2010 (with exception of purchases of municipal customers). According to the Federal State Statistic Service, the average price reduction in 2008 was about 7.63%, just as 2 years later – 4.09%.

Thus, the substitution of the highest bid–lowest bid scoring rule and highest/lowest bid scoring rule, which was valid till the November of 2009 too, with linear scoring rule without threshold dealt a heavy blow to the Russian public procurement system.

Finally, we will discuss the assumption of the existence of two strategies of suppliers in the conditions of the linear scoring rule without threshold, which was stated above. To study the reaction of the public procurement system on the incentives, generated by the rule, a statistical analysis of the results of contests, declared by the healthcare organizations of St. Petersburg in 2013, the last year of the rule line, has been performed. That year 143 contests were declared with the level of competition that reflected in the following table (the number of contests is relatively small because the public buyers must purchase all the drugs through the price auctions only) (Table 3).
Table 3

The level of competition in the contests of St. Petersburg healthcare organizations

Total number of contests (lots), from them


All bids were rejected


The selection stage passed single supplier


The selection stage passed two suppliers


The selection stage passed more than two suppliers


The statistical analysis of price discounts in the contests has revealed that the average price reduction was 5.28%, while in the 50% of contests, the price was reduced by less than 0.5% and 75% – less than 5%. In turn, 5% of contests gave the price reduction more than 31.76%.

Thus, we can conclude that in the more than 50% of contests, there were practically no price discounts, the winners of more than 5% of contests submitted dumping bids, and, correspondingly, on the market in question, public buyers are more inclined to refuse to reduce prices (Strategy 2, Fig. 5) than to the dumping (Strategy 1, Fig. 5).
Fig. 5

Two strategies of suppliers in the conditions of the linear scoring rule without threshold

Thus, we can see that the linear scoring rule without threshold cannot be recommended at all and, in the case of assessment of price bids, it is natural to introduce threshold and put it not less than upper level of abnormal bids (i.e., in the modern Russian public procurement legislation, the bids with price discount more than 25% are considered to be abnormal bids; hence, we can set up P thr equal to 0.75P 0).

The linear scoring rule without threshold was valid in the Russian Federation about 4 years and from the beginning of 2014 was substituted with highest/lowest bid scoring rule, which has applied for the assessment of the all tender criteria.


In this section, we will discuss the importance of the lessons that other countries could extract from the Russian experience in applying scoring rules. The paper’s findings can be significant for the developing countries with the similar institutional environment. On the other hand, the low competition in tenders, which in Russia is stem from corruption and/or weak regulatory impact assessment, in developed countries may be due to other causes: special structure of the market, state capture, etc.

There are some traps in applying highest bid–lowest bid scoring rule in the conditions of low competition in tenders (in particular, on the duopoly markets): weights of the criteria cannot exceed 0.5 and, respectively, the total number of criteria must be three or more. In this case, the mala fide agent, for example, cannot put w p > 0.5 and award contract to preselected low-cost supplier for kickbacks.

In turn, the linear scoring rule without threshold deprives the suppliers of incentive to price discounts or, contrary, forced them to dumping. In the framework of this rule, mala fide public buyer can easily award contract to preselected supplier for kickbacks or another reason, establishing the specific quality criterion adjusted for him. Thereby, the linear scoring rule without threshold cannot be recommended at all; it is natural to introduce threshold and put it equal to abnormal bid threshold or higher.

At the present time, the highest/lowest bid scoring rule is used in Russia; however it demands an additional time to identify advantages and disadvantages of its applying in the specific institutional environment.



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Authors and Affiliations

  1. 1.Graduate School of ManagementSaint Petersburg UniversitySaint PetersburgRussian Federation