Looking for best performers: a pilot study towards the evaluation of science parks

Abstract

Science Parks are complex institutions that aim at promoting innovation and entrepreneurship at local level. Their activities entertain a large set of stakeholders going from internal and external researchers to entrepreneurs, local level public administration and universities. As a consequence, their performances extends on a large set of dimensions affecting each other. This feature makes Science Parks particularly difficult to be properly compared. However, evaluating their performances in a comparable way may be important for at least three reasons: (1) to identify best practices in each activity and allow a faster diffusion of these practices, (2) to inform potential entrepreneurs about institutions better supporting start-ups birth and first stages and (3) to guide public policies in the distribution of funds and incentives. The multidimensional nature of Science Parks raises the problem of aggregating performances in simple indexes that can be accessed by stakeholders willing to compare different structures on the basis of their own preferences. This paper exploits a new dataset on Italian Science Parks to provide a pilot study towards this direction. In particular, we apply Choquet integral based Multi-Attribute Value Theory to elicit stakeholders’ preferences on different dimensions of Science Parks’ performances and construct a robust index allowing to rank them. This tool can be used to support the decision making process of multiple stakeholders looking for best (or worst) performers and allows to account both for subjective nature of the evaluation process and the interactions among decision attributes. Despite the present study employs only a limited number of respondents and performance measures, the procedure we present can be straightforwardly adapted to much richer environments.

Appendices

Appendix 1

In this Appendix, we present in greater details the technical features of the methodology involved in the evaluation of SPs. As we have seen, the application of MAVT based on the Choquet integral requires the identification of a capacity. In what follows, we firstly describe an alternative representation of the the Choquet integral, then we will formally specify the definition of the two indexes of aggregation which have been used in the analysis.

Any set function $\mu :\mathcal{P}(N)\rightarrow \mathbb {R}$ can be uniquely expressed in terms of its Möbius transform by:

$$\mu (T)=\sum _{S\subseteq T}m_{\mu }(S),\ \forall T\subseteq N$$

where the set function $m_{\mu }:\mathcal{P}(N)\rightarrow \mathbb {R}$ is called Möbius transform of a capacity $\mu$ and it is defined by:

$$m_{\mu }(S)=\sum _{T\subseteq S}(-1)^{s-t}\mu (T),\ \forall S\subseteq N$$

Now we can rewrite the Choquet integral in terms of the Möbius representation of a capacity. For any $u(x):=(u_{1}(x_{1}),\ldots ,u_{n}(x_{n}))\in \mathbb {R}$, the Choquet intergral of x w.r.t $\mu$ is given by:

$$C_{m_{\mu }}(u(x))=\sum _{T\subseteq N}m_{\mu }(T)\bigwedge _{i\in T}u_{i}(x_{i})$$

where $\bigwedge$ represents the minimum operator. The notation $C_{m_{\mu }}$ clarifies the fact that the Choquet integral is computed w.r.t. the Möbius transform of the capacity $\mu$.

As we have mentioned before, the concept of k-additivity can capture the trade-off between the complexity of the capacity and its modeling ability. A capacity $\mu$ on N, indeed, is totally defined by a reasonably large number of coefficients, i.e. $2^{n}-2$, and therefore such complexity ma result prohibitive in some applications.

Definition

Let $k\in 1,\ldots ,n$. A capacity $\mu$ on N is said to be k-additive if its Möbius representation satisfies $m_{\mu }(T)=0$ $\forall T\subseteq N$ such that $t\ge k$ and there exists at least one subset T of cardinality k such that $m_{\mu }(T)\ne 0$.

Clearly, the notion of 1-additivity coincides with that of additivity. Moreover, a capacity that is k-additive $(k<n)$ turns out to be completely defined by the knowledge of $\sum _{l=1}^{k}\left( {\begin{array}{c}n\\ l\end{array}}\right)$.

In order to clarify the behaviour of the Choquet integral as an aggregation operator, we relied on two different measures:

• The Importance Index

• The Interaction Index

Here, we will define them more formally.

As we have seen, the overall importance of an attribute $i\in N$ can be captured by means of its Shapley value which is formally defined by:

$$\begin{aligned} \phi (\mu ,i):=\sum _{T\subseteq N\setminus i}\frac{(n-t-1)!t!}{n!}[\mu (T\cup i)-\mu (T)] \end{aligned}$$

where for each subset of attribute $S\subseteq N$, $\mu (S)$ can be interpreted as the importance os S in the decision problem. Therefore, the Shapley value can be defined as a weighted average value of the marginal contribution $\mu (T\cup i)-\mu (T)$ of element i alone in all combinations.

Moreover, in terms of its Möbius transform the Shapley value takes a really simple form:

$$\begin{aligned} \phi (\mu ,i)=\sum _{T\ni i}\frac{1}{t}m_{\mu }(T) \end{aligned}$$

For what concerns the interaction index, instead, following Murofushi and Soneda (1993), we can consider the interaction index for i and j as the average value of this marginal interaction. Therefore, setting

$$\begin{aligned} (\Delta _{ij}\mu ):=\mu (T\cup ij)-\mu (T\cup i)-\mu (T\cup j)+\mu (T) \end{aligned}$$

we have that the interaction index of attributes i and j related to $\mu$ is given by:

$$\begin{aligned} I(\mu ,ij):=\sum _{T\subseteq N\setminus ij}(\Delta _{ij}\mu )(T) \end{aligned}$$

Clearly, such index takes a negative value as long as i and j are positively correlated and, as a consequence, a positive value when i and j are negative correlated. Furthermore, $I(\mu ,ij)\in [-1,1]\ \forall i,\, j\subseteq N$. The value 1(respectively, -1) denotes maximum complementarity (substitutivity) between attributes i and j.

Appendix 2

See Tables 10 and 11.

Table 10 Choquet integral for all actual science parks

Table 11 Choquet integral for all actual science parks