Decision Making for New Technology: A Multi-Actor, Multi-Objective Method

Abstract

Technology managers increasingly face problems of group decision. The scale and complexity of research, development, and alliance efforts in emerging fields of technology mandate a correspondingly sophisticated form of group coordination. Choices made include the selection of projects, the choice of investment alternatives, and the formation of technology licensing agreements. Multi-criteria decision analysis (MCDA) methods are often used to help decision makers in such situations. This chapter explores an approach closely related to MCDA, known as exchange modeling. Exchange modeling incorporates actor preferences, and assumptions about the play of the game, to better examine the resulting preferences of groups. The advantage of this method is that the results provide an improved prescription for strategy, given the constraints of preferences and the existing alliance structures. The model is motivated based upon the needs of technology managers in new, converging fields of technology. The model is formally analyzed using operations research techniques.

Reprinted from Technological Forecasting and Social Change, 76 (1), Cunnigham and van der Lei, Decision making for new technology: A multi-actor, multi-objective method, 26–38, (2009), with permission from Elsevier.

Appendices

Appendix A: Mathematical Derivations

The following derivations are due to Coleman [13–15]. A collection of n actors exercise control over m goods. This is expressed in a C matrix dimensioned n by m. Control by a given actor is normalized to 1.00 without loss of generality. Similarly, actors have interest in a set of goods which may not be the same goods over which they have control. This matrix is represented by X, and for convenience is transposed and therefore dimensioned m by n. We may further scale these matrices without loss of generality, so that the sum total of control across each actor sums to 1.00, and the sum of control of interest across each good also sums to 1.00.

The utility of each actor for receiving control over goods is expressed in the following equation, where utility is an n by 1 matrix (Eq. 9.1). For notational convenience we suspend the subscript on n actors, noting that the same optimization problem applies to each actor. Elements of the matrix are represented using subscripted, lower case letters (c and x respectively).

$$ U = \mathop \prod \limits_{i = 1}^{m} c_{i}^{{x_{i} }} $$

(9.1)

We hypothesize a final set of market values, determining the final exchange valuation of each good. This matrix, V is an m by 1 matrix. Actors have resources R which are proportional to the final valuation of their goods times their control. We may now cast the decision problem of the actors as follows:

$$ \begin{array}{*{20}l} {{\text{maximize }}({\text{U}}),} \\ {{\text{with respect to c}}} \\ {{\text{subject to R}} = {\text{cv}}} \\ \end{array}$$

(9.2)

All actors in the system maximize their utility with respect to their decision to exchange control with other actors. However they are subject to a budget, as they are limited to a sum total of exchanges which are equal to their resources. This is reflected in the budget constraint of Eq. (9.2).

The problem may be solved using a Lagrangian, as shown below in Eqs. (9.3a–c). The problem reduces to m equations (one for each actor), plus an additional equation to calculate the Lagrangian multiplier Eq. (9.3c).

$$ L = U + \lambda (r - c_{i} v_{i} ) $$

(9.3a)

$$ \frac{\partial L}{{\partial c_{i} }} = \frac{{ x_{i} }}{{ c_{i} }}U - \lambda v_{i} $$

(9.3b)

$$ \frac{\partial L}{\partial \lambda } = r - c_{i} v_{i} $$

(9.3c)

The optimization equation implies that, at equilibrium exchange, the ratio of the marginal utilities is equal to the ratio of going market rates for the good (v). Additional linear algebra calculations allows further derivation of the following equations. Equation (9.4a) shows how the stationary value of r is a function of actor control and interest, and Eq. (9.4b) shows the equivalent for market rates. Full derivations of these standing equations are available in Coleman [14].

$$ r = {\text{CXr}} $$

(9.4a)

$$ v = {\text{XCv}} $$

(9.4b)

Marsden further elaborated the model to include network constraints of trade, where the matrix A is an n by n matrix indicating the social structure of the exchange network. Trades permitted by the network structure of the model are indicated by a 1 in the matrix; trades not permitted by network structure are indicated by a 0. This model too has a potential Markov chain solution Eqs. (9.5a and b).

$$ r \, = {\text{ rA}} $$

(9.5a)

$$ r \, = {\text{ CXrA}} $$

(9.5b)

Also determined by these equations is the exchange rate (v) for goods in the political or economic exchange. This is the dual problem to determining individual actor resources. As noted earlier, this exchange rate is significant across a number of models of group-decision analysis. Worked example below provides additional mathematical details about the solution of these Eqs. (9.4a, b and 9.5a, b).

As a side note, it is interesting to note that Web search engines calculate the significance of any given page in terms of its “exchange” of hyperlinks with other significant pages on the Internet. This model, embodied in the Google search engine, is fundamentally similar to the Coleman and Marsden models [46].

Appendix B: Worked Example

Tables A.1 and A.2 show a hypothetical control matrix for 12 microelectronics firms. This corresponds to the C matrix in standard exchange models. Quantities in the table are normalized by column, so that for instance the sum total of “wireless expertise” is summed to 100 %. Quantities in this table might be estimated by research and development indicators (such as patenting).

Table A.1 Exchange participants

Table A.2 Control matrix (matrix C)

Table A.3 shows a hypothetical interest matrix for 12 microelectronics firms. This corresponds to the X matrix in standard exchange models. Note that the table, as shown, is transposed. Quantities in the table are normalized by row, so that for instance the sum total of interest of alliance partner “A” is summed to 100 %. Quantities in this table might be estimated through interviews, yearly reports, or industrial classification schemes. A traditional MCDA approach might also be incorporated here.

Table A.3 Interest matrix (matrix X, transposed)

The example in Sect. 9.4 was run both with complete access to partners, and with limited access to partners. Table A.4 shows a hypothetical alliance structure for this analysis.

Table A.4 Alliance structure

The matrix is normalized so that each alliance partner spends a proportional amount of trading with each of its peers. This is the matrix used for the exchange analysis (Table A.5).

Table A.5 Alliance structure (matrix A)

Calculations of stable exchange rates and actor resources proceeds as discussed previously in mathematical derivations above. The resultant eigenvalue problem may be solved using the power method. Since all three of the matrices (C, X, A) are interpretable as probabilities, the problem may be formulated as a Markov chain, and then solved for an equilibrium vectors using a linear system of equations. This is the approach used herein.

Rearranging the equations derived from the actor decision problem (Eq. 9.6a), we have the resultant linear system of Eq. (9.6b), subject to the constraint that the stable probability vector must sum to 1 (Eq. 9.6c).

$$ r \, = {\text{ CXr}} $$

(9.6a)

$$ \left( {I - {\text{CX}}} \right) \, r \, = 0 $$

(9.6b)

$$ \sum r \, = { 1} $$

(9.6c)