Traditional BCC-DEA and Zero-Sum Gains DEA Methodology

DEA is a linear programming model that identifies an efficient frontier, which consists of efficient decision-making units (DMUs). Efficient DMUs are those units for which no other DMUs are able to generate at least the same amount of each output under given inputs (Charnes et al., 1978). The efficiency score reflects the ability of firms to generate the maximum outputs under a given level of inputs.

3.1.1.1 Traditional BCC-DEA Model

DMU_i represents the object unit that is attempting to maximise its output. All DMUs in the same year constitute the reference set used to construct the efficiency frontier for each DMU_i. The aim of the traditional DEA model is to make the less efficient object unit at least as efficient as the others by increasing its output. For each DMUi the efficiency score (i) is obtained from a measure of the ratio of all outputs over all inputs. Charnes et al. (1978) develop the constant-returns-to-scale (CRS) DEA model as below:

1

>0 represent input and output data for the j-th DMU with the ranges for j, k, and m indicated in (1); N is the number of DMUs; x_j^k is the amount of the k-th input consumed by the j-th DMU; y_j^m is the amount of the m-th output produced by the j-th DMU; and u_m and v_k are output and input weights assigned to the m-th output and the k-th input, respectively.

One problem with this above ratio form is that the number of solutions is infinite - e.g., if (u^*_m,v^*_k)is a solution, then (cu cv_m^*, ^*_k) is another solution, where c is a constant. In order to avoid this problem, an output-oriented DEA model, which is to achieve the efficient DMU by a radial expansion in outputs, can impose the constraint

1

Banker et al. (1984) extend the constant returns to scale (CRS) DEA model to a variable returns to scale (VRS) situation. The dual solution of the traditional output-oriented BCC-DEA model using duality expressed by Coelli et al. (2005) to

measure the efficiency scoreifor DMUiis shown as:

wherei depicts the inverse of the efficiency score of DMU_i; the efficiency scoreiof DMUi is 1/i; N is the number of DMUs; K and M are, respectively, the numbers of inputs and outputs; xjk

is the amount of the k-th input consumed by the j-th DMU; yjm

is the amount of the m-th output produced by the j-th DMU; and j is each efficient DMU’s individual share in the definition of the target for DMUi.

The BCC-DEA model here measures the firm-level efficiency score (i) in the securities industry. An SF (as a DMU in the DEA model) that is pursuing more market share naturally means that other SFs lose some market share, because the total market share is 100%. Accordingly, this constant sum of output is unable to use the traditional BCC-DEA model, in which the output of any given DMU is not influenced by the output of the others, to assess the efficiency score. This is our motivation for adopting the ZSG-DEA model to measure the efficiency scores of SFs.

3.1.1.2 Zero-Sum Gains DEA Model

The ZSG-DEA model assesses the efficiency score provided that the sum of outputs is constant. Lins et al. (2003) indicate that this is similar to a zero-sum game whereby how much is won by a player is lost by one or more of the other players.

The equal output reduction strategy is generated to measure the efficiency score

iyi



( 1

iR

iR

   ) for DMUi in equation (4) using duality expressed shown below and is

graphically represented using a simple case involving one input, x, and one output, y, in Figure 3-1:

where the term iR is the inverse of the efficiency score of the ZSG-DEA model with

iR  1; and the efficiency score iR of DMU_i is the inverse of iR (iR =1/iR) in the ZSG-DEA model. The termy_i^m(_iR-1), representing losses of the other DMUj (j ≠ i), must have one DMUito gain y_i^m(_iR -1) output units.

FIGURE 3-1. Graphical Representation of the Equal Output Reduction Method y

This model here causes some DMUs to have a negative output after replacing the output as the reduction coefficient. A simple example in Appendix A illustrates an unreasonable case in which an equal output reduction under a zero-sum game

generates a negative output. Hence, provided that

( -1) min ( ), 1,...,

m m

i iR j

y   y m M , this equal output reduction strategy can apply.

To avoid this major weakness, Lins et al. (2003) further develop the proportional output reduction strategy for any given DMUi using the ratio ( -1)

units, and the losses of the other DMUs are proportional to their levels of output.

The condition that the sum of the losses is equal to the gains of DMU_istill holds.

Figure 3-2 represents the ZSG-DEA frontier created by this proportional reduction strategy and the BCC-DEA frontier using a simple case involving one input and one output. DMU_igains y_i^m(_iR-1) output units, and the losses of other DMUs are proportional to their respective levels of output, which is ( -1)

( )

output yj of DMUj is larger than those of other DMUs, then the output reduction ( -1)

 is also larger than those of the others, and vice versa. Model (5)

substitutes model (4) for the proportional output reduction strategy in measuring the efficiency score ( 1

iR

iR

   ) of DMUias:

iyi

However, Lins et al. (2003) report that obtaining results based on this non-linear programming problem is very labour-consuming in particular because of the large number of variables. The model is thus simplified by having only a single output (m

=1). Appendix A provides an example to explain the computational steps of the proportional output reduction strategy.

FIGURE 3-2. Graphical Representation of the Proportional Output Reduction Method

The following theorem holds under a single output ZSG-DEA proportional reduction strategy:

LGSS Theorem (Lins et al., 2003). The target for a DMU to reach the efficiency y

frontier in a ZSG-DEA proportional output reduction strategy model equals the same target in the traditional BCC-DEA model multiplied by the reduction coefficient

(1-( -1)

Owing to this theorem, equation (6) below holds.

1

The efficiency score of the ZSG-DEA model is obtained from equation (7):

( - ) 2

In this research, due to the fact that the sum of the total market share in percentage terms is 100, Y is always 100 and equation (7) above can be expressed as equation (8): equals its value in the traditional BCC-DEA model. This ZSG-DEA model is then applied to measure the efficiency score of SFs when the market share in percentage terms always sums up to 100.