2. The descr iption of problem
3.4 The Group Aggr egation Utility Function
(
0≤uj xj ≤ , 0≤ pj≤1, 0≤uij(xij)≤1, and 0≤ pij≤1, E(u(x)') may be larger than 1.
Consequently, the utility must be normalized. Based on the utility theorem (Keeney and Raiffa, 1993), the utility normalization can be represented as Eq. (3).
j i x u p x
u x p
u p x
u p
x u p x
u p x
u ij ij
ij i i ij
ij i ij
ij i ij
ij i ij
ij ij ij
ij , max{ ( )} min{ ( )}, ,
)}
( { min )}
( { max
)}
( { min ) ( )
*( × ≠ × ∀
×
−
×
×
−
×
= (3)
Where u*(xij) is the normalized utility value.
Since 0≤u(xij)≤1, 0≤ pij ≤1, so 0≤uij*(xij)≤1, if max{ ( )} min{ ij ( ij)}
ij i i ij
x u p x
u
p × = × , then
0 )
*( ij = ij x
u . The normalized utility value ranges between 0 and 1. The special case for events with the single attribute outcome case is i=1.
3.3 Utility Function of Individual Negotiator s
Based on the definition of risk states and utility normalization as proposed above, the utility assessment model of negotiators can now be developed.
From the multi-attribute utility theory (Keeney and Raffia, 1993) and multi-attribute risk utility function (Seo, 1990); the utility assessment model related to multi-attribute outcome and the state expressed as Eq. (4) can be developed.
For negotiator q, uq(u*ij(xij))=u*ij(xij)<E(u*(x)) (4) Where u*ij(xij) is the normalized utility value of Eq. (3); uq(uij*(xij))is the utility value of a negotiator for a given event; and E(u*(x)) is the normalized expected utility value.
Because u*ij(xij) is the normalized utility value, the value of uq(u*ij(xij)) is between 0 and 1. However, if the value of uij*(xij) is less than E(u*(x)), the negotiator believes an event at state sj and attribute outcome xij involves some risk. Eq. (4) illustrates the risk assessment for an individual negotiator of a given event.
The above demonstrates that risk assessment and identification of an event is determined by all the negotiators. Additionally, incorporating the utility function of individuals into group aggregation utility function is important. Furthermore, the risk of an event can be assessed based on its utility.
3.4 The Group Aggr egation Utility Function
This section presents the utility function of attribute outcome for group negotiators of an event, as well as the group aggregation utility function. The risk assessment of an event is developed based on these.
(i). The Concept of Multi-Attr ibute Utility (MAU) Function
The multi-attribute utility theory proposed by Keeney and Raiffa (1973, 1993) had been broadly applied in decision-making (Seo and Sakawa, 1985), choice behavior (Tzeng,
et al, 1989) and other related research topics (Bosel et al., 1997). The MAU model is based on the utility theory, which can be divided into the additive and multiplicative models. The MAU model makes three assumptions, including (1) the total number of attributes should be no below two, (2) the preference of the decision-maker should be independent, and (3) the utility of the decision-maker is independent of their preference. The multi-attribute utility function can be expressed as Eq. (5).
) the event attribute; U(x) is the multi-attribute utility function; ki is the relative weighting value of attribute i, 0≤ki≤1; and k is the scale constant.
Eq. (5) is the generalized representation of MAU model, when ∑ =1 i
ki and the MAU is an additive model. Otherwise, ∑ ≠1
i ki , and the MAU is a multiplicative model.
(ii). The Multi-Attr ibute Utility Function of Negotiator s
As described in Section 2, the process of concession contract negotiation is a kind of group participation; which determines the primary and secondary event through group decision-making. This section, develops the utility function of group negotiators based upon the individual negotiator utility function. To assess the risk or non-risk state, the multi-attribute utility function of negotiators developed here is based on Eq. (5).
We assume there are Q negotiators in the BOT Company’s team, q=1,2,...,Q and the utility function of negotiator q is expressed as uq(uij*(xij)). Additionally, the utility, i.e., negotiator preference is assumed to be independent. The multi-attribute utility function of negotiators can then be expressed as follows.
))
Where GUq,ij denotes the utility value of the group negotiators for attribute outcome xij
at state sj of a given event and , j=1,2,..,n, i=1,2,...,m；while kq represents the relative weighting value of a negotiator, where 0≤kq≤1.
Since 0≤uq(uij*(xij))≤1 and 0≤kq≤1, 0≤GUq,ij ≤1. Eq. (6) is the mixed model for group negotiator utility. Let E(uq(uij*(xij))) be the expected utility value for all attributes of an event. When GUq,ij<E(uq(u*ij(xij))), we know that the negotiation group believes a risk exists for an event at state sj and attribute outcome xij. Otherwise, no risk exists for an event at state sj and attribute outcome xij. When negotiator utility, event attribute and state are independent, Eq. (6) is the additive model of MAU and the utility weighting value
kqof the negotiator can be solved by the weighting method, which was developed by Tzeng et al. (1989) and is shown in Eq. (7).
=
−
=
−
∑
+ + +
+ 1
) ) ( ( )
) ( ( )
) ( ( )
) (
( * * * ** 1 1 * * 1 1 * **
Q q
q
ij ij q q ij
ij q q ij
ij q q ij
ij q q
k
x u u k x
u u k x
u u k x
u u k
(7)
Where uq(uij*(xij)*)is the maximum value of the negotiator’s (q) utility for a given event at attribute outcomexij；uq(u*ij(xij)**) is the minimum value of the negotiator’s (q) utility for a given event at attribute outcomexij；uq+1(uij*(xij)*) is the maximum value of the negotiator’s (q+1) utility for a given event at attribute outcomexij；uq+1(uij*(xij)**) is the minimum value of the negotiator’s (q+1) utility for a given event at attribute outcomexijand kqis the relative weighting value of the negotiatorq.
If weighting value kqis incorporated into Eq. (6), the utility assessment of the negotiation group of a given event at each state can be obtained. This utility value represents mutual assessment result of the negotiation team for a specific state of an event.
Restated, the negotiation group can reach a consensus regarding the assessment of states of an event.
(iii). The Aggregation Utility Function of the Negotiation Group
This section develops the risk assessment model for the negotiation group of a given event; while the aggregation utility value of the negotiation group for the risk and non-risk states of an event is integrated by the concept of minimum distance in utility value between the risk and non risk states.
Section 3.4 (ii) provides the utility value of the negotiation group of an event at a given state, which the utility value can be distinguish the risk state or non-risk state.
Applying GUq,ij obtained from Eq. (6), which the value of GUq,ij can be ranked from 0 to 1. Let sj be the variable of the horizontal axial of a given event, and let GUq,ijbe the variable of the vertical axial. Based on the risk state defined herein, the utility value of the non-risk state exceeds that of the risk state. Ranking the state by GUq,ij from 0 to 1 distinguishes the risk state and non-risk state of a given event. By multiplying kq with the utility value of both states, the aggregation utility function of the negotiation group can be obtained. This utility value results from integrating all negotiators’ assessments of a given event at various states.
Following the assumptions made in section 3.1, GUq,ijcan be obtained through Eqs. (6) and (7). Based on the value of GUq,ij, the state of an event can be distinguished into risk state and non-risk state. Let GUUqu,ijbe the maximum utility value of the negotiation group of an event in the non-risk state; let GUUqλ,ij be the minimum utility value of the negotiation group for an event in the non-risk state; let GRUqu,ij be the maximum utility value of the negotiation group of an event in the risk state; and let GRUqλ,ij be the minimum utility value of the negotiation group of an event in the risk state. Finally, GU is the aggregation utility function of the negotiation group, which defined as Eq. (8). And figure 3 presents a conceptual diagram of this process.
) Wherekqis the utility weighting value of the negotiatorq.
Figure 3. Aggregating utility in risk state and non-risk state
Since 0≤GUUqu,ij ≤1，0≤GUUqλ,ij ≤1，0≤GRUqu,ij ≤1, and 0≤GRUqλ,ij ≤1，the GU value is between 0 and 1. The denominator of Eq. (8) is the utility range for the negotiation group of a given event. (GUUqu,ij−GUUqλ,ij) is the utility difference of the negotiation group of a given event in the non-risk state, while (GRUqu,ij −GRUqλ,ij) is the utility difference of the negotiation group of a given event in the risk state. However, utility value exists in both the risk and non-risk states of a given event. To integrate the utility value and consider utility weighting value of the negotiator at various states, the distance conception is applied to demonstrate the magnitude of the utility at various states.
The numerator of Eq. (8) is the distance differential between the non-risk and risk states, which shows the magnitude of the difference between two different states and falls between 0 and 1. This process is intended to integrate the assessment results of the negotiator toward utility at various states, and obtain the aggregation utility GU of the negotiation group of a given event. By integrating utility distance and utility weight, a single utility value GU, can be obtained, which represents the consensus of the negotiation group regarding the risk or non-risk state of a given event. When GU<E(U*), the negotiation group believes a given event is risky. Meanwhile, when GU≥E(U*), the event is considered a risky. E(U*) is the expected utility value of the negotiation group of a given event, ∑ ∑ ∑