Proposal Method
5.4 Searching for Pareto-Optimal Solutions
5.4.4 Approximation to the Pareto frontier
The negotiating parties are not always satisfied with one candidate for an agree-ment, but want to explore all possible solutions and then negotiate over them. This is understandable, since different solutions divide the additional gain in different ratios between the parties. Therefore we will generate here an approximation to the whole Pareto-optimal frontier. The approximation is obtained by generating Pareto-optimal solutions starting from six different reference points that are con-vex combinations of the DMs’ optima on the decision set. In a two-DM case, this way of choosing the reference point guarantees the Pareto-optimality of solutions.
Numerical examples show that most of the solutions are Pareto-optimal even in the case where several parties are negotiating (see Section 3.3 in Heiskanen, 2001).
The reference points, solutions, and number of iterations required are shown in Table 5.5. The number of iterations needed varied from 8 to 14, the average being 10.83. At most of the generated solutions, sulfur emissions were consider-ably reduced at the Kola and Karelia region, the St. Petersburg region, and Estonia.
On the other hand, the nitrogen emission of Kola and Karelia was always almost at its highest level. The approximation for the Pareto-optimal frontier made here is, of course, very rough. However, we did not want to make a more accurate ap-proximation since in practice it is very difficult and laborious to obtain preference information from the DMs.
In Table 5.6 we show the values of the countries’ value functions at the refer-ence points and at the solution points given in Table 5.5. We also calculated the
106 Pirja Heiskanen Table 5.5. Reference points (ri), corresponding solutions (si) and iterations needed when an approximation to the Pareto frontier was generated.
SO2 NOx
FIN K&K STP EST FIN K&K STP EST Iterations
r1 124 284 78 105 122 52 101 44 14
s1 137 266 72 87 121 86 86 49
r2 108 331 88 105 107 60 117 44 8
s2 118 354 71 87 111 86 104 51
r3 93 379 98 105 91 69 133 44 12
s3 103 364 76 87 91 85 122 47
r4 108 284 78 122 107 52 101 51 10
s4 132 282 68 96 107 84 87 65
r5 93 331 88 122 91 60 117 51 10
s5 108 360 74 87 92 86 94 62
r6 93 284 78 140 91 52 101 59 11
s6 111 285 68 121 90 84 86 72
Table 5.6. The values of the countries’ value functions at the reference points (r) and at the solution points (s).
Gain Cost Gain Cost
k uk(r1) uk(s1) (%) (M euro/yr) uk(r2) uk(s2) (%) (M euro/yr)
FIN 1,349 1,340 67 196 1,463 1,444 42 304
EST 123 118 17 69 121 117 19 67
RUS 510 490 50 223 456 442 65 157
Gain Cost Gain Cost
k uk(r3) uk(s3) (%) (M euro/yr) uk(r4) uk(s4) (%) (M euro/yr)
FIN 1,596 1,568 12 492 1,445 1,426 46 294
EST 119 118 17 74 105 99 51 39
RUS 409 397 80 123 515 503 45 220
Gain Cost Gain Cost
k uk(r5) uk(s5) (%) (M euro/yr) uk(r6) uk(s6) (%) (M euro/yr)
FIN 1,578 1,549 17 465 1,561 1,554 16 473
EST 104 99 51 48 91 84 76 21
RUS 460 443 65 171 521 509 44 221
percentage of gain achieved by moving from the status-quo solution x¯to the solu-tions:= sicalculated by Equation (5.9) as well as the additional abatement costs (M euro/year) for reducing emissions from the status-quo level.
Table 5.6 shows that each country gains by moving from the status-quo solution to any of the solutions. Depending on the solution, the gain for Finland varies
Efficient Alternatives in a Transboundary Air Pollution Negotiation 107 between 12% and 67%, the gain for Estonia varies between 17% and 76%, and the gain for Russia varies between 44% and 80%. One can also see that Finland almost always pays most of the emission abatement costs, even though emissions are reduced much more elsewhere. This situation results from the high marginal costs for emission abatements in Finland, observable in Figure 5.1 and Figure 5.2.
5.5 Discussion
In this chapter we tested the constraint proposal method with realistic data. Both the model and the problem settings were real and the value functions were estimated on the basis of the data. We assumed the DMs’ answers to be rational and accurate, which, of course, may not be the case in the real world.
The numerical study indicates the success of the constraint proposal method in finding efficient agreements dominating the status-quo solution of the negotia-tion. All the solutions were individually rational and considerable joint gains were achieved by moving to the solution from the status quo. At most of the generated solutions, the sulfur emissions of the Kola and Karelia region, the St. Petersburg region, and Estonia were considerably reduced. On the other hand, the nitrogen emission of Kola and Karelia was at its highest level. This indicates that, starting from the status-quo solution, the reductions in sulfur emissions are more important than those of nitrogen. If the countries become more concerned about the envi-ronmental damage in neighboring countries, emissions in Russia should be further reduced.
These results are important, since one must always evaluate models for decision support carefully before approaching the true DMs. Preliminary tests help point out possible shortcomings and improve the method before the real DMs, whose time and expertise are valuable resources, are involved in the process.
Acknowledgments
I want to thank Professors Markku Kallio and Pekka Korhonen for their idea to study the transboundary air pollution problem as well as Professor Seppo Salo for his comments. I am also grateful to the personnel of the Transboundary Air Pol-lution project at IIASA for clarifying discussions and for providing me with the RAINS software. This work has been financially supported by grants from the Ella and Georg Ehrnrooth’s Foundation, the KAUTE Foundation, Yrjö Jahnsson Foundation, and the Foundation of the Helsinki School of Economics.
108 Pirja Heiskanen
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