Path’s Name

Path’s Set

$\left(w_1,w_2\right)$

$F_j^{\ast }\left(p,w\right)$

${\overline{F}}_j^{\ast }\left(p\right)$

ME

MME

E

I

P₁

1-2-8-16-20

(0.05, 0.95)

(16, 11)

1.3333

1.2222

0.3333

0.2222

0.3333

180

173

P₂

1-3-8-16-20

(0.10, 0.90)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

200

195

P₃

1-3-8-16-20

(0.15, 0.85)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

276

270

P₄

1-3-8-16-20

(0.20, 0.80)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

278

273

P₅

1-3-8-16-20

(0.25, 0.75)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

486

475

P₆

1-3-8-16-20

(0.30, 0.70)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

463

452

P₇

1-3-8-16-20

(0.35, 0.65)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

254

249

P₈

1-3-8-9-15-20

(0.40, 0.60)

(17, 10)

1.4167

1.1111

0.4167

0.1111

0.4167

271

262

P₉

1-3-8-16-20

(0.45, 0.65)

(17, 9)

1.4167

1.0000

0.4167

0.0000

0.4167

227

225

P₁₀

1-4-7-17-18-20

(0.50, 0.50)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

350

335

P₁₁

1-4-7-17-18-20

(0.55, 0.45)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

285

279

P₁₂

1-4-7-17-18-20

(0.60, 0.40)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

390

381

P₁₃

1-4-7-17-18-20

(0.65, 0.35)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

240

233

P₁₄

1-4-7-17-18-20

(0.70, 0.30)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.444

414

400

P₁₅

1-4-7-17-18-20

(0.75, 0.25)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

332

325

P₁₆

1-4-7-17-18-20

(0.80, 0.20)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

330

320

P₁₇

1-4-7-17-18-20

(0.85, 0.15)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

463

450

P₁₈

1-4-7-17-18-20

(0.90, 0.10)

(12, 13 )

1.0000

1.4444

0.0000

0.4444

0.4444

360

352

P₁₉

1-4-7-17-18-20

(0.95, 0.05)

(12, 13)

1.0000

1.4444

0.0000

0.4444

0.4444

170

163

The utopian distance and utopian social cost are $f_1^{\ast }\left(p\right)=12$ and $f_2^{\ast }\left(p\right)=9$ respectively.