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Decision making under uncertainty

Outlook

•  Payo matrix

•  Max-min method (pessimist)

•  Max-max method (op+mist)

•  Mixed op+mist-pessimist method (Hurwitz criterion)

•  Minimizing regret

•  Cases

Payo Matrix

Alternatives,		States of environment (hypotheses),
Ai, i=1,…,I			Zh, h=1,…,H
	Z1		Z2	…	Zh
A1	e11		e12	…	e1H
A2	e21		e22	…	e2H
…	…		…	eih	…
AI	eI1		eI2	…	eIH

Each poten+al ac+on (alterna+ve) Ai relates to several possible states of nature Z.

eih – expected payo (u7lity, performance, total score) for (of) the alterna+ve Ai and the occured hypothesis Zh.

eih = wAi (Zh )

Payo Matrix (Example)

Expected revenues for investments in the commodoty markets A1, A2 and A3, thousand $

				Political situation (P)

	stable, p		unstable, p		stable,		p	unstable, p
Alternative
Alternative				Competition grade (Q)
Markets,
Markets,	weak,	q	strong, q		strong,		q	weak, q
Ai	weak,	q	strong, q		strong,		q	weak, q
				Hypotheses, Zh

	Z1	pq	Z2	pq	Z3	pq		Z4 pq
A1	530			460		240		220

A2	490			390		300		270

A3	575			420		260		190

P = { p, p}, where	Q = {q, q}, where
p − stable,	q − strong,
p − unstable.	q − weak.	43
		43

-Min Method

e(A*) =

• 

is a decision rule used in the

and

for maximizing the minimum beneﬁt.

• 

It selects the ac+on with the largest worst reward.

Political

stable,

unstable, p

weak,

strong,

min eij

240

460

220

260

420

190

270

A2 is the best

Case: Farming strategy (pessimist)

A farmer is planning corn, wheat and barley. His proﬁt depends on the weather. What to seed?

	Wet	Dry

Corn	$100	-$10

Wheat	$70	$40

Barley	$80	$35

Max-Max Method (op+mist strategy)

e(A*) = maxmaxeih

i h

Max-Max is used for maximizing the maximum beneﬁt. It selects the ac+on with the largest best reward.

				Political situation (P)

	stable,		p	unstable, p		stable,		p	unstable, p
Alternative				Competition grade (Q)
markets
markets	weak,		q	strong,	q	strong,		q	weak,	q
	weak,		q	strong,	q	strong,		q	weak,	q
					Hypothesis (Z)

	Z1	pq		Z2	pq	Z3	pq		Z4 pq
A1		240		460			530		220

A2		300		390			490		270

max eij

530

490

e( A*) 575

A3 is the best alternative !

Case: Farming strategy (op+mist)

A farmer is planning corn, wheat and barley. His proﬁt depends on the weather. What to seed?

	Wet	Dry

Corn	$100	-$10

Wheat	$70	$40

Barley	$80	$35

Mixed Op+mist-Pessimist Method

(Hurwitz criterion)

e( A*) = α max maxeih	+ (1−α) max min eih	= max[α maxeih		+ (1−α) mineih ]
i h	i	i	h	h

•  This criterion is a combina+on of the Max-Min and Max-Max rules.

•  Hurwitz proposed a weigh+ng coe cient „alpha“ selected from the interval from 0 to 1

– the coe cient of op+mism.

•  This criterion stands for an intermediate posi+on between the pessimism and op+mism

Political

(P)

stable,

unstable, p

stable,

unstable, p

Alternative

Competition grade (Q)

markets

weak,

strong,

weak,

Hypothesis (Z)

Z3 pq

Z4 pq

min eij

max eij

j220

j 530

240

460

220

300

390

270

490

260

420

190

575

Hurwitz scores for α = 0.5 :		If	α = 0	=>	pessimism
f(A1) = 0.5530 + (1-0.5)220 =	375	If	α = 1	=>	optimism
f(A2) = 0.5490 + (1-0.5)270 =	380
f(A3) = 0.5575 + (1-0.5)190 =	382.5	e( A*)

A3 is the best alternative !

Mixed Op+mist-Pessimist Method (Hurwitz criterion)

Hurwitz scores:

f ( A1) = α 530 + (1−α) 220; f ( A2) = α 490 + (1−α) 270; f ( A3) = α 575 + (1−α) 190;

		Alterna3ves
Alpha	A1		A2	A3
0	220		270	190
0,1	251		292	228,5
0,2	282		314	267
0,3	313		336	305,5
0,4	344		358	344
0,5	375		380	382,5
0,6	406		402	421
0,7	437		424	459,5
0,8	468		446	498
0,9	499		468	536,5
1	530		490	575

Sensi3vity of alterna3ves to α

		560
		510
		460
	Payos	410
	Payos	360
		360

310

260

210

160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Coe cient in Hurwitz must be selected in acoordance with the DM‘s considera+ons: the more dangeros the situa+on is, the safer the decision maker

wants to go, and the larger value of this coe cient he takes.

Minimizing Regret

This strategy is used if a player (DM) wants to decrease her/his loss

1)  Find the maximal u+lity (payo ) value for each state of nature:

eh * = max{eih} = max{eih		Zh }
2)  Set up loss matrix (regret matrix)	–	subtract the numbers in each column from the largest
number in that column:		Payo matrix

w(eih ) = eh * −eih ;

3)  Pick the ac+on with minimal regret:

e( A*) = min max w(eih ).

i h

Regret matrix

Alternative			Hypotheses (Z)
markets
markets	Z1	pq	Z2 pq	Z3	pq	Z4 pq
	Z1	pq	Z2 pq	Z3	pq	Z4 pq
A1		240	460		530	220

A2		300	390		490	270

A3		260	420		575	190

Max, eh *		300			575
Max, eh *		300	460		575	270

Alternative			Hypotheses (Z)
markets
				pq Z3 pq	Z4 pq
	Z1	pq	Z2				Max, w(e )
	Z1		Z2				Max, w(e )
							ih
A1		60	0	45	50	60

A2	0	70	85	0	85	A1 is the best alternative !
A3	40	40	0	80	80
						50

				Min, e( A*)	60
				Min, e( A*)	60

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