The DTREE Procedure

Example 7.2 Oil Wildcatter’s Problem in Risk-Averse Setting

Continuing with the oil wildcatter’s problem, suppose that in addition to possibly buying insurance to spread the risk of the venture, the wildcatter is considering sharing the risk by selling a portion of this venture to other investors. Now, the decision he faces is whether to buy insurance or not and what percentage of the investment to divest. Again, assume that the wildcatter is risk averse with a risk tolerance of $1,200,000. Notice that in the program that follows the 'Divestment' decision includes possibilities of no divestment to 100% divestment in 10% increments.

/* -- create the STAGEIN= data set                 -- */
data Dtoils4;
format _STNAME_ $12. _OUTCOM_ $15. _SUCCES_ $12.;
input _STNAME_ $ _STTYPE_ $ _OUTCOM_ $
    _SUCCES_ $ ;
datalines;
Divestment      Decision    No_Divestment      Insurance
.               .           10%_Divestment     Insurance
.               .           20%_Divestment     Insurance
.               .           30%_Divestment     Insurance
.               .           40%_Divestment     Insurance
.               .           50%_Divestment     Insurance
.               .           60%_Divestment     Insurance
.               .           70%_Divestment     Insurance
.               .           80%_Divestment     Insurance
.               .           90%_Divestment     Insurance
.               .           100%_Divestment    .
Insurance       Decision    Buy_Insurance      Cost
.               .           Do_Not_Buy         Cost
Cost            Chance      Low                Oil_Deposit
.               .           Fair               Oil_Deposit
.               .           High               Oil_Deposit
Oil_Deposit     Chance      Dry                .
.               .           Wet                .
.               .           Soaking            .
;

The probabilities associated with the uncertain events are given in the PROBIN= data set named Dtoilp4. Except for the order of the variables in this data set, it is the same as the Dtoilp1 data set used in the section Introductory Example.

/* -- create the PROBIN= data set                         -- */
data Dtoilp4;
input _EVENT1 $ _PROB1  _EVENT3 $ _PROB3 ;
datalines;
Low         0.2     Dry         0.5
Fair        0.6     Wet         0.3
High        0.2     Soaking     0.2
;

/* -- create the PAYOFFS= data set                        -- */
data Dtoilu4(drop=i j k l);
length _STATE1-_STATE4 $16. ;
format _VALUE_ dollar12.0;

/* define and initialize arrays */
array DIVEST{11}  $16. _TEMPORARY_ ('No_Divestment',
                                    '10%_Divestment',
                                    '20%_Divestment',
                                    '30%_Divestment',
                                    '40%_Divestment',
                                    '50%_Divestment',
                                    '60%_Divestment',
                                    '70%_Divestment',
                                    '80%_Divestment',
                                    '90%_Divestment',
                                    '100%_Divestment'  );
array INSUR{3} $16.    _TEMPORARY_ ('Do_Not_Buy',
                                    'Buy_Insurance',
                                    ' '                );
array COST{4} $        _TEMPORARY_ ('Low',
                                    'Fair',
                                    'High',
                                    ' '                );
array DEPOSIT{4} $     _TEMPORARY_ ('Dry',
                                    'Wet',
                                    'Soaking',
                                    ' '                 );

do i=1 to 10;            /* loop for each divestment */
   _STATE1=DIVEST{i};

   /*
    * determine the percentage of ownership retained
    * for this scenario
    */
   PCT=1.0-((i-1)*0.1);

   do j=1 to 2;       /* loop for insurance decision */
      _STATE2=INSUR{j};

      /*
       * determine the premium need to pay for this
       * scenario
       */
      if _STATE2='Buy_Insurance' then PREMIUM=130000;
      else                            PREMIUM=0;

      do k=1 to 3;        /* loop for each well cost */
         _STATE3=COST{k};

         /* determine the cost for this scenario */
         if      _STATE3='Low' then  _COST_=150000;
         else if _STATE3='Fair' then _COST_=300000;
         else                        _COST_=500000;

         do l=1 to 3;  /* loop for each deposit type */
            _STATE4=DEPOSIT{l};

            /*
             * determine the oil deposit and the
             * corresponding net payoff for this scenario
             */
            if      _STATE4='Dry' then _PAYOFF_=0;
            else if _STATE4='Wet' then _PAYOFF_=700000;
            else                       _PAYOFF_=1200000;

            /* determine redeem received for this scenario */
            if _STATE2='Buy_Insurance' and _STATE4='Dry' then
                 REDEEM=200000;
            else REDEEM=0;

            /* calculate the net return for this scenario */
            _VALUE_=(_PAYOFF_-_COST_-PREMIUM+REDEEM)*PCT;

            /* drop unneeded variables */
            drop _COST_ _PAYOFF_ PREMIUM REDEEM PCT;

            /* output this record */
            output;
         end;
      end;
   end;
end;

/* output an observation for the scenario 100%_Divestment */
_STATE1=DIVEST{11};
_STATE2=INSUR{3};
_STATE3=COST{4};
_STATE4=DEPOSIT{4};
_VALUE_=0;
output;

run;

The Dtoilu4 data set for this problem, which contains 181 observations and 5 variables, is displayed in Output 7.2.1.

Output 7.2.1: Payoffs of the Oil Wildcatter’s Problem with Risk Sharing

Oil Wildcatter's Problem

The Payoffs

Obs	_STATE1	_STATE2	_STATE3	_STATE4	_VALUE_
1	No_Divestment	Do_Not_Buy	Low	Dry	$-150,000
2	No_Divestment	Do_Not_Buy	Low	Wet	$550,000
3	No_Divestment	Do_Not_Buy	Low	Soaking	$1,050,000
4	No_Divestment	Do_Not_Buy	Fair	Dry	$-300,000
5	No_Divestment	Do_Not_Buy	Fair	Wet	$400,000
6	No_Divestment	Do_Not_Buy	Fair	Soaking	$900,000
7	No_Divestment	Do_Not_Buy	High	Dry	$-500,000
8	No_Divestment	Do_Not_Buy	High	Wet	$200,000
9	No_Divestment	Do_Not_Buy	High	Soaking	$700,000
10	No_Divestment	Buy_Insurance	Low	Dry	$-80,000
11	No_Divestment	Buy_Insurance	Low	Wet	$420,000
12	No_Divestment	Buy_Insurance	Low	Soaking	$920,000
13	No_Divestment	Buy_Insurance	Fair	Dry	$-230,000
14	No_Divestment	Buy_Insurance	Fair	Wet	$270,000
15	No_Divestment	Buy_Insurance	Fair	Soaking	$770,000
16	No_Divestment	Buy_Insurance	High	Dry	$-430,000
17	No_Divestment	Buy_Insurance	High	Wet	$70,000
18	No_Divestment	Buy_Insurance	High	Soaking	$570,000
19	10%_Divestment	Do_Not_Buy	Low	Dry	$-135,000
20	10%_Divestment	Do_Not_Buy	Low	Wet	$495,000
21	10%_Divestment	Do_Not_Buy	Low	Soaking	$945,000
22	10%_Divestment	Do_Not_Buy	Fair	Dry	$-270,000
23	10%_Divestment	Do_Not_Buy	Fair	Wet	$360,000
24	10%_Divestment	Do_Not_Buy	Fair	Soaking	$810,000
25	10%_Divestment	Do_Not_Buy	High	Dry	$-450,000
26	10%_Divestment	Do_Not_Buy	High	Wet	$180,000
27	10%_Divestment	Do_Not_Buy	High	Soaking	$630,000
28	10%_Divestment	Buy_Insurance	Low	Dry	$-72,000
29	10%_Divestment	Buy_Insurance	Low	Wet	$378,000
30	10%_Divestment	Buy_Insurance	Low	Soaking	$828,000
31	10%_Divestment	Buy_Insurance	Fair	Dry	$-207,000
32	10%_Divestment	Buy_Insurance	Fair	Wet	$243,000
33	10%_Divestment	Buy_Insurance	Fair	Soaking	$693,000
34	10%_Divestment	Buy_Insurance	High	Dry	$-387,000
35	10%_Divestment	Buy_Insurance	High	Wet	$63,000
36	10%_Divestment	Buy_Insurance	High	Soaking	$513,000
37	20%_Divestment	Do_Not_Buy	Low	Dry	$-120,000
38	20%_Divestment	Do_Not_Buy	Low	Wet	$440,000
39	20%_Divestment	Do_Not_Buy	Low	Soaking	$840,000
40	20%_Divestment	Do_Not_Buy	Fair	Dry	$-240,000
41	20%_Divestment	Do_Not_Buy	Fair	Wet	$320,000
42	20%_Divestment	Do_Not_Buy	Fair	Soaking	$720,000
43	20%_Divestment	Do_Not_Buy	High	Dry	$-400,000
44	20%_Divestment	Do_Not_Buy	High	Wet	$160,000
45	20%_Divestment	Do_Not_Buy	High	Soaking	$560,000
46	20%_Divestment	Buy_Insurance	Low	Dry	$-64,000
47	20%_Divestment	Buy_Insurance	Low	Wet	$336,000
48	20%_Divestment	Buy_Insurance	Low	Soaking	$736,000
49	20%_Divestment	Buy_Insurance	Fair	Dry	$-184,000
50	20%_Divestment	Buy_Insurance	Fair	Wet	$216,000
51	20%_Divestment	Buy_Insurance	Fair	Soaking	$616,000
52	20%_Divestment	Buy_Insurance	High	Dry	$-344,000
53	20%_Divestment	Buy_Insurance	High	Wet	$56,000
54	20%_Divestment	Buy_Insurance	High	Soaking	$456,000
55	30%_Divestment	Do_Not_Buy	Low	Dry	$-105,000
56	30%_Divestment	Do_Not_Buy	Low	Wet	$385,000
57	30%_Divestment	Do_Not_Buy	Low	Soaking	$735,000
58	30%_Divestment	Do_Not_Buy	Fair	Dry	$-210,000
59	30%_Divestment	Do_Not_Buy	Fair	Wet	$280,000
60	30%_Divestment	Do_Not_Buy	Fair	Soaking	$630,000
61	30%_Divestment	Do_Not_Buy	High	Dry	$-350,000
62	30%_Divestment	Do_Not_Buy	High	Wet	$140,000
63	30%_Divestment	Do_Not_Buy	High	Soaking	$490,000
64	30%_Divestment	Buy_Insurance	Low	Dry	$-56,000
65	30%_Divestment	Buy_Insurance	Low	Wet	$294,000
66	30%_Divestment	Buy_Insurance	Low	Soaking	$644,000
67	30%_Divestment	Buy_Insurance	Fair	Dry	$-161,000
68	30%_Divestment	Buy_Insurance	Fair	Wet	$189,000
69	30%_Divestment	Buy_Insurance	Fair	Soaking	$539,000
70	30%_Divestment	Buy_Insurance	High	Dry	$-301,000
71	30%_Divestment	Buy_Insurance	High	Wet	$49,000
72	30%_Divestment	Buy_Insurance	High	Soaking	$399,000
73	40%_Divestment	Do_Not_Buy	Low	Dry	$-90,000
74	40%_Divestment	Do_Not_Buy	Low	Wet	$330,000
75	40%_Divestment	Do_Not_Buy	Low	Soaking	$630,000
76	40%_Divestment	Do_Not_Buy	Fair	Dry	$-180,000
77	40%_Divestment	Do_Not_Buy	Fair	Wet	$240,000
78	40%_Divestment	Do_Not_Buy	Fair	Soaking	$540,000
79	40%_Divestment	Do_Not_Buy	High	Dry	$-300,000
80	40%_Divestment	Do_Not_Buy	High	Wet	$120,000
81	40%_Divestment	Do_Not_Buy	High	Soaking	$420,000
82	40%_Divestment	Buy_Insurance	Low	Dry	$-48,000
83	40%_Divestment	Buy_Insurance	Low	Wet	$252,000
84	40%_Divestment	Buy_Insurance	Low	Soaking	$552,000
85	40%_Divestment	Buy_Insurance	Fair	Dry	$-138,000
86	40%_Divestment	Buy_Insurance	Fair	Wet	$162,000
87	40%_Divestment	Buy_Insurance	Fair	Soaking	$462,000
88	40%_Divestment	Buy_Insurance	High	Dry	$-258,000
89	40%_Divestment	Buy_Insurance	High	Wet	$42,000
90	40%_Divestment	Buy_Insurance	High	Soaking	$342,000
91	50%_Divestment	Do_Not_Buy	Low	Dry	$-75,000
92	50%_Divestment	Do_Not_Buy	Low	Wet	$275,000
93	50%_Divestment	Do_Not_Buy	Low	Soaking	$525,000
94	50%_Divestment	Do_Not_Buy	Fair	Dry	$-150,000
95	50%_Divestment	Do_Not_Buy	Fair	Wet	$200,000
96	50%_Divestment	Do_Not_Buy	Fair	Soaking	$450,000
97	50%_Divestment	Do_Not_Buy	High	Dry	$-250,000
98	50%_Divestment	Do_Not_Buy	High	Wet	$100,000
99	50%_Divestment	Do_Not_Buy	High	Soaking	$350,000
100	50%_Divestment	Buy_Insurance	Low	Dry	$-40,000
101	50%_Divestment	Buy_Insurance	Low	Wet	$210,000
102	50%_Divestment	Buy_Insurance	Low	Soaking	$460,000
103	50%_Divestment	Buy_Insurance	Fair	Dry	$-115,000
104	50%_Divestment	Buy_Insurance	Fair	Wet	$135,000
105	50%_Divestment	Buy_Insurance	Fair	Soaking	$385,000
106	50%_Divestment	Buy_Insurance	High	Dry	$-215,000
107	50%_Divestment	Buy_Insurance	High	Wet	$35,000
108	50%_Divestment	Buy_Insurance	High	Soaking	$285,000
109	60%_Divestment	Do_Not_Buy	Low	Dry	$-60,000
110	60%_Divestment	Do_Not_Buy	Low	Wet	$220,000
111	60%_Divestment	Do_Not_Buy	Low	Soaking	$420,000
112	60%_Divestment	Do_Not_Buy	Fair	Dry	$-120,000
113	60%_Divestment	Do_Not_Buy	Fair	Wet	$160,000
114	60%_Divestment	Do_Not_Buy	Fair	Soaking	$360,000
115	60%_Divestment	Do_Not_Buy	High	Dry	$-200,000
116	60%_Divestment	Do_Not_Buy	High	Wet	$80,000
117	60%_Divestment	Do_Not_Buy	High	Soaking	$280,000
118	60%_Divestment	Buy_Insurance	Low	Dry	$-32,000
119	60%_Divestment	Buy_Insurance	Low	Wet	$168,000
120	60%_Divestment	Buy_Insurance	Low	Soaking	$368,000
121	60%_Divestment	Buy_Insurance	Fair	Dry	$-92,000
122	60%_Divestment	Buy_Insurance	Fair	Wet	$108,000
123	60%_Divestment	Buy_Insurance	Fair	Soaking	$308,000
124	60%_Divestment	Buy_Insurance	High	Dry	$-172,000
125	60%_Divestment	Buy_Insurance	High	Wet	$28,000
126	60%_Divestment	Buy_Insurance	High	Soaking	$228,000
127	70%_Divestment	Do_Not_Buy	Low	Dry	$-45,000
128	70%_Divestment	Do_Not_Buy	Low	Wet	$165,000
129	70%_Divestment	Do_Not_Buy	Low	Soaking	$315,000
130	70%_Divestment	Do_Not_Buy	Fair	Dry	$-90,000
131	70%_Divestment	Do_Not_Buy	Fair	Wet	$120,000
132	70%_Divestment	Do_Not_Buy	Fair	Soaking	$270,000
133	70%_Divestment	Do_Not_Buy	High	Dry	$-150,000
134	70%_Divestment	Do_Not_Buy	High	Wet	$60,000
135	70%_Divestment	Do_Not_Buy	High	Soaking	$210,000
136	70%_Divestment	Buy_Insurance	Low	Dry	$-24,000
137	70%_Divestment	Buy_Insurance	Low	Wet	$126,000
138	70%_Divestment	Buy_Insurance	Low	Soaking	$276,000
139	70%_Divestment	Buy_Insurance	Fair	Dry	$-69,000
140	70%_Divestment	Buy_Insurance	Fair	Wet	$81,000
141	70%_Divestment	Buy_Insurance	Fair	Soaking	$231,000
142	70%_Divestment	Buy_Insurance	High	Dry	$-129,000
143	70%_Divestment	Buy_Insurance	High	Wet	$21,000
144	70%_Divestment	Buy_Insurance	High	Soaking	$171,000
145	80%_Divestment	Do_Not_Buy	Low	Dry	$-30,000
146	80%_Divestment	Do_Not_Buy	Low	Wet	$110,000
147	80%_Divestment	Do_Not_Buy	Low	Soaking	$210,000
148	80%_Divestment	Do_Not_Buy	Fair	Dry	$-60,000
149	80%_Divestment	Do_Not_Buy	Fair	Wet	$80,000
150	80%_Divestment	Do_Not_Buy	Fair	Soaking	$180,000
151	80%_Divestment	Do_Not_Buy	High	Dry	$-100,000
152	80%_Divestment	Do_Not_Buy	High	Wet	$40,000
153	80%_Divestment	Do_Not_Buy	High	Soaking	$140,000
154	80%_Divestment	Buy_Insurance	Low	Dry	$-16,000
155	80%_Divestment	Buy_Insurance	Low	Wet	$84,000
156	80%_Divestment	Buy_Insurance	Low	Soaking	$184,000
157	80%_Divestment	Buy_Insurance	Fair	Dry	$-46,000
158	80%_Divestment	Buy_Insurance	Fair	Wet	$54,000
159	80%_Divestment	Buy_Insurance	Fair	Soaking	$154,000
160	80%_Divestment	Buy_Insurance	High	Dry	$-86,000
161	80%_Divestment	Buy_Insurance	High	Wet	$14,000
162	80%_Divestment	Buy_Insurance	High	Soaking	$114,000
163	90%_Divestment	Do_Not_Buy	Low	Dry	$-15,000
164	90%_Divestment	Do_Not_Buy	Low	Wet	$55,000
165	90%_Divestment	Do_Not_Buy	Low	Soaking	$105,000
166	90%_Divestment	Do_Not_Buy	Fair	Dry	$-30,000
167	90%_Divestment	Do_Not_Buy	Fair	Wet	$40,000
168	90%_Divestment	Do_Not_Buy	Fair	Soaking	$90,000
169	90%_Divestment	Do_Not_Buy	High	Dry	$-50,000
170	90%_Divestment	Do_Not_Buy	High	Wet	$20,000
171	90%_Divestment	Do_Not_Buy	High	Soaking	$70,000
172	90%_Divestment	Buy_Insurance	Low	Dry	$-8,000
173	90%_Divestment	Buy_Insurance	Low	Wet	$42,000
174	90%_Divestment	Buy_Insurance	Low	Soaking	$92,000
175	90%_Divestment	Buy_Insurance	Fair	Dry	$-23,000
176	90%_Divestment	Buy_Insurance	Fair	Wet	$27,000
177	90%_Divestment	Buy_Insurance	Fair	Soaking	$77,000
178	90%_Divestment	Buy_Insurance	High	Dry	$-43,000
179	90%_Divestment	Buy_Insurance	High	Wet	$7,000
180	90%_Divestment	Buy_Insurance	High	Soaking	$57,000
181	100%_Divestment				$0

The optimal decisions for this problem can be identified by invoking PROC DTREE and using the SUMMARY statement as follows:

title "Oil Wildcatter's Problem";
proc dtree stagein=Dtoils4
           probin=Dtoilp4
           payoffs=Dtoilu4
           criterion=maxce rt=1200000
           nowarning;
   evaluate;
   summary / target=Divestment;
   summary / target=Insurance;
quit;

The optimal decision summaries in Output 7.2.2 and Output 7.2.3 show the optimal strategy for the wildcatter.

The wildcatter should sell 30% of his investment to other companies and reject the insurance policy offered to him.
The insurance policy should be accepted only if the decision to not divest is made.
If the decision to buy the insurance policy is made, then it is optimal to divest 10% of the venture.

Output 7.2.2: Summary of the Oil Wildcatter’s Problem for DIVESTMENT

Oil Wildcatter's Problem

The DTREE Procedure

Optimal Decision Summary

Order of Stages
Stage	Type
Divestment	Decision
Insurance	Decision
Cost	Chance
Oil_Deposit	Chance
_ENDST_	End

Decision Parameters
Decision Criterion:	Maximize Certain Equivalent Value (MAXCE)
Risk Tolerance:	$1,200,000
Optimal Decision Yields:	$50,104

Optimal Decision Policy
Up to Stage Divestment
Alternatives or Outcomes	Cumulative Reward	Evaluating Value
No_Divestment		$45,728
10%_Divestment		$48,021
20%_Divestment		$49,907
30%_Divestment		$50,104*
40%_Divestment		$48,558
50%_Divestment		$45,219
60%_Divestment		$40,036
70%_Divestment		$32,965
80%_Divestment		$23,961
90%_Divestment		$12,985
100%_Divestment		$0

Output 7.2.3: Summary of the Oil Wildcatter’s Problem for INSURANCE

Oil Wildcatter's Problem

The DTREE Procedure

Optimal Decision Summary

Order of Stages
Stage	Type
Divestment	Decision
Insurance	Decision
Cost	Chance
Oil_Deposit	Chance
_ENDST_	End

Decision Parameters
Decision Criterion:	Maximize Certain Equivalent Value (MAXCE)
Risk Tolerance:	$1,200,000
Optimal Decision Yields:	$50,104

Optimal Decision Policy
Up to Stage Insurance
Alternatives or Outcomes		Cumulative Reward	Evaluating Value
No_Divestment	Buy_Insurance		$45,728*
No_Divestment	Do_Not_Buy		$44,499
10%_Divestment	Buy_Insurance		$46,552
10%_Divestment	Do_Not_Buy		$48,021*
20%_Divestment	Buy_Insurance		$46,257
20%_Divestment	Do_Not_Buy		$49,907*
30%_Divestment	Buy_Insurance		$44,812
30%_Divestment	Do_Not_Buy		$50,104*
40%_Divestment	Buy_Insurance		$42,186
40%_Divestment	Do_Not_Buy		$48,558*
50%_Divestment	Buy_Insurance		$38,350
50%_Divestment	Do_Not_Buy		$45,219*
60%_Divestment	Buy_Insurance		$33,273
60%_Divestment	Do_Not_Buy		$40,036*
70%_Divestment	Buy_Insurance		$26,927
70%_Divestment	Do_Not_Buy		$32,965*
80%_Divestment	Buy_Insurance		$19,284
80%_Divestment	Do_Not_Buy		$23,961*
90%_Divestment	Buy_Insurance		$10,317
90%_Divestment	Do_Not_Buy		$12,985*

This information can be illustrated graphically using the GPLOT procedure. Output 7.2.4, produced by the PROC GPLOT statements shown in the following code, provides a clear picture of the effects of the divestment possibilities and the insurance options.

   /*  create a data set for the return corresponds to each */
   /*  divestment possibilities and the insurance options   */
data Data2g;
   input  INSURE DIVEST VALUE;
   datalines;
       1       0   45728
       0       0   44499
       1      10   46552
       0      10   48021
       1      20   46257
       0      20   49907
       1      30   44812
       0      30   50104
       1      40   42186
       0      40   48558
       1      50   38350
       0      50   45219
       1      60   33273
       0      60   40036
       1      70   26927
       0      70   32965
       1      80   19284
       0      80   23961
       1      90   10317
       0      90   12985
       1     100       0
       0     100       0
;

   /* -- define a format for INSURE variable             -- */
proc format;
   value sample 0='Do_Not_Buy' 1='Buy_Insurance';
run;

   /* -- define title                                    -- */
title h=3 "Oil Wildcatter's Problem";
     
   /* define legend                                      -- */
legend1 frame cframe=white label=none
        cborder=black position=center ;

   /* define symbol characteristics of the data points      */
   /* and the interpolation line for returns vs divestment  */
   /* when INSURE=0                                         */

   /* define symbol characteristics of the data points      */
   /* and the interpolation line for returns vs divestment  */
   /*  when INSURE=1                                        */

   /* -- define axis characteristics                     -- */
axis1 minor=none label=('Divestment (in percentage)');
axis2 minor=none label=(angle=90 rotate=0 'Certainty Equivalent');

   /* set graphics options                                  */
goptions htext=1.5;

   /* plot VALUE vs DIVEST using INSURE as third variable   */
proc gplot data=Data2g ;
   plot VALUE*DIVEST=INSURE / haxis=axis1
                              vaxis=axis2
                              legend=legend1
                              name="dt2"
                              frame 
                              cframe=white ;
   format INSURE SAMPLE.;
run;

quit;

Note that the data input into the Data2g data set is obtained from the optimal decision summary as in Output 7.2.3. The value 1 of the INSURE variable represents the alternative 'Buy_Insurance' and the value 0 represents the alternative 'Do_Not_Buy'.

Output 7.2.4: Returns of the Oil Wildcatter’s Problem