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Selecting the best baseball cards for investment
0895-7177192 $5.00 + 0.00 Copyright@ 1992 Pergamon Press Ltd
Mathl. Comput. Modelling Vol. 16, No. 10, pp. 135-142, 1992 Printed in Great Britain. All rights reserved
SELECTING
THE BEST BASEBALL
CARDS
FOR INVESTMENT
A. G. LADDE George Washington University School of Business and Public Management Washington, DC 20052, U.S.A. (Received
October
1991)
Abstract-Baseball cards can be analyzed for investment purposes like any other investments. There are four criteria that the investor must consider about each of the cards. This paper describes an investment model that utilizes a data base to store specific info-tion about the selected cards. Then, by using regression analysis and the Analytic Hierarchy Process (AHP), it is possible to select the card that has the best potential of increasing.
INTRODUCTION For the past several years, popular investments have included securities and real estate. Today, many have discovered the value of baseball cards, and Doescher notes that big-time investors can see the potential for appreciation [l]. Like other investments, in-depth analysis can be used to predict future growths of particular cards based on historical data. This analysis requires the use of the Analytic Hierarchy Process because of its ability to interpret qualitative data through ratio scales [2]. The overall goal of this problem is to select the baseball card (given a finite set of cards) that has the best potential for growth.
DATA
COLLECTION
To organize the information of the six chosen baseball cards, a data base is created using DBASE III PLUS ( see Figure 1). Each record corresponds to a different player, and each field designates a different aspect of the player and his respective rookie card. The cards in question are the rookie cards of the prospective players, but the cost of the rookie card refers to the price of the card at the end of year 1990. Since several companies manufacture baseball cards (the main ones being Topps, Score, Donruss and Upper Deck), the price used for this report is the average market price. The “NA” in a field means that those categories do not apply to the player. The last two fields MODELCST (model cost) and PREMIUM (premium) will be explained when we discuss the premium criterion. The RETRIEVE/LIST option gave a hard copy of the file, and the last two fields were inserted at later time than the other fields, using the MODIFY/DATABASE FILE option. The data were entered in these two fields using the UPDATE/EDIT option. STRUCTURING
THE
HIERARCHY
In order to achieve our objective, an Expert Choice model is developed, so that the AHP can be used in making the card selection. The AHP allows one “to elicit systematic judgements from unstructured information” [3]. The four criteria for making the selection are the year the player’s card entered the market, the premium (current cost minus model cost), the popularity of the player in the future and the player’s ability in the future. These criteria are entered beneath the goal node and the alternatives (the six players’ names) are entered beneath each criterion (see Figure 2). Typeset by A&-‘&X 135
A.G. LADDE
136 Record# 1 2 3 4 5 6
NAME Ben McDonald Juan Gonzales John Olerud Ellis Burks Benito Santiago Eric Lindros
TEAM Baltimore Orioles Texas Rangers Toronto Blue Jays Boston Red Sox San Diego Padres Toronto Blue Jays
Record#
NAME Ben McDonald Juan Gonzales John Olerud Ellis Burks Benito Santiago Eric Lindros
COST90 4.00 4.50 6.00 4.00 8.00 5.00
1 2 3 4 5 6
MODELCST 4.32 5.51 5.01 5.43 5.49 5.64
ROOXIECD 1990 1990 1990 1987 1986 1990
HR NA 17 0 12 16 -
RBI BA NA NA .150 0 375 61 :303 62 . 236 -
W 0 NA NA NA NA NA
K
EJ?J,
0 NA NA NA NA NA
0.00 NA NA NA NA NA
PREMUIM -0.32 -1.01 +0.89 -1.43 +2.51 -0.64
Figure 1. To select
the card
that has the best
L 0.0701I -MCDONALD L 0.166 -GONZALES L 0.166 -0LERUD L 0.166 -BUP.ES L 0.069 -SANTIAGO L 0.046 -LINDROS L 0.388
ABILITY BURES GONZALES LINDROS MCDONALD OLERUD POPULAR PREMIUM SANTIAGO YEARS
---------------------
-MCDONALD L 0.092 -GONZALES L 0.247 -0LERUD L 0.051 -BURKS L 0.457 -SANTIAGO L 0.026 -LINDROS L 0.127
potential
of increasing
L 0.633( -MCDONALD L 0.384 -GONZALES L 0.084 -0LERUD L 0.132 -BURES L 0.296 -SANTIAGO L 0.042 -LINDROS L 0.061
-MCDONALD L 0.091 -GONZALES L 0.267 -0LERUD L 0.059 -BURRS L 0.390 -SANTIAGO L 0.163 -LINDROS L 0.030
Ability of the player in the future Ellis Burks, Boston Red Sox (rookie year - 1987) Juan Gonzales, Texas Rangers (rookie year - 1990) Eric Lindros, Toronto Blue Jays (rookie year - not applicable) Ben Mcdonald, Baltimore Orioles (rookie year - 1990) John Olerud, Toronto Blue Jays (rookie year - 1990) Popularity of the player in the future Premium - the current cost minus the model cost Benito Santiago, San Diego Padres (rookie year - 1986) Number of years played Figure 2.
ANALYSIS
OF
THE
CRITERIA
Before performing pairwise comparisons (the foundations of the AHP) on the criteria, each criterion must be analyzed [4]. The first criterion, the year the player’s first card entered the baseball card market, can be understood by looking at the data base file (Figure 1). The particular field in question is the year of the player’s rookie card (ROOKIECD). For the most part, the listed year is also the player’s first full year in professional baseball. The only exceptions are Juan Gonzales, who played the majority of the 1990 season in the minor league’s, and Eric Lindros, a hot prospect who was still in high school. By not playing any major league baseball, that player’s card can increase due to speculation on his ability. Even though Lindros’ premier sport is hockey, his baseball card is valued highly. Generally, the more years played, the less likely the player’s card will increase significantly.
Best baseball JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURXS SANTIAGO LINDROS
GONZALES 5.0
AND
PRIORITIES GOAL
GONZALES
OLERUD BURES SANTIAGO LINDROS
WITH
137 RESPECT
BURXS 6.0 3.0 4.0
OLERUD (Z)
Matrix entry indicates that ROW element 5 STRONGLY 3 MODERATELY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD
cards TO
SANTIAGO 3.0 1.0 3.0 1.0
LINDROS 5.0 2.0 ii:!; .
is 7wRY
STRONGLY
9 EXTREMELY
:Ben McDonald, Baltimore Orioles (rookie Year - 1990) :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.119 GONZALES 0.215 OLERUD 0.056 BURKS 0.083 SANTIAGO 0.094 LINDROS INCONSISTENCY
RATIO
= 0.046
Figure 3. Next, the premium criterion requires that we first calculate the model cost and then subtract it from the actual (1990) cost. In doing this, we use a regression equation to estimate what a card should be selling for, given past relationships, and then we compare this value to the 1990 selling price. The difference between these two costs is the premium. The following regression equation allows us to find the model cost of each player.
Y = bo + blzl + bzx2,
where, bo = the y-intercept (value of y when x1 = 0 and 22 = 0); bl = the coefficient of the player’s proven ability priority value; b2 = the coefficient of the player’s popularity priority value. Data for the model are derived from each player’s proven ability and popularity. Priorities for the cards are determined by separate Expert Choice models. One goal is to evaluate the baseball player’s popularity. The six alternatives are the players. Since no criteria exist, the alternatives are compared based on preference, because relative worth is more appropriate than absolute worth “since economic value is a relative matter” [5]. For example, with regard to the goal, the question is how preferred is Ben McDonald to each of the other players. The relative preference (or importance) is based on a scale from one (equally preferred (important)) to nine (extremely preferred (important)). The same scale was utilized by Saaty in several of his judgments studies utilizing pairwise comparisons [S]. Each alternative (or criterion) is compared to all of the other alternatives (or criteria). A 6 x 6 matrix of pairwise-comparison scores is produced, since there are six alternatives for this goal, and priority values are assigned to each alternative. The judgements
138
A.G.
LADDE
JUDGMENTS AND PRIORITIES WITH RESPECT TO GOAL MCDONALD MCDONALD GONZALES OLERUD BURES SANTIAGO LINDROS
GONZALES OLERUD (3.0) (2.0) (2.0)
BURKS
Matrix entry indicates that ROW element is 7TRY 5 STRONGLY 1 EQUALLY 3 MODERATELY more PREFERABLE than COLUMN element unless enclosed in parenthesis. MCDONALD GONZALES OLERUD BURKS SANTIAGO LINDROS
SANTIAGO (6.0) (4.9) (4.0) 3.0
LINDROS 3.0 3.0 4.0 8.0 6.0
STRONGLY
9 EXTREMELY
7.0) 6.0) i 5.0)
:Ben McDonald, Baltimore Orioles (rookie year - 1990) :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.053 MCDONALD 0.076 GONZALES 0.113 OLERUD 0.460 BURXS 0.267 SANTIAGO 0.032 LINDROS
INCONSISTENCY RATIO = 0.052 Figure4.
1990
cost(S) 4 4.5 6 4 8 5
ability popular 0.433 0.053 0.119 0.076 0.113 0.215 0.46 0.056 0.267 0.083 0.094 0.032
Regression Output: Constant 6.031224 Std Err of Y Est 1.88715 R Squared 0.100294 No. of Observations 6 Degrees of Freedom 3 X Coefficient(s) Std Err of Coef.
-0.84212 -3.04438 5.88728 6.920272
Figure 5.
and resulting priorities are exhibited in Figure 3. The best player in 1990 based on popularity is Ben McDonald, with a priority value of 0.433. The evaluation of the best 1990 player based on proven ability goes through the same procedure as for popularity. The only difference is that each player’s proven ability can be assessed after looking at each player’s statistics with respect to the number of games played, before making the pairwise comparisons. In this case, this can be done without using criteria because of the chosen sample. If the players had been equal in terms of the number of games played and the number of seasons in the major leagues, then the use of criteria would have been the better method. The 1990 statistics of each player can be found in the data base file created earlier (Figure 1). Again,
139
Bestbaseballcards JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURXS SANTIAGO LINDROS
AND
GONZALES 3.0
PRIORITIES GOAL
RESPECT
BURKS (?8)
OLERUD 3.0 1.0
(4.0)
TO
SANTIAGO 5.0 3.0 3.0 5.0
LINDROS 4.0 1.0 4.0 (:::)
Matrix entry indicates that ROW element 3 MODERATELY 5 STRONGLY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD GONZALES OLERUD BURKS SANTIAGO LINDROS
WITH
is 7TRY
STRONGLY
9 EXTREMELY
:Ben McDonald, Baltimore Orioles (rookie year - 1990) Texas Rangers (rookie year - 1990) :Juan Gonzales, :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Bulks, Boston Red Sox (rookie year - 1987) San Diego Padres (rookie year - 1986) :Benito Santiago, :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.343 MCDONALD 0.100 GONZALES 0.134 OLERUD 0.305 BURKS 0.046 SANTIAGO 0.072 LINDROS INCONSISTENCY
RATIO
= 0.054
Figure 6. after completion of the AHP, the best 1990 player based on proven ability is Ellis Burks with a 0.460 priority value. Along with Burks, the other players’ priority values are in Figure 4. Since we know the priority values for popularity and proven ability, we can calculate the coefficients of the regression equation. The independent variables are proven ability and popularity, and the dependent variable is cost (see Figure 5). The resulting multiple regression equation is
y = 6.03 - 0.84 xl - 3.84 x2. To calculate the model cost, we substitute the priority values obtained from Expert Choice into the corresponding variables x1 and x2. The model costs for each player are in the data base file (Figure 1). The premium tells us whether the cards are sold at a premium [(actual cost model cost) > 0] or at a discount [(actual cost - model cost) < 01. The premiums are also in the data base file, in the field labeled PREMIUM. The third criterion, the projected popularity of the player, is analyzed. This can be accomplished by using Expert Choice, where our goal is to select the best player in the future based on popularity, and the resulting priorities for each player are in Figure 6. The same process is utilized on the fourth criterion, the projected ability of the player (see Figure 7), but in this case, our goal is to choose the best player in the future based on ability. Since all four criteria have been examined, we can do pairwise comparisons with respect to relative importance. Here, we are interested in finding out how much better one criterion is than the others in predicting growth. To judge the relative importance of projected popularity
A.G. LADDE
140 JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURES SANTIAGO LINDROS
AND PRIORITIES GOAL
GONZALES (3.8)
OLERUD 1.0 4.0
RESPECT
BURES (5.0) (3.8) (5.0)
Matrix entry indicates that ROW element 3 MODERATELY 5 STRONGLY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD GONZALES OLERUD BUPXS SANTIAGO LINDROS
WITH
TO
SANTIAGO (3.0) (:::) 3.0
LINDROS 4.0 5.0 4.0 7.0 5.0
is 7-Y
STRONGLY
9 EXTREMELY
Baltimore Orioles (rookie year - 1990) :Ben McDonald, :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.076 MCDONALD 0.261 GONZALES 0.071 OLERUD 0.402 BURXS 0.158 SANTIAGO 0.032 LINDROS INCONSISTENCY
RATIO
= 0.082
Figure 7. Speculation cost(S) 4 4.5 6 4 8 5
future popularity 0.343 0.1 0.134 0.305 0.046 0.072
on Popularity
Regression Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom X Coefficient(s)
Output: 6.671422 1.235892 0.485498 6 4 -8.52853
Figure 8. Speculation
cost(S) 4 4.5 6 4 8 5
future ability 0.076 0.261 0.071 0.402 0.158 0.032
on Ability
Regression Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom X Coefficient(s)
Output: 5.773215 1.650063 0.082877 6 4 -3.13929
Figure 9.
to projected ability, cost (see Figure 8), cost (see Figure 9). which measures the
we do a regression analysis on projected popularity with respect to current and another regression analysis on projected ability with respect to current In both cases, current cost is the dependent variable. We desire the R2, variation in the dependent variable (current cost, in these two cases) that
Best baseball cards
141
To select the card that has the best potential of increasing Sorted Details for Sorted Synthesis of Leaf Nodes with respect to GOAL LEVEL 4 LEVEL 2 LEVEL 3 LEVEL 5 LEVEL1 _-_____ ------_ __-____ ---_--_ ------POPULAR
&ARS
ABILITY
PREMIUM
~0.633 MCDONALD BURKS OLERUD GONZALES LINDROS SANTIAGO
=0.243 =0.188 =0.083 =0.053 =0.039 =0.026
LINDROS MCDONALD GONZALES OLERUD BURKS SANTIAGO
=0.077 =0.033 =0.033 =0.033 =0.014 =0.009
BURKS GONZALES SANTIAGO MCDONALD OLERUD LINDROS
=0.038 -0.026 =0.016 =0.009 =0.006 =0.003
BURXS GONZALES LINDROS MCDONALD OLERUD SANTIAGO
=0.032 =0.017 =0.009 =0.006 =0.004 =0.002
-0.199
=0.098
=0.070
Figure10.
To select the card that has the best potential Of increasing Sorted Synthesis of Leaf Nodes with respect to OVERALL INCONSISTENCY INDEX =
GOAL
0.05
MCDONALD 0.292 BUP.KS
0.272
GONZALES 0.130 LINDROS
0.128
OLERUD
0.126
SANTIAGO 0.053 ===== 1.000
BURKS GONZALES LINDROS MCDONALD OLERUD SANTIAGO
-------------
Ellis Btirks,Boston Red Sox (rookie year - 1987) Juan Gonzales, Texas Rangers (rookie year - 1990) Eric Lindros, Toronto Blue Jays (rookie year - not applicable) Ben Mcdonald, Baltimore Orioles (rookie year - 1990) John Olerud, Toronto Blue Jays (rookie year - 1990) Benito Santiago, San Diego Padres (rookie year - 1986) Figure11.
is explained by the regression equation. Since the R2 for projected popularity and current cost (0.485) is greater than the R2 associated with projected ability and current cost, we can say that, with respect to our goal, the future popularity of a player is more important than his ability in the future. In making the other judgments based on relative importance it is not necessary to do any further calculations. Then, we must do pairwise comparisons with respect to the relative preference of the six alternatives under each criterion. Here, we look to select the card which can be best characterized by each criterion’s influence on the card’s price. The results are in Figure 10. After solving this Expert Choice model, the card which has the best potential for
142
A.G.
LADDE
growth is a rookie Ben McDonald with a priority value of 0.291. Ellis Burks is a close second with a priority of 0.276. The priorities of the cards are in Figure 11. CONCLUSIONS This study shows that Ben McDonald and Ellis Burks are the two top cards. Since the scores for these two cards are relatively close, both cards should be considered for any portfolio. Two of the other four cards should not be ignored and should be considered in the future: Eric Lindros and Juan Gonzales, since they are still teenagers and have not had a legitimate shot in professional baseball. The AHP can be a useful instrument to solve a variety of problems. Bahmani and Blumberg used the AHP to solve a complex marketing problem, and the AHP aided in dealing with this investment problem [7]. Collecting baseball cards was a pleasurable childhood diversion, but today the investment aspect of baseball cards has grafted some of the complex financial analysis associated with the securities market onto this simple pastime. REFERENCES 1. W.F. Doescher, Investing in baseball cards, D d B Reports 37, 10-11 (1989). 2. W.C. Wedley, Combining qualitative and quantitative factors-An analytic hierarchy approach, SocioEconomic Planning Sciences 24, 57-64 (1990). 3. B. Liu and S. Xu, Development of the theory and methodology of the Analytic Hierarchial Process and its applications in China, Math. Model. 9, 17%183 (1987). 4. F. Zahedi, The Analytic Hierarchial Process-A survey of the method and its applications, Interfacea 16, 96-108 (1986). 5. E.H. Forman, Relative vs. absolute worth, Math. Model. 9, 195-202 (1987). 6. T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, (1980). 7. N. Bahmani and H. Blumberg, Consumer preferences and reactive adaptation to a corporate solution of the over-the-counter medication dilemma, Math. Model. 9, 293-298 (1987).
Mathl. Comput. Modelling Vol. 16, No. 10, pp. 135-142, 1992 Printed in Great Britain. All rights reserved
SELECTING
THE BEST BASEBALL
CARDS
FOR INVESTMENT
A. G. LADDE George Washington University School of Business and Public Management Washington, DC 20052, U.S.A. (Received
October
1991)
Abstract-Baseball cards can be analyzed for investment purposes like any other investments. There are four criteria that the investor must consider about each of the cards. This paper describes an investment model that utilizes a data base to store specific info-tion about the selected cards. Then, by using regression analysis and the Analytic Hierarchy Process (AHP), it is possible to select the card that has the best potential of increasing.
INTRODUCTION For the past several years, popular investments have included securities and real estate. Today, many have discovered the value of baseball cards, and Doescher notes that big-time investors can see the potential for appreciation [l]. Like other investments, in-depth analysis can be used to predict future growths of particular cards based on historical data. This analysis requires the use of the Analytic Hierarchy Process because of its ability to interpret qualitative data through ratio scales [2]. The overall goal of this problem is to select the baseball card (given a finite set of cards) that has the best potential for growth.
DATA
COLLECTION
To organize the information of the six chosen baseball cards, a data base is created using DBASE III PLUS ( see Figure 1). Each record corresponds to a different player, and each field designates a different aspect of the player and his respective rookie card. The cards in question are the rookie cards of the prospective players, but the cost of the rookie card refers to the price of the card at the end of year 1990. Since several companies manufacture baseball cards (the main ones being Topps, Score, Donruss and Upper Deck), the price used for this report is the average market price. The “NA” in a field means that those categories do not apply to the player. The last two fields MODELCST (model cost) and PREMIUM (premium) will be explained when we discuss the premium criterion. The RETRIEVE/LIST option gave a hard copy of the file, and the last two fields were inserted at later time than the other fields, using the MODIFY/DATABASE FILE option. The data were entered in these two fields using the UPDATE/EDIT option. STRUCTURING
THE
HIERARCHY
In order to achieve our objective, an Expert Choice model is developed, so that the AHP can be used in making the card selection. The AHP allows one “to elicit systematic judgements from unstructured information” [3]. The four criteria for making the selection are the year the player’s card entered the market, the premium (current cost minus model cost), the popularity of the player in the future and the player’s ability in the future. These criteria are entered beneath the goal node and the alternatives (the six players’ names) are entered beneath each criterion (see Figure 2). Typeset by A&-‘&X 135
A.G. LADDE
136 Record# 1 2 3 4 5 6
NAME Ben McDonald Juan Gonzales John Olerud Ellis Burks Benito Santiago Eric Lindros
TEAM Baltimore Orioles Texas Rangers Toronto Blue Jays Boston Red Sox San Diego Padres Toronto Blue Jays
Record#
NAME Ben McDonald Juan Gonzales John Olerud Ellis Burks Benito Santiago Eric Lindros
COST90 4.00 4.50 6.00 4.00 8.00 5.00
1 2 3 4 5 6
MODELCST 4.32 5.51 5.01 5.43 5.49 5.64
ROOXIECD 1990 1990 1990 1987 1986 1990
HR NA 17 0 12 16 -
RBI BA NA NA .150 0 375 61 :303 62 . 236 -
W 0 NA NA NA NA NA
K
EJ?J,
0 NA NA NA NA NA
0.00 NA NA NA NA NA
PREMUIM -0.32 -1.01 +0.89 -1.43 +2.51 -0.64
Figure 1. To select
the card
that has the best
L 0.0701I -MCDONALD L 0.166 -GONZALES L 0.166 -0LERUD L 0.166 -BUP.ES L 0.069 -SANTIAGO L 0.046 -LINDROS L 0.388
ABILITY BURES GONZALES LINDROS MCDONALD OLERUD POPULAR PREMIUM SANTIAGO YEARS
---------------------
-MCDONALD L 0.092 -GONZALES L 0.247 -0LERUD L 0.051 -BURKS L 0.457 -SANTIAGO L 0.026 -LINDROS L 0.127
potential
of increasing
L 0.633( -MCDONALD L 0.384 -GONZALES L 0.084 -0LERUD L 0.132 -BURES L 0.296 -SANTIAGO L 0.042 -LINDROS L 0.061
-MCDONALD L 0.091 -GONZALES L 0.267 -0LERUD L 0.059 -BURRS L 0.390 -SANTIAGO L 0.163 -LINDROS L 0.030
Ability of the player in the future Ellis Burks, Boston Red Sox (rookie year - 1987) Juan Gonzales, Texas Rangers (rookie year - 1990) Eric Lindros, Toronto Blue Jays (rookie year - not applicable) Ben Mcdonald, Baltimore Orioles (rookie year - 1990) John Olerud, Toronto Blue Jays (rookie year - 1990) Popularity of the player in the future Premium - the current cost minus the model cost Benito Santiago, San Diego Padres (rookie year - 1986) Number of years played Figure 2.
ANALYSIS
OF
THE
CRITERIA
Before performing pairwise comparisons (the foundations of the AHP) on the criteria, each criterion must be analyzed [4]. The first criterion, the year the player’s first card entered the baseball card market, can be understood by looking at the data base file (Figure 1). The particular field in question is the year of the player’s rookie card (ROOKIECD). For the most part, the listed year is also the player’s first full year in professional baseball. The only exceptions are Juan Gonzales, who played the majority of the 1990 season in the minor league’s, and Eric Lindros, a hot prospect who was still in high school. By not playing any major league baseball, that player’s card can increase due to speculation on his ability. Even though Lindros’ premier sport is hockey, his baseball card is valued highly. Generally, the more years played, the less likely the player’s card will increase significantly.
Best baseball JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURXS SANTIAGO LINDROS
GONZALES 5.0
AND
PRIORITIES GOAL
GONZALES
OLERUD BURES SANTIAGO LINDROS
WITH
137 RESPECT
BURXS 6.0 3.0 4.0
OLERUD (Z)
Matrix entry indicates that ROW element 5 STRONGLY 3 MODERATELY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD
cards TO
SANTIAGO 3.0 1.0 3.0 1.0
LINDROS 5.0 2.0 ii:!; .
is 7wRY
STRONGLY
9 EXTREMELY
:Ben McDonald, Baltimore Orioles (rookie Year - 1990) :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.119 GONZALES 0.215 OLERUD 0.056 BURKS 0.083 SANTIAGO 0.094 LINDROS INCONSISTENCY
RATIO
= 0.046
Figure 3. Next, the premium criterion requires that we first calculate the model cost and then subtract it from the actual (1990) cost. In doing this, we use a regression equation to estimate what a card should be selling for, given past relationships, and then we compare this value to the 1990 selling price. The difference between these two costs is the premium. The following regression equation allows us to find the model cost of each player.
Y = bo + blzl + bzx2,
where, bo = the y-intercept (value of y when x1 = 0 and 22 = 0); bl = the coefficient of the player’s proven ability priority value; b2 = the coefficient of the player’s popularity priority value. Data for the model are derived from each player’s proven ability and popularity. Priorities for the cards are determined by separate Expert Choice models. One goal is to evaluate the baseball player’s popularity. The six alternatives are the players. Since no criteria exist, the alternatives are compared based on preference, because relative worth is more appropriate than absolute worth “since economic value is a relative matter” [5]. For example, with regard to the goal, the question is how preferred is Ben McDonald to each of the other players. The relative preference (or importance) is based on a scale from one (equally preferred (important)) to nine (extremely preferred (important)). The same scale was utilized by Saaty in several of his judgments studies utilizing pairwise comparisons [S]. Each alternative (or criterion) is compared to all of the other alternatives (or criteria). A 6 x 6 matrix of pairwise-comparison scores is produced, since there are six alternatives for this goal, and priority values are assigned to each alternative. The judgements
138
A.G.
LADDE
JUDGMENTS AND PRIORITIES WITH RESPECT TO GOAL MCDONALD MCDONALD GONZALES OLERUD BURES SANTIAGO LINDROS
GONZALES OLERUD (3.0) (2.0) (2.0)
BURKS
Matrix entry indicates that ROW element is 7TRY 5 STRONGLY 1 EQUALLY 3 MODERATELY more PREFERABLE than COLUMN element unless enclosed in parenthesis. MCDONALD GONZALES OLERUD BURKS SANTIAGO LINDROS
SANTIAGO (6.0) (4.9) (4.0) 3.0
LINDROS 3.0 3.0 4.0 8.0 6.0
STRONGLY
9 EXTREMELY
7.0) 6.0) i 5.0)
:Ben McDonald, Baltimore Orioles (rookie year - 1990) :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.053 MCDONALD 0.076 GONZALES 0.113 OLERUD 0.460 BURXS 0.267 SANTIAGO 0.032 LINDROS
INCONSISTENCY RATIO = 0.052 Figure4.
1990
cost(S) 4 4.5 6 4 8 5
ability popular 0.433 0.053 0.119 0.076 0.113 0.215 0.46 0.056 0.267 0.083 0.094 0.032
Regression Output: Constant 6.031224 Std Err of Y Est 1.88715 R Squared 0.100294 No. of Observations 6 Degrees of Freedom 3 X Coefficient(s) Std Err of Coef.
-0.84212 -3.04438 5.88728 6.920272
Figure 5.
and resulting priorities are exhibited in Figure 3. The best player in 1990 based on popularity is Ben McDonald, with a priority value of 0.433. The evaluation of the best 1990 player based on proven ability goes through the same procedure as for popularity. The only difference is that each player’s proven ability can be assessed after looking at each player’s statistics with respect to the number of games played, before making the pairwise comparisons. In this case, this can be done without using criteria because of the chosen sample. If the players had been equal in terms of the number of games played and the number of seasons in the major leagues, then the use of criteria would have been the better method. The 1990 statistics of each player can be found in the data base file created earlier (Figure 1). Again,
139
Bestbaseballcards JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURXS SANTIAGO LINDROS
AND
GONZALES 3.0
PRIORITIES GOAL
RESPECT
BURKS (?8)
OLERUD 3.0 1.0
(4.0)
TO
SANTIAGO 5.0 3.0 3.0 5.0
LINDROS 4.0 1.0 4.0 (:::)
Matrix entry indicates that ROW element 3 MODERATELY 5 STRONGLY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD GONZALES OLERUD BURKS SANTIAGO LINDROS
WITH
is 7TRY
STRONGLY
9 EXTREMELY
:Ben McDonald, Baltimore Orioles (rookie year - 1990) Texas Rangers (rookie year - 1990) :Juan Gonzales, :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Bulks, Boston Red Sox (rookie year - 1987) San Diego Padres (rookie year - 1986) :Benito Santiago, :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.343 MCDONALD 0.100 GONZALES 0.134 OLERUD 0.305 BURKS 0.046 SANTIAGO 0.072 LINDROS INCONSISTENCY
RATIO
= 0.054
Figure 6. after completion of the AHP, the best 1990 player based on proven ability is Ellis Burks with a 0.460 priority value. Along with Burks, the other players’ priority values are in Figure 4. Since we know the priority values for popularity and proven ability, we can calculate the coefficients of the regression equation. The independent variables are proven ability and popularity, and the dependent variable is cost (see Figure 5). The resulting multiple regression equation is
y = 6.03 - 0.84 xl - 3.84 x2. To calculate the model cost, we substitute the priority values obtained from Expert Choice into the corresponding variables x1 and x2. The model costs for each player are in the data base file (Figure 1). The premium tells us whether the cards are sold at a premium [(actual cost model cost) > 0] or at a discount [(actual cost - model cost) < 01. The premiums are also in the data base file, in the field labeled PREMIUM. The third criterion, the projected popularity of the player, is analyzed. This can be accomplished by using Expert Choice, where our goal is to select the best player in the future based on popularity, and the resulting priorities for each player are in Figure 6. The same process is utilized on the fourth criterion, the projected ability of the player (see Figure 7), but in this case, our goal is to choose the best player in the future based on ability. Since all four criteria have been examined, we can do pairwise comparisons with respect to relative importance. Here, we are interested in finding out how much better one criterion is than the others in predicting growth. To judge the relative importance of projected popularity
A.G. LADDE
140 JUDGMENTS
MCDONALD MCDONALD GONZALES OLERUD BURES SANTIAGO LINDROS
AND PRIORITIES GOAL
GONZALES (3.8)
OLERUD 1.0 4.0
RESPECT
BURES (5.0) (3.8) (5.0)
Matrix entry indicates that ROW element 3 MODERATELY 5 STRONGLY 1 EQUALLY more PREFERABLE than COLUMN element unless enclosed in parenthesis.
MCDONALD GONZALES OLERUD BUPXS SANTIAGO LINDROS
WITH
TO
SANTIAGO (3.0) (:::) 3.0
LINDROS 4.0 5.0 4.0 7.0 5.0
is 7-Y
STRONGLY
9 EXTREMELY
Baltimore Orioles (rookie year - 1990) :Ben McDonald, :Juan Gonzales, Texas Rangers (rookie year - 1990) :John Olerud, Toronto Blue Jays (rookie year - 1990) :Ellis Burks, Boston Red Sox (rookie year - 1987) :Benito Santiago, San Diego Padres (rookie year - 1986) :Eric Lindros, Toronto Blue Jays (rookie year - not applicable)
0.076 MCDONALD 0.261 GONZALES 0.071 OLERUD 0.402 BURXS 0.158 SANTIAGO 0.032 LINDROS INCONSISTENCY
RATIO
= 0.082
Figure 7. Speculation cost(S) 4 4.5 6 4 8 5
future popularity 0.343 0.1 0.134 0.305 0.046 0.072
on Popularity
Regression Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom X Coefficient(s)
Output: 6.671422 1.235892 0.485498 6 4 -8.52853
Figure 8. Speculation
cost(S) 4 4.5 6 4 8 5
future ability 0.076 0.261 0.071 0.402 0.158 0.032
on Ability
Regression Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom X Coefficient(s)
Output: 5.773215 1.650063 0.082877 6 4 -3.13929
Figure 9.
to projected ability, cost (see Figure 8), cost (see Figure 9). which measures the
we do a regression analysis on projected popularity with respect to current and another regression analysis on projected ability with respect to current In both cases, current cost is the dependent variable. We desire the R2, variation in the dependent variable (current cost, in these two cases) that
Best baseball cards
141
To select the card that has the best potential of increasing Sorted Details for Sorted Synthesis of Leaf Nodes with respect to GOAL LEVEL 4 LEVEL 2 LEVEL 3 LEVEL 5 LEVEL1 _-_____ ------_ __-____ ---_--_ ------POPULAR
&ARS
ABILITY
PREMIUM
~0.633 MCDONALD BURKS OLERUD GONZALES LINDROS SANTIAGO
=0.243 =0.188 =0.083 =0.053 =0.039 =0.026
LINDROS MCDONALD GONZALES OLERUD BURKS SANTIAGO
=0.077 =0.033 =0.033 =0.033 =0.014 =0.009
BURKS GONZALES SANTIAGO MCDONALD OLERUD LINDROS
=0.038 -0.026 =0.016 =0.009 =0.006 =0.003
BURXS GONZALES LINDROS MCDONALD OLERUD SANTIAGO
=0.032 =0.017 =0.009 =0.006 =0.004 =0.002
-0.199
=0.098
=0.070
Figure10.
To select the card that has the best potential Of increasing Sorted Synthesis of Leaf Nodes with respect to OVERALL INCONSISTENCY INDEX =
GOAL
0.05
MCDONALD 0.292 BUP.KS
0.272
GONZALES 0.130 LINDROS
0.128
OLERUD
0.126
SANTIAGO 0.053 ===== 1.000
BURKS GONZALES LINDROS MCDONALD OLERUD SANTIAGO
-------------
Ellis Btirks,Boston Red Sox (rookie year - 1987) Juan Gonzales, Texas Rangers (rookie year - 1990) Eric Lindros, Toronto Blue Jays (rookie year - not applicable) Ben Mcdonald, Baltimore Orioles (rookie year - 1990) John Olerud, Toronto Blue Jays (rookie year - 1990) Benito Santiago, San Diego Padres (rookie year - 1986) Figure11.
is explained by the regression equation. Since the R2 for projected popularity and current cost (0.485) is greater than the R2 associated with projected ability and current cost, we can say that, with respect to our goal, the future popularity of a player is more important than his ability in the future. In making the other judgments based on relative importance it is not necessary to do any further calculations. Then, we must do pairwise comparisons with respect to the relative preference of the six alternatives under each criterion. Here, we look to select the card which can be best characterized by each criterion’s influence on the card’s price. The results are in Figure 10. After solving this Expert Choice model, the card which has the best potential for
142
A.G.
LADDE
growth is a rookie Ben McDonald with a priority value of 0.291. Ellis Burks is a close second with a priority of 0.276. The priorities of the cards are in Figure 11. CONCLUSIONS This study shows that Ben McDonald and Ellis Burks are the two top cards. Since the scores for these two cards are relatively close, both cards should be considered for any portfolio. Two of the other four cards should not be ignored and should be considered in the future: Eric Lindros and Juan Gonzales, since they are still teenagers and have not had a legitimate shot in professional baseball. The AHP can be a useful instrument to solve a variety of problems. Bahmani and Blumberg used the AHP to solve a complex marketing problem, and the AHP aided in dealing with this investment problem [7]. Collecting baseball cards was a pleasurable childhood diversion, but today the investment aspect of baseball cards has grafted some of the complex financial analysis associated with the securities market onto this simple pastime. REFERENCES 1. W.F. Doescher, Investing in baseball cards, D d B Reports 37, 10-11 (1989). 2. W.C. Wedley, Combining qualitative and quantitative factors-An analytic hierarchy approach, SocioEconomic Planning Sciences 24, 57-64 (1990). 3. B. Liu and S. Xu, Development of the theory and methodology of the Analytic Hierarchial Process and its applications in China, Math. Model. 9, 17%183 (1987). 4. F. Zahedi, The Analytic Hierarchial Process-A survey of the method and its applications, Interfacea 16, 96-108 (1986). 5. E.H. Forman, Relative vs. absolute worth, Math. Model. 9, 195-202 (1987). 6. T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, (1980). 7. N. Bahmani and H. Blumberg, Consumer preferences and reactive adaptation to a corporate solution of the over-the-counter medication dilemma, Math. Model. 9, 293-298 (1987).