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Strategic behavior and the persistence of discrimination in professional baseball
Marhematica/ Social Sciences 26 (1993) 299-315 0165-4896/93/$06.00 0 1993 - Elsevier Science
299 Publishers
B.V. All rights
reserved
Strategic behavior and the persistence discrimination in professional baseball
of
Van Kolpin * ‘, Larry D. Singe11 Jr. Depurtment
of Economics,
University
of Oregon,
Eugene.
OR 97403-1285,
USA
Communicated by M. Kaneko Received October 1992 Revised
February
1993
Abstract This paper examines labor-market discrimination in major-league model. We find that strategic interaction may have played a pivotal
baseball using a game-theoretic role in the history of discrimina-
tion in the major leagues. Our results offer insight into the collapse of the color barrier league’s subsequent evolution. We also show that changes in institutional features such procedures, parity, and the number of competing teams can influence discriminatory behavior that would go undetected without formal game-theoretic analysis. Key words:
Major
leagues;
Baseball;
Game-theoretic
and the as draft in ways
model
Introduction Discriminatory hiring practices appear to be a pervasive and persistent aspect of the labor market in many industries. Numerous social programs, e.g. affirmative action, have been designed in hopes of mitigating this behavior. However, to be effective such policies must account for the manner in which the economic environment can induce discrimination. We examine this issue in the context of major-league baseball by constructing a game-theoretic model of the labor market for rookie (entry level) players. This model provides a framework to examine how league operating procedures affected discrimination in the past and how recent modifications in these procedures could affect incentives to discriminate in the future. This paper differs from traditional studies of discrimination (e.g.. Becker, 1957; Arrow, 1973) in that we find employers may actively pursue discriminatory hiring policies even when their preferences are not racially biased. Two other papers have examined this possibility from somewhat different perspectives. Kaneko and * Corresponding author. * The authors acknowledge
the valuable
suggestions
of Mamoru
Kaneko
300
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Kimura (1992) developed an intriguing model in which no member of society is endowed with prejudicial preferences. Their analysis revealed that while discrimination may not arise in a standard Nash equilibrium framework, it may manifest itself as a stationary state of a stable convention. Kolpin and Singe11 (1992) found that in the presence of asymmetric or incomplete information, Nash equilibrium may lead firms to discriminate against a segment of the workforce that they actually prefer. Our paper differs from these works in that if focuses exclusively on subgame perfect equilibria of games with both complete and symmetric information. The use of a professional sports league as a context for study of labor market behavior is not unique, e.g. DeBrock and Roth (1981) and Atkinson, Stanley and Tschirhart (1988). For our purposes, major-league baseball exemplifies an industry experiencing a distinct transition from an exclusively white to an integrated workforce. It is important to note, however, that our model provides a methodology to analyze employment discrimination in other industries as well. Indeed, many industries have an implicit ‘draft’ in which there is a widely recognized ‘pecking order’ amongst firms for new job candidates, e.g. the academic labor market. As a consequence, our results have direct analogs that apply outside the confines of major-league baseball. The main body of our paper is organized as follows. Section 2 derives a series of theoretical results examining three historical phenomena: (i) the persistence of discrimination prior to the onset of integration; (ii) the eventual demise of the prohibition on black players often referred to as the color barrier; and (iii) the reluctance of individual teams to discontinue exclusive white hiring policies. Section 3 investigates the modern reverse-order draft, parity, and league expansion. These factors are assessed both in terms of their immediate impact on discriminatory behavior and their role in the league’s long run evolution. The Appendix contains our formal proofs.
2. Breaking
the color harrier
Despite a wealth of talent available in the Negro leagues, no black players were hired in major-league baseball until 1947, when the employment of Jackie Robinson broke the color barrier. The primary objective of this section is to introduce an elementary model of the professional baseball labor market and to use this model as a context for examining major-league employment policies on and around 1947. In particular, we investigate three historical phenomena: (i) the persistence of league wide discrimination prior to the entry of black players; (ii) the eventual collapse of the color barrier; and (iii) the persistence of team level discrimination even after integration was initiated. Formal proofs of our results appear in the Appendix. The formal structure of drafting games is characterized by the strategy, talent, and preference components of the model. First consider strategic structure. Let
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n} represent the set of teams and P the finite set of rookie players. Each team drafts (selects) a single player from amongst those not previously drafted. This process proceeds in a sequence beginning with team 1, followed by team 2, etc. For each i E N let II(i) denote the set of all non-repeating sequences of i-l elements in P. Thus II(i) denotes the set of all possible selections by team i’s predecessors and a strategy for team i is simply a function assigning to each rr(i) E II(i) a player not found in the sequence r(i). For each i E N, let S, denote the strategy set for team i. Finally, let S = XiEN Si represent the space of pure strategy profiles and for each s E S, let p,(s) represent the player ultimately selected by team i. Figure 1 depicts the extensive form when the teams N = { 1,2} draft players from the pool P = {a, b, c}. The labeled branches represent possible draft selections at each information set and endpoints denote the realized outcomes of this process. For instance, if s * is defined by ~:=a, s;(a)= b, sT(b)=a, and s;(c) = a, then p,(s*) = a and p*(s*) = b as represented by the outcome z,. Talent of both teams and rookies is assumed to be a unidimensional attribute. For each i E N, team i is endowed with a positive level of veteran talent denoted by Ti. Each p E P is likewise endowed with positive talent denoted by t(p). Team talent can be thought of as the average level of talent amongst a team’s players; thus a new average is formed after the draft is complete. To be precise, if s E S represents the strategies implemented then team i’s after-draft talent becomes Th e weights of S/9 and l/9 are (arbitrarily) T,(s) = (819)T; + (1/9)t(p,(s)). chosen to represent eight veteran players and one rookie player on a team of nine. aTeam preferences are influenced by both success and racial bias. Each team’s expected success depends directly on its relative talent. Given s E S, of team i defeating team j on the ‘;(‘)‘LT;(‘) + Tj(s)l re P resents the probability baseball field. Assuming every team plays each of its II - 1 opponents an equal number of times, it follows that team i’s expected ratio of victories to games played is given by y(s) = [ll(n - l)] cj+i T,(s) /[ T,(s) + T,(s)]. Racial characteristics of rookie players are identified through the binary function r : P+ (0, 1). A player p E P is white (black) if r(p) = 0 (r(p) = 1). Preferences over strategy
N={l,...,
Team 2
Z
z
2
Fig. 1. Extensive
z
3
z
4
z
5
form of a sample
Z
6
drafting
game.
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profiles are defined through a utility function of the form U,(S) s V,(S) 6,r( p,(s)), 6, 2 0. The disutility realized from hiring a black player is represented by 6,. As such, we shall refer to ai as team i’s discrimination coefficient. Let us return to the example depicted in Fig. 1 and assume that veteran and rookie talent is represented by T, = 1, Tz = 0.8, t(u) = 1, t(b) = 0.9, and t(c) = 0.8; while racial characteristics of players and discrimination coefficients of teams are characterized by r(a) = 1, r(b) = 0, r(c) = 0, and 6, = SZ = 0.002. Table 1 depicts the team talent levels, expected victory ratios, and final utilities realized at each possible outcome of the draft. For instance, if S* is the strategy profile defined by s”; = a, s:(a) = b, s;(b) = a, and S;(C) = a, then z, is the outcome realized; T,(s”) = (8/9)T, + (1/9)t(u) = 1 and T?(s*) = (819) T, + (1/9)t(b) = 0.811; V,(s’“) = 111.811 = 0.552 and V&*) = 0.811/1.811 = 0.448; u,(P) = V,(s*) - 6, = 0.55 and z.Q(s*) = V2(s*) = 0.448. As the reader may verify, the strategy profile s* is in fact a subgame perfect equilibrium for the draft game constructed. Note that drafting games satisfy perfect information as each team is fully informed of the players drafted prior their own draft choice. Games with perfect information ‘almost always’ have a unique subgame perfect equilibrium in pure strategies (e.g. Myerson, 1991, theorem 4.7). Throughout our analysis we consider only pure strategy subgame perfect equilibria. We define an abstract league to be a set of teams N = { 1, . . . , n} and pool of players P endowed with the racial characteristics r. Varying N, P, or r yields distinct abstract leagues. Each such abstract league generates a space of drafting games which are uniquely determined by the data (Ti)itN, (Gl)itN, and (t( P))~~~. Consequently, a given abstract league can be thought of as a topological subspace of a 2n + (#P) dimensional Euclidean space. A number of our theoretical results examine the sensitivity of team behavior to perturbations in these parameters. With this in mind, we define a subclass of games to be large if it has a non-empty interior. Thus, with exception of its ‘boundary’, games within a large subclass are robust to small changes in parameter values.
Table 1 After-draft
=I 22 z 2, 2, =I,
talent
levels, expected
victory
ratios,
and final utilities
in a sample
draft game
T, (~1
Tds)
v,(s)
V2(s)
UI(S)
4s)
1
0.811 0.8 0.822 0.8 0.822 0.811
0.552 0.556 0.546 0.553 0.543 0.547
0.448 0.444 0.454 0.447 0.457 0.453
0.55 0.554 0.546 0.553 0.543 0.547
0.448 0.444 0.452 0.447 0.455 0.453
1 O.Y89 0.989 0.978 0.978
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Definition 2.1. Given a drafting game G, i E N, and s E S, we shall say that: (a) Team i is racist (non-racist) if 6, > 0 (8; = 0). (b) Team i racially discriminates if there exists a p E P such that p is not drafted prior to p,(s), r(p) # r( p,(s)), and t(p) > t( p,(s)). Whether or not a team is racist is a matter of exogenously given preferences. A team is racist when the hiring of a black player in of itself inflicts diminished realized utility. Discrimination, however, is a matter of choice. Racial discrimination occurs when a player is drafted ahead of an even more talented player belonging to a different race. (For instance, a mediocre white player drafted ahead of a highly talented black player.) As noted in our introduction, traditional models of labor-market discrimination suggest that employers pursue discriminatory hiring only if they are endowed with prejudicial preferences. One important implication of our model is that employers may exhibit openly discriminatory behavior even when they are not racist. Theorem 2.1. Given any abstract league and any team i not choosing last in the draft, the subclass of games in which team i is non-racist and yet racially discriminates in perfect equilibrium is large in the subspace of games for which team i is non-racist. Theorem 2.1 implies that not only may non-racist teams racially discriminate in equilibrium, this behavior may be robust to parameter modification even when the team in question remains non-racist. Consequently, the prevalence of discriminatory hiring in the major leagues prior to 1947 need not be considered anomalous even if some team owners did not have prejudicial preferences. The logic behind this conclusion is straightforward. The gains from hiring a relatively talented black player may be offset by the fact that such a choice supplements the talent pool of white players for subsequent racist teams. Hiring a white player may thus be an effective means of reducing the after-draft talent of competitors and thereby improving one’s own relative standing. Several key features prevailed in the major-league employment policies preceding 1947. First, only white players were hired. Second, teams strove to hire the best (white) players available. In effect, teams with the earliest draft picks drafted the best (white) players. Third, the best black player was more talented than at least some of the white players employed. A fourth feature was not directly observable, but likely present. When a team considered the ramifications of not drafting the best available white player, it did not expect opponents with subsequent draft choices to respond by drafting players of lesser talent than originally intended. Opposing teams were expected to either draft the best available players from the ‘talent enhanced’ pool of white players, or to cross the color barrier in search of even greater talent. These features are formally characterized by w-equilibria as defined below.
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Definition 2.2. A perfect equilibrium (1) Y(p,(s)) = 0 for all i E N,
s is an w-equilibrium
299-315
if:
(2) t( p,(s)> 2 t( (3) max{t(P) I 4~) = 11 >min{t(p,(s)) Ii E W (4) t(p,(s\s;)) z- t(pi(s)) for all i,j E N, i #j, and S; E S,. Pj(s)>
if
i
After over 50 years of exclusively hiring white players the color barrier’s collapse presumably caught many by surprise, suggesting major-league baseball experienced temporary disequilibrium. To examine such disequilibrium behavior we hypothesize a perturbation in a team’s preferences that is favorable to black employment and unobservable to the remainder of the league. We wish to identify the team with the lowest ‘threshold of discrimination’, that which requires the smallest perturbation in preferences to induce the hiring of a black player. Definition 2.3. Given s E S, a team i E N, and F I> 0 let B,(F, S) be the set of all s: E S, which maximize r/l(sls:) - (Sl - &)Y(p,(s\s:)). If s is a perfect equilibrium and Y(p,(s)) = 0 then team i’s threshold of discrimination is defined by T,(S) = min{ e 2 0 1r( p,(s\s:)) = 1 for some s: E Bi(&, s)}. The set B,(E, s) consists of the best responses for team i to the strategy profile s when its disutility from hiring a black player has diminished by e. The value T,(S) is the smallest perturbation in team i’s discrimination coefficient sufficient to induce it to hire a black player when other teams were unaware of i’s change of preferences, i.e. team i’s threshold of discrimination. Theorem 2.2. Suppose all teams have the same positive discrimination coeficient and team n (who chooses last in the draft) has the lowest veteran talent. Then ifs is an w-equilibrium it follows that team n has the lowest threshold of discrimination. In the years preceding 1965, league operating procedures offered rich (successful) teams a decided advantage in the hiring of new payers. In effect, the league operated under a direct-order draft in which the best teams were able to hire the most talented players while weaker teams were forced to hire from amongst the remainder. Suppose that when modeling the state of affairs in major-league baseball prior to 1947, one further assumes that teams were uniform in their ‘taste for discrimination’. It follows Theorem 2.2 implies the weakest team has the lowest threshold of discrimination; suggesting the weakest team is also the most likely to break the color barrier. Even if teams are not uniformly racist, our proof reveals that the weakest team still has the largest ‘victory incentive’ to cross the color barrier; thereby lending legitimacy to the application of this conclusion in more general settings. The contention that less successful teams were likely candidates to hire the first black players is largely supported by the empirical evidence. For example, the Brooklyn Dodgers were an unsuccessful team prior to the hiring of Jackie Robinson, winning only one pennant between 1900 and 1946.
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The final result of this section examines the persistence of team level discrimination after the league itself has initiated integration. As a historical example, the Boston Red Sox did not hire a single black player until 1959, 12 years after the color barrier collapsed. On the surface such behavior appears to suggest a higher degree of racism on the part of persistently discriminatory teams. Indeed, their relative position could be expected to erode as opponents not only capitalized on the freshly tapped reservoir of black rookies, but also retained these players as they matured into veterans. However, we find that the increasing competitiveness of opponents may actually strengthen a team’s resolve to discriminate, even if its prejudicial preferences are no different from those of opponents. In stating this result let pi E S, denote the strategy which by definition drafts the most talented black player from amongst those not previously drafted. i E N be a team that hires a white player in the perfect equilibrium s. Then team i’s threshold of discrimination is an increasing function of team j’s veteran talent whenever t( p,(s\p,)) + t( p,(@,)) 2 t( p,(s)) + t( pi(s)) and
Theorem
[T;(s\P;)
2.3. Let
+ T,(s\Pi)12/T,(s\P,)
< [T;(s)
+
‘(j(S)I”T~(S)*
A similar theorem may, of course, be constructed to characterize circumstances under which the increasing competitiveness of opponents will decrease a team’s threshold of discrimination. However, our main point is that there is a broad set of circumstances under which the erosion of a team’s relative standing may actually increase a team’s proclivity to discriminate. The conditions stated are derived from a direct application of differential calculus. The following provides a concrete numerical example. Example. Suppose the set of teams is N = {1,2}; team talent is characterized by T, = 1, T2 = x, 0.6 5 x 5 1.1; discrimination coefficients are 6, = 6, = 0.01; P = characteristics {WI> w2, b,, 6,) is the rookie player set; and rookie talent/radical are respectively represented by t(p) = 1, 0.6, 1.1, 1 and r(p) = 0, 0, 1, 1 for the unique perfect equilibrium s through backward P = w,, w2, b, , b,. Computing induction and applying Theorem 2.3 verifies that team l’s threshold of discrimination is an increasing function of T2 up to its theoretical limit of 1.1 (the highest talent of any player who may enter the league in the context of this example). The reader may note that Theorem 2.3 implicitly assumes that the increase in a competitor’s talent does not affect the game’s perfect equilibrium. While this is true of the example provided above, games certainly exist for which sufficiently large perturbations in veteran talent lead to new equilibria. In such scenarios, Theorem 2.3 may be interpreted as characterizing the ‘first-order’ effects of increases in the opposition’s veteran talent. Our qualitative conclusions remain unchanged provided these effects either dominate or are supplemented by the ‘second-order’ effects resulting from equilibrium shifts.
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3. The reverse-order
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draft and incentives
to discriminate
The sequential nature of the draft may appear to suggest that the order in which individual players are drafted depends only on the talent and racial characteristics of the player pool. In this section we argue that seemingly unrelated factors may affect not only the order in which players are hired, but also the severity of discrimination observed. Attention is initially focused on a comparison of the direct- and reverse-order drafts. Given that the major leagues first enforced a modern style reverse-order draft in 1965, this study forms a rigorous contrast between the pre- and post-1965 labor markets. Our analysis concludes with a series of ‘possibility theorems’ investigating the role league parity and league expansion play in observed discriminatory behavior. Our results reveal that the welfare implications of changes in league operating procedures require careful analysis of strategic behavior. From a broader perspective, our analysis suggests that government policies that subsidize or offer other preferential treatment to specific firms within an industry (thereby influencing the pecking order of firms in the labor market) may have the unintended effect of harming disadvantaged segments of the work force. Given a draft game G, implementation of the direct-order draft reorganizes the draft sequence so that the most talented team has first choice, second most talented team has second choice, etc. (Whenever T, = T, and i
be characterized by talent levels coefficients 6, = 6, = 6, = 0.0045.
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by the talent and The player pool P= {w,, w2, w3, b, , b,, b3} is characterized racial characteristics, t(p) = 1, 0.85, 0.7, 1.12, 1.01, 0.9 and r(p) = 0, 0, 0, 1, 1, 1 induction reveals that p,(s) = b,, for P = wl, w2, w3, b,, b2, b,. Backward p2(s) = w,, and p3(s) = b,, where s is the unique perfect equilibrium under the direct-order draft. Note that while there is individualized discrimination in that team 2 drafts the less talented white player w1 ahead of the more talented black player b,, there is no global discrimination as the three most talented players are employed. Backward induction also reveals that p3(s*) = b,, p2(s*) = wl, and p,(s*) = w2, where s* is the unique perfect equilibrium under the reverse-order draft. Thus the reverse-order draft induces global discrimination in addition to individualized discrimination by both teams 1 and 2 (the second most talented player, b,, is not drafted by any team). This example and our simplified repeated game framework can be used to verify that draft procedures may dramatically influence the long-term evolution of a league. Under a direct-order draft, league talent can be shown to converge over time to the distribution T, = 1.12, T2 = T, = 1. (Some fluctuation is introduced by the fact that occasionally, not more than every seventh year, the team with the last draft will choose w2 rather than b2.) Under the reverse-order draft, veteran talent converges to a distribution of approximately 0.9956, 0.9950, 0.9794. (The specific teams assigned to these values rotates from season to season.) As in the original single season game, 6, is unemployed in every season of the multiseason reverse-order draft. This example suggests that a reverse-order draft may induce more severe discriminatory behavior than a direct-order draft. Although this need not always be true, there exists an important class of circumstances under which the reverse-order draft is clearly detrimental to the welfare of black players. We shall say G is naive consistent if there exists 6 ~-0 such that in perfect equilibrium of both the direct- and reverse-order drafts; each team behaves as if its discrimination coefficient is 6 and it expects opponents to experience negligible improvement from the draft. Formally, the sequence of players drafted is identical to that which would result if each team chooses from amongst available players, that which uniquely maximizes lV;( p) - 6r( p); where W,(p) = [l l(n - 1)] Cjri [(a/ 9)Ti + (l/9)t(p)]/[(8/9)Tj + (1/9)t(p) + T,]. Naive consistency does not assume that teams naively ignore the impact of the draft on their opponents, only that their revealed behavior is consistent with this hypothesis. Theorem 3.1. If G is naive consistent then fewer black players are hired in perfect equilibrium under a reverse-order draft than under a direct-order draft. It is worth noting that the same conclusion may be derived under an alternative hypothesis. One may instead assume there exists a concave function u such that each team behaves as if it seeks to maximize u( T, + t(p)) - 6r( p)), subject to the availability of p. In other words, if equilibrium behavior leads teams to behave as if ‘utility’ is derived from absolute rather than relative talent; it then follows that the reverse-order draft is more prone to discrimination than a direct-order draft.
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It is interesting to note that Theorem 3.1 is supported by empirical evidence. For instance, Fig. 2 demonstrates that the participation of black players in majorleague baseball grew rapidly from 1947 to 196.5, a trend that was suspended in the years following 1965. Table 2 reveals that a decline in the black-white performance differentials was also reversed after the reverse-order draft was adopted, suggesting that relative standards for hiring black players had been raised. When the reverse-order draft was first implemented in 1965, one of its main objectives was to increase parity within the major leagues. With increased parity, pennant races become more competitive, fan interest increases, and (presumably) league profits rise. Consequently, revision of draft procedures may provide a mechanism for increasing league welfare. The ‘fact’ that a reverse-order draft enhances league parity may appear to be self evident as it ensures that the weakest teams may choose from amongst the best players. Nonetheless, we shall find that this ‘fact’ may well be fallacy when discriminatory preferences are present. In practice, the concept of parity may acquire a variety of subtly distinct meanings. We adopt a straightforward approach and say that parity is increased (decreased) if variance in after-draft talent necessarily falls (rises). Given a drafting game G and s E S let variance in after-draft talent be defined by C’,(S) = CIEN [Ti(s) - T(s)]*/n, where T(S) = CiEN T,(s)/n. We shall say that the drafting game G’ increases (decreases) parity relative to G if v:;,(s’) < g:;(s) (a:,(~‘) > a:;(s)) for all s and s’ perfect equilibria of G and G’ respectively. Theorem 3.2. Given any abstract league there is a large class of games for which the reverse-order draft decreases parity relative to the direct-order draft.
Percent
Black
47
52
57
62
67
72
77
82
87
Year Fig. 2. Participation Ebony, June issues,
of black players 1960-78.
in major-league
baseball.
----,
National
Baseball
Library;
-,
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Table 2 Performance
differentials
between
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black
and white
Batting
players
(excluding
in major-league
baseball
pitchers) Average
At bats
Batting average
309
299-315
Slugging average
Hits
per 550 at bats Home runs
Stolen bases
White
966,564
261
393
144
13
4
Black
80,702
280
455
154
20
10
White
511,463
251
380
138
14
4
Black
174,176
269
421
148
17
14
White
488,677
251
369
138
12
5
Black
237,343
272
419
149
16
16
1947-60
1961-68
1969-75
Pitching Per 9 innings pitched Innings pitched
Winning percentage
E.R.A.
Strikeouts
Hits allowed
Base on balls
White
290,047
0.499
3.98
4.29
8.93
3.66
Black
7,999
0.545
3.90
5.38
8.53
3.43
White
201,467
0.496
3.56
5.69
8.47
3.06
Black
13.553
0.522
3.39
6.44
8.02
3.44
White
211,287
0.493
3.63
5.35
8.63
3.36
Black
18,630
0.541
3.47
6.32
8.05
3.45
Neft and Cohen
(1977).
1947-60
1961-68
1969975
Source:
The essential logic behind this conclusion is straightforward. In the presence of prejudicial preferences, teams may no longer choose to draft players in decreasing sequence of talent. Thus a reverse-order draft may actually increase variability of after-draft talent. This conclusion is equally valid in a multi-season framework. To verify this assertion it suffices to use the example found in our formal proof (see Appendix) as a basis for repeated games under both a direct- and reverse-order draft. Equilibrium computation reveals that variance in after-draft talent converges to a value almost 300 times larger under the reverse-order draft compared to the direct-order draft.
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Even if increasing parity is an effective means for increasing league profits, ‘social planners’ may also be concerned with its implications for minority employment. The following theorem reveals that there is a significant class of games for which parity enhancement may in of itself lead to reduced employment of black players. Theorem 3.3. Given any abstract league with at least three teams, there is a large class of games for which increased parity can reduce employment of black players in perfect equilibrium. As an informal proof, note that when a team has relatively low talent, there is also a relatively large return to investment in additional talent. Thus a black player with high talent may be a team’s most desirable draft choice. If, however, each team’s talent becomes more nearly equalized, the return from additional talent may be relatively low and insufficient to cover the disutility realized from hiring a black player. To verify that this behavior may persist in the long run, consider a repeated reverse-order drafting game based on the initial conditions offered in our formal proof (see Appendix). Forward induction reveals that in equilibrium of this repeated game, a black player is hired in only the very first season. Thus, the league’s successful move toward parity effectively eliminates black players from the workforce. Effective 1993, two new teams will be introduced to the major leagues. The resulting impact on quality of play and league profits has been a source of much debate. There has been no discussion devoted to the effects expansion will have on the opportunities of players. Conventional wisdom would say that increasing the number of teams will increase employment of both white and black players. In the context of our model, league expansion occurs when new teams are introduced to a drafting game without modifying any other structure, i.e. the original teams retain their relative positions within the new draft order while rookie players and their characteristics are unchanged. Our final result reveals that league expansion may lead to decreased employment of minority players even in absolute terms. Theorem 3.4. Given any abstract league there is a large class of games for which league expansion can reduce employment of black players in perfect equilibrium. The logic behind this result is as follows. The introduction of weak teams into a league may actually decrease the marginal return from talent to the strong established teams. This effect may cause black players that are marginally more talented than their white counterparts to be passed over in the draft as a direct consequence of league expansion. This conclusion also applies in a multi-season context. Consider a modification of the example found in our formal proof (see Appendix). Suppose the expansion team’s discrimination coefficient is 6, = 0.025. Assuming other specifications are unchanged, it is easy to show that none of the
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311
teams will hire black players. In repeated play the league fluctuates over time between hiring two black players and none at all. It never returns to the pre-expansion status, in which two black players were hired in every season.
Appendix The following lemma proves useful. Given an abstract league denoted by r and N,, C_N, let T(N,,) = {G E r 16, = 0 for all i E N,}. Thus T(N,) represents the set of all drafting games in the abstract league r for which each team in N, is non-racist. Note that T(N,) = r if N, = 0. Lemma A. Let the abstract league r and N, C N be given. If G E r(N,) has a unique perfect equilibrium s *, then there exists an open set U C_r (N,) such that G E U and for each G’ E U, s* is a perfect equilibrium of G’. Proof. Consider the extensive form of G. If s* is the unique perfect equilibrium of G then for each i E N, sT uniquely specifies a best response to s* at every r(i) E II(i) (i.e. at every subgame). Thus in response to s*, team i strictly prefers the choices specified by ST to alternative courses of action. But at each s E S, V,(s) - 6;r( p,(s)) depends continuously on the parameters (T,),,,, IS,, and Consequently, sufficiently slight perturbations in these parameters do (t( P))pEP. not affect the strict preferences discussed above. It follows that for sufficiently small perturbations in (Tj)jtN, (6j),FN, and (t(p))pEPr s* remains a perfect equilibrium. 0 Before proceeding it is worthwhile commenting on the methodology used to prove the ‘possibility theorems’-Theorems 2.1, 3.2, 3.3, and 3.4. These proofs are constructive and based on routine examples. More ‘realistic’ examples can be designed to yield the same qualitative conclusions. However, as the conclusions themselves remain unchanged, we have chosen to avoid superfluous complexity. In a similar spirit, we have assumed that all drafting games have at least as many of both white and black rookie players as there are teams. Imposing this condition simplifies constructive proofs and does not diminish the generality of conclusions in any meaningful way. Indeed, it is difficult to argue that there is a shortage of athletes of either race willing to play professional baseball. Proof
of Theorem 2.1. Let r be an abstract league defined by N = {1,2}, and r(b)=O, 0, 1, 1 forp=w,, w, b,, b,. Let GETbe p = {WI, wq, b,,b,}, such that T, = 1, T2 = 0.7, 6, = 0, 6, = 0.02, and t(b) = 1, 0.5, 1.002, 1.001, for induction reveals G has a unique perfect equilibP = w,, W?, b, , 6,. Backwards rium s and p,(s) = w, , p*(s) = w2 thus both teams racially discriminate. Appealing to Lemma A we conclude there exists an open set U in r( { l}) containing G such that s is a perfect equilibrium for each G’ E U. We conclude the subclass of all
312
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games such that team 1 is non-racist and yet racially discriminates in perfect equilibrium is large in the subspace of games for which team 1 is non-racist. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. 0 Proof of Theorem 2.2. For each i E N define /3, E Sj by pl(,(i)) = argmax{t(p)l p@n(i) and r(p) = l} for each no n(i). s an w-equilibrium implies T,(S) = V,(s) - V,(s\p,) + 6. It follows that T,(S) > T,(S) if and only if V,(s\p,) - V,,(s) > r/l(s\&) - V,(s). For each i,j E N let y(s 1j) = Ti(s)/ [T;(s) + 7’,(s)]. We claim that for each i # ~1: 1n) - lqs 1n) 5 V,(s\/3,
qs\p,
1i)
- V,(s 1i)
(Al)
for each j # i, n; v((s\/?, 1j) - y(s I j) < Vn(s\pn 1j) - V,,(s 1j) . (AZ) As y((s\p,) - V,(S) is l/(n - 1) times the sum of the left hand sides of (Al) and (A2) and V,(s\p,,) - V,(s) is ll(n - 1) times the sum of the right hand sides, our conclusion thus follows directly from (Al) and (A2). To establish (Al) note that s an o-equilibrium implies ~(P~(s\P,~)) = t(~~(~\/3,))= th=max{t(p)Ir(p)=l}. For every jC5N let tj=t(pj(s)). s an w-equilibrium and the fact that n has the final draft pick implies t( p,(s\P,)) 2 t, and t( p,(s\p,)) = ti 2 t,. Thus (Al) is implied by, though not necessarily equivalent to, the following inequality: [ST, + t,]/[8T,
+ t, + 8T,, + t,] - [ST, + t,]/[ST, + t, + ST, + t,]
5 [8T, + tb]/[8Ti Rearranging
these
terms
[8T, + t,,]l[ST;
+ ti + ST,, + tb] - [8T, + t,]/[ST,
yields
the equivalent
the left hand
+ t, + ST, + t,] .
side of the inequality
[8T, + t, + tj - t,]/[ST, - [ST, + th]/[8Ti
Thus (Al) is verified. We now verify (A2). s an t( p,(s\p,)) 2 t, and t, 2 t,. Thus equivalent to, the following:
above
by:
+ ti + 8T,, + th] + ti + ST, +
t,,] .
w-equilibrium implies that (A2) is implied by, though
+ t, + 8T, + t,] - [ST, + t,]i[ST,
< [8T, + tJl[ST,
is dominated
+ t, + tj - t, + 8T, + t,]
= [ST, - 8T,, + ti - t,]/[8T,
[8Ti + t,]/[STi
inequality:
+ t, + ST, + t,] - [8T, + t,,]/[8T, + ti + 8T, + th]
5 [ST, - 8T, + ti - t,]/[ST, But ti 2 t, implies
+ tj + 8T, + t,] .
for each i f not necessarily
j,
+ t, + 8Tj + tj]
+ tj + ST, + th] - [ST,, + t,]i[ST,
+ tj + 8T, +
t,] .
V. Kolpin,
Rearranging
L. D. Singe11 Jr.
and simplifying
I Mathematical
algebraically
Social Sciences
26 (1993)
313
299-315
we see that this inequality
is equivalent
to: [8Tj + tj][tb - t,]/[ST, <[ST, This
final inequality
+ t, + ST, + t,][8T, + t, + ST, + t,]
+ tl][tb - t,]/[ST,
follows
directly
from
Proof of Theorem 2.3. First note that plicitly a function of T,. It follows: dq(s)/dT,
= -8[8Ti + 8[8T; +
Using
the fact that
+ t, + 8Tj + t,] .
Ti > T, and t,, > t,.
T;(S) = V,(S) - U;(s\/?,) + aj, which
+ t(p[(s))]/[ST,
+ t(p,(s))
0
is im-
+ ST, + t(pj(s))]*
t(Pi(S\Pi))]‘[8Ti + t(Pi(S\Pi)) + 8Tj + t(Pj(SVi))I*
of Ti(s) and rearranging
the definition
&-,(s)/dT,
+ t, + ST, + t,][8T,
the resulting
inequality,
.
it follows
>O if {[Ti(s\P,)
+ Tj(S\P,)]/[Ti(S)
+
Tj(S)I)*< T~(s\P~)‘T;(J).
(A31
Clearly (A3) is . e q uivalent to [ T,(s\&) + T,(s\~~)]~/T,(s\~~) < [T,(S) + T,(S)]‘/ T;(s), implying T,(S) is a locally increasing function of T, (local to the initial value of T,). To show that r,(s) is increasing for arbitrary increases in T,, (A3) implies that it suffices to verify that [ T,(s\p,) + T,(s\p,)] /[T,(s) + T,(s)] is everywhere weakly decreasing in T,. But for all T, > 0 a{[8Ti
+ t(Pi(S\Pi))
+8T,
+
t(Pj(SVi))I’[‘Tz+~(PI(s))+‘T,
+ t(P,(s))l)“Tj = s[t(p,(s))+ t(PiW)- t(p,WP,))- t(~,WiWW’i + t(~i(s>> + 8Tj the t(Pi(s))
latter +
+ t@(s))]*
inequality t(P,(S))’
following
50 from
the
fact
that
t(
p,(s\p,))
+
t(
pj(s\pi))
Z-
I7
Proof of Theorem 3.1. For convenience assume T, 2 T2 2 . . . P T, . Let s and s* be perfect equilibria under the direct- and reverse-order drafts respectively. Let q and p represent the number of black players hired in s and s*. Our conclusion is trivial if 7 = II so assume r] < p 5 12 and let A = max{i E N 1 r(pi(s) = 0}, i.e. the last team to hire a white player in a direct-order draft. It follows A 2 n - 7. Let 5 E N be the index of the team drafting the (7 + 1)th black player in the equilibrium s * , thus (‘Sn-T’A and T5?TA. Define wd = argmax {t(P) 14~) PF{P,(sL P,+,(s*)>>,
1r(p) = 1 and = 0 and PF’{P~(s), . . . , P,+,(S)>>, b, = argmax{t(p) . . . 2 P,-,(S))), w, = argmax{t(p) 14~) =O and P@~(P,(s*), ..., and b, = argmax{t(p) 14~) = 1 and pF’{p,(s*), . . , P~+~(s*>>>.
314
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L. D. Singe11 Jr.
I Mathematical
Social Sciences
26 (1993)
299-315
Naive consistency implies that: (1) wd is the (rz - v)th most talented white; (2) b, is the (77 + 1)th most talented black; (3) t(w,) 2 t(wd); and (4) t(bd) 2 t(b,). Note that r(p<(s*)) = 1 implies 6 < W,(b,) - W,(w,) and thus t(b,) > t(w,). But 6 > W,(b,) - W,(w,,) 2 W,(b,) - WA(w,) 2 W,(b,) - W,(w,), the first inequality follows from r( p,(s)) = 0, the second follows from (3) and (4), and the final inequality follows from the fact that T5 2 Th and t(b,) > t(w,). As W,(b,) W<(w,) cannot be both strictly larger and strictly smaller than 6, we must conclude Al.I 7. 0 Proof of Theorem 3.2. Let r be an abstract league defined by N = {1,2}, P={w,,w,,b,,b,},andr(p)=O,0,1,1forp=w,,w,,b,,b2.LetG~~such that T,=1, T,=0.5,6,=6,=0.015,andt(p)=1,0.5, l.l,lforp=w,, w2,b1, b,. Backward induction reveals [T,(s), T2(s)] = [l, 0.571 and [T,(s*), T2(s*)] = [l.Ol, 0.561; w h ere s and s* are respectively the unique perfect equilibria under the direct- and reverse-order drafts. It follows the reverse-order draft decreases parity relative to the direct-order draft. Note that T, > T,, thus sufficiently small perturbations in ( Ti)iEN do not affect the order in which teams choose under either a direct- or reverse-order draft. Thus Lemma A implies that sufficiently small changes in (T;)iEN, (tSoitN, and (t(~))~_ do not affect the equilibrium status of s and s*. As (Ti(~))IEN and (TL(~*))iEN are continuous in (Tl)rEN, we conclude there exists an open set U E r such that (&)iEN, and (4 dptP G E U and the reverse-order draft decreases parity relative to the direct-order draft for all G’ E U, i.e. there is a large class of games for which the reverse-order draft decreases parity relative to the direct-order draft. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. Cl Proof
of Theorem
3.3. Let
r
be an abstract
league
defined
by N = (1,2,3),
l,l, Iforp=w,,w,,w,,b,,b,, p = {w,, w2, wj, b,, b,, b,},andr(p)=O,O,O, 6,. Let G E r such that T, = 0.4, T, = T, = 1, 6, = 8, = 6, = 0.004, and t(p) = 0.902, 0.901, 0.9, 1.002, 1.001, 1 for p = w,, w?, w3, b,, b,, 6,. Backward induction reveals that p,(s) = b,, and p2(s) = wr, p3(s) = w2; where s is the unique perfect equilibrium of G. Lemma A implies there exists an open set U C r such that G E U and s is an equilibrium of G’ for each G’ E U. Note that if the distribution of veteran talent changes to T, = 0.8 for all i E N, backward induction reveals that only white players are hired in perfect equilibrium. We conclude there is a large class of games for which an increase in parity can lead to fewer black players being hired in perfect equilibrium. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. 0 Proof
of Theorem
3.4. Let
r
be
an
abstract
league
defined
by N = (1,2),
=O, O,O, 1, 1, 1 forp = wi, w2, wj, b,, b,, p = {W,> w2, w3, b,, b,, b3}, andr(p) 6,. Let G E r such that T, = 1, T2 = 1, S, = 6, = 0.0084, and t(p) = 0.702, 0.701,
V. Kolpin, L. D. Singe/l Jr.
I Mathematical Social Sciences 26 (1993)
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315
reveals 0.7, 1.002, 1.001, 1 for p = wl, w?, w3, b, , b2, b,. Backward induction that p,(s) = b,, p2(s) = b,; where s is the unique perfect equilibrium of G. Now let us suppose team 0 is introduced to G; where T, = 0.7 and 6,) = 0.0084. Further suppose team 0 is granted the first draft choice. Backward induction reveals that . where s* is the unique perfect equilibrium P&*) = b,, Pi = ~1, P&*) = ~2, of the expanded drafting game. Lemma A thus implies there is a large class of games for which expansion can lead to strictly fewer black players being hired in perfect equilibrium. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. I7
References S. Atkinson, L. Stanley and J. Tschirhart, Revenue sharing as an incentive in an agency problem: example from the national football league, Rand J. 19 (1988) 27-43. K. Arrow, The theory of discrimination, in: 0. Ashenfelter and A. Rees, eds., Discrimination
an in
Labor Markets (Princeton University Press, Princeton, 1973). G.S. Becker, The Economics of Discrimination (University of Chicago Press, Chicago, 1957). L.M. DeBrock and A. Roth, Strike two: labor-management negotiations in major league baseball, Bell J. 12 (1981) 413-425. M. Kaneko and T. Kimura, Conventions, social prejudices and discrimination: a festival game with merrymakers, Games Econom. Behav. 4 (1992) 511-527. V. Kolpin and L.D. Singell, Jr., Strategic discrimination, scholarly performance, and the gender composition of economics departments, Mimeo, University of Oregon (1992). R.B. Myerson, Game Theory Analysis of Conflict (Harvard University Press, Cambridge, 1991). D.S. Neft and R. Cohen (eds.), The Sports Encyclopedia: Baseball (St. Martin Press, New York, 1977).
299 Publishers
B.V. All rights
reserved
Strategic behavior and the persistence discrimination in professional baseball
of
Van Kolpin * ‘, Larry D. Singe11 Jr. Depurtment
of Economics,
University
of Oregon,
Eugene.
OR 97403-1285,
USA
Communicated by M. Kaneko Received October 1992 Revised
February
1993
Abstract This paper examines labor-market discrimination in major-league model. We find that strategic interaction may have played a pivotal
baseball using a game-theoretic role in the history of discrimina-
tion in the major leagues. Our results offer insight into the collapse of the color barrier league’s subsequent evolution. We also show that changes in institutional features such procedures, parity, and the number of competing teams can influence discriminatory behavior that would go undetected without formal game-theoretic analysis. Key words:
Major
leagues;
Baseball;
Game-theoretic
and the as draft in ways
model
Introduction Discriminatory hiring practices appear to be a pervasive and persistent aspect of the labor market in many industries. Numerous social programs, e.g. affirmative action, have been designed in hopes of mitigating this behavior. However, to be effective such policies must account for the manner in which the economic environment can induce discrimination. We examine this issue in the context of major-league baseball by constructing a game-theoretic model of the labor market for rookie (entry level) players. This model provides a framework to examine how league operating procedures affected discrimination in the past and how recent modifications in these procedures could affect incentives to discriminate in the future. This paper differs from traditional studies of discrimination (e.g.. Becker, 1957; Arrow, 1973) in that we find employers may actively pursue discriminatory hiring policies even when their preferences are not racially biased. Two other papers have examined this possibility from somewhat different perspectives. Kaneko and * Corresponding author. * The authors acknowledge
the valuable
suggestions
of Mamoru
Kaneko
300
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I Mathematical Social Sciences 26 (1993)
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Kimura (1992) developed an intriguing model in which no member of society is endowed with prejudicial preferences. Their analysis revealed that while discrimination may not arise in a standard Nash equilibrium framework, it may manifest itself as a stationary state of a stable convention. Kolpin and Singe11 (1992) found that in the presence of asymmetric or incomplete information, Nash equilibrium may lead firms to discriminate against a segment of the workforce that they actually prefer. Our paper differs from these works in that if focuses exclusively on subgame perfect equilibria of games with both complete and symmetric information. The use of a professional sports league as a context for study of labor market behavior is not unique, e.g. DeBrock and Roth (1981) and Atkinson, Stanley and Tschirhart (1988). For our purposes, major-league baseball exemplifies an industry experiencing a distinct transition from an exclusively white to an integrated workforce. It is important to note, however, that our model provides a methodology to analyze employment discrimination in other industries as well. Indeed, many industries have an implicit ‘draft’ in which there is a widely recognized ‘pecking order’ amongst firms for new job candidates, e.g. the academic labor market. As a consequence, our results have direct analogs that apply outside the confines of major-league baseball. The main body of our paper is organized as follows. Section 2 derives a series of theoretical results examining three historical phenomena: (i) the persistence of discrimination prior to the onset of integration; (ii) the eventual demise of the prohibition on black players often referred to as the color barrier; and (iii) the reluctance of individual teams to discontinue exclusive white hiring policies. Section 3 investigates the modern reverse-order draft, parity, and league expansion. These factors are assessed both in terms of their immediate impact on discriminatory behavior and their role in the league’s long run evolution. The Appendix contains our formal proofs.
2. Breaking
the color harrier
Despite a wealth of talent available in the Negro leagues, no black players were hired in major-league baseball until 1947, when the employment of Jackie Robinson broke the color barrier. The primary objective of this section is to introduce an elementary model of the professional baseball labor market and to use this model as a context for examining major-league employment policies on and around 1947. In particular, we investigate three historical phenomena: (i) the persistence of league wide discrimination prior to the entry of black players; (ii) the eventual collapse of the color barrier; and (iii) the persistence of team level discrimination even after integration was initiated. Formal proofs of our results appear in the Appendix. The formal structure of drafting games is characterized by the strategy, talent, and preference components of the model. First consider strategic structure. Let
V. Kolpin, L. D. Singe11 Jr.
I Mathematical Social Sciences 26 (lYY3)
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301
n} represent the set of teams and P the finite set of rookie players. Each team drafts (selects) a single player from amongst those not previously drafted. This process proceeds in a sequence beginning with team 1, followed by team 2, etc. For each i E N let II(i) denote the set of all non-repeating sequences of i-l elements in P. Thus II(i) denotes the set of all possible selections by team i’s predecessors and a strategy for team i is simply a function assigning to each rr(i) E II(i) a player not found in the sequence r(i). For each i E N, let S, denote the strategy set for team i. Finally, let S = XiEN Si represent the space of pure strategy profiles and for each s E S, let p,(s) represent the player ultimately selected by team i. Figure 1 depicts the extensive form when the teams N = { 1,2} draft players from the pool P = {a, b, c}. The labeled branches represent possible draft selections at each information set and endpoints denote the realized outcomes of this process. For instance, if s * is defined by ~:=a, s;(a)= b, sT(b)=a, and s;(c) = a, then p,(s*) = a and p*(s*) = b as represented by the outcome z,. Talent of both teams and rookies is assumed to be a unidimensional attribute. For each i E N, team i is endowed with a positive level of veteran talent denoted by Ti. Each p E P is likewise endowed with positive talent denoted by t(p). Team talent can be thought of as the average level of talent amongst a team’s players; thus a new average is formed after the draft is complete. To be precise, if s E S represents the strategies implemented then team i’s after-draft talent becomes Th e weights of S/9 and l/9 are (arbitrarily) T,(s) = (819)T; + (1/9)t(p,(s)). chosen to represent eight veteran players and one rookie player on a team of nine. aTeam preferences are influenced by both success and racial bias. Each team’s expected success depends directly on its relative talent. Given s E S, of team i defeating team j on the ‘;(‘)‘LT;(‘) + Tj(s)l re P resents the probability baseball field. Assuming every team plays each of its II - 1 opponents an equal number of times, it follows that team i’s expected ratio of victories to games played is given by y(s) = [ll(n - l)] cj+i T,(s) /[ T,(s) + T,(s)]. Racial characteristics of rookie players are identified through the binary function r : P+ (0, 1). A player p E P is white (black) if r(p) = 0 (r(p) = 1). Preferences over strategy
N={l,...,
Team 2
Z
z
2
Fig. 1. Extensive
z
3
z
4
z
5
form of a sample
Z
6
drafting
game.
V. Kolpin, L. D. Singe11 Jr.
302
I Mathematical Social Sciences 26 (1993)
299-315
profiles are defined through a utility function of the form U,(S) s V,(S) 6,r( p,(s)), 6, 2 0. The disutility realized from hiring a black player is represented by 6,. As such, we shall refer to ai as team i’s discrimination coefficient. Let us return to the example depicted in Fig. 1 and assume that veteran and rookie talent is represented by T, = 1, Tz = 0.8, t(u) = 1, t(b) = 0.9, and t(c) = 0.8; while racial characteristics of players and discrimination coefficients of teams are characterized by r(a) = 1, r(b) = 0, r(c) = 0, and 6, = SZ = 0.002. Table 1 depicts the team talent levels, expected victory ratios, and final utilities realized at each possible outcome of the draft. For instance, if S* is the strategy profile defined by s”; = a, s:(a) = b, s;(b) = a, and S;(C) = a, then z, is the outcome realized; T,(s”) = (8/9)T, + (1/9)t(u) = 1 and T?(s*) = (819) T, + (1/9)t(b) = 0.811; V,(s’“) = 111.811 = 0.552 and V&*) = 0.811/1.811 = 0.448; u,(P) = V,(s*) - 6, = 0.55 and z.Q(s*) = V2(s*) = 0.448. As the reader may verify, the strategy profile s* is in fact a subgame perfect equilibrium for the draft game constructed. Note that drafting games satisfy perfect information as each team is fully informed of the players drafted prior their own draft choice. Games with perfect information ‘almost always’ have a unique subgame perfect equilibrium in pure strategies (e.g. Myerson, 1991, theorem 4.7). Throughout our analysis we consider only pure strategy subgame perfect equilibria. We define an abstract league to be a set of teams N = { 1, . . . , n} and pool of players P endowed with the racial characteristics r. Varying N, P, or r yields distinct abstract leagues. Each such abstract league generates a space of drafting games which are uniquely determined by the data (Ti)itN, (Gl)itN, and (t( P))~~~. Consequently, a given abstract league can be thought of as a topological subspace of a 2n + (#P) dimensional Euclidean space. A number of our theoretical results examine the sensitivity of team behavior to perturbations in these parameters. With this in mind, we define a subclass of games to be large if it has a non-empty interior. Thus, with exception of its ‘boundary’, games within a large subclass are robust to small changes in parameter values.
Table 1 After-draft
=I 22 z 2, 2, =I,
talent
levels, expected
victory
ratios,
and final utilities
in a sample
draft game
T, (~1
Tds)
v,(s)
V2(s)
UI(S)
4s)
1
0.811 0.8 0.822 0.8 0.822 0.811
0.552 0.556 0.546 0.553 0.543 0.547
0.448 0.444 0.454 0.447 0.457 0.453
0.55 0.554 0.546 0.553 0.543 0.547
0.448 0.444 0.452 0.447 0.455 0.453
1 O.Y89 0.989 0.978 0.978
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303
Definition 2.1. Given a drafting game G, i E N, and s E S, we shall say that: (a) Team i is racist (non-racist) if 6, > 0 (8; = 0). (b) Team i racially discriminates if there exists a p E P such that p is not drafted prior to p,(s), r(p) # r( p,(s)), and t(p) > t( p,(s)). Whether or not a team is racist is a matter of exogenously given preferences. A team is racist when the hiring of a black player in of itself inflicts diminished realized utility. Discrimination, however, is a matter of choice. Racial discrimination occurs when a player is drafted ahead of an even more talented player belonging to a different race. (For instance, a mediocre white player drafted ahead of a highly talented black player.) As noted in our introduction, traditional models of labor-market discrimination suggest that employers pursue discriminatory hiring only if they are endowed with prejudicial preferences. One important implication of our model is that employers may exhibit openly discriminatory behavior even when they are not racist. Theorem 2.1. Given any abstract league and any team i not choosing last in the draft, the subclass of games in which team i is non-racist and yet racially discriminates in perfect equilibrium is large in the subspace of games for which team i is non-racist. Theorem 2.1 implies that not only may non-racist teams racially discriminate in equilibrium, this behavior may be robust to parameter modification even when the team in question remains non-racist. Consequently, the prevalence of discriminatory hiring in the major leagues prior to 1947 need not be considered anomalous even if some team owners did not have prejudicial preferences. The logic behind this conclusion is straightforward. The gains from hiring a relatively talented black player may be offset by the fact that such a choice supplements the talent pool of white players for subsequent racist teams. Hiring a white player may thus be an effective means of reducing the after-draft talent of competitors and thereby improving one’s own relative standing. Several key features prevailed in the major-league employment policies preceding 1947. First, only white players were hired. Second, teams strove to hire the best (white) players available. In effect, teams with the earliest draft picks drafted the best (white) players. Third, the best black player was more talented than at least some of the white players employed. A fourth feature was not directly observable, but likely present. When a team considered the ramifications of not drafting the best available white player, it did not expect opponents with subsequent draft choices to respond by drafting players of lesser talent than originally intended. Opposing teams were expected to either draft the best available players from the ‘talent enhanced’ pool of white players, or to cross the color barrier in search of even greater talent. These features are formally characterized by w-equilibria as defined below.
304
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I Mathematical Social Sciences 26 (lY93)
Definition 2.2. A perfect equilibrium (1) Y(p,(s)) = 0 for all i E N,
s is an w-equilibrium
299-315
if:
(2) t( p,(s)> 2 t( (3) max{t(P) I 4~) = 11 >min{t(p,(s)) Ii E W (4) t(p,(s\s;)) z- t(pi(s)) for all i,j E N, i #j, and S; E S,. Pj(s)>
if
i
After over 50 years of exclusively hiring white players the color barrier’s collapse presumably caught many by surprise, suggesting major-league baseball experienced temporary disequilibrium. To examine such disequilibrium behavior we hypothesize a perturbation in a team’s preferences that is favorable to black employment and unobservable to the remainder of the league. We wish to identify the team with the lowest ‘threshold of discrimination’, that which requires the smallest perturbation in preferences to induce the hiring of a black player. Definition 2.3. Given s E S, a team i E N, and F I> 0 let B,(F, S) be the set of all s: E S, which maximize r/l(sls:) - (Sl - &)Y(p,(s\s:)). If s is a perfect equilibrium and Y(p,(s)) = 0 then team i’s threshold of discrimination is defined by T,(S) = min{ e 2 0 1r( p,(s\s:)) = 1 for some s: E Bi(&, s)}. The set B,(E, s) consists of the best responses for team i to the strategy profile s when its disutility from hiring a black player has diminished by e. The value T,(S) is the smallest perturbation in team i’s discrimination coefficient sufficient to induce it to hire a black player when other teams were unaware of i’s change of preferences, i.e. team i’s threshold of discrimination. Theorem 2.2. Suppose all teams have the same positive discrimination coeficient and team n (who chooses last in the draft) has the lowest veteran talent. Then ifs is an w-equilibrium it follows that team n has the lowest threshold of discrimination. In the years preceding 1965, league operating procedures offered rich (successful) teams a decided advantage in the hiring of new payers. In effect, the league operated under a direct-order draft in which the best teams were able to hire the most talented players while weaker teams were forced to hire from amongst the remainder. Suppose that when modeling the state of affairs in major-league baseball prior to 1947, one further assumes that teams were uniform in their ‘taste for discrimination’. It follows Theorem 2.2 implies the weakest team has the lowest threshold of discrimination; suggesting the weakest team is also the most likely to break the color barrier. Even if teams are not uniformly racist, our proof reveals that the weakest team still has the largest ‘victory incentive’ to cross the color barrier; thereby lending legitimacy to the application of this conclusion in more general settings. The contention that less successful teams were likely candidates to hire the first black players is largely supported by the empirical evidence. For example, the Brooklyn Dodgers were an unsuccessful team prior to the hiring of Jackie Robinson, winning only one pennant between 1900 and 1946.
V. Kolpin, L. D. Singe11Jr. I MathematicalSocial Sciences 26 (1993) 299-315
305
The final result of this section examines the persistence of team level discrimination after the league itself has initiated integration. As a historical example, the Boston Red Sox did not hire a single black player until 1959, 12 years after the color barrier collapsed. On the surface such behavior appears to suggest a higher degree of racism on the part of persistently discriminatory teams. Indeed, their relative position could be expected to erode as opponents not only capitalized on the freshly tapped reservoir of black rookies, but also retained these players as they matured into veterans. However, we find that the increasing competitiveness of opponents may actually strengthen a team’s resolve to discriminate, even if its prejudicial preferences are no different from those of opponents. In stating this result let pi E S, denote the strategy which by definition drafts the most talented black player from amongst those not previously drafted. i E N be a team that hires a white player in the perfect equilibrium s. Then team i’s threshold of discrimination is an increasing function of team j’s veteran talent whenever t( p,(s\p,)) + t( p,(@,)) 2 t( p,(s)) + t( pi(s)) and
Theorem
[T;(s\P;)
2.3. Let
+ T,(s\Pi)12/T,(s\P,)
< [T;(s)
+
‘(j(S)I”T~(S)*
A similar theorem may, of course, be constructed to characterize circumstances under which the increasing competitiveness of opponents will decrease a team’s threshold of discrimination. However, our main point is that there is a broad set of circumstances under which the erosion of a team’s relative standing may actually increase a team’s proclivity to discriminate. The conditions stated are derived from a direct application of differential calculus. The following provides a concrete numerical example. Example. Suppose the set of teams is N = {1,2}; team talent is characterized by T, = 1, T2 = x, 0.6 5 x 5 1.1; discrimination coefficients are 6, = 6, = 0.01; P = characteristics {WI> w2, b,, 6,) is the rookie player set; and rookie talent/radical are respectively represented by t(p) = 1, 0.6, 1.1, 1 and r(p) = 0, 0, 1, 1 for the unique perfect equilibrium s through backward P = w,, w2, b, , b,. Computing induction and applying Theorem 2.3 verifies that team l’s threshold of discrimination is an increasing function of T2 up to its theoretical limit of 1.1 (the highest talent of any player who may enter the league in the context of this example). The reader may note that Theorem 2.3 implicitly assumes that the increase in a competitor’s talent does not affect the game’s perfect equilibrium. While this is true of the example provided above, games certainly exist for which sufficiently large perturbations in veteran talent lead to new equilibria. In such scenarios, Theorem 2.3 may be interpreted as characterizing the ‘first-order’ effects of increases in the opposition’s veteran talent. Our qualitative conclusions remain unchanged provided these effects either dominate or are supplemented by the ‘second-order’ effects resulting from equilibrium shifts.
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3. The reverse-order
I Mathematical Social Sciences 26 (1993 ) 299-315
draft and incentives
to discriminate
The sequential nature of the draft may appear to suggest that the order in which individual players are drafted depends only on the talent and racial characteristics of the player pool. In this section we argue that seemingly unrelated factors may affect not only the order in which players are hired, but also the severity of discrimination observed. Attention is initially focused on a comparison of the direct- and reverse-order drafts. Given that the major leagues first enforced a modern style reverse-order draft in 1965, this study forms a rigorous contrast between the pre- and post-1965 labor markets. Our analysis concludes with a series of ‘possibility theorems’ investigating the role league parity and league expansion play in observed discriminatory behavior. Our results reveal that the welfare implications of changes in league operating procedures require careful analysis of strategic behavior. From a broader perspective, our analysis suggests that government policies that subsidize or offer other preferential treatment to specific firms within an industry (thereby influencing the pecking order of firms in the labor market) may have the unintended effect of harming disadvantaged segments of the work force. Given a draft game G, implementation of the direct-order draft reorganizes the draft sequence so that the most talented team has first choice, second most talented team has second choice, etc. (Whenever T, = T, and i
be characterized by talent levels coefficients 6, = 6, = 6, = 0.0045.
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307
by the talent and The player pool P= {w,, w2, w3, b, , b,, b3} is characterized racial characteristics, t(p) = 1, 0.85, 0.7, 1.12, 1.01, 0.9 and r(p) = 0, 0, 0, 1, 1, 1 induction reveals that p,(s) = b,, for P = wl, w2, w3, b,, b2, b,. Backward p2(s) = w,, and p3(s) = b,, where s is the unique perfect equilibrium under the direct-order draft. Note that while there is individualized discrimination in that team 2 drafts the less talented white player w1 ahead of the more talented black player b,, there is no global discrimination as the three most talented players are employed. Backward induction also reveals that p3(s*) = b,, p2(s*) = wl, and p,(s*) = w2, where s* is the unique perfect equilibrium under the reverse-order draft. Thus the reverse-order draft induces global discrimination in addition to individualized discrimination by both teams 1 and 2 (the second most talented player, b,, is not drafted by any team). This example and our simplified repeated game framework can be used to verify that draft procedures may dramatically influence the long-term evolution of a league. Under a direct-order draft, league talent can be shown to converge over time to the distribution T, = 1.12, T2 = T, = 1. (Some fluctuation is introduced by the fact that occasionally, not more than every seventh year, the team with the last draft will choose w2 rather than b2.) Under the reverse-order draft, veteran talent converges to a distribution of approximately 0.9956, 0.9950, 0.9794. (The specific teams assigned to these values rotates from season to season.) As in the original single season game, 6, is unemployed in every season of the multiseason reverse-order draft. This example suggests that a reverse-order draft may induce more severe discriminatory behavior than a direct-order draft. Although this need not always be true, there exists an important class of circumstances under which the reverse-order draft is clearly detrimental to the welfare of black players. We shall say G is naive consistent if there exists 6 ~-0 such that in perfect equilibrium of both the direct- and reverse-order drafts; each team behaves as if its discrimination coefficient is 6 and it expects opponents to experience negligible improvement from the draft. Formally, the sequence of players drafted is identical to that which would result if each team chooses from amongst available players, that which uniquely maximizes lV;( p) - 6r( p); where W,(p) = [l l(n - 1)] Cjri [(a/ 9)Ti + (l/9)t(p)]/[(8/9)Tj + (1/9)t(p) + T,]. Naive consistency does not assume that teams naively ignore the impact of the draft on their opponents, only that their revealed behavior is consistent with this hypothesis. Theorem 3.1. If G is naive consistent then fewer black players are hired in perfect equilibrium under a reverse-order draft than under a direct-order draft. It is worth noting that the same conclusion may be derived under an alternative hypothesis. One may instead assume there exists a concave function u such that each team behaves as if it seeks to maximize u( T, + t(p)) - 6r( p)), subject to the availability of p. In other words, if equilibrium behavior leads teams to behave as if ‘utility’ is derived from absolute rather than relative talent; it then follows that the reverse-order draft is more prone to discrimination than a direct-order draft.
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It is interesting to note that Theorem 3.1 is supported by empirical evidence. For instance, Fig. 2 demonstrates that the participation of black players in majorleague baseball grew rapidly from 1947 to 196.5, a trend that was suspended in the years following 1965. Table 2 reveals that a decline in the black-white performance differentials was also reversed after the reverse-order draft was adopted, suggesting that relative standards for hiring black players had been raised. When the reverse-order draft was first implemented in 1965, one of its main objectives was to increase parity within the major leagues. With increased parity, pennant races become more competitive, fan interest increases, and (presumably) league profits rise. Consequently, revision of draft procedures may provide a mechanism for increasing league welfare. The ‘fact’ that a reverse-order draft enhances league parity may appear to be self evident as it ensures that the weakest teams may choose from amongst the best players. Nonetheless, we shall find that this ‘fact’ may well be fallacy when discriminatory preferences are present. In practice, the concept of parity may acquire a variety of subtly distinct meanings. We adopt a straightforward approach and say that parity is increased (decreased) if variance in after-draft talent necessarily falls (rises). Given a drafting game G and s E S let variance in after-draft talent be defined by C’,(S) = CIEN [Ti(s) - T(s)]*/n, where T(S) = CiEN T,(s)/n. We shall say that the drafting game G’ increases (decreases) parity relative to G if v:;,(s’) < g:;(s) (a:,(~‘) > a:;(s)) for all s and s’ perfect equilibria of G and G’ respectively. Theorem 3.2. Given any abstract league there is a large class of games for which the reverse-order draft decreases parity relative to the direct-order draft.
Percent
Black
47
52
57
62
67
72
77
82
87
Year Fig. 2. Participation Ebony, June issues,
of black players 1960-78.
in major-league
baseball.
----,
National
Baseball
Library;
-,
V. Kolpin, L. D. Singe11 Jr.
Table 2 Performance
differentials
between
I Mathematical Social Sciences 26 (1993)
black
and white
Batting
players
(excluding
in major-league
baseball
pitchers) Average
At bats
Batting average
309
299-315
Slugging average
Hits
per 550 at bats Home runs
Stolen bases
White
966,564
261
393
144
13
4
Black
80,702
280
455
154
20
10
White
511,463
251
380
138
14
4
Black
174,176
269
421
148
17
14
White
488,677
251
369
138
12
5
Black
237,343
272
419
149
16
16
1947-60
1961-68
1969-75
Pitching Per 9 innings pitched Innings pitched
Winning percentage
E.R.A.
Strikeouts
Hits allowed
Base on balls
White
290,047
0.499
3.98
4.29
8.93
3.66
Black
7,999
0.545
3.90
5.38
8.53
3.43
White
201,467
0.496
3.56
5.69
8.47
3.06
Black
13.553
0.522
3.39
6.44
8.02
3.44
White
211,287
0.493
3.63
5.35
8.63
3.36
Black
18,630
0.541
3.47
6.32
8.05
3.45
Neft and Cohen
(1977).
1947-60
1961-68
1969975
Source:
The essential logic behind this conclusion is straightforward. In the presence of prejudicial preferences, teams may no longer choose to draft players in decreasing sequence of talent. Thus a reverse-order draft may actually increase variability of after-draft talent. This conclusion is equally valid in a multi-season framework. To verify this assertion it suffices to use the example found in our formal proof (see Appendix) as a basis for repeated games under both a direct- and reverse-order draft. Equilibrium computation reveals that variance in after-draft talent converges to a value almost 300 times larger under the reverse-order draft compared to the direct-order draft.
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Even if increasing parity is an effective means for increasing league profits, ‘social planners’ may also be concerned with its implications for minority employment. The following theorem reveals that there is a significant class of games for which parity enhancement may in of itself lead to reduced employment of black players. Theorem 3.3. Given any abstract league with at least three teams, there is a large class of games for which increased parity can reduce employment of black players in perfect equilibrium. As an informal proof, note that when a team has relatively low talent, there is also a relatively large return to investment in additional talent. Thus a black player with high talent may be a team’s most desirable draft choice. If, however, each team’s talent becomes more nearly equalized, the return from additional talent may be relatively low and insufficient to cover the disutility realized from hiring a black player. To verify that this behavior may persist in the long run, consider a repeated reverse-order drafting game based on the initial conditions offered in our formal proof (see Appendix). Forward induction reveals that in equilibrium of this repeated game, a black player is hired in only the very first season. Thus, the league’s successful move toward parity effectively eliminates black players from the workforce. Effective 1993, two new teams will be introduced to the major leagues. The resulting impact on quality of play and league profits has been a source of much debate. There has been no discussion devoted to the effects expansion will have on the opportunities of players. Conventional wisdom would say that increasing the number of teams will increase employment of both white and black players. In the context of our model, league expansion occurs when new teams are introduced to a drafting game without modifying any other structure, i.e. the original teams retain their relative positions within the new draft order while rookie players and their characteristics are unchanged. Our final result reveals that league expansion may lead to decreased employment of minority players even in absolute terms. Theorem 3.4. Given any abstract league there is a large class of games for which league expansion can reduce employment of black players in perfect equilibrium. The logic behind this result is as follows. The introduction of weak teams into a league may actually decrease the marginal return from talent to the strong established teams. This effect may cause black players that are marginally more talented than their white counterparts to be passed over in the draft as a direct consequence of league expansion. This conclusion also applies in a multi-season context. Consider a modification of the example found in our formal proof (see Appendix). Suppose the expansion team’s discrimination coefficient is 6, = 0.025. Assuming other specifications are unchanged, it is easy to show that none of the
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teams will hire black players. In repeated play the league fluctuates over time between hiring two black players and none at all. It never returns to the pre-expansion status, in which two black players were hired in every season.
Appendix The following lemma proves useful. Given an abstract league denoted by r and N,, C_N, let T(N,,) = {G E r 16, = 0 for all i E N,}. Thus T(N,) represents the set of all drafting games in the abstract league r for which each team in N, is non-racist. Note that T(N,) = r if N, = 0. Lemma A. Let the abstract league r and N, C N be given. If G E r(N,) has a unique perfect equilibrium s *, then there exists an open set U C_r (N,) such that G E U and for each G’ E U, s* is a perfect equilibrium of G’. Proof. Consider the extensive form of G. If s* is the unique perfect equilibrium of G then for each i E N, sT uniquely specifies a best response to s* at every r(i) E II(i) (i.e. at every subgame). Thus in response to s*, team i strictly prefers the choices specified by ST to alternative courses of action. But at each s E S, V,(s) - 6;r( p,(s)) depends continuously on the parameters (T,),,,, IS,, and Consequently, sufficiently slight perturbations in these parameters do (t( P))pEP. not affect the strict preferences discussed above. It follows that for sufficiently small perturbations in (Tj)jtN, (6j),FN, and (t(p))pEPr s* remains a perfect equilibrium. 0 Before proceeding it is worthwhile commenting on the methodology used to prove the ‘possibility theorems’-Theorems 2.1, 3.2, 3.3, and 3.4. These proofs are constructive and based on routine examples. More ‘realistic’ examples can be designed to yield the same qualitative conclusions. However, as the conclusions themselves remain unchanged, we have chosen to avoid superfluous complexity. In a similar spirit, we have assumed that all drafting games have at least as many of both white and black rookie players as there are teams. Imposing this condition simplifies constructive proofs and does not diminish the generality of conclusions in any meaningful way. Indeed, it is difficult to argue that there is a shortage of athletes of either race willing to play professional baseball. Proof
of Theorem 2.1. Let r be an abstract league defined by N = {1,2}, and r(b)=O, 0, 1, 1 forp=w,, w, b,, b,. Let GETbe p = {WI, wq, b,,b,}, such that T, = 1, T2 = 0.7, 6, = 0, 6, = 0.02, and t(b) = 1, 0.5, 1.002, 1.001, for induction reveals G has a unique perfect equilibP = w,, W?, b, , 6,. Backwards rium s and p,(s) = w, , p*(s) = w2 thus both teams racially discriminate. Appealing to Lemma A we conclude there exists an open set U in r( { l}) containing G such that s is a perfect equilibrium for each G’ E U. We conclude the subclass of all
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games such that team 1 is non-racist and yet racially discriminates in perfect equilibrium is large in the subspace of games for which team 1 is non-racist. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. 0 Proof of Theorem 2.2. For each i E N define /3, E Sj by pl(,(i)) = argmax{t(p)l p@n(i) and r(p) = l} for each no n(i). s an w-equilibrium implies T,(S) = V,(s) - V,(s\p,) + 6. It follows that T,(S) > T,(S) if and only if V,(s\p,) - V,,(s) > r/l(s\&) - V,(s). For each i,j E N let y(s 1j) = Ti(s)/ [T;(s) + 7’,(s)]. We claim that for each i # ~1: 1n) - lqs 1n) 5 V,(s\/3,
qs\p,
1i)
- V,(s 1i)
(Al)
for each j # i, n; v((s\/?, 1j) - y(s I j) < Vn(s\pn 1j) - V,,(s 1j) . (AZ) As y((s\p,) - V,(S) is l/(n - 1) times the sum of the left hand sides of (Al) and (A2) and V,(s\p,,) - V,(s) is ll(n - 1) times the sum of the right hand sides, our conclusion thus follows directly from (Al) and (A2). To establish (Al) note that s an o-equilibrium implies ~(P~(s\P,~)) = t(~~(~\/3,))= th=max{t(p)Ir(p)=l}. For every jC5N let tj=t(pj(s)). s an w-equilibrium and the fact that n has the final draft pick implies t( p,(s\P,)) 2 t, and t( p,(s\p,)) = ti 2 t,. Thus (Al) is implied by, though not necessarily equivalent to, the following inequality: [ST, + t,]/[8T,
+ t, + 8T,, + t,] - [ST, + t,]/[ST, + t, + ST, + t,]
5 [8T, + tb]/[8Ti Rearranging
these
terms
[8T, + t,,]l[ST;
+ ti + ST,, + tb] - [8T, + t,]/[ST,
yields
the equivalent
the left hand
+ t, + ST, + t,] .
side of the inequality
[8T, + t, + tj - t,]/[ST, - [ST, + th]/[8Ti
Thus (Al) is verified. We now verify (A2). s an t( p,(s\p,)) 2 t, and t, 2 t,. Thus equivalent to, the following:
above
by:
+ ti + 8T,, + th] + ti + ST, +
t,,] .
w-equilibrium implies that (A2) is implied by, though
+ t, + 8T, + t,] - [ST, + t,]i[ST,
< [8T, + tJl[ST,
is dominated
+ t, + tj - t, + 8T, + t,]
= [ST, - 8T,, + ti - t,]/[8T,
[8Ti + t,]/[STi
inequality:
+ t, + ST, + t,] - [8T, + t,,]/[8T, + ti + 8T, + th]
5 [ST, - 8T, + ti - t,]/[ST, But ti 2 t, implies
+ tj + 8T, + t,] .
for each i f not necessarily
j,
+ t, + 8Tj + tj]
+ tj + ST, + th] - [ST,, + t,]i[ST,
+ tj + 8T, +
t,] .
V. Kolpin,
Rearranging
L. D. Singe11 Jr.
and simplifying
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algebraically
Social Sciences
26 (1993)
313
299-315
we see that this inequality
is equivalent
to: [8Tj + tj][tb - t,]/[ST, <[ST, This
final inequality
+ t, + ST, + t,][8T, + t, + ST, + t,]
+ tl][tb - t,]/[ST,
follows
directly
from
Proof of Theorem 2.3. First note that plicitly a function of T,. It follows: dq(s)/dT,
= -8[8Ti + 8[8T; +
Using
the fact that
+ t, + 8Tj + t,] .
Ti > T, and t,, > t,.
T;(S) = V,(S) - U;(s\/?,) + aj, which
+ t(p[(s))]/[ST,
+ t(p,(s))
0
is im-
+ ST, + t(pj(s))]*
t(Pi(S\Pi))]‘[8Ti + t(Pi(S\Pi)) + 8Tj + t(Pj(SVi))I*
of Ti(s) and rearranging
the definition
&-,(s)/dT,
+ t, + ST, + t,][8T,
the resulting
inequality,
.
it follows
>O if {[Ti(s\P,)
+ Tj(S\P,)]/[Ti(S)
+
Tj(S)I)*< T~(s\P~)‘T;(J).
(A31
Clearly (A3) is . e q uivalent to [ T,(s\&) + T,(s\~~)]~/T,(s\~~) < [T,(S) + T,(S)]‘/ T;(s), implying T,(S) is a locally increasing function of T, (local to the initial value of T,). To show that r,(s) is increasing for arbitrary increases in T,, (A3) implies that it suffices to verify that [ T,(s\p,) + T,(s\p,)] /[T,(s) + T,(s)] is everywhere weakly decreasing in T,. But for all T, > 0 a{[8Ti
+ t(Pi(S\Pi))
+8T,
+
t(Pj(SVi))I’[‘Tz+~(PI(s))+‘T,
+ t(P,(s))l)“Tj = s[t(p,(s))+ t(PiW)- t(p,WP,))- t(~,WiWW’i + t(~i(s>> + 8Tj the t(Pi(s))
latter +
+ t@(s))]*
inequality t(P,(S))’
following
50 from
the
fact
that
t(
p,(s\p,))
+
t(
pj(s\pi))
Z-
I7
Proof of Theorem 3.1. For convenience assume T, 2 T2 2 . . . P T, . Let s and s* be perfect equilibria under the direct- and reverse-order drafts respectively. Let q and p represent the number of black players hired in s and s*. Our conclusion is trivial if 7 = II so assume r] < p 5 12 and let A = max{i E N 1 r(pi(s) = 0}, i.e. the last team to hire a white player in a direct-order draft. It follows A 2 n - 7. Let 5 E N be the index of the team drafting the (7 + 1)th black player in the equilibrium s * , thus (‘Sn-T’A and T5?TA. Define wd = argmax {t(P) 14~) PF{P,(sL P,+,(s*)>>,
1r(p) = 1 and = 0 and PF’{P~(s), . . . , P,+,(S)>>, b, = argmax{t(p) . . . 2 P,-,(S))), w, = argmax{t(p) 14~) =O and P@~(P,(s*), ..., and b, = argmax{t(p) 14~) = 1 and pF’{p,(s*), . . , P~+~(s*>>>.
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26 (1993)
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Naive consistency implies that: (1) wd is the (rz - v)th most talented white; (2) b, is the (77 + 1)th most talented black; (3) t(w,) 2 t(wd); and (4) t(bd) 2 t(b,). Note that r(p<(s*)) = 1 implies 6 < W,(b,) - W,(w,) and thus t(b,) > t(w,). But 6 > W,(b,) - W,(w,,) 2 W,(b,) - WA(w,) 2 W,(b,) - W,(w,), the first inequality follows from r( p,(s)) = 0, the second follows from (3) and (4), and the final inequality follows from the fact that T5 2 Th and t(b,) > t(w,). As W,(b,) W<(w,) cannot be both strictly larger and strictly smaller than 6, we must conclude Al.I 7. 0 Proof of Theorem 3.2. Let r be an abstract league defined by N = {1,2}, P={w,,w,,b,,b,},andr(p)=O,0,1,1forp=w,,w,,b,,b2.LetG~~such that T,=1, T,=0.5,6,=6,=0.015,andt(p)=1,0.5, l.l,lforp=w,, w2,b1, b,. Backward induction reveals [T,(s), T2(s)] = [l, 0.571 and [T,(s*), T2(s*)] = [l.Ol, 0.561; w h ere s and s* are respectively the unique perfect equilibria under the direct- and reverse-order drafts. It follows the reverse-order draft decreases parity relative to the direct-order draft. Note that T, > T,, thus sufficiently small perturbations in ( Ti)iEN do not affect the order in which teams choose under either a direct- or reverse-order draft. Thus Lemma A implies that sufficiently small changes in (T;)iEN, (tSoitN, and (t(~))~_ do not affect the equilibrium status of s and s*. As (Ti(~))IEN and (TL(~*))iEN are continuous in (Tl)rEN, we conclude there exists an open set U E r such that (&)iEN, and (4 dptP G E U and the reverse-order draft decreases parity relative to the direct-order draft for all G’ E U, i.e. there is a large class of games for which the reverse-order draft decreases parity relative to the direct-order draft. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. Cl Proof
of Theorem
3.3. Let
r
be an abstract
league
defined
by N = (1,2,3),
l,l, Iforp=w,,w,,w,,b,,b,, p = {w,, w2, wj, b,, b,, b,},andr(p)=O,O,O, 6,. Let G E r such that T, = 0.4, T, = T, = 1, 6, = 8, = 6, = 0.004, and t(p) = 0.902, 0.901, 0.9, 1.002, 1.001, 1 for p = w,, w?, w3, b,, b,, 6,. Backward induction reveals that p,(s) = b,, and p2(s) = wr, p3(s) = w2; where s is the unique perfect equilibrium of G. Lemma A implies there exists an open set U C r such that G E U and s is an equilibrium of G’ for each G’ E U. Note that if the distribution of veteran talent changes to T, = 0.8 for all i E N, backward induction reveals that only white players are hired in perfect equilibrium. We conclude there is a large class of games for which an increase in parity can lead to fewer black players being hired in perfect equilibrium. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. 0 Proof
of Theorem
3.4. Let
r
be
an
abstract
league
defined
by N = (1,2),
=O, O,O, 1, 1, 1 forp = wi, w2, wj, b,, b,, p = {W,> w2, w3, b,, b,, b3}, andr(p) 6,. Let G E r such that T, = 1, T2 = 1, S, = 6, = 0.0084, and t(p) = 0.702, 0.701,
V. Kolpin, L. D. Singe/l Jr.
I Mathematical Social Sciences 26 (1993)
299-315
315
reveals 0.7, 1.002, 1.001, 1 for p = wl, w?, w3, b, , b2, b,. Backward induction that p,(s) = b,, p2(s) = b,; where s is the unique perfect equilibrium of G. Now let us suppose team 0 is introduced to G; where T, = 0.7 and 6,) = 0.0084. Further suppose team 0 is granted the first draft choice. Backward induction reveals that . where s* is the unique perfect equilibrium P&*) = b,, Pi = ~1, P&*) = ~2, of the expanded drafting game. Lemma A thus implies there is a large class of games for which expansion can lead to strictly fewer black players being hired in perfect equilibrium. Essential details of the example above can be generalized to apply to arbitrary abstract leagues. Details are available upon request. I7
References S. Atkinson, L. Stanley and J. Tschirhart, Revenue sharing as an incentive in an agency problem: example from the national football league, Rand J. 19 (1988) 27-43. K. Arrow, The theory of discrimination, in: 0. Ashenfelter and A. Rees, eds., Discrimination
an in
Labor Markets (Princeton University Press, Princeton, 1973). G.S. Becker, The Economics of Discrimination (University of Chicago Press, Chicago, 1957). L.M. DeBrock and A. Roth, Strike two: labor-management negotiations in major league baseball, Bell J. 12 (1981) 413-425. M. Kaneko and T. Kimura, Conventions, social prejudices and discrimination: a festival game with merrymakers, Games Econom. Behav. 4 (1992) 511-527. V. Kolpin and L.D. Singell, Jr., Strategic discrimination, scholarly performance, and the gender composition of economics departments, Mimeo, University of Oregon (1992). R.B. Myerson, Game Theory Analysis of Conflict (Harvard University Press, Cambridge, 1991). D.S. Neft and R. Cohen (eds.), The Sports Encyclopedia: Baseball (St. Martin Press, New York, 1977).