Equality of Opportunities and Welfare in a

Model of Job Allocation.

Pau Balart

June 10, 2012

Abstract

According to Roemer (1998), equality of opportunity must com-pensate disadvantaged individuals for circumstances beyond their con-trol but not for those under their responsability. We provide a theo-retical analysis of equality of opportunity that explicitly accounts forthis difference. To do this we need to include more than one source ofheterogeneity in the study of equality of opportunity. We show thatthe distinction between relevant and irrelevant circumstances is nottrivial, and the effects of equality of opportunity depends on it to agreat extent.

1 Introduction.

Under equality of opportunity individuals’ success must not depend on cir-

cumstances beyond their own control1. Such circumstances are called irrel-

evant, in the sense that they are ethically non-accountable. They must be

distinguished from relevant circumstances, which are legitimated to affect the

competition2. In order to provide equality of opportunity the former must

1 See Sen (1980), Dworkin (1981a, b), Cohen (1993) and Roemer (1998).2Gender or ethnicity are usually accepted as irrelevant circumstances, while preferences

are generally seen as a relevant one.

1

be compensated, but not the latter. Our objective is to provide a theoretical

analysis of equality of opportunity that explicitly accounts for this difference

and their effects under a competitive environment.

The analysis of equality of opportunity has often obviated the difference

between relevant and irrelevant circumstances. To take into account this

difference we need a model with more than one source of heterogeneity, in

order to distinguish each type of circumstance. In particular we consider dif-

ferences in preferences as a relevant source of inequality, while differences in

the ability to compete are assumed to arise from an irrelevant circumstance,

as gender or ethnicity. The inclusion of different sources of heterogeneity and

more than two individuals in a complete information framework gives cause

for the emergence of multiple equilibria that complicates the assessment of

the effects of equality of opportunity. We contribute to the literature analyz-

ing affirmative action effects on incentives to exert effort (Coate and Loury,

1993; Fryer and Loury, 2005a; Fu, 2006; Franke, 2007; Fain, 2009, Hickman,

2010; or Calsamiglia et al., 2010), accounting for the difference between rel-

evant and irrelevant circumstances. At the same time we study some other

considerations not included in previous works as allocation efficiency and

welfare.

Regardless of its fairness considerations, there is some controversy about

the effects of equality of opportunity on individuals’ performance. On one

hand, defenders of affirmative action argue that leveling the playing field

induces a more fierce competition that improves incentives to exert effort.

On the other hand, the detractors see affirmative action as a deterioration

of the requirements to win the competition, becoming less worthwhile mak-

2

ing effort. Economic literature goes beyond these arguments to provide an

analytical basis for the assessment of the consequences on effort of equality

of opportunity. Schotter and Weigelt (1992) and Calsamiglia et al. (2010)

use experiments to asses that affirmative action can improve incentives to

exert effort. Franke (2008) uses a rent-seeking contest with many players to

show that equality of opportunity often coincides with the effort maximizing

policy. Fu (2006) and Fain (2009) obtain similar results.

Our work is specially related to Fu (2006) because both model the com-

petition with an all-pay auction with complete information. However, we

extend Fu’s environment by adding individuals, prizes and sources of het-

erogeneity. A simple framework with four individuals and two sources of

heterogeneity is sufficient to generate less categorical conclusions about in-

dividuals’ incentives to exert effort. With the inclusion of a relevant source

of heterogeneity, effort could increase or decrease depending on whether the

competition is more or less fierce after equalizing opportunities. When there

is a unique source of heterogeneity, equality of opportunity fosters an in-

creased competition because candidates are more matched. However, with

two sources of heterogeneity it might be the case that the competition is

more biased after equalization because of the influence of uncompensated

relevant circumstances. Moreover, the emergence of multiple equilibria after

equalization, adds more ambiguity to the final effects of affirmative action.

In a recent and enlightening work, Hickman (2010) also find ambiguous

effects of affirmative action. He uses a multiple-object all-pay auction with

a continuum of agents that takes into account differences between relevant

3

and irrelevant circumstances3. Hickman’s remarkable point is that not all

low ability are in a competitive disadvantage, hence some targeted but ad-

vantaged individuals may be over-favored after affirmative action, which can

have a negative impact on effort. In our case all low ability individuals’ are

in fact in a competitive disadvantage (we assume that the irrelevant circum-

stance has more influence than the relevant one), but some of them may end

up in a competitive advantage because preferences are the unique source of

inequality after equalization. The approach used here is much less general

than the one in Hickman. However, we give some insights on the individ-

uals’ decision on whether participating or not in the competition, which is

one of the potential effects of affirmative action. Instead Hickman’s analy-

sis only concerns individuals that have already decided to take part in the

competition.

Accounting for preferences as an acceptable source of inequality (rele-

vant circumstance) allows to pay attention to an issue that has been largely

excluded from the analysis of equality of opportunity effects: allocation effi-

ciency. If preferences are considered as a relevant circumstance, then equaliz-

ing opportunities may increase valuations’ influence on final allocation. This

rises the probability of allocating positions to those individuals that value

them more, which induces potential improvements on total welfare. Allocat-

ing indivisible goods without a medium of exchange is a well known problem

(Shapley and Scarf, 1974). In these cases, competition does not guaran-

3Although Hickman does not use this terminology his analysis implicitly accounts forit. He includes two sources of heterogeneity. The first one concerns on socio-economicbackground while the second deals with individuals’ intrinsic disutility of exerting effort.He compensates the former but not the latter.

4

tee efficiency in the allocation4. Hylland and Zeckhauser (1979) discuss the

problem of efficiency when allocating individuals to position. Che and Gale

(2007) also argue that “selling the good at the market-clearing price may not

guarantee an efficient allocation”. This is analogous to what happens in job

allocation. If any group is disfavored by some circumstance as could be race

or gender then this group is less likely to obtain one of the better positions al-

though valuating it more highly. Here we are concerned to include allocation

efficiency in an auctioneering approach to equality of opportunity5.

Our findings present equality of opportunity not only as a way of obtain-

ing a fairer allocation, but also as a potential way of increasing welfare. If

the improvement in workers’ welfare arising from the more efficient alloca-

tion takes places without harming incentives to exert effort, then affirmative

action results in positive effects on total welfare. The improvement in total

welfare arises without the necessity of using a total welfare function that

gives special weight to most disfavored individuals6.

4This literature understands efficiency in allocation as allocating indivisible objects tohigher valuation individuals. With a transferable utility, this corresponds to the usualPareto notion. However, with some indivisible objects as jobs, or college seats it is notstraightforward the existence of a transferable utility. Although, not being possible todirectly sell positions, we assume that utility can be transferred by some other mean astaxation, hence we also care about efficiency in allocation.

5 See the seminal paper of William Vickrrey in 1961 for a study about the strategicincentives of individuals in auction contexts that leads to efficient allocation of objects.

6 Some theories of social justice associates more compensated allocations with gains intotal welfare (Rawls, 1971 or Sen, 1979). Generally this type of conclusion arise becausethey consider a specific functional forms for total welfare including ethical considerationor individual utility functions that provides a better outcome when there is a balanceddistribution of resources. In our case we have used a utilitarian approach for aggregatingwelfare. The utilitarian total welfare function only includes the weak ethical requirementof considering everybody’s welfare equally important. However, we are improving thesituation of disfavored individuals, so our results will still hold under other types of totalwelfare functions more concerned with disadvantaged individuals’ welfare.

5

Clark and Riis (1998) developed a complete information multiple-object

all-pay auction. Their framework fits well with the situation we want to rep-

resent here with the difference that after equalization we may deal with some

individuals with common valuation which gives cause for the emergence of

multiple equilibria. Since effort is continuous and the number of prizes dis-

crete there is no equilibrium in pure strategies, hence our results arise from

the equilibrium mixed strategies of the model7. Another senseful way to

represent the competition we are describing here is by some other allocation

mechanisms with unobservable effort as rent-seeking contests (Tullock, 1980)

or tournaments (Lazear and Rosen, 1981). Although non-observability of ef-

fort is an appealing property, including it involves the necessity of introducing

some assumptions on the source of uncertainty. These assumptions play an

important role in the expected allocation of prizes. Since the expected alloca-

tion is one of our focuses we renounce considering an imperfectly observable

effort.

We consider two different ways of equalizing opportunities. The first

one is quotas compensation which arises by the application of Roemer’s al-

gorithm. It consists in creating parallel competitions for advantaged and

disadvantaged individuals, respectively. We analyze it because it is the most

extended way of implementing equality of opportunity. The other type of

compensation that we apply is not very extended but has relevant theoret-

ical support. It consists in modifying the valuation of individuals’ perfor-

7 Fu (2006) provides a reasoning on the suitability of the mixed strategy with reallife behavior based on a Bayesian interpretation. According to Harsanyi (1973), almostall mixed strategy equilibrium can be approximated by the equilibrium of a nearby gamewith incomplete information.

6

mance in order to correct the bias in the competition. We refer to it as

returns to effort compensation. According to Calsamiglia (2009) it is the

unique way of providing equality of opportunity in a decentralized environ-

ment. This equalization is less extended than quotas, but we can see it for

instance in the public procurement auctions in California (Krasnokutskaya

and Seim, 2005). Hickman (2010) uses his stylized framework to compare

different compensation policies. He pays special attention to lump-sum and

quotas compensation, showing that the latter provides greater incentives to

exert effort than the former. In our case the multiplicity of equilibria arising

after returns to effort compensation prevents to obtain a solid conclusion.

Incentives to exert effort, expected allocation and welfare depends to a great

extent on the resulting equilibrium.

In section 2, we introduce the model. Then we characterize the equilib-

rium in the non-equalized competition (section 3) and under each type of

equalization (section 4).

In section 4 we compare the previous outcomes in order to compare the

effects of different compensations and a non-equalized competition.

2 The Model.

We use the model in Clark and Riis (1998) for competition over more than one

prize to characterize our framework. In a competition for jobs the employer

allocates a position to the most prepared candidates. In order to compete

for the better jobs the aspirants have to expend effort, making them more

eligible. In the case that they are not elected, they don’t recover the cost

7

of their preparation. This situation is well represented by an all-pay auction

with multiple prizes. Better positions are assigned to those individuals that

have presented higher bids (effort) and their costs are not refunded in case

of not winning.

We consider two types of jobs, that we call good and bad jobs, respec-

tively8. Everybody prefers the better jobs rather than the bad ones or in

other words, everybody agrees about which positions are better.

Call N the set of individuals that compete for positions. We classify

these individuals into four different types arising from the combination of

two binary sources of heterogeneity. The first source of heterogeneity is the

valuation of good jobs and the second one arises from individuals’ ability to

compete. These characteristics are common knowledge.

Heterogeneity in valuation only concerns good jobs. Everybody values

equally the less preferred jobs, assuming without loss of generality that its

value is zero. Agents differ in the valuations of the better jobs. We use a

large V to indicate individuals with a high valuation and v for those with

a relatively lower. We represent the value of the good job as GJj, where

j = V, v denotes individuals’ type according to valuation. We denote GJv =

W and GJV = βW , with β > 1, so β accounts for relative differences in

valuation.

The second source of heterogeneity consists in differences in ability to

compete. Because of past discrimination or an adverse social background

some individuals performs worse although exerting the same effort. We de-

note with A those individuals with a high ability to compete and with a those

8 Alternatively we can also interpret bad jobs as the situation of being unemployed,and everybody prefers working rather than being unemployed

8

with a low ability. The bid function of advantaged individuals is increased in

a multiplicative way, bA = φeA with φ > 1 while for disadvantaged types is

just their effort, ba = ea. We assume that this difference in the ability to com-

pete comes from an ethically non-accountable circumstance (i.e. irrelevant

circumstance).

By combining both sources of heterogeneity we have four different types

of agents. We use a two-character subscript, i.e. jk, in order to refer to each

type, we use j = v, V to denote the individuals type with respect to their

preferences and k = a, A to indicate whether the individual is favored or un-

favored with respect to the irrelevant circumstance. We assume that there is

one agent of each type. Therefore the set of players is N = {V A, vA, V a, va}.

We also assume that there are two prizes (denoting by n the number of prizes,

n = 2) that is, there are two vacancies in each type of job.

With respect to the heterogeneity parameters we assume that φ > β.

This means that heterogeneity in ability is more relevant than heterogene-

ity in preferences. Then, individuals with low ability are in a competitive

disadvantage independently on their valuation.

As we have said individuals expend effort in order to be eligible for better

jobs. Effort is a continuous variable, such that ejk ∈ <+. We assume that the

cost of effort is linear, and equal to the exerted effort, i.e. C(ejk) = ejk. Effort

transforms into bids. The competition is an all pay auction: first players

simultaneously make their bids. Only the two top bidders obtain a good job.

If two players make the same bid we assume that the probability of winning

is the same for both players. Independently of winning or not individuals

are not refunded for their effort. Hence their payoff can be expressed in the

9

following terms:

Ujk(ejk) =

GJj − C(ejk) if bjk is one of the two higher bids1pGJj − C(ejk) if jk ties in the second higher bid with p− 1 others

−C(ejk) otherwise

Clark and Riis (1998) characterize the equilibrium for a multiple-prize all-

pay auction. Following them we denote Gjk(bjk) the probability distribution

function that individual jk wins when he bids bjk and all of the other players

follow their equilibrium strategies. Then we can write the expected utility of

bidding bjk as:

EUjk(ejk) = Gjk(bjk)GJj − C(ejk)

where j = {V, v} and k = {A, a}. We can use the bid function in order

to write the cost of effort as a function of bids. In that case we have an

ability-type specific cost function, i.e. C(e(bjk)) = Ck(bjk). In particular,

C(bjA) = 1φbjA and C(bja) = bja for j = V, v. Then we rewrite expected

utility in terms of bids:

EUjk(bjk) = Gjk(bjk)GJj − Ck(bjk)

with j = V, v and k = A, a. Individuals maximize their expected utility.

3 No Equalization

First of all we analyze the outcome of an uncompensated competition. A

non-equalized competition treats equally all-candidates. However, no dis-

crimination or equal treatment is not enough to guarantee equality of op-

portunity. An uncompensated competition is biased in favor of advantaged

10

candidates. We want to find the final allocation of positions and computing

the welfare level that arises from it. Denoting by Σjk the set of mixed strate-

gies of individual jk over his bids’ support [bljk, bujk] ⊂ <+ we can define the

game as follows:

Definition 1. The non-equalized competition outcome is the equilibrium of

the following game: (N,Σ,U).

Solving a non-equalized competition is equivalent to solve a sealed bid all-

pay auction with two prizes and four players with strictly different valuations

and complete information9. The solution to this kind of game has been

proposed in Clark and Riis (1998).

Proposition 1. Under a non-equalized competition, there is a unique equi-

librium. Individuals play the following mixed strategies:

FV A(bV A) = 1−√

βW−bV AW with bV A ∈ [(β − 1)W,βW ]

FvA(bvA) =

1 if βW ≤ bvA

1−√

βW−bvA

β2Wif (β − 1)W ≤ bvA < βW

1− βW−bvAβW if 0 ≤ bvA < (β − 1)W

0 if bvA < 0

FV a(bV a) =

1 if βW ≤ bV a

1−√

βW−bV a

φ2Wif (β − 1)W ≤ bV a < βW

1− βW−bV aφW if 0 ≤ bV a < (β − 1)W

0 if bV a < 0

9In fact we are solving a model with only two strictly different valuations but also

with differences in cost. An affine transformation of expected utilities reveals that this is

analogous to a situation with four different valuations, since individuals behave identically.

11

With the following payoffs for each individual:

UV A = βW (1− 1φ) UvA = W (1− β

φ) UV a = 0 Uva = 0

The expected probabilities of obtaining one of the best jobs are:

PV A = 1− 13βφ PvA = 1− β

2φ + 16βφ PV a = 1

3βφ + β2φ −

16βφ PV a = 0

And the expected level of effort of each player:

eV A = βW ( 1φ −

13βphi) evA = βW ( 1

2φ + 16β2φ

) eV a = (1+3β2)W6φ eva = 0

We can directly apply the work developed by Clark and Riis (1998) to

characterize this equilibrium. The unique difference is that our model con-

siders a difference in cost of bids instead in valuations. However an affine

transformation of expected utility of high ability individuals reveals that we

can treat the heterogeneity in cost equivalently to a heterogeneity in valua-

tions without affecting individuals’ behavior.

Here we obtain the principal variables that we will use in our analysis.

We will compare them with the ones that arises after compensation in order

to obtain conclusions about effects of equality of opportunity.

4 Equalizing Opportunities

We assume that the heterogeneity in ability comes from an irrelevant circum-

stance. There is some group that because of some circumstance beyond their

control is more able to compete, producing more attractive candidatures with

their effort. For instance, we may want that belonging to a minority may

not influence individuals’ personal success. However, because of some differ-

ences in communication habits, dress style or non verbal communication, the

12

minority effort may produce a less profitable output for the principal. No

discrimination is not sufficient to correct minority disadvantage. We need af-

firmative action in order to balance the competition in favor to the minority.

However, this may represent a cost for the employer, since minority effort is

less productive for him.

Heterogeneity in valuations is a relevant circumstance, then it shouldn’t

be compensated. Since we only assume two different sources of heterogeneity,

in an equalized competition valuations are the unique source of heterogeneity.

In this section we want to compute the outcomes of competitions modified

to satisfy equality of opportunity. We explore the consequences of two dif-

ferent equalization policies: returns to effort and quotas. Equalizing returns

to effort consists in eliminating differences in bids when individuals make

the same effort level. We talk about quotas compensation when applying

the algorithm proposed by Roemer (1998), consisting in equalizing outcomes

with respect to the relative degree of effort.

The first we should do is to specify each type of compensation and argue

how we should modify our benchmark competition in order to represent the

equalized competitions.

4.1 Equalizing returns to effort.

To equalize returns to effort, all individuals that exert the same effort, must

obtain the same returns, i.e. the same probabilities of winning10.

We model returns to effort compensation policy τ( ) as a function of the

10If we had differences in costs of effort instead of in the bid function, this is equivalentto say that they must obtain the same probabilities of winning when they inccur into thesame disutility.

13

initial bid function and denote by b the compensated bids, where b = τ(b).

Σ is the set of mixed strategy under returns to effort compensation.

Definition 2. A returns to effort compensation (RE) is a policy such that

bA = τ(bA) = bA for advantaged individuals and ba = τ(ba) = φba for disad-

vantaged individuals.

We define a returns to effort equalized competition as the all pay auc-

tion that assigns positions to individuals according to the following game:

(N, Σ, U)

Returns to effort compensation consists in improving the valuation of

disadvantaged candidates’ bids. By applying it we eliminate one source of

heterogeneity hence the set of individuals becomes more homogenous. We

have two pairs of identical players, denote: I = [V, V ′, v, v′].

Proposition 2. Under Returns to effort compensation (RE) there is a con-

tinuum of mixed strategies equilibria.

1. In the asymmetric mixed strategies’ equilibrium with three active play-

ers, distribution functions are:

FV (bV ) = FV ′(bV ′) = 1−√

φW−bVφW with supports bV , bV ′ ∈ [0, φW ]

Fv(bv) =

1−√

φW−bvφβ2W

if v′ is inactive

0 if v′ is active

Fv′(bv′) =

1−√

φW−bv′φβ2W

if v is inactive

0 if v is active

with supports bv, bv′ ∈ [0, φW ].

14

The following payoffs for each individual:

UV = UV ′ = (β − 1)W Uv = Uv′ = 0

The expected probabilities of obtaining one of the best jobs are the next ones:

PV = PV ′ = 1− 13β

Pv =

23β if v′ is inactive

0 if v′ is active

Pv′ =

23β if v is inactive

0 if v is active

And expected effort levels:

eV = eV ′ =23W

ev =

23β if v′ is inactive

0 if v′ is active

ev′ =

23β if v is inactive

0 if v is active

2. In the symmetric mixed strategies’ equilibrium with four active players:

Payoffs are:

UV = UV ′ = (β − 1)W Uv = Uv′ = 0

The expected probabilities of obtaining one of the best jobs are the next ones:

15

PV = PV ′ = 2(2+√

(β−1)β)−1

6β Pv = Pv′ = 12 −

2(2+√

(β−1)β)−1

6β

The expected level of effort of each player:

eV = eV ′ = 16W (9−6β+ 2

√(β − 1)β) ev = ev′ = (1

2 −2(2+

√β−1+β)−16β )W

We have characterized the two extreme outcomes of this game; the sym-

metric equilibrium with all players bidding in the same supports and the

asymmetric equilibrium with an inactive player. As it arises from Baye et

al. (1990) there is a continuum of equilibria, with one of the low valuation

candidates bidding in the same interval than high valuation types and the

remaining one bidding continuously in an interval with an arbitrary lower

support greater than zero and concentrating a mass of probability at zero.

The proof is in the appendix. We will focus our analysis in the extreme out-

comes and we skip the continuum of intermediate equilibria. We will refer to

the two cases we study as the symmetric and asymmetric one, respectively.

4.2 Quotas compensation.

In Roemer 98 he asserts that if a group of individuals with a similar irrelevant

circumstance exerts less effort than another group, they should not be held

accountable for their generalized lower levels of effort. To correct this it is

necessary to establish an intra-group competition in which individuals only

compete against their fellows. We call this policy a quotas compensation.

In our model this means to separate players into two parallel competitions,

one for the favored individuals and another for the unfavored ones.

Definition 3. Quotas compensation (Q) consists in:

16

1. Take the set N = {V A, vA, V a, va} and make the following partition:

NA = {V A, vA} and Na = {V a, va}.

2. Assign one of the prizes to NA and the other to Na, then apply an

all-pay-auction to each group.

3. The highest bidder of each group obtains a prize.

Proposition 3. Under quotas compensation there is a unique equilibrium.

Individuals play the following mixed strategies:

FV A(bV A) = 1− φW−bV A

φWFvA(bvA) = 1− φW−bvA

φβWwith supports

bjA ∈ [0, φW ]

FV a(bV a) = 1− W−bV a

WFva(bva) = 1− W−bva

βWwith supports bja ∈ [0,W ]

With the following payoffs for each individual:

UV A = (β − 1)W UvA = 0 UV a = (β − 1)W Uva = 0

The expected probabilities of obtaining one of the best jobs are the following:

PV A = 1− 12β

PvA = 12β

PV a = 1− 12β

PV a = 12β

The expected level of effort of each player:

eV A = 12W evA = 1

2βW eV a = 1

2W eva = 1

2βW

The outcome of this proposition arises immediately by applying the sealed-

bid all-pay auction with a unique prize, incomplete information and common

valuation characterized in Hillman and Riley (1989) or in Baye et al. (1990).

17

5 Allocation, Welfare and Effort.

By using the outcomes generated in the previous section we can study the

effects of applying equality of opportunity. A simple framework with four

individuals, two sources of heterogeneity and multiple prizes is sufficient to

generate a multiplicity of equilibria. This generates ambiguous conclusions

on the consequences of equality of opportunity.

First, we focus on the comparison between different ways of providing

equality of opportunity. To our knowledge Hickman (2010) is the unique

theoretical work that compares different equalization methods (Calsamiglia

et al., 2010 also do it experimentally). Hickman pays special attention to

the comparison of lump-sum compensation (Michigan system) and quotas.

Instead here we compare quotas system with returns to effort compensation11.

Our conclusions depends on a large extent on the type of equilibrium that

arises after returns to effort compensation.

In particular we compare the effects of this two forms of equality of oppor-

tunity on allocation efficiency, total effort, workers welfare and total welfare.

In order to compute the probability of achieving an efficient allocation we

must compute the probability of allocating high valuation individuals to the

better jobs. For total effort we aggregate the individuals’ effort in previ-

ous propositions. For workers welfare we add workers’ payoff under each

equalization outcome12.

11Both lump-sum (Michigan system) and returns to effort compensation are particularcases of what Hickman called admission preference systems.

12Alternatively, we can also consider an auctioneer that only captures the bids of hired

individuals. This interpretation fits well with selection processes that requires a very

specific preparation, useless for other recruiters. Auctioneer’s payoff in this case is the

18

Proposition 4.

1. For a β sufficiently large the allocation is more efficient with returns to

effort compensation than with quotas.

2. Total effort after quotas compensation is lower than in the asymmetric

equilibrium of a returns to effort compensation but greater than in the

symmetric equilibrium.

3. Workers’ welfare is the same under all equalization policies.

4. Total welfare after quotas compensation is lower than in the asymmetric

equilibrium of a returns to effort compensation but greater than in the

symmetric equilibrium.

The proof of this proposition arises from the comparison of the outcomes

in the previous sections. It is left in the appendix. All policies are equivalent

in terms of workers welfare, but produce different outcomes in terms of all

other aspects. However we can not argue that any policy is superior to the

other. Their relative outcomes are absolutley conditional on the equilibrium

that arises after returns to effort compensation.

In the asymmetric equilibrium of a returns to effort compensation we ob-

tain a more efficient allocation than in quotas compensation. Instead, in the

symmetric equilibrium we have four active players. This makes more diffi-

cult to obtain an efficient allocation. We need preferences to be sufficiently

relevant in order to produce a more efficient allocation in the symmetric

equilibrium than in the quotas compensation.

sum of winners’ expected bids:∑

jk∈N E[Gjk(bjk)bjk].

19

Incentives to exert effort depend on how balanced is the competition as

well as on the number of players. Quotas compensation creates two asym-

metric competitions with two individuals. This makes more likely for each

type of high ability individual to win their respective competition, and re-

duces incentives to exert effort with respect to the asymmetric returns to

effort compensation. On the other hand, in the symmetric equilibrium we

have a biased competition with four individuals. Probabilities of winning

decrease for low ability individuals, because they are competing against a

two high ability types and against a low ability individual.

Since all policies provide the same level of workers’ welfare the effects on

total welfare depends on differences in the auctioneer’s payoff. Then total

welfare is greater in the equilibria with better incentives to exert effort.

We want to compare not only differences between different compensation

policies but also with respect to the uncompensated case.

Proposition 5.

1. If β is sufficiently large, then the expected allocation under equality of

opportunity is more efficient than in a non-equalized competition.

2. Total effort is greater in a non-equalized competition than in the asym-

metric equilibrium of a returns to effort compensation13 if φ ≤ 1−β+9β2+3β3

8β+4

and than in an equilibrium with quotas compensation if φ ≤ 1−β+9β2+3β3

6β+6.

If β is sufficiently large, then total welfare is greater in a non-equalized

competition than in the symmetric equilibrium of a returns to effort

compensation.

13This is assuming that the inactive individual is va. The condition is more restrictive ifinstead we consider vA as the inactive player, however results do not change significantly

20

3. If β is large enough, workers welfare is greater in compensation than

without equalizing opportunities.

4. Total welfare is greater with a non-equalized competition than in the

asymmetric equilibrium of a returns to effort compensation if φ ≤3β2+12β+1)

8(β+1)+ 1

8

√57β4−72β3−26β2+40β+1

(1+β)2and than in equilibrium after quo-

tas compensation if φ ≤ 3β2+13β−2)6(β+1)

+ 16

√45β4−30β3+25β2−40β+4

(1+β)2. If β is

sufficiently large, then total welfare is greater with a non-equalized com-

petition than in the symmetric equilibrium of a returns to effort com-

pensation.

We can proceed as in the proof of the previous proposition to obtain

these results, therefore we have skipped it. Statement one of this proposition

shows that a more efficient allocation may arise after equalization. This is

always true in the asymmetric equilibrium of a returns to effort compensation,

since preferences are more relevant in determining the final allocation than

in the a non-equalized competition case. In the symmetric equilibrium and

in quotas compensation there is an additional low valuation active player

than in the a non-equalized competition that makes more difficult to obtain

an efficient allocation. Then, for observing an improvement in allocation

efficiency, preferences should have an influence large enough to compensate

the presence of an additional low valuation active player.

By evaluating the effects on effort, results are ambiguous because of the

heterogeneity in preferences. If the irrelevant circumstance was the unique

source of heterogeneity, equality of opportunity induces a more balanced

competition with a positive effect on incentives to exert effort. Instead, with

21

two heterogeneities compensation do not necessarily provide a more matched

competition. If in the uncompensated case relevant circumstances counter-

balanced some of the asymmetry arising from irrelevant differences, it can

arise a more biased competition after compensation with a negative effects

on incentives to exert effort. As we can see in the proposition’s expression,

this depends on the relative size of β and φ. If differences coming from the

irrelevant circumstance are large enough relatively to preferences, equality of

opportunity positively affects total effort. The intuition is the following: if

φ is relatively large, the irrelevant circumstance induces a huge bias, that is

mitigated once we implement equality of opportunity. On the other hand, if

differences in preferences were relatively larger, heterogeneity in ability can-

cels out some of their influence on the final outcome. When this is the case,

compensation has a negative impact on incentives to exert effort.

Comparing the net utility of workers we observe by comparison that their

welfare level increase with compensation if β > 3φ2+φ

14. In order to guarantee

an increase in workers welfare is enough with a condition that only concerns

preferences’ parameter. If this is the case, the gain from a more efficient

allocation is sufficently large to generate an increase in workers’ welfare.

Finally total welfare depends on the relationship between workers’ wel-

fare and effort incentives. On one hand, if β is sufficiently large equality

of opportunity increases workers’ welfare, but on the other hand it may de-

crease incentives to exert effort. If the first effect exceeds the second then

total welfare increases after equalization. This may happen under quotas

and in the asymmetric equilibrium of a returns to effort compensation if the

14Notice that this always holds for β > 3.

22

negatives effects on effort are small enough (β sufficiently small with respect

to φ). However, this never happens in the symmetric equilibrium of a returns

to effort compensation if β is great enough.

Again we see that conclusions depend in a great extent not only in the

way of providing equality of opportunity but also on the multiplicity of equi-

librium that may rise after it.

6 Conclusions

As Roemer points out in his seminal book Equality of Opportunities “the

most common objections to the Eop policy are these: first, (...) providing

low incentives to members of those (disadvantaged) types to increase their

effort, and second, that it will be socially inefficient in sending too many re-

sources on disadvantages types” Some recent economic literature neutralizes

the first critique assessing the effort enhancement effects of equality of op-

portunity. However, in Hickman (2010) and in the present job we can see

that these conclusions are not always true and depend on how balanced is

the competition after equalization. With respect to the potential inefficien-

cies of affirmative action, we show that in fact it may have positive effects

on total welfare arising from a more efficient allocation. If workers’ welfare

increases without harming disproportionately the incentives to exert effort

we may observe an improvement in total welfare.

We show that a slight extension of Fu (2006), including the distinction

between relevant and irrelevant sources of inequality, is sufficient to van-

ish certain effort enhancing effects of equality of opportunity. Instead, in our

23

framework this depends on whether the competitors are more or less matched

after equalization. The presence of differences in preferences gives compet-

itive advanatage to some candidates, which may provie a more asymmetric

competition after compensation, with negative consequences on incentives to

exert effort.

Hickman (2010) presents quotas as a better way of providing equality of

opportunity than a lump-sum compensation (Michigan system). Here we do

a similar analysis between returns to effort and quotas systems. The multi-

plicity of equilibria suggest to be cautious in the extraction of conclusions.

Their effects on incentives to exert effort, expected allocation and welfare

depends to a great extent on the resulting equilibrium.

Finally, previous literature with a unique source of heterogeneity argues

that firms or institutions trying to maximize the bids of individuals should

adopt an equalization admission policy. In our case this is no longer true

(auctioneer payoff could increase or decrease after equalization). This has a

direct policy implication. According to the conclusions with a unique het-

erogeneity we may expect firms or institutions to implement equalization

policies with the unique aim for maximizing total effort of their candidates.

However many times equality of opportunity and affirmative action policies

are only carried on if there is some law enforcement. The present work fits

well with this reality and gives some arguments in terms of total welfare of

why public authorities may have reasons to require firms or institutions to

apply affirmative action.

24

7 Appendix

7.1 Proof of Proposition 3.2.

We can proceed similar to Clark and Riis (1998) to find the equilibrium

of a multiple-prize all-pay auction with complete information. The unique

consists in the fact that here there are individuals with an identical valuation.

In this sense the proof is close to Baye et al. (1990).

We have four individuals competing for two positions or prizes, notice that

since we have removed heterogeneity in ability we have two couples of iden-

tical individuals that differ only in their valuations, that is I = {V, V ′, v, v′}

with V = V ′ > v = v′. Hence, under returns to effort compensation we

only need to use one subscript, i ∈ I. Since we are considering the all-pay

auction after returns to effort compensation, we deal with compensated bids b

Step1: The number of players with a bid higher than zero is at least three.

First of all denote by m the number of players that bid something strictly

greater than zero with positive probability, from now on we will call them

the active players. Notice that to have an equilibrium necessarily m > 2

(i.e. the number of active players is at least the number of prizes, n = 2).

Otherwise, nobody has incentives to bid a positive amount, which is never

an equilibrium.

Step2: Mixed strategy equilibrium distribution functions are continuous above

zero.

Assume on the contrary that individual i bids b′ with a strictly positive mass

25

of probability. Then for any j 6= i, limε→0 Gj(b′ − ε) < Gj(b′)15, ε ∈ (0, τ ]

with τ sufficiently small. This implies that player j 6= i never plays in the

interval [b′ − ε, b′] so then for the player i, Gi(b′) = Gi(b′ − ε) which implies

that ui(b′ − ε) > ui(b

′), hence b′ can never be in the strategy’s supports of

player i reaching a contradiction.

Step3: bui = bu = maxi{bui } for at least three individuals.

bui denotes individual i’s strategy upper support. Then, at least three indi-

viduals coincides in their strategies’ upper support (bu). Otherwise if only

two players bid bu with a strictly positive probability they win with certainty.

But denoting bu′

the second highest upper support and because the continu-

ity of F (b) above zero, G(bu′) = G(bu) = 1 which implies that u(bu) < u(bu

′),

hence bu cannot be in the support of any player.

Step 4: Gi(bli) ≥ 0 (with > if bu < GJi).

By continuity of F ( ) above zero, and the definition of bu, necessarily G(bu) =

1. Denote by bli the individual i’s strategy lower support, then:

ui(bli) ≥ ui(b

u)

G(bli)GJi − bli ≥ GJi − bu

Gi(bli) ≥ 1− bu − bliW

≥ 1− bu

GJi≥ 0

and notice that if GJi > bu then last inequality strictly holds.

15Gj( ) is a linear function of other players’ equilibrium distribution functions Fi( ) with

i 6= j, so then if for some player i, Fi( ) is discontinuous (i.e. limb→b′+ Fj(b)) = Fj(b′),

then a linear function of it (i.e.Gj( )) must also be discontinuous

26

Step 5: bli = bl = min{bli} for at least three individuals.

Otherwise denote by bl′> 0 the second minor lower support. Then if only

two or less players have bl as their lower support notice that any player with

bli = bl, bidding any b′i ∈ [bl, bl′] implies that G(b′) = 0, which induces a neg-

ative expected utility. Hence b′i > 0 cannot be in the supports of the mixed

strategy equilibrium distribution function, which is in contradiction with the

continuity of it above zero.

Step 6: Two players bid the lowest support with strictly positive probabil-

ity, that is F (bl) > 0 for at least two players.

By step 3, bui = bu for at least three individuals. Then bu must be lower than

the third maximum affordable compensated bid (i.e. the one that provides

an expected utility equal to zero when the probability of winning is equal

to one). In model notation, bu ≤ φW , which joint with step 4 means that

Gi(bli) > 0 for i = V, V ′. By step 5 of this proof at least three individuals

have bli = bl, so then necessary one of the individuals with higher prefer-

ences, V, V ′ has bli = bl as his lower support. Since GV (blV ), GV (blV ′) > 0 and

the lower support is bl for at least one high-valuation individual, necessarily,

GV (bl) > 0 or GV ′(bl) > 0, for at least one of them. In order to exist one

player with strictly positive probability to win by bidding the lower support

we need that at least two individuals concentrate a mass of probability in

the lowest support Fi(bl) > 0.

Step 7: Two players bid something strictly greater than the lowest support

with certainty, that is Fi(bl) = 0 at least two players.

27

We know, from step 6, that Fi(bl) > 0 for at least two players. Notice that

if Fi(bl) > 0 for an additional player then nobody has incentives to bid bl

because limb→bl+Gi(b) > Gi(bl), which is incompatible with Step 6.

Step 8: In equilibrium bu = φW

Since Fi(bl) = 0 for 2 players, this means that Gj(bl) = 0 for the remaining

two and hence their expected utility is zero when bidding bl or any other

bid in their strategies’ support. By Step 4 of this proof the individuals with

a zero probability of winning after bidding the lowest support must be low

valuation ones, so Gv(bl) = Gv′(bl) = 0 Then necessarily bl = 0, otherwise low

valuation individuals would obtain negative utility. By step 3, at least one of

the low valuation players has bui = bu. Given that their expected payoffs are

zero for any bid in their equilibrium supports, we can substitute G(bu) = 1

in their expected utility EUv(bu) = Gv(bu)W − bu

φand equalizing it to zero

we obtain bu = φW

Step 9 : Continuum of equilibria:

From previous steps high valuation players are always active and low valu-

ation one’s bid zero with a positive mass of probability. By step 3, one of

them is active for sure, and then must play with a continuous distribution

of probabilities over the supports [0, φW ]. Assume without loss of generality

that v is always active. Following the same reasoning as Theorem 5 in Baye

et al. (1990) the action of v′ determines a continuum of equilibria. On one

hand when Fv′(0) = 1, v′ is inactive an asymmetric equilibrium with only

three active agents arise. On the other hand he can mimic individual h strat-

28

egy. Then arises a symmetric equilibrium with four active players. Finally,

following Baye et al. (1990) there is a continuum of intermediate situations

with player v′ bidding 0 with a positive probability and bidding Fv′(b) = 0

∀b such that 0 < b < b and Fv′(b) = Fv(b) ∀b ≥ b, where b is some arbitrary

value such that b ∈ [0, φW ]. We characterize the two extreme cases of a

symmetric and an asymmetric with inactive player equilibria.

Step 10 : Symmetric Equilibrium:

When both low valuation individuals bid continuously in the supports [0, φW ]

the symmetric equilibrium arises. Using the fact that Gi(φW ) = 1 for any

i ∈ I and since in mixed strategy payoffs are the same for any bid in the

individuals’ support:

βW −W = Gi(bi)βW −biφ

Gi(bi) = 1− φW − biβφW

for i = V, V ′ and

Gi(bi)W −biφ

= 0

Gi(bi) = 1− φW − biφW

for i = v, v′

Since equilibrium distribution functions are symmetric for low preference

players, in equilibrium this must be also true for higher preferences ones, since

we know that their supports and their conditional probabilities of winning

are the same. We can find the equilibrium distribution functions by solving

29

the following system of equations:

1−W − biβW

= 1−(1−Fv(bi))2FV (bi)−2(1−Fv(bi))(1−FV (bi))Fv(bi)−(1−FV (bi))(1−Fv(bi))2

for i = V, V ′

1−W − biβW

= 1−(1−FV (b′i))2Fv(b′i)−2(1−FV (b′i))(1−Fv(b′i))FV (b′i)−(1−FV (b′i))

2(1−Fv(b′i))

for i = v, v′

Unfortunately, we can not find a closed form solution for this system.

However, it is not necessary for our analysis.

Step 11 : Asymmetric Equilibrium:

Proceeding similarly than before in the asymmetric equilibria with three

active players’ equilibrium distribution functions we find:

GV (bV ) = 1− φW − bVβφW

= 1− (1− FV ′(bV ))(1− Fv(bV ))

GV ′(bV ′) = 1− φW − bV ′

βφW= 1− (1− FV (bV ′))(1− Fv(bV ′))

Gv(bv) = 1− φW − bvφW

= 1− (1− FV (bv))(1− FV ′(bv))

Finally we can compute each of the items shown in the proposition.

• Net Utilities.

Since the payoff is the same for any bi in the equilibrium strategy

supports we can use the winning probabilities for a given bid obtained

before, in order to compute expected net utilities.

EUi = Gi(bi)GJi − Ci(bi)

30

for all i ∈ I.

Substituting bu for φW and Ci( ) for each player cost function we find

the payoffs stated in the proposition.

• Total probabilities of allocation.

In each type of equilibrium we obtain different probabilities of alloca-

tion and different expected efforts.

In order to find such a probabilities first of all notice that the distri-

bution functions conditional on bidding greater than individual s 6= i

greater support (i.e. bi ≥ bls) are symmetric, that is, Fi(b|b ≥ bli) =

Fs(b)16.Using this we know that when bidding in the same interval all

agents have the same probabilities of winning, then it is easy to compute

expected winning probabilities. Using that bli = 0, in the asymmetric

case we can find:

PV = PV ′ = 1− 1

3(1− Fv(0)) = 1− 1

3β

Pv = 2− PV − PV ′ =2

3β

In the symmetric case, it is a bit more difficult, since we do not have

a closed form solution for distribution functions. However, we know

that GV (0) = GV ′(0) = Fv(0)Fv′(0). Then since we know that GV (0) =

GV ′(0) = 1− 1β

and in a symmetric equilibrium, Fv(0) = Fv′(0) = Fo(0),

we can find that Fo(0)2 = 1 − 1β. Then Fo(0) =

√1− 1

β. Using Fo(0)

16Showing this is immediately by applying Fi(b|b ≥ bls) = Fi(b)−Fi(bls)

1−Fi(bls)

and substituting

bli by the expression found above.

31

we can find the symmetric expected probability of winning of each

candidate:

PV = PV ′ = Fo(0)2 + 223Fo(0)(1− Fo(0)) +

12

(1− Fo(0))2 =

=2(2 +

√(β − 1)β)− 1

6β

And for low valuation individuals:

Pv = Pv′ =1

2(1− PV − PV ′) =

1

2−

2(2 +√

(β − 1)β)− 1

6β

• Effort level. Finally we can compute individuals’ effort level by de-

ducing net utilities from gross utilities. For gross utilities we need to

multiply the probabilities of winning the prize time the valuation of the

prize without deducing the cost of the bid. Formula gross utility:

EU i = PiW

for all i ∈ I

ei = EU i − EUi

for all i ∈ I In the asymmetric case with three active players:

eV = eV ′ = (1− 13β

)βW − (β − 1)W = 23W

ev = 23βW

32

7.2 Proof of Proposition 3.4.

1. In order to compute the probability of achieving an efficient allocation

we must compute the probability of allocating high valuation individ-

uals to the better jobs. In an asymmetric equilibrium of a returns to

effort compensation, the probability that both high valuation individu-

als wins is 1−Pv, i.e. the probability that the low ability individual do

not win. Although do not having a closed form solution for equilibrium

distribution functions in the symmetric equilibrium of a returns to ef-

fort compensation we can find the expected probability of an efficient

allocation. We can proceed similarly as in the development of high

valuation individuals’ expected probabilities of winning in the proof of

proposition 2. Then the probability of an efficient allocation in the

symmetric equilibrium of a returns to effort compensation is:

Fo(0)2 + 21

3Fo(0)(1− Fo(0)) +

1

6(1− Fo(0))2 =

=2(2 +

√(β − 1)β)− 1

6β

Finally after quotas compensation, since both competitions are inde-

pendent the probability of an efficient allocation is PV A PV a = (1− 12β

)2.

By comparing all these values we observe that in an asymmetric equilib-

rium of a returns to effort compensation the probability of an efficient

allocation always exceeds that in a quotas compensation. Instead in

the symmetric equilibrium case we need a β large enough.

2. To compare total effort we need to aggregate all individuals’ effort

found in propositions 2 and 3. Statement three arises automatically by

33

comparing these magnitudes.

3. We can see from propositions 2 and 3 that the sum of workers’ payoff

is the same in both cases.

4. Total welfare takes into account workers’ welfare and principal’s pay-

off. The payoff of the principal is equal to the sum of uncompensated

bids, i.e.∑

i bi. Since workers’ welfare is the same under all types of

equalization, the sum of individuals’ bids determines the differences in

total welfare.

References

[1] M.R. Baye, D. Kovenock, and C.G. de Vries. The all-pay auction with com-

plete information. Economic Theory, 8(2):291–305, 1996.

[2] C. Calsamiglia. Decentralizing Equality Of Opportunity. International Eco-

nomic Review, 50(1):273–290, 2009.

[3] C. Calsamiglia, J. Franke, P. Rey-Biel, U.A. de Barcelona, and TU Dortmund.

The Incentive Effects of Affirmative Action in a Real-Effort Tournament.

University Autonoma Barcelona, Working Paper, 2010.

[4] D.J. Clark and C. Riis. Competition over more than one prize. The American

Economic Review, 88(1):276–289, 1998.

[5] S. Coate and G.C. Loury. Will affirmative-action policies eliminate negative

stereotypes? The American Economic Review, 83(5):1220–1240, 1993.

[6] G.A. Cohen. Equality of what? On welfare, goods and capabilities. The

quality of life, 9:29, 1993.

34

[7] R. Dworkin. What is equality? Part 1: Equality of welfare. Philosophy and

public affairs, 10(3):185–246, 1981.

[8] R. Dworkin. What is equality? Part 2: Equality of resources. Philosophy and

Public Affairs, 10(4):283–345, 1981.

[9] J.R. Fain. Affirmative action can increase effort. Journal of Labor Research,

30(2):168–175, 2009.

[10] J. Franke. Does Affirmative Action Reduce Effort Incentives?: A Contest

Game Analysis. RUB, Dep. of Economics, 2010.

[11] R.G. Fryer Jr and G.C. Loury. Affirmative action and its mythology. The

Journal of Economic Perspectives, 19(3):147–162, 2005.

[12] Q. Fu. A theory of affirmative action in college admissions. Economic Inquiry,

44(3):420, 2006.

[13] John C. Harsanyi. Games with randomly disturbed payoffs: A new rationale

for mixed-strategy equilibrium points. International Journal of Game Theory,

2:1–23, 1973.

[14] B. R. Hickman. Effort, Achievement Gaps and Affirmative Action: A New

Look at College Admissions. University of Iowa. Mimeo.

[15] A.L. Hillman and J.G. Riley. Politically Contestable Rents and Transfers.

Economics & Politics, 1(1):17–39, 1989.

[16] A. Hylland and R. Zeckhauser. The efficient allocation of individuals to po-

sitions. The Journal of Political Economy, 87(2):293–314, 1979.

[17] E. Krasnokutskaya and K. Seim. Bid preference programs and participation

in highway procurement auctions. American Economic Review.

35

[18] E.P. Lazear and S. Rosen. Rank-order tournaments as optimum labor con-

tracts. The Journal of Political Economy, 89(5):841, 1981.

[19] J. Rawls. A theory ofjustice. Cambridge, MA: Harvard UP, 1971.

[20] J.E. Roemer. Equality of opportunity. Harvard University Press, 1998.

[21] A. Schotter and K. Weigelt. Asymmetric tournaments, equal opportunity

laws, and affirmative action: Some experimental results. The Quarterly Jour-

nal of Economics, 107(2):511, 1992.

[22] A. Sen. Utilitarianism and welfarism. The Journal of Philosophy, 76(9):463–

489, 1979.

[23] A. Sen. Equality of what? in S. McMurrin (ed.), Tanner Lectures on Human

Values, 1980.

[24] L. Shapley and H. Scarf. On cores and indivisibility* 1. Journal of mathe-

matical economics, 1(1):23–37, 1974.

[25] G. Tullock. Efficient rent seeking. Toward a Theory of the Rent-Seeking

Society, page 109.

[26] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders.

The Journal of finance, 16(1):8–37, 1961.

36