Group Compositions and Contracts

We define five competitive groups {g₀, g₁, g₂, g₃, g₄}, composed of 3 agents each. The composition of all five groups is shown in Table 3.2. As a benchmark, the homoge-neous group g0 is formed where agents are identical in ability aM and uncertainty σM

described by i0(aM, σM), j0(aM, σM), and k0(aM, σM). Group g1 comprises the most common form of heterogeneity modeled in the literature where agents vary in ability only i₁(a_L, σ_M), j₁(a_M, σ_M), and k₁(a_H, σ_M). Less common, agents within group g₂ vary only by uncertainty i₂(a_M, σ_L), j₂(a_M, σ_M), and k₂(a_M, σ_H).

Table 3.2: Group compositions of heterogeneous backgrounds

Group Composition Agent i^∗ Agent j^∗ Agent k^∗

0 Homogeneous i0(aM, σM) j0(aM, σM) ko(aM, σM) 1 Heterogeneous Ability i1(aL, σM) j1(aM, σM) k1(aH, σM) 2 Heterogeneous Uncertainty i2(aM, σL) j2(aM, σM) k2(aM, σH) 3 Ability ∝ Uncertainty i3(aL, σL) j3(aM, σM) k3(aH, σH) 4 Ability ∝ Precision i4(aL, σH) j4(aM, σM) k4(aH, σL)

* Ability: {aL, aM, aH} = {10, 15, 30}, Std. Dev.: {σL, σM, σH} = {13.5, 20.3, 40.6}

Groups g₃ and g₄ represent previously unexplored cases in the experimental litera-ture in that agents vary in both dimensions of ability and variance. Group g3 models the composition of agents typically found in the production of goods where the abil-ity to produce a given level of output is positively correlated with the variance of idiosyncratic uncertainty by i₃(a_L, σ_L), j₃(a_M, σ_M), and k₃(a_H, σ_H). Broiler tourna-ments are a widely known example. Group g₄ depicts the composition of agents found in service industries or sports competitions such as golf where top performers are also characterized with relatively high precision i₄(a_L, σ_H), j₄(a_M, σ_M), and k₄(a_H, σ_L).

When agents are heterogeneous, relative compensation contracts must account for

µ^∗_i, µ^∗_j, µ^∗_k

Group 0: Homogeneous f_i(a_M, σ_M), f_j(a_M, σ_M), f_k(a_M, σ_M)

0 50 100 µ

µ^∗_i µ^∗_j µ^∗_k Group 1: Heterogeneous Ability

f_i(a_L, σ_M) f_j(a_M, σ_M) f_k(a_H, σ_M)

0 50 100 µ

µ^∗_i, µ^∗_j, µ^∗_k

Group 2: Heterogeneous U ncertainty f_i(a_M, σ_L)

f_j(a_M, σ_M)

f_k(a_M, σ_H)

0 50 100 µ

µ^∗_i µ^∗_j µ^∗_k Group 3: Ability ∝ U ncertainty

f_i(a_L, σ_L)

f_j(a_M, σ_M)

f_k(a_H, σ_H)

0 50 100 µ

µ^∗_i µ^∗_j µ^∗_k Group 4: Ability ∝ P recision f_i(a_L, σ_H)

f_j(a_M, σ_M) f_k(a_H, σ_L)

0 50 100 µ

Figure 3.1: Group density graphs for socially optimal strategies

so, the group compositions described in Table 3.2 are graphed in Figure 3.1 at the efficient Nash equilibrium. The uniform density functions f overlap in equilibrium for each group setting. The mean of each density is centered at the efficient Nash equilibrium effort level chosen by the agent and the density limits are ±√

3σ_v where v ∈ {L, M, H}.

The parameter values are selected to satisfy several design constraints. As shown in Table 3.1, ability levels in the experiment have the following relationship 3aL = 2aM = aH. Likewise, standard deviations in the experiment have the following rela-tionship 3σ_L = 2σ_M = σ_H. The ratios are kept the same so that ability and uncer-tainty scale proportionally in g₃ and inversely proportional in g₄ in order to examine how well multiplicative handicaps perform independent of offset handicaps. Abilities are chosen so that strategies in equilibrium that yield efficient effort were sufficiently separated in the decision interval [0, 100]. As depicted in Figure 3.1, the ability lev-els {a_L, a_M, a_H} = {10, 15, 30} generate effort levels µ^∗_i, µ^∗_j, µ^∗_k = {20, 30, 60} when agents work efficiently. Second, the standard deviations were made just large enough to satisfy the second order conditions and the global participation constraint so that all tournaments have a theoretical Nash equilibrium {σ_L, σ_M, σ_H} = {13.5, 20.3, 40.6}.

Table 3.3: Tournament contracts

Contract Groups Description Efficient φ h Payoffs

H Homogeneous, g0 Homogeneous Yes No No Identical^∗

A g1, g2, g3, g4 Equal access to opportunity No No No Identical^∗ B g1, g2, g3, g4 Equal expected earnings, E [u] Yes No Yes Individual C g2, g3, g4 Equal E [u] and chance to win Yes Yes Yes Individual

* The payoffs are the same for all types and for all groups, including the homogeneous group

To study concepts of fairness and efficiency, we evaluate the performance of the groups and agents in three contracts and one benchmark. The tournament contracts

are listed in Table 3.3. The first and second contracts do not use handicaps. The third contract incorporates offset handicaps h = {h_i, h_j, h_k} to induce efficient effort µ^∗ =

µ^∗_i, µ^∗_j, µ^∗_k . The fourth contract employs both offset and multiplicative handicaps φ = {φ_i, φ_j, φ_k}.

Contract H is a benchmark case and exhibits all the properties of fairness: equal access to opportunity, equal expected earnings, and equal chance to win. H is designed as an incentive for agents in the homogeneous group g0 to work efficiently. Since g0

is homogeneous, handicaps are unnecessary, the probability of winning for each agent is P = ¹₃, and expected earnings are equal to the fixed payment X.

In Contract A, the firm offers agents in the heterogeneous groups equality in pay structure with no handicaps φ = {1, 1, 1} and h = {0, 0, 0}. The contract is identical to the payoffs offered in Contract H (i.e. all agents in groups g₁, g₂, g₃, g₄ are offered the same winning and losing payoffs). A contest offering equal payoffs is common practice in labor settings; however, because agents are heterogeneous, the best response functions lead to non-uniform strategies, unequal probabilities of winning, and lower than efficient effort. Therefore, expected earnings are unique and no longer equal to X. The concept of fairness in Contract A is equal access of opportunity; even though agents are different, they all have access to the same contest. Differences in ability and uncertainty are ignored in policy.

In Contract B, the firm incorporates only offset handicaps to provide the incentive for all agents to perform efficiently. The Multiplicative handicaps are not used, φ = {1, 1, 1}. All agents in Contract B have fairness in expected earnings equal to X.

The offset handicap provides the firm with an extra degree of freedom to create the proper incentives for agents to generate efficient effort. Offset handicaps are set h = {40, 20, 0} to compensate for differences in expected efficient effort when

µ^∗_i 6= µ^∗_j 6= µ^∗_k in groups g₁, g₃, and g₄. Since ability is the same for agents in group g₂, handicaps are unnecessary h = {0, 0, 0} to adjust for differences in effort. To equate expected earnings to X, individual payoff contracts are offered to each agent by adjusting winning and losing payoffs based on individual attributes of ability and uncertainty. The firm does not have enough degrees of freedom in Contract B to equate the probability of winning between agents in g₂, g₃, and g₄ because the offset handicap is not sufficient to compensate for the differences in uncertainty. However, since uncertainty in g1 is i.i.d., agents have the same probability of winning for this one case in Contract B.

In the final Contract C, the firm induces efficiency, equalizes expected outcomes, and equalizes the probability of winning. This mechanism is unique in the literature and uses a combination of multiplier and offset handicaps as needed to achieve the firm’s design goals for each group. Differences in risk contribute to inequalities in expected earnings and the probability of winning in Contract A. To eliminate this effect, the multiplicative handicap is used in Contract C to equate the distributions of uncertainty between agents. The agents endowed with the standard deviations {σ_L, σ_M, σ_H} correspond to the set of handicaps φ = {3, 2, 1}. The agent with the smallest variance in distribution receives the largest multiple. Since r^∗_i = φ_iµ^∗_i+ φ_iε_i+ h_i, the handicap φ has the effect of scaling both the error term and the effort for the agent.

This is a desirable property in g₃ because ability is proportional to uncertainty.

The specific cost function in this experiment induces the same proportion in equi-librium effort when agents are working efficiently; therefore in g₃, the multiplicative handicap is sufficient to equate the distributions and effort simultaneously such that the offset handicap is not needed h = {0, 0, 0}. In the other heterogeneous groups

g₂, and g₄, h is used to compensate for the scaling effect of φ. Evaluating the expec-tation of the rank rule E [r_i] = r, the offset handicap for all agents were calculated h_i = r − φ_iµ^∗_i. To avoid inducing loss aversion, the offset handicaps were normalized so that h_i ≥ 0 by choosing r = maxφ_iµ^∗_i, φ_jµ^∗_j, φ_jµ^∗_j and the agent of the group with the smallest offset handicap received h_i = 0. In g₁, the handicap φ_i = 1 since the variance is identical across agents. In fact, the optimal contract for group g₁ is to use only h as in Contract B; hence g1 is omitted from Contract C.