Promotion as Tournament
Examples
 promotion from being an associate in a law firm to the partner of
the firm brings about a substantial pay raise, but the productivity of
the lawyer does not increase overnight
 ditto for promotion from assistant professorship to associate
professorship, though the pay raise is not substantial
 salesperson of the year award
 promotion from vicepresident to president of the
companyusually
associated with substantial changes in job duties
The tournament model
Suppose there are two workers, Joe and Katie. Let
q_{j} = m_{j} +
e_{j}
where q is output, m is Joe's effort, and e is a luck
component. The cost of effort is C = C(m).
Similarly, let
q_{k} = m_{k} +
e_{k}
The value of each unit of
output is p. Assume workers are risk neutral.
Efficiency can be induced by paying workers a piece rate of p dollars
per unit of output. Workers who choose the piece rate will choose the
optimal effort level m* such that p = C'(m*).
But piece rate is not the only compensation system that can lead to
efficiency. A suitably designed tournament can also produce
the same results. In particular, suppose there is an implicit contest
between Joe and Katie. If one worker produces more output than the
other, the first worker gets W_{h}
while the second gets W_{l}. One can think of
W_{h}  W_{l} as the
"prize" of winning. The prize of winning can take the form of a bonus
or a promotion. Also note that we assume that the outcome of the
contest depends only on the ranking of output but not on the actual
output levels of the two workershence the name "rankorder
tournaments."
Consider the choice problem for Joe. Let
P = probability that Joe wins
Joe chooses m_{j} to maximize
P W_{h} + (1P) W_{l} 
C(m_{j})
The firstorder condition is
(W_{h}  W_{l})∂
P/∂ m_{j}
 C'(m_{1}) = 0
Now, Joe
wins the tournament if
and only if q_{j} >
q_{k}. This can be written as
m_{j} + e_{j} >
m_{k} + e_{k}. Let
η = e_{k}  e_{j},
and let G() and g() represent the c.d.f. and p.d.f. of η. The
probability that Joe wins the contest is
P =
Pr[ m_{j}  m_{k} > η ]
= G(m_{j}  m_{k})
Hence,
∂ P/∂ m_{j} =
g(m_{j}  m_{k})
If Joe and Katie are identical, then
m_{j} =
m_{k}. This allows us to simplify the
firstorder condition into
(W_{h}  W_{l})
g(0) 
C'(m_{j}) = 0
From this, we can conclude
 The higher is the prize spread B =
W_{h}  W_{l}, the
higher is the effort level. (It is the prize spread rather than the
level of the prize that motivate workers.)
 Effort is higher the higher is g(0).
The magnitude of g(0) depends on the variance of the luck component.
If the luck component has a large variance, then a marginal increase
in effort may only increase the chance of winning slightly (g(0) is
small), and therefore the tournament may not be very effective in
encouraging effort.
 pure lottery
 promotion in US firms versus Japanese firms
 promotion in new industries versus established industries
Optimal wage structure
What determines the optimal prize structure? Let m(B) be the effort
chosen by the workers when the prize spread is B. The value of the
firm is
p m(B)  C(m(B))
The optimal B must satisfy
(p  C'(m(B))) dm(B)/dB = 0
Since dm(B)/dB > 0, this equation implies that we must have
p  C'(m(B)) = 0.
Comparing this equation with the firstorder condition for the
workers' choice, we see that optimality can be achieved by setting
(W_{h}  W_{l})g(0) = p
When the prizes are chosen in this way, we have m(B) = m*.
Furthermore, the zero profit condition implies that the value of
output must be equal to the value of wage payments. So
2 p m* = W_{h} + W_{l}
These two equations allow us to solve for the equilibrium prize
structure. In particular, we have
W_{h} = p m* + p/(2g(0))
W_{l} = p m*  p/(2g(0))
Several observations follow:
 The optimal spread is B* = p/g(0). Thus, the optimal spread is
larger if p is larger or g(0) is smaller.
 The outputs of the workers are not equal to their wages. Yet the
equilibrium is fully efficient.
 The same outcome can also be achieved by giving a bonus to workers
whose output exceed a certain fixed standard. However, firms would
have an incentive to claim that the standard has not been met.
Absolute versus relative performance
 One advantage of the tournament over piece rates is that it
requires only ordinal but not cardinal measure of output. In
management level jobs, a cardinal measure of output is sometimes
simply impossible.
 Another advantage of relative performance evaluation is that it
eliminates that component of luck which is common to all contestants
(e.g., weather, the state of the economy,
overly generous or mean evaluators).
 One disadvantage of relative performance evaluation is that it could
invite collusion among contestants.
 Alternatively, workers may also become too competitive among
themselves.
Empirical evidence
 The same golfer achieve better scores on the same course when
prize spread is greater.
 Absenteeism falls when firms give larger raises upon promotion.
 Do professors slack off once they obtain tenure?
Implications for organizational design
Consider the following organization chart of a firm:
CEO: o $350000
EVP: o o o o o o $300000
VP: o o o o o o o o o o o o $150000
What is wrong with this organizational structure?
Heterogeneous work force
 When worker ability is not symmetric, the effort level of both
may fall. (Why?)
 If workers can change their effort during the contest, workers may
slack off when they find that the distance between the contestants is
too far.
 The firm may want to group workers in such a way that they may
have similar chances of winning within their group.
 The firm may want to provide some reward at every level of
promotion to prevent workers from giving up.
 External recruitment may dilute the incentives of internal
contestants.
 Promotion is often associated with a different kind of tasks to
be performed. To the extent that tournaments select a more able
worker (instead of just a worker who puts in more effort), it performs
the selection function as well.
 But workers may also be promoted to the level of their
incompetencethe Peter principle.

Lecture Notes
