Agenda
Lecture 11: Moral Hazard in Teams: The Problem
of Free-Riding
24th August 2005
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Model
Part I
The Model
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Model
Assumptions
One principal (P) and n agents (i = 1, ..., n);
Each agent chooses unobservable effort ai with ai ∈ [0, 1],
effort cost amount to ψi (ai ) = ai2 ;
All agents together produce joint output
P
Q = Q(a1 , a2 , ..., an ) = a1 + a2 + ... + an = ni=1 ai ,
joint output is measurable and observable;
The production function Q(.) generally satisfies the following
assumptions:
∂Q(.)
∂ai > 0, i.e., team output Q
ai chosen by agent i,
∂ 2 Q(.)
≤ 0, i.e., marginal team
∂ai2
increases with increasing effort
output decreases (or stays at
least the same in case of “= 0”) with increasing effort ai , e.g.,
the first working hour generates higher marginal output (no
less in case of “= 0”) than the last one,
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Model
Assumptions concerning Q(.) continued ...
∂ 2 Q(.)
∂ai ∂aj
≥ 0, i.e., agent j’s input has positive marginal effect (or
no in case of ”= 0”) on agent i’s marginal productivity gives agent i’s marginal productivity,
impact on i’s marginal productivity);
∂ 2 Q(.)
∂ai ∂aj
∂Q(.)
∂ai
gives agent j’s
Principal and agents are assumed to be risk-neutral;
We define a partnership (or a sharing-rule) to be
output-contingent compensations for each agent i = 1, ..., n
w (Q) = [w1 (Q), ..., wn (Q)]
Pn
such that i=1 wi (Q) ≤ Q.
I.e., Q is shared among all team members i = 1, ..., n such
that the sum of all shares is no larger than Q itself.
In our example, we choose
the equal-split-sharing-rule with
P
wi (Q) = n1 · Q and ni=1 wi (Q) = Q with which the whole
team output is evenly shared.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Example
Part II
The Example
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Example
Our Little Class Room Experiment
i
ai
1 0
2 .4
3 .7
4 .7
what if
1 .2
2 .4
3 .7
4 .7
Q
wi (Q) = ψi (ai ) =
Πi =
1
2
ai
wi (Q) − ai2
n ·Q
1.8
.45
0
.45
1.8
.45
.16
.29
1.8
.45
.49
-.04
1.8
.45
.49
-.04
agent i = 1 chose ai = .2?
2
.5
.04
.46
2
.5
.16
.34
2
.5
.49
.01
2
.5
.49
.01
Agent 1 changes effort level from 0 to .2;
Individual marginal return (.05) higher than marginal cost (.04);
Thus, increasing the level of effort is individually profitable and
enhances social efficiency.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
The Example
i
ai
Q
wi (Q) = ψi (ai ) =
1
ai2
n ·Q
1 .2 2
.5
.04
2 .4
2
.5
.16
3 .7
2
.5
.49
4 .7
2
.5
.49
∗
what if agent i chose ai = .4?
1 .4 2.2
.55
.16
2 .4 2.2
.55
.16
3 .7 2.2
.55
.49
4 .7 2.2
.55
.49
Πi =
wi (Q) − ai2
.46
.34
.01
.01
.39
.39
.06
.06
Agent 1 changes effort level from .2 to .4;
Individual marginal return (.05) lower than marginal cost (.12).
Thus, increasing the level of effort is individually not profitable.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Part III
Results and Conclusions
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
The Individual Payoff-maximizing Decision
Each agent i = 1, ..., n maximizes his individual payoff Πi with
respect to his effort decision ai :
n
1 X
max Πi = wi (Q) − Ψi (ai ) = ·
ai − ai2
ai
n
i=1
∂Πi
1 .
1
.
= − 2 · ai = 0 ⇔ = 2 · ai
∂ai
n
n
The individual marginal return is n1 , the individual marginal cost is
2ai . Each agent i chooses his effort ai by equating marginal return
and marginal cost.
1
2n
These are the NASH-equilibrium (NE) effort-levels in the team
game.
⇒ âi =
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
The Socially Efficient Effort-levels
A social planner maximizes the whole team payoff Π with respect
to every single effort decision ai for i = 1, ..., n (this situation can
be viewed as a principal allocating observable efforts in the best
possible way):
max Π =
ai
n
X
i=1
wi (Q) −
n
X
Ψi (ai ) = Q −
i=1
n
X
i=1
ai2 =
n
X
i=1
ai −
n
X
ai2
i=1
∂Π
.
.
= 1 − 2 · ai = 0 ⇔ 1 = 2 · ai
∂ai
The marginal return for the team as a whole amounts to 1, the
individual marginal cost for each agent is 2 · ai . Each effort ai is
chosen by equating marginal return and marginal cost.
1
2
All external effects are internalized!
⇒ ai∗ =
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Definition: External Effects
An external effect is a change in payoffs of one agent through the
decision of another agent, the decision-maker.
An external effect can be negative or positive.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Why do External Effects arise?
In the team game, with the equal-split-sharing-rule, agent i’s
effort generates output that all other agents also benefit from
(positive externality).
Such an externality arises when no measure of individual
performance is available.
(If individual performance/output was measurable, an incentive
compatible contract could be written!)
Thus, rewarding one agent for raising aggregate output means:
rewarding this agent’s higher output, and also
rewarding the efforts of all other agents higher!
Remember, the effort cost are to be borne individually!
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Can the Agents be Motivated for Choosing Socially
Efficient Efforts?
The first order condition in the social optimum is:
.
1 = 2 · ai∗
The first order condition in the NASH-equilibrium is:
1 .
= 2 · âi
n
By offering an individual marginal return of 1 instead of n1 , each
agent i can be motivated to choose an effort level âi = ai∗ .
This means, that every team member must be rewarded with the
whole team output Q.
P
However, in this case the budget constraint ni=1 wi (Q) = Q is
not satisfied.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Result under Budget Balanced Sharing-Rule
Thus, with
unobservable effort and
a budget balanced sharing-rule (b. bal. sh-r. meaning that
the whole Q is shared among the team members),
the socially efficient (first-best) situation (with observable efforts)
is not attainable, even though all team members were better off
compared to the (self-enforcing) NASH-equilibrium outcome.
The team game is an n-persons prisoners’ dilemma. The NE is not
Pareto-optimal. It is Pareto-dominated by the social optimum.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Changing the Sharing Rule.
Now P
assume a sharing-rule w (Q) = [w1 (Q), ..., wn (Q)]
with ni=1 wi (Q) ≤ Q such that
wi (Q) =
1
Q
n
if Q ≥ Q ∗ , and
wi (Q) = 0
if Q < Q ∗ .
Thus, every input is “wasted” if the social optimum is not attained.
NE?
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
NE With the Altered Sharing Rule
There are two NASH-equilibria:
âi = ai∗ for all i, and
âi = 0 for all i.
(Check by yourself that - given all others choose the NE strategy to respond with the NE strategy is a best answer, i.e., if all others
choose ai∗ then ai∗ is the best answer, if all others choose 0 then 0
is the best answer.)
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Result under Budget Breaking Sharing-Rule
Thus, with
unobservable effort and
a budget breaking sharing-rule (b. break. sh.-r. meaning a not
budget balanced sh.-r., i.e., Q is either exceeded or not always
fully exhausted),
the socially efficient (first-best) situation is attainable through a
NE, even though a second NE exists in which all agents choose
effort of 0.
Possible problems?
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Problems under Budget Breaking Sharing-Rule
The coordination on the Pareto-efficient NE is difficult. One
free-rider is enough to “destroy” the efforts of the other team
members.
Collusions are possible, if not likely.
E.g.,
P colludes with one agent,
agent i chooses ai < ai∗ ,
the result is Q < Q ∗ and wi (Q) = 0 for all agents,
thus, P gets Q,
and: i and P can share Q.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
Results under Budget Balance
Each team member chooses a socially inefficient effort level
1
âi = 2n
< 12 = ai∗ if he is not provided with his full marginal
return from his effort.
If he was provided with the full marginal return from his
effort, then the budget constraint was not satisfied.
The NASH-equilibrium effort âi =
team.
1
2n
is all lower the larger the
The lower the share the lower the NE effort level.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding
Results and Conclusions
General Results
Free-riding occurs because each team member, by choosing
higher effort than zero, generates positive external effects the
others profit from (or: can free-ride on).
See the above “Result under Budget Balanced Sharing-rule”.
See the above “Result under Budget Breaking Sharing-rule”.
Lecture 11: Moral Hazard in Teams: The Problem of Free-Riding