Assuming a continuum of agents makes the intuition clearer and is in keeping with related work such as Bowles et al. [2009]. However, as the population of agents under consideration may be a school class, a village or some other such population, the model’s applicability to discrete, finite populations is an important matter. In this section we analyse the finite analogue of the model presented in the previous
q
1 1
u
Figure 6.3: The left hand graph just shows the probability of success given effort as it varies with the fraction of agents mh putting in effort e. The right hand figure
shows the expected utility depending on the choice of the agent, given the measure of agents putting in the effort. The line of constant utility is for no effort, the other shows how expected utility varies with the proportion putting in effort.
q
1 1
E(u)
Figure 6.4: This figure is the linear special case of figure 6.3. It allows for clearer analysis but, at least in terms of equilibria, the results should be qualitatively the same.
section.
We assume that we have agentsi∈ {1, . . . , I}and as before the probability of success q depends on the fraction of agents who put in effort eh. Letting this
fraction berh we explore the finite versions of our model. In the below analysis we
assume we have a sufficient number of agents for the model to be meaningful; so we have strictly positive fractions of agents in each partition (as required for identifying Nash equilibria).
6.4.1 Complementarity and Competitive scenarios
The complementarity scenario is now straightforward. Generically we cannot have a fraction of agentsrh putting in efforthsuch thatrh= ρ(ue+ul
h−ul)+ul; so the unstable
Nash equilibrium will not exist. This leaves us with up to two Nash equilibria which will be on the boundaries, that isrh= 0 and rh= 1. The later exists ifρuh ≥ul.
The competitive model is more interesting for a finite set of agents. Assuming
uhβ > ul and ul > uh(β−ρ) then generically there exists a least proportion ˜rh of
agents strictly larger than β(uh−ul)−eh
ρ(uh−ul) .
Proposition 5. The proportion r˜h is the unique Nash Equilibrium for the compet-
itive model, when uhβ > ul and ul> uh(β−ρ).
Proof. At ˜rh there are two classes of agents, those currently putting in effort eh
and those not. Consider unilateral deviations for each class. For those putting in effort eh their expected utility is (β −ρr˜h)uh > (β −ρmh)uh = ul. That is their
current expected utility is greater than if they deviate. Now consider an agent with effort e = 0. Their current (expected) utility is ul; if they deviate it will be
(β−ρ(Ir˜h+ 1)/I)uh but by definition ˜rhis the least proportion of agents larger than
β(uh−ul)−eh
ρ(uh−ul) so the expected utility upon deviation will be less thanul. Therefore ˜rh
is a Nash equilibrium.
For any rh > r˜h the expected utility upon deviation for those agents who
do put in effort eh is strictly larger than their current utility (so it cannot be a
Nash equilibrium) and for rh <r˜h the same holds for agents who are not putting
in effort eh. Therefore ˜rh is the unique Nash equilibrium when uhβ > ul and
ul> uh(β−ρ).
6.4.2 Benchmark scenario
We turn now to our benchmark model for a finite set of agents. For this we will have up to two Nash equilibria. Again, generically, the unstable Nash equilibrium will not exist (as it would require a specific proportion of agents to be possible). As
with the complementarity case we will have a Nash equilibrium at rh = 0. There
may be a further Nash equilibrium whenI is large enough anduhβ−nρ2 > ul and
ul > uh(β −ρ2). Generically there exists a least proportion ˜rh of agents strictly
larger than β(uh−ul)−eh
ρ(uh−ul) , this is a second Nash equilibrium.
Proposition 6. For the finite version of the benchmark model with large enough I
there are up to two Nash equilibria. One where rh = 0 and a second at ˜rh.
Proof. Therh= 0 case is straightforward as we only need to consider deviations of
any agent from no effort to eh. But for I large enough the expected utility upon
deviationρ1uh+ (1−ρ1)ul−eh < ul.
Now consider ˜rh. The key point to note is that for Nash equilibrium char-
acterisation we consider only unilateral deviations. ForI large enough we are con- sidering a situation that is locally similar to that in proposition 5; so in terms of unilateral deviations (and hence Nash equilibrium) the same result holds.