The decision maker collects information by acquiring a signal S ∈ S that has state-dependent distribution Fz(·). Upon observing S = s, the optimal verdict is
dFG(s)}all realizations of S that leads to an acquittal decision. Similarly, denote
SC = {s: dFI(s) < dFG(s)} all realizations of S that leads to conviction. We assume that the measure of S \(SA∪ SC) is zero under any state, so that the decision maker is (almost) never indifferent between the two verdicts. Denote
pz,v = IP[S ∈ Sv |z] the probability of making the decision v ∈ {A, C}, given
statez. Then the probability of the correct decision is 12(pI,A +pG,C)and therefore
the expected utility from signal S is 12(pI,A+pG,C)Q+R. When the decision
maker does not use the signal, his expected utility is Q2 +R. Thus, the quality of signalS can be summarized by
1
2(pI,A+pG,C)− 1 2.
We assume that the decision maker can increase the quality by paying more for the signal. Formally, the quality of the signal is an increasing function of cost, h(c). Thus, the expected utility is
1
2 +h(c)
Q+R−c
and the decision maker chooses costc >0. The first order condition is
h0(c)Q= 1. (4.11)
To guarantee that the solution to (4.11) exists, is unique and maximizes the expected utility, we assume
Assumption 4.3. h: (0,+∞) →
0,12 is such that h(0) = 0, lim c→0h
0(c) = +∞,
lim c→+∞h
0(c) = 0,h00(c)<0.
Given that general model, we impose the following definition of overconfidence:
Definition 4.2. Anη-typedecision maker perceives the quality of the signal being
h(ηc)while payingc.
Consider theη-type decision maker. His expected utility from signalSis
1
2+h(ηc)
Q+R−c. (4.12)
Example (Normal distribution)WhenS ≡Xt ∼ N(µzt, σ2t), its quality is equal
tof σt2 , where f(ρ) = √1 π √ρ 2 Z 0 e−x2dx. (4.13)
Figure 4.1: Functionf(ρ) = √1π
√ρ 2
R
0
e−x2dxand its derivative.
Then a linear cost functionc(t) = κ·timpliesh(c) = f κσc2
. Figure 4.1 shows thath(c)satisfies Assumption 4.3. Moreover, it is easy to see that Definition 4.2 is the analog of Definition 4.1 for the static case.
Optimizing (4.12) overc >0, we get
c > 0 : ηh0(ηc)Q= 1. (4.14)
Treating the solution to (4.14) as a function of the overconfidence levelη, we have
c0(η) =−h
0(ηc) +ηch00(ηc)
η2h00(ηc) . (4.15)
A highercmeans higher probability of the correct decision,12+h(c). As we increase the level of overconfidenceη,cincreases if and only ifh0(ηc)+ηch00(ηc)is positive. From Assumption 4.3,h0(ηc)is always positive, whileηch00(ηc)is always negative. The termh0(ηc)corresponds to thefirst force: higher effective costηcincreases the quality of the signal becausehis increasing. The termηch00(ηc)corresponds to the
second force: higher effective costηcdecreases the marginal benefit of information becauseh0 is decreasing.
The total effect is captured by the behavior of functionxh0(x), which derivative is equal toh0(x) +xh00(x). If it increases at pointx=ηc(η), then increasing the level of overconfidence makes the final decision better. If it decreases, the final decision becomes worse with increased overconfidence.
Note the interpretation of xh0(x), x = ηc, as the marginal benefit of information. The first x corresponds to the first force, which increases function xh0(x) as we
increasexthrough the level of overconfidence η. The secondxcorresponds to the second force, which decreases functionxh0(x).
Without making any additional assumptions, it is hard to say more about the behavior ofc(η). One interesting special case is when the following assumptions holds:
Assumption 4.4. There exists c >ˆ 0 such that ch0(c) increases for c < ˆc and it decreases forc >ˆc.
This assumption says that when the amount of collected information is below some threshold, the second force is weaker than the first force, and vice versa. Recall that the second force comes from changing the benefit of already collected information. As we increase the amount of collected information is small, this force becomes stronger. On the other hand, the first force comes from changing the benefit of the marginal information piece and therefore it depends on the amount of collected information only through the non-stationary properties of the information flow. Assumption 4.4 says that the information flow is stationary enough to make sure that there is a unique threshold such that the second force prevails if and only if the amount of collected information is above that threshold.
Under Assumption 4.4, we can prove that there exists a unique optimal level of overconfidenceη∗(which can be less than 1, which corresponds to underconfidence) such that more overconfidence is good below that level and it is bad for allηabove
η∗. Formally:
Theorem 4.3. The probability of choosing the correct action is increasing inη ∈
(0, η∗)and it is decreasing inη∈(η∗,+∞), whereη∗ = Qh10(ˆc). See Appendix C.3 for the proof.
Example (Normal distribution)
WhenS ≡Xt∼ N (µzt, σ2t), functionch0(c) = κσc2f
0 c
κσ2
satisfies Assumption 4.4 (see Figure 4.2). In that caseη∗ = 2
√
2eπκσ2
Q .
Here is an example where Assumption 4.4 is violated.
Example (Binary distribution)SupposeS ∈ {I, G},P[S =I |z =I] =P[S =G|z =G]≥
1
2. Then its qualityh =P[S =z |z]− 1
2. This means that the binary distribution
Figure 4.2: Functionρf0(ρ) = e −ρ
2√ρ
2√2π defines how the marginal benefit of informa-
tion changes with the level of overconfidence under normal distribution assumption.
1. Suppose the agent payment is some decreasing function of the variance of the random variable1(S =z), for example,
c=−log (4P[S =z |z] (1−P[S =z |z])).
Thenh(c) =P[S =z |z]−12 = 12
√
1−e−c
satisfies both Assumptions 4.3 and 4.4.
2. Functionh(c) = 2c+1csin(c)
1 4 −1
satisfies Assumption 4.3 but not As- sumption 4.4. See Figure 4.3.
Note that the optimal level of overconfidence is decreasing inQ. This means that overconfidence is bad when the benefit from choosing the correct action is high. Intuitively, when the benefit from choosing the correct action is very low, the rational agent collects very little information. A distortion in his incentives by increasing the perceived quality of information always has a positive effect. Indeed, an increase in the quality of already collected information (the second force) does not have a large effect since the amount of this information is small. So, overconfidence is good for low Q. On the other hand, when the benefit from choosing the correct action is very high, the rational agent collects a lot of information. This means that
Figure 4.3: Functionsh0(c)andch0(c), whereh(c) = 2c+1csin(c)
1 4 −1
.
the second force has a lot of power as it works with a large amount of information. Consequently, overconfidence is bad in this case.
At the end of this section we give an example of a model when only the first force is active.
Binary Information Acquisition Decision
Suppose the decision maker can choose only between two values,c∈ {0,c}¯ . This describes the situation when the agent simply has to decide whether to acquire information or not. In this case the optimality condition (4.14) should be changed to
c= ¯c ⇔ h(ηc¯)Q >¯c. (4.16) Definition 4.3. The maximum cost the decision maker is ready to pay for the signal is called thewillingness to pay.
Given condition (4.16) and Assumption 4.3, the willingness to pay is equal to
c >0 : h(ηc)Q=c, ηh0(ηc)Q <1. (4.17) Treating the solution to (4.17) as a function of the overconfidence levelη, we have
c0(η)>0.
Theorem 4.4. The willingness to pay is increasing inη.
See Appendix C.4 for the proof.
In this model, there is no “already collected information” since the choice is binary, either buy the signal or not. Thus, the second force is absent here and Theorem 4.4 result is driven by the first force.