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1.Show that the monotone likelihood ratio condition (f(x)/g(x) increasing in x, where x is a real number) implies first-order stochastic dominance (F(x) £ G(x) for all x).

2.Let q = q1 or q2 denote the state of nature in a two state world. Let p0 = Pr(q = q1). Let f(p) be an arbitrary distribution on the unit interval such that òpf(p)dp = p0. Show that f(p) can be construed as a distribution over posterior probabilities of p, stemming from an experiment with outcomes y, a prior probability p = p0 and two likelihood functions p(y|q1) and p(y|q2). [It may be easier to solve the problem if you assume that f(p) consists of two mass points so that the experiment will have only two relevant outcomes, say, yL and yH.]

3. An agent has to decide between two actions a1 and a2, uncertain of the prevailing state of nature s, which can be either s

. The agent’s payoff as a function of the action and the state of nature is as follows:

or s

The prior probability of state s1 is p = Pr(s1).

(a)Give a graphical representation of the agent’s decision problem as a function of the probability of state s1. If p = (.4), what is the agent’s optimal decision?

(b)Let the agent have access to a signal y before taking an action. Assume y has two possible outcomes with the following likelihoods

How valuable is this information system if l

= ½? How valuable is it if l1 = ½ and l2 = 0?

(c)Show that, irrespectively of the prior, the information system l1 = ½ and

= 0 is preferred to the information system

Solutions

1 Solution 1

Definition 1 MLRP: g(x) is increasing in x.

½a

½b and

b, where a and b are any numbers between 0 and 1. (Hint: show that latter system is garbling of former).

Definition 2 FOSD: F (x) ≤ G(x)∀x

Show that MLRP ⇒ FOSD

Proof: MLRP implies that there exists x0 such that or f (x) < g(x)∀x. But since f (x)dx = g

= 1, neither of these cases are possible.

We have two cases:

)

not then

2 Solution 2

You want to design an experiment, that is a random variable Y (that takes value in [0;1], for simplicity, and is characterized by the joint distribution p(q,y) ) such that the distribution of posteriors generated by this experiment is given by f(p). In the “experiment” the posterior will be given by Pr(q1|y) and the probability that this posterior arises is simply Pr(y) . In the statement, the probability that posterior p arises would be given by f(p). Therefore, we’d like to take Pr(q1|y)=p and Pr(y)=f(p) (for y=p) which directly defines p(q1,y)=pf(p) ;Pr(q2|y)=1-p ; p(q2,y)=(1-p)f(p).

Does such a random variable exists ?

We have p(q,y)≥0 for all y,q and

p(q,y)dydq =ò p(q1,y)dy+ò p(q2,y)dy

pf(p)dp+ò(1-p)f(p)dp (by construction, since p(q1,y)=pf(p) for y=p)

+ (1-p0 ) =1 (by hypothesis)

Therefore such an experiment exists (we can construct a random variable Y such that the joint distribution of (Y,θ) is p(q,y) since p is non negative and sums up to 1) and the prior is given by Pr(θ1)=ò p(q1,y)dy =p0 as wanted

We can then define the likelihood functions using Bayes rule and we have Pr(y|q1)= Pr(q1| y)Pr(y)/Pr(q1)=pf(p)/p0 and similarly for Pr(y|θ2) . You can then directly verify that the experiment defined by the outcome y, these likelihood functions and the prior p0 generate posteriors distributed according to f.

1.We are given u (a, s) therefore we can derive

indicates the maximum expected utility that the agent can reach if faced with probability of

equal to p):

At p = 0.4 the optimal decision is a

2. We are given the likelihood matrix:

since:

The probability of observing the two signals is:

Let’s calculate the posterior probabilities:

If

1 = λ2 = 2 the information system has no value because the same signal y1 is as likely to

appear in the two states of nature. Whenever λ1 = λ2 the information system has no value.

If λ1 = 2 and λ2 = 0 the the likelihood matrix is the following:

Posteriors in this case are:

And probabilities of the two signals are:

— When observe y1:

therefore the optimal choice as a function of the signal is:

Hence:

therefore the optimal choice as a function of the signal is:

We can now calculate the value of information as:

3.We

Blackwell

gives general conditions under which one information system 2 2 is preferred to another. So we just have to prove that the information system ¡

know that the

theorem

We have to find a Markov matrix M :

such that the following relationship between the likelihood matrices holds:

You can verify that such matrix M exists and is equal to the following:

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