Construct by defining the functions q and ti as required by item (ii) in Proposition

3.1. We can prove the result by showing that truth telling is a Bayesian equilibrium of

the game. Suppose it were not. If type ?i prefers to report that her type is ?

i , then the

same type ?i prefers to deviate from ?, and to play the strategy that ? prescribes for

?

i in . Hence ? is not a Bayesian equilibrium of . -

Proposition 3.1 shows that in the setup that we have described we can, without

loss of generality, restrict our attention to the case in which the seller chooses a di-

rect mechanism and proposes to agents that they report their types truthfully. Note,

however, that it is crucial to this construction that we have neglected problems of mul-

tiple equilibria by assuming that agents follow the seller's proposal provided that it is

an equilibrium and provided that it gives them expected utility of at least zero. This

is crucial because the equivalent direct mechanism that is constructed in the proof of

Proposition 3.1 might have Bayesian Nash equilibria other than truthtelling, and there

is no reason why these equilibria should be equivalent to any Bayesian Nash equilib-

rium of the indirect mechanism . Depending on how equilibria are selected, one or

the other mechanism might be strictly preferred by the seller in that case.

The revelation principle greatly simplifies our search for optimal mechanisms. We

can restrict our attention to direct mechanisms in which it is a Bayesian equilibrium

that everyone always reports their type truthfully and in which every type's expected

utility is at least zero. We want to define these properties of a direct mechanism

formally. For this we introduce additional notation.

We denote by ?-i the vector of all types except player i's type. We define

-i ? [ ?, ?]N-1. We denote by F-i the cumulative distribution of ?-i, and we denote

by f-i the density of F-i. Given a direct mechanism, we define for each agent i ? I a

function Q i : [ ?, ?] ? [0, 1] by setting

Q i(?i) = -

-i

qi(?i, ?-i)f-i(?-i) d?-i. (3.1)

I only need help with parts 4 and 5. I got an EV of 335.5 for 2. A decision maker's risk tolerance is assessed at $1,210. Assume that this risk-averse individual's preferences can be modeled with an exponential utility function. 1) Compute the following utilities: U($1,000) U($800) U(SO) U(-$1,250) 2) Determine the expected utility for an investment with the following pay-off distribution: P($1,000) = 0.33 P($800) = 0.21 P(SO) = 0.33 P(-$1,250) = 0.13 3) Compute the exact certainty equivalent for the investment and associated risk premium. 4) Find the approximate certainty equivalent using the expected value and variance of above pay-off distribution. 5) Another investment possibility has an expected value of $2,400 and standard deviation of $300. Find the approximate certainty equivalent for this investment.