I'm currently looking at this course in economic game theory.
However, when attempting this example:
In this question you are asked to price a simplified version of mortgage-backed securities. A banker lends money to $n$ homeowners, where each homeowner signs a mortgage contract. According to the mortgage contract, the homeowner is to pay the lender 1 million dollar, but he may go bankrupt with probability $p$, in which case there will be no payment. There is also an investor who can buy a contract in which case he would receive the payment from the homeowner who has signed the contract. The utility function $u$ of the investor is given by $u(x) = −\text{exp}(−\alpha x)$, where $x$ is the net change in his wealth.
I could not seem to understand their solution:
He pays a price $P$ if and only if $E[u(x−P)] ≥ u(0)$, i.e., $−(1 −p)\text{exp}(−\alpha (1 −P )) −p \text{exp}(−\alpha (0 −P)) ≥ −1$
Could someone please explain to me how they arrived at this?
I thought the function would've been:
$$-\text{exp}(\alpha(x-P)) \ge -1$$
Help would be greatly appreciated.
The point is that the investor does not know the value of $x$ when buying the contract, so $x$ cannot be part of the decision criterium.
If the investor is lucky, the contract will be repaid and the investor will make a profit of $1-P$. If he is unlucky, the homeowner will go bankrupt and the investor will lose $P$. The expected utility of buying the contract is $$(1-p)\cdot u(1-P) + p\cdot u(-P)$$ which is equal to the left hand side of your answer.
If the investor does not buy the contract, the utility is $u(0)=-1$, which is your right hand side.