You are suing for company for 100000, which the company refuses to pay, but they are willing to settle for 35000. You give a counter-offer of 65000, which you think there is 40% change of the company accepting. If they refuse the counter-offer, the case goes into court that you think you have 50% of winning. If you win, you gain the 100000 or if you lose, you end up paying 55000 in legal fees. Which path you should take, and what is the expected value of perfect information for this problem.
I think there are two scenarios (accepting 35000 or countering), that I think have the expected values of:
\begin{align} EV_{\text{accept}} &= 1\times35000=35000\\ EMV=EV_{\text{counter}} &= \overbrace{0.4\times65000}^{\text{counter is accepted}} + \overbrace{(1-0.4)[0.5\times100000-0.5\times55000]}^{\text{case goes to court}} = 39500 \end{align} So you should choose to counter and not accept if you are risk neutral. But I have trouble understanding the value of perfect information. Wiki states that the value of perfect information is the difference between the (maximum) expected value with out information and with information. Would the expected value WITH information then be:
\begin{align} EV|PI &= \sum_j p_j (\operatorname{max}_i R_{ij}), \\ &= 1.0\times\operatorname{max}\{35000,0\} + 0.4\times\operatorname{max}\{65000,0\}+0.6\times \operatorname{max}\{ 0.5\times100000, -0.5\times55000\},\\ &= 91000. \end{align}
Then the value of perfect information would be. But I cannot somehow intuitively confirm if its true or not.. \begin{equation} EVPI =EV|PI - EMV = 91000-39500 = 56000. \end{equation}
Edit: I'm actually not sure if the expected monetary value for the counter offer is calculated correctly.
The value of the option to counteroffer (relative to accepting settlement) with perfect information on what the result would be is:
40% of the time: raise settlement value by +65000 - 35000 = +30000
30% of the time: lawsuit award - refused settlement = +100000 -35000 = +65000
30% of the time: avoided lawsuit penalty + avoided settlement loss = +55000 + 35000 = +80000
The expected value of that is in the tens of thousands.
Without the information, the expected value of a counteroffer is calculated as +4500 in the question, or about 10 times lower. The difference between the higher and the lower expected value is the value of the perfect information (assuming all decisions are made based on expected value).