Game theory maths

Question

I have a question that I can only partly solve. You bid for an company that you can increase the value by $80\%$

• A company is valued somewhere between $20M$ and $120M$ uniformly distributed. Thus, $\text{exp(Value)} = 70M$

• Probability $60M$ bid succeed is $(60-20)/(120-20) = 0.40$

• What is the expected profit at time bid? Therefore, at the point of time when you make your bid of $60M$.

I have calculated $\text{exp(Value)}=(60/100) =0.60$; $70M(0.60) = 42M$; $42M(1+80%) = 75.6M$. Is this correct? I presume the $60\%$ from the 60M bid divided by $100M$ range.

What should you bid if yo want to maximize you expected profit? This is an algebraic expression for expected profit as seen from the point in time when you are making your bid. Choose $X$ to maximize this expression. This needs to be an quadratic equation, but I'm not sure where to start.

Help appreciated.

Answer 1

As I read the question and comments, the following needs to be added

• The company is valued by its current owners somewhere between $20M$ and $120M$ uniformly distributed
• If you make a bid for it, the owners will sell if your bid exceeds their valuation
• If you buy the company, you will increase its value by $80\%$ and sell it at the new value

As you say, the chance your $60M$ bid is successful is $0.40$. If the bid is successful the value is between $20M$ and $60M$ uniformly distributed. After your increase, the value will be between $36M$ and $108M$ uniformly distributed. The mean of this is $72M$.

Your expected profit is then $0.60 \cdot 0$ (if your bid fails) plus $0.40\cdot (72M-60M)=4.8M$