Polar, Inc. and ILEN are the two largest companies that produce and sell database software. They each have been negotiating to buy DSOFT, the third largest company in the database software market. Polar currently has 50% of the world market for database software, ILEN has 35%, DSOFT has 10%, and there are several other smaller companies that share the remaining 5% of the market for database software. Financial and market analysts at Polar have estimated the value of DSOFT to be $300 million in net worth. Throughout the preliminary negotiations, DSOFT has made it clear that they will not accept a purchase offer below $300 million. Jacob Pratt, the CEO of Polar, figures that acquiring DSOFT will make Polar the dominant player in the industry as Polar would then have 60% of the market. In addition, Jacob knows that DSOFT has been developing a new product that has tremendous earnings potential. Jacob has estimated that the new product would increase the net worth of DSOFT by an additional $300 million with probability 0.50, by an additional $150 million with probability 0.30, or have no impact on net worth with probability 0.20. To simplify matters, Jacob has decided to consider three possible strategies regarding the possible purchase of DSOFT: (i) Make a "high" offer of $400 million; (ii) make a "low" offer of $320 million; or (iii) make no offer at all. If he pursues this third strategy (making no offer), Jacob is certain that ILEN will buy DSOFT. If Polar makes an offer to DSOFT (either a "high" or a "low" offer), Jacob figures that ILEN will increase the offer further. He is uncertain about what ILEN will offer in this case, but he has made the following intelligent estimates of possible outcomes: ILEN would increase Polar's offer by 10% with probability 0.30, by 20% with probability 0.40, or by 30% with probability 0.30. If ILEN were to make such an offer, Jacob would then need to decide whether he would make a final offer to DSOFT. His thinking is that after ILEN makes a counter-offer, Polar would either withdraw from the bidding, match the counter-offer, or make a final offer at 10% above ILEN's counter-offer. If Polar's offer and ILEN's offer are identical, Jacob estimates that the probability that DSOFT will accept Polar's final offer is 0.40 ; however, if Polar counters with an offer which is 10% higher than ILEN's offer, Jacob estimates that DSOFT will accept Polar's final offer with probability 0.60.
(a) Structure Polar's acquisition offer problem as a decision tree.
(b) Solve for the optimal decision strategy for Polar.