A More Complex Decision Tree Analysis

In $1984,$ Penzoil and Getty Oil agreed to a merger. However, before the deal was closed, Texaco offered Getty a better price. So Gordon Getty backed out of the Penzoil deal and sold to Texaco. Penzoil immediately sued, won the case, and was awarded $$11\mathrm{.}1$ billion. A court order reduced the judgment to $$2$ billion, but interest and penalties drove the total back$$up to $$10\mathrm{.}3$ billion. James Kinnear, Texaco's chief executive officer, said he would fight the case all the way up to the U$.$S$.$ Supreme Court because, he argued, Penzoil had not followed Security and Exchange Commission regulations when negotiating with Getty. In $1987,$ just before Penzoil was to begin filing liens against Texaco, Texaco offered to give Penzoil $$2$ billion to settle the entire case. Hugh Liedke, chairman of Penzoil, indicated that his advisors told him a settlement between $$3$ billion and $$5$ billion would be fair.

What should Liedke do$?$ Two possible decision alternative are $(1)$ he could accept the $$2$ billion or $(2)$ he could turn it down. The third possible alternative is that he counteroffers $$5$ billion. If he does counteroffer, he judges that Texaco will either accept the counteroffer with probability $0\mathrm{.}17,$ refuse the counteroffer with probability $0\mathrm{.}5,$ or counter back in the amount of $$3\mathrm{.}5$ billion with probability $0\mathrm{.}33.$ If Texaco does counter back, Liedke will then have the decision of whether to refuse or accept the counteroffer. Liedke assumes that if he simply turns down the $$2$ billion with no counteroffer, if Texaco refuses his counteroffer, or if he refuses their return counteroffer, the matter will end up in court. If it does go to court, he judges that there is $0\mathrm{.}1$ probability Penzoil will be awarded $$10\mathrm{.}3$ billion, a $0\mathrm{.}5$ probability they will be awarded $$4$ billion, and a $0\mathrm{.}4$ probability they will get nothing.utility