$4.$ Individual Problems $15-5$

Every year, management and labor renegotiate a new employment contract by sending their proposals to an arbitrator, who chooses the best proposal $($effectively giving one side or the other $\backslash (\backslash$$ $1\backslash )$ million$).$ Each side can choose to hire, or not hire, an expensive labor lawyer $($at a cost of $\backslash (\backslash$$ $100,000\backslash ))$ who is effective at preparing the proposal in the best light. If neither hires a lawyer or if both hire lawyers, each side can expect to win about half the time. If only one side hires a lawyer, it can expect to win nine tenths, or $0\mathrm{.}9,$ of the time.

Use the given information to fill in the expected payoff, in dollars, for each cell in the matrix. $($Hint: To find the expected payoff, multiply the probability of winning by the dollar amount of the payoff. Be sure to account for lawyer costs, which are incurred with certainty if a lawyer is hired.$)$

The Nash equilibrium for this game is for Management to

a lawyer, and for Labor to

a lawyer.