Asymmetric War of Attrition

Two firms contest a prize. Firm A values it at 6 and Firm B values it at 12. Time is continuous. Each firm pays 1 per hour for as long as both remain in the contest. The first firm to quit loses (it has paid only its waiting costs), and the other firm receives its prize value at that moment, net of the waiting costs it has paid. Each firm chooses a quitting time, simultaneously and in advance. Consider the Nash equilibrium in which both firms use continuous quitting-time distributions with positive density on all of [0, ∞) (so neither quits immediately with positive probability). What is the probability that Firm A, the one with the lower valuation, wins? Give a reduced fraction.

• combinations
• nash equilibrium

Source: War of attrition after Maynard Smith (1974); asymmetric-value equilibrium as in Hendricks, Weiss and Wilson (1988). Statement written for AxiomIQ.