Hedging with risk for game options in discrete time

We study the problems of efficient hedging of game (Israeli) options when the initial capital in the portfolio is less than the fair option price. In this case a perfect hedging is impossible and one can only try to minimise the risk (which can be defined in different ways) of having not enough funds in the portfolio to pay the required amount at the excercise time. We solve the minimization problems and find via dynamical programming appropriate efficient hedging strategies for discrete time game options in multinomial markets. The approach and some of the results are new also for standard American options.