Error estimates for binomial approximations of game options

We justify and give error estimates for binomial approximations of game (Israeli) options in the Black-Scholes market with Lipschitz continuous path dependent payoffs which are new also for usual American style options. We show also that rational (optimal) exercise times and hedging self-financing portfolios of binomial approximations yield for game options in the Black-Scholes market "nearly" rational exercise times and "nearly" hedging self-financing portfolios with small average shortfalls and initial capitals close to fair prices of the options. The estimates rely on strong invariance principle type approximations via the Skorokhod embedding.