We investigate the error exponent of exclusive-or multiple-access channels (XMAC) where the receiver wants to reconstruct the exclusive-or of incoming messages from two transmitters. We consider both the multiple access channel (MAC) strategy and the network coding (NC) strategy. In our analysis, the cutoff rate is used as a performance measure. Assuming a Gaussian XMAC and binary phase-shift keying (BPSK) modulation at each node, we show that the MAC strategy performs better than the NC strategy in the low rate region while the NC strategy performs better in the high rate region.