Contestants can either assess their own resource-holding potential relative to their opponent (mutual assessment) or rely solely on the assessment of their own fighting ability (self-assessment). To discriminate between these possibilities, we staged dyadic territorial contests between 'size-matched' male swordtails. These contests consist of a combination of ritualized displays and direct fighting. Although size differences were small, winners were larger than losers and smaller fish tended to be winners only when the size difference was negligible. Body size, however, did not influence contest duration and there was no increase in contest duration with mean body size; thus, there is no support for self-assessment in these animals. We also examined the effects of the sword, which comprises a sexually selected extension used in female choice that reduces swimming efficiency but increases acceleration. The length of the sword (adjusted for body size) did not differ between winners and losers; however, losers conceded earlier if the opponent had a large sword for its body size but this decision was independent of the loser's own sword length. Losers thus assessed the swords of winners, which precludes self-assessment; however, because winners appeared not to assess the swords of losers, this does not fully support the idea of mutual assessment.