LAS Controller described a ground conflict event when traffic issued a 'give way' instruction to another taxing aircraft out in front of the same; causing a hard break stop; the reporter suggests pilots be more vigilant to ATC clearances.

Narrative

I was working Ground Control East. Air Carrier X had crossed Runway 25R at A5 and was instructed to 'Taxi via B; C6; to the ramp' by the Local Control 1 Controller. Air Carrier Y called me for taxi around the same time; and was told 'give way to Air Carrier X right to left; Runway 25R; Taxi via B;' to which the pilot acknowledged. I then received a call for taxi on the other side of the airport. As I turned around to observe the new aircraft's position; I heard the Local Control 1 Controller say 'hey tell Air Carrier Y to give way to Air Carrier B.' As I looked to see what the problem was; I noticed Air Carrier Y had taxied out in front of Air Carrier X at a high rate of speed; causing Air Carrier X to hit the brakes pretty hard. At the same time Air Carrier Y asked me 'were we supposed to go in front of Air Carrier X?' I replied 'negative; you were instructed to give way.' Air Carrier Y acted like nothing was wrong and simply said 'oh we must have missed that.' Air Carrier X complained to the Local Control 1 Controller and later called the supervisor in the Tower Cab. Pilots need to listen carefully to instructions; and ask for clarification before acting upon them if they are unsure what was heard.