||Our study concerns bargaining behavior in situation where one party is in a strong position than the other. We investigate both the nature of the tradeoff the favored party makes between pursuing one's strategic advantage and giving weight to other players' concern for fairness, and the tradeoff the disadvantage player makes between pursuing a fair outcome from a disadvantaged position and the cost of that pursuit. In particular, we hypothesize that the degree to which strategically strong players attempt to exploit their strategic advantage depends on how much it costs them to do so. In addition, the degree to which weak players persist in seeking "fairness" is also a function of how much it costs them. Students negotiated in pairs over the division of $HK50 using a finite horizon, fixed-cost (per rejection) alternating offer rule. Each pair consisted of a high-cost and a low-cost bargainer. In accordance with the hypothesis, the willingness of the high-cost bargainers to demand fairness and to persist in their demands was a function of how much it cost them to do so, and the degree to which the low cost bargainers attempted to exploit their strategic advantages depended on their own cost of rejection. It can be said then that "fairness" has a price and that the higher its price, the lower the "demand" for it.