The aim of this paper is to propose a solution method for 2-player zero sum games with fuzzy payoffs, which takes different risk attitudes of the players into account. In the proposed model, players' risk attitudes are modeled with alpha-cut concept of fuzzy logic, and the game is solved by multi-objective linear programming approach. The model is implemented on an online marketing problem to show its applicability and use. The game solutions for both players are found by optimizing specific alpha-cuts that represent the risk levels of players and given as the game value and corresponding efficient mixed strategies. The results indicated that the proposed model provides a flexible decision-making environment to the players for their different risk attitudes.