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This paper develops a theory of grade games based on the grade payoff. The grade payoff is defined as the grade rank of play's decision choices against the choices of other players. Then, the concept of grade games is formulated and the solutions such as Nash equilibria and mixed strategies equilibria for the grade games are proposed. Because the information of the ranks of payoffs and the difference between them are included in the grade payoffs, the mixed strategies equilibria can be analyze different from ordinal game theory. Furthermore, it has been proved that the mixed strategy equilibrium of the corresponding grade game will converge to the mixed strategy equilibrium of cardinal game in zero-sum case. Finally, some examples are given to show the results. [ABSTRACT FROM AUTHOR]