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We address the allocation of subcarriers, bits and power among the base stations (BSs) of downlink multi-cell OFDMA systems using the non-cooperative game theoretic approach. The utility function of a player is defined as the difference between the revenue generated from transmitting the bits and the cost of the power consumed. Unlike the commonly adopted water-filling approach where continuous bit loading is used, we consider assigning integer number of bits to subcarriers. We address the existence of pure Nash equilibrium (NE) of the static games for two-cell modulation adaptive and three-cell single modulation OFDMA systems. Theorems related to the existence of pure NE for the two games are developed. Using these results, we show that the existence of a stable solution for the OFDMA games cannot be guaranteed if minimum rate requirements must be satisfied.