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9.2.1 Aloha-Based Protocol Medium Access... > Nash Equilibria - Pg. 249

9.2 Access Control Games for Multiple Access Collision Channels 249 1 1 and the corresponding utility is K (1 - K ) K-1 . The sum of utilities or social welfare is 1 (1 - K ) K-1 , which goes to 1 when K goes to the infinity. For this reason, the bound e 1 e is called the asymptotic capacity of the system. It can be checked that all results given so far hold in the m + 1-action case when m + 1 > 2. Nash Equilibria First, note that from the Nash existence theorem (see Chapter 2), the existence of at least one mixed Nash equilibrium is guaranteed, since the game G K,m is finite. In fact, the number of mixed Nash equilibria is infinite: all mixed strategy profiles for which one component is 1 and the others are arbitrary mixed strategies (x i , 1 - x i ) are mixed Nash equilibria. The game G K,m has also pure Nash equilibria. Indeed, all the pure strategy Pareto optimal solutions derived in the previous section are pure Nash equilibria. In Table 9.1 the 2-transmitter 2-action access control game is represented in a bi-matrix form. Pareto optimal solutions and pure Nash equilibria are highlighted. Stackelberg Solutions In this section we consider a Stackelberg formulation (see Chapter 2) of the two- transmitter two-action access control game when only pure strategies are allowed. The game is known to have at least one Stackelberg solution in pure strategies. More