182	8. The Superframe as Single Stage Game

Nash equilibria but Pareto efficient, a coordinated change of action among all players is necessary. How this coordination is established is discussed in the context of behaviors and strategies in the following.

8.3.1Bargaining Domain

The concept of Pareto efficiency can be illustrated in what is referred to as bargaining domain. In the bargaining domain of two players, the payoff of one player is drawn against the payoff of the other player for any action profile that may be demanded by the two players. The resulting figure is a pattern of payoffs that depends on the requirements of the players. Figure 8.5, p. 183, shows an example of a bargaining domain of the SSG, where resulting payoffs for both players are indicated for a number of action profiles. In this example, the game has one Nash equilibrium that is not Pareto efficient. This is the case because there are other action profiles that lead to higher payoffs for at least one of the players, compared to the payoffs in Nash equilibrium.

A helpful method to illustrate the efficiency of Nash equilibria is the set of payoffs in the bargaining domain with higher payoff results than in the Nash equilibria for one of the players, and the same payoff result as in the Nash equilibria for the other players. This is illustrated as a line in the bargaining domain that is referred to as Pareto boundary in the following. The payoffs observed in Nash equilibria in the SSG are located on this Pareto boundary. See Zbigniew and Mason (1996) for a discussion of Pareto boundaries.

The Pareto boundary in the example of Figure 8.5 indicates that there are action profiles that result in higher payoffs (outcomes) for both players than what is achieved in the Nash equilibrium. Any action profile that leads to payoffs outside that boundary, are more Pareto efficient than any Nash equilibrium that may exist in the game. However, these profiles are not Nash equilibria, they cannot be achieved through rational behavior. At least one player has the intention to unilaterally change its action as part of its rational behavior, when these Pareto efficient profiles have been selected before.

8.3 ParetoTP PT Efficiency Analysis, and Behaviors

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Figure 8.5: Bargaining domain of an example of an SSG. Indicated are the resulting payoffs of both players for a number of action profiles, i.e., demands. This game has one Nash equilibrium that is not Pareto efficient: there are action profiles that result for both players in higher payoffs than what is achieved in the Nash equilibrium.

One Nash equilibrium exists in the shown example. When multiple Nash equilibria exist, the Pareto boundary is given as the combined set of all action profiles that lead to payoffs higher than the payoffs in any Nash equilibrium.

To achieve Pareto efficiency, it is required that players cross the Pareto boundary, for example as a result of mutual cooperation in case the Nash equilibrium itself is not Pareto efficient. Once that boundary is crossed, there may be still action profiles that lead to higher payoffs than others, i.e., some action profiles may be more Pareto efficient than others. The Figure 8.5 further shows the line where both players observe the same payoff, which is referred to as action profile that leads to a fair share of radio resources.

In such outcomes, both players achieve the same payoffs according to their individual requirements, but not necessarily the same QoS. It is said that the interacting players achieve the same level of satisfaction when the resulting outcome of a SSG lies at this line. In symmetric games, where both players have exactly the same QoS requirements, the Nash equilibrium lies on this line.

Any action profile that improves the outcome of the SSG in the sense that, compared to the payoffs in Nash equilibria, the total sum of the payoffs of all players