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quire that players know the QoS requirement of their opponents. Because of the coexistence approach without interworking, i.e., without information exchange, players observe actions of their opponent player and with the help of a prediction method the demands of their opponent player, but do not have access to the QoS requirements their opponent player is attempting to achieve. The prediction method to determine the demand of the opponent player from the observations is explained in Section 7.4.

It is assumed that one common utility function is used by all players, where the individual players do not know the parameters of the utility functions of their respective opponents.

For the game approach to be successfully applied, all involved radio systems have to follow it.

It is not the intention of this approach to allow a CCHC to improve its performance against legacy stations, i.e., stations that do not follow the game approach. However, upon detecting a co-located legacy station, a CCHC does model this opponent as a so-called myopic, persistent player, which does not consider any utility, but attempts to allocate its requirement. There are strategies that optimize the utility against such players, as will be shown in Chapter 9.

7.2.3.2Preference and Behavior

The utility U i	represents the preference relation \i of player i in the sense	that
U i( a ) ≥U i( b )	whenever a \i b , with a,b Ai . In this case it is said that	a is

preferred over b, a weakly dominates b. IfU i( a ) >U i( b ) , a strictly dominates b, i.e., a i b , a,b Ai . The binary operator ” \i ” can be understood as a function from Α to Ai (Debreu 1959:6).

A behavior that a player actually is following is based on such preferences. For example, a player may attempt to maximize its utility within the current stage n by selecting a demand that optimizes its own utility. In contrast, it may prefer to select a demand that allows the opponent player to achieve a certain utility as well. Various kinds of behavior are introduced in Section 8.3.

7.2.4Payoff, Response and Equilibrium

So far, the utility was introduced as a value that represents the QoS a player observes. Because the observed utility is dependent on the actions of all involved players, another term is introduced that highlights this dependency.

148	7. The Game Model

Figure 7.6: Utility U i of a player i vs. observation (Θobsi , ∆obsi ) , with (Θreqi , ∆reqi ) = (0.4,0.04) . There is no impact of any opponent player -i in this example, the player i observes its de-

mands. To demand the requirement is the action that maximizes the observed utility.

The resulting utility as the outcome of an SSG at stage n is called the observed payoffV i :

V i( a( n )) :=( ai( n ),a−i( n )) → U i( n ), a( n ) Α =×	Αi	(7.16)
i N

Here, a( n ) =( ai( n ), a−i( n )) is a vector of actions, which is also referred to as action profile. The principle of an observed payoff completes the SSG model. The payoff describes what a player receives as utility for selecting an action, as function of the demand of all involved players. It denotes the single stage payoff of the player i as function of the actions, or demands, of all involved players i, -i. Figure 7.7 shows the payoff V i( a( n )) of player i for the utility function presented above. In this example, the same requirements as before are assumed, whereas player -i demands( Θdem−i = 0.4, ∆dem−i = 0.04 ) . Comparing the payoff in Figure 7.7 with the utility in Figure 7.6, the mutual impact of the allocation processes of the players can be clearly seen. The optimal action of player i, i.e., the action that maximizes the observed payoff, differs significantly from its requirement, i.e., the action that maximizes the utility when no other player demands any resources.

In contrast to decision problems, in a game each player has only partial control over the environment. The payoff of each player depends not only upon its actions but also upon the actions of other players. As consequence, when demanding resources, a player must therefore act by taking into account the possible demand of resources of its opponent. This may be interpreted as response or

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reaction to the opponent’s actions. A best response is the action that maximizes the player’s expected payoff. A player that selects this action is in this thesis referred to as acting rationally by responding to the correct expectation about its

opponent player’s actions. Figure 7.7 shows the payoff		of	player i in stage n,
V i ( a( n )) =V i ( ai ( n ),a−i( n )) , where in this example		it is	assumed	that the
opponent player takes	the action a−i( n ) =( 0.4,0.04 ) . When players			interact,
there is another payoff	V −i( a( n )) for the opponent player -i, which also de-

pends on the action of both players. This payoff is not shown in Figure 7.7. In an SSG without history, and without information exchange between players, the players cannot estimate the upcoming actions of their opponents. What action their opponent player will take is not known to any player.

For successful QoS support, demands must be selected, i.e., actions must be taken, that allow a minimum level of observed QoS, i.e., a minimum payoff, regardless what action the opponent takes. This leads to inefficient usage of resources. In contrast, if there is mutual knowledge about the opponents, this can be used to coordinate the actions. Such knowledge may be taken from the history of interaction of earlier games (as it is assumed in this thesis), or through dedicated information exchange (for example via a common spectrum coordination channel).

Figure 7.7: Payoff V i ( a(n )) of player i vs. demand ai = (Θi		, ∆i	) , with the same re-
quirements as before, (Θreqi , ∆reqi	dem	dem
	) = (0.4,0.04) . The player i does not observe its require-

ments and does not observe its demands due to the implications of the actions taken by

the opponent player -i. Indicated is the optimal action ai * = (Θdemi * = 0.4, ∆demi * = 0.027) . In this example, the opponent player -i demands (Θdem−i = 0.4, ∆dem−i = 0.04) . The figure is based on the analytical model of the SSG, as described in Section 8.1.