200	9. Coexisting Wireless LANs as Multi Stage Game

The payoffs indicated in Figure 9.11 show the advantage of the cooperative behavior about the persistent, as well as about the rational best response behavior, especially for player 2. By crossing the Pareto boundary, when playing cooperatively, the players achieve higher payoffs than before. However, the result that cooperation outperforms the rational behavior is valid in the analyzed example here, but it is not necessarily a valid assumption in general.

The left subfigure of Figure 9.12 shows the delays of resource allocations. The figure indicates that due to the shorter resource allocations of both players, the delays are smaller than before, which of course is beneficial in terms of observed resource allocation periods and observed share of capacity.

In contrast, Figure 9.12, right subfigure, clearly indicates that it cannot be generally stated, that it is always beneficial for a player to cooperate. As described already, only if the Nash equilibrium is not Pareto efficient, it is worth for the interacting players to cooperate. However, it depends on the requirements and utility functions if a Nash equilibrium is Pareto efficient. The fact that it is not always preferable for a player to cooperate can be seen in the right subfigure of Figure 9.12, where the results shown in the case of small requirements for share of capacity imply that mutual cooperation results in smaller payoffs compared to what can be achieved by rational behavior. The players achieve high payoffs by playing their best responses as part of the rational behavior, as discussed in the previous section, and as illustrated in Figure 9.9. A Nash equilibrium in such a game, may it be unique or not, is probably Pareto efficient, and in such a situation, rational behavior will be preferred over cooperation by a player.

9.2.2.3Conclusion

Three static strategies, the simple persistent STRAT-P, the payoff maximizing STRAT-B, and the cooperative STRAT-C have been compared with each other. In summary, the following conclusions can be drawn from the analysis. A player that does not meet its requirements because of the allocation process and the competition with another player shall not allocate resources as defined by the QoS requirements; it shall in contrast play rational and attempt to maximize its payoff:

U i( BEH−B ) ≥U i( BEH−P )

STRAT−B \i STRAT−P, i Ν.

The rational behavior weakly dominates the persistent behavior, however, only if resource allocations by the opponent player have implications on the resulting payoffs.

9.2 Static Strategies

201

Cooperation is not always beneficial. Depending on the requirements, there may exist an action profile that is not a Nash equilibrium profile, but results in higher payoffs for at least one player. In such a game, where Nash equilibria are not Pareto efficient, cooperation is a desirable action profile:

U i( BEH−C ) ≥U i( BEH−B )

STRAT−C \i STRAT−B, i Ν.

In any other case, playing rational will lead to a Nash equilibrium point, which is Pareto efficient, and thus rational behavior is preferred.

The QoS requirements of opponents are not known to the players. Therefore, without any history of interactions, players cannot assess if an outcome of a game is Pareto efficient or not. Further, it is difficult for players to assess if their opponent players are cooperating or not. Once a player concludes that cooperative behavior results in better outcomes (in higher resulting payoffs) than rational behaviors, it has to ensure that the opponent player will cooperate as well in future stages of the MSG.

Dynamic strategies are discussed in Section 9.3. Dynamic strategies allow players to switch between behaviors dynamically, as part of the strategies. In dynamic games, i.e., in games where players apply dynamic strategies, payoffs and Nash equilibria are calculated as defined in Section 9.1 for the whole duration of the MSG. The defective behavior BEH-D is used by players to coordinate the actions, and to prevent the opponent from not playing cooperatively once cooperation has been reached, as is shown in the next section.

202	9. Coexisting Wireless LANs as Multi Stage Game

9.2.2.4Results: Two Overlapping BSS Coordinated by CCHCs, Without Dynamic Behavior Adaptation: STRAT-P Against STRAT-P

1	simulated				required		1			required
	(solid line)				observed			simulated		observed
					demanded			(solid line)		demanded
0.6							0.6
Θ1						↑	Θ2			↑
	analyt. apprx.		some variations		arrow indicates that
	(dashed line		because of EDCF		players generally
	not visible)				attempt to maximize			analyt. apprx.
0					throughput		0	(dashed line)
0.1	simulated	analyt. apprx.			required		0.1	simulated		required
	(solid line)				observed max					observed max
		(dashed line)							analyt. apprx.
								(solid line)
					demanded				(dashed line)	demanded
∆1						↓	∆2			↓
0.04				arrow indicates that
							0.023
				players generally
				attempt to minimize
0				delays			0

0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4	0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
			time (s), SFDUR = 200ms										time (s), SFDUR = 200ms

solid lines: analytics, dashed lines: YouShi simulation results

Figure 9.4: Required, demanded, and observed QoS parameters for player 1 (left) and player 2 (right). Top figures show the Θ parameter, bottom figures show the ∆ parameters.

	1
	1,2 ∆
	U	required resource allocation intervals
	Utilities
		cannot be achieved, player 1 observes
		some advantages
	0	pl 1
		pl 2

defect pl 1 pl 2

persist

both players demand coop requirements throughout

all stages (BEH−P)

best

0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
			time (s), SFDUR = 200ms

		1	player 1 achieves the
				pl 1
			required share of capacity
				pl 2
	1,2	Θ	results vary because of
	U		random nature of EDCF
	Utilities
			player 2 does not
		0	achieve its requirements
		1

1,2 Θ∆				in total, player 1 observes a								pl 1
U				higher payoff than player 2 when								pl 2
Utilities				both demand their requirements
	0.4
	0.2
	0
	0.2		1	1.8	2.6	3.4	4.2	5	5.8	6.6		7.4
					time (s), SFDUR = 200ms

Figure 9.5: Selected behaviors throughout the game (bottom left). Resulting payoffs

(=observed utilities) for player 1 and player 2: U i			(top left) ,	U i	(top right), and the total
payoff V i =U i =U i	U i	∆		Θ
		(bottom right), i =1, 2 .
Θ	∆

9.2 Static Strategies

100

(CDF)
Prob(delay>x)	10−1
	10−2

0 2

player 2 observes		pl 1
larger delays because		pl 2
of longer allocations		EDCF
by player 1
Θ1	= 0.60
obs
Θ2	= 0.39
obs

EDCF share = 0.00

4

6

8

10

203

x = resource allocation delay (TXOP delay) [ms]

Figure 9.6: Probabilities of delay of resource allocation attempts, TXOP delay (left), and throughput and payoff results for varying requirements (right).

9.2.2.5Results: Two Overlapping BSS Coordinated by CCHCs, Playing the Best Response: STRAT-B against STRAT-B

1

0.6

Θ1

0

required	as both players play their	1
observed
demanded	best responses, the demands
	converge into Nash equilibrium (NE)	0.6
	↑
		Θ2
sim.	apprx.
observed thrp. decreases, because now the
opponent player 2 plays its best response		0

now demanding high thrp. when converging into NE

		this player gains from playing
		the best response
			↑
		sim.	required
	apprx.		observed
			demanded

0.1							required				0.1							required
							observed max											observed max
							demanded						apprx.					demanded
												sim.
∆1									↓	∆2										↓
0.04
				sim.		apprx.		demanding NE			0.023
																		demanding NE
0											0
0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4		0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
			time (s), SFDUR = 200ms											time (s), SFDUR = 200ms

solid lines: analytics, dashed lines: YouShi simulation results

Figure 9.7: Required, demanded, and observed QoS parameters for player 1 (left) and player 2 (right). Top figures show the Θ parameter, bottom figures show the ∆ parameters.

204	9. Coexisting Wireless LANs as Multi Stage Game

1

Utilities U∆1,2

0

defect

persist

coop

best

0.2 1

in NE, the required resource allocation intervals are sufficiently met in this game

pl 1 pl 2

pl 1 pl 2

after 2s, both players change their behavior from BEH−P to BEH−B independently, then attempting to improve their individual payoffs

1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
	time (s), SFDUR = 200ms

		1
		1,2 Θ
		U
		Utilities	player 1 and 2 now equally share capacity
			and suffer both from the allocation process
			pl 1
		0	pl 2
		1	pl 1
			pl 2
	1,2 Θ∆		in total, player 2 gains and player 1
			suffers from playing the best responses
	U
	Utilities	0.4	neither player is able to improve its outcome

0.2by unilaterally changing its behavior from what

is demanded after the process converged into NE

0

0.2

1

1.8

2.6

3.4

4.2

5

5.8

6.6

7.4

time (s), SFDUR = 200ms

Figure 9.8: Selected behaviors throughout the game (bottom left). Resulting payoffs

(=observed utilities) for player 1 and player 2: U i			(top left) ,	U i	(top right), and the total
payoff V i =U i =U i	U i	∆		Θ
		(bottom right), i =1, 2 .
Θ	∆

Prob(delay>x) (CDF)

100
		both players now		pl 1
				pl 2
		attempt to improve
				EDCF
		their payoffs
10−1	before		Θobs1	= 0.47
	(persist)		Θ2	= 0.51
			obs
			EDCF share = 0.01
10−2	2	4	6	8	10
0

x = resource allocation delay (TXOP delay) [ms]

shares (thrp.) and payoffs per player

1

best payoff in the presense

of rational opp.

0.8 share pl.1

share pl.2 share EDCF

0.6 payoff pl.1 payoff pl.2

0.4

0.2

no EDCF because of lower priority

0	0.2	0.3	0.4	0.5	0.6	0.7	0.8
0.1

req. Θ per player

Figure 9.9: Probabilities of delay of resource allocation attempts, TXOP delay (left), and throughput and payoff results for varying requirements (right).

9.2 Static Strategies

205

9.2.2.6Results: Two Overlapping BSS Coordinated by CCHCs, Playing both Cooperatively: STRAT-C against STRAT-C

1	required		1		required
apprx.
(not visible)	observed				observed
sim.	demanded				demanded
			0.6
0.6
Θ1		↑	Θ2		↑
			sim.	apprx.
0			0
0.1	required		0.1		required
	observed max				observed max
sim.	demanded			apprx.	demanded
apprx.		↓	∆2	sim.	↓
∆1
0.04
			0.023

0											0
0.2		1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4	0.2		1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
				time (s), SFDUR = 200ms											time (s), SFDUR = 200ms

solid lines: analytics, dashed lines: YouShi simulation results

Figure 9.10: Required, demanded, and observed QoS parameters for player 1 (left) and player 2 (right). Top figures show the Θ parameter, bottom figures show the ∆ parameters.

1
	required resource alloction
1,2 ∆	intervals are achieved by
	both players when cooperating
Utilities U
	pl 1
0	pl 2
defect	pl 1
	pl 2

persist

coop

both players cooperate after 2s

best

0.2	1	1.8	2.6	3.4	4.2	5	5.8	6.6	7.4
			time (s), SFDUR = 200ms

	1
1,2		requirements for thrp are
	Θ	nearly met in cooperation
Utilities U
		pl 1
	0	pl 2
	1

1,2 Θ∆					in total, payoffs are higher in cooperation
U					than in NE, therefor the NE is not Pareto
Utilities					efficient in this example
	0.4
	0.2
												pl 1
	0											pl 2
	0.2		1	1.8	2.6	3.4	4.2	5	5.8	6.6		7.4
					time (s), SFDUR = 200ms

Figure 9.11: Selected behaviors throughout the game (bottom left). Resulting payoffs

(=observed utilities) for player 1 and player 2: U i			(top left) ,	U i	(top right), and the total
payoff V i =U i =U i	U i	∆		Θ
		(bottom right), i =1, 2 .
Θ	∆