1 00:00:00,670 --> 00:00:03,662 Now back to our power control program, is that power 2 00:00:03,662 --> 00:00:07,280 control is competition and competition can be made as a game. 3 00:00:07,280 --> 00:00:09,990 So what kind of game are we talking about here? 4 00:00:09,990 --> 00:00:12,957 Well, let's identify this set of players first, 5 00:00:12,957 --> 00:00:16,062 well that's the logical link or the transceiver pair 6 00:00:16,062 --> 00:00:17,240 [UNKNOWN]. 7 00:00:17,240 --> 00:00:20,000 Let's identify the strategy space here. 8 00:00:20,000 --> 00:00:22,652 It becomes a little more complicated, in the two examples we 9 00:00:22,652 --> 00:00:26,590 just saw, the strategy space is pretty simple, it's just two choices. 10 00:00:26,590 --> 00:00:31,606 But here, is a continuum of the power levels for each 11 00:00:31,606 --> 00:00:36,871 user, I such that, the target SIR gamma is achieved. 12 00:00:39,760 --> 00:00:44,380 Now as you can see, the strategy space, set so defined is a set of P's. 13 00:00:45,580 --> 00:00:50,290 Is determined in part by what the other users P's might be. 14 00:00:50,290 --> 00:00:52,342 Cause that would determine how easy or hard 15 00:00:52,342 --> 00:00:54,370 is it for you to achieve the target, SIR. 16 00:00:54,370 --> 00:00:59,270 So it's through the coupling of strategy space, instead of the coupling of payoff 17 00:00:59,270 --> 00:01:02,140 function that we see the interference phenomena 18 00:01:02,140 --> 00:01:04,660 relfected in the power control game here, 19 00:01:04,660 --> 00:01:07,560 because the payoff function is very simple. 20 00:01:07,560 --> 00:01:11,172 You, basically want to look at, a cost function, 21 00:01:11,172 --> 00:01:13,720 that is Pi, I want to make it small. 22 00:01:15,670 --> 00:01:18,670 So that is the definition of the game. 23 00:01:18,670 --> 00:01:22,310 And here are two simple but important facts. 24 00:01:22,310 --> 00:01:25,820 Fact number one, you easily verify for yourself, is 25 00:01:25,820 --> 00:01:29,490 that the DPC algorithm, what is the algorithm again? 26 00:01:29,490 --> 00:01:34,942 It says that the transmit power for each transmitter I at time T plus 1 27 00:01:34,942 --> 00:01:40,582 is the current transmit power times the ratio between the target SIR and 28 00:01:40,582 --> 00:01:46,240 the current SIR at time T. And this goes on for all the users. 29 00:01:47,530 --> 00:01:51,119 This actually is the best response strategy 30 00:01:51,119 --> 00:01:55,450 in response to all the other players' strategies. 31 00:01:55,450 --> 00:01:59,190 Whatever they might be, they are reflected through the SIR value. 32 00:02:01,710 --> 00:02:05,781 And by changing my power level this way, changing my strategy this way 33 00:02:05,781 --> 00:02:10,350 to the next time slot, I am basically executing the best response strategy. 34 00:02:12,140 --> 00:02:17,260 The second thing you can check is a monotonicity of the strategy space. 35 00:02:17,260 --> 00:02:18,820 What do I mean by that? 36 00:02:18,820 --> 00:02:22,486 By that, I mean that if all the other users make 37 00:02:22,486 --> 00:02:27,250 their powers, Pj, j not equal to me I am user I. 38 00:02:27,250 --> 00:02:32,450 Smaller okay, make these smaller then my strategy space, 39 00:02:32,450 --> 00:02:37,800 set of Ps such that the gammas can be achieved will be bigger. 40 00:02:40,420 --> 00:02:44,032 Now, this is obvious once I stated right, because 41 00:02:44,032 --> 00:02:48,484 If you reduce your transmit power, I have less interference, 42 00:02:48,484 --> 00:02:50,920 so I have more choices of P a smaller 43 00:02:50,920 --> 00:02:53,900 value of a P could give me my target gamma. 44 00:02:55,890 --> 00:02:59,766 And this is what I mean about monotonicity of the strategy space, it 45 00:02:59,766 --> 00:03:03,620 turns out that there is a special name for this kind of game. 46 00:03:03,620 --> 00:03:05,596 And subject to some technical condition that 47 00:03:05,596 --> 00:03:07,140 holds in this case. 48 00:03:07,140 --> 00:03:13,980 We say that best response strategies, iterations of best responses, 49 00:03:13,980 --> 00:03:19,990 will converge to an equilibrium, to a natural equilibrium. 50 00:03:21,860 --> 00:03:24,512 Provided the strategy space is for all the user 51 00:03:24,512 --> 00:03:27,440 is a monotonic in the sense that we just described. 52 00:03:28,670 --> 00:03:30,750 So, we wont have time to go through the 53 00:03:30,750 --> 00:03:34,845 proof of that, but this is one way in addition to do the optimization method 54 00:03:34,845 --> 00:03:37,185 that we will go through in advance material 55 00:03:37,185 --> 00:03:39,935 part of the lecture to prove convergence of. 56 00:03:39,935 --> 00:03:40,410 [SOUND] 57 00:03:40,410 --> 00:03:43,860 Distributed, power control algorithm.