Now back to our power control program, is that power
control is competition and competition can be made as a game.
So what kind of game are we talking about here?
Well, let's identify this set of players first,
well that's the logical link or the transceiver pair
[UNKNOWN].
Let's identify the strategy space here.
It becomes a little more complicated, in the two examples we
just saw, the strategy space is pretty simple, it's just two choices.
But here, is a continuum of the power levels for each
user, I such that, the target SIR gamma is achieved.
Now as you can see, the strategy space, set so defined is a set of P's.
Is determined in part by what the other users P's might be.
Cause that would determine how easy or hard
is it for you to achieve the target, SIR.
So it's through the coupling of strategy space, instead of the coupling of payoff
function that we see the interference phenomena
relfected in the power control game here,
because the payoff function is very simple.
You, basically want to look at, a cost function,
that is Pi, I want to make it small.
So that is the definition of the game.
And here are two simple but important facts.
Fact number one, you easily verify for yourself, is
that the DPC algorithm, what is the algorithm again?
It says that the transmit power for each transmitter I at time T plus 1
is the current transmit power times the ratio between the target SIR and
the current SIR at time T. And this goes on for all the users.
This actually is the best response strategy
in response to all the other players' strategies.
Whatever they might be, they are reflected through the SIR value.
And by changing my power level this way, changing my strategy this way
to the next time slot, I am basically executing the best response strategy.
The second thing you can check is a monotonicity of the strategy space.
What do I mean by that?
By that, I mean that if all the other users make
their powers, Pj, j not equal to me I am user I.
Smaller okay, make these smaller then my strategy space,
set of Ps such that the gammas can be achieved will be bigger.
Now, this is obvious once I stated right, because
If you reduce your transmit power, I have less interference,
so I have more choices of P a smaller
value of a P could give me my target gamma.
And this is what I mean about monotonicity of the strategy space, it
turns out that there is a special name for this kind of game.
And subject to some technical condition that
holds in this case.
We say that best response strategies, iterations of best responses,
will converge to an equilibrium, to a natural equilibrium.
Provided the strategy space is for all the user
is a monotonic in the sense that we just described.
So, we wont have time to go through the
proof of that, but this is one way in addition to do the optimization method
that we will go through in advance material
part of the lecture to prove convergence of.
[SOUND]
Distributed, power control algorithm.