5. Continuing with the diffusion model from Chapter 19, recall that the threshol

Question

5.

Continuing with the diffusion model from Chapter 19, recall that the threshold q was derived from a coordination game that each node plays with each of its neighbors. Specifically, if nodes v and w are each trying to decide whether to choose behaviors A and B, then:

• if v and w both adopt behavior A, they each get a payoff of a > 0;

• if they both adopt B, they each get a payoff of b > 0; and

• if they adopt opposite behaviors, they each get a payoff of 0.

The total payoff for any one node is determined by adding up the payoffs it gets from the coordination game with each neighbor. Let’s now consider a slightly more general version of the model, in which the payoff for choosing opposite behaviors is not 0, but some small positive number x. Specifically, suppose we replace the third point above with:

• if they adopt opposite behaviors, they each get a payoff of x, where x is a positive number that is less than both a and b.

Here’s the question: in this variant of the model with these more general payoffs, is each node’s decision still based on a threshold rule? Specifically, is it possible to write down a formula for a threshold q, in terms of the three quantities a, b, and x, so that each node v will adopt behavior A if at least a q fraction of its neighbors are adopting A, and it will adopt B otherwise?

in your answer, either provide a formula for a threshold q in terms of a, b, and x; or else explain why in this more general model, a node’s decision can’t be expressed as a threshold in this way ?

Explanation / Answer

the General formula for Network Coordinate game for the equation to diffusion model theory with Graph G=(V,W).

where V=>rows and W=>Columns

now PayOff Matrix can be rewritten as follows for a, b and c

coordinationg Game G=(A,B) with respect to v rows and w columns

in this given set of rowsv and columns w , we can consider following behaviours as Neighbours adopt A and for some adopt B.

The relation between the payoff matrices values a and b are:

in this representation of that a p fraction of V neighbours have behaviour A and a (1?p) fraction have behaviour B i.e. if V has neighbours then pd adopt A and (1?p)d adopt B

in this methods, for a given constaints, which gets payoff of pda value and if its choosen as B which gets a payoff of (1?p)db pda ? (1?p)db i.e. p?b/a+b

Cascading behaviour : In a typical Network wide coordinated Game like this, ther e are two posssibilities either every one adopts A and on the other hand remaining adopts B. fro the origin point of network till the the end in order to have a equilibrium between the coexistance of A and Bin some parts of As adoption B is also adopted..

hence byconsidering all set of Inititial adopter (A,B) which will start a new behaviour A while every other nodeswill starts with other one i.e. behaviour B. these nodes are repeatedly evalute the descision to switch from B to A using thereshold of q.

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