(Single Player Utility Sample Clauses

(Single Player Utility. We define G’s utility of bribing an agent a ∈ A, uG(a), by the change in probability that the topic definitively wins after the agent a is successfully bribed. We define uG(i) as the change in probability of G winning given that a currently blue node with highest influence (it might not be unique) in partition Pi has been chosen, i.e., uG(i) = maxa∈Pi {uG(a)}. Let B be an external agent that seeks to bribe a green agent to increase the probability of the topic being rejected. We define uB(a) analogously and uB(i) as the change in probability of B winning given that (one of) the most influential green nodes in partition Pi have been chosen. Before seeking to answer Questions D3 and D4 relating to the two briber problem, we first define the following notation. n-AF: we say an AF is an n-AF iff (1) A has only one SCC; (2) AF is bipartite with partitions Pfor and Pag; and (3) if the greatest common divisor of the length of all cycles in A is n. In particular, an n-AF is also a n-partite AF . For example, in the AF depicted in Figure 4.6, the greatest common divisor is 2, thus it is a 2-AF . P1, . . . , Pn: are the partitions of an n-AF such that Pi(K) ⊂ Pfor iff i is odd (and Pi(K) ⊂ Pag iff i is even). Note that an n-AF is both bipartite (with partitions Pfor and Pag) and n-partite (with partitions P1, . . . , Pn). gi (resp. bi): is the highest influence among agents currently coloured green (resp. blue) in partition Pi, i.e, gi = max a∈Pi and S(a)=1 bi = max a∈Pi and S(a)=0 {µ(a)} (4.19) {µ(a)} (4.20) I Θ^g : is the product of Θi, 0 ≤ i ≤ n such that i ∈/ I, where I ⊂ {1, . . . , n}. e.g., k=1 Θ^g = Qn Θk, or Θ^g = n k=1,k/=i Θk. We define Θ^b , b^I , and g^I analogously. I We omit the curly brackets from the set I (e.g., Θi,j) for readability. Example 4.5.3. In the example in Section 4.5.1, we have a 2-AF , partitions P1 506 506 506 506 2783 5566 and P2, g1 = 138 , g2 = 101 , b1 = 174 , b2 = 147 , Θ^g = 325 , and finally Θ^b = 2091 .‌
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