For the threshold utility analysis, a matrix of hypothetical utility weights (u i ) for the TOX and REL health states was constructed by varying utility from 0 to 1 by 0.25, resulting in 25 combinations, relative to the utility of TWiST.
A threshold utility analysis was performed to show treatment comparisons of Q-TWiST for a matrix of possible utility weight combinations where μTOX and μREL are varied from 0 to 1 relative to μTWIST=1.
The Q-TWiST difference is clinically meaningful and was statistically significant across an entire matrix of possible utility weights.
We combine the utility matrix of Table 1 and the utility matrix of Table 2 to obtain Table 3 whose components are expected utilities of malicious type relay and cooperative type relay.
In fact, for a given n and m, the product of any two of these invertible matrices is an injective matrix (the utility of which we have not yet established).
Table 2 illustrates the utility matrix of attacker and source on relay R i with probability 1 - μ i t k being cooperative at stage t k.
Substitute α I I i + α S S i by A i, Table 1 illustrates the utility matrix of attacker and source on relay R i with probability μ i t k being malicious at stage t k.
Table 1 Utility matrix of attacker and source on relay R i with probability μ i t k malicious at stage t k Attack Not attack Select -(1 - 2a - C a )A i,(1 - 2a - C a )A i -(b C f + C m )A i,0 Not select -A i,(1 - C a )A i 0,0 Table 2 Utility matrix of attacker and source on relay R i with probability 1 - μ i t k cooperative at stage t k Not attack Select -(b C f + C m )A i,0 Not select 0,0.
Table 3 Utility matrix of attacker and source on relay R i at stage t k after Harsanyi transformation Attack, Not attack, Not attack* Not attack* Select - μ i t k ( 1 - 2 a + C m ) A i - ( 1 - μ i t k ) × ( b C f + C m ) A i, μ i t k ( 1 - 2 a - C a ) A i -(b C f + C m ) A i,0 Not select - μ i t k A i, μ i t k ( 1 - C a ) A i 0,0.
From these matrices I estimated, using PAUP, standard measures of utility, defined for the purpose of this study as properties of the matrices measured prior to analysis [19].
