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 1 : mark 1 2 : 3 : \chapter{Proofs} 4 : 5 : 6 : 7 : 8 : 9 : \begin{figure}[ht] 10 : \includegraphics[width=4.5cm]{figs/coop} 11 : %\psfig{file=coop.eps,width=4.5cm} 12 : \caption{Mutual coordination by third-party mediation.\label{coop}} 13 : \end{figure} 14 : \bigskip 15 : \begin{lemma}[Observably consistent promises]\label{th1} 16 : Two promises made to a third-party node, $n_3$, by two other nodes 17 : $n_1$ and $n_2$, are consistent (see fig. \ref{coop}), i.e. 18 : \beq 19 : \stackrel{b}{n_1\rightarrow n_3} ~\Leftrightarrow~ \stackrel{b}{n_2\rightarrow n_3}, 20 : \eeq 21 : if and only if 22 : the two nodes additionally promise one another cooperation $C(b)$ 23 : in the matter of $b$. 24 : \end{lemma} 25 : \bigskip 26 : \begin{proof} 27 : Consistency implies 28 : \beq 29 : \stackrel{b}{n_1\rightarrow n_3} ~\Leftrightarrow~ \stackrel{b}{n_2\rightarrow n_3}\label{xxx}, 30 : \eeq 31 : but from, eqn. \ref{xx} we may write 32 : \beq 33 : \stackrel{b}{n_1\rightarrow n_3} &\Leftarrow& \stackrel{C(b)}{n_1\rightarrow n_2}\otimes\stackrel{b}{n_2 \rightarrow n_3}\\ 34 : \stackrel{b}{n_2\rightarrow n_3} &\Leftarrow& \stackrel{C(b)}{n_2\rightarrow n_1}\otimes\stackrel{b}{n_1 \rightarrow n_3}. 35 : \eeq 36 : Thus, using eqn. (\ref{xxx}) on both sides of these equations, we derive 37 : \beq 38 : \stackrel{C(b)}{n_2\rightarrow n_1} ~\Leftrightarrow~ \stackrel{C(b)}{n_1\rightarrow n_2}. 39 : \eeq 40 : \end{proof} 41 : \bigskip 42 : Moreover, since the collaborative promise implies the existence of 43 : a promise of type $b$ to a common third party, the reverse direction is 44 : also true. 45 : By measuring $p$ from both $n_1$ and $n_2$, $n_3$ acts as a judge of 46 : their compliance with the mutual agreements between them (see 47 : fig. \ref{mm}). This allows the basis of a theory of measurement or 48 : observation in collaborative networks.