Abstract
The notion of a system carries the idea of interconnected parts that have interactions. In this chapter, we present some methods for analyzing systems comprised of several agents, making decisions in an order that is not fixed in advance.
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Notes
- 1.
We avoid the notation \(\mathcal {A}\) for the state of Nature \(\sigma \)-field, because there might be a visual confusion with the set \(A\) of agents.
- 2.
Our definition of precedence is directly inspired by the one that Ho and Chu give in their 1974 paper [83], and not by the one they provide in their 1972 paper [81] (which rather relates to the subsystem relation to be discussed in Sect. 9.4.2).
- 3.
That is, \(\mathfrak {S}\) is antisymmetric: \(\forall (\alpha ,\beta ) \in A^2\), \(\overline{ \{ \alpha \}} = \overline{ \{ \beta \}} \Rightarrow \alpha = \beta \) or, in other words, if two agents generate the same subsystem, they must be equal.
- 4.
That is, \(\alpha \not \in \langle \alpha \rangle _{\mathfrak {P}}^k\), \(\forall \alpha \in A\), \(\forall k \ge 1\).
- 5.
Bounded is for the sake of simplicity, so that integrability with respect to the probability \(\mathbb {P}\) is easily established. A nonnegative criterion would also do.
- 6.
See Footnote 5 in p. 286.
- 7.
For this, map the reduced history space \(\mathbb {U}_{A'} \times \varOmega \) into the full history space \( \mathbb {U}_{A} \times \varOmega \) by associating with each reduced history \((u_{A'},\omega ) \in \mathbb {U}_{A'} \times \varOmega \) the image \(S_{\bar{\lambda }}(\omega )\) of the solution map, where the policy \(\bar{\lambda }\) is given by the constant mapping \(\bar{\lambda }_\beta \equiv u_\beta \) for \(\beta \in A'\), and by the policies \(\bar{\lambda }_\beta = \lambda _\beta \) for \(\beta \in C\). Then, the reciprocal images of the information fields \(\mathcal {I}_{\beta }\) for \(\beta \in A'\) provide information fields \(\mathcal {I}'_{\beta }\) over the reduced history space \(\mathbb {U}_{A'} \times \varOmega \).
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Carpentier, P., Chancelier, JP., Cohen, G., De Lara, M. (2015). Multi-Agent Decision Problems. In: Stochastic Multi-Stage Optimization. Probability Theory and Stochastic Modelling, vol 75. Springer, Cham. https://doi.org/10.1007/978-3-319-18138-7_9
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DOI: https://doi.org/10.1007/978-3-319-18138-7_9
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