Some notes after reading:
- Han, Z., Niyato, D., Saad, W., Başar, T., & Hjørungnes, A. (2011). Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511895043
Introduction
Def
A coalitional game (or a game in coalitional form) is defined by the pair (N, v), where N is the set of players, and v is a mapping that determines the payoffs that these players receive in the game.
Def
the characteristic function of a coalitional game with transferable utility is a function v over the real line defined a $v: 2^N -> \R$ with $v(\phi) = 0$
Characteristic form
the value of a coalition depends solely on the members of that coalition, with no dependence on how the players in N\S are structured
TU games
⇒ The TU property implies that the total utility represented by this real number can be divided in any manner between the coalition members
NTU games
⇒ The payoff that each player in a coalition S receives depends on the joint actions that the players of coalition S select
Partition form
⇒ with “strong” dependence on how the players in N\S are structured
Example
Canonical (coalitional) games
It is assumed that when forming a larger coalition the players cannot do worse than by acting alone (non- cooperatively)
Players always tend to form the Grand coalition N
- Finding a payoff allocation that guarantees that no group of players has an incentive to leave the grand coalition (having a stable grand coalition)
- Assessing the gains that the grand coalition can achieve as well as the fairness criteria that must be used for distributing these gains (having a fair grand coalition)
The core as the solution
The Shapley value
The nucleolus
Coalition-formation games
Study the network coalitional structure
- Which coalitions will form?
- What is the optimal coalition size?
- How does the network structure evolve over time?
- How can we assess the structure’s characteristics? ⇒ generally not superadditive and can support both the characteristic-form and partition-form models (in TU or NTU) ⇒ the presence of a cost for forming coalitions.
Static and dynamic
Applications
