Multiagent Systems

Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 215/731
3. Decision Making (1) 2. Non-coop Games
Let X consist of all mixed strategy profiles and let f be the
best response set: f(〈s 1 , . . . , s n 〉) is the set of all best
responses 〈s ′ 1, . . . , s ′ n〉 (where s ′ i is players i best response
to 〈s 1 , . . . , s i−1 , s i+1 , . . . s n 〉).
Why is that a subset of an n-dimensional Euclidean
space?
A mixed strategy over k actions (pure strategies) is a
k − 1 dimensional simplex (namely the one satisfying
∑ k
i=1 p i = 1). Therefore X is the cartesian product of n
simplices. X is compact and convex (why?).
The function f satisfies the remaining properties listed
above. Thus there is a fixed point and this fixed point is a
Nash equilibrium.