I am trying to learn game theory and I am really confused about common knowledge and events. There is a question in one of my notebooks. Consider the following information structure between three agents:
- $Ω = \lbrace α, β, γ, δ, e, µ, ρ, σ, τ, ω\rbrace$
- $P_1 = \lbrace\lbrace α\rbrace, \lbrace β, γ\rbrace, \lbraceδ, e, τ, ω\rbrace, \lbrace µ, ρ, σ\rbrace\rbrace$;
- $P_2 = \lbrace\lbrace α\rbrace, \lbrace β\rbrace, \lbrace γ, δ, e, µ, σ\rbrace, \lbrace ρ, τ, ω\rbrace\rbrace$;
- $P_3 = \lbrace\lbrace α, β, γ\rbrace, \lbrace δ, e\rbrace, \lbrace µ, σ\rbrace, \lbrace ρ, τ, ω\rbrace\rbrace$
- Each individual holds a uniform prior on $Ω$. An event is said to be a $p$ evident event if whenever it happens every agent believes that it happens with at least $p$ probability. Find all $1/2$ evident events.
I cannot understand what "an event happens" means. I know I should use information partitions but how? Should I do the same thing when we find probability of an event happens when true state of the world is $w$. I would appreciate some help.