Assignment:
Let Γ be a game in extensive form. The agent-form game derived from Γ is a strategic-form game where each player i in Γ is split into several players: for each information set Ui ∈ Ui of player i we define a player (i, Ui) in the agent-form game. Thus, if each player i has ki information sets in Γ, then there are players in the agent-form game. The set of strategies of player (i, Ui) is A(Ui). There is a bijection between the set of strategy vectors in the game Γ and the set of strategy vectors in the agent-form game: the strategy vector σ = (σi)i∈N in Γ corresponds to the strategy vector (σi(Ui)){i∈N ,Ui∈Ui} in the agent-form game. The payoff function of player (i, Ui) in the agent-form game is the payoff function of player i in the game Γ. Prove that if σ = (σi)i∈N is a Nash equilibrium in the game Γ, then the strategy vector (σi(Ui)){i∈N ,Ui∈Ui} is a Nash equilibrium in the agent-form game derived from Γ.
Provide complete and step by step solution for the question and show calculations and use formulas.