Eco514-Game Theory Lecture 14: General Extensive Games Marciano siniscalchi November 10. 1999 Introduction By and large, I will follow OR, Chapters 1l and 12, so I will keep these notes to a minimum. J Games with observed actions and payoff uncertainty Not all dynamic models of strategic interaction fit within the category of games with observed actions we have developed in the previous lectures. In particular, no allowance was made for payoff uncertainty On the other hand, it is very easy to augment the basic model to allow for it. I am going follow OR very closely here Definition 1 An extensive-form game with observed actions and incomplete information (OR: a Bayesian extensive game with observed actions)is a tupleT=(N, A, H, P, Z, (0i, Pi, UdieN) where N, is a set of players a is a set of actions: H is a collection of finite and countable sequences of vectors of elements of A, such that i)0∈H; (i)(a2,,.a)∈ H implies(a1,,a)∈ h for all e<k; i)fh=(a1,,a,)and(a1,…,a0)∈ h for all k≥1, then h∈H (al, ...,ak, a) H for all a E A. Also let X=H\Z. All infinite histories are terma nd Z is the set of terminal histories: that is,(a P: X=N is the player correspondence ei is a set of payoff types; write e=ilen ei Pi is a probability distribution over i; for i j, Pi and p; are independent U=(Ui)iEN: Z x Lien 0i-R is the payoff function, associating a vector of payoffs te every terminal history and every profile of payoff typesEco514—Game Theory Lecture 14: General Extensive Games Marciano Siniscalchi November 10, 1999 Introduction [By and large, I will follow OR, Chapters 11 and 12, so I will keep these notes to a minimum.] Games with observed actions and payoff uncertainty Not all dynamic models of strategic interaction fit within the category of games with observed actions we have developed in the previous lectures. In particular, no allowance was made for payoff uncertainty. On the other hand, it is very easy to augment the basic model to allow for it. I am going to follow OR very closely here: Definition 1 An extensive-form game with observed actions and incomplete information (OR: a Bayesian extensive game with observed actions) is a tuple Γ = (N, A, H, P, Z,(Θi , pi , Ui)i∈N ) where: N, is a set of players; A is a set of actions; H is a collection of finite and countable sequences of vectors of elements of A, such that: (i) ∅ ∈ H; (ii) (a 1 , . . . , ak ) ∈ H implies (a 1 , . . . , a` ) ∈ H for all ` < k; (iii) If h = (a 1 , . . . , ak , . . .) and (a 1 , . . . , ak ) ∈ H for all k ≥ 1, then h ∈ H. Z is the set of terminal histories: that is, (a 1 , . . . , ak ) ∈ Z iff (a 1 , . . . , ak ) ∈ H and (a 1 , . . . , ak , a) 6∈ H for all a ∈ A. Also let X = H \ Z. All infinite histories are terminal. P : X ⇒ N is the player correspondence. Θi is a set of payoff types; write Θ = Q i∈N Θi . pi is a probability distribution over Θi ; for i 6= j, pi and pj are independent. U = (Ui)i∈N : Z × Q i∈N Θi → R is the payoff function, associating a vector of payoffs to every terminal history and every profile of payoff types. 1