Dynamic Psychological Games: Omitted Material

This manuscript collects proofs and other material omitted from Dynamic Psychological Games, mimeo, November 2007 (DPG). To make it more self-contained, key de…nitions and results of the main paper are included. For the convenience of readers of DPG, the numbers of equations and statements coincide with DPG. 1 Extensive forms with observable actions Here we provide a more complete de…nition of …nite extensive forms with observable actions and related concepts used in results and proofs below. Fix a …nite player set N and …nite action sets Ai (i 2 N). Let A = Q i2N Ai. A history of length ` is a …nite sequence of action pro…les h = (a; :::; a) 2 A. History h = (a; :::; a) precedes h = (a; :::; a), written h h, if h is a pre…x of h, i.e. k < ` and (a; :::; a) = (a; :::; a). In this case, we also write h = (h; a; :::; a). The empty sequence (the history with zero length) is denoted h. By convention h precedes every proper history. A …nite extensive form with observable actions is a structure hN;Hi where H fhg [ SL `=1A ` is a …nite set of histories with the following properties:1 h 2 H: 8h 2 H, if h h then h 2 H. 8h 2 H, fa 2 A : (h; a) 2 Hg = Q i2N Ai(h) where Ai(h) = ( ai 2 Ai : 9a i 2 Y j 6=i Aj; (h; (ai; a i)) 2 H ) is the set of possible actions of player i at history h. Note that hH; i is a tree with distinguished root h; the symmetric closure of is denoted by .2 We let Z = fh 2 H : Q i2N Ai(h) = ;g denote the set of terminal (or complete) histories. We can now de…ne the following derived elements: Si = fsi = (si;h)h2H 2 (Ai) : 8h 2 HnZ; si;h 2 Ai(h)g is the set of strategies of player i, S = Qn i=1 Si, S i = Q j2Nnfig Sj. : S ! Z is the path function, that is, z = (a; :::; a) = (s) i¤ a = (si;h0)i2N ,8t 2 f1; :::K 1g, a = (si;(a1;:::;at))i2N . For any h 2 H, S(h) is the set of strategy pro…les consistent with h, i.e., S(h) = fs 2 S : h (s)g. Since past actions are observed, it follows that S(h) = Qn i=1 Si(h), where Si(h) is the projection of S(h) on Si. 1Cf. Osborne & Rubinstein (1994, Chapter 6). 2Thus, h h0 i¤ either h h0 or h = h0.