Anonymous sequential games: Existence and characterization of equilibria

Summary In this paper we consider Anonymous Sequential Games with Aggregate Uncertainty. We prove existence of equilibrium when there is a general state space representing aggregate uncertainty. When the economy is stationary and the underlying process governing aggregate uncertainty Markov, we provide Markov representations of the equilibria.Table of notation Agents' characteristics space ( ) - A Action space of each agent (aA) - Y Y = x A - Aggregate distribution on agents' characteristics - (X) Space of probability measures onX - C(X) Space of continuous functions onX - _X Family of Borel sets ofX - State space of aggregate uncertainty ( ) - x_t=1 aggregate uncertainty for the infinite game - = (₁,₂,...,_t,...) - ^{t t} (₁, ₂,..., _t) - L₁(^t,C ×A),v_t Normed space of measurable functions from^t toC( x A) - 8o(^t,( x A)) Space of measurable functions from^tto( x A) - X^t X^t= x_s=1^tX - _X^t Borel field onX^t - v Distribution on - v_t Marginal distribution of v on^t - v(^t)((¦^t)) Conditional distribution ongiven^t - v^t(^s)(v_t(¦^s)) Conditional distribution on^tgiven^s(wheres) - _t Periodt distributional strategy - Distributional strategy for all periods =(₁,₂,...,_t,...) - _t Transition process for agents' types - (_t,^t,y)(P_t+1(, _t,^t,y)) Transition function associated with_t - u_t Utility function - V_t(, a, , ^t) Value function for each collection (, a, , ^t) - W_t(, , ^t) Value function given optimal action a - C() Consistency correspondence. Distributions consistent with and characteristics transition functions - B() Best response correspondence (which also satisfy consistency) - E Set of equilibrium distributional strategies - x_t=1(^t, (x A)) - S Expanded state space for Markov construction - (, a, ) Value function for Markov construction - P(_t^*, _ty)(P(, _t^*, _t,_y)) Invariant characteristics transition function for Markov gameWe wish to acknowledge very helpful conversations with C. d'Aspremont, B. Lipman, A. McLennan and J-F. Mertens. The financial support of the SSHRCC and the ARC at Queen's University is gratefully acknowledged. This paper was begun while the first author visited CORE. The financial support of CORE and the excellent research environment is gratefully acknowledged. The usual disclaimer applies.