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.
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Abbreviations
- Λ :
-
Agents' characteristics space (α∈ Λ)
- A:
-
Action space of each agent (a∈A)
- Y :
-
Y = Λ x A
- μ :
-
Aggregate distribution on agents' characteristics
- ℳ(X) :
-
Space of probability measures onX
- C(X) :
-
Space of continuous functions onX
- ℬ X :
-
Family of Borel sets ofX
- Θ :
-
State space of aggregate uncertainty (θ∈ Θ)
- Θ ∞ :
-
Θ∞ ≡ x ∞t=1 Θ aggregate uncertainty for the infinite game
- θ ∞ :
-
θ∞ = (θ1,θ2,...,θt,...)∈Θ∞
- θ t :
-
θt ≡ (θ1, θ2,..., θt)
- L1(Θt,CΛ ×A),v t :
-
Normed space of measurable functions fromΘ t toC(Λ x A)
- 8o(Θt,ℳ(Λ x A)):
-
Space of measurable functions fromΘ ttoℳ(Λ x A)
- Xt :
-
Xt= x ts=1 X
- ℬ t X :
-
Borel field onX t
- v :
-
Distribution onΘ ∞
- vt :
-
Marginal distribution of v onΘ t
- v(θt)(υ(•¦θt)):
-
Conditional distribution onΘ ∞givenθ t
- vt(θs)(vt(•¦θs)):
-
Conditional distribution onΘ tgivenΘ s(wheres<t)
- τ t :
-
“Periodt” distributional strategy
- τ :
-
Distributional strategy for all periodsτ =(τ1,τ2,...,τt,...)
- ξ t :
-
Transition process for agents' types
- ℱ(τ t,θt,y)(P t+1(•, τ t ,θt,y)):
-
Transition function associated withξ t
- u t :
-
Utility function
- V t (α, a, τ, θt):
-
Value function for each collection (α, a, τ, θ t)
- W t (α, τ, θt):
-
Value function given optimal action a
- C(τ):
-
Consistency correspondence. Distributions consistent withτ and characteristics transition functions
- B(τ):
-
Best response correspondence (which also satisfy consistency)
- Eμ:
-
Set of equilibrium distributional strategies
- ℳ ∞ :
-
x ∞t=1 ℱ(Θ t, ℳ(Λx A))
- S:
-
Expanded state space for Markov construction
- υ(α, a, θ):
-
Value function for Markov construction
- P(τ * t , θ t y)(P(•, τ * t , θ t , y )):
-
Invariant characteristics transition function for Markov game
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We 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.
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Bergin, J., Bernhardt, D. Anonymous sequential games: Existence and characterization of equilibria. Econ Theory 5, 461–489 (1995). https://doi.org/10.1007/BF01212329
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DOI: https://doi.org/10.1007/BF01212329