Discounted Stochastic Games with Voluntary Transfers

This paper studies discounted stochastic games perfect or imperfect public monitoring and the opportunity to conduct voluntary monetary transfers. We show that for all discount factors every public perfect equilibrium payoff can be implemented with a simple class of equilibria that have a stationary structure on the equilibrium path and optimal penal codes with a stick and carrot structure. We develop algorithms that exactly compute or approximate the set of equilibrium payoffs and find simple equilibria that implement these payoffs.Stochastic games, Monetary transfers, Computation, Imperfect public monitoring, Public perfect equilibria

Exact Algorithms for Solving Stochastic Games

Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games

Corrigendum to “Discounted stochastic games with no stationary Nash equilibrium: two examples”

Levy (2013) presented examples of discounted stochastic games that do not have stationary equilibria. The second named author has pointed out that one of these examples is incorrect. In addition to describing the details of this error, this note presents a new example by the first named author that succeeds in demonstrating that discounted stochastic games with absolutely continuous transitions can fail to have stationary equilibria

On K-Class discounted stochastic games

For a discounted stochastic game with an uncountable state space and compact metric action spaces, we show that if the measurable-selection-valued, Nash payoff selection correspondence of the underlying one-shot game contains a sub-correspondence having the K- limit property (i.e., if the Nash payoff selection sub-correspondence contains its K-limits and therefore is a K correspondence), then the discounted stochastic game has a stationary Markov equilibrium. Our key result is a new fixed point theorem for measurable-selection-valued correspondences having the K-limit property. We also show that if the discounted stochastic game is noisy (Duggan, 2012), or if the underlying probability space satisfies the G-nonatomic condition of Rokhlin (1949) and Dynkin and Evstigneev (1976) (and therefore satisfies the coaser transition kernel condition of He and Sun, 2014), then the Nash payoff selection correspondence contains a sub-correspondence having the K-limit property

Discounted Stochastic Games with Voluntary Transfers

This paper studies discounted stochastic games perfect or imperfect public monitoring and the opportunity to conduct voluntary monetary transfers. We show that for all discount factors every public perfect equilibrium payoff can be implemented with a simple class of equilibria that have a stationary structure on the equilibrium path and optimal penal codes with a stick and carrot structure. We develop algorithms that exactly compute or approximate the set of equilibrium payoffs and find simple equilibria that implement these payoffs

Stationary Markov equilibria for approximable discounted stochastic games

We identify a new class of uncountable-compact discounted stochastic games for which existence of stationary Markov equilibria can be established and we prove two new existence results for this class. Our approach to proving existence in both cases is new – with both proofs being based upon continuous approximation methods. For our first result we use approximation methods involving measurable-selection-valued continuous functions to establish a new fixed point result for Nash payoff selection correspondences - and more generally for measurable-selection-valued correspondences having nonconvex values. For our second result, we again use approximation methods, but this time involving player action-profile-valued continuous functions to establish a new measurable selection result for upper Caratheodory Nash payoff correspondences. Because conditions which guarantee approximability - the presence of sub-correspondences taking contractible values (or more generally, Rd-values) - are the very conditions which rule out Nash equilibria homeomorphic to the unit circle, we conjecture that for uncountable-compact discounted stochastic games, the approximable class is the widest class for which existence of stationary Markov equilibria can be established

Stationary Markov equilibria for K-class discounted stochastic games

For a discounted stochastic game with an uncountable state space and compact metric action spaces, we show that if the measurable-selection-valued, Nash payoff selection correspondence of the underlying one-shot game contains a sub-correspondence having the K-limit property (i.e., if the Nash payoff selection sub-correspondence contains its K-limits and therefore is a K correspondence), then the discounted stochastic game has a stationary Markov equilibrium. Our key result is a new fixed point theorem for measurable-selection-valued correspondences having the K-limit property. We also show that if the discounted stochastic game is noisy (Duggan, 2012), or if the underlying probability space satisfies the G-nonatomic condition of Rokhlin (1949) and Dynkin and Evstigneev (1976) (and therefore satisfies the coaser transition kernel condition of He and Sun, 2014), then the Nash payoff selection correspondence contains a sub-correspondence having the K-limit property