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Common Prior Type Spaces In Which Payoff Types and Belief Types Are Independent

Common prior type spaces in which for each agent the agent’s payoff type and the agent’s belief type are independent deserve attention as the polar opposites of common prior type spaces in which agents’ beliefs determine their preferences - a class of type spaces whose special properties are much studied. We find a necessary and sufficient condition for the independence of each agent’s payoff type and belief type. Different agents’ payoff types must be independent. Agents may hold payoff irrelevant information. The payoff irrelevant signals that agents receive may be correlated with each other, but they must be jointly independent of all agents’ payoff types. We conclude that type spaces with independent payoff types, as commonly used in game theory and mechanism design, constitute, up to payoff irrelevant information, the class of all type spaces in which payoff types and belief types are independent for each agent.