Questions: Bayesian Games (Games of Incomplete Information)

Question 1

In a Bayesian game, what is the distinction between a player's 'type' and their 'strategy'?

• A. A type is the action a player chooses in the game; a strategy is their long-run plan across multiple games.
• B. A type is private information the player holds (such as their valuation or cost); a strategy is a function mapping each possible type to an action.
• C. A type is a player's role or identity; a strategy is their belief about what other players will do.
• D. Types and strategies are interchangeable — both describe what a player will do given their information.

Answer: (B)

Type captures what a player privately knows (e.g., how much they value an item, their production cost, their risk tolerance). Strategy in a Bayesian game is a complete contingent plan: it specifies what action to take for every type the player might be. This is more complex than in complete-information games, where a strategy is just an action choice.

Question 2

In a Bayesian Nash equilibrium, each player's optimal strategy requires knowing the actual types of most other players.

• T. True
• F. False

Answer: False

Players do NOT observe other players' types — that is the whole point of incomplete information. Instead, each player maximizes their *expected* payoff, averaging over the possible types of others weighted by their prior beliefs. The equilibrium requires that each type's strategy be optimal given those beliefs, not given knowledge of actual types.

Question 3

Describe a real-world situation that can be modeled as a Bayesian game and identify what 'types' represent in that context.

Model answer: A first-price sealed-bid auction is a canonical example. Each bidder is a player, and their type is their private valuation for the item being auctioned. Bidders choose how much to bid (their strategy) without knowing others' valuations. A Bayesian Nash equilibrium describes a bidding function — a rule mapping each valuation to a bid — such that no bidder can improve their expected payoff by deviating.

The auction example clearly illustrates the key features: private information (only you know your value), beliefs (you have a distribution over others' values), and strategies that are type-contingent (your bid depends on your value). Other examples include entry games with private costs, insurance markets with private risk levels, and bilateral trade with privately known gains from trade.