Harsanyi Transformation

Transforms games of incomplete information into games of imperfect information via a type space.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

An incomplete-information game is reformulated as a Bayesian game $\langle N,\{\Theta_i\},p,\{A_i\},\{u_i\}\rangle$ with Nature drawing types $\theta\sim p$ and revealing $\theta_i$ to player $i$ .

Under the common-prior assumption, all players share $p$ ; heterogeneous priors yield generalized Bayesian games.

Intuition

Turns "I don't know what you know" into "I know the distribution of what you might know".

Enables equilibrium analysis of asymmetric-information games.

Worked example

Auctions with privately known valuations are modeled as Bayesian games via the Harsanyi transformation.

Signaling games and reputation models routinely apply it.

The math

Harsanyi (1967-68) Econometrica trilogy; the common-prior assumption is often called the Harsanyi doctrine.

Mertens-Zamir constructed the universal type space ensuring a canonical model exists.

Where it is used

Auction theory, insurance, finance with asymmetric information, and mechanism design.

Essential conceptual tool in modern IO and information economics.

More in Game theory

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