A single-file study note on Bayesian games, written in the same style as the dynamic-games page above. The focus is conceptual completeness, readable mathematics, and worked examples that connect directly to the cases discussed earlier: Harsanyi transformation, discrete Bayesian reasoning, first-price auctions, the signaling extension, and why electricity markets are often modeled with Bayesian games.
A Bayesian game is a game in which at least one payoff-relevant feature is not publicly known. The unknown object is encoded as a type. Each player knows their own type, but may not know the types of the others.
In an ordinary Nash game, player i compares actions given what others do. In a Bayesian game, player i compares actions given what others do and what others might privately be.
\[ \text{Ordinary game: } u_i(a_i,a_{-i}) \qquad \text{Bayesian game: } u_i(a_i,a_{-i};t_i,t_{-i}). \]| Concept | Meaning | Why it matters |
|---|---|---|
| Complete information | Everyone knows the game structure and every player's payoff function. | This is the benchmark in ordinary normal-form analysis. |
| Incomplete information | At least one payoff-relevant variable is privately known or unknown to some players. | This is exactly the setting modeled by Bayesian games. |
| Perfect information | Whenever a player moves, they observe the entire past history of play. | This is about observed actions, not private types. |
| Imperfect information | At some decision points, a player does not observe all earlier actions. | A game can be Bayesian and still be imperfect-information or perfect-information depending on the timing. |
A type can represent any privately known payoff determinant: a buyer's valuation, a firm's marginal cost, a worker's productivity, an insurer's risk class, or a generator's true operating cost.
Players do not know the true type profile, so they attach probabilities to possible types and choose actions that maximize expected payoff under those beliefs.
A standard finite Bayesian game can be written as
\[ G=\bigl(N,(A_i)_{i\in N},(T_i)_{i\in N},p,(u_i)_{i\in N}\bigr). \]The ingredients are:
A pure Bayesian strategy for player \(i\) is a function
\[ s_i:T_i\to A_i. \]Interpretation: for each possible type, the strategy specifies what action that type takes.
A mixed Bayesian strategy assigns a distribution over actions to each type:
\[ \sigma_i(\cdot\mid t_i)\in \Delta(A_i). \]Interpretation: type \(t_i\) may randomize over actions.
If types can be correlated, player \(i\)'s belief about the other players' types after observing their own type \(t_i\) is obtained by conditioning the common prior:
When types are independent, this simplifies substantially because knowing your own type does not change the distribution of the others' types.
A privately known characteristic, such as cost or valuation.
A probability assessment over the unknown types of the other players.
A full plan that tells each type what action to choose.
Harsanyi's idea is to convert incomplete information into an ordinary extensive-form game by adding a move by Nature at the beginning. Nature draws the type profile, each player observes their own type, and play then continues with those types fixed in the background.
Instead of saying, "player i does not know player j's true cost," we say:
A Bayesian Nash equilibrium (BNE) is a strategy profile in which every type of every player is choosing a best response, given beliefs about other players' types and given the strategy functions of the other players.
Given a pure-strategy profile \(s=(s_1,\dots,s_n)\), player \(i\)'s expected payoff conditional on type \(t_i\) is
\[ U_i(s\mid t_i) = \sum_{t_{-i}\in T_{-i}} \mu_i(t_{-i}\mid t_i) \,u_i\bigl(s_i(t_i),s_{-i}(t_{-i});t_i,t_{-i}\bigr). \]If types are continuous, the sum becomes an integral:
\[ U_i(s\mid t_i) = \int_{T_{-i}} u_i\bigl(s_i(t_i),s_{-i}(t_{-i});t_i,t_{-i}\bigr) \,d\mu_i(t_{-i}\mid t_i). \]A strategy profile \(s^*\) is a Bayesian Nash equilibrium if for every player \(i\), every type \(t_i\), and every feasible alternative action \(a_i\in A_i\),
\[ U_i(s^*\mid t_i) \ge \sum_{t_{-i}\in T_{-i}} \mu_i(t_{-i}\mid t_i) \,u_i\bigl(a_i,s^*_{-i}(t_{-i});t_i,t_{-i}\bigr). \]So each type must be best responding in expected-value terms.
Ordinary Nash equilibrium asks whether an action is a best response to others' actions. BNE asks whether a type-contingent action is a best response to others' type-contingent strategies under beliefs about the unknown type profile.
The extra object is not a new optimization method. It is the player's belief about hidden types. Once the belief is specified, the computation is expected-utility maximization.
This example supports the earlier discussion of Harsanyi transformation and Bayesian reasoning. A seller posts a price and a buyer privately knows whether they are a high-value or low-value type.
To be formally complete, the buyer's strategy is a mapping from type and observed price to a response:
\[ s_B:\{H,L\}\times\{5,9\}\to\{\text{Buy},\text{Not Buy}\}. \]| Buyer type | If P = 5 | If P = 9 | Best response pattern |
|---|---|---|---|
| H with value 10 | Buy payoff = 10 - 5 = 5 | Buy payoff = 10 - 9 = 1 | Buy at both prices |
| L with value 4 | Buy payoff = 4 - 5 = -1 | Buy payoff = 4 - 9 = -5 | Reject at both prices |
The seller knows that only type \(H\) buys. Therefore the sale probability is \(0.6\) under either feasible price, and expected profit is:
This is the most useful continuous-type Bayesian example for developing technique. Each bidder privately knows their valuation and submits a bid simultaneously. The highest bidder wins and pays their own bid.
Fix a bidder with true valuation \(v\). If they bid as if they were type \(z\), they submit bid \(b(z)\). Since the bid function is strictly increasing, they win exactly when every rival valuation is below \(z\). Thus the winning probability is \(F(z)^{n-1}\), so expected profit is
In equilibrium, type \(v\) should choose \(z=v\). Differentiate \(\Pi(z;v)\) with respect to \(z\) and set \(z=v\):
At \(z=v\), the equilibrium condition becomes
\[ -b'(v)F(v)^{n-1} + \bigl(v-b(v)\bigr)(n-1)F(v)^{n-2}f(v)=0. \]Using the integrating factor \(F(v)^{n-1}\), one obtains
\[ \frac{d}{dv}\Bigl(F(v)^{n-1}b(v)\Bigr) = (n-1)vf(v)F(v)^{n-2}. \]Imposing the natural boundary condition \(b(0)=0\), the symmetric equilibrium bid function is
If valuations are uniform on \([0,1]\), then \(F(v)=v\) and \(f(v)=1\). Substituting into the formula yields
Static Bayesian games treat types as hidden and actions as direct responses to those hidden types. Signaling games add an extra strategic layer: actions themselves can reveal, conceal, or distort beliefs about type.
| Object | Static Bayesian game | Signaling game |
|---|---|---|
| Unknown variable | Type is privately known | Type is privately known |
| Timing | Usually simultaneous or one-stage incomplete information | A sender moves first, then a receiver updates beliefs and responds |
| Main equilibrium object | Type-contingent strategies | Type-contingent signals, posterior beliefs, and receiver actions |
| New issue | Expected-payoff maximization under hidden types | Actions change beliefs, and beliefs change later actions |
If the sender sends signal \(m\), the receiver forms a posterior
\[ \mu(t\mid m). \]Now the strategic chain is
\[ \text{signal } m \to \text{belief } \mu(\cdot\mid m) \to \text{receiver action} \to \text{sender payoff}. \]A player's action now has two effects at once: a direct payoff effect and an informational effect. The sender asks not only "what do I gain from this move?" but also "what will the receiver infer from this move?"
Separating equilibrium: different types choose different signals. Pooling equilibrium: different types choose the same signal. The distinction is about whether information is revealed endogenously.
Electricity markets are a natural environment for Bayesian modeling because they combine private information, strategic bids, and market-clearing rules under physical network constraints.
Generators know more than rivals do about marginal costs, outage risk, start-up constraints, fuel positions, ramping capability, or the true flexibility of their assets.
Participants submit offers and quantities while anticipating the bids of competitors and the clearing rule used by the market operator.
Rivals do not observe each other's true marginal costs, start-up costs, or real-time capability with full precision. Modeling everyone as fully informed can erase exactly the strategic uncertainty that matters.
They are most useful when the research question is strategic bidding under private information. They are not the only tool in electricity economics; optimization, stochastic programming, and equilibrium-with-network models remain central for many other questions.
| Concept | What it adds | What to compute |
|---|---|---|
| Bayesian game | Private payoff-relevant information | Type spaces, beliefs, and type-contingent strategies |
| Harsanyi transformation | Turns hidden characteristics into Nature's initial move | The induced type structure and information pattern |
| Bayesian Nash equilibrium | Best-response optimality for every type | Expected payoff conditional on own type |
| Continuous-type auction analysis | Functional strategy rather than a single action | The equilibrium bid function \(b(v)\) |
| Signaling extension | Actions affect beliefs and later responses | Signals, posterior beliefs, and receiver best responses |