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You are pricing against a competitor whose costs you cannot see. They might be a lean operation that can undercut you, or a bloated one that cannot. You have to set your output now, against a rival who could be either.

Bayesian games and Bayes-Nash

Real strategic settings are full of things you do not know: a rival’s cost, a bidder’s value, a partner’s preferences. Harsanyi’s move was to treat that uncertainty as a draw of the other player’s type from a known distribution, so you can best-respond to the odds even when you cannot see the outcome.

From earlier lessons

Two quick ones before we start.

In Cournot competition, a firm's best-response output falls when the rival's output...

A rational agent facing uncertainty maximizes...

Predict, then play

Demand is $P = 12 - Q$ . Your cost is zero. Your rival is low-cost (cost 0) with some probability, otherwise high-cost (cost 6), and you cannot tell which. Set your output; the two rival types sit at their equilibrium responses. Then move the prior.

Why did that happen?

Take a guess, then reveal the reason

You cannot see the rival’s cost, so you cannot tailor your output to it. What you can do is form the expected rival output, averaging over the types, and best-respond to that. A low-cost rival produces more, dragging the price down, so the more likely they are low-cost, the more crowded the market you expect, and the less you should produce. You are best-responding to a distribution, not to a realized type.

The math

Definition, Bayesian game and Bayes-Nash equilibrium

A Bayesian game adds a set of types for each player, a prior distribution over them, and payoffs that depend on the type profile. A strategy specifies an action for each of your own types. A Bayes-Nash equilibrium is a profile in which every type of every player is best-responding to the distribution of the others’ types and their strategies.

In this Cournot example, profit is linear in the rival’s quantity once your own quantity is fixed. That is why firm 1 can respond to expected rival output. In a nonlinear game, replacing an uncertain action by its mean need not give the same expected payoff.

Firm 2 of type with cost $c$ best-responds to $q_1$ : maximizing $(12 - q_1 - q_2 - c)q_2$ gives

q_2(c) = \frac{12 - q_1 - c}{2}.

Firm 1 does not see $c$ , so it best-responds to the expected rival output, taking that number as fixed: $q_1 = (12 - \mathbb{E}[q_2])/2$ . The two conditions have to hold at once. With the rival low-cost ( $c=0$ ) with probability $\mu$ and high-cost ( $c=6$ ) otherwise, the type-by-type reactions give $\mathbb{E}[q_2] = 3 + 3\mu - q_1/2$ , and substituting that into firm 1’s best response solves the system:

q_1 = \frac{12 - (3 + 3\mu - q_1/2)}{2} \quad\Longrightarrow\quad q_1 = 6 - 2\mu.

At $\mu = \tfrac12$ , $q_1 = 5$ , with $q_2(\text{low}) = 3.5$ and $q_2(\text{high}) = 0.5$ . As $\mu \to 1$ , $q_1 \to 4$ : a rival more likely to be low-cost crowds the market, so firm 1 pulls back.

Work it out

Solve the Bayes-Nash equilibrium

Each rival type best-responds: $q_2(c) = (12 - q_1 - c)/2$ . At $\mu = \tfrac12$ , the expected rival output is $\mathbb{E}[q_2] = \tfrac12\cdot\tfrac{12 - q_1}{2} + \tfrac12\cdot\tfrac{6 - q_1}{2} = 4.5 - q_1/2$ .

Firm 1’s first-order condition, treating $\mathbb{E}[q_2]$ as its rival’s output, is $q_1 = (12 - \mathbb{E}[q_2])/2$ . Substituting, $q_1 = (12 - 4.5 + q_1/2)/2 = 3.75 + q_1/4$ , so $q_1 = 5$ . Then $q_2(\text{low}) = (12 - 5)/2 = 3.5$ and $q_2(\text{high}) = (12 - 5 - 6)/2 = 0.5$ . Firm 1 never learns the cost; it just plays against the average.

Key idea

Under incomplete information you best-respond to the distribution of the other player’s types, not to any single one. Each of your own types plays its own best response, and all of them together form the Bayes-Nash equilibrium.

Check yourself

Check yourselfapply

You cannot see whether your rival is low- or high-cost. What do you best-respond to?

Check yourselfanalyse

As the rival becomes more likely to be low-cost, firm 1's equilibrium output moves from 5 toward 4. Why down?

Check yourselfbrings back: expected utility

In what sense is Bayes-Nash just Nash equilibrium applied with expected utility?

Sources

Harsanyi, J. C. (1967-68). Games with incomplete information played by Bayesian players, parts I-III. Management Science, 14(3, 5, 7).