The Fog — The Game, Plate vii

The opponent’s move is difficult enough to foresee when his purposes are known. The deeper uncertainty begins when the same move could serve different purposes: a seller may hold out because the object is valuable to him, because he thinks it valuable to us, or because he expects another buyer tomorrow. Each explanation describes an intelligible calculation, yet each calls for a different reply. The earlier games often supplied the players with a common view of the payoffs and left them to choose their strategies. In the fog, part of that view is itself concealed. We must reason about what another person will do while admitting that we cannot see everything he has reason to want. The discipline is to make that ignorance explicit enough to enter the calculation.

Harsanyi gave the hidden differences a mathematical address by describing players through their types. A type can contain a cost, a valuation, a piece of information, or the beliefs associated with that information; it need not be a personality or an enduring disposition. Each player knows his own type, while his uncertainty about the others is represented by probabilities. In the common-prior formulation, those probabilities arise by conditioning a shared distribution on the information each possesses. The device is often pictured as nature drawing the types before play, but the draw can be a representation rather than a literal event. A strategy now says what to do for each possible type one might be. The opponent remains incompletely known, while the model supplies an agreed account of the possibilities and their relation to what each player knows.¹

1. John C. Harsanyi, Games with Incomplete Information Played by “Bayesian” Players, Part I (1967), §§1–7, pp. 163–182. Types include information and payoff-relevant attributes; §§4–5 state the consistency condition behind a common basic distribution. Section 6 gives alternative interpretations of the hypothetical random mechanism. Incomplete information about the game differs from imperfect observation of earlier moves.

An invented entry game makes the distinction concrete. A newcomer earns two if an incumbent accommodates entry, loses three if it fights, and earns zero by staying out. The incumbent privately knows which of two types it is. After entry, the tough type earns two from fighting and one from accommodating; the ordinary type earns minus two from fighting and one from accommodating. Either type earns four if entry never occurs. Fighting therefore serves one type’s interests, accommodation the other’s. If the newcomer assigns probability one quarter to the tough type, entry yields three quarters of a unit in expectation: three quarters times two, less one quarter times three. At probability one half, the same calculation yields minus one half, and staying out is preferable. In general, entry pays when the probability of toughness is below two fifths. The relevant prediction joins the newcomer’s belief to conduct that each incumbent type actually has reason to choose.²

2. The game and all payoffs are invented; each player maximizes expected payoff, and the incumbent observes entry before responding. The entrant’s return is 2 − 5p. Harsanyi, Part II: Bayesian Equilibrium Points (1968), §8, pp. 320–322, defines equilibrium through conditional best replies by types, subject to a probability-zero qualification. The example explicitly supplies optimal responses after entry; it is not a claim of a unique equilibrium of every representation.

⁂

Evidence can change the decision without ever revealing the incumbent’s type with certainty. Keep the invented prior at one quarter, and suppose an external inspection produces a warning with probability four fifths when the incumbent is tough and one fifth when it is ordinary. These likelihoods are stipulated, and the warning has no effect on either type’s payoffs. Toughness accompanied by a warning has probability one fifth; an ordinary incumbent accompanied by a warning has probability three twentieths. Conditional on seeing the warning, the probability of toughness is therefore four sevenths. Entry’s expected return becomes minus six sevenths, so the newcomer now stays out. A warning is neither proof nor an empty gesture. Its force comes from how much more likely it is under one explanation than the other, together with how plausible those explanations were before it arrived.³

3. This external signal is another invented assumption. Bayes’s rule gives (¼ × ⅘) ÷ [(¼ × ⅘) + (¾ × ⅕)] = 4/7, and 2 − 5(4/7) = −6/7. Harsanyi, Part I, §§4–6, pp. 172–180, develops the relation between a joint distribution and conditional beliefs. The signal is not an empirical estimate of an incumbent’s resolve.

A declaration chosen by the incumbent requires a further calculation, because its likelihood comes from a strategy. Both types in the example would prefer to keep the newcomer out, since four exceeds either type’s return after entry. If both announce that they will fight with probability one, hearing the announcement leaves the probability of toughness at one quarter. The words cannot separate types that use them identically. This establishes the information content of the stipulated announcement pattern; a complete account of the message game would also have to specify responses to other messages and check deviations. Where an action is more attractive to one type than another, it may carry information, but its expense alone supplies no certificate of honesty. The question is whether the proposed pattern of actions, the beliefs drawn from it and the replies to those beliefs can sustain one another.⁴

4. The pooling announcement calculation is invented: equal likelihoods leave the prior unchanged. The discussion applies the type-dependent strategy and conditional-best-reply framework in Harsanyi, Part II, §8. It does not assert that every costly signal separates types, or supply a complete equilibrium selection for a signaling game.

The resulting equilibrium is a discipline of consistency whose conclusions remain answerable to its premises. It does not tell us where to find the right prior, whether the listed types exhaust the relevant possibilities, or which equilibrium actual players will reach when several satisfy the conditions. Harsanyi’s common-distribution construction itself requires a compatibility among beliefs; subjective probabilities need not arrive already fitted to it. Even an internally consistent model can omit a motive that matters. The newcomer who treats toughness as a concern for market share will misread an incumbent whose overriding purpose is to preserve a reputation elsewhere. That observation does not make probabilities useless. It makes comparison among plausible models part of their use: which uncertainty changes the decision, which evidence would resolve it, and how far the recommendation survives a different account of the opponent.⁵

5. Harsanyi, Part I, §§4–7, distinguishes a player’s assessment, the consistency requirement and alternative representations. Part II, §8, gives an equilibrium existence result under its conditions, not general uniqueness. The comparison of models in the paragraph is the plate’s inference about using conditional predictions.

We play against unseen aims by giving the possible aims enough definition that their consequences can be compared. A threat then has to be read beside the interests it might serve; a concession beside what each type would sacrifice by making it; an apparent misunderstanding beside the information that could make it reasonable. None of this grants access to another mind. It supplies a way of distinguishing a belief supported by conduct from a story merely repeated about that conduct. The fog need not lift entirely for a decision to become better founded. It is enough to know which hidden difference would change the best reply, what evidence bears on that difference, and what remains uncertain after the evidence is heard. Strategic judgment begins to mature when it can state those limits without ceasing to act.