How incomplete probability information can steer decisions

How much does a recommendation tell you if you do not know how it relates to the recommendations everyone else receives? In a November 10, 2026 revision of an arXiv preprint, Francesco Giordano of HEC Paris studies how an information provider can exploit precisely that uncertainty. The work presents mathematical results, not experiments showing how people actually behave.

Its central distinction is between the process that really generates messages and the process recipients believe generates them. Disclosed facts must be accurate, but need not identify the true process. Under the paper’s assumptions, that leaves room to steer decisions without announcing false statistics.

Filling in the missing probabilities

The model starts with a finite game: several players choose from limited sets of actions, and their payoffs depend on everyone’s choices. A mediator chooses private messages, their joint probability distribution, and which statistics about that distribution to disclose publicly.

Players then select one probability distribution consistent with those statistics. They update it after receiving their own message and choose an action that maximizes their expected payoff.

The leading belief-selection rule is maximum Shannon entropy: choose the distribution with the greatest entropy among those matching the disclosed information. If only each player’s separate message frequencies are known, this rule selects an independent distribution, potentially missing real correlations. If only the possible message combinations are known, it assigns them equal probabilities.

Giordano also studies a broader class of rules. They produce a unique belief, treat message combinations symmetrically under relabeling, and never assign zero probability to an event unless the disclosed facts require it. This last property supplies a crucial constraint: something that really happens cannot be considered impossible.

A boundary beyond correlated equilibrium

The benchmark is a correlated equilibrium. Imagine a mediator sending action recommendations through a fully understood random process. The recommendations form a correlated equilibrium when nobody expects to gain by deviating after seeing their own recommendation.

With partial disclosure, actual outcomes can fall outside that benchmark. But they cannot be arbitrary. Every action combination that occurs must receive positive probability in at least one correlated equilibrium of the game. The paper calls distributions satisfying this support restriction “jointly coherent.” Their probabilities need not themselves satisfy the correlated-equilibrium incentives.

For games whose payoffs are rational numbers, the paper characterizes exactly which outcomes can be implemented: all and only the jointly coherent ones. For general real-valued payoffs, it establishes necessity and a weaker sufficiency result allowing arbitrarily small incentives to deviate. That approximation concerns players’ incentives, not merely closeness to a target distribution.

The constructive proof expands the message space with multiple messages that lead to the same action. Carefully choosing which message combinations are ruled out makes recipients’ selected beliefs support optimal behavior, while the true process generates the desired outcome. Disclosure need only identify a set of combinations that never occurs.

For rational-payoff games, the same boundary survives an extension in which incomplete knowledge concerns messages and actions together, including when the strategy profile is not disclosed.

Why the message vocabulary matters

The broad characterization relies on freedom to create messages. Splitting one message into several action-equivalent versions can change the selected belief: equal treatment of message combinations does not necessarily mean equal treatment of actions.

When messages are restricted to direct action recommendations, that flexibility disappears. Implementability then depends on the particular belief-selection rule. The paper gives an exact condition: some correlated-equilibrium belief must cover every outcome that actually occurs and minimize the loss along the line through that belief and the target distribution, within the probability simplex. Merely belonging to the jointly coherent set is no longer enough.

A judge who always convicts

A single-agent example makes the mechanism concrete. In the paper’s stylized decision problem, a judge knows that guilt has prior probability one-third. Correct decisions have equal rewards, so at a belief of one-half guilt, conviction and acquittal are equally good.

The true process always sends the first of two messages. The disclosure reveals the prior and that guilty defendants never receive the second message, but leaves innocent defendants’ message frequencies unspecified.

The selected belief divides innocent cases equally between the two messages. Under that belief, the first message implies a one-half chance of guilt. Choosing conviction when indifferent therefore produces conviction every time, although the actual guilt frequency remains one-third.

More generally, with rational payoffs and a rational full-support prior, the designer can induce any action that is optimal under some belief assigning positive probability to every state.

These are conditional theoretical possibilities, not findings about courts or platforms. The model assumes specific belief rules and extensive control over messages and disclosure. Learning from repeated observations remains an open extension. Its lesson is narrower: truthful partial statistics need not make the beliefs guiding decisions accurate.