Experiment 13 Common Knowledge
Does Everyone Know That Everyone Knows?
Send private messages and acknowledgments, then compare them with an ideal public announcement.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Make a Prediction
“I Got Your Message. Did You Get Mine?”
Two teams can start a shared project. The fact they need is “the plan is ready.” Team A knows that fact. Team B has not heard it. Starting together earns each team 4 points. Starting alone loses 6 points. Waiting earns 0. You control Team A’s final choice; Team B follows the knowledge rule below.
Send a private message. If it arrives, the recipient can send an acknowledgment, a reply confirming receipt. Every message, including a reply, can get lost. The sender cannot directly see delivery. The log is visible to you as an outside observer, not to both teams.
2. Make Your Choice
3. Reveal the Incentives
A Fact Can Have Layers
At level 1, a team knows the plan is ready. At level 2, it knows that both teams know the plan is ready. At level 3, it knows that both teams have level 2 knowledge. Each next level adds another “everyone knows that” layer.
| Knowledge level | Team A | Team B | Both teams |
|---|
The Newest Private Knowledge Chain
Common knowledge means the fact survives every number of “everyone knows that” layers. Seeing “Yes” in all six displayed rows does not establish it. A seventh layer, an eighth layer, and every later layer matter too.
4. Change One Assumption
How Much Certainty Does the Rule Require?
Changing a setting restarts the exchange. A seed is a number that makes the same random delivery sequence repeat. Each run permits 12 successful private messages. A failed message ends the reply chain. Private delivery always has some chance of failure; the public announcement is a separate, idealized channel.
The public assumption is strong: both teams hear the same announcement, know that both hear it, and know this at every further level. A web page, email, read receipt, or meeting does not automatically meet that assumption in real life.
Work It Out, Step by Step
- Before messages: A knows the plan. B does not. Only A has level 1 knowledge.
- After the first delivery: B knows the plan and knows A knows it, because A sent it. But A cannot rule out a lost message. Both have level 1, while only B has level 2.
- After a reply reaches A: A now knows B received the plan. Both have level 2. B cannot tell whether its reply arrived, so only A has level 3.
- Another reply adds another layer. It never removes uncertainty about the latest private delivery. However long a finite chain becomes, a further layer is still missing.
- Separate knowledge from the decision. Under the level 1 rule, a delivered first message makes B act. You can choose to act too and earn 4 each, for \(4+4=8\) combined points. That successful coordination does not turn the private exchange into common knowledge.
A rule tells the simulated team when to act; it is not a proof that acting is the best possible choice. Real teams may accept risk, use deadlines, or rely on repeated practice. This lab studies what they know, not the best way to handle every uncertain project.
For experts: formal model and assumptions
Reading the Knowledge Symbols
- \(F\), \(K_A F\), and \(K_B F\)In words: “the fact,” “A knows the fact,” and “B knows the fact.”
- F stands for “the plan is ready.” K is a knowledge operator: an instruction to ask what a team can be certain of. The lower A or B identifies the team. These are statements that can be true or false, not numbers to multiply.
- \(E F\) and \(E^2 F\)In words: “everyone knows the fact” and “everyone knows that everyone knows the fact.”
- E means everyone. The raised 2 repeats the knowledge instruction twice; it does not square a number. Here “everyone” includes only A and B. See preferences and knowledge.
- \(\land\), \(=\), and parenthesesIn words: “and,” “is defined here as,” and “the statement inside.”
- The pointed symbol \(\land\) requires both statements to be true. In the definitions below, the equals sign names an equivalent statement. Parentheses group a statement so that the outside knowledge instruction applies to all of it.
- \(C F\)In words: “the fact is common knowledge.”
- C says that E F, E squared F, and every later finite repetition hold. A check of only the first six levels is insufficient.
In words: “Everyone knows the fact when A knows it and B knows it. Two levels mean everyone knows that the first everyone-knows statement is true.” A team has the displayed level n when it knows that the preceding everyone-knows level holds; level 1 starts with the fact itself.
A Finite Model of Possible Situations
A possible world here is simply a complete situation consistent with what a team has observed. Worlds include a not-ready situation and ready situations with different lengths of successful alternating message delivery. The actual fact is ready. A distinguishes not-ready from ready and records replies received from B. B records messages received from A. Teams remember their observations and reason without mistakes. They do not see the outside observer’s log.
With no reply, A cannot distinguish zero deliveries from one. After one incoming message, B cannot distinguish one delivery from two. After one incoming reply, A cannot distinguish two deliveries from three. The same pattern continues. A knows a statement only if it is true in every world A cannot distinguish; the rule is identical for B.
The computation includes the not-ready world and possible chains of successful deliveries from zero through the current successful count plus eight. It computes the displayed six levels by repeatedly checking both teams’ indistinguishable worlds. For common knowledge, it follows every uncertainty link of either team. Every finite private delivery chain remains connected to B’s original uncertainty about readiness, so common knowledge is false. The extra worlds avoid making the displayed boundary look like certainty; the finite model illustrates the chain rather than proving a general theorem about every communication system.
An ideal public announcement restricts both teams to ready worlds, with that restriction itself publicly known. All remaining worlds satisfy F, so common knowledge holds. Changing the private delivery probability changes sampled outcomes, not which deliveries are logically possible. The level-based action policy is assumed and is not derived as a Bayesian equilibrium. A Bayesian equilibrium would additionally check whether each team’s action maximizes its average payoff given its beliefs about uncertainty.
Concept references: Halpern and Moses on knowledge and unreliable communication and Cornell’s lecture on coordination and common knowledge.
5. Take It Somewhere Else
Is a Read Receipt Enough?
A release manager emails two teams that a change is ready. Both read it. Does that guarantee each knows the other has read it, knows the other knows that, and will act at the agreed time?
Reveal the reasoning
No. Those are different claims. A read receipt adds information but may itself be delayed or missed. Teams often solve the practical problem with a shared system, an explicit action rule, a deadline, and a safe fallback. That can support useful coordination without pretending that a finite exchange creates unlimited certainty.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.