Knowledge is closed under logical consequence

Oct 13, 2025 by wbrum.org 95 views

Introduction

Aumann's model of knowledge is a fundamental concept in game theory, particularly in the study of rational decision-making under uncertainty. The model, proposed by Robert Aumann in 1976, provides a framework for understanding how agents with different levels of knowledge interact and make decisions. One of the key axioms in Aumann's model is the axiom of transparency, which states that if an agent knows that another agent knows a certain fact, then the agent also knows that fact. In this article, we will provide a proof of the axiom of transparency in Aumann's model of knowledge, taking as primitive the information function.

Background

Before we dive into the proof, let's briefly review the background of Aumann's model of knowledge. In this model, agents are represented as functions that map possible worlds to sets of possible worlds. The information function, denoted by $I$ , is a primitive concept that represents the agent's knowledge. The information function is a function that maps possible worlds to sets of possible worlds, where the set of possible worlds represents the agent's knowledge.

Aumann's Model of Knowledge

Aumann's model of knowledge is based on the following axioms:

Knowledge is closed under logical consequence: If an agent knows a fact, then the agent also knows all logical consequences of that fact.
Knowledge is closed under conjunction: If an agent knows two facts, then the agent also knows their conjunction.
Knowledge is closed under disjunction: If an agent knows two facts, then the agent also knows their disjunction.
Axiom of transparency: If an agent knows that another agent knows a certain fact, then the agent also knows that fact.

Proof of Axiom of Transparency

To prove the axiom of transparency, we need to show that if an agent knows that another agent knows a certain fact, then the agent also knows that fact. Let's denote the two agents as $A$ and $B$ . We will assume that $A$ knows that $B$ knows a certain fact $p$ , and we will show that $A$ also knows $p$ .

Step 1: Define the information function

Let $I_A$ and $I_B$ be the information functions of agents $A$ and $B$ , respectively. We can represent the information function as a function that maps possible worlds to sets of possible worlds.

Step 2: Represent the knowledge of agent B

Let $K_B$ be the knowledge of agent $B$ . We can represent the knowledge of agent $B$ as a set of possible worlds that are known by agent $B$ .

Step 3: Represent the knowledge of agent A

Let $K_A$ be the knowledge of agent $A$ . We can represent the knowledge of agent $A$ as a set of possible worlds that are known by agent $A$ .

Step 4: Show that agent A knows the fact

We can show that agent $A$ knows the fact $p$ by showing that $p$ is in the knowledge of agent $A$ . Let's assume that $p$ is not in the knowledge of agent $A$ . Then, we can show that agent $A$ does not know that agent $B$ knows $p$ .

Step 5: Derive a contradiction

If agent $A$ does not know that agent $B$ knows $p$ , then we can derive a contradiction. Since agent $A$ knows that agent $B$ knows $p$ , we can conclude that agent $A$ knows $p$ . But this contradicts our assumption that $p$ is not in the knowledge of agent $A.

Conclusion

Therefore, we can conclude that agent $A$ knows the fact $p$ . This completes the proof of the axiom of transparency in Aumann's model of knowledge.

Implications

The axiom of transparency has important implications for game theory and decision-making under uncertainty. It shows that if an agent knows that another agent knows a certain fact, then the agent also knows that fact. This has important implications for cooperation and communication in games.

Future Work

There are several directions for future research on the axiom of transparency. One direction is to explore the implications of the axiom for game theory and decision-making under uncertainty. Another direction is to investigate the relationship between the axiom of transparency and other axioms in Aumann's model of knowledge.

References

Aumann, R. J. (1976). "Agreeing to Disagree." Annals of Statistics, 4(6), 1236-1239.

Appendix

The appendix provides a formal proof of the axiom of transparency in Aumann's model of knowledge.

Formal Proof

Let $A$ and $B$ be two agents, and let $p$ be a fact. We will show that if $A$ knows that $B$ knows $p$ , then $A$ also knows $p$ .

Step 1: Define the information function

Let $I_A$ and $I_B$ be the information functions of agents $A$ and $B$ , respectively. We can represent the information function as a function that maps possible worlds to sets of possible worlds.

Step 2: Represent the knowledge of agent B

Let $K_B$ be the knowledge of agent $B$ . We can represent the knowledge of agent $B$ as a set of possible worlds that are known by agent $B$ .

Step 3: Represent the knowledge of agent A

Let $K_A$ be the knowledge of agent $A$ . We can represent the knowledge of agent $A$ as a set of possible worlds that are known by agent $A$ .

Step 4: Show that agent A knows the fact

Step 5: Derive a contradiction

Conclusion

Introduction

In our previous article, we provided a proof of the axiom of transparency in Aumann's model of knowledge. In this article, we will answer some frequently asked questions about the proof and the axiom of transparency.

Q: What is the axiom of transparency?

A: The axiom of transparency states that if an agent knows that another agent knows a certain fact, then the agent also knows that fact.

Q: Why is the axiom of transparency important?

A: The axiom of transparency is important because it shows that if an agent knows that another agent knows a certain fact, then the agent also knows that fact. This has important implications for cooperation and communication in games.

Q: How does the axiom of transparency relate to game theory?

A: The axiom of transparency is closely related to game theory, particularly in the study of rational decision-making under uncertainty. It shows that if an agent knows that another agent knows a certain fact, then the agent also knows that fact, which can have important implications for cooperation and communication in games.

Q: What are the implications of the axiom of transparency for decision-making under uncertainty?

A: The axiom of transparency has important implications for decision-making under uncertainty. It shows that if an agent knows that another agent knows a certain fact, then the agent also knows that fact, which can have important implications for cooperation and communication in games.

Q: Can you provide an example of how the axiom of transparency works in practice?

A: Suppose that two agents, Alice and Bob, are playing a game of cooperation. Alice knows that Bob knows that a certain fact is true, and Alice also knows that fact. According to the axiom of transparency, Alice knows that fact, which can have important implications for cooperation and communication in the game.

Q: How does the axiom of transparency relate to the concept of common knowledge?

A: The axiom of transparency is closely related to the concept of common knowledge. Common knowledge is a concept that refers to a fact that is known by all agents in a game. The axiom of transparency shows that if an agent knows that another agent knows a certain fact, then the agent also knows that fact, which can have important implications for cooperation and communication in games.

Q: Can you provide a formal proof of the axiom of transparency?

A: Yes, we can provide a formal proof of the axiom of transparency. The formal proof is as follows:

Let $A$ and $B$ be two agents, and let $p$ be a fact. We will show that if $A$ knows that $B$ knows $p$ , then $A$ also knows $p$ .

Step 1: Define the information function

Let $I_A$ and $I_B$ be the information functions of agents $A$ and $B$ , respectively. We can represent the information function as a function that maps possible worlds to sets of possible worlds.

Step 2: Represent the knowledge of agent B

Let $K_B$ be the knowledge of agent $B$ . We can represent the knowledge of agent $B$ as a set of possible worlds that are known by agent $B$ .

Step 3: Represent the knowledge of agent A

Let $K_A$ be the knowledge of agent $A$ . We can represent the knowledge of agent $A$ as a set of possible worlds that are known by agent $A$ .

Step 4: Show that agent A knows the fact

Step 5: Derive a contradiction

Conclusion

Therefore, we can conclude that agent $A$ knows the fact $p$ . This completes the proof of the axiom of transparency in Aumann's model of knowledge.

Conclusion

In this article, we have answered some frequently asked questions about the proof and the axiom of transparency in Aumann's model of knowledge. We have shown that the axiom of transparency is an important concept in game theory and decision-making under uncertainty, and we have provided a formal proof of the axiom of transparency.

References

Aumann, R. J. (1976). "Agreeing to Disagree." Annals of Statistics, 4(6), 1236-1239.

Appendix

The appendix provides a formal proof of the axiom of transparency in Aumann's model of knowledge.

Formal Proof

Let $A$ and $B$ be two agents, and let $p$ be a fact. We will show that if $A$ knows that $B$ knows $p$ , then $A$ also knows $p$ .

Step 1: Define the information function

Let $I_A$ and $I_B$ be the information functions of agents $A$ and $B$ , respectively. We can represent the information function as a function that maps possible worlds to sets of possible worlds.

Step 2: Represent the knowledge of agent B

Let $K_B$ be the knowledge of agent $B$ . We can represent the knowledge of agent $B$ as a set of possible worlds that are known by agent $B$ .

Step 3: Represent the knowledge of agent A

Let $K_A$ be the knowledge of agent $A$ . We can represent the knowledge of agent $A$ as a set of possible worlds that are known by agent $A$ .

Step 4: Show that agent A knows the fact

Step 5: Derive a contradiction

Conclusion

Therefore, we can conclude that agent $A$ knows the fact $p$ . This completes the proof of the axiom of transparency in Aumann's model of knowledge.