Context of the three number problem

Relevance for multi-agent technology and real world situations

The three number problem is a good example of a situation in which multi-agent principles can be applied. A very general feature of this problem is that it's about a situation in which there is knowledge distributed amongst several agents. Only by combining their own knowledge with the knowledge of others, are the agents able to reacht the right conclusion: what numbers do the three agents have pasted on their face.

The challenge that this multi-agent problem poses is not in the field of knowledge transfer (making it a communication protocol problem), but rather in the field of logical deduction by multiple agents. It seems that part of the problem is also that the messages must be stated as general as possible, without stating specific knowledge of the domain. E.g. the agents only state whether they know the answer to the problem or not and they do not state the knowledge that they already know (the numbers of the other agents).

This method for stating information seems relevant for situations in which a group of agents wants to know more information without stating explicitly what they already know. An example might be gossiping about other persons (you want to share spicy information without being the first one that leaked information). But also in situations in which very private information is exchanged, the method for solving the three number problem might be relevant. For instance, when the AIVD tries to discuss who might be a criminal out of a set of arrested persons, without stating too explicit who they think might be the criminal. In this case the problem would not be "who is a sum-agent and who is a term-agent" but rather be "who is a criminal agent and who is no criminal agent".

However, implementing the three number problem in these contexts directly would be rather tricky. This is caused by the fact that in the three number problem there is a clear relationship between the numerical values of the three agents: the values of two of the (term) agents add up to the value of the third (sum) agent. In the case of deciding who is a criminal, there isn't (usually) a clear relationship between several agents. As a consequence, there are no explicit and clear domain constraints that can be applied in solving the problem.

A more complex three number problem

An important feature of the three number problem is that the domain space is constrained: only the numbers one to ten are used, and all numbers that can be used can be ordered from smallest to largest number (from 0 to 10). In a larger domain, the number of possible solutions to problems might increase, and it would be also more difficult to exclude certain combinations. For instance, if the domain ranged from 0 to 15, agent A couldn't conclude "I know the numbers" in the situation that's described in our solution section.

So extending the domain would be a challenge for the current model. An example of such an extention is shown in figure 1. The figure shows the common knowledge of the agents after all three have stated once that they don't know the numbers as was the case in the normal problem. In this extended version, though, the numbers are limited at 15 instead of 10. Note that there are a lot of worlds in which two agents consider other worlds as possible, but that in all cases there is at least one agent that does know what the solution is, for instance agent A in situation 4.

Figure 1: Possible worlds when the x,y,z <= 15.Click to enlarge.

Comparison with human reasoning

The current implementation differs somehow from the way in which humans solve the three numbers problem. A human would first of all perceive the numbers of the other two agents, and then decide what number he would be if he was the term-agent and what number he would be if he was the sum-agent. On the basis of these numbers, he would reason about which situations the other agents might think of as possible, and which situations they might not think of.

For example: if the numbers were <4,2,6>, agent A would perceive numbers 2 and 6, and would consider the situation <4,2,6> (term-agent) and <8,2,6> (sum-agent) as possible. He would use this knowledge to infer that agent B for instance might think of the following situations as possible: <4,2,6> (B is term-agent), <4,10,6> (B is sum-agent), <8,2,6> (B is term-agent). Using this knowledge, the agent wouldn't reason about other cases (except for the ones that agent C might think of as possible).

In contrast, our computational model thinks of all possibilities, even the ones in which numbers are involved that none of the agents might think of in the current situation (for instance <1,5,7> in the above case). The model doesn't limit the search space using the constraints that are provided by perception. These constraints are only implicitly implemented in the statements that the agents make. If they say "I don't know the numbers", they can only say this, if they have made observations of the other agents their numbers and are in doubt of their own number (although the agent would always make the claim "I don't know the numbers" if he didn't perceive at all, however we exclude this case for our smart and fully logical agents).

As a result, the computational model has a lot more information to process then humans have most of the time. This processing of all information is necessary however, because the model only wants to reason using knowledge that all agents have. And because every agent has different observations (and there is no full overlap in numbers), reasoning about all possible sets in the domain is the only solution to this problem.

Humans however tend to reason non-logical, focusing only on their current setting. They would switch to more logical reasoning if that was given as a constraint, or they might switch to this mode if they couldn't solve the puzzle using some initial strategy of their own. Nonetheless, using the fully logical reasoning of representation of all knowledge might be very difficult for agents that can't store all possibilities in their memory.