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Human beings are more productive in groups than individually, possibly due to the fact that there is a limit to how much one human brain can improve itself in terms of speed of computation and areas of expertise.

By contrast, if a machine with general-purpose artificial intelligence is created and then assigned a task, would it be possible that the machine will be able to better accomplish its task by continuously improving its own computational power and mastery of various skills, as opposed to collaborating with other agents (whether copies of itself, other AI's, or even humans)?

In other words, would an AGI ever need to collaborate, or would it always be able to achieve its goals alone?

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In my answer, I have often switched between AGI and ASI for reference. This is fine as an AGI will reach ASI as it is optimizing itself and learning.
I think it is not only important by necessary that AGI and ASI are of collaborative nature. Nick Bostrom, in his book Superintelligence: Paths, Dangers, Strategies in Chapter 10 described three ways in which an ASI might function:

As an oracle, which answers nearly any question posed to it with accuracy, including complex questions that humans cannot easily answer—i.e. How can I manufacture a more efficient car engine? Google is a primitive type of oracle.
As a genie, which executes any high-level command it’s given—Use a molecular assembler to build a new and more efficient kind of car engine—and then awaits its next command.
As a sovereign, which is assigned a broad and open-ended pursuit and allowed to operate in the world freely, making its own decisions about how best to proceed—Invent a faster, cheaper, and safer way than cars for humans to privately transport themselves.

This is when and if AGI and ASI are of controlled manner and their output are as expected. By controlled I mean they don't start seeing human as a threat and start eliminating the human race. More on this here. As you can see, all the above ASI are of collaborative nature. Either they are collaborating with humans or they need to collaborate with other systems.

Now to your answer:
First, collaborative nature is of great use in terms of efficiency and performance. This is the reason why Distributed Systems are being made. We even have distributed OS now. Also, Modular approach in coding/ developments and huge success of Object Oriented Model are proofs of advantages of using collaboration among different entities.
If you think about it even the AGI is collaborating and using resources from other places in some way. As your AGI is learning, it is gaining information from the internet. It reads the information and tries to structure is accordingly (This will depend on its neural schema) and create knowledge (or something valuable) for itself. It is collaborating using network protocols with the outer world (other systems). If it doesn't collaborate then the firewall of the system might not allow it to use the service. Different services on the internet require a different set of protocols to be followed. So, if AGI wants to communicate for information it needs to follow those protocols. This way the AGI will learn to collaborate with different entities even at an early stage.
From AGI point of view communicating a web server and communicating another AGI machine is very similar. As we, humans, don't store all the information in our brain; similarly AGI won't find it efficient to store all the information within. Not all information is needed all the time. The memory hierarchy is proof of that. Even if AGI is made to store all the information within, with time it will figure out how inefficient it is and will re-program itself to only hold very vital information and use the internet for less frequently used information.

I will like to add one more thing to this. Let us begin with the human analogy first. We, as humans, collaborate with other humans. But we also collaborate with our different body parts. Like, what about the collaboration within us. Collaboration (or coordination) between our legs, hands, our body, and mind, etc. This leads us to the question of who as a person we are, our whole body, or just our brain. If I remove my hand and replace it with an organic implant, would I be still me?
Similarly, what is AGI? Is it the whole structure, or just the code. Does the RAM, ROM, hard disks also a part of AGI? If you think the hardware is not the part of AGI then your AGI is co-ordinating with these devices too, using certain protocols. To some level, this is collaboration too.

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Communications is expensive. It requires a communication channel, a protocol and of course time. Communications is also limited to the expressivity defined by the protocol. Note also that agents may compete over resources, or may have contradictory goals, so in some situations they may try to mislead each other.

On the other hand - computational power and memory are limited, so multiple agents may solve computation-intensive or memory-intensive problems together better/faster than each single agent can. Different agents may have different sensors, and mobile agents may have information about different parts of their realm, so by sharing knowledge they may have more complete information and make better decisions. Goals may also be time-bounded, and rewards may be time-dependent. Sometimes working together means greater rewards.

In summary, there may be situations where collaboration is beneficial, and there may be some situations where collaboration is essential to achieve one's goal or to achieve a common goal.

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