Arbitrary can mean the opposite of arbitrary.
Historically, a decision was described as arbitrary if it depended on somebody's judgement. We still have the word arbitrator which is a person appointed, usually by the agreement of both parties to a dispute, to judge the matter and make a decision.
Constitutional theories about the role of government, as they developed in the seventeenth and eighteenth centuries, contrasted models of government in which the king had absolute discretion to do in any matter whatever he wished, with models in which the king, and everybody else, was required to obey the law. Every citizen should be treated equally and fairly, according to mutually understood laws. All should be equal under the law. The government must act according to commonly accepted standards.
The word arbitrary was increasingly used to mean the exercise of power contrary to law and/or custom. Broadly the US placed most emphasis on law and the British Empire most emphasis on custom, but both were held in contrast to the absolute right of government to do as it thought best. The US Declaration of Independence, the English Bill of Rights and the Bonfire Prayers recited by the Church of England on the anniversary of the Gunpowder Plot (and landing of William III), all use arbitrary in the sense of uncontrolled power.
It is this sense of things happening in a way not governed by law, custom or principle which corresponds to the use of arbitrary today in the sense of purely random, that is not in accordance with any discernible rationale.
The Cambridge dictionary includes a definition of arbitrary as
based on chance rather than being planned or based on reason.
and gives an example
Did you have a reason for choosing your destination or was it arbitrary?
Here the word arbitrary mean without reason of any kind.
Longmans's dictionary includes the following example
This effectively enables the Home Office to make arbitrary decisions, deporting people as they see fit, without any independent inquiry.
Here the Home Office (a UK government department) is said to be able to deport people as they see fit - that is not according to any publically accepted criteria, but as they see fit, which is to say that they have their own criteria. This does not mean they are random, but we do not know why they deport one person and choose to overlook another.
If a company decides to make some of its staff redundant and, as a matter of fact, all the ones made redundant are over 55, then the management may claim that the decision was purely arbitrary. It is purely by random chance that the people made redundant all happen to be over 55. However they might well be accused of making an arbitrary decision, a decision intentionally flouting discrimination laws and the norms of society. In this scenario arbitrary is the opposite of arbitrary.
Your friend's decision to oppose some people joining the game, and allow others may not be based on any accepted principle or custom. In that case he is acting as something of an autocrat, decreeing who he will or will not play with. His proposal of no two people in the same workplace could be described as arbitrary in the sense it does not derive from any accepted principle or custom, but merely from his own discretion. It does not appear to be purely ransom though, he seems to have his own reasons which may be to maximise his chances of winning by not allowing in people who know each other well, and encouraging in those who don't. He is arbitrarily using his influence, but he is not making arbitrary decisions in the sense of random decisions.
Of course I am not attempting to arbitrate as to whether your friend's non-random strategy to increase his chances of winning by making discretionary decisions is, or is not, an arbitrary abuse of his influence. It is, after all, a game of Diplomacy you are arranging.