A company rules Mars
With companies becoming arguably more influential players on the global stage than some nation-states, it is perceivable that Mars will be under the rule of, for example, Elon Musk's SpaceX.
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The Argument
With the success of SpaceX and its development of reusable rockets, a private company has accomplished more than governmental organizations have been able to regarding space colonization. In addition, aside from isolated individuals, companies like SpaceX are the only ones seriously talking about--and arguably soon capable of--settling Mars.
Because of this, private companies seem to be the best option for leading a Martian government--they are the ones getting the process started. If nothing else, a private company will likely enter into an agreement with governing forces, adopting leadership responsibilities due to the absence of government involvement.
Counter arguments
The fact that a company is powerful and innovative does not automatically make it qualified to lead a planet. Also, just because private companies are the current leaders in plans to settle Mars does not mean that governmental organizations like NASA will not join the effort in the future. If Earth governments and companies settle Mars as a joint effort rather than simply the latter on their own, it is debatable which of the two should lead the newly settled planet.
Premises
[P1] A private company is currently leading efforts to colonize Mars.
[P2] For this reason, it is rational to conclude that a private company will govern Mars.
Rejecting the premises
Further Reading
More on the SpaceX plan to colonize Mars: https://www.geekwire.com/2016/spacex-elon-musk-colonize-mars/