Aftermath of a Presidential Death Match

Question

This world is so ridiculous that it's very existence seemed hard to justify without being 100% subjective. I struggled for a while with a premise to root the story in something. And I have to say, I'm happy with it now, so without further ado, here it is:

Premise

My world follows a catastrophic world war that killed off 80% of the world's population and redefined the geopolitical landscape. A powerful, mind-altering drug was used all over the world to boost endurance, concentration and reduce the need for sleep. Everyone used it from soldiers, militia and guerrilla groups to the generals and all the head honchos. Unfortunately, moderate mental disorders were an invariable side-effect. It proved not to be an utterly crippling disorder in most cases, but rather predisposed people to extreme risk-taking and poor judgement. And yes, the course of history took an absurd turn:

As the world leaders grew weary of the complicated logistics of supplying the small disparate pockets of fighters that yet live and fight on across the world, an international summit was held. Despite the fact that the world leaders all had mental disorders, they managed to agree on something: All of the world powers had agreed to disarm their entire arsenals including navy fleets, air planes and tanks. An estimated 15% of the world's military didn't get the memo as they were off-grid, in deep reconnaissance or other-wise unreachable. They were to be informed later, and now the deed is done. 85% of the world's military was destroyed or disarmed. But the peace dove's were nowhere to be seen...

The world leaders spoke:"Enough is enough. Now, we're going to fight like men." Lo, it was not the doves of peace but the eagles of battle that soared up, latched talons and fought gloriously in the sky for all to see. This was the omen for what was to come. Instead of long-drawn out attrition, a Presidential Death-Match was set in motion. It was hand-to-hand combat; no weapons of any sort, and it was fought to the death. Some of the world leaders may be too demented to realize what they signed up for, but they fought nonetheless.

Question: How can the winner(s) of the Presidential Death-Match stake their claim to the spoils of war?

The most glaring problem with staking their claim is the fact they have just put themselves at the losing end of the arms race. With all of their national arsenals long gone, they can only strut into town bare-chest and bare-fist. Meanwhile, where the fighting rages on, the 15% of the armed military could shoot them down the "sissy" way. The world leaders seem to be thinking on a different wave-length, I'm not sure it's possible for them to stake any claim at all.

Further Clarification

• For clarity, the 85% of unarmed troops will be referred to as the "Unarmed Military" and the 15% of armed troops will be referred to as the "Armed Military"
• mental disorder is permanent, makes people prone to high-risk taking behavior and overall poor judgement. Not outright debilitating though
• Spoils of war includes: natural resources, territorial claims, ect
• The Unarmed Military whose president lost in the Death Match are good sports and will not stand in the way of the victorious Presidents staking their claims. Though I do not specify whether they are obligated to actively help their conquerors in conquering them. I will leave this fairly open.
• The 15% Armed Military who did not witness the valiant performance of the Death Match Presidents should be considered dangerous
• The presidents who lost the Death Match are dead
• Answers can employ any number of tactics such as: winning hearts and minds, warfare designed for fighting a superior force (bare-fist to fully armed), ect

Answer 1

This politics StackExchange question makes it clear why having leaders fight is impractical in real life, however real life isn't nearly as exciting as this world.

I see two options for how these world leaders can take charge of the world after they win:

More Reasonable Option: The off-the-grid militaries that weren't disarmed still follow the chain of command. At some point the chain of command of the Armed Military would have entered into the Unarmed Military. Since the Unarmed Military was willing to follow the leaders' call to disarm, it's reasonable that they also follow the leaders' order to respect the winner of the Death Match. Therefore, they will be able to order the Armed Military to stand down, especially if there is some proof of agreement between the leaders prior to the Death Match. A soldier is compelled by law to disobey an illegal order, but is trained to obey any other order.

More Fun Option: Forget the WWE and all the fake wrestling, this world just got a new tier of world-wide celebrity fighters. The Death Match was obviously televised, because who wouldn't want to watch Angela Merkel suplex Shinzō Abe, right before Trudeau enters the ring with a steel chair? The winning leaders would have higher approval ratings than they ever had pre-Death Match, and any attempts by the Armed Military to go against the will of a match would result in mass riots by adoring fans. Personally, my money's on the rise of the Icelandic Empire.

Answer 2

The mind altering drug (or a different one used on subordinates) also causes a very enhanced sense of loyalty to the chain of command or the government leader specifically.

The underlings are so loyal that prior to the death match a pronouncement by their glorious leader to give loyalty to the winner and not interfere are enough for them to gladly comply.

If all of the leaders make these announcements, and possibly recordings for the off grid forces (the death match would of course be televised, yes?), this would make claiming the winnings easy for the winner. The drug induced loyalty would make them the new leader of all groups involved in the death match.