Today's episode of Doctor Who featured an interesting dilemma, centered around an object called the "Osgood Box". I'm not well-versed enough in game theory to recognize it as an existing problem, but it seems to me that principles of game theory could potentially be applied to it.

The Setup: There are two factions, currently at cease-fire (Humans and Zygons). One rebel of one faction (Zygon) is attempting to break the cease-fire and ignite a worldwide war. Zygons are shapeshifters, and there are twenty million of them hidden among the Human population of Earth. Most Zygons just want to live in peace pretending to be Humans, and most Humans don't know Zygons exist at all.

The Zygon rebel commander and U.N.I.T.'s Chief Scientific Officer face off with two boxes between them. Each box has two buttons, labeled "Truth" and "Consequences".

The Doctor informs them that one of the two buttons in front of the U.N.I.T. CSO will release a gas that will create a chain reaction in Earth's atmosphere and kill every Zygon on the planet, and the other will detonate a nuke directly underneath them, annihilating London in the process (as well as all of the characters in the room). One of the buttons in front of the Zygon rebel commander will cause all Zygons on the planet to revert to their natural form for an hour (revealing them to Humanity at large and kickstarting the war), while the other will prevent any Zygon on Earth from shifting out of their current human form ever again.

The Question: Given this information about what the four buttons will do,

  • Should the U.N.I.T. CSO press either button in front of her, and if so is there enough information to get a better than 50% chance at stopping the war by pressing a button?
  • Should the Zygon rebel commander press either button in front of her, and if so is there enough information to get a better than 50% chance at starting the war by pressing a button?
  • If either representative of her species chooses to press a button, should her opponent change her decision about pressing a button and/or which button to press?

Resolution in the Show: (Spoilers for DW fans who have not watched the episode)

The Doctor has setup the situation to try and keep the peace, explaining that the two boxes are a microcosm of every single war ever waged. The U.N.I.T. CSO opts to press neither button, and the Doctor monologues at the Zygon rebel commander for a bit, before she comes to the conclusion that neither box does anything at all. None of the four buttons are pressed.

The U.N.I.T. CSO expresses incredulity at the situation, and the Doctor tells her "that's what you said the last 15 times," before wiping her memory of the events.

  • $\begingroup$ they should swap boxes, or alternatively, ask each other what button they would press $\endgroup$ – JMP Nov 8 '15 at 3:48

Excepting the show's resolution, and addressing the problem:

Both boxes are, with exception to color, visibly identical, bearing two labelled buttons. The button labels are irrelevant, as both terms could be considered differently relative to the user. In the case of the Zygon, "truth" can mean either exposure of Zygons or the Doctor's view of both races as equalized, thus enforcing one uniform appearance. In the case of U.N.I.T., "consequences" can refer easily to both situations.

This places, all other information being mute, the two users at a state of apparent mathematical equality, but they are not equal. The muddling factor? The boxes do not perform identically.

From the position of U.N.I.T., neither choice in front of the user is preferable. The annihilation of 20 million aliens in human form would not go unnoticed and possibly lead to world panic and chaos, with decades of therapy for the PTSD from watching ones friends turned inside out. The other option is suicide and, while it might slow the threat on the other side of the table, it would only weaken humanity's resistance to that threat when a new leader stepped forward. The best option is to push nothing.

From the position of the Zygon, the exact object they desire is one of the two options. This is why any convincing is actually needed. The Zygon is the only one whose personal conviction holds much sway. The second option is only marginally a penalty, as it allows her race to survive, albeit trapped in human form.

The puzzle actually depends on U.N.I.T., and U.N.I.T. is the primary influence over the Zygon. If the Zygon presses a button and is not immediately transformed, U.N.I.T. has won. If they are transformed, U.N.I.T. must press a button and either immediately win the war or destroy leaders of both sides and a lot of the enemy with them to give humanity a fighting chance. U.N.I.T. will not allow a transformation to go unanswered and we can assume combinations of transformation without U.N.I.T. pressing a button can be discarded.

Realistically, this means that the logical combinations are:

  • Zygon transforms, U.N.I.T. suicides and takes out Zygon leadership, both sides lose.
  • Zygon trapped, U.N.I.T. does nothing, U.N.I.T. wins
  • Zygon transforms, U.N.I.T. releases agent and kills all Zygons, U.N.I.T. wins.
  • Zygon does nothing, U.N.I.T. suicides and takes out Zygon leadership, Zygons win.
  • Zygon does nothing, U.N.I.T. releases agent and kills all Zygons, U.N.I.T. wins.

If we include the combination where the Zygons transform and U.N.I.T. does nothing, and account for the vastly outnumbered Zygons who already admit they would lose the war, there is only a 33% chance of Zygon victory. If we remove that for the aforementioned reasons, there is only a 20% chance of victory. From this, even the calculus is that the Zygon should not press a button at all, and that if she does, that U.N.I.T. must press a button in response.

NOTE: This is on the presumption both sides may press a button. This can be assumed only because it is not expressly forbidden by the experiment as stated. If it were, then desire to preempt the other would come into play. We can assume this is not so, as both parties would then be likely to strike buttons immediately.

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