Blackmail, Nukes and the Prisoner's Dilemma

by Stuart_Armstrong1 min read10th Mar 201020 comments

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Coordination / CooperationPrisoner's DilemmaBlackmail / ExtortionGame TheoryHumor
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This example (and the whole method for modelling blackmail) are due to Eliezer. I have just recast them in my own words.

We join our friends, the Countess of Rectitude and Baron Chastity, in bed together. Having surmounted their recent difficulties (she paid him, by the way), they decide to relax with a good old game of prisoner's dilemma. The payoff matrix is as usual:

(Baron, Countess)
Cooperate
Defect
Cooperate
(3,3) (0,5)
Defect
(5,0) (1,1)

Were they both standard game theorists, they would both defect, and the payoff would be (1,1). But recall that the baron occupies an epistemic vantage over the countess. While the countess only gets to choose her own action, he can choose from among four more general tactics:

  1. (Countess C, Countess D)→(Baron D, Baron C)   "contrarian" : do the opposite of what she does
  2. (Countess C, Countess D)→(Baron C, Baron C)   "trusting soul" : always cooperate
  3. (Countess C, Countess D)→(Baron D, Baron D)   "bastard" : always defect
  4. (Countess C, Countess D)→(Baron C, Baron D)   "copycat" : do whatever she does

Recall that he counterfactually considers what the countess would do in each case, while assuming that the countess considers his decision a fixed fact about the universe. Were he to adopt the contrarian tactic, she would maximise her utility by defecting, giving a payoff of (0,5). Similarly, she would defect in both trusting soul and bastard, giving payoffs of (0,5) and (1,1) respectively. If he goes for copycat, on the other hand, she will cooperate, giving a payoff of (3,3).

Thus when one player occupies a superior epistemic vantage over the other, they can do better than standard game theorists, and manage to both cooperate.

"Isn't it wonderful," gushed the Countess, pocketing her 3 utilitons and lighting a cigarette, "how we can do such marvellously unexpected things when your position is over mine?"

Next to the bed, the butler had absent-mindedly left a pair of nuclear bombs, along with the champagne flutes. The countess and the baron each picked up one of the nukes, and the wily baron proposed that they play another round of prisoner's dilemma. With the added option of setting off a nuke, the game payoff matrix looks like:

(Baron, Countess)
Cooperate
Defect
Nuke!
Cooperate
 (3,3) (0,5) -∞
Defect
(5,0) (1,1)
-∞
Nuke!  -∞ -∞ -∞

In this new setup, the baron has the choice of nine tactics, corresponding to the ways the countess' three possible actions map to his three possible actions. The most interesting of his tactics is the following:

  • (Countess C, Countess D, Countess N) -> (Baron D, Baron N, Baron N)  : "Nuclear blackmail"

The baron, quite simply, is threatening to nuke the pair of them if the countess doesn't cooperate with him - but he's going to defect if she does cooperate. Under this assumption, the countess can choose to cooperate with payout (5,0), or defect or nuke, both with payoff -∞.

To maximise her utility, she must therefore cooperate with the baron, even though she will get nothing out of this. Since the baron cannot make more utility than 5 in this game, Nuclear blackmail will be the tactic he will choose to implement, and thus the payoff will be (5,0).

With the addition of nukes, the blackmailer has gone from being able to force cooperation, to being able to force his preferred option.

"Out, out!" the countess said, propelling the baron away from her with a few well aimed kicks. "On second thoughts, I don't want you over me any more!"

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Marriage as nukes (a fun analogy): Consider a man and a woman in a relationship. The man says "I love my freedom, but it hurts so much to see her with another man..." That is, suppose that the opportunity to cheat on the woman without too much consequence offers the man less utility than he loses when the woman cheats on him. Suppose the woman feels similarly. This is precisely a prisoner's dilemma: self interest harms the "opponent" more than it benefits the player.

Solution: you each sacrifice a little utility to buy a marriage license which ensures sufficient punishment for infidelity to prevent either of you from doing it. You lose your freedom, but you gain the more valuable asset of partner's fidelity. I.e., you buy nukes :)

(I once used this to argue against someone who said "Marriage is always pointless.")

This is not properly analogous to buying nukes. It's changing the payoff matrix - the expected payoff of "cheat" becomes negative. The same mechanism where the victim must trigger massive negative utility for both parties is not present, or at least is not made present simply by a better contract. The mechanism of punishment would (almost certainly) transfer assets or rights from the cheater to the wronged partner. This would be done through a prenuptial agreement, not a marriage license; the former is a binding contract, the latter is basically just a piece of paper you need to get stamped before getting married.

This is not an unusual arrangement, as one can easily write into a prenuptial agreement an infidelity clause so that, upon divorce, the wronged partner is entitled to a greater share of community property, or to a fixed-sum payment per incident/extra partner. This means that each partner knows that cheating will give their partner a strong incentive to divorce them, and the cost of divorce will increase dramatically. One could actually specify a lot in such contracts (though not child custody); I remember hearing of a pro athlete who had a contract in his prenup such that if his wife gained more than thirty pounds, he could divorce her and she would get nothing/almost nothing - though whether the court would uphold that is anyone's guess. The failure of people to actually use prenuptial agreements is itself a fascinating study in real human decision theory.

Certainly, what you describe is not analogous to a weapon, because of the transfer of utility. The analogy fits better in a society where either 1) adultery is a huge disgrace to the offender, possibly involving punishment (the nuke only harms one side), or 2) adultery is a huge shame to both sides (the nuke harms everyone, like in the scenario posted). Though clearly not universal, the existence of this analogy is enough for me to call it "fun" ... Just in case you had any doubt, I surely do think this is a terrible view of marriage, for both societal and game theoretic reasons!

a marriage license, which ensures sufficient punishment for infidelity

It does?

Heheh, in response, I have edited out the comma in that sentence :)

[-][anonymous]11y 0

May somebody dare to explain this to a non-native speaker? The simplified English grammar I got taught just said " which", and " that", which is obviously false. Also, with my German intuition the sentence is equivalent, both with and without comma; I cannot sense any semantic difference.

The rule is that nonrestrictive relative clauses are separated by a comma, while restrictive relative clauses are not. There is an additional "rule" that which is only used in non-restrictive clauses and that only used in restrictive clauses, which is probably the source of the rule you learned. But this rule is the same sort of nonsense as not separating infinitives and not ending sentences with a preposition, that is it is based on someones idea how the language should work rather than any observation how it does work and hence it does not match the intuitions of a native speaker.

In German the difference between a restrictive and a nonrestrictive clause is not defined the presence or absence of the comma and there appears to be no easy and straightforward rule. (Faustregel: Bestimmtes Bezugsnomen -> erläuternder Relativsatz, unbestimmtes Bezugsnomen -> einschränkender Relativsatz)

Compare:

  • Sie streben ein Eherecht an, das eine hinreichende Abschreckung gegen Ehebruch darstellt. [restrictive]

  • Unser Eherecht, das eine hinreichende Abschreckung gegen Ehebruch darstellt, findet seinen Ursprung in dem Bestreben... [nonrestrictive]

The rule is that nonrestrictive relative clauses are separated by a comma, while restrictive relative clauses are not.

And that that is used only in restrictive clauses. Geoff Pullum describes this as "overwhelmingly complied with by everyone".

Example:

The banana, which is my favorite fruit, is yellow.

*The banana, that is my favorite fruit, is yellow.

And that that is used only in restrictive clauses. Geoff Pullum describes this as "overwhelmingly complied with by everyone".

Ah, I thought that was the case and couldn't think of any counter examples, but I wasn't completely sure and since the clause that started this sub-thread used which and I definitely knew the reverse was not true I didn't mention it.

The quoted version says something about a marriage license, and explains that marriage licenses ensure punishments... . The corrected version (without commas) says something about a marriage license which ensures such punishment, but makes no general statements about marriage licenses.

[-][anonymous]11y 0

So, a comma determines whether a property applies to an instance or to the general class of some thing? Wow.

If syntax didn't affect semantics, it'd be useless.

[-][anonymous]11y 0

Not in general. It separates the sentence into multiple parts. I don't know enough grammar words to explain it better, but the original version meant almost the same as

... a marriage license (which ensures sufficient punishment for infidelity) ...

Up-vote for the Epic Grammar Nazi! >D

The most interesting of his tactics is the following:

  • (Countess C, Countess D, Countess N) -> (Baron D, Baron N, Baron N) : "Nuclear blackmail"

This is more interesting. It also results in Epic Win for the Baron...

  • (Countess C, Countess D, Countess N) -> (Baron D, Baron N, Baron C) : "Please nuke me"

"Please nuke me"

I'd prefer to call that one "Make my day."

I don't understand the difference. 'Please nuke me' and 'nuclear blackmail' seem to be identical except the baron Cooperates if the Countess Nukes in the 'Please' scenario - but cooperation has the same payoff as the Baron nuking right back.

Stuart, I for one appreciate your attempt to explain the game theory + decision theory hybrid our people are working on, and I think you are doing a good job so far.

Thanks!

I apologise for any mistakes made in the first post (overuse of rational and lack of attention to Godel-type problem).

[-][anonymous]11y 1

The most interesting of his tactics is the following:

  • (Countess C, Countess D, Countess N) -> (Baron D, Baron N, Baron N) : "Nuclear blackmail"

This is more interesting. It also results in Epic Win for the Baron...

* (Countess C, Countess D, Countess N) -> (Baron D, Baron N, Baron C) : "Please nuke me"