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:
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:
- (Countess C, Countess D)→(Baron D, Baron C) "contrarian" : do the opposite of what she does
- (Countess C, Countess D)→(Baron C, Baron C) "trusting soul" : always cooperate
- (Countess C, Countess D)→(Baron D, Baron D) "bastard" : always defect
- (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:
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!"