This game has multiple Nash equilibria and cheap talk is allowed, so correlated equilibria are possible. Here's how you implement a correlated equilibrium if your opponent is smart enough:
"We have two minutes to talk, right? I'm going to ask you to flip a coin (visibly to both of us) at the last possible moment, the exact second where we must cease talking. If the coin comes up heads, I promise I'll cooperate, you can just go ahead and claim the whole prize. If the coin comes up tails, I promise I'll defect. Please cooperate in this case, because you have nothing to gain by defecting, and anyway the arrangement is fair, isn't it?"
This neglects diminishing marginal utility - few people would actually prefer a 50% chance at everything to a 100% chance at half of it. It does solve the coordination problem, though. Interesting approach.
he merely has the option to hurt him out of spite.
Merely? Never underestimate the power of Spite.
The changed payoff matrix makes this unlike the Prisoner's Dilemma even without the addition of communication; more like a restricted bargaining game. One noteworthy difference from the Prisoner's Dilemma is that this game lacks a pure Nash equilibrium.
Edit: Apparently not quite; see below.
The usual definition of Nash equilibrium requires only ≤, not <, so Defect-Defect, Cooperate-Defect and Defect-Cooperate are Nash equilibria (and pure) but not "strong" Nash equilibria. You want this definition, because games need not have strong Nash equilibria, even if you allow mixed strategies.
(Apparently the game is called "weak prisoner's dilemma" in the literature).
I love the bit at the end where Ibrahim (market trader) says it's "the hardest money I've ever had to work for" and Nick (charity worker) jokes "he obviously hasn't worked in the charity sector to try and get money", then the look on Nick's face when Ibrahim says he's going to respray his yacht!
I felt that Nick displayed a good mix of hot and cold rationality.
The next contestant needs to say:
"I'm going to choose steal. If you choose split, I'll give you 25 percent after the show. I promise."
I don't think that's gonna work even a tenth as well as a promise of 50 percent. Promises work on the basis of the kind of honor that's also correlated to concepts of justice and fairness.
Once someone has already effectively proclaimed themselves to be unfair and dishonorable, their promises would seem worthless. At that point the other contestant would be even more likely than before to choose "steal" for the purposes of retribution.
WTF why did he (the precommiter) choose split after all?
The WHOLE point is to change it from a difficult decision with close outcomes to one where split is the only choice with positive utility (the chance that other person will keep their word).
WTF why did he (the precommiter) choose split after all?
If the other guy (Abraham) had chosen "steal", the supposed precommitter (Nick) would not have gotten anything either way, under any scenario.
If Abraham had chosen "split", it was a time-saver to have the gameshow hosts divide the money between them, than to gift half of it afterwards to him.
So Nick's whole ploy was to effectively scare Abraham into saying "split" -- by making him think that contrary to normal expectations it was saying "steal" that would ensure he would get nothing. Once Nick had convinced him of that, there was no longer any need to say "steal": Nick had either managed to convince him or he hadn't.
Nick sacrifices credibility for future claimed precommitments of course.
Nick sacrifices credibility for future claimed precommitments of course.
He sacrifices credibility in future threats against people, but maintains credibility in future promises to act in others' benefit just as much as if he had decided to steal and then give Abraham half the money. This latter credibility is probably much more useful in most real situations.
no no no.
The moment I found out I was going to be on this show I would obtain two notarized contracts.
When the time comes to deliberate I whip out the first contract. It states that if I choose "split" I must donate $10k to the KKK + any prize money I get (the $10k at least is held in trust).
Then I ask my opponent to split, I whip out a second contract stating that any and all prize money I receive is going 100% to medical research. His choice now has nothing to do with money for me or him, only if the television studio keeps the money or if it goes to medical research.
I hope this is novel enough to land me a talk show appearance where I pimp my ebook on using cognitive science and game theory to improve your life.
And then your opponent still steals, gets all the money, and nothing goes to medical research. woopty doo.
No, Romeo chooses steal. If his opponent also chooses steal (in spite of Romeo's credible commitment to choosing steal himself), the opponent does not get any money.
To be a little technical, this is not actually prisoners dilemma, because they are allowed to communicate. The whole point of prisoners dilemma is that they cannot communicate, and thus, must choose to cooperate or defect based upon their knowledge of nash's equilibrium alone.
Although this is honestly quite an interesting solution to this kind of problem. I'll be using it the next time I'm offered a situation similar to this.
To be even more technical, "Prisoner's Dilemma" is actually used as a generic term in game theory. It refers to the set of two-player games with this kind of payoff matrix (see here). The classic prisoners dilemma also adds in the inability to communicate (as well as a bunch of backstory which isn't relevant to the math), but not all prisoners dilemmas need to follow that pattern.
In game theory terms, I'm not sure communication would do much for a one-off prisoner's dilemma.
I can say, without hindsight interfering, that this strategy would not have worked on me. Because I can explain exactly what I was thinking as it happened.
You see, when I see someone alter the rules of a game, my instinct is that they are trying to do so for their own gain, and thus are not altruistic. Thus I immediately assumed the promise was a lie (which was right), and that he would not be splitting the money with me (which was wrong).
The question then becomes rather simple. My choices are to choose SPLIT, receive $0, and reward the treachery, or choose STEAL, still receive $0, and punish the treachery. Obviously, the latter is more valuable to me.
Now, in the short time required, I did not have time to check if my assumptions were correct. But let's say they aren't. The most likely way for me to be wrong would not be that he wasn't lying, but that he was going to choose SPLIT. Well, that's still a winning outcome for me. And if I feel guilty for winning all the money, I can always split after the fact with him. So that's not a problem either. The only option that is a possible problem is if he's telling the entire truth. But I see this as highly unlikely, as what does he have to gain from splitting after the fact rather than just using the balls?
I honestly was surprised that this worked. I actually thought the other guy was foolish for choosing SPLIT until the reveal. As I do not know the other possible solutions I cannot say the first guy's solution was rational, but I am fairly confident in saying the second guy's decision was not.
split is not 0. It is some probability he will give you money out of gratitude + the probability he is lying and will actually choose split.
I found this to be a very interesting method of dealing with a modified Prisoner's Dilemma. In this situation, if both players cooperate they split a cash prize, but if one defects he gets the entire prize. The difference from the normal prisoner's dilemma is that if both defect, neither gets anything, so a player gains nothing by defecting if he knows his opponent will defect; he merely has the option to hurt him out of spite. Watch and see how one player deals with this.
http://www.youtube.com/watch?v=S0qjK3TWZE8