$2 is the optimal strategy if the other person is either an idiot or a game theorist. I’m still wondering if there’s a difference.
I question the result. With a $2 answer, my possible outcomes are $2 (if the other person is an idiot or game theorist) or $4 (if they’re not).
With a $100 answer, my possible outcomes range from $2 to $100.
With a $99 answer, they range from $2 to $101.
With a $98 answer, they range from $2 to $100.
…
Of course, $2 is the correct answer if we’ve all played this game a zillion times, like tic-tac-toe. It’s a great example where game theory gives an answer that would be idiotic in the real world. Which is the whole point.
Actually, that bit isn’t necessary for the game theory outcome.
The game theory outcome is simply the result of doing the math. The math shows that if you adopt the $2 strategy and play against every other possible strategy, you’ll come out ahead of any other strategy.
However, I don’t assum I’m playing against all other strategies. I’ll assume that the other guy wants as much money as possible and is more interested in the big win than the little win, and I’d bet accordingly. I’d guess most people would pick $100, so I’d pick $99.
That’s the whole point of the dilemma: game theory doesn’t provide what is likely to really be the best strategy, even if the players are sharp.
Your opponent picks a strategy. You choose the optimal counter-strategy. They then switch to the optimal counter-strategy to that. You switch to the optimal counter-strategy to that. And so on and so on. If this process ever stabilizes, what you end up with is a Nash equilibrium. It’s not meant to be the best outcome for everyone, it’s just meant to be the equilibrium outcome. (And in the case of this particular game, the aforementioned process is guaranteed to hit equilibrium after at most 99 steps; for example, if the opponent’s original strategy was to always pick $100, your optimal counter-strategy would be to always pick $99, and their optimal counter-strategy would be to always pick $98, and so on down till you settle into the ($2, $2)-groove)
[You can question the relevance to various applications of iterating the process of producing optimal counter-strategies, of course, but it was never about “How can I defeat an idiot?”. Picking an optimal counter-strategy is the opposite of idiotic.]
I can see why the decision to bet $2 is supposedly rational - saying $2 is the only way to ensure that you get any money at all in a scenario when the opponent can ensure that you get $0 if you give any other amount. (This invalidates SaintCad’s estimated value. If we stipulate a rational second person then they choose $2 100% of the time. The estimated value can only make sense if it’s used to predict how many people are irrational.)
That said, I’d still be one of those people with a number in the 90s or 100, even when I understand that I’m handing my opponent a chance to stiff me. $2 is enough that I’m willing to throw away some rational safety for a chance at a much higher return, hoping that the other person is also irrational. $2 is a rounding error - it’ll barely buy me a cup of coffee.
In order to convince me to be “rational” you’d have to change the amount of money. If you multiplied everything by 1,000 then I’d have a more rational behavior. $2,000 is enough for me to get serious about “protecting my money.”
I suppose I should have said iterated prisoner’s dilemma.
If the goal is to beat your opponent, then sure, a $2 bet is optimal. However, in that case it’s nonsensical to use money as the medium of betting. $4 is not better than $100, period. Now, 4 points might be better than 100 points, it depends on what your opponent has, but that clears up the whole dilemma lickety split and isn’t so “interesting.”
Exactly wrong. $2 is the optimal answer if you only play once. $100 is the optimal answer if played zillions of times.
There is a close analog with the Prisoner’s Dilemma, where defecting is the optimal solution for a one-shot game, but where if played multiple times (called the “iterated” Prisoner’s Dilemma) there are strategies that result in continued cooperation (one of the best, and simplest, is called tit-for-tat).
An iterated game means you have a chance to test if your partner is likely to betray you or not. Two honest partners will quickly converge to a system where they are both maximizing their returns. That’s not possible with a one-shot game like what’s described.
But in this case, it is idiotic, especially if the game is played only once against any given opponent. I agree with Buck that this does highlight a weakness in treating the Nash Equilibrium as the “optimal strategy”.
Good point, except I disagree with it being optimal if you play only once. It’s not the optimal strategy, it’s just the Nash Equilibrium.
The optimum strategy is the one that nets you the most money.
The key is you shouldn’t follow any strategy that assumes the other guy is going to follow a different strategy. Assume he’s your mirror image and will do whatever you’re doing.
So a strategy of “I’ll underbid and he won’t” is not a real strategy. The two real options are you both try to underbid and you both choose not to underbid.
If you both try to underbid the other guy, you both end up with two dollars. If you both forego underbidding, you both end up with a hundred dollars. It’s pretty obvious not underbidding is the optimum strategy.
For what it’s worth, you are essentially noting that against the “mixed” strategy of picking each option equally often (1/99 of the time picking $2, 1/99 of the time picking $3, etc.), picking $2 is not a very good counter-strategy. Indeed, as you note, it is the worst counter-strategy to that particular one, netting only $394/99 (just under $4) on average.
This is all true. But so what? Every strategy has some against which it fares very poorly. If your strategy is to pick $96, then you’ll do even worse against my strategy of picking $2 (netting $0 on average).
There’s nothing particularly significant about the particular strategy “Pick each option equally often”, or its optimal counter-strategies. It’s just one particular point among many.
$2 is a better strategy only if your opponent will bid $2 almost all the time. If he’ll bid $100 even 5% of the time, $100 comes out ahead.
But like st. Cad said, if you make the stakes sufficiently high, you can get to the low bid. If you made it $2 million - $100 million, I’d go with the $2.
Note 1: $2 is also the best strategy if your opponent never bids higher than $3.
This may seem like a pedantic observation, but it’s important in the following way:
Note 2: You should never pick $100. $100 is a terrible strategy in comparison to $99. The latter will do just as good whenever the opponent bids less than $99, and will do better whenever the opponent bids $99 or $100.
Note 3: If you know your opponent is never going to pick $100, mind you, then you should never pick $99. $99 is a terrible strategy in comparison to $98, under those conditions. The latter will do just as good whenever the opponent bids less than $98, and will do better whenever the opponent bids $98 or $99.
Note 4: If you know your opponent is never going to pick $100 or $99, then you should never pick $98. $98 is a terrible strategy in comparison to $97, under those conditions. The latter will do just as good whenever the opponent bids less than $97, and will do better whenever the opponent bids $97 or $98.
Note 5: I assume you see where this is going?
Those $2 lower limits are actually $0. If your opponent nominates $2 and you nominate anything higher, you get $0. (I’m fairly sure you knew that and it was just a typo.)
What’s the utility function that corresponds to this? It’s not a game in the game theory sense unless you have the utilities written out pretty explicitly. If you can specify a function that leads to this behavior, then we have something to talk about, but otherwise it just seems like idle speculation to me.
The problem with this type of strategizing is that you’re always supposed to assume that everyone involved is just as rational and self-interested as you are. You are (heh heh) Indistinguishable.
So you can’t say things like $2 is also the best strategy if your opponent never bids higher than $3 or $99 will do better when your opponent bids less than $99, and will do better whenever your opponent bids $99 or $100. Because the premise of the scenario is that your opponent thinks of all the same possibilities you do and will arrive at the same conclusion you arrive at.
So you should go into these scenarios thinking that whatever number you end up writing, your opponent will end up writing down the exact same number. And if you try to change your number at the last second, he’ll be changing his number at the last second too. No matter how you juggle strategies, if the two of you are equally rational, you’re going to end up with the same number in the end.
The solution then is to look at the scenario and pick out the number that will give you the maximum reward if both people pick it. And knowing your opponent is your equal in rationality, you can count on him arriving at the same conclusion and picking the same number.
That’s certainly a possible analysis. That’s what they call superrationality, and would indeed lead to you preferring $100 to $2.
(But it doesn’t make the particular excerpts “$2 is also the best strategy if your opponent never bids higher than $3” or “$99 will do better when your opponent bids less than $99, and will do better whenever your opponent bids $99 or $100” into false observations. It just makes them irrelevant. It does invalidate the reasoning “$99 is better than $100 because $99 fares at least as well as $100 against any particular counter-strategy, and fares even better against some”, by tossing out the idea of the opponent acting independently and reducing the situation to a one-player game.)
