What would you say, then, to a variant on the same game, where for Player 1, the bonuses/penalties are $2, as usual, but for Player 2, the bonuses/penalties are $3?
Now, there’s an asymmetry in the game between the players, and there’s no particular reason for them to assume the other one acts exactly the same way. It’s hard to see how to invoke superrationality. But once again, by the same reasoning as before, $99 weakly dominates $100 for both players, and so on and so on, so that the unique Nash equilibrium ends up being ($2, $2).
I’ll admit that complicates it. But ultimately it’s still the same thing. You can’t base a strategy on out-thinking the other player when the premise is you’re both equally rational. Assume that whatever strategy you come up with, the other guy will know what you’re doing and take it into account in his strategy and vice versa.
Think of it as if you’re writing a program to play the game and then loading the program on to two separate computers. You’re never going to be able to write a program that will allow one computer to outwit another computer running the same program.
That’s precisely the idea that leads to the Nash equilibrium. Note that this is different from superrationality, which is instead “Assume that whatever strategy you use, the other guy will use the same strategy”.
Consider also this game: we remove the penalty for the player who announces the higher value, and instead pay a $2 bonus to both players if they disagree (on top of the smaller value). For simplicity’s sake, let’s also suppose the values the players can choose from are $0 or $1.
This is again a symmetric game. Naively, one might suppose superrationality would lead to both players picking $1 and getting a $1 payoff. Actually, one can do even better with mixed (i.e., randomized) strategies: if both players adopt the strategy “Pick $1 with probability 2/3 and $0 with probability 1/3”, then they’ll get mean payoffs of $4/3 ~= $1.33.
That is a Nash equilibrium, but there are also two other ones which don’t require randomization: at ($0, $1) and ($1, $0), both players get payoffs of $2.
In this case, superrationality leads to a result which is rather worse for everyone than suitable Nash equilibria.
Is there actually a solution to this? Because once you start from the $100, and work your way down to $2, it seems that you would work your way up again. If you figure that the other person will say $2, as would you, you can then say $3. But then you can think that the other person would say $3, and, instead, say $4. And so on, back up to $100, then down to $2, etc.
If you figure that the other person will say $2, you shouldn’t say $3: saying $3 will net you only $1 (because of the $2 penalty for saying the larger value), while saying $2 will net you $2.
Similarly, if you figure that the other person will say $3, you shouldn’t say $4: saying $4 will net you only $2 (because of the $2 penalty for saying the larger value), while saying $2 would net you $4 (because of the $2 bonus for saying the lower value).
In general, if you know what the opponent will pick, your best move is to go 1 below that and collect the $2 bonus (ending up making $1 more than what your opponent selected). However, you can’t do this in the one case where your opponent picks the minimum value, in which case your best move is to tie them (anything else would just result in you earning the same minimum, minus the $2 penalty.)
I like this thought, and I think there is relevance from this exercise to the way we behave as human beings.
I note that Bush the Elder’s “Thousand Points of Light” was widely mocked by the Left, so I’m not so sure it’s right v left thing; right wingers supposedly have much higher rates of charitable contributions, for example. If we as humans really did behave charitably, we wouldn’t need government social programs. The right-left debate is about government v private as the owner of social stewardship, and in that debate the right is clearly on the side of “being of good heart” as the mechanism by which social stewardship is executed. The left, not trusting us to be spontaneously charitable enough, wants government to redistribute by fiat. (I don’t necessarily disagree with what you say about the Market side of the equation; just saying it’s not a neat right-left thing here.)
But two humans can do better when they are willing to take the risk that the other human being will also put a common good ahead of a personal good.
Our problem is not so much trusting ourselves to be of good heart as it is trusting the other guy to be of good heart.
So the real dilemma, which this exercise neatly demonstrates, is the expectation that we will unilaterally act in the common interest knowing that this is not the likely behaviour for all.
We must be the change we want to see in the world, even knowing we often will lose the upper hand for any given round, in order to avoid the tragedy of the commons (in which we all lose).
One of the reasons that the game fails is that it measures only dollars and ignores risk considerations. Most people don’t value all dollar sums proportionally. Two examples:
I give you the chance to pay 1 penny for a 1/100,000 chance of winning $900. Technically, it’s a losing game, but most people would play because they don’t give a crap about the penny.
(Just think about the lottery compared to the Traveler - people are giving up $2 all the time for the chance at getting more)
I offer you the chance to play a game. I flip a coin. If it comes up heads, I double your possessions (you get another car, another house, double the amount in your bank account, etc.). If it comes up tails, I take all of your possessions (so now you have no house, no car, no money, etc.) BUT I buy you a nice steak dinner and give you twenty bucks. This is clearly a winning deal, but no one would play it because they’re not interested in trading the risks involved, and they don’t value the first x dollars the same way they value the next x dollars.
Of course, someone with no house, no car, and only twenty bucks might pay it. And that’s the point, not only does each person value each dollar differently, but every person values dollars differently, and has their own curve that they apply to the value of money and the associated risk. Until you incorporate those, you can’t solve the equation.
That’s true, but it has no relevance to the Nash equilibrium analysis for this game, which, as I demonstrated above, in this case depends only on the ordinal properties of payoffs.
Fine, then scale everything up till you do distinguish all the prize differences. Instead of dollars as the basic unit, use thousands of dollars, or whatever. Any 102 different points on your personal preference spectrum will do, to serve as the values 1 through 102. [And if you don’t have 102 different points on your personal spectrum, well, we’ll play with whatever you’ve got.]
But that gets to the point I was making in the OP. I do not believe humans act irrationally merely because they make a choice that is sub-optimal according to game theory.
I understand your point and there’s nothing wrong with your math or logic per se - the problem is that people don’t try to optimize for dollars, they optimize for value and the relationship between dollars and value is very non-linear - and I don’t just mean curved, there are step changes all over the place.
If there are three games, one where you pay $10 for a 60% chance of winning $20, one where you pay $100,000 for a 60% chance of winning $200,000, and one where you pay $100 for a 0.06% of winning 200,000 - they all have equal expected payoff ratios. But people will correctly treat each of the games very differently and different people will each treat the games differently than other people, and, again, they will be correct in doing so.
Neither do I. What in the message you quoted implied otherwise?
Sure. But what is the relevance of that to this game? Since the reasoning is purely ordinal, the only relevant nonlinearity would be failure to distinguish between certain prize differences, but I think it’s perfectly reasonable to take the spirit of the game to be such that the points are stand-ins for whatever prizes the players can be assumed to distinguish between.
(I think everyone is reading me as far more of a “‘game theory rationality’ is the right account of rationality” guy than I am. My thread history will back me up when I say I do not at all feel that way. I just think the interesting Nash equilibrium thing here has been dismissed by many too quickly, with improper appreciation. There’s something interesting and noteworthy to the argument that no one should ever pick $100 because it’s weakly dominated, and, over time, we could expect that everyone would become familiar with this and no one would ever pick $100, which would then make $99 weakly dominated, and so on and so on. I don’t want that line of reasoning to be handwaved away too glibly; it’s not nonsense.)
Absolutely agree, and I think that idea can be expressed if we change the game to the following:
Each of the travellers can choose between $100 and $110, in increments of $1. If you tie, you both get the stated amount. If you come in under the other person, you get $15 over the lowest bid (yours), if you come in over the other person, you get $15 less than the lowest bid (theirs). In this version of the game, it should become more obvious that the lowest number is the appropriate choice, because we should expect a linear relationship across the range of outcomes and we could also assume that the other person would have a linear view of the values across the range of outcomes (which is equally important, if not more so, than our own valuation of the possible outcomes).
The one item that I wanted to mention to you was that, in the original scenario, what if you were aware that 99% of other people playing would prefer a 1% chance to receive $100 compared to a 100% chance to recieve $2 - would that change the way that you played the game?
Chief Pedant, I don’t think the argument you’re making supports what you think it does. In this game, if the players can communicate beforehand and make a binding agreement between them (say, a contract with a significant penalty clause), they can both get significantly better results: In such a case, they could both pick $100 with confidence that the other player won’t undercut them. This is certainly a far better outcome than the Nash equilibrium, and it’s also a better outcome for at least one player, and generally for both, than the real-world case where people do mostly trust each other and pick in the mid to high 90s. So the binding agreement between the players to regulate their behavior is a good thing. But this is exactly what a government is: A binding agreement between players to regulate their behavior. So situations like this, to the extent they occur in the real world, make the case that social stewardship should be handled by the government.
It also shows why government intervention to prevent oligopolistic collusion is appropriate (there are better, more direct examles, but this shows how a ‘deal’ can cause deviation from a market result and result in higher (and less efficicient) costs for consumers).
I partly agree with what you write here, and should explain my earlier, over-simplified post. As Chronos implies, it is the “left” which understands that in practice, government help is often needed to help enforce the Golden Rule. Some individuals on the “right” may be of better “heart” and expect more from their fellows.
Where my own overly-terse posts confuse is that I often neglect to mention the vast gulf between right-wing opinion-makers and their right-wing followers. IMO, they are often on the opposite sides of “morality” with the former practicing mass deception on the latter.