Golden Balls: Split or Steal

Reading through this article's statistics I saw something very interesting: "Surprisingly, we find little evidence that contestants' propensity to cooperate depends positively on the likelihood that their opponent will cooperate. While an opponent's promise to cooperate is a strong predictor of her actual choice, contestants appear not to be more likely to cooperate if their opponent might be expected to cooperate." Going further, "While a promise is a strong signal of cooperation (according to the statistics those who make a promise are 31% more likely to cooperate), contestants whose opponents made a promise do not have a higher propensity to choose split. In fact, as Model 7 also shows, if an opponent promises to be cooperative, the other player even displays a marginally significant decrease in the likelihood of choosing "split". Though it's hard to say how accurate the results are based on various factors it is nonetheless very interesting and is the basis of my claim.

What I am claiming is that the more certain a player is about their opponent choosing "split" the more likely they would "steal". Or in other words the more uncertain one is about their opponent's decisions the more likely they would "split". Now logically this is quite unclear because it's as though we are saying (in extreme senses) if you know your opponent is going to "split" you are going to "steal" for sure, and if you have absolutely no idea then you would be more likely to "split". But why does this uncertainty play such a role in one's decisions? If you are planning to "steal" if you know your opponent would "split" why should it change your decision if you did not know your opponent would do so? As long as there is even a 1% chance of your opponent to "split" it should still be better for you to "steal".

I asked another friend, who had not studied game theory, in a sort of manipulated way what he would do. I gave him the brief overview of the Golden Balls scenario. I asked him what he would do given that situation and was completely unsure of what the opponent would do. He said he would most likely "share". Then I asked him what would he do if he knew the opponent would "split" he said "steal" because 100% > 50%. Now was this result because he lacked knowledge in game theory? Perhaps. But in a way this decision in a sense is rational. I also gave him the typical situation and he said always "steal" because it always gives you a better result. Either way as for the previous answers I want to compare it a game of matching:

Basically two players (strangers who have not discussed beforehand) have the choice of Left, Middle, Up, Down. If LL, MM, UU they get a utility of (5 , 5). If DD they get a utility of (4 , 4). Any other combination and they both get 0.

Game theory does not really have a solution to this game, perhaps a mixed strategy with less probability assigned to down. In this game the players want to cooperate as their opponent's gain is their own. But without prior discussion and such how can we choose the best method? Going down must be a quite reasonable if not the best choice because out of all the other combinations it is the only unique one (even though utility-wise it is the worst combination out of the three). So in a way in the Golden Balls scenario it feels as though it is possible that people could have used this sort of rationalization because "split" is the only choice that guarantees some sort of plus (even if it is not themselves). The decision of "steal" on the other hand has occurrences where (if both "steal") there would be zero reward to both players. 


Of course if you look at this it's strange because by reasoning in such a way it does not increase the probability your opponent would "split". But no matter how I look at it, it seems as though the fear of your opponent to "steal", in a way almost induces people try to avoid a (steal, steal) situation in which if at the least if they "split" not all of the money deteriorates (even if there is a chance they do not get anything). Perhaps another reasoning behind it could be that in a way by making a move you are hoping your opponent does the same. Again this is sort a dubious idea because logically speaking we do not have some sort of brain control waves in which by making a decision it sends signals to your opponent. But in a way two rational players would look at a chunk of money and by all means would want to avoid the situation of (0%, 0%). The concept to keep the money alive perhaps may influence our decision to choose "split". And this reasoning all comes from the uncertainty of what another player does. In a way you can if you know your opponent would "split" you would "steal" to gain the extra 50% (this is not necessarily true as you may want to share with your opponent or something), and if you know your opponent would "steal" you would "steal" too because if you are going to go down you can at least take him down (this perhaps is inaccurate as you actually might find more "splits" for example because then at least you can look like a good person, but the counter argument of looking like a sucker goes against this too- in the end it's really hard to say but for arguments sake assume "steal" if "steal"). So assuming that your opponent plays in such ways it only seems logical to do "steal" but the uncertainty of your opponent's decision gives one the fear of the potential of playing "steal", And the threat / potential of doing "steal" or "split" has a larger impact than the decision itself or a promise and thus gives a higher likeliness of playing "split" despite the fact that "steal" is rationally the best option. 

So this leads me to the discussion before the two actually make their decision. It puzzled me at first to the reason why they even did this. It should not really matter what your opponent says to you (because you have no ways of making some sort of behind-the-doors agreement) your choice should already be set on in a way, on your values. Thus this leaves me with a conclusion that strategy-wise the best thing to do is during the discussion make it as unclear as possible on what you would do. I mean this does vary among people, for example, if you know that your opponent is a good, trusting person you may instead want to make it seem clear that you would "split", but otherwise in my opinion (and according to the statistics perhaps) by making it so that your opponent has absolutely no idea what you would do, they would play "split".

Golden Balls Introduction

This post is just a general introduction to my next post and thus not necessary to read (if anyone is reading anyways). The idea behind this post also came from a friend who was contemplating on discussing game theory for his paper and these are mine / our conclusions to the situation. Of course his paper is focused on a completely different aspect.

I have somewhat recently came across a clip on Youtube which displayed a very interesting example of the prisoner's dilemma. The basic summary of this is there are two players who are faced with a decision to split or steal a certain amount of cash (average of $20,000 source). If they both split they get 50% each, if one steals and the other splits the one who steals get's all the money while the other gets none. In the situation that they both steal they both get nothing, unlike in the "typical" model where there is some sort of consolation prize.

 

 (From now on I would be referring to the model on the right as the typical model)

The difference in this case is that instead of having strictly dominating strategies we only see weakly dominated strategies as each player is indifferent between stealing and splitting when their opponent steals. 

In the theoretical standpoint we would see the same results between the two, where both players steal and end up getting nothing or 10%. In both cases there is an obvious Pareto improvement (where at least one player can benefit without harming the other), which is (split, split) gaining 50% as opposed to 0% or 50%. But in the Golden Balls situation we see that (steal, split) or (split, steal) is a Pareto improvement too. For both situation we see that by playing through game theory the result is an unfavorable one.

Now looking at it in a psychological perspective is quite interesting. In both models compare the choices one player can make considering that the other player would split. With this viewpoint we are comparing 100% if steal with 50% if split. Game theory tells us 100% > 50% that's your decision. But the reasoning behind this can really only be said to be as greed. Whether there is a negative connotation behind that or not cannot really be said as it is almost as if asking someone would you rather have $100,000 or $50,000 (if you do not consider your opponent). To me more interesting though is the decision made assuming your opponent steals. In the Golden Balls model we see that a player should be indifferent between stealing and splitting as they would get nothing no matter what they do. But in the typical model stealing would allow you to have 10% rather than nothing at all. In this sense it, stealing almost acts as a defense mechanism. In this post (link), I discuss extreme situations of the prisoner's dilemma where such a defense has a much larger impact. So in a way, the decision maker in Golden Balls is really only looking at whether to split or steal based on their opponent splitting. In that perspective we see that you are looking at two categories of people (ie. the benevolent sharing kind, and the greedy). Of course there are various other factors which I would discuss later. In the case of the typical model though I would say that being unsure of your opponent stealing is a likely reaction (without considering the extra money you get if they split) because it defends you in the case that your opponent steals. With that in mind, there should be a large number of the "benevolent" people using steal thus if tested we should see a higher % of steals in the typical situation compared to golden balls.

Out of the 574 final contestants, 53% of them chose to split. (link source) To me this is a very surprising number in that it is really quite high. There are quite a few somewhat obvious reasoning that can be drawn. Just as one would gain some sort of sense of well-being when offering their seat to the elderly, the decision for a player to do split can add the gains to themselves based on making their opponent happy. Of course in such a situation it probably only has that effect when the other player splits too, as I would definitely have some sense of added disutility given I trusted my opponent yet he backstabbed me. Another idea is that being on television you do not want to be seen as an asshole by stealing. The article mentions there is a counter effect in which you also do not want to be seen as a "sucker" by splitting. But to add to that I also feel as though given the situation of the cash prize, having your amount of money doubled should not really play too much on your image. In a way you can almost say it gives you a positive image in the sense that you outsmarted your opponent (of course this is dubious, you may also be seen as a psychopath). There is also probably the group of people who can be said to be genuinely good and would share no matter the situation as even if their opponent steals the result is nothing anyways.

But the reason I started this blog, was because it seems as though the most curious aspect of uncertainty may be the one that has the largest impact on the decision to split. While this is unclear and perhaps counter logical (because it shouldn't matter whether you know what your opponent would do as you really are only considering what they would do if split) yet in a way can be rational. I discuss this idea in this blog post (link).