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    Game Theory - The Art Of Acting Rational
    By Johannes Koelman | December 13th 2009 02:17 AM | 74 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune

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    You are in a game show with nineteen other players. You don't know the other players, you can't see them, and you can't communicate with them. The game you are in is called 'Greed!', and is straightforward to explain. You are asked to write down a whole dollar amount in the range $1 - $1,000,000 on a piece of paper. You will be paid the amount you asked for if it is deemed to be 'non-greedy'. Whether your request is indeed 'non-greedy' will be decided once all twenty request have been received by the host of the show. Your requested amount will be labeled 'non-greedy' if no other player has asked for less, and at least one player has asked for more.

    How do you play?

    This is your chance to go home with a life-changing sum of money. But... you have to balance your greed with the risk of getting underbid. You better think this through carefully.

    Let's assume you go for the maximum. Well, that is not really a rational move, isn't it? No-one can overbid you, so your bid can impossibly be classified as 'non-greedy'. No matter what the others do, you are guaranteed to go home empty-handed.

    So let's avoid this suckers strategy and ask for one dollar less: $999,999. That will win you a handsome amount if no-one underbids you, and at least one person asks for more. But how likely is such an outcome? Surely the other players will have reached insights similar to yours. And if all other players draw the same conclusions as you do, no-one will have written down a higher amount. So either all other players ask for the same amount as you do, or one or more players underbid you. In both cases you go home empty-handed.

    Apparently, you have to bid less than $999,999. And again, when making this downward adjustment, you can not prevent your competitors from following the same logic. We see a pattern emerging. Continuing this iterative line of reasoning will take you and your competitors in free fall towards the strategy of requesting the smallest permissible amount: $1. This is the outcome predicted by game theory: the game 'Greed!' forces rational players towards a strategy of requesting the smallest amount. If all players follow this logic, they all end up writing down an amount of $1, and all of them go home empty-handed. Yet, no player can do better by deviating single-handedly from this strategy: selecting any higher amount will not yield a better return if the other players stick to their choice of $1. In game-theory jargon, such a strategy combination for which none of the players feels an incentive to deviate, is referred to as a Nash equilibrium.

    John Nash

    So what do you do? Will you indeed write down the amount $1?

    I doubt it.

    The game 'Greed!' relies on deep iterative reasoning to demonstrate the Nash equilibrium. Other games with these characteristics such as the game Guess 2/3 of the average and Traveler's dilemma invariably lead to experimental results that deviate markedly from the game-theoretical (Nash equilibrium) predictions. Some researchers argue that the failure of standard game theory to reproduce experimental outcomes for such games necessitate a re-definition of what constitutes rational behavior. Kaushik Basu, inventor of the game 'Traveler's dilemma', writes in Scientific American:

    “When playing [traveler's dilemma], people consistently reject the rational choice. In fact, by acting illogically, they end up reaping a larger reward--an outcome that demands a new kind of formal reasoning.”

    and:

    “The [Traveller's dilemma] highlights a logical paradox of rationality [..] This paradox has led some to question the value of game theory in general, while others have suggested that a new kind of reasoning is required to understand how it can be quite rational ultimately to make non-rational choices.”

    Douglas Hofstadter, in his book "Metamagical Themas" argues that humans have the potential to transcend game-theoretical rationality and can act 'superrationally'.

    Such remarks ignore the fact that the concept 'Nash equilibrium strategy' is not necessarily synonymous to 'optimal play'. A Nash equilibrium can define an optimum, but only as a defensive strategy against stiff competition. More specifically: Nash equilibria are hardly ever maximally exploitive. A Nash equilibrium strategy guards against any possible competition including the fiercest, and thereby tends to fail taking advantage of sub-optimum strategies followed by competitors. Achieving maximally exploitive play generally requires deviating from the Nash strategy, and allowing for defensive leaks in ones own strategy. Poker players know this all too well. Nash equilibrium strategies are the optimal strategies – and a perfectly rational choice - against the strongest competition. Yet, against weaker competition, deviating from Nash equilibrium strategies becomes a perfectly rational choice. There is absolutely no 'logical paradox' hidden in these facts.

    So how you should play the game 'Greed!'? This depends very much on your perception who you are competing with. Without any information on the group of opponents, you assume randomly selected competitors, and make model assumptions on their capability to iterate towards the Nash equilibrium. Behavioral studies on games like Guess 2/3 of the average suggest that significant fractions of randomly selected candidates fail to iterate more than one step. In the game 'Greed!' this forces you to a rational choice of selecting amounts well above the Nash equilibrium value of $1.

    However, if you know that a large fraction of your competitors are game theoreticians, mathematicians and others with likely full insight into the theoretical and strategic aspects of the game, you better adopt your choice to a value closer to the Nash value of $1. Note that even under these circumstances it is not necessarily rational to select the precise Nash value. For the Nash strategy to be optimum, your opponents not only need to be optimally skilled at the game, they also need to be convinced that all their competitors are equally skilled players who know they are playing against optimally skilled opponents... etc. A circumstance that is seldomly true in many-player games.



    Assuming 19 opponents selected randomly from the readers of this column, how would you play the game 'Greed!'?


    ------------------------------------------------

    More Hammock Physicist articles: The Big Picture. The Largest Distance between Two Points. What you didn't know about E=mc2. Telepathy and the Quantum".


    Comments

    Gerhard Adam
    It would seem that you have to assess the probability that someone in the group will write down a value greater than $1 and the probability that someone will write down $1. 

    If someone enters a value greater than $1, then the minimum amount can be a winner.  Similarly if there is someone that puts down $1, then all others will be losers.

    So depending on the perceived probability of those two events it becomes a matter of assessing how the other players view their own sense of "greed".  In other words, they would likely consider that too large a value is to readily undercut, while too low a value (while potentially being a winner, may be less than optimal).  After all, there is little risk in the game since the worst consequence is merely leaving with $0, which is what you started with.  Therefore I suspect that most people would try to maximize their return profitably without becoming too "greedy".

    Mundus vult decipi
    The risk in this game is huge. If 19 players select $999999 and one player $1000000 (the superrational solution), then an average participant would pocket $949999. In the Nash solution (all players selecting $1) nobsody earns a penny.

    I would expect reality to be much closer to what Nash predicts. I go for the Adams optimum: $42.

    I'd like to think I would bid $1m. I mean, sure I'd lose, but I'd make a lot of people happy doing so.

    I would try $4998. Doing so I will have no regrets in case of loosing or winning.

    Yes, same here, $4000-5000 and for the same reason.

    I would choose $100 for the same reason.

    I would feel even less worse than you about losing because $100 is less than $5000, but if I won it's just enough to making me feel a little bit happy. Probably I'd spend the money on dinner and a movie.

    If all my opponents are Terry Pratchett Fans then $7
    If all my opponents are fans of Douglas Adams, then $41
    If all my opponents are Satanists, $665
    If all my opponents are bankers, $999,999

    Best way for this to work is to trust the other people to not be idiots and get greedy for a guaranteed $1 and that collusion will happen afterwards.

    Ideally everybody asks for a large amount 999,999 and one person asks for 1M. But who asks for 1M? What we need is a random assignment of who asks for how much that has a very very small chance of everybody asking for the same amount. Flipping a fair coin for the choice will work quite nicely. Doing some quick calculations in R shows that each persons expected winnings to be 499,998.5.
    Here is the r-code for the calculation:
    total <- 0
    for( i in 1:19 ){
    total <- total + dbinom(i, 20, .5)*i*999999
    }
    personal.winnings <- total/20

    Now what if we let the coin be biased towards choosing 999,999? Doing the same calculation as above shows the the optimal weight is somewhere near .85. This has an expect winnings per player of 811239.7.

    If all players are perfectly rational... that is exactly what they would do, because the other option leaves them winning a lousy $1.

    you said: "If all players are perfectly rational... that is exactly what they would do, because the other option leaves them winning a lousy $1."

    what you meant: "If all players are perfectly rational and trusting... that is exactly what they would do, because the other option leaves them winning a lousy $1."

    that's quite a difference in a game such as this. is there any reason any player actually has to split the winnings after the "rational" choices have been made? no.

    "If all players are perfectly rational... that is exactly what they would do, because the other option leaves them winning a lousy $1." ...... What you describe is Hofstadter's 'superrational' strategy.
    Interesting to see the marked difference in results between the selfish Nash strategy, and Hofstadter's wishful thinking approach. I would never play according to Hofstadter. And neither according to Nash. As the author says: being rational is an art, not a science.

    How exactly are you going to manage that strategy with no ability to communicate with your fellow players? I mean, if we could communicate, I would immediately volunteer to ask for $1M, and ask for $50K from each other player as reward for helping them out.

    I must agree. I would follow this plan or similar: randomly choosing either $999.999 or $1.000.000,
    following the categorical imperative, and be prepared to share with anyone choosing $1.000.000.

    Note that I would EXPECT to gain nothing in either case, people being what they are.
    The thing is, when I go home with nothing, I would know that I stood by my principles,
    that I did the correct thing, and that given the choice again, I would do the same thing.
    (Narcissistic self-righteousness FTW!!!)

    Unless the winner bid the same as someone else, though, he will always ask himself,
    with regret, whether he couldn't have bid higher (or so I will tell myself).

    Of course, the optimal solution would be to negotiate, have one person bid $1m, and all others bid $999,999. Then, just split the payout evenly amongst all participants. Net return for 100 ppl: $989,999.

    > Of course, the optimal solution would be to negotiate, have one person bid $1m, and all others bid $999,999. Then, just split the payout evenly amongst all participants. Net return for 100 ppl: $989,999.

    Optimal for the group maybe. Optimal for me is to get all you suckers to agree to the above and then for me to bid $999,998 and keep it all.

    How is that optimal for you? You gained $49,999 and pissed off 19 people. 19 people who would argue that the game was rigged and get your winnings revoked. It's much better to maintain goodwill and be a good sport in the long run.

    19 people who you probably will never have to talk to again? So what if I offend them, I've gained an extra $50k! Sorry, but I think your strategy is almost mindlessly optimistic.

    The problem with this is other people are likely to have the same idea or get suspicious and may underbid you... so you're back to the original dilemma unless everyone can just agree to work together. Also... if you don't feel guilty after depriving 19 people of a life changing about of money just so you can get a tiny bit extra, you're probably a sociopath. I'm not trying to be offensive... it's just true.

    The risk isn't huge at all, since there is no chance of leaving with less than you came with.

    I'd go for the amount of principal I have left on my mortgage. I either pay my mortgage off or I get entertained by watching someone else get some amount of money.

    You still run the risk of *not* winning $ 999,999.

    I run that risk every day. And I always lose! Every time!

    (I like the mortgage theory but I'd probably bet $4990. Or $1m. It really is almost a matter of whimsy.)

    That was my thought, I'd probably ask for enough money to pay off all my loans and credit cards, and then get mad at myself because I forgot to ask for enough for a train ticket home.

    Gerhard Adam
    You don't know the other players, you can't see them, and you can't communicate with them.
    Negotiation is not an option.  The whole point of such a game is that each player possesses imperfect knowledge.
    Mundus vult decipi
    AFAIK a Nash Equilibrium only refers to a situation where players do not change their strategy *** after having been told of the other player's strategies ***. Is this incorrect?

    I think this article explains it well: the Nash equilibrium is the particular choice of strategies (one for each player) such that no single player is tempted to deviate from the particular strategy selected. The player might be tempted to deviate from the Nash strategy provided he can form a coalition, but on his own he can't improve his situation.

    Adorable picture of Nash (:

    Should there not be a mixed strategy that mixes over 1 to 1 million uniformly? This has to have a greater expected payoff than 1$.

    No. You can easily see why:
    If you play a mixed strategy in which the highest amount included is $N, then rather than playing the same mixed strategy, it is better for me to play the same mixed strategy with the amount $N excluded (and thereby increasing the probabilities of the lower amounts). But.... in that case it is better for you to.... etc.

    i would go with the number of the taxicab i took to the game show. $1729

    I saw you coming. The taxi number was 6789.

    For those not in on the joke: Web search for 1729 taxi..

    "I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

    I've read the Wiki about this number.. it's crazy!

    Even if everyone plays at the Nash equilibrium and bids "1", everyone goes home empty-handed. So there is no way you even have a single dollar safe. So we are not looking for a strategy that does better than 1$, but a strategy that does better than $0. The average payoff is of course maximized if everyone picks randomly a number not much below 1 million. (So as to avoid everyone bidding the same number, but the lowest number still being high.) But if you are greedy and are looking only at your own payoff, you can beat that strategy by going somewhat farther below 1 million. I guess it all depends on how envious you think you'd be if you end up not winning. So pick a number somewhere around your "bearable envy threshold". Or a number a little below what you estimate your competitors' envy thresholds to be. OK, after all that waffling: 45.000

    Everyone goes for as much as needed to pay for their next vacation.

    I would assume the other players were not mathematicians or economists and therefore that upon reading the problem statement they would interpret greedy to mean 'more than a fair share' and non-greedy to mean 'a fair share or less,' regardless of the strict terms or the minmaxability of the game. I would also suppose the other players would be unhappy to underbid others with a number that would be insignificant to all. Therefore I would choose a random number between the least amount that would feel significant (let's say a month of my household income) and the most that anyone would agree is fair (1,000,000/19).

    I'd pick $800,001

    The assumption implicit here is that the competition is against the other 19 players. However, if, as I do, we take the view that the proponent(s) of the contest are the competion, then going with the $1,000,000 option is the way to go. I walk away with nothing (no worse off), and te satisfaction of knowing they walk away with anything from -$999,999 *19 to some lesser number.
    Serves them right for coming up with unpleasant competions, I say!

    Brilliant.

    What annoys me is that game theory always misses that there is a silent participant; the one who is fronting with the money in this case. If the game is viewed as having 2 players, the banker and the contestants, then as a group the best course of action is to pick a random high value (999,980 - 1,000,000) and the banker stands to lose the most. This is perhaps why game theory breaks down and explains irrational altruism. Which reminds me, I would love to recast "survival of the fittest" to "the survival of the most cooperative".

    Dammit. I agree with Fadingtoblack. I'd hit submit and went AFK, but had erroneously placed a comma in my email address, so having corrected that I resubmitted only to find someone has already made my point.

    It is so pathetic to see economists clining onto puffed up talk of "rationality" like this when the philosophical support for such a view has been shown lacking for very long. It indicates how baseless and isolated modern day economics has become. Please, STFU and do your Hume reading assignments properly.

    Hi all,

    Seriously, going to play a game like this one only to win 1$ makes no sense at all.
    I'd prefer to put 500$ and have a chance to win them (cause let's be honest, if there's no arrangement between players beforehand then you'll never win 999 999$). If I put 500$ and get beaten by a guy who put 1$, who's gonna look more stupid? the guy who earned a dollar and is happy about it or the guy who didn't earn 500?

    But that only IMHO ;)

    If the objective is to merely win, go for $1. No where is it stated you are required to push for a maximum amount achievable. If you go home with $1 in your pocket, you've still won infinitely % more than if you would go home with $0.

    Yes true...

    But my greed tells me there's no point in winning just 1$ and that therefore i should try to get at least 500$... that's why the game is so well thought.

    Given the "most players don't iterate more than once" assumption, I'd bet for 249.998 - that is "put myself in the first 'safer' half of the range, then again, then down one to win over other overcompensators. And some".

    I'd very much love to see an actual simulation of the game... and please let me know who am I supposed to give my home address to have the win delivered there.

    I'd play $1, if only to prove all others wrong ;-)

    The earlier poster who did the math has it right -- the optimal strategy for people who have read this article (and the comments) is to either bid $1m or $999,999 at random, bidding $1m about 15% of the time (flip 3 coins, if you get 3 heads, bid $1m).

    If you are one of the altruists who ends up bidding $1m, then your fellow players ought to compensate you (nothing in the rules against this after the fact!); if not, they are publicly branded as untrustworthy bastards.

    Similarly, defectors who bid $999,998 are revealed to untrustworthy and scumsucking bastards.

    Note that the longer the show is on, the more clear and widely known the optimal strategy becomes, which means the show becomes boring. Expect it to last 13 episodes or less.

    This game is actually very close to the Prisoner's dilemma:
    http://en.wikipedia.org/wiki/Prisoner's_dilemma

    What if the game were changed slightly: each contestant is given a different maximum $ amount. Ranging from 1-5 million, each of the 20 people gets a random prize amount disclosed to them. Then, you don't know what everyone else's range is. Obviously this only works for a group that doesn't go in knowing the "rules" and each person would assume that everyone else had the same "top prize."

    Also, you'll never find a group of 20 people where 1 bids $1m and the rest $1 less. Since they can't communicate, the person bidding $1m would be left with nothing and allow everyone else to win. As we know, the tragedy of the commons dictates that everyone will play for themselves, not the other players.

    And why not make it so the payout is only under the condition that no one has bid <= your amount? Then, in a tie, the show wins, not the players. Thus, bidding $1 is just as bad as $1m as 2 people bidding $1 is a loss for the players.

    clearly 1,000,000 would be the least greedy because you are almost assured to walk with nothing. Any play to actually win would be greedy.

    That's as simple as it can get... aahah

    This would be once in life time experience.
    Smart people are generally richer and probably can take risk and not be under $50k.
    Poor aggressive is similar.
    There probably is some poor timid who are looking for car or something in $15000+ range.
    There might be some that are influenced by lucky numbers such as 711 777 888 etc. or round number or just below it.

    So, I would probably pick $7600 to be below $7777.

    Anyone below that are probably truly needy and deserve it, or just want to win.

    Funny, I was thinking in the vicinity of $5000 myself. Any less than that and it isn't enough money to make a significant impact on my life. I would likely regret asking for $1000 or less and finding out I could have gotten much more than that, but at the $5000 range it would feel like I made a relatively safe choice that still resulted in some real money.

    A factor here that isn't taken into consideration by the original example of game theory is the value of the reward itself--winning can't be assumed to be the sole objective with winning $1 being better than not winning $500. There is a minimum threshold where participating in the game isn't even worth the reward in a cost-benefit analysis (especially when you consider psychological distress from regretting a decision that was made--say, asking for $100 and finding out you could have had $300,000). When you factor this in, even if you're sure you're playing against very skilled opponents, the minimum acceptable amount is probably far greater than $1 for anyone who's actually concerned with the reward rather than with victory in itself.

    100% agreed. Thank you for posting this. So few people seem to take into account "game-show theory", which is that there is a minimum acceptable threshold of money that must be won for it to be considered a success, and that this threshold is different for each contestant. The best strategy (in my mind) is to decide what your personal minimum threshold is (for me it would also be around $5000, like the previous poster said) and bid that much exactly. If you win that much money, you are satisfied with the outcome, and if you lose it is also acceptable, because the only way you could have won that game was to guarantee that you'd be unhappy with the results (by bidding less than your minimum threshold). I never see anyone talk about this theory and in my mind is hugely influential on the way people play game shows.

    Consider the example of a show like Deal or No Deal. If you are at the point where the banker is offering you a life-changing amount of money (say $100,000) and there's only one case left above that, in my mind you must take the deal. You are currently guaranteed a life-changing amount of money, but if that higher case gets removed, you lose that guarantee. What's strange is that so many people that are contestants don't seem to understand this!

    I believe the goal of these types of game shows is to try and get yourself to a point where you are guaranteed a life-changing amount of money, and then never take any big risks to jeopardize that guarantee. Different people will argue what defines a "big risk", but I'm ok with that.

    If everyone bids 1,000,000, then everyone is equal, no one can be deemed greedy. All 20 players should bid the maximum to assure everyone receives the highest potential payout.

    Greed! is a popular television show in this fictional world. Assuming it has been on-air a few times and people are familiar with the concept, I would put down $800k to $900k. Somebody is going to put down $1,000,000 and hope for a small cut of the winnings from the other 19 players (which I would oblige). Unfortunately, somebody else of the twenty players will also put down $1,000,000, meaning I'll owe both players a small cut.

    But if no player feels confident enough that they'll get a cut of the winnings to put down the $1 million bid, and 20 players put down $999,999, they'll all lose!

    Will the group maximize it's potential winnings, or will too many (or none of the) players go for the selfless act to ensure the other players win?

    But here's the catch: I have no faith in the mental capabilities of complete strangers. Furthermore, I assume complete strangers have no faith in my own mental capabilities. Because we all consider each other dumb idiots, we'll all be trying to outsmart each other by assuming the worst. Therefore, we'll all win because none of us trusts anyone else to do the right thing (bid $1 million), but it's assumed somebody is stupid enough to believe someone else will bid very high in order to maximize their winnings off the moron that bets $999,999. Therefore, all the bids will be below $999,999, but still incredibly high.

    Which is why I would bid between $800k and $900k. Because we're all idiots.

    The bet on this game isn't sufficient exciting. Let's add the rule that all the loosers will be killed. Also, not writting a number will only add you to loosers list.

    I think everybody will bid $1 with cross fingers. At second thought, considering myself doomed, I would try at least to offer more chances to my opponents. I would bid $13. Still my chances to win are not zero!

    $4.

    Theoretically, the lowest possible winning bid is $2, because at least one person needs to bid lower than you (i.e., $1). That said, bidding $1 makes no sense as then no one can bid lower than you. So if you assume (!) all participants know that fact, then no one will bid $2 and instead they'll bid $3 at the lowest. If $3 is then the lowest practical losing bid, $4 becomes the lowest practical winning bid.

    tiffany said "I do remember reading that our brains would prefer to punish someone we feel is acting unfair and make no money than getting some money but less than them."

    I came across this most recently in Matt Ridley's book "The Origin of Virtue", which spends a lot of time on game theory and exercises just like this one. Aside from the matter of fairness, people also have a particular -- not strictly rational in mathematical terms -- sense of value. When $1,000,000 is theoretically possible most people would find small sums worse in some sense than nothing -- better to bet big and get nothing than to take something minimal. That's irrational but very visible in the world. Take another case : the losers of the Superbowl (or other sports championship) are often vilified to a greater degree than the teams that lost earlier in the playoffs. Logically, they're second best, but there's a feeling (emotional/psychological) that being *so* close to winning and then not doing so is worse than getting booted 2 rounds earlier. That factor alone tosses the pure Nash result right out the window.

    I think that the responses bidding low are absurd. If you did this experiment repeatedly, (assuming fresh co-bidders) and bid $500,000, you would have to lose 500,000 times to do as poorly as the $1 people (assuming they won each and every time). The potential amount of the reward must have something to do with one's bid. So must the number of bidders! Only one has to bid the full $999,999 for you to be able to get $999,998. I would bid at least $499,999 on the theory that surely some brave soul, out of 20, would have the moxie and the normality not to consider $500,000 too greedy when $1,000,000 is on offer.

    Gerhard Adam
    I'm not sure people are actually reading the conditions of this "game".  The point is that no one must ask for more and that no one ask for less in order to be a winner.  In addition, it is a one time game with no communications.

    Therefore all the numbers selected need to be within a range where you are effectively the lowest value and yet high enough to make the payoff worthwhile (greater than $1).  It isn't simply a matter of one person being greater, but also all the players that may be lower that determine who the winner is.  If we presume that people are willing to take a risk by selecting a value greater than $1, the goal is to select the highest value and still be the lowest value asked for.  In effect, as was mentioned earlier, it plays into our individual sense of fairness.  So if we assume that $50,000 is a fair distribution, then the individual that selects $49,999 would be the lowest reasonable value in a fair split.  However, many have already indicated far lower values which, once again, illustrates the problem when there is imperfect information regarding the other players.

    Even if the game were iterated, it is unlikely that it would appreciably change the results since such additional information would simply cause the strategies to converge to some value minus $1 to produce winners.  Inevitably this would also lead to $1.
    Mundus vult decipi
    From watching man-on-the-street interviews, I'm pretty convinced that none of the 19 other people in the room are game theorists or mathematicians or economists. Therefore, game theory is useless. What you need is Schmuck Theory.

    Schmuck Theory states that people will balance their ability to screw-over others with the possible gain/loss related to that action. For example, the experiment where one player picks the split of some amount of money, and the other person gets to accept the split or reject it, in which case both get nothing. We know that the 2nd person will reject the split if it's "too low," even though it means nothing for either of them. That's Schmuck Theory in action. If you split $100 by giving me $1, I'm going to think, "What a schmuck. It's worth it for me to schmuck you out of $99 for a measly $1, you total dill-weed." Get it up to around $20, though... and although you might think it unfair, you'd probably not pay $20 to out-schmuck someone $80.

    In the case of this contest, you need to determine what the average, lowest amount of money someone would take to out-schmuck a bunch of other people out of what they think others would bid.

    Nobody's going to bid $1 million. If they did, they're a super-schmuck, and the rules make it impossible for that guy to win. Hah! Eat that. Now... would anybody be stupid/schmucky enough to bid $900,000. Yeah, maybe. But you can count on somebody else to underbid that out of 20 regular schmucks. So... what's the lowest amount I'd see as worthwhile, that has a very good chance of out-shmucking everyone else, but where I won't be too pissed if somebody else over-schmucks me?

    For me, it's $1,000. If everybody else bids like, $100,000 or more... what a bunch of schmucks. It's worth it to me to go home with a grand just to stick it to those guys. On the other hand, if somebody bids $10... OK, fine. You want it more than I do. If you need $10 that much, go ahead. Schmuck. No loss on my part.

    So it's not about winning the optimal amount. It's not about greed. It's an equation about the value of sticking it to the other guy.

    The smuck theory is one of the best model of reality. We go as low as, the amount of money is still worthwile. The winner has the lowest level of expectation. If somebody can elaborate a real statistic on series of random selected opponents, I guess the medium outcome of statistic will be above $1, but bellow $100. Between every 19 opponents there will be most the times somebody poor and inteligent.

    Suggest someone write a quick bit of code and lets all have a play??

    That sounds like the worst idea for a game show, ever.
    Somebody is probably working on a reality version as I type this.

    Thanks for the interesting article.

    I've posted an excerpt, along with a link to the article, on my game design blog, the Handy Vandal's Almanac:

    http://handyvandal.com/2009/12/game-theory-the-art-of-acting-rational/

    In poker you can assess the competition after playing a few hands. Here you know nothing of your opponents, you cannot communicate with them and therefore have no means to form any opinions about their abilities based on your assessment of them. Given these conditions you have no criteria to base a decision of asking for more than $1.00. You don't even have enough information for this to be considered a risk assessment. It's a shot in the dark at best. The most you're going to walk away with is $1.00 and possibly nothing. So I say, the logical thing to do is save on the gas money, get a nice bottle of wine, rent a movie and forget about the stupid contest altogether. LOL

    I always liked John Nash's work on astrophysics better anyway. :-þ hahaha
    logicman
    Rational: of or pertaining to ratios.
    Ratiocination: thinking, the weighing up or balancing of ideas against each other.

    On weighing up the strategies, and knowing nothing whatsoever about my competitors, I would focus on the only useful fact available: the name of the game, 'greed'.  I would go for $999 on the assumption that going for 1/1000th of the available pot would probably not be considered greedy by most people.
    I find this article funny. Take 20 random people off the streets and, of course, you aren't going get the game theory results. There is a hidden assumption in the process...that everyone fully understands the consequences of their actions and can logically determine the final conclusion based on the consequences.

    Do psychologists actually work with real people? You'll be lucky to get a group where 50% of the people understand the process well enough to make that kind of decision...thereby introducing what appears to be randomness into the system. They have their own thoughts and rationales for their decisions, but it won't be a strictly logical / mathematical approach to be sure.

    ROFL!!!.....Quite true!
    rational is subjective to the player...if a high tech ceo was invited to play, he wouldn't care if he bet the near the max amount

    Instead of wondering what others might put (because you simply don't know) I instead attack the problem by asking, "what is the most money am Iwilling to walk away from? How much can I afford to walk away from factoring in my investment of time?" There isn't much time involved. I would be annoyed if I woke up tomorrow $500 less than I could have had so perhaps for me the optimal guess would be 500. The publicity of winning is bonus. Wait. The publicity is worth $500 even to me..

    answer: $1

    To be honest, if everyone was a perfectly rational person in my game, I would bid the full million. If everyone reaches the Nash solution, and I bid the full amount, at least some people are getting money out of it, rather than no one.

    I am not sure, but haven't you used the iterated elimination of weakly dominated strategies? In such a way you have eliminated many other Nash Equilibria. Assume everybody else but one player plays 1 and the one guy plays whatever grater than one then this is deviation proof state of the world and thus it is indeed Nash equilibrium.

    The first amount that came to my mind was $100. It's small enough, that if I lose, no big deal, and if I win, no big deal, but probably worth the effort and time cost in playing. It's roughly close to a days work and median American salary. So I think it depends alot on the financial situation of the player. For example, a billionaire would not care about the game at all, but to a severely impoverished third world citizen, any amount would be supremely excellent, thus bidding $1 seems the only rational choice for them. However, they are so impoverished, that it might not matter to them as well.