i am not redefining rational in any way. how is the strategy of betting zero optimal when it gives you a zero payoff with certainty? in what way are you better off by following that strategy so that you can claim it is dominant?
Because if you start backward induction at the original amount of 1MM then each strategy is dominant by offering $1 less than the expected amount you believe will be the lowest. As each strategy dominates the preceeding amount, you inevitably end up at zero. While it does no dominate the strategy of not even participating, any strategy of offering any more is strictly dominated while offering $1 is weakly dominated. At least that is what I can rehash from game theory, though I will admit it was 5 years ago.
i disagree. the goal of this strategy should be to maximize your payoff, not to win the game at any cost and regardless of the outcome. there is nothing that contradicts game theory principles in this analysis. by choosing zero, you are guaranteeing that you will not lose the game and also that you will have a zero payoff. choosing higher than zero allows for an outcome in which you can possibly get a non-zero payoff. a guaranteed zero payoff is weaker than a probable non-zero payoff any way you look at it
I didnt say choose zero, I said it would stop at $1. If you cant model the amount you are choosing, then you cant use it as a dominant strategy. The post you are referring to says “it would be a good idea to choose xxxx” and you cant justify this on a problem set in game theory. You have to use the rules given to you to arrive at an outcome. While it may make sense, you would fail this question if you were to try to rationalize with an arbitrary number and no exact steps to arrive at your conclusion. Unfortunately this is a non repeated game and thus does not have a dominant strategy that pays off. But this is common in single round games.
i am in no way arguing with the logic of the backward induction that you are applying here, or that it leads to the conclusion “choose $1”. but one may need to differentiate between a strategy whose goal is to optimize your chances of winning the game vs a strategy whose goal is to maximize your monetary payoff. if your dominant strategy for winning the game guarantees you a zero payoff, it may still be suboptimal with respect to maxmizing your payoff. i don’t see that as a contradiction
Sure if you can quantify how you are picking your strategy, then there may be a different number. IE the participants dont act rationally, you know their marginal utility of each dollar, etc. But in this case, the rules of the game stated no extra information on the participants, just the assumption that they are rational as is the case with every game and player participation. If you add extra assumptions, then you may be able to make a case for a different outcome. Based on the information give, backward induction appears to be the most likely model and everyone choosing $1 is the outcome. In real life, I would not expect this outcome, but according to game theory, this is it.
I agree with BizBanker. You guys need to pick up a game theory book.
Greetings, Sanka’s Mom.
Brian Fellows Wrote: ------------------------------------------------------- > I agree with BizBanker. You guys need to pick up a > game theory book. Thank you. Its good to know my 1 year of PhD econ allows me to win over some support on a CFA thread. I dont know whether to cry or ask for a refund.
game theory only works when there is a payout for the null outcome. both say Y, payoff = $1 one says Y, other says N, payoff = $10 for Y, $0 for N both say N, payoff = $5 this is where use of a “rational” strategy works because you can lock in an expected value that is higher. if all outcomes based on rational thought = $0, game theory does not apply and it becomes a hand of poker, semi-unpredictable human action, semi-expected probabilities. A: if in the event that someone says $1, then you’re max payoff is $0. B: if in the event that nobody says $1, they will likely say something that is much higher than zero (likely between than $44,999 and $54,999 based on what people perceive as “moral”). your plan: if A happens and someone chooses $1, you get $0. if B happens and nobody chooses $1, you get $1. my plan: if A happens and someone chooses $1, I get $0. if B happens and nobody chooses $1, I get say $40,000. utility of money comes into factor for this…
No game theory works when the definitions of game theory are met, players, time, payoffs, rules, assumptions of players, and one more I forgot. The outcome is not favorable but still can be defined by game theory. You are welcome to offer extra assumptions, but that was not done in the OP so classic game theory applies. If you add extra assumptions, nonlinear utility of money function most likely defined by some ln function, then you can change the strategy, but again this was not offered. Again, because the outcome does not make sense does not mean that you can throw out game theory as the predominant model. A classic example is chess, by game theory definition this game should be won by the first move but as we see it is not due to faulty assumptions and limited rationality. By the definition here, every $$ amount is strongly dominated by an amount less than that amount.
you can create a general assumption for the utility of money based on society’s average utility of money as we know nothing of our sample. only by adding this, would we come to a reasonable entry for this game. putting $1 is unreasonable and unacceptable based on my personal utility of money.
Again, changing the assumptions of the game. If any of you still talk to an econ professor, ask them from the original post if $1 is not the answer to the question, not any extra assumptions made after the post but from the original post. Creating a general utility theory of money only really comes into play in macro, never into game theory.
BizBanker Wrote: ------------------------------------------------------- > Again, changing the assumptions of the game. If > any of you still talk to an econ professor, ask > them from the original post if $1 is not the > answer to the question, not any extra assumptions > made after the post but from the original post. > Creating a general utility theory of money only > really comes into play in macro, never into game > theory. This is an honest question, not an attack on your answer, but aren’t you also changing the assumptions of the game by assuming that game show participants are going to behave as rational economists? I’m not that familiar with game theory, but I think the game’s participants would have to be rational for it to apply. I doubt 20 game show contestants would fit the bill unless they were taken from the econ department at Princeton.
higgmond Wrote: ------------------------------------------------------- > BizBanker Wrote: > -------------------------------------------------- > ----- > > Again, changing the assumptions of the game. If > > any of you still talk to an econ professor, ask > > them from the original post if $1 is not the > > answer to the question, not any extra > assumptions > > made after the post but from the original post. > > Creating a general utility theory of money only > > really comes into play in macro, never into > game > > theory. > > This is an honest question, not an attack on your > answer, but aren’t you also changing the > assumptions of the game by assuming that game show > participants are going to behave as rational > economists? I’m not that familiar with game > theory, but I think the game’s participants would > have to be rational for it to apply. I doubt 20 > game show contestants would fit the bill unless > they were taken from the econ department at > Princeton. No it is implicately assumed that participants behave rationally unless otherwise stated. Adding heterogenous utility functions would make this game more or less unsolvable by myself or any non PHD economist. Declining marginal utility is not even assumed in most basic game theory applications, though there is some evidence for it in TV game shows and it is assumed as a General Theory of individuals.
higgmond Wrote: ------------------------------------------------------- > BizBanker Wrote: > -------------------------------------------------- > ----- > > Again, changing the assumptions of the game. If > > any of you still talk to an econ professor, ask > > them from the original post if $1 is not the > > answer to the question, not any extra > assumptions > > made after the post but from the original post. > > Creating a general utility theory of money only > > really comes into play in macro, never into > game > > theory. > > This is an honest question, not an attack on your > answer, but aren’t you also changing the > assumptions of the game by assuming that game show > participants are going to behave as rational > economists? I’m not that familiar with game > theory, but I think the game’s participants would > have to be rational for it to apply. I doubt 20 > game show contestants would fit the bill unless > they were taken from the econ department at > Princeton. No it is implicitly assumed that participants behave rationally unless otherwise stated. Adding heterogenous utility functions would make this game more or less unsolvable by myself or any non PHD economist. Declining marginal utility is not even assumed in most basic game theory applications, though there is some evidence for it in TV game shows and it is assumed as a General Theory of individuals
Actually, the shape of your utility curve is irrelevant to this argument unless it is not strictly monotonic.
The only way I could see the utility curve issue coming in is if there was a potential that 19 participants had wealth and a marginal utility such that instead of $1
Professors answer to this question email cut and pasted below. Sorry, I guess Im a last word freak and have to justify the hell I spent in grad school: how are you? sorry for the delay ins response; things have been hectic. i think the strategy you mention is right. everyone bids 1 and no one gets any money. there is no mixed strategy, as can be seen because of the dominance. best, gary > Hi Professor. If I might pick your brain for a brief minute. I am > having a discussion with a friend about the actual strategy to the > following game. > > *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?* > > I said that backward induction would lead us to bidding a dollar as > any other bid is strongly dominated by X-1. Being so far out of school > and not actively studying game theory, I was hoping you might be so > kind as to offer your opinion. Time permitting of course, your input > would be greatly appreciated.
i agree with your own logic and explanation of backward induction, because it makes sense, not cause some professor says so. he might be the brightest guy but there are also tons of dumba$$ professors all over the place. proofs by credentials and authority don’t apply