Is Game Theory Applicable Here?

A friend of mine and I were discussing a fairly simple game: you are presented the opportunity to take either $240 straight up or to play a game where 25% of the time you will win $1,000. You may only play the game once.

My friend aruged that the expected value for the game is $250 if you take the shot at $1000 and only $240 if you do not, and therefore you should take the chance. I completely understand, but...

Seeing as it is a one time event, I argued that it was not necessarily advantageous to play for the long run. Therefore, it is not unreasonable to sacrifice $10 in expected value in order to play a game you cannot lose rather than one that you are a huge underdog to win.

I understand that one's decision might be influenced by how much money they have to start, or if the game had different values (I feel most people would rather take 240K straight up than take a 1-in-4 on $1 million, even if it is mathematically incorrect), but in general is my logic faulty?

[pzhon]

/images/graemlins/diamond.gif Game theory is completely irrelevant.

/images/graemlins/diamond.gif EV makes sense even for events that only happen once. Many people find this counterintuitive.

/images/graemlins/diamond.gif It is ok to have a nonlinear value for money, which can lead you to being rationally risk averse. If the value you place on having $x is log x, you should accept the the gamble if you have at least $9038.19, and decline if you have less.

/images/graemlins/diamond.gif "it is not unreasonable to sacrifice $10 in expected value in order to play a game you cannot lose rather than one that you are a huge underdog to win" sounds overly emotional, and does not easily fit into a consistent system for analyzing gambles. Many people are willing to pay for short-term certainty. This is an easily exploited weakness in poker.

I'm not too familiar with game theory, that's just what he called it. Sorry if i erred.

Your last point about exploting my idea as a weakness in poker makes sense, but in poker you can consistently make this decision over and over again and play for the long term. I would not even bring up this question in a poker game, my question was about a one time scenario.

Thanks for the other stuff though.

[ThinkQuick]

pzhon's summary is great

you can use the search function to find other similar threads.. the last one I remember was 'deal or no deal' I think.

He isn't saying that you are a weak poker player. He's agreeing with you in fact that its ok to sacrifice some EV for guaranteed money if the money means something to you. I personally may not give up any EV for only 250/240 bucks, it may have to be more.
now take this lesson and ensure that you don't play above your bankroll in order to allow you to keep pushing all edges

[AaronBrown]

In addition to phzon's excellent reply, I would add that you have to be careful once you deviate from maximizing expected value. It can be rational to deviate, but most people wander into inconsistency and error when they do. Here is one famous example known as Allais' paradox (for which Maurice Allais won the 1988 Nobel Prize). The poker adaptation in my own.

(A) You’re at the final table of a Poker tournament with two other entrants left. There is a $2,500,000 first prize, $500,000 second prize but no third prize. You have the middle stack, the woman on your right has ten times your stack, the guy on your left is down to a chip and a chair. You think there is a 10% chance you will win, an 89% chance you will take second and a 1% chance you will take third. The other players offer a split. You get $500,000. The chip leader gets $2,500,000 and will compensate the short stack out of that. Do you take the split?

(B) Same tournament and prizes, but you now have the short stack. You figure you have no chance at all to win, an 11% chance of picking up the $500,000 and 89% chance of getting nothing. The chip leader offers to settle for second place, taking $500,000 and her chips off the table. The middle stack says he’ll do it if you give up 10% of your chips, then play out for first place or nothing. With this deal, you figure to have a 90% chance of ending up with nothing, and a 10% chance of winning $2,500,000.

First answer honestly what you would do in each situation, then look more closely and I'll bet you've made completely inconsistent decisions in the two cases.

[WhiteWolf]

[ QUOTE ]
In addition to phzon's excellent reply, I would add that you have to be careful once you deviate from maximizing expected value. It can be rational to deviate, but most people wander into inconsistency and error when they do. Here is one famous example known as Allais' paradox (for which Maurice Allais won the 1988 Nobel Prize). The poker adaptation in my own.

(A) You’re at the final table of a Poker tournament with two other entrants left. There is a $2,500,000 first prize, $500,000 second prize but no third prize. You have the middle stack, the woman on your right has ten times your stack, the guy on your left is down to a chip and a chair. You think there is a 10% chance you will win, an 89% chance you will take second and a 1% chance you will take third. The other players offer a split. You get $500,000. The chip leader gets $2,500,000 and will compensate the short stack out of that. Do you take the split?

(B) Same tournament and prizes, but you now have the short stack. You figure you have no chance at all to win, an 11% chance of picking up the $500,000 and 89% chance of getting nothing. The chip leader offers to settle for second place, taking $500,000 and her chips off the table. The middle stack says he’ll do it if you give up 10% of your chips, then play out for first place or nothing. With this deal, you figure to have a 90% chance of ending up with nothing, and a 10% chance of winning $2,500,000.

First answer honestly what you would do in each situation, then look more closely and I'll bet you've made completely inconsistent decisions in the two cases.

[/ QUOTE ]

Am I missing something - this seems trivially easy?

Situation A - no deal (EV of 695000) beats out deal (EV of 500000).

Situation B - deal (EV of 250000) beats out no deal (EV of 55000).

[Chipp Leider]

You might wanna look into Certainty Equivalent.

[ohnonotthat]

Aaron, or is this simply a complex (or maybe not so complex) case of utility theory ?

I'm going to now read all the responses - both to the original thread and to your tourney settlement question - but before doing so I'll state that I've taken the worst of it many times in order to minimize the chance of disaster.

I posess the resources, albeit just barely /images/graemlins/frown.gif, to post a bond in lieu of paying for liability coverage on my car but I'd never entertain the thought of actually doing so.

[ohnonotthat]

It IS trivially easy unless your girlfriend/fiance/wife is standing over you with that "WTF are you thinking /images/graemlins/mad.gif" look in her eye.

If you've ever seen that look you know NOTHING is trivial when facing it. /images/graemlins/grin.gif

[ohnonotthat]

got a link to that ?

It's likely to be over my head but I can dream of the day I'll understand it. /images/graemlins/confused.gif

[AaronBrown]

It's trivially easy if you make decisions by expected value, but most people don't. Almost everyone accepts both deals. This has been measured by poll and by real choices, in some cases with significant amounts of money at stake. It has been verified in many cultures and situations.

In A, you give up a 10% chance to make $2 million (converting $500,000 to $2,500,000) in order to eliminate a 1% chance of losing $500,000 (going from $500,000 to zero).

In B, you give up a 1% chance of getting $500,000 instead of zero, in order to get a 10% chance of getting $2,500,000 instead of $500,000.

Here is a good link (http://mathworld.wolfram.com/AllaisParadox.html) to the paradox (without my poker gloss).

My point is that if you don't use expected value to make decisions, it's easy to find yourself making inconsistent decisions.

I think the poker example is not so good for this board because people here tend to think in EV terms at least wrt poker. "Who Wants to be a Millionaire" provides a good example of Allais' Paradox. Suppose you are at the $32000 question and use your last lifeline the 50/50 giving you 2 possible choices. You have absolutely no clue which one is the correct answer. You can walk away with a guaranteed $16000 or can take a guess where 50% of the time you win 32000 and 50% of the time you win $1000. Guessing has a higher expected value especially if you consider that a correct guess leads to more money potential. Yet, how often do people walk away with the $16,000. And how often would you walk away instead of guess.

The Allais "paradox" isn't an example of people behaving incorrectly, it's an example of a problem with the standard economic way of analyzing decisions under uncertainty (ie assume a concave utility function to account for risk aversion). Realisticly, people aren't bankrolled for life. You're going to have to deviate from maximizing EV once you get outside of the realm of poker, and it's not entirely clear what the best way to do that is. I'm not comfortable with calling something that occurs so frequently as the preference reversal problem (Allais paradox) a reasoning error.

[ebroil]

No, this is not an example of Allais paradox. This is an example of utility theory. People frequently makes choices that have a smaller EV because their marginal utility for money increases at a decreasing rate. That is, having 16,000 for sure has a greater utility than having a possibility of 32,000, even though that option has a (slightly) higher EV.

Allais paradox occurs when people make choices that are inconsistent with expected utility theory. The example you provided is simply an example of utility theory.

This is correct. The first example appeared to be a correct version of the "paradox" though I didn't read it very carefully. The original question is appropriately answered by utility theory, as is the millionair question. I'd certainly take the 240 straight up rather than the 1/4 chance at 1000 mostly because 240 is a lot of money in graduate school. My main point is that it's not entirely clear that people who exhibit preference reversals are making inconsistent decisions, just decisions inconsistent with expected utility theory (specifically the independence axiom).

[AaronBrown]

Now we have gone full circle, as happens so often in message boards. The original post asked if game theory could justify sacrificing expected value. phzon gave an excellent reply explaining it had nothing to do with game theory, it was utility theory that sometimes argued for accepting less than the maximum expected value.

I seconded that post, but added you have to be careful deviating from maximum expected value as a criterion. It is quite easy to construct examples in which people make decisions that conflict with utility theory, Allais' Paradox is one simple way. No one said the original question was Allais' Paradox, or any paradox at all.

So I think we all agree. People often sacrifice expected value. While there are some simple explanations of why they might do so, the explanations do not correlate well with real human behavior. In fact, it's easy to find behavior that does not correlate with any imaginable rational calculation. So if you're tempted to sacrifice expected value, it's prudent to examine the decision closely before you make it.

Right. I agree with what you're saying. And I agree that people ought to closely examine why they are deviating from maximizing EV. I just find it moderately distressing when some economists point to the Allais paradox as an example of people behaving incorrectly simply because it deviates from our theories of what they ought to do. I think it was a good thing for you to bring up the Allais paradox so that people are careful when they think about these issues, I just wanted to provide a different perspective that preference reversals aren't necessarily wrong, just incosistent with our theories.