Re: observation selection effects
John M wrote: Dear Kory, your argument pushed me off balance. I checked your table and found it absolutely true. Then it occurred to me that you made the same assumption as in my post shortly prior to yours: a priviledge of "ME" to switch, barring the others. I continued your table to situations when the #2 player is switching and then when #3 is doing it - all the way to all 3 of us did switch and found that such extension of the case returns the so called 'probability' to the uncalculable (especially if there are more than 3 players) like a many - many body problem. Cheers John Why would it matter if the other players switch? Based on the description of the game at http://tinyurl.com/4oses I thought the "winning flip" was determined solely by what each player's original flip was, not what their final bet was. In other words, if two players get heads and the other gets a tails, then the winning flip is automatically tails, even if the two players who got heads switch their bet to tails. Assuming this is true, it's pretty easy to see why it's better to switch--although it makes sense to say the winning flip is equally likely to be heads or tails *before* anyone flips, seeing the result of your own coinflip gives you additional information about what the winning flip is likely to be. If I get heads, I know the only possible way for the winning flip to be heads would be if both the other players got tails, whereas the winning flip will be tails if the other two got heads *or* if one got heads and the other got tails. Jesse
Re: observation selection effects
At 02:57 PM 10/10/2004, John M wrote: Then it occurred to me that you made the same assumption as in my post shortly prior to yours: a priviledge of "ME" to switch, barring the others. I think this pinpoints one of the confusions that's muddying up this discussion. Under the Flip-Flop rules as they were presented, the Winning Flip is determined before people switch, and the Winning Flip doesn't change based on how people switch. In that scenario, my table is correct, and there is no paradox. We can also consider the variant in which the Winning Flip is determined after people decide whether or not to switch. But that game is functionally identical to the game where there is no coin-toss at all - everyone just freely chooses Heads or Tails, then the Winning Flip is determined and the winners are paid. Flipping a coin, looking at it, and then deciding whether or not to switch it is identical to simply picking heads or tails! The coin-flips only matter in the first variant, where they determine the Winning Flip *before* people make their choices. In this variant, it doesn't matter whether you switch or not (i.e. whether you choose heads or tails) - you are more likely to lose than win. We can use the same 3-player table we've been discussing to see that there are eight possible outcomes, and you only win in two of them. Once again, there's no paradox, although you might *feel* like there is one. You might reason that the Winning Flip is equally likely to be heads or tails, so no matter which one you pick, your odds of winning will be 50/50. What's missing from this logic is the recognition that no matter what you pick, your choice will automatically decrease the chances of that side being in the minority. -- Kory
Re: observation selection effects
At 04:47 PM 10/10/2004, Jesse Mazer wrote: If I get heads, I know the only possible way for the winning flip to be heads would be if both the other players got tails, whereas the winning flip will be tails if the other two got heads *or* if one got heads and the other got tails. I agree with this, but I want to add a subtle point: it's correct to switch *even if I haven't looked at my own coin*. That's because, despite the fact that I don't know whether my own coin is heads or tails, I know that, whichever it is, it's more likely to be in the majority than the minority. That's not to say that *nobody* needs to look at my coin. In order to determine whether or not my choice to "switch" puts me in the heads or the tails group, *someone's* going to have to look at my coin. But the rules of the game allow me to pass off this act of looking to someone else - they essentially allow me to tell the casino worker "hey, take a look at my coin, will ya, and assign me to the opposite". The important point is that I can safely pass off this instruction to switch without even knowing the result of my coin-flip, because I know that, whatever my coin is, it's more likely to be in the majority group. If we change the rules of the game slightly, and say that, instead of choosing whether or not to "switch", you have to actually choose "heads" or "tails", then, of course, you yourself do need to see the result of your own coin-flip. -- Kory
re: observation selection effects
You're right, as was discussed last week. It seems I clicked on the wrong thing in my email program and have re-sent an old post. My apologies for taking up the bandwidth! --Stathis From: Kory Heath <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Subject: re: observation selection effects Date: Sat, 09 Oct 2004 18:17:50 -0400 At 10:35 AM 10/9/2004, Stathis Papaioannou wrote: From the point of view of typical player, it would seem that there is not: the Winning Flip is as likely to be heads as tails, and if he played the game repeatedly over time, he should expect to break even, whether he switches in the final step or not. That's not correct. While it's true that the Winning Flip is as likely to be heads as tails, it's not true that I'm as likely to be in the winning group as the loosing group. Look at the case when there are only three players. There are eight possible outcomes: Me: H Player 1: H Player 2: H - WF: T Me: H Player 1: H Player 2: T - WF: T Me: H Player 1: T Player 2: H - WF: T Me: H Player 1: T Player 2: T - WF: H Me: T Player 1: H Player 2: H - WF: T Me: T Player 1: H Player 2: T - WF: H Me: T Player 1: T Player 2: H - WF: H Me: T Player 1: T Player 2: T - WF: H I am in the winning group in only two out of these eight cases. So my chances of winning if I don't switch are 1/4, and my chances of winning if I do switch are 3/4. There's no paradox here. -- Kory _ Enter our Mobile Babe Search and win big! http://ninemsn.com.au/babesearch
Re: observation selection effects
At 07:17 PM 10/10/2004, Kory Heath wrote: We can also consider the variant in which the Winning Flip is determined after people decide whether or not to switch. In a follow-up to my own post, I should point out that your winning chances in this game depend on how your opponents are playing. If all of your opponents are playing randomly, then you have a negative expectation no matter what you do. If your opponents are not playing randomly, then you may be able to exploit patterns in their play to generate a positive expectation. -- Kory
The FLip Flop Game
In one version of Flip Flop, each of an odd number of players simply flips a coin. The majority result, heads or tails, pays the casino $1 each while the minority result gets paid $2 each. Based on these rules, I worked out Kory's tables for 3, 5, 7 and 9 players. The results show that the player's expectation changes according to how many players there are. For example, if there are 3 players then the long-term odds are that each game costs each player 25 cents. If there are 5 players, the average cost goes down to 6.3 cents per game. If there are 7 players, they make on the average 3.1 cents per game. If there are 9 players they make about 9 cents per game. It isn't clear to me why this should be so. Norman - Original Message - From: "Kory Heath" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Sunday, October 10, 2004 5:32 PM Subject: Re: observation selection effects At 07:17 PM 10/10/2004, Kory Heath wrote: >We can also consider the variant in which the Winning Flip is determined >after people decide whether or not to switch. In a follow-up to my own post, I should point out that your winning chances in this game depend on how your opponents are playing. If all of your opponents are playing randomly, then you have a negative expectation no matter what you do. If your opponents are not playing randomly, then you may be able to exploit patterns in their play to generate a positive expectation. -- Kory
