Two Plus Two Poker Forums What is the optimal strategy for this game?
 Register FAQ Search Today's Posts Mark Forums Read Video Directory TwoPlusTwo.com

 Notices

 Science, Math, and Philosophy Discussions regarding science, math, and/or philosophy.

 07-29-2012, 01:42 PM #16 Carpal \'Tunnel   Join Date: Dec 2003 Posts: 6,124 Re: What is the optimal strategy for this game? If A randomizes his coins, a, and his totals guess, x = a,a+1,a+2,a+3, then he wins 1/4 of the time. But I think B can then do better. If B sees A has guessed x=6 then B knows A's number is 3. B then knows what to guess and wins 3/4 of the time (whenever B's number is not 3). If B sees x=5 then B knows A's number is not 0 or 1. If B sees x= 0 then B knows A's number is 0. If B sees x=1 then B knows A's number is not 2 or 3. A's guess gives information to B for all guesses other than 3. PairTheBoard
 07-29-2012, 02:34 PM #17 grinder   Join Date: Dec 2007 Posts: 421 Re: What is the optimal strategy for this game? Ooops. I overlooked something. I'll recheck and post. Last edited by Rhaegar; 07-29-2012 at 02:42 PM.
 07-29-2012, 02:48 PM #18 grinder   Join Date: Dec 2007 Posts: 421 Re: What is the optimal strategy for this game? OK, I was right the first time. For a second I thought I had overlooked something, when I had actually factored it in. First I'll make a note to myself that there are 4 combinations of coin choices for each player and the 16 total combinations are : 0+0 = 0 (1 comb for 0) 0+1 = 1+ 0 = 1 (2comb for 1) 0 + 2 = 2+ 0 = 1 + 1 (3comb for 2) 0 + 3 = 3 + 0 = 1+ 2 = 2 + 1 (4comb for 3) 1 + 3 = 3 + 1 = 2 + 2 (3comb for 4) 2 + 3 = 3 + 2 (2comb for 5) 3 + 3 (1comb for 6) If both players randomize equally between 0,1,2 and 3, playerA is justified in guessing 3 with 1/4 certainty. Can player B randomize in such a way as to make 3 less possible if A is randomized? NO. The chance is always 1/4 if player 1 randomizes perfectly. Then.. Player B has to pick between 2 and 4 with 3/16 chance. Can either player improve on this? I'd say no. If player A chooses 2 or 4 he not only sacrifices 4/16 for 3/16, he also allows player B to choose 3 for his gain. There is absolutely no gain by picking any particular number of 0 1 2 or 3, as long as you randomize. If you take a look you'll notice that in the calculations for the sum total of 3, the numbers 0 1 2 and 3 appear each 2 times, so there is no difference. Edit: I had missed one thing - if B has picked 0 he has to pick 2, if he has picked 3 he has to pick 4. FINAL ANSWER : Optimal strategy is to randomize the initial pick perfectly for both players and for A to choose always 3 and for B to always choose 2 or 4 (and if he has chosen 0 or 3 he has no such choice). There is no mixed strategy involved, since initial picks don't influence the sum total. P.S. That's one ****ty game. P.P.S. I can't believe how slow you guys are. Only EvilSteve kind of solved it, but erroneously enough for me to claim being the first. If I find any of you have picked on my latest thread you'll never hear the end of it! Last edited by Rhaegar; 07-29-2012 at 03:12 PM.
07-29-2012, 03:15 PM   #19
grinder

Join Date: Dec 2007
Posts: 421
Re: What is the optimal strategy for this game?

Actually, now that I'm rereading what EvilSteve said, he is quite wrong.

Quote:
 Originally Posted by Rikers on random choices guessing 3 has a 4/16 chance and any other pick has less then that so player A should always say 3. Player A has a first move advantage over player B since Player B has to say a different score that obv. has a smaller chance to win. ok, I just saw I overlooked that you know your hand... :P
This, "knowing your hand" thing tilted me hard, because I disproved its significance immediately and then thought I hadn't for quite some time. Man, am I stupid.

Last edited by Rhaegar; 07-29-2012 at 03:40 PM.

 07-29-2012, 03:31 PM #20 grinder   Join Date: Dec 2007 Posts: 421 Re: What is the optimal strategy for this game? I should add one more thing. This is an optimal solution. This means that we assume no knowledge of opponent's frequencies. If we have sufficient knowledge of opponent's tendencies we can pick the number immediately. A certain mathematical formula can be specified which would judge whether you go for optimal or for a guess. I'm not going to do that (I'm not even sure I could; I'm way too rusty) If there's a 26-25-25-24 distribution, you still go optimal. If there's 50-50-0-0 distribution and you have the zeroes giving you your optimal number, then you pick one of the 50s.
07-29-2012, 06:08 PM   #21
Carpal \'Tunnel

Join Date: Dec 2003
Posts: 6,124
Re: What is the optimal strategy for this game?

Quote:
 Originally Posted by Rhaegar OK, I was right the first time. For a second I thought I had overlooked something, when I had actually factored it in. First I'll make a note to myself that there are 4 combinations of coin choices for each player and the 16 total combinations are : 0+0 = 0 (1 comb for 0) 0+1 = 1+ 0 = 1 (2comb for 1) 0 + 2 = 2+ 0 = 1 + 1 (3comb for 2) 0 + 3 = 3 + 0 = 1+ 2 = 2 + 1 (4comb for 3) 1 + 3 = 3 + 1 = 2 + 2 (3comb for 4) 2 + 3 = 3 + 2 (2comb for 5) 3 + 3 (1comb for 6) If both players randomize equally between 0,1,2 and 3, playerA is justified in guessing 3 with 1/4 certainty. Can player B randomize in such a way as to make 3 less possible if A is randomized? NO. The chance is always 1/4 if player 1 randomizes perfectly. Then.. Player B has to pick between 2 and 4 with 3/16 chance. Can either player improve on this? I'd say no.[/B]

I think B can do better than 3/16 against A's strategy. B can still randomize his coins. But if he sees his coin number is 0 he guesses 2, with 1/4 chance A's coin number is 2. If B sees his coin number is 3 he guesses 4 with 1/4 chance A's coin number is 1. If B sees his coin number is 1 or 2 he can guess either 2 or 4 with 1/4 chance A's coin number is what he needs to be right.

PairTheBoard

 07-29-2012, 06:33 PM #22 grinder   Join Date: Dec 2007 Posts: 421 Re: What is the optimal strategy for this game? Ah. Wait a minute. Yes, I was wrong. A must choose 3, because he doesn't give up any information doing so. B can choose any of 3 options and they all have same value. I feel like an idiot now. So, there's plenty of room for leveling. A must call 3 as to not give up information, but B can play RPS and try to guess in any of his 4 spots. Yup, I'm an idiot. It's been confirmed and now official. At least I'm content that everyone else felt confused by it. I guess I got mislead by the overall numbers, to miss something so obvious. Last edited by Rhaegar; 07-29-2012 at 06:49 PM.
07-31-2012, 08:57 AM   #23
old hand

Join Date: Jul 2007
Location: Losing at Omahaha
Posts: 1,483
Re: What is the optimal strategy for this game?

Quote:
 Originally Posted by Rhaegar P.P.S. I can't believe how slow you guys are. Only EvilSteve kind of solved it, but erroneously enough for me to claim being the first. If I find any of you have picked on my latest thread you'll never hear the end of it!
What is wrong about my answer (1st on the page) and the others similar to it?

Quote:
 Originally Posted by PairTheBoard I think B can do better than 3/16 against A's strategy. B can still randomize his coins. But if he sees his coin number is 0 he guesses 2, with 1/4 chance A's coin number is 2. If B sees his coin number is 3 he guesses 4 with 1/4 chance A's coin number is 1. If B sees his coin number is 1 or 2 he can guess either 2 or 4 with 1/4 chance A's coin number is what he needs to be right.
I fail to see how B improves on his 1/4 chance of winning?

07-31-2012, 12:30 PM   #24
Carpal \'Tunnel

Join Date: Mar 2006
Location: party fight subchampion
Posts: 12,061
Re: What is the optimal strategy for this game?

Well first Rhaegar thought I "kind of solved it", then decided I was "quite wrong", but apparently never noticed that I was using a different interpretation of the rules and therefore wasn't addressing the same problem at all.

Quote:
 Originally Posted by EvilSteve I need clarification on the rules. Do you have to guess the exact number to win and there's no winner if neither player guessed correctly, or do you win by guessing closer to the actual number than your opponent (and a tie if you're both equally close)? For the following I'll assume the win condition is being the closest.
But yeah, I'm pretty sure acehole60 gave the correct strategy in the first response, based on the consensus rule set where there's no winner unless somebody guesses the total exactly.

 07-31-2012, 01:19 PM #25 Pooh-Bah     Join Date: Aug 2006 Location: you got it Posts: 4,006 Re: What is the optimal strategy for this game? Apologies for the ambiguity in the OP. Banzai knows more about the game than me, so I hope Post #15 cleared this up.
 07-31-2012, 05:21 PM #26 old hand   Join Date: Jun 2007 Posts: 1,557 Re: What is the optimal strategy for this game? Post #2 in the thread showed that A can guarantee doing as well as B. It remains to show that B can do as well as A. This is very unsurprising, but one explicit strategy for B to complete the proof of the "solution" of the game is: Choose 0,1,2,3 coins uniformly at random. (This means A can't win more than 1/4). Then (a) if A doesn't choose 3, then guess 3 without having to bother to look how at many coins you chose (b) if A does choose 3, then look at how many coins you chose and if you chose 0 guess 0; if you chose 1 guess 2; if you chose 2 guess 4; if you chose 3 guess 6. This gives B precisely a 1/4 chance of winning regardless of A's strategy - even if A has not chosen uniformly at random. Thus the solution is the game is equally fair to both players.
 07-31-2012, 05:37 PM #27 journeyman   Join Date: Sep 2004 Posts: 360 Re: What is the optimal strategy for this game? OP stated the game is played over several turns. If Player A detects player B is not at random choosing his 3 numbers: I guess player B becomes vulnerable for exploitation. So I guess Player B has to choose both coin and numbers at random: Coin = 0: 0,1,2 Coin = 1: 1,2,4 Coin = 2: 2,4,5 Coin = 3: 4,5,6 Player B should play: 1/12 -> 0 2/12 -> 1 3/12 -> 2 3/12 -> 4 2/12 -> 5 1/12 -> 6
 07-31-2012, 06:10 PM #28 journeyman   Join Date: Sep 2004 Posts: 360 Re: What is the optimal strategy for this game? Probably there are more optimal strategies for Player B (cfr Pokerfarian).
08-01-2012, 03:32 AM   #29
Carpal \'Tunnel

Join Date: Dec 2003
Posts: 6,124
Re: What is the optimal strategy for this game?

Quote:
 Originally Posted by acehole60 What is wrong about my answer (1st on the page) and the others similar to it? I fail to see how B improves on his 1/4 chance of winning?

Rhaeger was saying B could only do 3/16 so my reply to him was to show B could do 1/4.

I didn't understand what you were saying when I first read your first post. Having now gotten up to speed a little better I can see your first post made perfect sense and was right on.

PairTheBoard

 08-01-2012, 12:39 PM #30 grinder   Join Date: Dec 2007 Posts: 421 Re: What is the optimal strategy for this game? Yes, I was comically (tragically) wrong.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are Off Pingbacks are Off Refbacks are Off Forum Rules

All times are GMT -4. The time now is 03:07 PM.

 Contact Us - Two Plus Two Publishing LLC - Privacy Statement - Top