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 2020-12-23, 00:40 #1 ZFR     Feb 2008 Meath, Ireland 2678 Posts A game theory paradox puzzle So while analysing Mafia end game situations, I came across an interesting one that leads to a nice paradox: a player's expected payoff increases if he chooses a set of prizes all of which are smaller. Consider the following simple game: A box contains 3 white balls and 1 golden ball: WWWG. Player 1 can make the following choice: Remove a white ball from the box OR Offer draw (in which case no balls are removed from the box). Plyer 2 can then make the following choice: If a draw was offered he may choose to accept it. The game ends in a draw. If no draw was offered, or if he rejects the offer, he randomly selects a ball from the box. If he draws the golden ball he wins, otherwise Player 1 wins. The prize is 1$for winning, 0$ for losing and x\$ for a draw, where x is in the range [0,1]. Question: Assuming all players act rationally, for which value of x should Player 1 offer a draw? So the solution I came up with is as follows: If player 1 removes a ball, his chances of winning are 2/3. Therefore, if x is larger than 2/3 he should offer draw, which he knows will be accepted. His expeted payoff will be x. If x is smaller than 2/3, but larger than 1/4 he should remove a white ball. He will than have 2/3 expected payoff. However if x is smaller than 1/4, then he should offer draw! Because he knows that Player 2 will reject the offer, and so he will have 3/4 chance of winning! Player 2 choice is simple: he's going to accept any offer where x is greater than 1/4 and reject otherwise to maximise his own expected payoff. So, Player 1 should offer draw if x ∈ (0, 1/4) ∪ (2/3, 1] Between 0 and 1/4 his expected payoff is 3/4. Betwwen 1/4 and 2/3, expected payoff is 2/3. Between 2/3 and 1, his expected payoff is equal to that value. Which makes a nice paradox. Suppose the organiser of the game comes to Player 1 and tells him: You can have one of those sets of prizes: 1 for winning, 0.5 for drawing. OR 0.9 for winning, 0.2 for drawing. Please choose one of them, the rules won't change otherwise. Then Player 1 should choose the second option to maximise his expected payoff, despite both values being smaller! In the first case, his expected payoff is 0.6666..... In the second case, his expected payoff is 0.9 x 0.75 = 0.675 I was wondering if someone could go over this problem and tell me if I made any mistakes. And also if this is an example of a known paradox in game theory, and if so what is it called? Last fiddled with by ZFR on 2020-12-23 at 00:44
2020-12-24, 22:28   #2
uau

Jan 2017

32·17 Posts

Quote:
 Originally Posted by ZFR Which makes a nice paradox.
I don't think there is anything particularly paradoxical here. This is not a zero-sum game unless x = 1/2. It's not surprising that changes which result in player 2 switching strategy may help player 1, whether those changes are generally in direction of smaller prizes or not.

 2021-01-05, 10:17 #3 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 2·47·109 Posts Say we play the following (roulette-style) game: You draw a circle using 2 colors, say red and green, it doesn't matter if the two colors are contiguous or not, you can make it red, green, red, green, etc, or just three quarters red and one quarter green, etc, up to you. So, if we draw an imaginary radius, it hits either red circle, or green circle, but not both. The proportion of red and green is up to you. Then we put a rotating arrow in the middle of the circle and play the following game: I put money on the table, you rotate the arrow. If the arrow stops on the color my money is, and the color is dominant (more than half of the circle has that color), I take back my money and you pay me the same amount of money. If the arrow stops on the color my money is, and the color is not dominant (half or less of the circle has that color), I take back my money and you pay me the double amount of money I did bet. If the arrow stops on the color my money is not, you take my money. We can use the same circle to play many games, or you can re-draw the circle for every spin, up to you, but you must use 2 colors each time, you can't draw a monochrome circle, and you have to be careful with the splitting (continuous or not, it doesn't matter). Clearly, if you split it half-half, you will always lose. How do you draw the circle to maximize your chances? Last fiddled with by LaurV on 2021-01-05 at 10:17
2021-01-05, 10:53   #4
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

11001111100012 Posts

Quote:
 Originally Posted by LaurV How do you draw the circle to maximize your [profit]?

2021-01-05, 21:30   #5
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

2·3·1,801 Posts

Quote:
 Originally Posted by LaurV If the arrow stops on the color my money is,
What color your money is is a big unknown.

2021-01-05, 23:11   #6
Dr Sardonicus

Feb 2017
Nowhere

6,101 Posts

Quote:
 Originally Posted by LaurV How do you draw the circle to maximize your chances?
You draw the circle so that if the arrow lands on green, your money is suddenly in 20-baht notes, and if it lands on red, your money is suddenly in 100-baht notes.

2021-01-05, 23:20   #7
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

2A3616 Posts

Quote:
 Originally Posted by Dr Sardonicus You draw the circle so that if the arrow lands on green, your money is suddenly in 20-baht notes, and if it lands on red, your money is suddenly in 100-baht notes.
Canadian 1's were green and the 2's were red.

 2021-01-06, 00:32 #8 Dr Sardonicus     Feb 2017 Nowhere 10111110101012 Posts Draw the circle so that you (claim you) are red-green colorblind. Then you (claim you) win every time.
2021-01-06, 03:39   #10
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

29·229 Posts

Quote:
 Originally Posted by LaurV ... the best play for you is to keep the crayons in your pocket.
Quote:
 Originally Posted by WarGames movie ... the only winning move is not to play.

2021-01-06, 13:40   #11
Dr Sardonicus

Feb 2017
Nowhere

6,101 Posts

Quote:
 Originally Posted by LaurV Haha, this thread unexpectedly took an odd turn, due to my bad English and phrasing The idea of my puzzle was to see how many of you think out of the box, and/or dream about "gaming the system"
I had no problem with your English or phrasing. In my case, there is another reason. (Tip o' the hat to retina): I merely assumed that his diagnosis "keep your crayons in your pocket" was correct.

And, consequently, that the only way to play and win was to cheat.

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