Voluntary contributions

[Fioravante Patrone1]

You are going to play the following game.

You are one among a group of six players. The players (you included) don't know each other. Moreover, you are put in a cubiculum with a screen in front. Communication will take place only via the screen and keyboard.
At the end of the game, everyone will leave independently, without seeing each other. So, not only you did not know each other before the game, but the same will be true also after you have played.

Everyone receives 20 euro.
He can decide how much of these euro to put in a common pot. The money that he will not put in the pot will remain his.
The decision will be communicated via keyboard, without knowing the choices made by the other five participants.
All of the money collected in the pot will be multiplied by three and then divided evenly among the six participants.

How much do you put in the common pot? (The answers allowed must be an integer number ranging from 0 to 20, extremes included)


Please DO NOT ANSWER before Sunday at noon. Leave room to potential participants to ask questions, further information.
Please answer using the "Spoiler" to allow participants to give their answer without knowing your choice.

[wedge]

dear professor Patrone, how does the keyboard fit in the game? can everyone write everything on it?
what's the meaning of communicating the decision via keyboard if everyone has no access to others' choices?

[Fioravante Patrone1]

"wedge":dear professor Patrone

"wedge":how does the keyboard fit in the game? can everyone write everything on it?
what's the meaning of communicating the decision via keyboard if everyone has no access to others' choices?

the main point is that each one of the six players will express his choice privately and in an anonymous way
the secrecy of his choice is guaranteed by the fact that it will be typed by him on a keyboard (so that he cannot be identified by the other players)
every player will have a reserved space (like cabins for elections) with his keyboard
the choices of every one will be recorded by a computer, so that the (small!) computations needed to find how much has earned each of the players

[wedge]

thanks, everything is clear to me now (except the solution)

is heavy-combinatorics required?

[Fioravante Patrone1]

you "must" choose an integer number between 0 and 20

that you base your choice on:
- heavy combinatorics
- gut feelings
- bright intuitions and powerful theorems
- how many birds you saw today
is up to you!

[fields1]

Do we have to justify our answer? Maybe it's better we do not explain our choice to avoid influence on the other players...

[wedge]

Here I am!

It's a matter of generosity and trust. If I leave x in the pot, I lose x/2 but everyone else gains x/2. The difference between my money before and my money after the game is in fact:

D = -x/2 + 5/2 y

where y is the average of others' contributions.
If I were extremely egoistic I would think "I'll put 0 in the pot, so I'll be sure of not losing anything"
But obviously if everyone puts 0, there's no possibility of profit.
So, do I trust other people to give me something? I suppose that everyone has an attitude similar to mine, so x is similar to y. Maybe that's not perfectly true, so I suppose that y may range from x-2 to x+2.
My aim is to gain at least 5. So, in the worst case where y=x-2,

D = -x/2 + 5/2 x-2 = 5

It means x=5. I'll put 5 in the pot. If each other player puts more than 3 in the pot, I'll gain more than 5. If I got wrong with everything and everyone deals with egoism, I'll lose only 2.5. In order not to lose money I just need each of them to put 1. I think a more negative conclusion is so remote! This strategy ensure me a gain of about 5 with a risk of 2.5

Fioravante, please leave a comment (spoiler-way). I'm curious!

edit: errore del mio English! non ho cambiato la risposta

[Fioravante Patrone1]

"fields":Do we have to justify our answer? Maybe it's better we do not explain our choice to avoid influence on the other players...

no, you don't have to justify you answer
but I strongly suggest that you do, so that the thread will be more interesting

please leave your answer and justifications (if any) hidden in the spoiler

[Fioravante Patrone1]

"wedge":
Fioravante, please leave a comment (spoiler-way). I'm curious!

I will comment extensively on the answers received. But later...
Enjoy the remaining part of this holiday!

[Kroldar]

Answer: $0$
In my opinion, the problem is to find a probability distribution for the other players. But I don't anything about the other players so I can't guess the probability distribution I would need. In this case, the maxmin strategy could be a good choice. I won't be wondered if also the other players think the same thing... in fact the combination of strategies $(0,0,0,0,0,0)$ is a Nash equilibrium for the game.

1 - wedge
2 - Kroldar
....
We need 4 other players

[fields1]

The best strategy is to put in the pot 0 euro.

Let us prove this assertion. Since other players' choices are independent of my choice (they cannot know it by the rules of the game), I can assume they have put in the common pot N euro. Let x be the money I put in the pot. Then I win N/2+20-x+x/2=N/2+20-x/2 euro. So the best is to choose x=0.

[cozzataddeo]

Here is my choice!

I put all 20 euro in the pot.
Some considerations:
- if each player puts x euro in the pot, each one receives 3*x euro after the game, therefore the best way to increase money is to put all in the pot, hoping that the other players are aware that this is the best choice (if they are rational, they should see that...);
- if the amount was more important (such as 1000 euro or more) I would never put all in the pot because the risk would be too high for me. Probably I would put only about 200 euro.

Have a good game!

[wedge]

two more players! let's go guys!

[Cheguevilla]

Here is my choice:

1

[_luca.barletta]

0

[cozzataddeo]

I understand that 0 is the Nash equilibirum but it's not the optimal choice. If the aim is increasing the amount of money, all can reach this goal by putting all 20 euro in the pot (payoff of 60 euro!!!).
On the oppostite, in this way each one has gained only 10 euro (while I lost them, damn!!! )...
If this wasn't a mathematical forum but, for instance, a literature or politic one, I think that people would have put much more money in the pot (disregarding completely Nash equilibrium) and each of them would have a greater payoff (this is only a supposition of mine, but I believe in it...).
Maybe too much reasoning doesn't lead always to the best result...

P.S.: I'm sure that it is evident, but I am completely ignorant in this matter...

[wedge]

I agree with CozzaTaddeo.
As I wrote (0,0,0,0,0,0) is the optimal choice for one who wants not to lose money.
If everyone wants to gain something, everyone gotta risk! That's the meaning of my answer. Maybe the strategy was ingenous and based on an assumption (each player wants to gain), but I tried to look for and justify a "third way" between 0 and the total risk.
Bye!

[Fioravante Patrone1]

PLAYER	NAME	    CHOICE FROM POT TOTAL FINAL MONEY FOR EACH PLAYER	
1	     wedge		    5	  13		28	
2	     Kroldar	     0	  13		33	
3	     fields		   0	  13		33	
4	     Cozza Taddeo  20	  13		13	
5	     Cheguevilla 	1	  13		32	
6	     Luca Barletta  0	  13		33	
	      POT:		    26			   172	TOTAL MONEY

the results are listed above

before going on, please let me know whether I have made any kind of mistake, or ask fo clarifications in case the table is not as clear as I believe...

[Kroldar]

I have a proposal: let's ask the administrators to give us the money we need, in order to play the game really!!

[wedge]

your table is ok, Fioravante

good proposal, Kroldar!

[Fioravante Patrone1]

"Kroldar":I have a proposal: let's ask the administrators to give us the money we need, in order to play the game really!!

Actually, money availability is a severe constraint that does not allow to make as much experimentation as it would be needed!!!

For science's sake, of course