Voluntary contributions
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.
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.
Risposte
Personal comments under the spoiler.
EDIT: I put comments under spoiler tag.
EDIT: I put comments under spoiler tag.
Sorry, I will continue to leave aside my "explanations".
I would like to ask a question to participants (and newcomers, if they like).
Having seen the results of this game, which would be your choice in case you were again in front of the same game to play?
For the participants and newcomers: would you mind to explain the reasons behind your choice?
For the participants to the first "round": in case you change you choice, would you mind to give reasons for your change?
As before, I suggest to answer using the "Spoiler"
I would like to ask a question to participants (and newcomers, if they like).
Having seen the results of this game, which would be your choice in case you were again in front of the same game to play?
For the participants and newcomers: would you mind to explain the reasons behind your choice?
For the participants to the first "round": in case you change you choice, would you mind to give reasons for your change?
As before, I suggest to answer using the "Spoiler"
Scusate l'italiano, c'è un equivoco di fondo: gli amministratori amministrano parole (e nemmeno tanto bene visto quello che si dice in altre parti di questo forum) non soldi. 
I'm interested to know the factor to multiply the money in the pot in order that will be convinient for everyone to leave some money in the pot.
I'm interested to know the factor to multiply the money in the pot in order that will be convinient for everyone to leave some money in the pot.
"Admin":
c'è un equivoco di fondo
gli amministratori amministrano parole ... non amministrano soldi.
dammn!
"Admin":
A meno che non inventiamo una moneta virtuale.
Uhm, with virtual money one can pay, at best, only virtual players!
"Fioravante Patrone":
Having seen the results of this game, which would be your choice in case you were again in front of the same game to play?
"Fioravante Patrone":
Having seen the results of this game, which would be your choice in case you were again in front of the same game to play?
Forgive my english!
"Fioravante Patrone":
Having seen the results of this game, which would be your choice in case you were again in front of the same game to play?
For the participants and newcomers: would you mind to explain the reasons behind your choice?
@fields
@wedge
Very interesting answers.
I think that there is enough to debate for all of the summer!
I will start answering first the choices to the first round. Then I will move to comments, and eventually to the "second round".
Just to fix some coordinates, if we assume that preferences of the players:
- depend only on the money that they get
- they like more money than less,
then we are facing a "free riding" problem, that arises typically when contributions to some public project are considered.
Our "camallo" (alias cheguevilla) has correctly detected the category of problems to which the example belongs.
To make more "concrete" the problem we have discussed, you can think that you are asked to make a contribution for, e.g., building a bridge to cross a river that divides your city into two parts.
The bridge is a typical "public good", with the characteristic that its "consumption" (better, the "consumption" of the services that it gives) is neither rival nor excludable. Is not rival since the fact that you use the bridge does not prevent other people from using it. It is not excludable, since you will have the right to use the bridge, independently on the fat that you contributed to it or not.
[As the example of the bridge shows, the rivarly/non rivaly and excludability/non excludability are rarely 0/1 variables. Congestion could bring rivarly, while you could use a toll system for the bridge, anyway].
As a last word (for today) I subscribe also the connection that cheguevilla has made with the prisoner's dilemma.
I think that there is enough to debate for all of the summer!
I will start answering first the choices to the first round. Then I will move to comments, and eventually to the "second round".
Just to fix some coordinates, if we assume that preferences of the players:
- depend only on the money that they get
- they like more money than less,
then we are facing a "free riding" problem, that arises typically when contributions to some public project are considered.
Our "camallo" (alias cheguevilla) has correctly detected the category of problems to which the example belongs.
To make more "concrete" the problem we have discussed, you can think that you are asked to make a contribution for, e.g., building a bridge to cross a river that divides your city into two parts.
The bridge is a typical "public good", with the characteristic that its "consumption" (better, the "consumption" of the services that it gives) is neither rival nor excludable. Is not rival since the fact that you use the bridge does not prevent other people from using it. It is not excludable, since you will have the right to use the bridge, independently on the fat that you contributed to it or not.
[As the example of the bridge shows, the rivarly/non rivaly and excludability/non excludability are rarely 0/1 variables. Congestion could bring rivarly, while you could use a toll system for the bridge, anyway].
As a last word (for today) I subscribe also the connection that cheguevilla has made with the prisoner's dilemma.
Maybe you have to open a new topic
Let's move to the formal analysis of the game.
First of all, I will describe the game form (see: https://www.matematicamente.it/f/viewtopic.php?t=20428, sorry for the Italian...)
We have six players. The strategy space for each one is ${0,1,2,\ldots,20}$.
For sake of convenience, I will denote by $X_i$ the strategy space of player $i$. Of course, all of the $X_i$ coincide with ${0,1,2,\ldots,20}$.
As $E$ we can take $RR^6$, even if we could make more parsimonious choices.
The function $h$ is defined as follows. Since $h$ take values in $RR^6$, it will be convenient to describe it as $(h_1,\ldots,h_6)$, where $h_i$ are real-valued functions.
We have:
$h_i(x_1,\ldots,x_6) = 20 - x_i + \frac{1}{2} \cdot \sum_{j=1}^6 x_j$.
Notice that the game form is completely independent of any assumption about the preferences of the players. So, in particular, it is independent of what I said in my previous post about the "meaning" of the game we are discussing.
In the next post I will discuss the preferences of the players, so that we shall be able to say something more interesting, perhaps.
First of all, I will describe the game form (see: https://www.matematicamente.it/f/viewtopic.php?t=20428, sorry for the Italian...)
We have six players. The strategy space for each one is ${0,1,2,\ldots,20}$.
For sake of convenience, I will denote by $X_i$ the strategy space of player $i$. Of course, all of the $X_i$ coincide with ${0,1,2,\ldots,20}$.
As $E$ we can take $RR^6$, even if we could make more parsimonious choices.
The function $h$ is defined as follows. Since $h$ take values in $RR^6$, it will be convenient to describe it as $(h_1,\ldots,h_6)$, where $h_i$ are real-valued functions.
We have:
$h_i(x_1,\ldots,x_6) = 20 - x_i + \frac{1}{2} \cdot \sum_{j=1}^6 x_j$.
Notice that the game form is completely independent of any assumption about the preferences of the players. So, in particular, it is independent of what I said in my previous post about the "meaning" of the game we are discussing.
In the next post I will discuss the preferences of the players, so that we shall be able to say something more interesting, perhaps.
In the last post I have described the game form.
Of course, I should say "a game form", since it is my choice as a modeler to identify what I consider the correct mathematical objects to use.
However, in this case it can be considered that the game form is more or less(*) "obliged", apart from obvious and trivial arbitrarynesses. Of course, considering $E = QQ^6$ is just a sample of what I mean.
So, let's move to preferences.
If we assume, as I said in a couple of post before, that the preferences of each player:
- depend only on the money that they get
and that players:
- they like more money than less,
Than it is immediate to show that, given whatever choice of the remaining players, the choice of putting $0$ in the pot is the best choice; i.e. is a strongly dominant choice.
That is, $(0,\ldots,0)$ not only is a Nash equilibrium, but each of its component strategies are strongly dominant.
Exactly as in the prisoner's dilemma, we are facing a very compelling "solution".
And, exactly as for the prisoner's dilemma, the result is inefficient.
This is the source of the enormous interest of such "games". Please notice that:
- these games are "toy games", true. But they represent situations that we face quite often!
- we have not assumed that players are stupid, crazy, or have strange beliefs abot their opponents (or co-players, better...)
I wait for comments. Please don't be too nasty
(*)
I feel obliged to discuss a little bit this "triviality"...
"more or less", since there is always a nontrivial arbitrary choice: which are the truly consequences of the "game" that is being played? It is the nightmare that will wake up in the middle of the night Cozza Taddeo, who has seen half of his money disappear? Or should we consider one of the straight men that put zero in the pot and will not be able to survive the humiliation of having shown to be so greedy to all of the community of this forum? Or it will be the fact that I will be banned forever from this forum after a burning discussion with one of them?
This is a issue that should be well known to economists (I bet it isn't!). An economic agent is assumed to have preferences on his possible consumption acts. But the preferences are on the consequences of consumption not on the consumptive act "per se". So, which are the truly relevant consequences of eating one apple instead of a pear?
Of course, I should say "a game form", since it is my choice as a modeler to identify what I consider the correct mathematical objects to use.
However, in this case it can be considered that the game form is more or less(*) "obliged", apart from obvious and trivial arbitrarynesses. Of course, considering $E = QQ^6$ is just a sample of what I mean.
So, let's move to preferences.
If we assume, as I said in a couple of post before, that the preferences of each player:
- depend only on the money that they get
and that players:
- they like more money than less,
Than it is immediate to show that, given whatever choice of the remaining players, the choice of putting $0$ in the pot is the best choice; i.e. is a strongly dominant choice.
That is, $(0,\ldots,0)$ not only is a Nash equilibrium, but each of its component strategies are strongly dominant.
Exactly as in the prisoner's dilemma, we are facing a very compelling "solution".
And, exactly as for the prisoner's dilemma, the result is inefficient.
This is the source of the enormous interest of such "games". Please notice that:
- these games are "toy games", true. But they represent situations that we face quite often!
- we have not assumed that players are stupid, crazy, or have strange beliefs abot their opponents (or co-players, better...)
I wait for comments. Please don't be too nasty
(*)
I feel obliged to discuss a little bit this "triviality"...
"more or less", since there is always a nontrivial arbitrary choice: which are the truly consequences of the "game" that is being played? It is the nightmare that will wake up in the middle of the night Cozza Taddeo, who has seen half of his money disappear? Or should we consider one of the straight men that put zero in the pot and will not be able to survive the humiliation of having shown to be so greedy to all of the community of this forum? Or it will be the fact that I will be banned forever from this forum after a burning discussion with one of them?
This is a issue that should be well known to economists (I bet it isn't!). An economic agent is assumed to have preferences on his possible consumption acts. But the preferences are on the consequences of consumption not on the consumptive act "per se". So, which are the truly relevant consequences of eating one apple instead of a pear?
Our "camallo" (alias cheguevilla) has correctly detected the category of problems to which the example belongs.Ohh, surprise surprise, maybe sometimes longshoremen are not that bad.
I will keep myself away from the discussion, at least for this moment.
I only suggest to pay attention in making general theories from abstract examples; this theory ALWAYS works, but we have to consider some particulars:
- players informations level.
- players personal utility function.
Thus, keeping in mind that the second is strictly depending from the former, don't be surprised if, sometimes, a choice may appear "illogical".
In this sense, somebody could trust on the good faith of other players and put 20.
Somebody other could have his's own satisfaction in making the interest of the community before than the personal one.
This is a strong demonstration of how, in many occasions, a cooperating community is better than a community based on individualism and competition.
Obviously, this last sentence, is a purely personal consideration.
I only suggest to pay attention in making general theories from abstract examples; this theory ALWAYS works, but we have to consider some particulars:
- players informations level.
- players personal utility function.
Thus, keeping in mind that the second is strictly depending from the former, don't be surprised if, sometimes, a choice may appear "illogical".
In this sense, somebody could trust on the good faith of other players and put 20.
Somebody other could have his's own satisfaction in making the interest of the community before than the personal one.
This is a strong demonstration of how, in many occasions, a cooperating community is better than a community based on individualism and competition.
Obviously, this last sentence, is a purely personal consideration.
I should like to propose a variation in rules of game:
- At the beginning each player receives 20 € and expresses his choice in a reserved area with his keyboard.
- The money collected in the pot will be multiplied by $3$.
- Each player who will have put an inferior amount to the average, he will have to play the difference and one penalty equal to the average (supposing $S_i i=1,2,...,n$ the chosen sum by the player $i$ , $AA S_i
How would you play in this case?
- At the beginning each player receives 20 € and expresses his choice in a reserved area with his keyboard.
- The money collected in the pot will be multiplied by $3$.
- Each player who will have put an inferior amount to the average, he will have to play the difference and one penalty equal to the average (supposing $S_i i=1,2,...,n$ the chosen sum by the player $i$ , $AA S_i
How would you play in this case?
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