Excreting gyan
P A R T 2 - A Unified Theory of Social Interaction
Hi. I've just given the worrrst exam in the worrrst way and I'm really pained about it. So I'm going to take this opportunity and time to vent the shit that’s been blocked in my head so I can feel better. This is the second article in a series that talks about a mathematical model for modeling social interaction patterns. If you've read part one, you're good to go.. otherwise you might wanna see part one first.. go ahead I'm waiting. (with due respect to Bulco aka. HB)
A very standard problem that sometimes comes up during optimization studies is the 'matching problem". This problem can be formulated as follows: There are 2N people in a closed room. The people are of different natures. Each person must be 'paired' with another person so as to form a couple. There is some way of quantifying the quality of the matching, say compatibility function that each of the 2N people have as an attribute. A 'good' match is said to have occurred when the result of a pre-defined function on the two compatibility functions of the individuals being paired has a high value. The aim of the problem is to provide an algorithm that results in the best matching overall. This is a utilitarian problem since it seeks to provide the maximum good for the maximum number of people. It is not viewed from the point of view of an individual that is part of the 2N-strong population. Instead it is modeled from the point of view of an entity that is external to the system (and also someone who wishes to do a good deed for no apparent selfish benefit, which in turn makes the model unrealistic). Anyway, a more practical problem would be to find the best match for yourself and give a shit about what happens to others. This is where our problem becomes interesting.
Assuming that the 2N individuals are rational, it is reasonable to assume that they would have figured out what we have figured out, namely that the aim of the problem from their point of view is to maximize their 'match quality' function given their own fixed compatibility functions. We now have a population of 2N 'intelligent' individuals who are aware of what they must achieve for themselves, regardless of what others achieve. This takes us to the formal question that this article aims to address: Must they now begin to use their own personal methods to go about achieving their respective goals independently? Or must they instead, use some form of co-operation within themselves and hope to achieve a more optimal solution.
The answer to this question, technically speaking, lies in the precise definition of what we consider to be 'more optimal'. We have said that for this practical version of our problem, we must ignore the 'utilitarian' version of optimality since it requires the direction by an (apparently) selfless entity external to the setup. The situation from the perspective of a single individual is similar to the one experienced by each prisoner in a Prisoner's Dilemma. The 'dilemma' that each individual faces is this: "Should I co-operate with others ? If I co-operate, I will earn a gmore-or-less guranteed stable payoff i.e. a fairly good match... on the other hand, I *might* do much better if I went all out on my own. It depends on what the others are doing. I don’t know what the others are doing. If they co-operate to a large degree and form a cartel, they’ll do pretty well for themselves and individuals who stay out alone may get crushed by group strategy. "
Based on the nature of the problem, the above line of reasoning assumes 'pay-offs' with the following mathematical structure:
1. Alone: Payoff Varies between B and C, depending on what others do
2. Cartel: Guaranteed payoff = A
B < A < C
Hence, the problem can me modeled as a 'multi-dimensional' Prisoner’s Dilemma. A simple Nash equilibrium exists fro the 2 prisoners case, therefore the extended problem should logically speaking, have a Nash equilibrium, namely that of co-operation for all individuals in the population. This 'equilibrium' would however come about when the problem is repeatedly run and past experiences are allowed to affect the decisions of individuals. (This closely models the real life situation of individuals seeking partners for a relationship ;) haha). In such a repeated trials model, individuals will realize that it is better to take a guaranteed payoff rather than lead an open-ended random-walk pay-off strategy. Moreover, a practical problem would be finite, meaning that each player would stop playing after a stage and his/her net payoff at that stage is his/her final payoff.
Hi. I've just given the worrrst exam in the worrrst way and I'm really pained about it. So I'm going to take this opportunity and time to vent the shit that’s been blocked in my head so I can feel better. This is the second article in a series that talks about a mathematical model for modeling social interaction patterns. If you've read part one, you're good to go.. otherwise you might wanna see part one first.. go ahead I'm waiting. (with due respect to Bulco aka. HB)
A very standard problem that sometimes comes up during optimization studies is the 'matching problem". This problem can be formulated as follows: There are 2N people in a closed room. The people are of different natures. Each person must be 'paired' with another person so as to form a couple. There is some way of quantifying the quality of the matching, say compatibility function that each of the 2N people have as an attribute. A 'good' match is said to have occurred when the result of a pre-defined function on the two compatibility functions of the individuals being paired has a high value. The aim of the problem is to provide an algorithm that results in the best matching overall. This is a utilitarian problem since it seeks to provide the maximum good for the maximum number of people. It is not viewed from the point of view of an individual that is part of the 2N-strong population. Instead it is modeled from the point of view of an entity that is external to the system (and also someone who wishes to do a good deed for no apparent selfish benefit, which in turn makes the model unrealistic). Anyway, a more practical problem would be to find the best match for yourself and give a shit about what happens to others. This is where our problem becomes interesting.
Assuming that the 2N individuals are rational, it is reasonable to assume that they would have figured out what we have figured out, namely that the aim of the problem from their point of view is to maximize their 'match quality' function given their own fixed compatibility functions. We now have a population of 2N 'intelligent' individuals who are aware of what they must achieve for themselves, regardless of what others achieve. This takes us to the formal question that this article aims to address: Must they now begin to use their own personal methods to go about achieving their respective goals independently? Or must they instead, use some form of co-operation within themselves and hope to achieve a more optimal solution.
The answer to this question, technically speaking, lies in the precise definition of what we consider to be 'more optimal'. We have said that for this practical version of our problem, we must ignore the 'utilitarian' version of optimality since it requires the direction by an (apparently) selfless entity external to the setup. The situation from the perspective of a single individual is similar to the one experienced by each prisoner in a Prisoner's Dilemma. The 'dilemma' that each individual faces is this: "Should I co-operate with others ? If I co-operate, I will earn a gmore-or-less guranteed stable payoff i.e. a fairly good match... on the other hand, I *might* do much better if I went all out on my own. It depends on what the others are doing. I don’t know what the others are doing. If they co-operate to a large degree and form a cartel, they’ll do pretty well for themselves and individuals who stay out alone may get crushed by group strategy. "
Based on the nature of the problem, the above line of reasoning assumes 'pay-offs' with the following mathematical structure:
1. Alone: Payoff Varies between B and C, depending on what others do
2. Cartel: Guaranteed payoff = A
B < A < C
Hence, the problem can me modeled as a 'multi-dimensional' Prisoner’s Dilemma. A simple Nash equilibrium exists fro the 2 prisoners case, therefore the extended problem should logically speaking, have a Nash equilibrium, namely that of co-operation for all individuals in the population. This 'equilibrium' would however come about when the problem is repeatedly run and past experiences are allowed to affect the decisions of individuals. (This closely models the real life situation of individuals seeking partners for a relationship ;) haha). In such a repeated trials model, individuals will realize that it is better to take a guaranteed payoff rather than lead an open-ended random-walk pay-off strategy. Moreover, a practical problem would be finite, meaning that each player would stop playing after a stage and his/her net payoff at that stage is his/her final payoff.
0 Comments:
Post a Comment
<< Home