# Doubts in modelling the Arthur(1991) paper: Designing economic agents that act like human agent

I am implementing Arthur's paper "Designing economic agents that act like human agent: A behavioral approach to bounded rationality" from 1991 in JAVA and I am having a problem understanding one part of the paper:

At one point it says: triggering actions randomly on the basis of their strengths and I don't know what this means.

The algorithm is described as follows:

1.) Actions are represented as $a = (1,2,...,A)$, these values represent the mean and all the actions have a fixed standard deviation. The mean of an action and its standard deviation help in plotting the normal distribution. The payoff of an action, which is unknown before is drawn after the action is chosen from the above described normal distribution.

2.) Ever action has a certain value of strength ($S(t)$) associated with it, the strength depends on prior beliefs (in my case it depends on prior skill set). The current sum of these strengths is C(t).

3.)At every time $t$ I calculate the probability vector $p(t) = S(t)/C(t)$

4.) Choose one action from the set according to the probability

5.) Update the strength of that action (point 3, page 354)

6.) Renormalized (point 4, page 354).

The problem occurs at point 4: choose an action according to the probability, what probability?

I don't know how to translate this statement into code.

• I find it surprising that in a paper with such a task, the specific step is described in such vague terms. By the way, if I take "random" with its "usual" informal meaning, I would translate it as "uniformly (equiprobably) distributed". But then, it is a contradiction to also take into account the "strength" of the action, whatever that may mean. This is a very interesting question about translating economic/behavioral theory into software code, but we need much more input from you (CONTD): – Alecos Papadopoulos May 10 '15 at 9:28
• (CONTD) I would suggest to go into some more detail about what these "actions" are, in what sense they have different "strengths" and whether the paper provides any other hint about the meaning of "random". And it would be good to also provide a link to the paper. – Alecos Papadopoulos May 10 '15 at 9:29
• The link for the paper, I am writing the precise details of action in the second comment that follows : multiagent.martinsewell.com/Arth91.pdf – Sabachi May 10 '15 at 10:39
• Please go to page page 354, under the section: A parametrized learning automation, a timeline description is provided which says what is done at every time 't'. Then, on page 355, 3rd paragraph starting with "The algorithm is nonlinear...." provides more detail about how an action is selected. I have translated most if it in codes (JAVA) but I have a feeling that I am making a small mistake in coding the randomness, i.e. the stochastic nature with which actions are selected at time 't'. (CONTD) – Sabachi May 10 '15 at 10:43
• Welcome to SE, Sabachi! Every question here should be as self contained as possible, so I tried to copy your comment into the question. Furthermore, I tried to make clear what you are asking. I hope you are fine with my edits, otherwise just reverse them or change them. – The Almighty Bob May 10 '15 at 11:52

Although I still don't understand what these "strengths" are all about, still, mechanically, step $4)$ is now clear.

At each point in time, there is a certain allocation of Strengths to actions, $\{S_i(t),\; i=1,2,...,A\}$. This allocation changes through time.

The allocation is turned into a relative frequency distribution by dividing each $S_i(t)$ by their sum $C(t)$.

But then

$$\sum_{i=1}^A \frac {S_i(t)}{C(t)} =1$$

and (I presume), $S_i(t) \geq 0,\; \forall i$. So indeed this relative frequency distribution can be treated as a probability distribution, and each $S_i(t)/C(t)$ can be treated as a probability that characterizes action $i$ in period $t$.

So you have to code this probability distribution, and then have the software "randomly draw" from it, as though this distribution was part of the distributions available in a "random number generator".

The floor is yours now.