A Finite-state Simulation of Economics Games

Running the simulation

To run five simulations in a row:

$ raco test -s five fsm0.rkt 

To run a single simulation:

$ racket -tm fsm0.rkt 

or

$ chmod +x fsm0.rkt 
$ ./fsm0.rkt 

Introduction

This program is a simple simulation of the tension between individuals focused on their own interest and the interests of the general population.

Individuals are represented by automata. A state in an automaton describes how an individual acts; the transitions from a state to another state explain how the individual interacts with some other individual. The simulation assumes that there is a fixed number of actions for the entire game. In general, each automata may differ in its number of states from every other automaton.

A classical game assumes that an individual may take one of two actions: "cooperate" or "defect". This choice dictates a two-by-two payoff matrix such as this one:

paypff matrix	cooperate	defect
cooperate	3,3	0,4
defect	4,0	1,1

That is, when a currently defecting individual meets on a currently cooperating one, the former receives four "credits" and the latter none.

Here are the transition matrices for three classical individuals:

• the selfless individual starts with the cooperative strategy and sticks to it, no matter what the other individual does:

  *       | cooperate | defect 
  

------------- | --------- | --------- cooperate | cooperate | cooperate defect | cooperate | cooperate

It is possible to represent this individual with a single state in which it always acts in a cooperative manner, and no matter how the other individual acts, it transitions to the very same state.

• the self-interested individual starts with the defecting strategy and sticks to it, no matter what the other individual does:

  *      | cooperate | defect 
  

------------ | --------- | ------------ cooperate | defect | defect defect | defect | defect

It is also possible to represent this individual with a single state in which it always defects, and no matter how the other individual acts, it transitions to the very same state.

• the tit-for-tat individual starts with the cooperative strategy but switches to defecting one when it encounters an individual that defects during an interaction. It switches back to a cooperative attitude after a cooperative interaction:

  •   | cooperate | defect 
    

----------- | --------- | ------------ cooperate | cooperate | defect defect | cooperate | defect

Representing such an individual calls for two states. In the first one it acts in a cooperative manner. If the transaction partner is cooperative, too, it transitions to the same state at the end of the transaction; otherwise it moves to the second state. In the second state, this individual defects but if the partner is cooperative it goes back to the first state; otherwise it stays in the same state.

The "Nash Demand Game" uses three actions: "high", "medium", "low". Each action represents what kind of share an individual wishes to grab from a shared pie: eight slices, five slices, two slices. Here is the typical payoff matrix:

paypff matrix	high	medium	low
high	0,0	0,0	8,2
medium	0,0	5,5	5,2
low	2,8	2,5	2,2

That is, when a "high" individual meets a "low" individual, the former receives eight slices of the pie and the latter two. In general, two individuals receive the desired slices if their total claims to the pie are below or equal to ten and nothing otherwise.

The "Ultimatum game" uses asymmetric actions. A proposer suggests a division of $10 between two players. The proposer may offer a "low" or a "fair" share, which represent two and five dollars, respectively. The responder may take one of two actions: accept or reject. If the responder accepts, both get the proposed share of the amount; otherwise neither gets anything. Here is the obvious payoff matrix for this game:

paypff matrix	low	fair
accept	2,8	5,5
reject	0,0	0,0

Current Implementation

The current implementation simulates random N-state, 2-action automata with the payoff matrix from above.

Interpretation

If per-automaton payoff trends towards three credits, the population consists of selfless, cooperating members. If the average payoff converges to one credit, the population consists of self-interested individuals; everybody defects everybody else.

Details of the Simulation

The simulation runs for c cycles with a population of size p. During each cycle, every individual interacts with one other individual for r rounds. The simulation program records the average payoff of all interactions. At the end of a cycle, s individuals (s << p) are replaced with s new ones. The replacements are cloned from individuals in the existing population. The choice process gives each individual a chance for cloning in proportion to its payoffs from the preceding round of simulation. Finally, the simulation resets all individuals in the "regenerated" population, that is, it eliminates the current payoff (100% estate tax) and it resets their starting strategy to the initial one.

Files

file	purpose
fsm0	is the top-level file
population	a data representation for a population
automata	a data representation for automata

Deprecated

V2 and VOO are deprecated versions of this repository.

Acknowledgment

The original code is Nguyen Linh Chi from the University of Trento, Italy. See github

Chi also reviewed several stages of this code development.

Chi, in turn, acknowledges a blog post of Tim Thornton

The original ideas are due to Axelrod (1987) with rule-based automata that have memory. Fogel (1993) showed how to use finite state machines.