The matching law describes how behavior allocation is related to reinforcement rates.
It suggests a proportional relationship between the behavior on one option and the reinforcement received.
Reinforcement is quantified using mathematical expressions.
Mathematical Representation of Matching Law
Behavior on option A is represented as b_a; reinforcement on option A is r_a.
The relationship can be expressed as:
rac{b_a}{b_a + b_b} = rac{r_a}{r_a + r_b}
This means the proportion of behavior on one choice matches the proportion of reinforcement for that choice compared to others.
Generalized Matching Law
The matching law can be extended further through generalized matching laws.
Use case example: Teacher’s attention toward students based on reinforcement.
More reinforcement from attentive students (e.g., smiles, nods) leads to more attention provided to them.
Experimental Evidence of Matching Law
In social settings, participants tend to interact with conversationalists who provide more positive reinforcement.
Example: During conversations, participants shift their gaze toward individuals who provide higher reinforcement (high nodding and engagement).
Linear Graph of Matching Law
If plotted, the relationship yields a linear graph showing relative rates of behavior vs. relative rates of reinforcement.
As reinforcement for option A increases, so does the relative rate of behavior allocated to it.
Variable Interval vs. Variable Ratio Schedules
Matching law is more pronounced under variable interval (VI) schedules than under fixed or variable ratio schedules.
Example scenario: One person offers $5 for a bracelet, another $20.
Most will choose to sell to the $20 payer to maximize reinforcement.
Under variable interval schedules, reinforcement is given at varying times (e.g., every 30 or 60 minutes).
Optimal Foraging Theory Comparison
The principle of matching law parallels the concept of optimal foraging theory in biology, which observes similar behavioral patterns in animals.
Predicting Behavior Allocation with Two Options
Example: Given two interactions:
Person A: VI 30 seconds -> 2 reinforcers/min
Person B: VI 60 seconds -> 1 reinforcer/min
Matching law predicts behavior would be 66% towards A and 33% towards B, based on reinforcement allocations.
Behavioral Implications for Training
Important to realize that behavior reflects choice influenced by available reinforcement rates.
Using the dog example: if a dog ignores commands, it may be receiving higher reinforcement from alternative behaviors.
To increase compliance, one needs to adjust reinforcement: either increase the reinforcement for desired behavior or decrease reinforcement for undesired behaviors.
Example of Behavior Choice
A scenario is utilized to illustrate:
Dog is trained with kibble at home but may ignore commands outdoors due to more appealing distractions (i.e., squirrels).
The implication is that higher quality or quantity of reinforcement is necessary to maintain expected behaviors.
Generalizing Matching Law to Multiple Options
The model can also be generalized to a quantitative law of effect.
For multiple options, the formula adjusts to:
rac{b_a}{b_a + b_{all}}
Where all other options are summed as extraneous reinforcement.
Hyperbolic Function in Matching Law
As reinforcement increases, behavioral response also increases but not linearly; it follows a hyperbolic relationship where after certain points, the rate of increase diminishes (asymptotic behavior).
The behavior graph remains asymptotic at a maximum value (k), with an inflection point indicating the onset of diminishing returns, tied back to reinforcement of extraneous environmental factors.
Reinforcement Parameters in Different Environments
Understanding individual circumstances can help predict behaviors:
Higher environmental reinforcements necessitate higher adjustments in desired behavior reinforcements.
Example: More engaging or competitive environments require increased reinforcement to sustain desired behaviors.
Conclusion
Matching law highlights that behavior allocations are based on relative reinforcement strength in different adjustable scales.
This understanding can aid interactions between trainers and clients, helping to strategize reinforcement effectively to maximize desired behaviors.